Xi, L. - FCF Research Group - University of Wisconsin–Madison

Transcription

NONLINEAR DYNAMICS AND
INSTABILITIES OF VISCOELASTIC FLUID
FLOWS
by
Li Xi
A dissertation submitted in partial
fulfillment of the requirements for the degree of
Doctor of Philosophy
(Chemical Engineering)
at the
UNIVERSITY OF WISCONSIN – MADISON
2009
i
Acknowledgments
Above all, I would like to thank my advisor Professor Michael D. Graham, for bringing
me into this area of research and exposing me to all these opportunities; and most
importantly, for his great passion of teaching, which has helped me grow in the past
five years in terms of both research capability and academic scholarship.
I am also indebted to Professor Fabian Waleffe, for many inspiring discussions that
benefited my research on viscoelastic turbulence. Same for Doctor John F. Gibson,
who generously shared his ChannelFlow code for Newtonian flows, based on which
my viscoelastic code was developed. He also offered some very helpful advice on my
numerical algorithm.
I have enjoyed working with many former and current members of the Graham group, including: Samartha G. Anekal, Yeng-Long Chen, Juan P. HernandezOrtiz, Aslin Izmitli, Pieter J. A. Janssen, Rajesh Khare, Wei Li, Mauricio Lopez,
Hongbo Ma, Pratik Pranay, Christopher G. Stoltz, Patrick T. Underhill and Yu Zhang.
Wei Li, in particular, offered many discussions in my first two year that helped me
greatly in understanding some quite obscure topics.
Finally, I am grateful to my family for the endless support they provided me
during all these years.
ii
Research projects presented in this dissertation are financially supported by the
National Science Foundation, and the Petroleum Research Fund, administered by the
American Chemical Society.
iii
Abstract
This dissertation focuses on the fluid dynamics of dilute polymer solutions, with an
emphasis on nonlinear flow behaviors and instabilities in different parameter regimes.
Even at a very low concentration, flexible polymer solutes can introduce strong viscoelasticity into the fluid, causing flow instability at very low Reynolds number. At
high Reynolds number, the coupling between inertial and elastic effects bring forth further complex dynamics. We study two representative problems in these two regimes,
respectively: elastic instabilities involving stagnation points at low Reynolds number,
and dynamics of viscoelastic turbulent flows at relatively high Reynolds number.
In the low Reynolds number case, interior stagnation point flows of viscoelastic
liquids arise in a wide variety of applications including extensional viscometry, polymer processing and microfluidics. Experimentally, these flows have long been known
to exhibit instabilities, but the mechanisms underlying them have not previously
been elucidated. We computationally demonstrate the existence of a supercritical
oscillatory instability of low-Reynolds number viscoelastic flow in a two-dimensional
cross-slot geometry. The fluctuations are closely associated with the “birefringent
strand” of highly stretched polymer chains associated with the outflow from the stagnation point at high Weissenberg number. Additionally, we describe the mechanism
iv
of instability, which arises from the coupling of flow with extensional stresses and
their steep gradients in the stagnation point region.
In turbulent flows, the observation that a minute amount of flexible polymers reduces turbulent friction drag has been long established. However, many aspects of
the drag reduction phenomenon are not well-understood; in particular, the existence
of the maximum drag reduction (MDR) asymptote, a universal upper limit of drag
reduction, remains a mystery. Our study focuses on the drag reduction phenomenon
in the plane Poiseuille geometry in a parameter regime close to the laminar-turbulent
transition. By minimizing the size of the periodic simulation box to the lower limit
for which turbulence persists, the essential self-sustaining turbulent motions are isolated. In these “minimal flow unit” (MFU) solutions, consistent with previous experiments, a series of qualitatively different stages are observed, including one showing
the universality of MDR: i.e. the mean flow is universal with respect to changing
polymer-related parameters. Before this stage, an additional transition exists between a relatively low degree (LDR) and a high degree (HDR) of drag reduction.
This transition occurs at about 13-15% of drag reduction, and is characterized by a
sudden increase in the minimal box size of sustaining turbulence, as well as many
qualitative changes in flow statistics. The observation of LDR–HDR transition at
less than 15% drag reduction shows for the first time that it is a qualitative transition instead of a quantitative effect of the amount of drag reduction. Spatiotemporal
flow structures change substantially upon this transition, suggesting that two distinct types of self-sustaining turbulence dynamics are observed. In LDR, similar as
Newtonian turbulence, the self-sustaining process involves one low-speed streak and
its surrounding streamwise vortices; after the LDR–HDR transition, multiple streaks
v
are present in the self-sustaining structure and complex intermittent behaviors of the
streaks are observed. This multistage scenario of LDR–HDR–MDR recovers all key
transitions commonly observed and studied at much higher Reynolds numbers.
The asymptotic upper-limit of drag reduction observed in MFU is much lower
than the experimentally-found MDR; however, an important progress has been made
toward the understanding of the latter. In all stages of transition, even in the Newtonian limit, we find intervals of “hibernating” turbulence that display many features
of the experimental MDR asymptote in polymer solutions: weak streamwise vortices,
nearly nonexistent streamwise variations and a mean velocity gradient that quantitatively matches experiments. As viscoelasticity increases, the frequency of these
intervals also increases, while the intervals themselves are unchanged, leading to flows
that increasingly resemble MDR. This observation would inspire future research that
might finally solve the puzzle of MDR.
vi
Contents
Acknowledgments
i
Abstract
iii
List of Figures
ix
List of Tables
xxiii
1 Overview: the scope of study
Part I
1
Dynamics at low Re: oscillatory instability in vis-
coelastic cross-slot flow
7
2 Introduction: elastic instabilities and viscoelastic stagnation-point
flows
8
3 Cross-slot geometry, governing equations and numerical methods
15
4 Results: viscoelastic cross-slot flow and its oscillatory instability
20
4.1
Steady states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
vii
4.2
Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
4.3
Instability mechanism
33
. . . . . . . . . . . . . . . . . . . . . . . . . .
5 Conclusions of Part I
48
6 Future work: nonlinear dynamics of viscoelastic fluid flows in complex geometries
Part II
50
Dynamics at high Re: viscoelastic turbulent flows
and drag reduction
56
7 Introduction: viscoelastic turbulent flows and polymer drag reduction
57
7.1
Fundamentals of polymer drag reduction . . . . . . . . . . . . . . . .
57
7.2
Previous direct numerical simulation (DNS) studies . . . . . . . . . .
61
7.3
Traveling waves and the nonlinear dynamics perspective of turbulence
64
7.4
Multistage transitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
7.5
About this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
8 DNS formulation and numerical method
77
8.1
Flow geometry and governing equations . . . . . . . . . . . . . . . . .
77
8.2
Numerical procedures . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
9 Methodology: minimal flow units (MFU)
83
10 Results: observations during multistage transitions
88
10.1 Overview of the multistage-transition scenario . . . . . . . . . . . . .
88
viii
10.2 Flow statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
10.3 Polymer conformation statistics . . . . . . . . . . . . . . . . . . . . . 109
10.4 Spatio-temporal structures . . . . . . . . . . . . . . . . . . . . . . . . 113
11 Toward an understanding of the dynamics: active and hibernating
turbulence
126
11.1 Intermittent dynamics in MFU . . . . . . . . . . . . . . . . . . . . . 126
11.2 Generalization to full-size turbulent flows: a preliminary investigation
12 Conclusions of Part II
143
150
13 Future work: dynamics of viscoelastic turbulence and drag reduction
in turbulent flows
154
13.1 Hibernation statistics: effect on the LDR–HDR transition . . . . . . . 155
13.2 A hypothetical dynamical-scenario
13.3 Development of methodology
. . . . . . . . . . . . . . . . . . . 156
. . . . . . . . . . . . . . . . . . . . . . 162
13.4 Further extensions: other drag-reduced turbulent flow systems . . . . 167
A Numerical algorithm for the direct numerical simulation of viscoelastic channel flow
Bibliography
172
185
ix
List of Figures
2.1
Symmetry-breaking instability in viscoelastic cross-slot flow (Arratia
et al. 2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
(a)
Dye convection pattern. . . . . . . . . . . . . . . . . . . . . . .
10
(b)
Contours of velocity magnitude (colors) and streamline (dark
lines) measured by particle image velocimetry (PIV) . . . . . .
10
3.1
Schematic of the cross-slot flow geometry. . . . . . . . . . . . . . . . .
16
4.1
Contour plots of steady state solution: Wi = 0.2 (only the central part
4.2
4.3
of the flow domain is shown). . . . . . . . . . . . . . . . . . . . . . .
22
(a)
kuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
(b)
∂ux /∂x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
(c)
tr(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Contour plots of steady state solution: Wi = 50 (only the central part
of the flow domain is shown). . . . . . . . . . . . . . . . . . . . . . .
23
(a)
kuk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
(b)
∂ux /∂x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
(c)
tr(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Profiles of tr(α) along y = 0. . . . . . . . . . . . . . . . . . . . . . . .
24
x
4.4
Profiles of tr(α) along x = 0 in the region very near the stagnation
point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
24
Effect of Wi on the size of the birefringent strand (tr(α) > 300 is
considered as the observable birefringence region). . . . . . . . . . . .
25
(a)
Birefringent strand width W . . . . . . . . . . . . . . . . . . . .
25
(b)
Birefringent strand length L. . . . . . . . . . . . . . . . . . . .
25
4.6
Profiles of ux along y = 0. . . . . . . . . . . . . . . . . . . . . . . . .
26
4.7
Average extension rate (∂ux /∂x)avg (averages taken in the domain
−0.1 < x < 0.1, −0.1 < y < 0.1). . . . . . . . . . . . . . . . . . . . . .
4.8
27
Evolution of the birefringence strand width W after a small initial
perturbation on the steady state; inset: enlarged view of 2500 6 t 6 3200. 30
4.9
Two dimensional projection of the dynamic trajectory from the steady
state to the periodic orbit at Wi = 66: ux at (0.5, 0) v.s. W . . . . . .
30
4.10 Left-hand axis: root-mean-square deviations of the birefringent strand
width W at periodic orbits, normalized by steady state values; righthand axis: oscillation periods. . . . . . . . . . . . . . . . . . . . . . .
31
4.11 Perturbation of the x-component of velocity, u0x with respect to steady
state at the periodic orbit: Wi = 66. The region shown is 0 < x < 1,
−1 < y < 0; the stagnation point is at the top-left corner. (To be
continued).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
(a)
t=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
(b)
t = 1.68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
(c)
t = 3.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
(d)
t = 5.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
xi
4.11 (Continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
(e)
t = 6.71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
(f)
t = 8.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
(g)
t = 10.06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
(h)
t = 11.74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.12 Perturbation of the y-component of velocity, u0y with respect to steady
state at the periodic orbit: Wi = 66. The region shown is 0 < x < 1,
−1 < y < 0; the stagnation point is at the top-left corner. (To be
continued).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
(a)
t=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
(b)
t = 1.68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
(c)
t = 3.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
(d)
t = 5.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.12 (Continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
(e)
t = 6.71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
(f)
t = 8.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
(g)
t = 10.06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
(h)
t = 11.74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
0
4.13 Perturbation of the xx-component of polymer conformation tensor, αxx
with respect to steady state at the periodic orbit: Wi = 66. The region
shown is 0 < x < 1, −1 < y < 0; the stagnation point is at the top-left
corner. The edge of the steady state birefringent strand is the line
y ≈ −0.05. (To be continued). . . . . . . . . . . . . . . . . . . . . . .
38
(a)
38
t=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
(b)
t = 1.68 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
(c)
t = 3.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
(d)
t = 5.03 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4.13 (Continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
(e)
t = 6.71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
(f)
t = 8.38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
(g)
t = 10.06 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
(h)
t = 11.74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.14 Time-dependent oscillations at (0, −0.05). Top view: perturbations of
variables normalized by steady-state quantities; bottom view: magnitudes of terms on RHS of Equation (4.4). . . . . . . . . . . . . . . . .
43
4.15 Schematic of instability mechanism (view of the lower half geometry).
Thick arrows represent net forces exerted by polymer molecules (dumbbells) on the fluid.
6.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
(a)
Thinning process of the birefringent strand. . . . . . . . . . . .
45
(b)
Re-thickening process of the birefringent strand. . . . . . . . .
45
The microfluidic flip-flop device (Groisman et al. 2003). . . . . . . . .
52
(a)
Overall Geometry. The auxiliary inlets (comp. 1 and comp. 2)
are for flow-rate measurement purpose. . . . . . . . . . . . . . .
(b)
Blowup near the intersection during the instability. Only fluids
from one of the two inlets are dyed. . . . . . . . . . . . . . . . .
7.1
52
52
Schamatic of the Prandtl-von Kármán plot. Thin vertical lines mark
the transition points on the typical experimental path shown as a thick
solid line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
xiii
7.2
Experimental data of maximum drag reduction (MDR) in pipe flow,
from different polymer solution systems and pipe sizes, plotted in the
Prandtl-von Kármán coordinates (Virk 1971, 1975). It can be shown
√
√
√
√
+
/ 2, Re f = 2Reτ (f in this plot is the friction
that 1/ f = Uavg
factor, which is denoted as Cf in this dissertation; Re in this plot is the
Reynolds number based on average velocity: Reavg ≡ ρUavg D/η, D is
the pipe diameter). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
59
Newtonian ECS solution at Re = 977 in plane Poiseuille flow; symmetric copies at both walls are shown. Slices show coutours of streamwise velocity, dark color for low velocity; the isosurface has a constant
streamwise vortex strength Q2D = 0.008, definition of Q2D is given
in Section 10.4. This plot is published by Li & Graham (2007); the
solution is originally discovered by Waleffe (2003). . . . . . . . . . . .
7.4
65
Dynamics of turbulence in the solution state space in a plan Couette
flow (Gibson et al. 2008). . . . . . . . . . . . . . . . . . . . . . . . . .
(a)
67
Dynamical trajectory of a turbulent transient in a MFU visualized in the state space using coordinates proposed by Gibson
et al. (2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b)
67
Same trajectory (dotted line) visualized in the context of TW
solutions (solid dots, except uLM , which is the laminar state) and
their unstable manifolds (solid lines). . . . . . . . . . . . . . . .
7.5
67
The Virk (1975) universal mean velocity profile for MDR in inner
+
scales: Umean
= 11.7 ln y + − 17.0. . . . . . . . . . . . . . . . . . . . . .
70
xiv
8.1
Schematic of the plane Poiseuille flow geometry: the box highlighted
in the center with dark-colored walls is the actual simulation box, surrounded by its periodic images. . . . . . . . . . . . . . . . . . . . . .
8.2
Schematic of the finitely-extensible nonlinear elastic (FENE) dumbbell
model for polymer molecules. . . . . . . . . . . . . . . . . . . . . . .
9.1
78
78
Summary of simulation results: “Turbulent” indicates that at least
one simulation run gives sustained turbulence within the given time
interval (Newtonian and β = 0.97, b = 5000). . . . . . . . . . . . . . .
85
10.1 Variations of the average streamwise velocity with Wi at different β
and b values (average taken in time and all three spatial dimensions);
the corresponding DR% is shown on the right ordinate. Solid symbols represent points in the asym-DR stage (defined in the text); the
horizontal dashed line is the average of all asym-DR points. . . . . . .
89
10.2 Mean velocity profiles (Newtonian and β = 0.97, b = 5000). . . . . . .
93
10.3 Spanwise box sizes used in this study for various parameters. Solid
symbols represent points in the asym-DR stage. . . . . . . . . . . . .
94
10.4 Variations of spanwise box size at different DR%. Solid symbols represent points in the asym-DR stage. . . . . . . . . . . . . . . . . . . .
96
10.5 Mean velocity profiles of 15 different asym-DR states (Wi: 27 ∼ 30
for β = 0.97, b = 5000, Wi: 32 ∼ 36 for β = 0.99, b = 10000 and
Wi: 40 ∼ 50 for β = 0.99, b = 5000). . . . . . . . . . . . . . . . . . . .
98
10.6 Deviations in mean velocity profile gradient from that of Newtonian
turbulence (β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23;
asym-DR: Wi = 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
xv
10.7 Magnitude of mean velocity profile gradient at y + = 40. Solid symbols
represent points in the asym-DR stage. . . . . . . . . . . . . . . . . . 100
10.8 Profiles of the Reynolds shear stress (Newtonian and β = 0.97, b =
5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR: Wi = 29. . . . . 101
10.9 Deviations in Reynolds shear stress profiles from that of Newtonian
turbulence (β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23;
asym-DR: Wi = 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10.10Magnitude of Reynolds shear stress at y + = 40. Solid symbols represent points in the asym-DR stage. . . . . . . . . . . . . . . . . . . . . 102
10.11Profiles of root-mean-square streamwise and wall-normal velocity fluctuations (Newtonian and β = 0.97, b = 5000). LDR: Wi = 17, 19;
HDR: Wi = 23; asym-DR: Wi = 29. . . . . . . . . . . . . . . . . . . . 104
10.12Profiles of root-mean-square wall-normal velocity fluctuations (Newtonian and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23;
asym-DR: Wi = 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.13Profiles of root-mean-square spanwise velocity fluctuations (Newtonian
and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asymDR: Wi = 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.14Profiles of root-mean-square velocity fluctuations and Reynolds shear
stress at 15 different asym-DR states (Wi: 27 ∼ 30 for β = 0.97,
b = 5000, Wi: 32 ∼ 36 for β = 0.99, b = 10000 and Wi: 40 ∼ 50 for
β = 0.99, b = 5000). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
xvi
10.15Normalized profiles of the trace of the polymer conformation tensor
(β = 0.97, b = 5000). Pre-onset: Wi = 16; LDR: Wi = 17, 19; HDR:
Wi = 23; asym-DR: Wi = 27, 29. . . . . . . . . . . . . . . . . . . . . 109
10.16Averaged trace of the polymer conformation tensor (average taken in
time and all three spatial dimensions). Solid symbols represent points
in the asym-DR stage. . . . . . . . . . . . . . . . . . . . . . . . . . . 110
10.17Position of the maximum in the tr(α) profile. Solid symbols represent
points in the asym-DR stage. . . . . . . . . . . . . . . . . . . . . . . 112
(a)
Dependence on DR% . . . . . . . . . . . . . . . . . . . . . . . . 112
(b)
Dependence on Wi . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.18Dynamics of the self-sustaining turbulent structures in a selected New+
tonian simulation (Re = 3600, L+
x = 360, Lz = 140). Top panel:
spatial-temporal patterns of the wall shear rate (∂vx /∂y taken at x = 0;
two periodic images are shown for each case. Note: the mean value is 2
owning to the fixed pressure gradient constraint.); bottom panel: (left
ordinate and thick line) spatially-averaged velocity and (right ordinate
and thin line) average wall shear rate (average taken in the z-direction
at x = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
xvii
10.19Dynamics of the self-sustaining turbulent structures in a selected LDR
simulation (Re = 3600, Wi = 19, β = 0.97, b = 5000, L+
x = 360,
L+
z = 150). Top panel: spatial-temporal patterns of the wall shear
rate (∂vx /∂y taken at x = 0; two periodic images are shown for each
case. Note: the mean value is 2 owning to the fixed pressure gradient
constraint.); bottom panel: (left ordinate and thick line) spatiallyaveraged velocity and (right ordinate and thin line) average wall shear
rate (average taken in the z-direction at x = 0). . . . . . . . . . . . . 116
10.20Dynamics of the self-sustaining turbulent structures in a selected HDR
simulation (Re = 3600, Wi = 23, β = 0.97, b = 5000, L+
x = 360,
L+
10.21Dynamics of the self-sustaining turbulent structures in a selected asymDR simulation (Re = 3600, Wi = 29, β = 0.97, b = 5000, L+
x = 360,
L+
xviii
10.22Typical snapshots of the flow field (Re = 3600, β = 0.97, b = 5000,
L+
x = 360). (Reg) denotes snapshots chosen from “regular” turbulence,
and (LS) denotes snapshots of “low-shear” events. Translucent sheets
are the isosurfaces of vx = 0.6vx,max ; opaque tubes are the isosurfaces of
Q2D = 0.3Q2D,max . The values of vx and Q2D for each plot is shown in
its caption. Note that (LS) states typically have much lower Q2D values
than (Reg) states. The bottom wall of each snapshot corresponds to
the wall shear rate patterns shown in Figures 10.18, 10.19, 10.20 and
10.21 at corresponding time. (To be continued). . . . . . . . . . . . . 119
(a)
Newtonian (Reg), L+
z = 140;
t = 8500, vx = 0.25, Q2D = 0.025. . . . . . . . . . . . . . . . . . 119
(b)
Newtonian (LS), L+
z = 140;
t = 4600, vx = 0.27, Q2D = 0.012. . . . . . . . . . . . . . . . . . 119
(c)
LDR (Reg): Wi = 19, L+
z = 150;
t = 5900, vx = 0.26, Q2D = 0.024. . . . . . . . . . . . . . . . . . 119
(d)
LDR (LS): Wi = 19, L+
z = 150;
t = 8200, vx = 0.29, Q2D = 0.0079. . . . . . . . . . . . . . . . . 119
10.22(Continued). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
(e)
HDR (Reg): Wi = 23, L+
z = 180;
t = 7700, vx = 0.31, Q2D = 0.026. . . . . . . . . . . . . . . . . . 120
(f)
HDR (LS): Wi = 23, L+
z = 180;
t = 7300, vx = 0.31, Q2D = 0.0089. . . . . . . . . . . . . . . . . 120
(g)
asym-DR (Reg): Wi = 29, L+
z = 250;
t = 8500, vx = 0.27, Q2D = 0.018. . . . . . . . . . . . . . . . . . 120
xix
(h)
asym-DR (LS): Wi = 29, L+
z = 250;
t = 8900, vx = 0.31, Q2D = 0.0050. . . . . . . . . . . . . . . . . 120
11.1 Mean shear rates at the walls (“b”–bottom, “t”–top) and bulk velocity
Ubulk as functions of time for typical segments of a Newtonian sim+
ulation run (Re = 3600, L+
x = 360, Lz = 140). Rectangular signals
in the middle panel indicate the hibernating periods at the wall of the
corresponding side, identified with the criterion explained in the text.
Dashed lines show the line h∂vx /∂yi = 1.80. Time average of h∂vx /∂yi
is 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
11.2 Mean shear rates at the walls (“b”–bottom, “t”–top) and bulk velocity
Ubulk as functions of time for typical segments of a high-Wi simulation
+
run (Re = 3600, Wi = 29, β = 0.97, b = 5000, L+
x = 360, Lz = 250).
Rectangular signals in the middle panel indicate the hibernating periods at the wall of the corresponding side, identified with the criterion
explained in the text. Dashed lines show the line h∂vx /∂yi = 1.80.
Time average of h∂vx /∂yi is 2. . . . . . . . . . . . . . . . . . . . . . . 129
11.3 Level of drag reduction and spanwise box size as functions of Wi (Newtonian and β = 0.97, b = 5000). . . . . . . . . . . . . . . . . . . . . . 130
11.4 Time scales (left ordinate) and fraction of time spent in hibernation
(right ordinate) as functions of Wi (Newtonian and β = 0.97, b =
5000): TA is the mean duration of active periods; TH is the mean
duration of hibernating periods; FH is the fraction of time spent in
hibernation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
xx
11.5 A hibernation event (200 6 t 6 600 in Figure 11.2). Thick black lines
are mean wall shear rates and bulk velocity Ubulk at Wi = 29. Thin
colored lines are from Newtonian simulations started at the corresponding colored dots, using velocity fields from the Wi = 29 simulation as
initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11.6 Instantaneous mean velocity profiles of selected instants before, during and after a typical hibernating period (marked with grid-lines in
Figure 11.5). Profiles for the bottom half of the channel are shown; superscript “*” represents variables nondimensionalized with inner scales
based on instantaneous mean shear-stress at the wall of the corresponding side. Black lines show important asymptotes: “viscous sublayer”,
∗
∗
= 2.44 ln y ∗ +5.2 (Pope 2000);
= y ∗ ; “Newtonian log-law”, Umean
Umean
∗
= 11.7 ln y ∗ − 17.0 (Virk 1975). . . . . . . . . . . 135
“Virk MDR”, Umean
11.7 Comparison between hibernation in Newtonian and high-Wi viscoelastic flows (the Newtonian simulation is the one starting from t = 260 in
Figure 11.5). Instantaneous mean velocity profiles for instants in hibernation (c) and after turbulence is reactivated (e) are show (marked
with grid-lines in Figure 11.5). Profiles for the bottom half of the
channel are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.8 Flow structures at selected instants before, during and after a typical hibernating period (marked with grid-lines in Figure 11.5). Green
sheets are isosurfaces vx = 0.3, pleats correspond to low-speed streaks;
red tubes are isosurfaces of Q2D = 0.02, Q2D is defined in Section 10.4.
Only the bottom half of the channel is shown. . . . . . . . . . . . . . 137
xxi
11.9 Instantaneous profiles of αxx (streamwise polymer deformation) for instants marked in Figure 11.5. Profiles for the bottom half of the channel
are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
11.10Instantaneous profiles of αyy (wall-normal polymer deformation) for
instants marked in Figure 11.5. Profiles for the bottom half of the
channel are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
11.11Instantaneous profiles of αzz (spanwise polymer deformation) for instants marked in Figure 11.5. Profiles for the bottom half of the channel are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
11.12Instantaneous profiles of Reynolds shear stresss for instants marked in
Figure 11.5. Profiles for the bottom half of the channel are shown. . . 140
11.13Flow structures of a typical snapshot in a full-size Newtonian simula+
tion (Re = 3600, L+
x = 4000, Lz = 800). Green sheet is the isosurface
of vx = 0.3; red tubes are isosurfaces of Q2D = 0.02. Only the bottom
half of the channel is shown. . . . . . . . . . . . . . . . . . . . . . . . 144
11.14Flow structures of a typical snapshot in a full-size viscoelastic simulation near MDR (Re = 3600, Wi = 80, β = 0.97, b = 5000, L+
x = 4000,
L+
z = 800). Green sheet is the isosurface of vx = 0.3; red tubes are
isosurfaces of Q2D = 0.02. Only the bottom half of the channel is shown.145
11.15Contours of streamwise velocity in a plane 25 wall units above the
bottom wall for the Newtonian snapshot shown in Figure 11.13 . . . . 146
11.16Contours of streamwise velocity in a plane 25 wall units above the bottom wall for the viscoelastic (near MDR) snapshot shown in Figure 11.14147
xxii
13.1 Schematic of near-transition turbulent dynamics: intermittent excursions toward certain saddle points and the laminar-turbulence edge
structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
13.2 Schematic of the edge-tracking method based on repeated bisection (Skufca
et al. 2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
xxiii
List of Tables
4.1
Terms on the right-hand side of Equation (4.4). . . . . . . . . . . . .
42
A.1 Numerical coefficients for the Adams-Bashforth/backward-differentiation
temporal discretization scheme with different orders-of-accuracy (Peyret
2002). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
1
Chapter 1
Overview: the scope of study
Viscoelastic fluids, materials that exhibit both viscous and elastic characteristics upon
deformation, have been an interesting object of study to researchers in various areas.
The most common type of fluids with viscoelasticity is polymeric liquids: melts and
solutions of polymers. Owning to the elasticity of polymer molecules, as well as the
nontrivial polymer-solvent and polymer-polymer interactions, behaviors of these liquids under certain flow conditions and deformations can be drastically different from
those of Newtonian fluids; some classical examples are discussed in Bird, Armstrong
& Hassager (1987). Another example of viscoelastic fluids is surfactant solutions with
worm-like micelles formed (Larson 1999, Walker 2001). Similar as polymer molecules,
these semi-flexible chain-like micelles can be deformed and reoriented by the flow; in
addition, the capability of dynamical break-up and reformation of the micellar structure, and the possibility of forming super-molecular aggregates under flow, make the
dynamics of surfactant solutions even more complicated (Cates & Candau 1990, Liu
& Pine 1996, Zakin et al. 1998, Butler 1999).
2
As suggested by the title of this dissertation, our study resides within the scope
of fluid dynamics of viscoelastic fluids. In particular, we are interested in nonlinear
behaviors of flowing viscoelastic fluids in different parameter regimes. For Newtonian
fluids, nonlinear flow behaviors are driven purely by inertia, the impact of which
can be measured with a single dimensionless parameter, the Reynolds number Re
(Re ≡ ρU l/η, here ρ and η represent the density and viscosity of the fluid, while
U and l are the characteristic flow velocity and length scale of the geometry) (Bird
et al. 2002). Significant nonlinear behaviors are only expected for Re > O(1); with
Re high enough, the flow eventually becomes fully turbulent. For viscoelastic fluids,
inertia is no longer the sole source of nonlinearity. Microscopic structures of the fluid,
including (take polymeric liquids for instance) individual polymer molecules as well
as high-order structures (e.g. clusters and networks) of polymers (in concentrated
solutions and melts), interact in a nontrivial way with the macroscopic momentum
and mass balances. Therefore nonlinearity can be significant even at very low Re.
Among many types of viscoelastic fluids mentioned above, we limit our attention
to dilute solutions of flexible linear polymer chains in this dissertation. By “dilute
solutions” we refer to those with concentration much lower than the overlap concentration (Rubinstein & Colby 2003): in these systems, polymer molecules are so
far apart from one another that they do not “feel” the existence of others; therefore
interactions between polymer molecules are negligible, and polymer dynamics under
flow is relatively simple. Individual polymer molecules can be oriented and stretched
by the flow; once the flow-induced strain is released, they have a tendency of relaxing toward the coiled configuration, which they prefer at equilibrium. During these
processes, when polymer molecules change configuration, they apply a drag force on
3
the solvent around; in terms of the macroscopic momentum balance, this polymer
feedback to the flow is described as an additional contribution to the stress, which
is well-captured by the FENE-P constitutive equation (Bird, Curtis, Armstrong &
Hassager 1987) for dilute solutions. These interactions between the macroscopic flow
and microscopic polymer dynamics introduce additional nonlinearity into the system.
Another dimensionless parameter, the Weissenberg number Wi ≡ λγ̇, is introduced,
which by definition is the time scale of the relaxation of polymer λ nondimensionlized
by the inverse of the characteristic strain rate γ̇ of the flow. When Wi > O(1), polymer relaxation lags significantly behind changes in fluid deformation, and the fluid
has a stronger “memory” effect, or elasticity, which could result in various nonlinear
behaviors and instabilities. Instabilities can occur even at extremely low Re, where
inertial effects are negligible; these types of instabilities driven completely by elasticity are very often mentioned as “purely-elastic” or “inertia-less” instabilities (Larson
1992, Shaqfeh 1996, Groisman & Steinberg 2000, Larson 2000). At high Re where
inertia by itself can trigger flow instabilities, the coupling between elastic and inertial
effects can cause intriguing nonlinear behaviors inaccessible with mere contribution
from either of them (Rodd et al. 2005). In particular, turbulent flows of viscoelastic fluids show qualitatively different dynamics at high Wi from that in Newtonian
turbulence (Xi & Graham 2009c).
There are of course numerous problems of interest in the whole parameter space
of Re and Wi. In this dissertation, we select two representative examples for case
study: one at the low Re limit and one at the high Re limit. In the low Re regime,
we choose a cross-slot geometry and study the instability mechanism involving stagnation points (Xi & Graham 2009b). Among different types of elastic instabilities
4
reported in various experimental conditions, the best-understood are those so-called
“hoop-stress” instabilities, which occurs in viscoelastic fluid flows with curved streamlines (Larson 1992, Pakdel & McKinley 1996, Shaqfeh 1996, Graham 1998). Meanwhile, instabilities are reported in flow geometries involving stagnation points as well,
including those in rheometric flows (Chow et al. 1988, Müller et al. 1988) and microfluidics (Arratia et al. 2006). However, understanding of these instabilities was
very limited. In the cross-slot geometry we study, a stagnation point is created in
the center; experimental research reported symmetry-breaking and oscillatory instabilities in flows of polymer solutions (Arratia et al. 2006). Our goal is to understand
these instabilities via numerical simulation, in the hope that the resulting mechanism
can be applicable to a wider range of instabilities involving stagnation points. On the
other hand, in terms of computational methodology, these problems are extremely
challenging: around a stagnation point there is typically a strong extensional flow
field, in which polymer molecules are highly stretched; fully resolving the stress field
without losing numerical stability is a difficult task. This makes the stagnation point
flow an excellent test problem for numerical methods. We are interested in developing
a generalizable method of computing viscoelastic fluid flows in complex geometries
(which is very common in microfluidic applications), using a finite element package,
Comsol Multiphysics; and the stagnation point flow naturally becomes a first problem
to look at.
For high Re, we are interested in viscoelastic turbulent flows, especially in the
regime where Re is close to but above the value of laminar-turbulence transition (Xi
& Graham 2009c,a). As a well-established experimental observation, flexible polymer
solutes at a very low concentration (O(10 ∼ 100) ppm) can reduce the friction drag
5
of turbulent flows by as much as 80% (Virk 1975, Graham 2004, White & Mungal
2008). This phenomenon is of obvious practical interest because of the potential
energy savings it can bring in fluid transport applications. On the theoretical side, this
problem lies between two challenging areas: turbulence and polymer dynamics, study
of which bears the prospect of advancing the knowledge in both of them. Despite the
long history of study (since its original discovery in the 1940s by Toms (1948, 1977)),
understanding of polymer drag reduction remains very limited, especially for systems
with high Wi and large extent of drag reduction. In particular, the existence of a
universal upper-limit of drag reduction (for a given Re), the “Virk maximum drag
reduction” (Virk 1975), remains a mystery.
Computer simulation has been proven an powerful tool of reproducing the full 3D
flow fields of turbulence in both Newtonian (Moin & Kim 1982, Kim et al. 1987) and
viscoelastic systems (Sureshkumar & Beris 1997, Dimitropoulos et al. 1998), which
makes a valuable supplement to experimental research where accurate measurement
of time-dependent 3D fields, especially the stress field, is very difficult. Most previous
computational studies in viscoelastic turbulence mainly focus on statistical descriptions of the flow. In this study, we take a nonlinear-dynamics approach and try to
understand the effect of polymer on individual coherent structures (Robinson 1991)
in turbulent flows. From a dynamical-system perspective, the temporal evolution of
a turbulent coherent structure is depicted as a complex transient trajectory (Jiménez
et al. 2005, Kerswell & Tutty 2007, Gibson et al. 2008) in the state space, built
around solution objects, such as traveling waves (TWs) (Nagata 1990, Waleffe 1998,
2001, 2003, Faisst & Eckhardt 2003, Wedin & Kerswell 2004). This view has benefited research in Newtonian turbulence greatly in the past 10 ∼ 15 years in terms
6
of understanding the self-sustaining mechanism of turbulence (Hamilton et al. 1995,
Waleffe 1997, Jiménez & Pinelli 1999) and the laminar-turbulence transition process
(see e.g. Skufca et al. (2006), Wang et al. (2007), Duguet et al. (2008)). Less has
been done on the viscoelastic turbulence side. Past work (Stone et al. 2002, Stone
& Graham 2003, Stone, Roy, Larson, Waleffe & Graham 2004, Li, Stone & Graham
2005, Li, Xi & Graham 2006, Li & Graham 2007) has studied the effects of polymer
on one class of traveling waves, the “exact coherent states” (Waleffe 1998), which,
according to above, correspond to the static building blocks of the trajectory. Focus of the current study is shifted to the dynamical side: we look at the dynamical
trajectory corresponding to the predominant coherent structures in near-transition
viscoelastic turbulence; and study the influence of polymer on temporal behaviors of
these structures. The goal is to interpret the dynamics of viscoelastic turbulence in
the context of the recent progresses made in Newtonian turbulence and viscoelastic
traveling waves reviewed above, and thus obtain a physical picture of the mechanism
of drag reduction
The following contents of this dissertation are thus divided into two independent
parts, each of which contains a review of previous studies, a summary of formulation
and methods, discussion of results, conclusions and a proposal for future research.
Although the scope covered by these two projects is only a small subset of the area
of viscoelastic fluid dynamics, these examples are representative enough that the
methodology we apply in these studies and the understanding we acquire about dynamics of viscoelastic fluid flows in different parameter regimes, can prospectively
impact a broader range of research.
7
Part I
Dynamics at low Re: oscillatory
instability in viscoelastic cross-slot
flow
8
Chapter 2
Introduction: elastic instabilities
and viscoelastic stagnation-point
flows
While Newtonian flows become unstable only at high Reynolds number Re, when
the inertial terms in momentum balance dominate, flows of viscoelastic fluids such as
polymer solutions and melts are known to have interesting instabilities and nonlinear
dynamical behaviors even at extremely low Re. These “purely-elastic” instabilities
arise in rheometry of complex fluids as well as in many other applications (Larson 1992, Shaqfeh 1996). Recent studies of viscoelastic flows in microfluidic devices
broaden the scope of these nonlinear dynamical problems in low-Re viscoelastic fluid
dynamics (Squires & Quake 2005). The small length scales in microfluidic devices
enable large shear rates, and thus high Wi (Weissenberg number, Wi ≡ λγ̇, where λ
is a characteristic time scale of the fluid and γ̇ is a characteristic shear rate of the
9
flow), at very low Re. Instabilities are not always undesirable, especially when the
accompanying flow modification is controllable and can thus be utilized in the design
and operation of microfluidic devices. Specifically, instabilities have been found and
flow-controlling logic elements have been designed in a series of microfluidic geometries, e.g. flow rectifier with anisotropic resistance (Groisman & Quake 2004), flip-flop
memory (Groisman et al. 2003) and nonlinear flow resistance (Groisman et al. 2003).
Another prospective application of these instabilities is to enhancement of mixing at
lab-on-a-chip length scales (Groisman & Steinberg 2001), where turbulent mixing is
absent due to small length scales and an alternative is needed.
The best understood of these instabilities are those that occur in viscometric flows
with curved streamlines: e.g. flows in Taylor-Couette (Muller et al. 1989), TaylorDean (Joo & Shaqfeh 1994), cone-and-plate (Magda & Larson 1988) and parallelplates (Magda & Larson 1988, Groisman & Steinberg 2000) flow geometries. In
these geometries, the primary source of instability is the coupling of normal stresses
with streamline curvature (i.e. the presence of “hoop stresses”), leading to radial
compressive forces that can drive instabilities (Magda & Larson 1988, Muller et al.
1989, Larson et al. 1990, Joo & Shaqfeh 1994, Pakdel & McKinley 1996, Shaqfeh
1996, Graham 1998). Similar mechanisms drive instabilities in viscoelastic free-surface
flows (Spiegelberg & McKinley 1996, Graham 2003).
Attention in this study focuses on a different class of flows, whose instabilities
are not well-understood – stagnation point flows, like those generated with opposedjet (Chow et al. 1988, Müller et al. 1988), cross-slot (Arratia et al. 2006), two-roll
mill (Ng & Leal 1993) and four-roll mill (Broadbent et al. 1978, Ng & Leal 1993)
devices. Figure 3.1 shows a schematic of a cross-slot geometry. A characteristic
10
(a) Dye convection pattern.
(b) Contours of velocity magnitude (colors) and
streamline (dark lines) measured by particle image
velocimetry (PIV)
Figure 2.1: Symmetry-breaking instability in viscoelastic cross-slot flow (Arratia et al.
2006).
phenomenon in these stagnation point flows is the formation of a narrow region of
fluid with high polymer stress extending downstream from the stagnation point. This
region can be observed in optical experiments as a bright birefringent “strand” with
the rest of the fluid dark (Harlen et al. 1990). Keller and coworkers (Chow et al.
1988, Müller et al. 1988) reported instabilities in stagnation point flows of semi-dilute
polymer solutions generated by an axisymmetric opposed-jet device. Specifically, for
a fixed polymer species and concentration, upon a critical extension rate (or critical
Wi) polymer chains become stretched by flow near the stagnation point and a sharp
uniform birefringent stand forms. The width of this birefringent strand increases with
increasing Wi until a stability limit is reached, beyond which the birefringent strand
becomes destabilized and changes in its morphology are observed. At higher Wi,
11
the flow pattern and birefringent strand become time-dependent. Recent tracer and
particle-tracking experiments of stagnation point flow in a micro-fabricated cross-slot
geometry by Arratia et al. (2006) show instabilities of dilute polymer solution at low
Re (< 10−2 ). In their experiments fluid from one of the two incoming channels is
dyed and a sharp and flat interface between dyed and undyed fluids is observed at
low Wi. Upon an onset value of Wi, this flow pattern loses its stability: spatial
symmetry is broken but the flow remains steady (Figure 2.1). The interface becomes
distorted in such a way that more than half of the dyed fluid goes to one of the
outgoing channels while more undyed fluid travels through the other. At even higher
Wi the flow becomes time-dependent and the direction of asymmetry flips between
two outgoing channels with time. Particle-tracking images in the time-dependent flow
pattern indicate the existence of vortical structures around the stagnation point.
Another class of stagnation point flows is associated with liquid-solid or liquid-gas
interfaces, such as flows passing submerged solid obstacles, around moving bubbles
or toward a free surface. For example, McKinley et al. (1993) reported a threedimensional steady cellular disturbances in the wake of a cylinder submerged in a
viscoelastic fluid. Around a falling sphere in viscoelastic fluids, fore-and-aft symmetry
of velocity field is broken and the velocity perturbation in the wake can be away from
the sphere, toward the sphere or a combination of the two depending on the polymer
solution (Hassager 1979, Bisgaard & Hassager 1982, Bisgaard 1983).
Remmelgas et al. (1999) computationally studied the stagnation point flow in a
cross-slot geometry with two different FENE (finitely-extensible nonlinear elastic)
dumbbell models. Using the two models, they studied the effects of configurationdependent friction coefficient on polymer relaxation and the shape of the birefringent
12
strand. Their simulation approach was restricted to relatively low Wi (O(1)) with
symmetry imposed on the centerlines of all channels. Harlen (2002) conducted simulations of a sedimenting sphere in a viscoelastic fluid to explore the wake behaviors. He
explained the experimental observations of both negative (velocity perturbation away
from the sphere) and extended (velocity perturbation toward the sphere) wakes in
terms of combined effects of the stretched polymer in the birefringent strand following
the stagnation point behind the sphere and the recoil outside of the strand. Neither
of these analyses directly addressed instabilities of these flows. In the recent work
of Poole et al. (2007), a stationary symmetry-breaking instability in the cross-slot
geometry has been predicted by conducting simulations using the upper-convected
Maxwell model. This instability is similar to the first steady symmetry-breaking instability in the experiments of Arratia et al. (2006). However, the question as to why
the flow field becomes time-dependent in different geometries involving stagnation
points still needs to be addressed.
Various approximate approaches have been taken in the past to obtain an understanding of the instabilities observed in experiments. Harris & Rallison (1993, 1994)
investigated the instabilities of the birefringent strand downstream of a free isolated
stagnation point through a simplified approach, in which polymer molecules are modeled as linear-locked dumbbells, which are fully stretched within a thin strand lying
along the centerline. Polymer molecules contribute a normal stress proportional to
the extension rate only when they are fully stretched (i.e. in the strand); otherwise the
flow is treated as Newtonian. The lubrication approximation is applied for the Newtonian region and the effects of birefringent strand are coupled into the problem through
point forces along the strand. Two instabilities are reported. At low Wi (≈ 1.2−1.7),
13
a varicose disturbance is linearly unstable, in which the width of birefringent strand
oscillates without breaking the symmetry of the flow pattern. At higher Wi another
instability is observed in which symmetry with respect to the extension axis breaks
and the birefringent strand becomes sinuous in shape and oscillatory with time, with
zero displacement at the stagnation point and increasing magnitude of displacement
downstream from it. Symmetry with respect to the inflow axis is always imposed.
The mechanism of these instabilities is explained: perturbations in the shape or position of the birefringent strand affect the stretching of incoming polymer molecules
such that they enhance the perturbation after they become fully stretched and merge
into the strand. This mechanism is close to the one we are about to present later
in this study with regard to the importance of flow kinematics and the extensional
stress. However, in their linear stability analysis with which the instability mechanism is investigated, the spatial dependence of the birefringent strand in the outflow
direction is neglected. Therefore although this factor is included in their numerical
simulation, it is not taken into consideration in their explanation of the instability.
As will be shown later, according to our simulations this spatial dependence of the
birefringent strand plays an important role.
In this study, we present numerical simulation results of viscoelastic stagnation
point flow in a two-dimensional cross-slot geometry. With increasing Wi, we observe
the formation and elongation of the birefringent strand across the stagnation point.
At high Wi, we find the occurrence of an oscillatory instability. These results resemble
the experimental observations of oscillatory birefringent width by Müller et al. (1988)
and the varicose instability predicted by Harris & Rallison (1994). By analyzing the
perturbations in both velocity and stress fields, a novel instability mechanism based
14
on normal stress effects and flow kinematics is identified.
15
Chapter 3
Cross-slot geometry, governing
equations and numerical methods
We consider a fourfold symmetric planar cross-slot geometry, as shown in Figure 3.1.
Flow enters from top and bottom and leaves from left and right. For laminar Newtonian flow, two incoming streams meet at the intersection of the cross, and each
of them splits evenly and goes into both outgoing channels, generating a stagnation
point at the origin near which an extensional flow exists. We use round corners at
the intersections of channel walls in order to avoid enormous stress gradients at the
corners, which cause numerical difficulties.
The momentum and mass balances are:
Re
∂u
+ u · ∇u
∂t
= −∇p + β∇2 u + (1 − β)
2
(∇ · τ p ) ,
Wi
(3.1)
∇ · u = 0.
(3.2)
Parameters in Equations (3.1) and (3.2) are defined as: Re ≡ ρU l/ (ηs + ηp ), Wi ≡
16
l
y
x
l
l
Figure 3.1: Schematic of the cross-slot flow geometry.
17
2λU/l and β ≡ ηs / (ηs + ηp ), where ρ is the fluid density, for a dilute polymer solution
we assume it to be the same as the solvent density; ηs is the solvent viscosity and
ηp is the polymer contribution to the shear viscosity at zero shear rate; U and l are
characteristic velocity and length scales of the flow. Here l is chosen to be the halfchannel width and the definition of U is based on the pressure drop applied between
the entrances and exits of the channel. Specifically, U is defined to be the centerline
velocity of a Newtonian plane Poiseuille flow under the same pressure drop in a
straight channel with length 20l, which is comparable to the lengths of streamlines
in the present geometry. According to this definition, the nondimensional pressure
drop in our simulation is fixed at 40 and the centerline Newtonian velocity in crossslot geometry is typically slightly lower than 1 since the extensional flow near the
stagnation point has a higher resistance than that in a straight channel. The polymer
contribution to the stress tensor is denoted τ p and is calculated with the FENE-P
constitutive equation (Bird, Curtis, Armstrong & Hassager 1987):
α
1−
tr(α)
b
Wi
+
2
∂α
b
T
+ u · ∇α − α · ∇u − (α · ∇u)
=
δ,
∂t
b+2
!
b+5
α
2
τp =
− 1−
δ .
b
b+2
1 − tr(α)
b
(3.3)
(3.4)
In Equations (3.3) and (3.4), polymer chains are modeled as FENE dumbbells (two
beads connected by a finitely-extensible-nonlinear-elastic spring). Here α ≡ hQQi
is the conformation tensor of the dumbbells where Q is the end-to-end vector of the
dumbbells and h·i represents an ensemble average. The parameter b determines the
maximum extension of the dumbbells: i.e. the upper limit of tr(α).
At the entrances and exits of the flow geometry, normal flow boundary conditions
18
are applied: i.e. t · u = 0 where t is the unit vector tangential to the boundary.
Pressure is set to be 40 at entrances and 0 at exits. No-slip boundary conditions are
applied at all other boundaries. Boundary conditions for stress are only needed at the
entrances, where the profile of α is set to be the same as that for a fully developed
pressure-driven flow in a straight channel with the same Wi. Unless otherwise noted,
several parameters are fixed for most of the results we report here: Re = 0.1, β = 0.95
and b = 1000, which means we focus on dilute solutions of long-chain polymers at
low Reynolds number.
The discrete elastic stress splitting (DEVSS) formulation (Baaijens et al. 1997,
Baaijens 1998) is applied in our simulation: i.e. a new variable Λ is introduced as
the rate of strain and a new equation is added into the equation system:
Λ = ∇u + ∇uT .
(3.5)
A numerical stabilization term γ∇ · ∇u + ∇uT − Λ is added to the right-handside of the momentum balance (Equation (3.1)), and it is worthwhile to point out
that this term is only nontrivial in the discretized formulation and does not change
the physical problem. In this term, γ is an adjustable parameter and γ = 1.0 is
used in our simulations. The velocity field u is interpolated with quadratic elements,
while pressure p, polymer conformation tensor α and rate of strain Λ are interpolated
with linear elements. Consistent with Baaijens’s conclusion (Baaijens 1998), DEVSS
greatly increases the upper limit of Wi achievable in our simulations. Quadrilateral
elements are used for all variables. Our experience shows that quadrilateral elements
have great advantages over triangular ones, yielding much better spatial smoothness
19
in the stress field at comparable degrees of freedom to be solved. Another merit
of quadrilateral elements is the capability of manual control over mesh grids. This
is extremely important when certain restrictions, such as symmetry, are required.
In our simulation, finer meshes are used within and around the intersection region
of the geometry, and the mesh is required to be symmetric with respect to both
axes. Within a horizontal band (−0.2 < y < 0.2) across the stagnation point, very
fine meshes are generated to capture the sharp stress gradient along the birefringent
strand. The streamline-upwind/Petrov-Galerkin (SUPG) method (Brooks & Hughes
1982) is applied in Equation (3.3) by replacing the usual Galerkin weighting function
w with w + δhu · ∇w/kuk, where h is the geometric average of the local mesh length
scales and δ is an adjustable parameter, set to δ = 0.3 in our simulations. This
formulation is implemented using the commercially available Comsol Multiphysics
software.
20
Chapter 4
Results: viscoelastic cross-slot flow
and its oscillatory instability
4.1
Steady states
Steady-state solutions are found for all Wi investigated (0.2 < Wi < 100) in our
study. For Wi 6 60 steady states are found by time integration and for those with
larger Wi Newton iteration (parameter continuation) is used because of possible loss
of stability, as we describe below. At low Wi the velocity field is virtually unaffected
by the polymer molecules. Velocity contours at Wi = 0.2 are plotted in Figure 4.1(a);
for clarity only part of the channel is shown. A stagnation point is found at the center
of the domain ((0, 0)). In both incoming and outgoing channels, the flow is almost the
same as pressure driven flow in a straight channel. No distinct difference can be observed for the incoming and outgoing directions in velocity field. Figure 4.1(b) shows
contours of extension rate at Wi = 0.2, in which a region dominated by extensional
21
flow is found near the stagnation point. High extension rate is also found near the
corners due to the no-slip walls. The magnitude of polymer stretching can be measured by the trace of its conformation tensor tr(α), and is plotted in Figure 4.1(c).
At low Wi, the extent to which polymers are deformed is barely noticeable, but it
can be clearly seen that polymers are primarily stretched in either the extensional
flow near the stagnation point and corners, or the shear flows near the walls. At
high Wi (Wi = 50, Figure 4.2), the situation is very different. Polymers are strongly
stretched by the extensional flow near the stagnation point and this stretching effect
by extensional flow overwhelms that of the shear flow. A distinct band of highly
stretched polymers (the birefringent strand) forms (Figure 4.2(c)). Since the polymer
relaxation time in this case is larger than the flow convection time from stagnation
point to the exits, this birefringent strand extends the whole length of the simulation domain. The resulting high polymer stress significantly affects the velocity
field (Figure 4.2(a)). Regions with reduced velocity extend much farther away in the
downstream directions of the stagnation point than in the low Wi case, especially
along the x-axis, where high polymer stress dominates. Correspondingly, a reduction
in the extension rate near the stagnation point is observed, most noticeably along the
birefringent strand (Figure 4.2(b)).
Figures 4.3 and 4.4 show profiles at various values of Wi of tr(α) along the outflow (x-axis) and inflow (y-axis) directions of this stagnation point (note the difference
in scales in the two plots). For increasing Wi the length of the region with highly
stretched polymer keeps increasing due to the increased relative relaxation time (Figure 4.3). In high Wi cases (Wi = 30 and Wi = 100), polymers are not fully relaxed
even when they reach the exit of the simulation domain. The cross-sectional view of
22
(a) kuk
(b) ∂ux /∂x
(c) tr(α)
Figure 4.1: Contour plots of steady state solution: Wi = 0.2 (only the central part of
the flow domain is shown).
23
(a) kuk
(b) ∂ux /∂x
(c) tr(α)
Figure 4.2: Contour plots of steady state solution: Wi = 50 (only the central part of
the flow domain is shown).
24
Figure 4.3: Profiles of tr(α) along y = 0.
Figure 4.4: Profiles of tr(α) along x = 0 in the region very near the stagnation point.
25
(a) Birefringent strand width W .
(b) Birefringent strand length L.
Figure 4.5: Effect of Wi on the size of the birefringent strand (tr(α) > 300 is considered as the observable birefringence region).
26
Figure 4.6: Profiles of ux along y = 0.
tr(α) profiles along the y-axis (Figure 4.4) show interesting non-monotonic behaviors.
Although the height of the profile (tr(α)max ) keeps increasing upon increasing Wi,
the width of the Wi = 100 case is smaller than that of Wi = 30, resulting in a steeper
transition section between low and high stretching regions. If we arbitrarily define
tr(α) > 300 as the observable birefringence region, the width W and the length L of
the birefringent strand (measured on the inflow and outflow axes, respectively) can
be plotted as functions of Wi, as in Figure 4.5 (values of L for Wi > 30 are not shown
since they exceed the length of the simulation domain). A clear non-monotonic trend
is observed in the plot of birefringence width, where W increases sharply at relatively
low Wi and peaks around Wi = 40. After that W decreases mildly but consistently
with further higher Wi. This non-monotonic trend is consistent with experimental
observations of birefringence in opposed-jet devices (Müller et al. 1988).
Similarly, a non-monotonicity is also found in the change of velocity field with Wi.
∂
27
Figure 4.7: Average extension rate (∂ux /∂x)avg (averages taken in the domain −0.1 <
x < 0.1, −0.1 < y < 0.1).
Velocity profiles along the outflow axis are plotted in Figure 4.6. Magnitude of the
outflow velocity is obviously reduced for high-Wi flows, consistent with the breakup
of fore-aft (along the streamlines) symmetry in velocity distributions observed in
Figure 4.2(a). Comparing the profiles of Wi = 5, Wi = 30 and Wi = 100, one
can find that this suppression of outgoing flow is also non-monotonic with increasing
Wi. Changes in velocity field affect the polymer stress field via changes in the strain
rate. Shown in Figure 4.7 is the value of extension rate, averaged within a box
around the stagnation point (−0.1 < x < 0.1, −0.1 < y < 0.1), as a function of Wi.
As Wi increases, the extension rate decreases at low Wi but increases at high Wi,
with a minimum found around Wi = 40. Besides, most of experimental results are
presented in terms of Deborah number (De), defined as the product of the polymer
relaxation time and an estimate of the extension rate near the stagnation point.
28
Noticing that the average (nondimensionlized) extension rate changes within a very
narrow range (around 0.55 ∼ 0.6), a conversion De = 0.3Wi can be adopted for
comparison of our results with experimental ones.
Some understanding of this non-monotonicity can be gained by looking at Figure 4.3. Here it can be seen that for Wi . 30, the birefringent strand is not yet “fully
developed” in the sense that the polymer stretching is not yet saturating near full
extension. Thus the evolution of the velocity field in this regime of Wi reflects the significant changes that occur in the stress field in this regime. At higher Wi, however,
the polymer stress field in the strand is saturating, and thus not changing significantly. Furthermore, at these high Weissenberg numbers, the relaxation of stress
downstream of the stagnation point diminishes, decreasing the gradient ∂τxx /∂x and
thus decreasing the effect of viscoelasticity on the flow near the stagnation point.
4.2
Periodic orbits
We turn now to the stability of the steady states that have just been described. Rather
than attempting to compute the eigenspectra of the linearization of the problem,
an exceedingly demanding task, we examine stability by direct time integration of
perturbed steady states. The perturbations take the form of slightly asymmetric
pressure profiles at the two entrances (0.1% maximum deviation from the steady state
value) that are applied for one time unit, then released. As an example, temporal
evolution of the birefringent strand width W starting from the perturbed steadystate at Wi = 66, measured on the inflow axis, is plotted in Figure 4.8. The system
stays near the steady-state solution for a long time (> 1000), before the tiny initial
29
perturbation grows to a noticeable extent. In the range 1000 < t < 3100, this
deviation oscillates with increasing amplitude; then it reaches a limit. After that
the system fluctuates around the steady state with a fixed magnitude and frequency,
i.e. it approaches a limiting cycle. Figure 4.9 shows a two dimensional projection of
the trajectory of the same process. Here the velocity magnitude at a point near the
stagnation point (0.5, 0) is plotted against the birefringent strand width W measured
on the inflow axis. The system starts at the steady state with W = 0.1593 and
ux |(0.5,0) = 0.2687 and spirals outward with time after the perturbation. Eventually
the trajectory merges into a cycle (the outer dark cycle in the Figure 4.9). This clearly
identifies the existence of a stable periodic orbit. Note the anti-correlation between
ux |(0.5,0) and W , i.e. when the flow speeds up near the stagnation point, the strand
thins and vice versa. Although a finite asymmetric perturbation has been introduced
in the simulation results presented here, it is worth to mention that in order to trigger
the instability, the initial perturbation does not have to be in this particular form,
nor does it have a finite threshold. We have tested another form of perturbation
in which we add zero-mean random noises of different orders-of-magnitude onto the
initial steady state solutions and the instability can always be observed.
Figure 4.10 shows the root-mean-square deviations over one period of W from its
steady-state values, normalized by the corresponding steady-state values Ws.s. , as a
function of Wi for all the cases where we found periodic orbits. Time integrations for
Wi > 74 did not converge due to the enormous stress gradient around the corners of
the no-slip walls and the consequent numerical oscillations downstream. Data points
for Wrms computed from our simulations are fitted with a function of the form a(Wi −
Wic )p , with p fixed at 1/2. Very good agreement is found for our simulation data with
30
Figure 4.8: Evolution of the birefringence strand width W after a small initial perturbation on the steady state; inset: enlarged view of 2500 6 t 6 3200.
Figure 4.9: Two dimensional projection of the dynamic trajectory from the steady
state to the periodic orbit at Wi = 66: ux at (0.5, 0) v.s. W .
31
Figure 4.10: Left-hand axis: root-mean-square deviations of the birefringent strand
width W at periodic orbits, normalized by steady state values; right-hand axis: oscillation periods.
the 1/2 power law, characteristic of a supercritical Hopf bifurcation (Guckenheimer &
Holmes 1983). The critical Weissenberg number Wic is identified to be 64.99 by this
fitting. Also shown in Figure 4.10 are periods of oscillations, where a slight decrease
with increasing Wi is found. This is interesting since it indicates that some time scale
other than the polymer relaxation time sets the period of oscillations.
Simulations have also been conducted at other values of β and b. Within the dilute
regime, Wic has a strong dependence on the polymer concentration (∝ (1 − β)) and
the bifurcation occurs at much higher Wi for more dilute solutions. (In the Newtonian
limit β → 1, Wic must diverge.) For example, for β = 0.96, Wic lies between 80 and
82. Simulations for lower β, i.e. higher concentration, are not feasible at this point
due to numerical instabilities. For b values not very far way from 1000, changing the b
32
parameter barely affects Wic . By changing the b parameter downward to 900, Wic is
almost unchanged. However, for further smaller b values, the dependence is stronger
and Wic increases with decreasing b.
As mentioned earlier, time-dependent instabilities have been observed in viscoelastic stagnation-point flows in both opposed-jet and cross-slot geometries. In particular,
the birefringent stability found by Müller et al. (1988) is very similar to the one reported in this study. In their optical experiments with semi-dilute aPS solutions, the
width of the birefringent strand oscillates rapidly between two values in a certain
range of extension rate. Compared with their experiments, as well as the asymptotic
model of Harris & Rallison (1994), our simulation predicts a higher critical Wi. This
could be as least partially attributed to the low concentration we are looking at. In
the cross-slot geometry, time-dependent oscillations are found for De > 12.5 (Arratia et al. 2006), which is of the same order-of-magnitude as what we have observed
(Dec ≈ 0.3Wic = 19.5). Although symmetry is not imposed in our simulations, we
do not observe any symmetry-breaking instability, which according the experiments
should occur at a much lower De. This might be related to the constant-pressure drop
constraint we applied between entrances and exits. In both the experiments (Arratia et al. 2006) and simulations (Poole et al. 2007) where asymmetry is observed,
there are no restrictions on the pressure at the boundaries, and the constant-flow rate
constraint is applied instead (at the flow entrances).
33
4.3
Instability mechanism
We turn now to the spatiotemporal structure of the instability and its underlying
physical mechanism. We will denote the deviations in velocity, pressure and stress
with primes, while steady-state values will be denoted with a superscript “s”:
u = us + u0 ,
(4.1)
p = ps + p0 ,
(4.2)
α = αs + α0 .
(4.3)
0
, respectively, at intervals of 1/8
Figures 4.11, 4.12 and 4.13 illustrate u0x , u0y and αxx
period, corresponding to the periodic orbit at a Weissenberg number close to the
bifurcation point (Wi = 66). Time starts from an arbitrarily chosen snapshot on the
periodic orbit and only a quarter of the region near the stagnation point is shown;
behavior in the rest of the domain can be inferred from the reflection symmetry across
the axes.
At the beginning of the cycle (Figure 4.11(a)), u0x is positive in the region very
close to the stagnation point while it is negative in most of the downstream region.
As time goes on, this positive deviation near the stagnation point grows into a “jet”,
a region of liquid moving downstream away from the stagnation point faster than
the steady-state velocity, as shown in Figures 4.11(b), 4.11(c) and 4.11(d). Correspondingly, by continuity, the inflow toward the stagnation point is also faster as
shown in Figures 4.12(a)–4.12(d). Note that very near the stagnation point deviations from steady state remain small. At the beginning of the second half of the cycle
(Figure 4.11(e)), the jet extends further downstream and grows to the full width of
34
(a) t = 0
(b) t = 1.68
(c) t = 3.35
(d) t = 5.03
Figure 4.11: Perturbation of the x-component of velocity, u0x with respect to steady
state at the periodic orbit: Wi = 66. The region shown is 0 < x < 1, −1 < y < 0;
the stagnation point is at the top-left corner. (To be continued).
35
(e) t = 6.71
(f) t = 8.38
(g) t = 10.06
(h) t = 11.74
Figure 4.11: (Continued).
36
(a) t = 0
(b) t = 1.68
(c) t = 3.35
(d) t = 5.03
Figure 4.12: Perturbation of the y-component of velocity, u0y with respect to steady
state at the periodic orbit: Wi = 66. The region shown is 0 < x < 1, −1 < y < 0;
the stagnation point is at the top-left corner. (To be continued).
37
(e) t = 6.71
(f) t = 8.38
(g) t = 10.06
(h) t = 11.74
38
(a) t = 0
(b) t = 1.68
(c) t = 3.35
(d) t = 5.03
0
Figure 4.13: Perturbation of the xx-component of polymer conformation tensor, αxx
with respect to steady state at the periodic orbit: Wi = 66. The region shown is
0 < x < 1, −1 < y < 0; the stagnation point is at the top-left corner. The edge of
the steady state birefringent strand is the line y ≈ −0.05. (To be continued).
39
(e) t = 6.71
(f) t = 8.38
(g) t = 10.06
(h) t = 11.74
40
the channel. Meanwhile, in the region closer to the stagnation point, velocity deviations drop (Figures 4.11(e), 4.12(e)) and start to change sign (Figures 4.11(f), 4.12(f)).
Consequently, the growth of the jet ends and a “wake”, a region of fluid moving slower
than the steady-state velocity, emerges from the stagnation point (Figures 4.11(f)–
4.11(h) and 4.12(f)– 4.12(h)). Similarly, as the wake grow larger, velocity deviations near the stagnation point change signs and a new cycle starts (Figures 4.11(a)
and 4.12(a)).
The velocity deviations are closely related with those of the stress field (Fig0
ure 4.13). Generally speaking, “jets” are accompanied by negative αxx
and thus
thinning of the birefringent strand and “wakes” are associated with the birefringent
thickening. The largest deviations are found at the edges of the birefringent strand
s
/∂y is largest. Note that deviations in the stress field are always small
where ∂αxx
along the centerline of the birefringent strand because there polymer molecules are
almost fully stretched and the huge spring force is sufficient to resist any perturbations.
One may notice the small spatial oscillations in the stress field deviations, characterized by alternating high and low stress stripes, along the outflow direction. These
oscillations, apparently unphysical and centered around zero, also exist along the birefringence strand in steady-state solutions, though they are not easy to see from the
contours in Figure 4.2(c) as they are overwhelmed by the high tr(α) in the birefringent
strand. Unfortunately, as shown by recent studies (Renardy 2006, Thomases & Shelley 2007), spatial non-smoothness is inevitable in numerical simulations of viscoelastic extensional flow upon certain Wi, owning to the singularities in stress gradients.
These singularities could not be fully resolved by any finite mesh size and this prob-
41
lem would always show up in numerical solutions of high Wi viscoelastic stagnation
point flows. However, we do not expect these oscillations to qualitatively affect our
observations for a couple of reasons. First, non-smoothness has been observed in our
simulation at Wi values much lower than the critical Wi of this instability. Second,
observable non-smoothness is always found some distance away from the stagnation
point in the downstream direction while the instability is dominated by the physics in
the close vicinity of the stagnation point and, since FENE-P is a convective equation,
we do not expect anything occurring downstream to affect upstream dynamics. Last,
and most importantly, simulations with different meshes display different mesh size
dependent stripes, while the nature of the instability remains virtually unchanged.
Insight into the mechanism of this instability can be gained by examining the
0
:
linearized equation for αxx
0
s
0
2
tr(α0 )
αxx
2
αxx
∂αxx
=−
−
2
∂t
Wi 1 − tr(αs )
Wi
tr(αs )
b 1− b
b
0
∂α0
∂αs
∂αs
∂αxx
− usy xx − u0x xx − u0y xx
∂x
∂y
∂x
∂y
0
s
s
0
s ∂ux
0 ∂ux
0 ∂ux
s ∂ux
+ 2αxy
+ 2αxx
+ 2αxy
.
+ 2αxx
∂x
∂y
∂x
∂y
− usx
(4.4)
In the following analysis, terms on the right-hand-side (RHS) of Equation 4.4 are
named “RHS∗”, where “∗” is determined by the order of appearance on the RHS.
Terms and their physical meanings are summarized in Table 4.1. To understand
the mechanism of the instability, magnitudes of these terms at the point (0, −0.05)
are plotted as a function of time during roughly a period in the bottom view of
Figure 4.14. Terms RHS3, RHS5, RHS8 and RHS10 are zero by symmetry and not
plotted. This position is right at the edge of the birefringent strand and as shown
42
Term
RHS1
RHS2
Formula
Physical Significance
α0xx
2
− Wi
tr(αs )
1− b
αsxx tr(α0 )
2
− Wi
tr(αs )
b 1−
(
b
0
RHS3 −usx ∂α∂xxx
0
RHS4 −usy ∂α∂yxx
s
RHS5 −u0x ∂α∂xxx
s
RHS6 −u0y ∂α∂yxx
0
s ∂ux
RHS7 2αxx
∂x
0
s ∂ux
RHS8 2αxy
∂y
s
0 ∂ux
RHS9 2αxx
∂x
s
0 ∂ux
RHS10 2αxy
∂y
)
Relaxation.
2
Relaxation.
Convection of conformation deviations by the
steady-state x-velocity.
Convection of conformation deviations by the
steady-state y-velocity.
Convection of the steady-state conformation by xvelocity deviations.
Convection of the steady-state conformation by yvelocity deviations.
Stretching caused by deviations in the extension
rate.
Stretching caused by deviations in the shear rate.
Stretching caused by deviations in the extensional
stress.
Stretching caused by deviations in the shear
stress.
Table 4.1: Terms on the right-hand side of Equation (4.4).
in Figure 4.13, it is also where significant deviations in the stress field are observed.
Time-dependent oscillations at other places, including off the symmetry axis x =
0, have also been checked and nothing that would qualitatively affect our analysis
was seen. Correspondingly, deviations in polymer conformation, inflow velocity and
extension rate, normalized by steady-state values, are plotted in the top view of
Figure 4.14.
Consistent with our earlier observations, deviations in the velocity field (u0y and
0
∂u0x /∂x) and deviations in stress field (αxx
) are opposite in sign for most of the time
within the period. Among the terms plotted, RHS4, RHS6, RHS7 and RHS9 are
much larger than the relaxation terms, RHS1 and RHS2, and dominate the dynamics.
(Relaxation terms are large at the very inner regions of the birefringent strand and
43
Figure 4.14: Time-dependent oscillations at (0, −0.05). Top view: perturbations of
variables normalized by steady-state quantities; bottom view: magnitudes of terms
on RHS of Equation (4.4).
44
that is why oscillations in the stress field there are barely noticeable.) Moreover,
0
RHS4, RHS6 and RHS9 are mostly in phase with αxx
and thus tend to enhance the
0
and hence damps the deviations. It is
deviations while RHS7 is out of phase with αxx
the joint effect of these competing destabilizing and stabilizing forces that gives the
oscillatory behavior of the system. Finally, notice that among the three destabilizing
terms, RHS6 is the one that leads the phase and thus guides the instability.
Based on these observations from Figure 4.14, a mechanism for the instability can
be proposed, which is illustrated schematically in Figure 4.15. At the beginning of the
cycle (t = 0), u0y is slightly above zero, indicating that the inflow speed is faster than
that in the steady state. As a consequence, RHS6 becomes negative first, followed by
RHS4 and RHS9. In particular, a faster incoming convective flow brings unstretched
polymer molecules toward the stagnation point (corresponding to RHS6), as depicted
in Figure 4.15(a). These polymer chains have less time to get stretched and when they
reach the edges of the birefringent strand (e.g. dumbbell B), they are less stretched
compared with the steady state. As a result, fluid around dumbbell B has lower stress
than at the steady state, corresponding to a thinning of the birefringent strand.
Meanwhile, since dumbbell B contains smaller spring forces than its downstream
neighbors A and A’, the net forces (thick arrows) exerted by polymer on the fluid
point outward, generating jets downstream from the stagnation point. (In other
words, when the stress at the center is lower, the net stress divergence points outward,
which increases momentum in the downstream directions.) By continuity, more fluid
has to be drawn toward the stagnation point and the initial deviation in u0y is then
enhanced. However, as the flow speeds up in the vicinity of the stagnation point, the
extension rate also starts to increase. This effect (corresponding to RHS7) tends to
45
(a) Thinning process of the birefringent strand.
(b) Re-thickening process of the birefringent strand.
Figure 4.15: Schematic of instability mechanism (view of the lower half geometry).
Thick arrows represent net forces exerted by polymer molecules (dumbbells) on the
fluid.
46
stretch polymer molecules more and stabilize the deviations, as shown in Figure 4.14.
Eventually this effect will be able to overcome that of RHS6 as well as RHS4 and
RHS9, and the stress near the stagnation point starts to increase after it passes the
minimum at around t = 3.5, which causes a re-thickening of the birefringent strand
as illustrated in Figure 4.15(b). By a similar argument as that above, dumbbell C has
higher spring forces than B and B’, the dumbbells which were passing near the center
when stress was at minimum, and the net polymer forces point inward, which starts
to suppress the jets. Inflow velocity decreases as the birefringent strand thickens,
and this gives incoming polymer molecules more time to be stretched, which further
thickens the birefringent strand. Eventually αxx will come back to the steady state
value at around t = 7.2. However, since all the deviations are not synchronized, a
negative deviation is found in uy ; and an identical analysis with opposite signs can
be made for the second half of the cycle.
Within this mechanism, a sharp edge of the birefringent strand, i.e. large magnitude of ∂αxx /∂y (∼ O(104 ) in our simulations), is required so that a small u0y can give
a sufficiently large RHS6 to drive the instability. This is made possible by the kinematics of the flow near the stagnation point, where the incoming polymer molecules
are strongly stretched within a short distance. Another similar effect is that stress
derivatives are stretched in the outgoing direction and thus greatly weakened as fluid
moves downstream; therefore the instability is dominated by physics in the vicinity of
the stagnation point. In the earlier mechanism for the so-called “varicose instability”,
given by Harris and Rallison (Harris & Rallison 1994), the importance of extensional
stress and flow kinematics, especially the role of the convection of incoming molecules,
was also recognized. However, the picture described in their work is not the same as
47
ours due to the simplifications in their model. Their linear stability analysis ignored
the x-dependence of the birefringent width while in our simulations, x-dependence of
the stress field is closely related to the changes in velocity field. Besides, their analysis
did not identify a restoring force for the deviations and the oscillatory behavior could
not be explained.
48
Chapter 5
Conclusions of Part I
Using a DEVSS/SUPG formulation of the finite element method, we are able to simulate viscoelastic stagnation point flow and obtain steady-state and time-dependent
solutions at high Wi. For Wi 1, a clear birefringent strand is observed. The width
of this birefringent strand increases with increasing Wi until Wi ≈ 40 after which it
declines gradually. This also results in a non-monotonic trend in the modification of
the velocity field.
At around Wi = 65 the steady state solution loses stability and a periodic orbit
becomes the attractor in phase space. Flow motion of the periodic orbit is characterized by time-dependent fluctuations, specifically, alternating positive (jet) and
negative (wake) deviations from the steady-state velocity in the regions downstream
of the stagnation point. A mechanism is proposed which, taking account of the interaction between velocity and stress fields, is able to explain the whole process of
the oscillatory instability. Extensional stresses and their gradients, as well as the
flow kinetics near the stagnation points, are identified as important factors in the
49
mechanism. This mechanism is different from that of the “hoop-stress” instabilities,
which occur in viscometric flows with curved streamlines, and we expect that this
mechanism could be extended to explain various instabilities occurring in viscoelastic
flows with stagnation points.
50
Chapter 6
Future work: nonlinear dynamics
of viscoelastic fluid flows in
complex geometries
With the method we develop in this study, we are able to obtain numerically-stable
and smooth solutions even for very high Wi. Based on the steady-state and timedependent solutions, we proposed a novel mechanism for an oscillatory instability
involving a stagnation point. Compared with the experimental results by Arratia
et al. (2006), we do not observe any symmetry-breaking in our simulations. Even
with asymmetric initial perturbations, time intergration would eventually lead to axissymmetric steady states or periodic orbits. A probable cause for this inconsistency
is the constant-pressure-drop constraint we applied between the entrances and exits.
The Arratia et al. (2006) experiments were performed under the constant-incomingflow-rate constraint, with no restriction on the pressure drop. Same constraint was
51
used in the simulation of Poole et al. (2007) where symmetry-breaking was observed
in their steady-state solutions of the Stokes equation (Navier-Stokes equation at the
Re → 0 limit (Deen 1998)) coupled with the “upper-convected Maxwell” (UCM)
constitutive equation (Bird, Armstrong & Hassager 1987). Mechanism of symmetrybreaking was not elucidated in that study, which we propose as the next goal of
our study on low-Re viscoelastic flows. Recall in Arratia et al. (2006) that steady
symmetry-breaking observed at moderate Wi would develop into a second instability
of fluctuating asymmetry at higher Wi. It is possible that the mechanism of the second
instability is a combination of the yet-unknown symmetry-breaking mechanism and
the oscillatory instability mechanism proposed in this study. Therefore understanding
the symmetry-breaking could be a key step toward the full understanding of both
instabilities.
Beyond the cross-slot flow, our method can be extended to many other flow geometries. The biggest advantage of the finite element method is that different flow
geometries can be implemented with minimal efforts. Since stagnation-point flows are
among the most difficult to simulate for viscoelastic fluids (stress field turns singular
at high Wi (Renardy 2006, Thomases & Shelley 2007, Becherer et al. 2009)), our
method should be numerically stable for a variety of geometries, an important merit
of a method designed for microfluidic applications.
As reviewed in Chapter 2, instabilities are observed in many different geometries
involving stagnation points. Compared with the relatively better-understood class of
“hoop-stress” instability, it is interesting to see if there is any commonness on the
mechanism level among instabilities in all these stagnation-point flows. One close
example is the so-called microfluidic “flip-flop” device (Groisman et al. 2003). Its
52
(a) Overall Geometry. The auxiliary inlets (comp. 1 and
comp. 2) are for flow-rate measurement purpose.
(b) Blowup near the intersection during the instability. Only
fluids from one of the two inlets are dyed.
Figure 6.1: The microfluidic flip-flop device (Groisman et al. 2003).
53
geometry (Figure 6.1(a)) is also of the cross-channel form, and is symmetric with
respect to the incoming axis, but asymmetric with respect to the outgoing axis. Two
incoming channels are different in width, and both connect to the intersection via a
contraction; after the intersection the flow diverges through two expansions toward the
exits. Metastable asymmetric states (Figure 6.1(b)) are observed at high Wi where
stream from either of the incoming channels almost all exits from one outgoing channel. This instability appears similar to the symmetry-breaking instability observed
by Arratia et al. (2006), but the contractions and expansions near the intersection
further complicate the problem. Besides flows with isolated stagnation points studied
here, instabilities in stagnation-point flows near solid-liquid interfaces may also share
similar mechanisms. These problems include many interesting nonlinear phenomena
in flows around immersed solid objects (Bisgaard & Hassager 1982, McKinley et al.
1993, Harlen 2002), where one stagnation point exists at the separatrix of streamline
in front of the object and another at the merging axis behind the object.
Viscoelastic fluid flows without stagnation points are of interest as well. For
dilute polymer solutions, strong nonlinear effects are expected in extensional flows.
Kinematics of many flow types encountered in microfluidics include both shear and
extension components (Pipe & McKinley 2009), e.g. flow-focusing (Oliveira et al.
2009), contraction and expansion (Groisman et al. 2003, Groisman & Quake 2004,
Rodd et al. 2005). Instabilities have been observed in many of them. Beyond the
scope of viscoelastic fluid dynamics, microfluidics has been applied extensively in
the manipulation and separation of individual bio-macromolecules (see, e.g. Perkins
et al. (1997), Dimalanta et al. (2004) and Chan et al. (2004)). Although the FENE-P
(dumbbell) model we use is too coarse-grained to capture certain degrees-of-freedom
54
in the dynamics of these molecules under flow, our method is still useful in terms of
obtaining a crude estimation of the stretching and orientation of these molecules, and
thus aid in the design of experiments and more refined computational studies.
On the methodology level, our current method is limited to 2D geometries. However, most microfluidic devices are built to be three-dimensional, i.e. the depth is
at the same order-of-magnitude as the width. Some applications even require a 3D
flow geometry to function: e.g. the microfluidic chaotic mixer (Stroock et al. 2002).
Dimensionality can affect nonlinear dynamics as well: although some instabilities are
be purely two-dimensional, such as the oscillatory instability studied here, many others need all three dimensions to develop. The “hoop-stress” instability (Larson et al.
1990, Pakdel & McKinley 1996, Shaqfeh 1996, Graham 1998) is one example (which
has also been applied in microfluidics for enhancing mixing (Groisman & Steinberg
2001)). Same for the inertio-elastic instabilities observed in the contraction-expansion
microchannel fabricated by Rodd et al. (2005), where streamlines are clearly overlapping and crossing in a 2D projection. Extending the current method to 3D geometries
would greatly expand its power of predicting nonlinear phenomena in microfluidics
(which unfortunately would also cost much more computational resource; the current
study is mostly performed on a desktop PC, and each simulation takes from a few
hours to one day).
Another extension to consider is to include liquid-liquid and liquid-gas interfaces
in the simulation. Multiphase flows are very commonly seen in microfluidic devices,
in the forms of drops, bubbles, free surfaces and immiscible streams (Anna et al. 2003,
Garstecki et al. 2004, Stone, Stroock & Ajdari 2004, Atencia & Beebe 2005, Squires &
Quake 2005). In the case of viscoelastic fluids, coupling between viscoelasticity and
55
interfacial dynamics can cause further complexity in nonlinear dynamics (see, e.g.
Arratia et al. (2008a,b), Sullivan et al. (2008)). To simulate these flows, in addition
to the relatively simple task of enabling free-slip or pratial-slip boundary conditions,
the main challenge is to include mobile boundaries in the computational model.
In summary, there are mainly three directions of utilizing and expanding the
achievements of the current study: (1) to further study the mechanisms of instabilities involving stagnation points, especially the symmetry-breaking instability and the
potential similarities among different types of stagnation-point flows; (2) to apply
the current numerical method directly to more general geometries and understand
different types of instabilities; (3) to improve the method and adapt it to the needs
of more general microfluidic applications.
56
Part II
Dynamics at high Re: viscoelastic
turbulent flows and drag reduction
57
Chapter 7
Introduction: viscoelastic
turbulent flows and polymer drag
reduction
7.1
Fundamentals of polymer drag reduction
It has been experimentally observed that by introducing a minute amount of flexible
polymers (at concentrations of O(10 − 100) ppm by weight or even lower) into a
turbulent flow, the turbulent friction drag can be substantially reduced (Virk 1975,
Graham 2004, White & Mungal 2008), resulting in a higher flow rate for a given
pressure drop. The percentage drop of the friction factor can be as high as 80%
in turbulent flows in straight pipe or channel geometries. Since its initial discovery
in the 1940s (Toms 1948, 1977), the phenomenon of polymer drag reduction has
been an active area of study due to its practical and theoretical significance. It
58
LT Trans.
La m
(Pois inar Flo
w
euille
's La
w)
U+avg
Laminar
PreOnset
Intermediate DR
tio
uc
ed t e)
R
o
rag pt
D ym
um 's As
m
xi irk
Ma (V
MDR
n Wi↑
nce
urbule
)
nian T rmán Law
to
w
e
á
N
n K
dtlvo
n
ra
(P
log(Reτ)
Figure 7.1: Schamatic of the Prandtl-von Kármán plot. Thin vertical lines mark the
transition points on the typical experimental path shown as a thick solid line.
can obviously be utilized to improve energy efficiency in various fluid transportation
applications. Moreover, unraveling the physical mechanism of the phenomenon in
terms of the complex interactions between turbulence and polymer molecules would
not only expand our knowledge of polymer dynamics in fluid flows, but also provide
additional insight into the nature of turbulence itself.
Bulk flow data obtained from drag reduction experiments are very often plotted
+
in Prandtl-von Kármán coordinates, i.e. a plot of average velocity Uavg
≡ Uavg /uτ
versus friction Reynolds number Reτ ≡ ρuτ l/η. (Here, ρ is the fluid density, η is
the total viscosity, and l is a characteristic length scale of the flow geometry; the
p
friction velocity uτ ≡ τw /ρ is a characteristic velocity scale for near-wall turbulence, where τw is the mean wall shear stress; the superscript “+” denotes quantities
nondimensionlized with inner scales, i.e. velocities scaled by uτ and lengths scaled
59
Figure 7.2: Experimental data of maximum drag reduction (MDR) in pipe flow,
from different polymer solution systems and pipe sizes, plotted in √
the Prandtl-von
√
+
Kármán coordinates (Virk 1971, 1975). It can be shown that 1/ f = Uavg
/ 2,
√
√
Re f = 2Reτ (f in this plot is the friction factor, which is denoted as Cf in
this dissertation; Re in this plot is the Reynolds number based on average velocity:
Reavg ≡ ρUavg D/η, D is the pipe diameter).
60
by η/ρuτ .) A schematic Prandtl-von Kármán plot for Newtonian and polymeric flow
is shown in Figure 7.1. In a typical experiment where the polymer solution system
and the pipe/channel size are fixed, measurements made under different Reτ are connected to form a line called an “experimental path”. Along an experimental path,
Re and Wi vary simultaneously, while their ratio, defined as the elasticity number
El ≡ Wi/Re, remains constant. (Here, Re ≡ ρU l/η is the Reynolds number using the
characteristic bulk flow velocity U as the velocity scale; Wi ≡ λγ̇ is the Weissenberg
number, which is the product of polymer relaxation time λ and a characteristic shear
rate γ̇. Note that γ̇ ∝ U/l, thus El ∝ λη/ρl2 is constant when the polymer solution system and flow geometry is fixed.) A typical experimental path is sketched in
Figure 7.1 as a thick solid line. With increasing Reτ , the flow system undergoes a
series of transitions among several qualitatively different stages, including: laminar
flow, laminar-turbulence transition, turbulence before the onset of drag reduction
(pre-onset), intermediate drag reduction and the maximum drag reduction (MDR).
The boundaries of each stage (i.e. the transition points) are marked with thin vertical lines for the experimental path denoted with the thick solid line. The last stage
(MDR) is so named because it is invariant with changing polymer species, molecular
weight, concentration and geometric-confinement length scale (pipe diameter or channel height) (Virk et al. 1967, Virk 1971, 1975, Graham 2004, White & Mungal 2008).
Experimental paths of different polymer solution systems and pipe (or channel) sizes
are sketched in dashed lines. Although changing the polymer solution system and the
confinement length scale, i.e. via changes in El, would affect the slope in the intermediate DR stage as well as the points of transition, all experimental paths collapse
into a single straight line after they reach the MDR stage (Figure 7.2). This line,
61
commonly referred to as the Virk’s MDR asymptote, sets the universal upper limit of
drag reduction when polymer is used as the drag-reducing agent. Note that once this
asymptote is reached, the friction drag is solely dependent on Reτ . This universality
of the MDR stage is perhaps the most intriguing problem in polymer drag reduction.
The study of polymer-induced drag reduction thus can be divided into several
important questions: (1) what is the mechanism by which polymers alter turbulence
and reduce drag; (2) what are the qualitative changes underlying these multistage
transitions; and in particular (3) why is there a universal upper limit on drag reduction
(MDR) and what is the nature of turbulence in that regime?
7.2
Previous direct numerical simulation (DNS)
studies
None of these questions has been completely answered to date; however, advances in
computer simulations of viscoelastic turbulent flows in the past decade have substantially advanced the understanding of drag reduction. Beris and coworkers pioneered
the direct numerical simulation (DNS) of viscoelastic turbulent flows (Sureshkumar &
Beris 1997, Dimitropoulos et al. 1998) using the FENE-P (Bird, Curtis, Armstrong &
Hassager 1987) constitutive equation. Most major experimental observations in the
intermediate drag reduction regime (after onset and before MDR), including the onset
of drag reduction, thickened buffer layer, wider streak spacing, and changes in the velocity fluctuations and Reynolds shear stress profiles, were qualitatively reproduced.
Since then DNS has been adopted as a powerful tool to access the details of velocity
and polymer stress fields, and thus to infer the mechanism by which polymers reduce
62
drag. By inspecting the instantaneous snapshots of velocity fluctuations and polymer
force fields, as well as the correlation between the two, De Angelis et al. (2002) claimed
that polymer suppresses turbulence by counteracting the velocity fluctuations. (This
mechanism is also predicted by Stone et al. (2002), Stone, Roy, Larson, Waleffe & Graham (2004) and Li & Graham (2007) with a different means, as we discuss below.)
Similar results on the velocity-polymer force correlations were reported by Dubief
et al. (2004, 2005), which showed that polymer forces are anti-correlated with velocity fluctuations in the transverse directions, while in the streamwise direction these
two quantities are positively correlated in the viscous sublayer and anti-correlated for
the rest of the channel. Based on this they suggested that polymer molecules suppress
the vortical motions and meanwhile are stretched by these near-wall vortices; when
they are convected toward the wall to the high-speed streaks during the “sweeping”
events, they release the energy back to the flow and thereby aid in the sustenance
of turbulence. Another common practice to interpret DNS data is to examine the
transport equations of kinetic energy and Reynolds stresses, and describe the effects
of polymer in terms of the changes it causes to different contributions to the energy
budgets. Min, Yoo, Choi & Joseph (2003) proposed that the kinetic energy of the
turbulent flow is transferred to elastic energy by stretching the polymer molecules
very close to the wall; these stretched molecules are lifted upward to the buffer and
log-law layers to release energy back to the flow. Ptasinski et al. (2003) evaluated the
budget of each component of the turbulent kinetic energy, and found that polymer
suppresses pressure fluctuations and thus impedes energy transfer among different
components via the pressure-rate of strain term in the Reynolds stress budgets.
The studies mentioned above primarily rely on the statistical representations of
63
the three-dimensional fields. Quantities being investigated are averaged in time as
well as in the two periodic dimensions, and the analyses are mostly based on mean
profiles with dependence on the wall-normal coordinate only. Although this approach
assures the statistical certainty of the results, it eliminates most of the structural
information about the turbulent flow motions. On the other hand, in the near wall
region turbulent flows are known to be dominated by coherent structures (Robinson
1991), where most drag reduction effects caused by polymer take place. Further
understanding of the interplay between turbulent structures and polymer dynamics
requires the capability of isolating the coherent structures from the complex turbulent
background.
Information about these coherent structures can be extracted from DNS solutions a posteriori. For example, the Karhunen-Loéve analysis (or proper orthogonal
decomposition) (Holmes et al. 1996, Pope 2000) has been applied to viscoelastic turbulent flows for this purpose (De Angelis et al. 2003, Housiadas et al. 2005). Given
a set of statistically independent snapshots from the time-dependent solution of the
turbulent flow, this method constructs a series of mutually orthogonal modes, or
eigen-states, which form an optimal decomposition of the original solution in the
sense that the leading modes always contain the largest amount of turbulent kinetic
energy. These studies showed that viscoelasticity modifies the turbulent flow by increasing the amount of energy carried by the leading modes, or the energy-containing
modes. However further study is still needed to connect this finding with the complex
process of the polymer-turbulence interactions. More recently, conditional averaging
has been used to sample the predominant structures around certain local events that
contribute substantially to the turbulent friction drag (Kim et al. 2007). These re-
64
sults confirmed that polymer inhibits vortical motions, both streamwise vortices in
the buffer layer and hairpin vortices further away from the wall, by applying forces
that counter them. This is consistent with many other studies (De Angelis et al. 2002,
Stone et al. 2002, Stone, Roy, Larson, Waleffe & Graham 2004, Dubief et al. 2005, Li
& Graham 2007). Using these sampled structures as the initial conditions for time
integration, evolution of the hairpin vortices was simulated (Kim et al. 2008), and
it was found that viscoelasticity not only suppresses the primary vortices but also
prevents secondary vortices from being created.
7.3
Traveling waves and the nonlinear dynamics
perspective of turbulence
In the past decade, the discovery of three-dimensional fully nonlinear relative steadystate solutions, or traveling wave (TW) solutions, to the Navier-Stokes equation,
made the a priori study of the coherent structures a reality. These solutions are
steady states of the Navier-Stokes equation typically in a reference frame moving at a
constant speed, and they are found in all canonical wall-bounded geometries for Newtonian turbulent flows (plane Couette, plane Poiseuille and pipe geometries) (Waleffe
1998, 2001, 2003, Faisst & Eckhardt 2003, Wedin & Kerswell 2004, Pringle & Kerswell 2007, Viswanath 2007). These TWs usually appear in the form of low-speed
streaks straddled by streamwise vortices, which closely (in both structure and length
scales) resemble the recurrent coherent structures in near wall turbulence. In particular, an optimal spanwise box size of 105.51 wall units was reported for the TW
solution found by Waleffe (2003) in the plane Poiseuille geometry (Figure 7.3), which
65
Figure 7.3: Newtonian ECS solution at Re = 977 in plane Poiseuille flow; symmetric
copies at both walls are shown. Slices show coutours of streamwise velocity, dark color
for low velocity; the isosurface has a constant streamwise vortex strength Q2D = 0.008,
definition of Q2D is given in Section 10.4. This plot is published by Li & Graham
(2007); the solution is originally discovered by Waleffe (2003).
is remarkably close to experimentally observed near-wall streak spacing of about 100
wall units (Smith & Metzler 1983). Transient structures that look very similar to
these solutions have been experimentally observed (Hof et al. 2004). In the context of
drag reduction, past work has examined one family of these TW solutions, the “exact
coherent states” (ECS) (Waleffe 1998, 2001, 2003) (see Figure 7.3), of viscoelastic
turbulent flows in both plane Couette (Stone et al. 2002, Stone & Graham 2003,
Stone, Roy, Larson, Waleffe & Graham 2004) and plane Poiseuille geometries (Li,
Stone & Graham 2005, Li, Xi & Graham 2006, Li & Graham 2007). Not only do the
viscoelastic ECS solutions show drag reduction compared with their Newtonian counterparts, they also capture many characteristics of drag-reduced turbulence, including
reduced vortical strength and changes in turbulence statistics. Consistent with DNS
66
results (De Angelis et al. 2002, Dubief et al. 2005, Kim et al. 2007), polymer influences the flow structures and causes drag reduction in ECS by counteracting velocity
fluctuations and vortical motions (Stone et al. 2002, Stone, Roy, Larson, Waleffe &
Graham 2004, Li & Graham 2007). Viscoelasticity also changes the minimal Re at
which ECS exist; under fixed Re and with high enough Wi, these solutions are totally
suppressed by the polymer (Stone, Roy, Larson, Waleffe & Graham 2004, Li, Xi &
Graham 2006, Li & Graham 2007). Based on these studies, a simple framework containing different stages of the ECS solutions in the parameter space, which includes
the laminar-turbulence transition, the onset of drag reduction and the annihilation of
ECS, was proposed (Li, Xi & Graham 2006, Li & Graham 2007). With the hypothesis
that the annihilation of ECS is linked with MDR, this framework covered most key
transitions in viscoelastic turbulent flows.
Although viscoelastic ECS solutions do provide new insight into the problem of
drag reduction, they are only fixed points (i.e. steady states) in the state space. On
the other hand, the dynamics of the coherent structures is a complex time-dependent
trajectory, so further investigation into the coherent turbulent motions requires the
study of transient solutions. DNS studies mentioned earlier belong to this category,
but in most of them periodic simulation boxes much larger than the characteristic
length scales of the coherent structures are used. Transient solutions obtained from
that approach typically involve a large number of coherent structures convoluted with
one another, and include the long range spatial correlations between them, which
makes the identification and analysis of individual coherent structures difficult. The
most straightforward way to isolate the transient solution corresponding to an individual coherent structure is the “minimal flow unit (MFU)” approach: i.e. by limiting
67
(a) Dynamical trajectory of a turbulent transient in a MFU visualized in the state space
using coordinates proposed by Gibson et al. (2008).
(b) Same trajectory (dotted line) visualized in the context of TW solutions (solid dots,
except uLM , which is the laminar state) and their unstable manifolds (solid lines).
Figure 7.4: Dynamics of turbulence in the solution state space in a plan Couette
flow (Gibson et al. 2008).
68
the simulation box to the smallest size that still sustains the turbulent motion, only
the very essential elements of the self-sustaining process of turbulence will be included in the simulation. This approach was first adopted by Jiménez & Moin (1991)
in Newtonian turbulent flows. The minimal spanwise box size in inner scales they
found, L+
z ≈ 100, is in very good agreement with the experimental measurement of
the streak spacing in the viscous sublayer (Smith & Metzler 1983), and this value
is insensitive to the change of Re. The minimal streamwise box size they reported
is dependent on Re and falls in the range of 250 . L+
x . 350, which is also consistent with experimental measurements of the streamwise structure spacings (Sankaran
et al. 1988). By comparing transient trajectories of MFU simulations with various
TW solutions in certain 2D projections of the state space, Jiménez et al. (2005) described the dynamical process of MFU in plane Poiseuille and Couette geometries
as a combination of relatively long-time stays in the vicinity of the TWs (“equilibrium”) and intermittent excursions away from these states (“bursting”). A different
result is obtained in pipe flows: Kerswell & Tutty (2007) proposed several correlation
functions as quantitative measurements of the distance between transient solutions
and TWs, and observed that the transient turbulent trajectories only visit the TWs
about 10% of the time and more complex objects in the state space, such as periodic
orbits, are necessary for a good approximation of the time-dependent solutions. In
the plane Couette geometry and using coordinates constructed with upper-branch
ECS solutions and symmetry arguments, Gibson et al. (2008) visualized the trajectories of MFU solutions together with the TW states and their unstable manifolds
in a geometrical view of the state space, with which the connection between transient solutions and the dynamical structure formed by TWs can been clearly seen
69
(Figure 7.4).
All these studies on MFU are focused on Newtonian turbulent flows. Despite the
simplicity behind the MFU idea, this approach has not been applied in the study
of viscoelastic turbulence and drag reduction, partially due to the additional degrees
of freedom in the parameter space when polymer is introduced. While Newtonian
flows can be characterized by a single parameter Re, this is no longer true in polymer
solutions where polymer species, molecular weight and concentration can also affect
the flow dynamics and hence minimal box sizes. To search for minimal box size
therefore becomes a highly computationally demanding task when variations in all
parameters are taken account of. Since the term MFU is often generalized by other
authors (e.g. Min, Yoo, Choi & Joseph (2003), Ptasinski et al. (2003), Dubief et al.
(2005)) to describe DNS in relatively small, but not necessarily minimal, boxes, here
we clarify that in this study, the term “minimal flow unit ” or MFU refers exclusively
to a flow determined via a size minimization process. That is, for each parameter
setting, different box sizes should be tested in order to determine a minimal size
at which turbulence persists. In this study, as we will discuss in Chapter 9, the
minimization process is only taken in the spanwise direction, while the streamwise
box size is fixed at the value of the Newtonian MFU. The goal of the current work is
to find the MFU of viscoelastic turbulence under a variety of parameters and observe
the transitions among different stages in terms of drag reduction behaviors.
70
Figure 7.5: The Virk (1975) universal mean velocity profile for MDR in inner scales:
+
Umean
= 11.7 ln y + − 17.0.
7.4
Multistage transitions
A classical picture of multistage transitions in viscoelastic turbulence includes: preonset turbulence, intermediate DR (after onset and before MDR) and MDR (Figure 7.1). Compared with the extensive studies of the intermediate DR regime summarized earlier, the research on MDR is very limited. Though there is a certain
degree of understanding of the phenomenon of how polymer additives reduce turbulent drag, the origin of the universal upper limit in the MDR stage remains very
poorly understood.
Early theory of Virk (1975) assumed that drag reduction only occurs in the buffer
layer; as viscoelasticity increases, thickness of this layer increases, and MDR is reached
when the buffer layer dominates the whole flow geometry. This view is similar to the
71
conclusion drawn from the elastic theory by Sreenivasan & White (2000), that at
MDR the length scale of turbulence structures affected by polymer is comparable
with that of the flow geometry, and indeed this view is consistent with the results
presented below. Based on these views, phenomenological models have been developed to predict mean velocity profiles, in which quantitative agreement with the Virk
MDR profile was reported (Benzi et al. 2006, Procaccia et al. 2008). These theories
achieved various levels of success in predicting many experimental results; however,
discrepancies are still found with some other observations, as discussed by White &
Mungal (2008). In addition, all these theoretical studies are based on average (in both
space and time) quantities, the lack of information in these models about turbulent coherent structures and their spatiotemporal behavior limits their ability to contribute
to a physical picture of the dynamics underlying experimental observations.
Among the few DNS studies on MDR, most efforts are dedicated to reproducing
the Virk mean velocity profile of MDR (Ptasinski et al. 2003, Dubief et al. 2005, Li,
Sureshkumar & Khomami 2006): i.e. they look for parameter settings under which the
mean velocity profile of DNS is the same as or close to that of experimentally observed
MDR at Re far from transition, which according to Virk (1971, 1975) is universal in
inner scales for a wide range of Re (Figure 7.5). The only exception to our knowledge
is the work of Min, Choi & Yoo (2003), where the convergence of DR% with increasing
Wi is used to identify MDR. In that study, DR% of several Wi are calculated with
other parameters held fixed, and the last two points on the high Wi end show almost
the same DR%. (The percentage of DR, DR% ≡ (Cf,s − Cf )/Cf,s × 100%, where
2
Cf ≡ 2τw /(ρUavg
) is the friction factor of the viscoelastic fluid flow, and Cf,s is the
friction factor of the flow of pure solvent.) As mentioned earlier, MDR is a stage
72
where the friction factor is only dependent on Re, and is unaffected by variations
in Wi and other polymer-related properties; therefore the problem of MDR is the
mechanism by which the same friction factor is preserved at fixed Re with changing
polymer parameters. This mechanism cannot be studied without simulation data at
MDR for a range of different parameter settings. Furthermore, whether one should
expect the same mean velocity profile in DNS studies as that of Virk is uncertain:
first, most experiments on MDR are conducted at relatively high Re, and the lack of
experimental measurements in the regime close to the laminar-turbulence transition
makes it hard to conclude whether the Virk profile is valid at Re comparable to those
in many DNS studies; second, the widely used FENE-P constitutive equation is a
highly simplified model for polymer molecules and how well it can quantitatively
predict the mean velocity at MDR is still unknown. In the simulations of Min, Choi
& Yoo (2003), the mean velocity profile after drag reduction reaches the limit at high
Wi is clearly lower than the Virk MDR profile; Dubief et al. (2005) also reported that
the Virk MDR profile is only obtained in a relatively small simulation box, and is
not found in large-box simulations; a small box is also used in the study of Ptasinski
et al. (2003). The only DNS study that predicts mean velocity profiles comparable
to Virk’s in large simulation boxes is Li, Sureshkumar & Khomami (2006). However,
they did not report the universal convergence of the mean velocity. In the present
work, we use the criterion that the friction factor converges with Wi to identify
regimes representing the asymptotic behavior of MDR.
In recent years, an additional distinction was noticed within the intermediate DR
regime between a low degree of drag reduction (LDR) and a high degree of drag
reduction (HDR). This difference was investigated by Warholic, Massah & Hanratty
73
(1999) in their channel (plane Poiseuille) flow experiments, where differences between
LDR and HDR appear in several flow statistical quantities, including: (1) mean velocity profile: LDR has the same log-law slope as Newtonian turbulent flows while
HDR shows larger slope of the log-law; (2) streamwise velocity fluctuation profile:
at LDR the magnitude of fluctuations (in inner scales) increases with DR% and the
location of the peak shifts away from the wall, while at HDR fluctuations are greatly
suppressed compared with Newtonian turbulent flows; (3) wall-normal velocity fluctuation profile: fluctuations are suppressed in both cases, but at LDR there is still a
recognizable maximum in the profile while at HDR the maximum is not observable;
(4) Reynolds shear stress profile: at LDR the Reynolds shear stress decreases with
DR but the profile retains the same slope as that of Newtonian turbulent flows at
large distance away from the wall, while at HDR the Reynolds shear stress is almost
zero across the channel and the slope farther away from the wall also changes significantly. Some of these differences have also been noted by several other groups
through both experiments (Ptasinski et al. 2003) and simulations (Min, Choi & Yoo
2003, Ptasinski et al. 2003, Li, Sureshkumar & Khomami 2006). Most authors tend
to treat these differences as quantitative effects of the percentage of drag reduction
DR%, and DR% ≈ 30% − 40% is commonly adopted as the separating point between
LDR and HDR.
7.5
About this study
As stated earlier, in this work we look for MFU solutions, i.e. transient solutions
containing the minimal self-sustaining structures, of viscoelastic turbulent flows. A
74
wide range of the parameter space is sampled in order to provide a complete picture
of the different stages in terms of drag reduction behaviors. Note that although experimental measurements are typically made following paths with constant El, along
which Re and Wi are varying simultaneously (ref. Figure 7.1), in this study (as in
many others, for example, Sureshkumar & Beris (1997), Min, Choi & Yoo (2003),
and Li, Sureshkumar & Khomami (2006)), we focus on the behavior as a function of
Wi while holding Re fixed. As shown by the vertical arrow in Figure 7.1, one can
still visit all different stages of transition, on different experimental paths, by varying
Wi under fixed Re; the advantage of doing so is that the MDR stage can be easily
identified as a plateau on the bulk flow rate versus Wi curve. Our results show that all
the stages of transition previously reported from both experiments and full-size DNS
studies, including pre-onset turbulence, LDR, HDR and a high-Wi regime showing
the asymptotic behavior of MDR, are observed in these transient solutions in MFUs.
In particular, we do identify a regime in which DR% converges with increasing Wi,
which should recall the experimental observation of MDR. We have varied all the
parameters (except Re) in the system and there is no observable difference in the
bulk flow rate with changing parameters once that asymptotic stage is reached. This
is to our knowledge the first report of a universal upper-limit of DR% in numerical
simulations, which matches the qualitative experimental hallmark of MDR: the bulk
flow rate is only a function of Re. In addition, all simulation results reported in this
study are obtained at a Re lower than any previously published DNS study, close to
the laminar-turbulence transition. The fact that all these key stages of viscoelastic
turbulence can be studied in the parameter regime close to the laminar-turbulence
transition, as predicted in earlier work (Li, Xi & Graham 2006, Li & Graham 2007), is
75
important not only from the computational point of view (computational cost grows
rapidly with increasing Re), but also in terms of the understanding of the turbulent
structures (at Re this low, the near-wall coherent structures dominate the whole flow
geometry and are easier to observe). We also need to mention that the highest DR%
reached in our simulations in only in the range of 20 − 30%, which is clearly below the
separating point between LDR and HDR identified in other studies. The fact that
the LDR–HDR transition exists under such low DR% indicates that it is a transition
between two qualitatively different stages during the drag reduction process instead
of a quantitative difference caused by the amount of drag reduction.
Following contents of this part are organized as follows. Chapter 8 summarizes the
mathematical formulation and numerical method. In Chapter 9 we discuss in detail
the process of finding minimal flow units. Discussion of our observations during these
transitions (Chapter 10) is divided into several sections: we start with an overview of
the multistage-transition scenario in MFU solutions for a variety of polymer-related
parameters (Section 10.1); then we present flow and polymer confirmation statistics (Sections 10.2, 10.3) at different stages for the sake of comparison with existing
publications; finally in Section 10.4 we study the spatiotemporal structure of the selfsustaining dynamics, which provides insight into the changes in turbulence dynamics
accompanying the multistage transitions.
Although the asymptotic MDR-like stage observed in Chapter 10 recover the universality of MDR, its degree of drag reduction is much lower than the experimentallyobserved Virk MDR. The Virk MDR profile can not be statistically reproduced in
our current MFU study. Nevertheless, a more careful inspection of the spatiotemporal dynamics of MFU does point a new direction of understanding the long-lasting
76
mystery of Virk MDR. Analysis of these results are included in Chapter 11, the motivation of which will only be clear after the discussions in Section 10.4. Conclusions
of our study on viscoelastic turbulence are summarized in Chapter 12, after which,
some discussions will be given about how these results, especially those reported in
Chapter 11, may lead to new levels of understanding of the Virk MDR, as well as
many other problems in drag-reduced turbulence (Chapter 12).
77
Chapter 8
DNS formulation and numerical
method
8.1
Flow geometry and governing equations
We consider a channel flow (plane Poiseuille, see Figure 8.1) geometry in which the
flow is driven by a constant mean pressure gradient. The x, y and z coordinates are
aligned with the streamwise, wall-normal and spanwise directions, respectively. The
no-slip boundary condition is applied at the walls and periodic boundary conditions
are adopted in the x and z directions; the periods in these directions are denoted Lx
and Lz . All lengths in the geometry are nondimensionalized with the half channel
height l of the channel and the velocity scale is the Newtonian laminar centerline
velocity U at the given pressure drop. Time t is scaled with l/U and pressure p with
78
Lx
Lz
y
2l
z
x
Figure 8.1: Schematic of the plane Poiseuille flow geometry: the box highlighted in
the center with dark-colored walls is the actual simulation box, surrounded by its
periodic images.
Stretching
Q0
Equilibrium
b /3
(
=
ax
1 /2
)
Q0
Qm
c
tra
e
R
tio
ce
or
F
n
s
Figure 8.2: Schematic of the finitely-extensible nonlinear elastic (FENE) dumbbell
model for polymer molecules.
79
ρU 2 . The conservation equations of momentum and mass give:
∂v
β 2
2 (1 − β)
+ v · ∇v = −∇p +
∇ v+
(∇ · τ p ) ,
∂t
Re
ReWi
(8.1)
∇ · v = 0.
(8.2)
Here, Re ≡ ρU l/(ηs + ηp ) (ρ is the total density of the fluid, and (ηs + ηp ) is the
total zero-shear rate viscosity; hereinafter, subscript “s” represents “solvent”, i.e.
the Newtonian fluids, and subscript “p” represents the polymer contribution) and
Wi ≡ 2λU/l, which is the product of the polymer relaxation time λ and the mean
wall shear rate. Under this definition, the friction Reynolds number, defined as Reτ ≡
√
ρuτ l/(ηs + ηp ), can be directly related to Re: i.e. Reτ = 2Re. The viscosity ratio
β ≡ ηs /(ηs + ηp ) is the ratio of the solvent viscosity and the total viscosity. For dilute
polymer solutions, 1 − β is approximately proportional to the polymer concentration.
The last term on the right-hand-side of Equation (8.1) captures the polymer effects
on the flow field, where the polymer stress tensor τ p is modeled by the FENE-P
constitutive equation (Bird, Curtis, Armstrong & Hassager 1987):
α
1−
tr(α)
b
Wi
+
2
∂α
b
T
+ v · ∇α − α · ∇v − (α · ∇v)
=
δ,
∂t
b+2
!
α
b+5
2
δ .
τp =
− 1−
b
b+2
1 − tr(α)
b
(8.3)
(8.4)
In Equations (8.3) and (8.4), polymer molecules are modeled as FENE dumbbells: two
beads connected by a finitely-extensible nonlinear elastic (FENE) spring (Figure 8.2).
The variable α is the nondimensional polymer conformation tensor α ≡ hQQi, where
Q is the end-to-end vector of the dumbbells. The parameter b defines the maximum
80
extensibility of the dumbbells; max (tr (α)) 6 b.
In this study, we fix Re = 3600 (Reτ = 84.85) and span the parameter space
at three different (β, b) pairs, (0.97, 5000), (0.99, 10000) and (0.99, 5000), with a
large range of Wi for each β and b. The importance of β and b becomes apparent
in considering the exetensibility parameter Ex, defined as the polymer contribution
to the steady-state stress in uniaxial extensional flow, in the high Wi limit. For the
FENE-P model, Ex = 2b(1 − β)/3β. For a dilute solution (1 − β 1), significant
effects of polymer on turbulence are only expected when Ex 1. For the three sets
of β and b given above, the values of Ex are 103.09, 67.34 and 33.67, respectively.
8.2
Numerical procedures
The coupled problem of Equations (8.1), (8.2), (8.3) and (8.4) is integrated in time
with a 3rd-order semi-implicit time-stepping algorithm: linear terms are updated
with the implicit 3rd-order backward differentiation method and nonlinear terms are
integrated with the explicit 3rd-order Adams-Bashforth method (Peyret 2002). The
continuity equation (Equation (8.2)) is coupled with the momentum balance (Equation (8.1)) with the influence matrix method (Canuto et al. 1988). The alternating
form is used to evaluate the inertia term in Equation (8.2): we switch between the
convection form v · ∇v and the divergence form ∇ · (vv) upon each time step (Zang
1991).
The Fourier-Chebyshev-Fourier spatial discretization is applied in all variables
and nonlinear terms are calculated with the collocation method. The numerical grid
spacing for the streamwise direction is δx+ = 8.57, and in the spanwise direction we
81
adjust the number of Fourier modes according to the varying box width (as discussed
in Chapter 9) to keep the grid spacing roughly constant, in the range of 5.0 6 δz+ 6
+
5.5; in the wall-normal direction 73 Chebyshev modes are used, which gives δy,min
=
+
= 3.7 at the channel center at Re = 3600. The time step
0.081 at the walls and δy,max
size is determined from the CFL stability condition: for the simulations reported in
this study, since the spatial grid spacing is fixed, a constant time step δt = 0.02 is
used.
An artificial diffusivity term 1/(ScRe)∇2 α is added to the right-hand side of
Equation (8.3), a common practice to improve numerical stability in pseudo-spectral
simulations of viscoelastic fluids (Sureshkumar & Beris 1997, Dimitropoulos et al.
1998, Housiadas & Beris 2003, Ptasinski et al. 2003, Housiadas et al. 2005, Li,
Sureshkumar & Khomami 2006, Kim et al. 2007). In this study, we use a fixed
value of the Schmidt number, Sc = 0.5, which gives a constant artificial diffusivity of 1/(ScRe) = 5.56 × 10−4 . The magnitude of this artificial diffusivity is at the
same order of those used by previous studies of other groups, typically O(10−4 ); an
additional diffusive term at this order of magnitude should not affect the numerical
solutions significantly while it helps to the numerical stability greatly. With the introduction of this term, an additional boundary condition is needed for Equation (8.3),
for which we used the solution without the artificial diffusivity (same as many other
DNS studies, e.g. Sureshkumar & Beris (1997)): i.e. we update the α values at the
walls without the artificial diffusivity term first; using these results as the boundary
values, we solve Equation (8.3) with the artificial diffusivity term added to update
the α field for the rest of the channel.
Numerical parameters listed above apply to most of the results in this part, with
82
exceptions of those discussed in Section 11.2. Detailed formulation for the numerical
scheme is provided in Appendix A.
83
Chapter 9
Methodology: minimal flow units
(MFU)
+
The dimensions of the simulation box (L+
x , Lz ) determine the longest wavelengths
captured in the numerical solutions. As introduced in Section 7.3, the MFU approach
finds the transient solutions of Equations (8.1), (8.2), (8.3) and (8.4) that correspond
to the self-sustaining coherent structures by finding the smallest box in which turbulent motions are sustained. Note that this minimal box size is in general a function of
all parameters in the system, i.e. Re, Wi, β and b, this minimization process has to be
performed for each different parameter combination. In Newtonian MFUs, a roughly
+
constant value L+
z ≈ 100 is found for different magnitudes of Re whereas Lx decreases
with increasing Re (Jiménez & Moin 1991). Experimentally measured steak spacings
in turbulent flows of polymer solutions are larger than the 100 wall units found for
Newtonian turbulent flows, and also increase with increasing DR% (Oldaker & Tiederman 1977, White et al. 2004). This observation is consistent with large-box DNS
84
results, where the length scales of spanwise spatial correlation functions increase with
increasing DR% (Sureshkumar & Beris 1997, De Angelis et al. 2003, Li, Sureshkumar
& Khomami 2006). Therefore, L+
z larger than that of the Newtonian MFUs is expected in our search for viscoelastic MFUs. Viscoelasticity increases the correlation
length scales in the streamwise direction as well. In particular, Li, Sureshkumar &
Khomami (2006) reported that the streamwise correlation length is increased by more
than an order of magnitude when DR% increases from 0 to 60% or more. As a result,
a significantly longer simulation box is required to capture all these long-range correlations at high DR%. Consistently, the optimal length scales, in both streamwise and
spanwise directions, of viscoelastic ECS solutions increase with increasing Wi (Li &
Graham 2007).
A rigorous search of MFUs should consider the parameter dependence of both L+
x
and L+
z , a task involving impractically large number of simulation runs. In this study,
+
we fix L+
x = 360 and focus on the variation of Lz only. Although both length scales
depend on parameters, L+
z is arguably the quantity of more interest: the dominant
structures at the Re we study are the streamwise streaks and the streamwise vortices
aligned alongside them, thus L+
z directly restricts the streak spacing and the size of
the vortices whereas L+
x only imposes a periodicity in the longitudinal direction. The
fact that we are able to find sustained turbulence in various stages of transitions at
fixed L+
x = 360, which is in the range of Newtonian-MFU streamwise sizes (Jiménez
& Moin 1991), indicates that the minimal streamwise box size may not change as
much as the streamwise correlation length does.
Note that there is not a widely-accepted definition of “sustained turbulence”;
in fact, the question of whether turbulence sustains indefinitely after the laminar-
85
350
Laminar
Turbulent
300
L+
z
250
200
150
↓ Single-wall Turbulence ↓
100
0
5
10
15
Wi
20
25
30
35
Figure 9.1: Summary of simulation results: “Turbulent” indicates that at least one
simulation run gives sustained turbulence within the given time interval (Newtonian
and β = 0.97, b = 5000).
turbulence transition or eventually decays after some long but finite life time is still
subject to controversy (Hof et al. 2006, Willis & Kerswell 2007). Here we take a
pragmatic approach to this issue by checking the persistence of turbulent motion
within a fixed time interval. In all results reported in this study, we use a statistically
converged MFU solution at an adjacent parameter (typically with a slightly different
Wi and/or L+
z ) as the initial condition, and declare that sustained turbulence is found
if the turbulent motions do not decay after 12000 time units, which is at the same order
as but larger than the longest time scale in the system (O(Re)). Figure 9.1 summarizes
our results with Newtonian runs and viscoelastic runs at β = 0.97, b = 5000. With the
exception of one Newtonian run where we use L+
z = 105.51 (this is the size of the ECS
solution when it starts to appear in a “optimal” box (Waleffe 2003)), at each Wi we
86
+
test different L+
z with an increment of ∆Lz = 10, and whether sustained turbulence
is found or not is recorded with filled and open symbols, respectively. Consistent
with all previous studies, L+
z of MFU has an obvious dependence on Wi and increases
almost monotonically with Wi. There is some roughness on the boundary between
the regions where turbulence persists and where it does not, however, this seeming
inconsistency is a natural consequence of the sensitivity of near-transition turbulence
to initial conditions: with the same parameters and box size, some initial conditions
will laminarize and some will not. Nevertheless, the laminar-turbulent boundary in
Figure 9.1 should still serve as a reasonable estimate of the smallest spanwise size
of the self-sustained turbulent motions. Similarly, some simulation runs with box
sizes larger than the minimal values still laminarize, especially at the high Wi side.
Results reported in the rest of this study are primarily from simulation runs with
the minimal L+
z , i.e. on the boundary of filled and open symbols in Figure 9.1.
+
The exceptions are those with L+
z < 140, where Lz = 140 is used instead of the
actual minimal values, because it is found that at Re close to the laminar-turbulence
transition, when L+
z is relatively small, turbulence very often tends to sustain near
only one wall of the channel, while near the other wall the flow is almost laminar. One
explanation is when Re is very low, the size of the coherent structure is comparable to
and sometimes larger than the half channel height, so that the channel is geometrically
not high enough to accommodate structures at both walls. This kind of “single-wall
turbulence” was also reported by Jiménez & Moin (1991) at Re near the laminar–
turbulence transition, and is highly undesirable in our study since the flow statistics
are strongly biased by the laminar side. Empirically, we find that this problem does
+
+
not show up for L+
z > 140. This truncation of Lz at the low Lz limit would of course
87
render our simulations there inconsistent with the definition of MFU; fortunately, as
shown later, this problem does not affect the viscoelastic turbulence after the onset
of drag reduction, the regime we are most interested in.
88
Chapter 10
Results: observations during
multistage transitions
10.1
Overview of the multistage-transition scenario
In this section, we present MFU simulation results of viscoelastic flows at various
parameters. Most of the results are presented in the form of statistical averages
(averages in time as well as in either the x and z dimensions or all three spatial
coordinates depending on the figure). Each viscoelastic simulation run is 12000 timeunits long, and we discard the solution of the first 4000 time units to avoid any
possible initial-condition dependence. Temporal averages are taken in the last 8000
time units of each simulation run. The Newtonian simulation is 20000 time units long
and the last 16000 time units are included in the statistics.
The foremost quantity of interest with regard to drag reduction is the average
streamwise velocity, as plotted in Figure 10.1 against Wi at different β and b. As we
89
0.38
30.0%
0.37
Asymptotic Upper Limit of DR
25.0%
DR Onset
0.36
20.0%
15.0%
0.34
DR%
Uavg
0.35
10.0%
0.33
β = 0.97, b = 5000, Ex = 103.1
β = 0.99, b = 10000, Ex = 67.3 5.0%
β = 0.99, b = 5000, Ex = 33.7
0.32
0.0%
Newtonian
0.31
0
5
10
15
20
25
Wi
30
35
40
45
50
55
Figure 10.1: Variations of the average streamwise velocity with Wi at different β and
b values (average taken in time and all three spatial dimensions); the corresponding
DR% is shown on the right ordinate. Solid symbols represent points in the asym-DR
stage (defined in the text); the horizontal dashed line is the average of all asym-DR
points.
90
report the simulation runs with the minimum L+
z that sustains turbulence as long
as L+
z > 140, the box size for different data points in Figure 10.1 are in general
different; the specific box size used for each data point is reported in Figure 10.3.
The corresponding amount of drag reduction, measured in terms of the percentage
drop of the friction factor DR%, is marked on the right ordinate. The error bars
on the plot show the error estimates of the time-averaged quantity with the blockaveraging method (Flyvbjerg & Petersen 1989). All three curves from different β
and b are qualitatively similar and here we start by taking the β = 0.97, b = 5000
curve as an example. At Wi 6 16, Uavg remains at the same level as the Newtonian
turbulent flow, which apparently belongs to the pre-onset turbulence stage. After the
onset, DR% increases monotonically with Wi until Wi > 27, where it starts to level
off and converges to a limit. Within the range 27 6 Wi 6 30, Uavg is approximately
independent of Wi. Recall in Section 7.4 that MDR is identified by the convergence
of the friction factor (subject to the statistical fluctuations in the data), and thus
Uavg in this plot, upon increasing Wi, this range of Wi hence corresponds to the
MDR stage for β = 0.97, b = 5000. As discussed below, one main difference of this
asymptotic upper-limit of drag reduction from the experimentally-observed MDR,
is that its mean velocity profile is much lower than the Virk (1975) MDR profile.
For the convenience of discussion, we will refer to this stage as “asym-DR” in the
following text, instead of MDR, despite that it recovers the most important feature
of the latter. Simulation runs with Wi > 30 all eventually become laminar within
the 12000 time-unit interval, regardless of the L+
z chosen. On the remaining two
curves, β = 0.99, b = 10000 and β = 0.99, b = 5000, the onset of drag reduction also
occurs at about Wionset & 16, but the increasing slope with Wi is different. The trend
91
of changing slope is consistent with changes in the extensibility number; higher Ex
corresponds to steeper rise of Uavg after onset. At the high Wi end, asym-DR stages
can be identified in both curves, at 32 6 Wi 6 36 and 40 6 Wi 6 50 respectively,
after which the flow laminarizes. There is no discernible difference in Uavg among the
asym-DR stages for all three curves. Despite the range of parameters, all of them give
DR% ≈ 26%, i.e. the friction drag at asym-DR is constant for a given Re in spite of
variations in Wi, β and b. This is to our knowledge the first time this universal aspect
of MDR is reproduced in numerical simulations. Figure 10.1 summarizes the whole
data set we will present and discuss in the rest of this chapter, from which we clearly
see that major components of the transitions in viscoelastic turbulent flows, including
the pre-onset stage, intermediate DR and a universal asymptotic upper limit of drag
reduction, are well-captured by the transient solutions in MFUs, even at Re very close
to the laminar-turbulence transition.
Housiadas & Beris (2003) reported full-size DNS results at Reτ = 125 (Re =
7812.5), β = 0.9, b = 900 (Ex = 66.67) and various Wi up to 125. With these
parameters, the onset occurs at Wionset ≈ 6, smaller than but within the same order
of magnitude as our estimation in MFUs. Studies on ECS solutions (Li, Xi & Graham
2006, Li & Graham 2007) predict that Wionset = O(10) and decreases slowly with
increasing Re. As to the dependence of drag reduction on Wi, Housiadas & Beris
(2003) found that Uavg increases monotonically with Wi for the whole range of Wi they
studied; however, the slope drops greatly at Wi ≈ 50. They did not see a complete
convergence of Uavg for the range of Wi they studied. It is unclear with which stage
in our MFU solutions their slowly-growing stage (50 . Wi 6 125) matches. Since
DR% keeps on increasing, it should be naturally categorized in the intermediate
92
DR stage (before asym-DR); however, we do not observe any appreciable change in
the increment slope against increasing Wi in our study. One possibility is that this
slowly-growing stage does not exist at the Re we study; since our Re is lower and
our simulations only include the structures in the buffer layer and part of the loglaw layer, the decrease in slope may be related to those structures excluded in our
simulation but included in theirs. Meanwhile we can not disprove the other possibility
that this stage corresponds to our asym-DR stage. Since our asym-DR stages only
span limited ranges of Wi, even if there indeed is a weak dependence on Wi, the
actual difference in Uavg must be small. Although our time average is taken in the
interval of 8000 time units, longer than most previous studies, the interval is still at
the same order of the longest time scale (O(Re)) in the system. Therefore our error
bars are not small enough to let us discern subtle changes in Uavg , if any exist. To
further reduce the errors would require increasing the length of each simulation run
by multiple times, which is practically infeasible in terms of the computational cost.
Nevertheless, since we have multiple (4 − 6) points for each β and b, where we do not
see any consistent dependence on Wi, for the present we will treat these solutions as
time-dependent coherent structures with the same DR%.
The mean velocity profiles of several typical points on the β = 0.97, b = 5000
curve in Figure 10.1 are plotted in Figure 10.2; for comparison, the asymptotic lines
of the viscous sublayer (U + = y + ), the log-law layer of Newtonian turbulent flows
(U + = 2.44 ln y + + 5.2) (Pope 2000) and the universal profile of MDR summarized
by Virk (U + = 11.7 ln y + − 17.0) (Virk 1975), are also shown on the same plot. All
profiles from our simulations collapse well on that of viscous sublayer at y + 6 5.
Further away from the wall, the Newtonian profile deviates from the U + = y + line
93
20
Newtonian, L+
z = 140
18
Wi = 16.0, L+
z = 140
Wi = 17.0, L+
z = 150
Wi = 19.0, L+
z = 150
14
Wi = 23.0, L+
z = 180
12
Wi = 27.0, L+
z = 210
U+
16
10
Wi = 29.0, L+
z = 250
8
6
Viscous Sublayer
4
Log-law for Newtonian Flows
Virk’s MDR Proﬁle
2
0 0
10
1
10
y+
Figure 10.2: Mean velocity profiles (Newtonian and β = 0.97, b = 5000).
in the buffer layer. Even though Re is too low in the present simulations for the
log-law layer to be fully developed, the Newtonian profile still lies very close to the
semi-empirical log-law at y + & 50. Among the viscoelastic cases, except that of
Wi = 16 which belongs to the pre-onset stage, the mean velocity profiles are all
elevated compared to the Newtonian case outside the viscous sublayer. The last two
curves, Wi = 27 and Wi = 29, are selected from the asym-DR stage and they collapse
well onto each other, although they are still notably lower than the Virk MDR profile.
We will further discuss the mean velocity profiles in Section 10.2.
In Figure 9.1 we presented the dependence of MFU box sizes on Wi at β = 0.97,
b = 5000; in Figure 10.3 we show the L+
z values for all data points in Figure 10.1.
For the reason explained in Chapter 9, we use a minimum of L+
z = 140 if the actual
minimal box size is smaller than this value. This truncation affects at most up to
94
260
β = 0.97, b = 5000
240
β = 0.99, b = 10000
β = 0.99, b = 5000
L+
z
220
200
180
160
140
0
5
10
15
20
25
Wi
30
35
40
45
50
55
Figure 10.3: Spanwise box sizes used in this study for various parameters. Solid
symbols represent points in the asym-DR stage.
Wi 6 16 for β = 0.97, b = 5000 and β = 0.99, b = 10000, and Wi 6 24 for
β = 0.99, b = 5000, which mostly belongs to the pre-onset stage. At higher Wi, L+
z
is larger than 140 and should faithfully reflect the size of the minimal self-sustaining
coherent structures, which increases with increasing Wi with some uncertainty owing
to the initial-condition dependence (Chapter 9). A somewhat surprising finding is
that this trend persists in the asym-DR stage: L+
z changes with Wi despite the
converged mean velocity profile and flow rate. This result suggests that different
points within the asym-DR stage in Figure 10.1 are distinguishable from one other,
i.e. they are not identical solutions, but rather different dynamical structures with
the same average velocity. We will further examine the similarities and differences
among these solutions at the asym-DR stage in the later sections of this chapter.
Comparing results at different β and b, the L+
z in the asym-DR stage are close in
95
magnitude, and they all fall into the range of 200 − 260, about twice the size of a
Newtonian MFU.
A natural question one may raise is about the legitimacy of comparing Uavg from
different parameters (Wi, β, b) in Figure 10.1 while the box size is changing. We would
like to address this point by first pointing out that for DNS study in general, unlike Re,
Wi, β or b, L+
z itself is not a physically meaningful input parameter for the simulation
that can be adjusted independently; it is indeed an artificial restriction, same as L+
x,
on the spatial periodicity of the solution. Therefore the most straightforward way of
treating the box size is to make it much larger than the longest possible correlation
in the physical system; changing box size in that regime will have no effect on the
statistics of the solution. This is what we have referred to as “full-size” DNS. For box
size smaller than that, the concern of solution statistics being affected is alway present.
There is no reason to believe that a fixed L+
z would be better than a varying one in
this sense. On the contrary, intrinsic length scales exist in turbulent flows, over which
coherent structures are spatially recurrent; in Newtonian turbulence, this length scale
is about 100 wall units in the spanwise direction as reviewed in Section 7.4. Small-box
simulations should take advantage of this inherent spatial-periodicity: box-sizes that
are multiples of these length scales should in principle minimize the box-size effects.
In viscoelastic flows, these length scales vary with polymer properties, and box size
should be adjusted accordingly. Finally, remember that the purpose of using MFU is
to isolate individual coherent structures and analyze their dynamics, while inevitably
sacrificing statistical accuracy to some extent. It is a model-based approach subject
to future verifications in full-size DNS.
Previous discussions focus on the dependence of the bulk flow rate and the length
96
260
β = 0.97, b = 5000
β = 0.99, b = 10000
240
β = 0.99, b = 5000
220
Asymptotic Upper Limit
of DR: 26.0%
L+
z
HDR
200
180
160
LDR
140
0
5
10
15
DR%
20
25
30
Figure 10.4: Variations of spanwise box size at different DR%. Solid symbols represent
points in the asym-DR stage.
scales of MFUs on various parameters (Wi, β, b); here we examine the existence of
possible structure-flow rate correlations by plotting L+
z against DR% in Figure 10.4. It
is interesting to note that the dependence of L+
z on DR% is insensitive to the changes
in β and b: data points from different β and b roughly fall onto a single relationship,
i.e. for any given DR% before the asym-DR stage, the corresponding values of L+
z
for different β and b are very close to one another. Note that the step size in our
+
MFU search is ∆L+
z = 10; the discrepancies among the Lz ∼ DR% relationships of
different β or b are smaller than the methodological uncertainty. Figure 10.4 shows
that the structural length scale increases monotonically after the onset of DR until
+
asym-DR is reached at DR% ≈ 26%, where L+
z seems to diverge: i.e. Lz increases
with approximately constant DR% and eventually turbulence does not sustain even
at larger boxes.
97
Within the intermediate DR stage, an additional transition can be identified at
DR% ≈ 13% − 15% by a sharp change in the slope of the increasing L+
z with DR%.
Note that in Figure 9.1, L+
z is about 140 wall units at the onset of DR, and after
that L+
z only increases by about 10 wall units when DR% reaches ∼ 13 − 14%. From
DR% ≈ 15% to just before asym-DR (DR% ≈ 25%), L+
z increases from ∼ 160
to ∼ 200 wall units. This transition divides the intermediate DR stage into two
parts, to which we will refer as LDR and HDR in the following text. As mentioned
in Section 7.4, the terms LDR and HDR are commonly used by other authors for
DR% . 35% and DR% & 35% respectively, whereas in this study they are used
to describe a qualitative transition within the intermediate stage. This transition is
further discussed below.
In summary, we have found transient viscoelastic turbulence solutions in MFU at
various Wi, β and b at a Re close to the laminar-turbulent transition, each of which
lasts > 12000 units in time. By studying the parameter-dependence of the bulk flow
Uavg and the structural length scale L+
z , the whole multistage transition sequence,
including pre-onset, LDR, HDR and an asym-DR stage corresponding to the MDR
regime, where DR% reaches a universal upper limit, is observed, even though the
highest DR% we observe is less than 30%.
10.2
Flow statistics
We start our discussion of turbulent flow statistics by revisiting the mean velocity
profiles in Figure 10.2. The six viscoelastic runs shown in that plot are selected from
the pre-onset (Wi = 16), LDR (Wi = 17, Wi = 19), HDR (Wi = 23) and asym-
98
20
18
16
Viscous Sublayer
Virk’s MDR Proﬁle
14
U+
12
10
8
6
4
2
0 0
10
1
10
y+
Figure 10.5: Mean velocity profiles of 15 different asym-DR states (Wi: 27 ∼ 30 for
β = 0.97, b = 5000, Wi: 32 ∼ 36 for β = 0.99, b = 10000 and Wi: 40 ∼ 50 for
β = 0.99, b = 5000).
DR (Wi = 27, Wi = 29) stages, respectively. The two curves at asym-DR overlap
each other. In Figure 10.5, mean velocity profiles of all runs in the asym-DR stage,
including those of other Wi not shown in Figure 10.2, and those at different β and
b, are plotted together. All these profiles from different Wi, β and b collapse well
onto a single curve. This profile is clearly lower than Virk’s MDR profile, but is
universal to different polymer properties in our simulations. Within the intermediate
DR stage (Figure 10.2), there is also a difference between LDR and HDR. The two
LDR profiles (Wi = 17 and Wi = 19), although shifted upward compared with the
Newtonian profile, still keep roughly the same slope in the log-law layer. Most of the
drag reduction occurs in the buffer layer, while the log-law layer seems unaffected and
stays parallel with the Newtonian log-law, which is thus described as the “Newtonian
99
0.12
Newtonian, L+
z = 140
0.1
Wi = 17.0, L+
z = 150
Wi = 19.0, L+
z = 150
dU + dU +
dy+ -( dy + )newt
0.08
Wi = 23.0, L+
z = 180
Wi = 29.0, L+
z = 250
0.06
0.04
0.02
0
−0.02
0
10
20
30
40
y+
50
60
70
80
Figure 10.6: Deviations in mean velocity profile gradient from that of Newtonian
turbulence (β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR:
Wi = 29.
plug” by Virk (1975). In the HDR stage (Wi = 23), consistent with the experimental
observations of Warholic, Massah & Hanratty (1999), a change in the log-law slope
can also be noticed, although it is not as large as those reported at higher Re, where
DR% is much higher. The log-law slope of the HDR profile is higher and lies between
that of the Newtonian turbulence and that of asym-DR.
To see this difference more clearly, in Figure 10.6 we plot deviations in the gradients of the mean velocity profiles from that of the Newtonian profile for several
selected runs. Note that with the constant-pressure-drop constraint, the mean wall
shear stress should be the same for all runs; in Figure 10.6 the mean shear rate values
at y + = 0 of viscoelastic solutions are slightly higher than that of the Newtonian
solution owing to the shear-thinning effect. Beyond the viscous sublayer, drag reduc-
100
0.1
β = 0.97, b = 5000
β = 0.99, b = 10000
dU + /dy+ |y+ =40
0.09
β = 0.99, b = 5000
0.08
HDR
0.07
0.06
LDR
0.05
0.04
0
5
10
15
DR%
20
25
30
Figure 10.7: Magnitude of mean velocity profile gradient at y + = 40. Solid symbols
represent points in the asym-DR stage.
tion is reflected in the increase of the gradient. For LDR (Wi = 17, 19), this increase
is mainly localized in the buffer layer and a reflection of the curves can be noticed
at y + ≈ 40 after which the deviations are rather small. In HDR and asym-DR, the
change of gradient is large and clear across the channel, except in the viscous sublayer
(y + 6 5) where no big change is expected. This difference is not specific to the conditions shown in Figures 10.2 and 10.6; it also exists between LDR and HDR when
β = 0.99, b = 10000 and β = 0.99, b = 5000. In Figure 10.7 we plot the magnitude
of dU + /dy + , measured at y + = 40, versus DR% for all MFU runs. The dependence
of mean velocity profile gradient on DR% is roughly the same (within statistical uncertainty) at different values of β and b. A distinction in this trend can be noticed
between relatively low and high DR%: significant increase of the gradient above the
buffer layer is only observed at DR% & 14%, before which change in the gradient is
101
Newtonian, L+
z = 140
Wi = 17.0, L+
z = 150
0.6
Wi = 19.0, L+
z = 150
Wi = 23.0, L+
z = 180
0.5
-vx vy /u2τ
Wi = 29.0, L+
z = 250
0.4
0.3
0.2
0.1
0
0
10
20
30
40
y+
50
60
70
80
Figure 10.8: Profiles of the Reynolds shear stress (Newtonian and β = 0.97, b = 5000).
LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR: Wi = 29.
small. This change well coincide with the LDR–HDR transition as identified from
Figure 10.4.
Recall that in Section 10.1, we defined the stages of LDR and HDR according to
the sudden change in the L+
z vs. DR% relationship; here we demonstrated that this
transition corresponds well to the change in the log-law slope observed by other groups
between low DR% and high DR% at higher Re (Warholic, Massah & Hanratty 1999,
Min, Choi & Yoo 2003, Ptasinski et al. 2003, Li, Sureshkumar & Khomami 2006).
This is why we choose to use the terms “LDR” and “HDR”, notwithstanding that
our highest DR% is less than 30%. The fact that this transition can be observed at
DR% ≈ 13 − 15% suggests that this corresponds to a qualitative transition in the
process of drag reduction instead of the quantitative effect of DR%. Consequently,
we also expect that the DR% of the LDR–HDR transition should be a function of Re.
102
0.02
-vx vy /u2τ + (vx vy /u2τ )newt
0
−0.02
−0.04
−0.06
−0.08
Newtonian, L+
z = 140
Wi = 17.0, L+
z = 150
−0.1
Wi = 19.0, L+
z = 150
−0.12
Wi = 23.0, L+
z = 180
Wi = 29.0, L+
z = 250
−0.14
−0.16
0
10
20
30
40
y+
50
60
70
80
Figure 10.9: Deviations in Reynolds shear stress profiles from that of Newtonian
turbulence (β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR:
Wi = 29.
0.49
0.47
-vx vy /u2τ |y+ =40
LDR
0.45
HDR
0.43
0.41
β = 0.97, b = 5000
β = 0.99, b = 10000
β = 0.99, b = 5000
0.39
0
5
10
15
DR%
20
25
30
Figure 10.10: Magnitude of Reynolds shear stress at y + = 40. Solid symbols represent
103
Similarly, a distinctive change also occurs in the Reynolds shear stress profiles
during the LDR–HDR transition. As shown in Figure 10.8, Reynolds shear stress is
suppressed with increasing drag reduction. Comparing the profiles of LDR (Wi = 17,
19) and HDR, asym-DR (Wi = 23, 29), one can notice that at LDR, −vx0 vy0 /u2τ is
suppressed mainly in the buffer layer (5 . y + . 30), and in the region y + & 40 the
deviation is barely noticeable; whereas at HDR and asym-DR, suppression is observed
even near the center. Deviations of −vx0 vy0 /u2τ with respect to the Newtonian profile
are plotted in Figure 10.9, where this difference is clearer: in HDR and asym-DR,
magnitude of deviation is substantial across the entire channel except the viscous
sublayer. This distinction between local and global suppression of the Reynolds shear
stress is also observed at the LDR–HDR transitions at the other values of β and b we
studied. As shown in Figure 10.10, at y + = 40 (above the buffer layer), Reynolds sheer
stress is substantially suppressed only after the LDR–HDR transition, which occurs at
DR% ≈ 13 − 15%. Warholic, Massah & Hanratty (1999) reported that the magnitude
of Reynolds shear stress is significantly lower in HDR than in Newtonian flow, and it
eventually drops to almost zero at DR% > 60%. While in some other studies, nonzero (though still significantly smaller than the Newtonian case) Reynolds shear stress
was reported even for cases with more than 70% drag reduction (Warholic, Massah
& Hanratty 1999, Min, Choi & Yoo 2003, Ptasinski et al. 2003, Li, Sureshkumar
& Khomami 2006). Based on our study, these seemingly contradicting results can
be well reconciled: Figures 10.8 and 10.10 show that −vx0 vy0 /u2τ remains at the same
order of magnitude as the Newtonian value even in our asym-DR stage. Therefore, the
quantitative magnitude of −vx0 vy0 /u2τ is not the key difference between LDR and HDR;
it instead might be affected by both DR% and Re. It is the location where −vx0 vy0 /u2τ
104
3
2.5
1.5
vx2
1/2
/uτ
2
Newtonian, L+
z = 140
1
Wi = 17.0, L+
z = 150
Wi = 19.0, L+
z = 150
0.5
Wi = 23.0, L+
z = 180
Wi = 29.0, L+
z = 250
0
0
10
20
30
40
y+
50
60
70
80
Figure 10.11: Profiles of root-mean-square streamwise and wall-normal velocity fluctuations (Newtonian and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23;
asym-DR: Wi = 29.
is suppressed that qualitatively indicates the transition. Indeed, despite the difference
in the magnitude of Reynolds shear stress reported in those studies (Warholic, Massah
& Hanratty 1999, Min, Choi & Yoo 2003, Ptasinski et al. 2003, Li, Sureshkumar &
Khomami 2006), one common observation is Reynolds shear stress is substantially
suppressed near the channel center only after the LDR–HDR regime. This agreement
is yet another evidence that this transition, initially identified in the L+
z vs. DR%
plot (Figure 10.4), corresponds to the LDR–HDR transition observed in other studies
at much higher Re.
The root-mean-square velocity fluctuation profiles are shown in Figures 10.11,
10.12 and 10.13. After the onset of drag reduction, the streamwise velocity fluctuations (Figure 10.11) increase with Wi until asym-DR is reached; meanwhile the peak
105
0.7
0.6
0.4
vy2
1/2
/uτ
0.5
0.3
Newtonian, L+
z = 140
Wi = 17.0, L+
z = 150
0.2
Wi = 19.0, L+
z = 150
Wi = 23.0, L+
z = 180
0.1
0
0
Wi = 29.0, L+
z = 250
10
20
30
40
y+
50
60
70
80
Figure 10.12: Profiles of root-mean-square wall-normal velocity fluctuations (Newtonian and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR:
Wi = 29.
0.9
0.8
0.7
0.5
1/2
/uτ
0.6
vz2
0.4
Newtonian, L+
z = 140
Wi = 17.0, L+
z = 150
0.3
Wi = 19.0, L+
z = 150
0.2
Wi = 23.0, L+
z = 180
Wi = 29.0, L+
z = 250
0.1
0
0
10
20
30
40
y+
50
60
70
80
Figure 10.13: Profiles of root-mean-square spanwise velocity fluctuations (Newtonian
and β = 0.97, b = 5000). LDR: Wi = 17, 19; HDR: Wi = 23; asym-DR: Wi = 29.
106
of the profile moves away from the wall, reflecting the thickening of the buffer layer.
Both the wall-normal (Figure 10.12) and spanwise (Figure 10.13) velocity fluctuations
are suppressed with increasing Wi.
As to the LDR–HDR transition, the spanwise velocity fluctuation profiles show
most notable differences between these two stages. In Figure 10.13, the LDR profiles
resemble that of the Newtonian turbulence in the shape, though they are lower in
the magnitude. In particular, one can notice two bulges at y + ≈ 16 and y + ≈ 46
between which the curves are concave. This subtle concavity is absent in the HDR
and asym-DR stages, and in those stages this part of the curve is roughly straight.
Therefore, unlike turbulence in LDR where the spanwise velocity fluctuations are almost uniformly suppressed across the channel, in HDR and asym-DR stages, more
suppression occurs in the buffer layer and the lower edge of the log-law layer. This is
also observed in data at other values of β and b, but has not previously been reported
in the literature. Meanwhile, Warholic, Massah & Hanratty (1999) reported experimentally that there is a maximum in the wall-normal velocity fluctuation profiles
when DR% . 35% whereas when DR% is high, the maximum becomes unrecognizable. It is unclear though whether this is a quantitative effect of the substantially
suppressed wall-normal velocity fluctuations, as at high DR% their vy02
1/2
/uτ magni-
tude is one order of magnitude smaller than that of the Newtonian profile, and the
noise of measurements can be comparable with the actual velocity fluctuation. In our
results (Figure 10.12), there is a very subtle maximum at y + ≈ 45 in the Newtonian
profile as well. As Wi increases, this bulge decreases in height and shrinks in size,
with the lower edge moving away from the wall. At the HDR and asym-DR stages,
the profile is almost flat after the initial uprising region near the wall, and the bulge
107
becomes unrecognizable. This effect, however, is not as obvious as the changes in the
spanwise velocity fluctuations.
It has also been reported experimentally that notable differences can be observed
in the streamwise velocity fluctuations between low DR% and high DR% (Warholic,
Massah & Hanratty 1999): when DR% . 35%, vx02
1/2
/uτ increases with DR% and
the peak of the profile moves away from the wall; at high DR%, vx02
1/2
/uτ is greatly
suppressed compared with the Newtonian flows. However, as shown in Figure 10.11,
this non-monotonicity is not observed in our MFU simulations; instead, our vx02
1/2
/uτ
profiles at different stages all follow the former (low DR%) case in experiments. DNS
studies from other groups reported contradictory results on whether or not this nonmonotonic trend exists in streamwise velocity fluctuations (Ptasinski et al. 2003, Min,
Choi & Yoo 2003, Li, Sureshkumar & Khomami 2006). The origin and significance of
this discrepancy are not understood, but the fact that in those studies, comparisons
between different DR% were made under different constraints (constant flow rate
vs. constant pressure drop) may have contributed to the complexity in this issue.
Our observation (that the trend is monotonic) is consistent with Li, Sureshkumar &
Khomami (2006), where the constant-pressure-drop constraint was also applied.
We have shown earlier that in the asym-DR stage, the mean velocity profiles converge to a single curve (Figure 10.5); here we resume the discussion of the turbulence
statistics in this stage. In Figure 10.14 we plot the RMS velocity fluctuations (left ordinate) and Reynolds shear stress (right ordinate) profiles for all the simulation runs
in the asym-DR stage (corresponding to the solid data points in Figure 10.1) with a
variety of Wi, β and b. The profiles of wall-normal and spanwise velocity fluctuations
converge for different parameters. The situation of the streamwise component is a bit
108
3.5
0.8
0.7
vx
3
0.6
2.5
-vx vy
1.5
0.4
-vx vy /u2τ
v 2
1/2
/uτ
0.5
2
0.3
1
vz
0.2
0.5
0.1
vy
0
0
10
20
30
40
y+
50
60
70
80
0
Figure 10.14: Profiles of root-mean-square velocity fluctuations and Reynolds shear
stress at 15 different asym-DR states (Wi: 27 ∼ 30 for β = 0.97, b = 5000, Wi: 32 ∼ 36
for β = 0.99, b = 10000 and Wi: 40 ∼ 50 for β = 0.99, b = 5000).
complicated: the profiles from different parameters are very close to one another near
the wall and reach maxima at very similar values in the buffer layer; while beyond
the buffer layer, they spread out. To detect any possible parameter dependence of
vx02
1/2
/uτ , we have examined the distributions of its magnitudes with respect to Wi, β
and b. Even though the vx02
1/2
/uτ profiles do not merge in the asym-DR stage, there
is no identifiable trend of dependence of vx02
1/2
/uτ on any of the parameters: vx02
1/2
/uτ
neither increases nor decreases with increasing Wi consistently in the asym-DR stage,
and the same applies for the other two parameters (β and b). Therefore we believe
that this dispersion of vx02
1/2
/uτ profiles in Figure 10.14 is a result of statistical uncer-
tainty: it might take much longer simulation runs to obtain reliable averages on the
streamwise velocity fluctuations than many other quantities we have discussed. As
109
0.25
Wi = 16.0, L+
z = 140
Wi = 17.0, L+
z = 150
Wi = 19.0, L+
z = 150
0.2
Wi = 23.0, L+
z = 180
Wi = 27.0, L+
z = 210
0.15
tr(α)/b
Wi = 29.0, L+
z = 250
0.1
0.05
0
0
10
20
30
40
y+
50
60
70
80
Figure 10.15: Normalized profiles of the trace of the polymer conformation tensor
(β = 0.97, b = 5000). Pre-onset: Wi = 16; LDR: Wi = 17, 19; HDR: Wi = 23;
asym-DR: Wi = 27, 29.
to the Reynolds shear stress, the convergence is very good over most of the channel
except in a small region near the maxima of the profiles at y + ≈ 30; this discrepancy,
as we have also examined, is again due to statistical uncertainty.
10.3
Polymer conformation statistics
We turn now to the statistics of the polymer conformation tensor. Figure 10.15 shows
the mean profiles of the trace of the polymer conformation tensor α, which physically
corresponds to the square of the end-to-end distance of the polymer chains, normalized
by its upper limit b, for several selected Wi with β = 0.97 and b = 5000. Perhaps the
most interesting observation is that although it is expected that polymers are more
highly stretched as Wi increases, this trend goes on in the asym-DR stage. The two
110
0.4
β = 0.97, b = 5000
β = 0.99, b = 10000
tr(α)avg /b
0.3
β = 0.99, b = 5000
0.2
0.1
0
0
5
10
15
20
25
Wi
30
35
40
45
50
55
Figure 10.16: Averaged trace of the polymer conformation tensor (average taken in
time and all three spatial dimensions). Solid symbols represent points in the asym-DR
stage.
curves belonging to the asym-DR stage in Figure 10.15 do not overlap: i.e. tr(α) keeps
on increasing with Wi even though the mean velocity (as well as many other velocity
statistical quantities) converges. This trend is confirmed in Figure 10.16, where the
average tr(α) normalized by b is plotted against Wi for all simulation runs reported
here. Data points in the asym-DR stage are filled. For every β and b, tr(α)avg /b
increases monotonically with Wi: the slope is relatively low at Wi ∼ O(1); after
the onset of drag reduction (Wi & 16), the curves are steeper and tr(α)avg /b roughly
rises in straight lines; tr(α)avg /b continues to increase at approximately constant slope
even after asym-DR is reached. In addition, the ranges of tr(α)avg /b at the asym-DR
stages of different β or b are far apart from one another even though their Uavg is very
close: for example, at β = 0.99 and b = 10000, tr(α)avg /b is more than twice as large
111
as that of β = 0.99 and b = 5000, and almost three times the magnitude at β = 0.97
and b = 5000. Similar to our findings, Housiadas & Beris (2003) reported in their
DNS studies that while the increase of mean velocity slows down at high Wi, tr(α)
continues to increase with Wi at about the same rate.
Another observation from Figure 10.15 is that the profile changes shape with
increasing Wi. At relatively low Wi, tr(α) decreases monotonically with distance
away from the wall y + . At higher Wi, the profile becomes non-monotonic with a
maximum some distance from the wall (in the buffer layer). This distance increases
with increasing Wi. This observation can be explained kinematically. The process of
near-wall polymer stretching is a combined effect of shear flow in the viscous sublayer
and extensional flow in the buffer layer. The former is relatively more effective in
stretching polymers at low Wi and the latter dominates at higher Wi; consequently
the maximum location reflects the shift of the dominant kinematic effect.
Although the separation between the maximum location of tr(α) and the wall
in Figure 10.15 might be thought to coincide with the LDR–HDR transition, this
agreement is totally fortuitous: unlike the changes in turbulent flow statistics we
studied earlier, this accordance between the Wi where the maximum shifts away
from the wall and the Wi at the LDR–HDR transition is specific to the choice of
β = 0.97 and b = 5000. In Figure 10.17 we plot the location of the maximum of
tr(α) against DR% and Wi, respectively, for all β and b we studied. One can see
from Figure 10.17(a) that although the detachment of the maxmum from the wall
occurs at DR% ≈ 15% for β = 0.97 and b = 5000, close to the LDR–HDR transition,
it takes place at much lower DR% under other β and b (far before the LDR–HDR
transition). Meanwhile, the dependence on W i for different β and b is very close. This
112
12
β = 0.97, b = 5000
y+ of the maximum of tr(α)
10
β = 0.99, b = 10000
β = 0.99, b = 5000
8
6
4
2
0
−5
0
5
10
DR%
15
20
25
30
(a) Dependence on DR%
y+ of the maximum of tr(α)
12
10
8
6
4
β = 0.97, b = 5000
β = 0.99, b = 10000
2
0
0
β = 0.99, b = 5000
5
10
15
20
25
Wi
30
35
40
45
50
55
(b) Dependence on Wi
Figure 10.17: Position of the maximum in the tr(α) profile. Solid symbols represent
113
is consistent with the above explanation that this displacement of the maximum is a
Wi-effect: polymer react to different local kinematics differently with increasing Wi;
the small differences between data from different β and b values is accounted for by
differences in local strain rates at the same Wi. The lack of correlation between the
maximum location of tr(α) profiles and DR%, and the increasing tr(α) in the asymDR stage where the mean velocity converges, suggest that the mean deformation of
polymer chains is a process independent of the transitions among LDR, HDR and
asym-DR (or MDR in experiments).
Polymers exert their influence on the flow field through the polymer force term,
f p = 2(1 − β)/(ReWi)(∇ · τ p ). Consequently, one might intuitively expect f p to
saturate in the asym-DR stage, instead of α or τ p , so that polymer would contribute
equally to the momentum balance (Equation (8.1)) despite the differences in the
magnitude of polymer stress. However f p profiles do not converge in the asymDR stage either, although the discrepancies of f p among different parameters are
significantly smaller than those of tr(α).
10.4
Spatio-temporal structures
Above we discussed statistical representations of the velocity and polymer conformation fields of MFU solutions during the multistage transitions. As MFU solutions
contain the structural information of the essential self-sustaining process of turbulence, we study here the spatial and temporal images of these transient structures.
Figures 10.18, 10.19, 10.20 and 10.21 show the spatial-temporal patterns in z and t
of the shear rate ∂vx /∂y at the lower wall y = −1, at fixed streamwise location of
114
x = 0, taken from one selected run for each of Newtonian turbulence, LDR, HDR and
asym-DR. The choice of x is arbitrary since the system is translation-invariant in x.
The distribution in the z direction of the wall shear rate is recorded every time unit
and plotted in color-scale in the axes of t and z + . A length of 8000 time units of each
simulation run, after turbulence reaches the statitistically-steady range, is included
in the plot. To aid interpretation, two periods in z are shown. Along with the wall
shear rate patterns, the spatially-averaged velocity Ubulk is also shown. Note that the
time-dependence of Ubulk is physically meaningful only in minimal flow units; in a
full-size DNS solution, the spatial average of any quantity should in principle be the
same as the ensemble average, and should be invariant with time. Also plotted is the
z average of the wall shear rate h∂vx /∂yiz as a function of time; note that the time
average of this quantity is 2 owing to the fixed pressure gradient constraint.
Figure 10.22 shows representative snapshots of the velocity field during different stages. Two snapshots are selected for each simulation run that is shown in
Figures 10.18, 10.19, 10.20 and 10.21, marked with (Reg) and (LS) in their captions according to the criterion to be discussed below. In each of them, isosurfaces for two quantities are plotted in a 3D view of the simulation box. The flat
translucent sheets with pleats are isosurfaces of streamwise velocity vx , taken at
the magnitude of 0.6vx,max , where vx,max is the maximum value of vx in the domain for the given snapshot. The pleats correspond to low-speed streaks, where
slowly-moving fluid near the wall is lifted upward toward the center. The tube-like
objects with opaque dark colors are the isosurfaces of a measure of the streamwisevortex strength Q2D , whose definition we now describe. We apply a modified version
of the Q-criterion of vortex identification (Jeong & Hussain 1995, Dubief & Del-
Figure 10.18: Dynamics of the self-sustaining turbulent structures in a selected Newtonian simulation (Re =
+
3600, L+
x = 360, Lz = 140). Top panel: spatial-temporal patterns of the wall shear rate (∂vx /∂y taken at x = 0;
two periodic images are shown for each case. Note: the mean value is 2 owning to the fixed pressure gradient
constraint.); bottom panel: (left ordinate and thick line) spatially-averaged velocity and (right ordinate and thin
line) average wall shear rate (average taken in the z-direction at x = 0).
115
Figure 10.19: Dynamics of the self-sustaining turbulent structures in a selected LDR simulation (Re = 3600,
+
Wi = 19, β = 0.97, b = 5000, L+
x = 360, Lz = 150). Top panel: spatial-temporal patterns of the wall shear rate
(∂vx /∂y taken at x = 0; two periodic images are shown for each case. Note: the mean value is 2 owning to the
fixed pressure gradient constraint.); bottom panel: (left ordinate and thick line) spatially-averaged velocity and
(right ordinate and thin line) average wall shear rate (average taken in the z-direction at x = 0).
116
Figure 10.20: Dynamics of the self-sustaining turbulent structures in a selected HDR simulation (Re = 3600,
+
Wi = 23, β = 0.97, b = 5000, L+
117
Figure 10.21: Dynamics of the self-sustaining turbulent structures in a selected asym-DR simulation (Re = 3600,
+
Wi = 29, β = 0.97, b = 5000, L+
118
119
(a) Newtonian (Reg), L+
z = 140;
t = 8500, vx = 0.25, Q2D = 0.025.
(b) Newtonian (LS), L+
z = 140;
t = 4600, vx = 0.27, Q2D = 0.012.
(c) LDR (Reg): Wi = 19, L+
z = 150;
t = 5900, vx = 0.26, Q2D = 0.024.
(d) LDR (LS): Wi = 19, L+
z = 150;
t = 8200, vx = 0.29, Q2D = 0.0079.
Figure 10.22: Typical snapshots of the flow field (Re = 3600, β = 0.97, b = 5000,
L+
x = 360). (Reg) denotes snapshots chosen from “regular” turbulence, and (LS)
denotes snapshots of “low-shear” events. Translucent sheets are the isosurfaces of
vx = 0.6vx,max ; opaque tubes are the isosurfaces of Q2D = 0.3Q2D,max . The values
of vx and Q2D for each plot is shown in its caption. Note that (LS) states typically
have much lower Q2D values than (Reg) states. The bottom wall of each snapshot
corresponds to the wall shear rate patterns shown in Figures 10.18, 10.19, 10.20 and
10.21 at corresponding time. (To be continued).
120
(e) HDR (Reg): Wi = 23, L+
z = 180;
t = 7700, vx = 0.31, Q2D = 0.026.
(f) HDR (LS): Wi = 23, L+
z = 180;
t = 7300, vx = 0.31, Q2D = 0.0089.
(g) asym-DR (Reg): Wi = 29, L+
z = 250;
t = 8500, vx = 0.27, Q2D = 0.018.
(h) asym-DR (LS): Wi = 29, L+
z = 250;
t = 8900, vx = 0.31, Q2D = 0.0050.
121
cayre 2000, Wu et al. 2005): i.e. by comparing the magnitudes of the vorticity
tensor and the rate-of-strain tensor, one can identify the local regions manifesting
strong vortical motions. For low Re, the buffer layer structure dominates the turbulence, so we use the Q-criterion in the y-z 2D plane only to focus on vortices
aligned along the mean flow direction. Specifically, we compute the 2D versions
of the rate-of-strain tensor Γ2D ≡ (1/2)(∇v 2D + ∇v T2D ) and the vorticity tensor
Ω2D ≡ (1/2)(∇v 2D − ∇v T2D ), where ∇v 2D ≡ (∂vy /∂y, ∂vz /∂y; ∂vy /∂z, ∂vz /∂z); then
calculate the quantity Q2D ≡ (1/2)(kΩ2D k2 − kΓ2D k2 ). Positive magnitudes of Q2D
would indicate regions having streamwise vortices; in Figure 10.22 the isosurfaces of
Q2D = 0.3Q2D,max are shown, where Q2D,max is the maximum value of Q2D in the
domain for the given snapshot. Note that this varies substantially among different
snapshots; the isosurface value for each image is reported in the caption.
A typical coherent structure of Newtonian turbulence contains a pair of streamwise vortices staggered alongside one sinuous low-speed streak, e.g. the structure at
the bottom wall of Figure 10.22(a). The dynamics around a single streak is sufficient
to make a self-sustaining process: the vortices on different sides of the streak rotate
in opposite directions so that the low-speed fluid near the wall between them is lifted
upward, forming the streak; instabilities of the streak will bring forth streamwise
dependence into its morphology, which through nonlinear interactions further maintains the vortices (Hamilton et al. 1995, Waleffe 1997, Jiménez & Pinelli 1999). In
Figures 10.18, 10.19, 10.20 and 10.21, low-speed streaks correspond to minima of the
wall shear rate distributions in the z direction, which in contour plots are observed
as dark stripes. The Newtonian MFU solution (Figure 10.18) contains one almost
continuous streak during the whole time range shown, which confirms that a self-
122
sustaining process involving one streak (and the vortices around it), lasting for a very
long life-time, dominates the dynamics of the transient solution. With the translation invariance in z, the streak is not bound to any position and is free to drift in the
spanwise direction. However, there are still certain periods (e.g. 6200 . t . 6800 and
7900 . t . 8600) when the streak appears to be quiescent and stays with the same
z location for a fairly large amount of time; in some other time intervals the streak
can be very active and move rapidly in the transverse direction (e.g. 5000 . t . 6200
and 7300 . t . 7900). The LDR stage (Figure 10.19) is qualitatively similar to the
Newtonian case with one continuous streak dominating the dynamics for a long time
period. In the particular case we show, there is only one break point, at t ≈ 7400,
where the first streak decays and meanwhile a second streak is growing. The minimal
spanwise box size to sustain turbulence is however slightly larger, which indicates that
the self-sustaining coherent structure is wider in size, resulting in an increase of streak
spacing. In HDR, as shown in Figure 10.20, the number of streaks in the minimal
box varies between one and two, and complex dynamics are seen from time to time.
These dynamics are also evident in asym-DR (Figure 10.21) where more frequently it
involves two streaks although a single streak can sometimes also been found. These
complex activities and dynamics of the streaks are observed through various events
that change the topology of the streak patterns, including: emergence of new streaks
(e.g. t ≈ 11300, z + ≈ 120 in Figure 10.20 and t ≈ 6500, z + ≈ 30 in Figure 10.21);
decay of existing streaks (e.g. t ≈ 8200, z + ≈ 25 in Figure 10.20); merger of multiple
(typically two) streaks into one (e.g. t ≈ 6600, z + ≈ 160 in Figure 10.21) and division
of one streak into multiple streaks (e.g. t ≈ 9600, z + ≈ 125 in Figure 10.21). This
transition from single-streak dynamics to multiple-streak dynamics at the LDR–HDR
123
transition suggests that the underlying self-sustaining mechanism of turbulence may
have changed; complex dynamics involving interactions between streaks might be
essential in sustaining turbulent motions in HDR and asym-DR stages.
Recall in Figure 10.4 that when the LDR–HDR transition occurs, the dependence
of L+
z on DR% undergoes an abrupt transition; this can be interpreted based on
the observations in Figures 10.18, 10.19, 10.20 and 10.21. In the LDR stage, the
underlying self-sustaining process is qualitatively the same as the Newtonian turbulence, which involves the nonlinear interactions between a single low-speed streak
and the streamwise vortices on its both sides (Hamilton et al. 1995, Waleffe 1997,
Jiménez & Pinelli 1999). Viscoelasticity reduces the drag by weakening the vortical
motions (Li, Xi & Graham 2006, Li & Graham 2007) and the increase of L+
z is caused
merely by the enlargement of the coherent structures (Li & Graham 2007). After the
LDR–HDR transition, viscoelasticity is strong enough to suppress the “Newtonian”
coherent structures (as predicted by earlier ECS study of Li, Xi & Graham (2006)
and Li & Graham (2007)), and the process involving a single isolated streak cannot
sustain turbulence for a very long time (see the relatively shorter streak segments
in Figures 10.20, 10.21). As a result a new self-sustaining process involving interstreak interactions arises, the details of which have yet to be elucidated. Therefore
the increase of L+
z in the HDR stage involves both the contribution from the enlarged
structure by viscoelasticity, and the extra room needed to accommodate more streaks.
As to the turbulent dynamics reflected by the evolutions of Ubulk and the mean
wall shear rate (bottom panels of Figures 10.18, 10.19, 10.20 and 10.21), one interesting observation is that there are certain moments in the self-sustaining process when
the change of Ubulk can be inferred by the shear rate at the wall. Specifically, during
124
these moments, the wall shear rate is low in magnitude and its curve remains relatively smooth for O(100) time units; meanwhile the mean velocity increases steadily.
Examples of these events include: t ≈ 4400 of Figure 10.18, t ≈ 4200, 4900 and 8000
of Figure 10.19, t ≈ 6200, 6900, 7300, 8800 and 10100 of Figure 10.20 and t ≈ 5500,
5800, 8100, 8800, 9700, 11000, 11300 and 11600 of Figure 10.21. By comparing these
temporal evolution plots with the spatial-temporal wall shear rate patterns (top panels of Figures 10.18, 10.19, 10.20 and 10.21), one may find that these events usually
correspond to the moments when the patterns are blurry: i.e. the wall shear rate
has relatively small variance in both space and time. Besides, these events appear
to occur more often as DR% increases; to quantify their frequency of occurrence,
and its dependence on Wi, simulations much longer in time are required, which will
be presented in Chapter 11. To a first approximation, the correlation between bulk
velocity and the wall shear rate can be interpreted as such: since the driving force of
the flow, the mean pressure gradient, is fixed, the change of the total momentum in
the flow unit is mainly determined by the rate momentum is consumed at the wall;
when shear rate at the wall is low, there is less momentum being transferred to the
wall by viscous shear stress, which makes it easier to accumulate momentum in the
flow unit and increase the mean velocity.
In the 3D views of velocity fields shown in Figure 10.22, one of the two snapshots
presented for each run are taken from one of these “low-shear” events, marked as (LS)
in the caption; and the other is from a regular turbulence cycle, marked as (Reg).
The typical snapshot of “regular” Newtonian turbulence (Figure 10.22(a)) has been
discussed above. At LDR (Figure 10.22(c)), the structure is qualitatively similar
with one sinuous streak near each wall surrounded by streamwise vortices. At HDR
125
(Figure 10.22(e)) and asym-DR (Figure 10.22(g)), this type of streak-vortex structure is still observed, though very often two streaks can be observed near each wall.
Compared with these snapshots of “regular” turbulence (Figures 10.22(a), 10.22(c),
10.22(e), 10.22(g)), those taken during the “low-shear” events (Figures 10.22(b),
10.22(d), 10.22(f), 10.22(h)) in general have much lower vortex strength, as reflected
by lower Q2D magnitudes. Meanwhile, the streaks are less wavy in shape: the xdependence of the streak morphology is weak. (This would explain the increased
smoothness of the h∂vx /∂yiz v.s. t curve: since average is taken only in the z-direction
at a fixed x position, dependence of ∂vx /∂y on x will be reflected in temporal fluctuations owning to the convection of flow structures). As discussed above, these
“low-shear” events occur more frequently as DR% increases, therefore we expect that
in a full-size system the probability of observing relatively straight streaks is higher in
HDR and asym-DR (or MDR in experiments) stages, while at lower DR% the streaks
should be mostly wavy. This is consistent with the observation by Li, Sureshkumar
& Khomami (2006) in full-size DNS that long straight streaks are more predominant
when the HDR regime is reached. The nature of these “low-shear” events is as yet
unclear. How these events are triggered and what roles they play in the self-sustaining
processes of turbulence will be important for further understanding of drag reduction
by polymers. Some further investigation into this dynamical feature of turbulence in
MFUs will be presented in Chapter 11.
126
Chapter 11
Toward an understanding of the
dynamics: active and hibernating
turbulence
11.1
Intermittent dynamics in MFU
Observations in Chapter 10 demonstrate that MFU solutions almost (with the exception of the Virk MDR profile in time-averages) fully recover the qualitative transitions previously reported in experiments and full-size DNS studies, most of which
were conducted at much higher Re. Since MFU contains isolated individual coherent structures only, instead of populations of them, temporal intermittency of the
self-sustaining process is more readily identifiable via this approach. Attempting to
interpret these transitions in terms of temporal dynamics in MFU is the natural next
step to take. In particular, spatiotemporal structures presented in Section 10.4 show
127
intermittent time periods with substantially low magnitudes of wall shear rate in all
stages of transition. These periods occur much more frequently in HDR and asymDR stages, where complex dynamics are also observed in the self-sustaining process.
Motivated by these results, we further focus on these periods in this chapter. Results
presented below will show that although in the current MFU study we are not able
reach comparable level of drag reduction with experimental MDR, new directions
that might lead to a final understanding of the mechanism of MDR will be pointed
out based on the study of these dynamics.
Figures 11.1 and 11.2 show time series of instantaneous bulk average velocity Ubulk
and area-averaged shear rate h∂vx /∂yi at the top and bottom walls for Newtonian
flow and viscoelastic flow at Wi = 29 (where DR% = 26 and L+
z has increased from
140 to 250; results in this chapter are all obtained at Re = 3600, β = 0.97, b = 5000
and L+
z = 360), respectively. (Note that here average for ∂vx /∂y is taken in both
x and z directions). In the Newtonian case one occasionally observes long-lasting
periods during which the shear rate at one or both walls is substantially lower than
the average value of 2 – for example the time interval 5000 < t < 5500. By momentum conservation, the bulk velocity increases during these periods. For reasons
that will emerge as the discussion proceeds, these periods will be termed “hibernation”. Turbulence outside these periods will be termed “active”. As Wi increases,
it is observed that hibernation periods become increasingly frequent – since the bulk
velocity increases during these periods their high frequency contributes substantially
to drag reduction. Note that the flow does not closely approach the laminar state
(Ubulk = 2/3) during hibernation periods.
To systematically identify hibernation events, two criteria are used: (1) area-
∂vx /∂yt
0.4
2
1
0
2
3
1000
2000
t
3000
4000
5000
6000
Figure 11.1: Mean shear rates at the walls (“b”–bottom, “t”–top) and bulk velocity Ubulk as functions of time for
+
typical segments of a Newtonian simulation run (Re = 3600, L+
x = 360, Lz = 140). Rectangular signals in the
middle panel indicate the hibernating periods at the wall of the corresponding side, identified with the criterion
explained in the text. Dashed lines show the line h∂vx /∂yi = 1.80. Time average of h∂vx /∂yi is 2.
∂vx /∂yb
0.35
Ubulk
3
128
∂vx /∂yt
0.4
2
1
0
2
3
1000
2000
t
3000
4000
5000
6000
Figure 11.2: Mean shear rates at the walls (“b”–bottom, “t”–top) and bulk velocity Ubulk as functions of time for
+
typical segments of a high-Wi simulation run (Re = 3600, Wi = 29, β = 0.97, b = 5000, L+
x = 360, Lz = 250).
Rectangular signals in the middle panel indicate the hibernating periods at the wall of the corresponding side,
identified with the criterion explained in the text. Dashed lines show the line h∂vx /∂yi = 1.80. Time average of
h∂vx /∂yi is 2.
∂vx /∂yb
0.35
Ubulk
3
129
130
30
260
25
240
220
20
L+
z
DR%
200
15
180
160
10
140
5
120
0
0
5
10
15
Wi
20
25
100
30
Figure 11.3: Level of drag reduction and spanwise box size as functions of Wi (Newtonian and β = 0.97, b = 5000).
averaged wall shear rate at one or both walls drops below a cutoff value h∂vx /∂yi|cutoff =
1.8; and (2) it stays there for longer than a certain amount of time ∆tcutoff = 50. Hibernating periods identified with these criteria are shown in the middle (bulk velocity)
panels of Figures 11.1 and 11.2 as rectangular signals, on the top or bottom of the
plot according to the wall(s) on which the criterion is satisfied. This criteria is so
chosen since it captures the main phenomenological characteristics of these periods.
Although the choices of cutoff values are to some extent arbitrary, we have found
that changing these values within a reasonable range does not qualitatively affect the
following discussion.
With these periods clearly identified, we can now quantify the dependence of their
frequency and duration on viscoelasticity. Since turbulence hibernation occurs intermittently, and the time scales between two adjacent occurrences can be rather long,
131
1200
0.3
1000
0.25
TA
TH
600
FH
0.2
FH
Time Scales
800
400
0.15
200
0
0
5
10
15
Wi
20
25
0.1
30
Figure 11.4: Time scales (left ordinate) and fraction of time spent in hibernation
(right ordinate) as functions of Wi (Newtonian and β = 0.97, b = 5000): TA is the
mean duration of active periods; TH is the mean duration of hibernating periods; FH
is the fraction of time spent in hibernation.
132
especially for Newtonian and LDR turbulence, much longer simulations (compared
with results in Chapter 10) are needed for satisfactory statistics. Extended amount
of MFU simulations are thus performed for selected Wi and fixed values of β = 0.97,
b = 5000. These results are presented in Figure 11.4, as functions of Wi, in terms
of: the mean duration of the hibernation periods TH , mean duration of active periods
TA , and fraction of time spent in hibernation FH . The corresponding DR% and box
size (set to be the same as in Chapter 10) are shown in Figure 11.3 for reference.
For each Wi, multiple runs with independent initial conditions are included in the
average; each of them lasts for a minimum of 8000 time units after the statisticallyconverging regime is reached; the total amount of time included in each data point
ranges from 32000 to 148600 time units (O(10Re) or longer). Error bars for TA and
TH are computed assuming that each transition between active and hibernating periods are independent from one another; error bars for DR% and FH are estimated
with the block-averaging method (Flyvbjerg & Petersen 1989) with a fixed block size
of 4000 time units (O(Re)).
Several important observations emerge from these results. First, the average duration TH of a hibernating period is almost completely insensitive to Wi. In contrast,
the average time TA between two neighboring hibernating periods decreases substantially after onset of drag reduction. Accordingly, the fraction of time spent in
hibernation is determined only by TA , since TH does not depend on Wi. These results indicate that in the high Wi regime, viscoelasticity compresses the lifetime of an
active turbulence interval, facilitating the occurrence of hibernation, while having no
effect on hibernation itself. The net outcome is hibernation becomes an increasinglysignificant component of the overall turbulent dynamics in the high-Wi regime. Also
133
∂vx /∂yt
3
a
c
b
e
d
2
Ubulk
0.40
0.35
∂vx /∂yb
3
2
1
200
250
300
350
400
t
450
500
550
600
Figure 11.5: A hibernation event (200 6 t 6 600 in Figure 11.2). Thick black lines
are mean wall shear rates and bulk velocity Ubulk at Wi = 29. Thin colored lines are
from Newtonian simulations started at the corresponding colored dots, using velocity
fields from the Wi = 29 simulation as initial conditions.
noted in Figure 11.4 is that substantial decrease in TA , and thus increase in FH , are
only observed at Wi > 19 (obviously higher than Wionset , which is below 16 as shown
in Figure 11.3), which coincides with the value where LDR–HDR transition occurs as
reported in Chapter 10. This suggests that those qualitative changes in flow statistics
between LDR and HDR might be linked with the increased frequency of hibernation.
Further investigation of the effect of hibernation on flow statistics is proposed for our
future work (Section 13.1).
The insensitivity of TH to Wi suggests that flow during hibernation does not
strongly stretch polymer molecules.
Indeed, as shown below (Figure 11.10), at
Wi = 29 the peak value of hαyy i, which is closely associated with streamwise vortex suppression (Stone, Roy, Larson, Waleffe & Graham 2004, Li & Graham 2007,
134
Procaccia et al. 2008), drops from about 210 in active turbulence to about 5 during
hibernation, a 40-fold reduction. These results suggest that hibernation should be
very similar in the Newtonian and viscoelastic cases. To test this possibility, velocity
fields from time instants before and during a hibernation event at Wi = 29 were used
as initial conditions for a Newtonian simulation, the trajectories of which were then
compared with those from the original viscoelastic simulation. Figure 11.5 illustrates
the original viscoelastic trajectory (thick black line) as well as Newtonian trajectories
(colors) started at various times. For the Newtonian run starting before any sign
of hibernation is observed (t = 205), active turbulence is sustained. However, the
runs started from later times show that once the system begins to enter hibernation,
removing the polymer stress does not cause turbulence to revert to an active state,
although the depth and duration of hibernation are weakly dependent on the time at
which their initial conditions are taken from the original viscoelastic run. In short,
while polymer increases the probability of entering hibernation, it has little effect on
flow within the hibernation region itself.
To better understand the above results, we examine more closely the hibernating period shown in Figure 11.5. Several time instants are selected as marked: (a)
is an instant right before turbulence enters hibernation; (b) is one on the path toward hibernation; (c) and (d) are within hibernation; (e) is after turbulence becomes
reactivated. Figure 11.6 shows instantaneous area-averaged velocity profiles in the
bottom half of the channel for these instants, plotted in inner units based on the instantaneous wall shear stress at the bottom wall (denoted by the superscript ∗ rather
than + ). In active turbulence (a and e), the profiles fluctuate substantially. Profiles
for instants completely in hibernation (c and d) are fundamentally different. In par-
135
25
20
Viscous sublayer
Newtonian log-law
Virk MDR
∗
Umean
15
10
a
b
c
d
e
5
0 0
10
1
10
y∗
Figure 11.6: Instantaneous mean velocity profiles of selected instants before, during
and after a typical hibernating period (marked with grid-lines in Figure 11.5). Profiles
for the bottom half of the channel are shown; superscript “*” represents variables
nondimensionalized with inner scales based on instantaneous mean shear-stress at
the wall of the corresponding side. Black lines show important asymptotes: “viscous
∗
∗
sublayer”, Umean
= y ∗ ; “Newtonian log-law”, Umean
= 2.44 ln y ∗ + 5.2 (Pope 2000);
∗
∗
“Virk MDR”, Umean = 11.7 ln y − 17.0 (Virk 1975).
136
25
W i = 29, instant (c)
20
W i = 29, instant (e)
Newtonian, instant (c)
Newtonian, instant (e)
∗
Umean
15
10
Viscous Sublayer
5
Virk’s MDR Asymptote
0 0
10
1
10
y∗
Figure 11.7: Comparison between hibernation in Newtonian and high-Wi viscoelastic
flows (the Newtonian simulation is the one starting from t = 260 in Figure 11.5).
Instantaneous mean velocity profiles for instants in hibernation (c) and after turbulence is reactivated (e) are show (marked with grid-lines in Figure 11.5). Profiles for
the bottom half of the channel are shown.
137
Figure 11.8: Flow structures at selected instants before, during and after a typical
hibernating period (marked with grid-lines in Figure 11.5). Green sheets are isosurfaces vx = 0.3, pleats correspond to low-speed streaks; red tubes are isosurfaces of
Q2D = 0.02, Q2D is defined in Section 10.4. Only the bottom half of the channel is
shown.
ticular, in the range 15 . y ∗ . 40, both profiles show a clear log-law relationship
with a slope very close to the MDR asymptotic slope of 11.7 reported by Virk (1975)
(also shown on the plot). The Newtonian hibernation periods are very similar and
the Virk MDR slope is observed there as well. In Figure 11.7 we can see that at instant (c) mean velocity profiles are almost indistinguishable between the Newtonian
and W i = 29 simulations. Their difference becomes noticeable only after turbulence
returns to active periods.
138
1400
1200
a
b
c
d
e
1000
αxx
800
600
400
200
0
0
10
20
30
40
y∗
50
60
70
80
90
Figure 11.9: Instantaneous profiles of αxx (streamwise polymer deformation) for instants marked in Figure 11.5. Profiles for the bottom half of the channel are shown.
Figure 11.8 shows flow structures corresponding to these time instants. Within
active periods ((a) and (e)), turbulence shows highly 3D coherent structures consisting of streamwise vortices and low-speed streaks (Jiménez & Moin 1991, Robinson
1991, Waleffe 1997). During hibernation ((c) and (d), also (b)), streamwise vortices
are significantly weaker; low-speed streaks are still observed, but are weak and only
weakly dependent on x. Weak streamwise vorticity and three-dimensionality are also
distinct characteristics of flow in the MDR regime (Virk 1975, White et al. 2004,
Housiadas et al. 2005, Li, Sureshkumar & Khomami 2006, White & Mungal 2008).
The weak effect of viscoelasticity on hibernating turbulence may lie in its nearly
streamwise-invariant kinematics. In the limiting case of a streamwise invariant steady
flow, material lines cannot stretch exponentially (Ottino 1989); accordingly, polymer
stretch in such a flow will not be substantial. As shown in Figures 11.10 and 11.11,
139
250
a
b
c
d
e
200
αyy
150
100
50
0
0
10
20
30
40
y∗
50
60
70
80
90
Figure 11.10: Instantaneous profiles of αyy (wall-normal polymer deformation) for
instants marked in Figure 11.5. Profiles for the bottom half of the channel are shown.
250
a
b
c
d
e
200
αzz
150
100
50
0
0
10
20
30
40
y∗
50
60
70
80
90
Figure 11.11: Instantaneous profiles of αzz (spanwise polymer deformation) for instants marked in Figure 11.5. Profiles for the bottom half of the channel are shown.
140
1.6
a
b
c
d
e
1.4
-< vx∗ vy∗ >
1.2
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
y∗
50
60
70
80
90
Figure 11.12: Instantaneous profiles of Reynolds shear stresss for instants marked in
Figure 11.5. Profiles for the bottom half of the channel are shown.
compared with active periods, in hibernation intervals polymer is almost undeformed
in the transverse directions; while in active turbulence, transverse polymer deformation is known to suppress streamwise vortices (Stone, Roy, Larson, Waleffe & Graham
2004, Dubief et al. 2005, Li & Graham 2007, Procaccia et al. 2008). During hibernation, deformation is noticeable only in the streamwise direction (Figure 11.9), the
direction of mean flow. Unlike in active periods, αxx profile in hibernation is monotonic, and decreases with distance from the wall, which reflects the distribution of
the mean shear rate.
Finally, the Reynolds shear stress during hibernation drops to very low values
relative to active turbulence (Figure 11.12); the peak value during hibernation is
about 0.3 compared to values near unity in active turbulence. Again, this result
is consistent with observations near and in the MDR regime (Warholic, Massah &
141
Hanratty 1999, Warholic et al. 2001, Ptasinski et al. 2001, 2003).
The qualitative picture that emerges from these simulations is thus the following. Active turbulence generates substantial stretching of polymer molecules. The
resulting stresses act to suppress this turbulence and drive the flow toward a very
weakly turbulent hibernating regime. During hibernation the polymer molecules are
no longer strongly stretched and they relax toward their equilibrium conformations.
Eventually the hibernation ends, as new turbulent fluctuations begin to grow, and
the system transits back into active turbulence. The active turbulence again stretches
polymer chains and the (stochastic) cycle repeats.
In this picture, experimental observations in which the Virk MDR mean velocity
profile is found correspond to a limiting situation – not achieved at the low Reynolds
number and small boxes studied here – where the fraction of time and space occupied
by active turbulence becomes small enough that the hibernating regime dominates
the statistics. Active turbulence cannot vanish entirely, because it is known experimentally (Warholic, Massah & Hanratty 1999, Ptasinski et al. 2003) that on average,
the polymer molecules carry a substantial fraction of the mean shear stress (they
must, if the time averaged Reynolds shear stress is to be small), and since hibernating turbulence does not stretch polymers, some active turbulence must remain. These
considerations lead to a picture of turbulence in the MDR regime as a state in which
hibernating turbulence is the norm, and active turbulence arises intermittently in
space and time only to be suppressed by the polymer stretching that it induces. MDR
is asymptotically independent of polymer properties because hibernating turbulence,
which dominates the statistics of MDR, is fundamentally a Newtonian phenomenon.
Only a study of turbulence in MFUs would allow for a picture this clean to emerge:
142
in a larger flow domain there are likely to be some regions where the turbulence is
active and some where it is hibernating, but without knowing in advance about these
regions they would be difficult to identify.
The present study focused on MFU flows at low Reynolds number, where there
is not yet a large separation between inner and outer scales. This approach allowed
the collection and analysis of an extensive data set in a regime where flow structures
are relatively simple. Remarkably, even this regime displays clear signatures of the
features of MDR commonly associated with higher Reynolds numbers. Future work
should use simulations at high Re to carefully evaluate the hypothesized picture just
presented.
In addition, attention must be focused on the hibernating turbulence phenomenon.
Recently, Waleffe has identified a class of nonlinear traveling wave solutions to the
Navier-Stokes equations in the plane Couette and Poiseuille geometries that share
many characteristics with hibernating turbulence, specifically weak streamwise vortices and weak streamwise dependence (Wang et al. 2007). Indeed, at least one family
of these solutions has vanishing streamwise dependence as Re → ∞. These states are
saddle points in phase space and it may be that hibernating turbulence is a trajectory
moving transiently in the vicinity of one of these saddles.
These hypotheses, that MDR turbulence is fundamentally hibernating turbulence,
and that hibernating turbulence is closely related to nonlinear traveling wave structures in Newtonian flow, point toward a fundamentally new direction for research in
the field of turbulent drag reduction by additives, which will be the topic of our future
work (Chapter 13).
143
11.2
Generalization to full-size turbulent flows: a
preliminary investigation
Although the above scenario of the transition toward MDR depends on temporal intermittency observed in individual coherent structures isolated by the MFU approach,
it is verifiable in experiments and full-size DNS studies. A full-size turbulent flow (in
either experiments or DNS) consists of a large population of coherent structures, each
of which evolves through the same (at least qualitatively) series of dynamical phases
observed in MFU, such as the active and hibernating intervals discussed above. In
general, evolution of these structures is not in phase with one another: for a given
instant, some of them may show characteristics of active turbulence, while others
might be in hibernation. Therefore, spatial intermittency would be expected in snapshots of these flow fields. The fraction of time spent in hibernation for one coherent
structure in a long-time simulation FH (plotted in Figure 11.4), would be reflected in
the fraction of coherent structures caught in hibernation within a randomly picked
snapshot of a sufficiently large simulation box at the statistically-steady regime (i.e.
fraction of space where hibernation is observed).
Some preliminary results of full-size DNS are presented here to illustrate this
interchangeability between temporal and spatial intermittency. These simulations
+
are performed in a periodic box with L+
x = 4000 and Lz = 800. According to
previous studies (Dubief et al. 2004, Li, Sureshkumar & Khomami 2006), with a box
size at this magnitude, effects of finite box size on flow statistics are negligible. This
simulation box is significantly larger, in both streamwise and spanwise directions,
than those used in the MFU study presented above, and should accommodate a
Figure 11.13: Flow structures of a typical snapshot in a full-size Newtonian simulation (Re = 3600, L+
x = 4000,
L+
=
800).
Green
sheet
is
the
isosurface
of
v
=
0.3;
red
tubes
are
isosurfaces
of
Q
=
0.02.
Only
the
bottom
x
2D
z
half of the channel is shown.
144
Figure 11.14: Flow structures of a typical snapshot in a full-size viscoelastic simulation near MDR (Re = 3600,
+
Wi = 80, β = 0.97, b = 5000, L+
x = 4000, Lz = 800). Green sheet is the isosurface of vx = 0.3; red tubes are
isosurfaces of Q2D = 0.02. Only the bottom half of the channel is shown.
145
Figure 11.15: Contours of streamwise velocity in a plane 25 wall units above the bottom wall for the Newtonian
snapshot shown in Figure 11.13
146
Figure 11.16: Contours of streamwise velocity in a plane 25 wall units above the bottom wall for the viscoelastic
(near MDR) snapshot shown in Figure 11.14
147
148
number of coherent structures. A 240 × 73 × 90 numerical grid is used for spatial
discretization, this corresponds to δx+ = 16.67 and δz+ = 8.89. Time step is in the
range of 0.03125 6 δt 6 0.04. Since the spatial and temporal resolutions are both
lower (still at the same level as those in previous studies, e.g. Jiménez & Moin (1991),
Housiadas & Beris (2003), Housiadas et al. (2005) and Li, Sureshkumar & Khomami
(2006)) than those used in the MFU study (Section 8.2), a slightly lower Schmidt
number (corresponding to a larger artificial diffusivity value) of Sc = 0.03 is used.
Figure 11.13 is a snapshot of a typical full-size Newtonian turbulent flow field (only
the bottom half of the channel is shown). Characteristic streak-vortex structures are
observed across the whole domain. Each pair of them are not identical, but they are
qualitatively similar. Almost all of them show features of active turbulence, including
strong vortical motions and wavy streaks. Streak structures are better observed in
the contour plot (Figure 11.15) of streamwise velocity in the x-z plane, taken at a
position within the buffer layer (y + = 25). Alternating high- and low-speed streaks
are clearly identified, most of which contain wrinkles and other features at relatively
small wavelengths (O(100)).
For viscoelastic turbulent flows near MDR, flow structures are very different. Figures 11.14 and 11.16 are a snapshot taken from a high-Wi run; the DR% for this
snapshot is 57% (DR% for Virk (1975) MDR is slightly above 60% at this Re). In
most of the domain, streaks are elongated and regulated, only very weak streamwisedependence is observed; intensity of streamwise vortices is lower than the isosurface
level and these vortices in the regions with weak streak-waviness are thus not visualized. These observations should recall the characteristics of hibernating turbulence.
As noted above, polymer cannot stabilize hibernating turbulence, hence intermittent
149
occurrence of active turbulence is expected. Indeed, scattered patches of structures
resembling active turbulence, i.e. those showing strong vortices and wavy streaks, are
still observed.
Further analysis of this spatial intermittency is due in our future work. The
important message from results in this chapter is that the answer to the four-decadelong puzzle of MDR might lie in the spatial and temporal intermittency in turbulent
flows, which has been screened out in most previous studies that typically focused on
spatially and temporally averaged profiles.
150
Chapter 12
Conclusions of Part II
In this study, we study viscoelastic turbulent flows under a variety of conditions.
These solutions are obtained from the minimal flow unit approach and represent the
essential coherent structures for the self-sustaining process of turbulent motions. The
box size is minimized in the spanwise direction with fixed streamwise wavelength. The
minimal box size to sustain turbulence increases with increasing Wi for fixed β and b,
and the correlation between this length scale and the bulk flow rate is approximately
universal with respect to varying β and b at fixed Re (Figure 10.4). At a Re close to
the laminar–turbulence transition, all key stages of transition, reported previously in
experiments and simulations at much higher Re, are observed in the MFU solutions,
including pre-onset turbulence, LDR, HDR and an asym-DR stage that reproduces
the universal aspect of experimental MDR. The onset of drag reduction (transition
between the pre-onset and LDR stages) is observed at Wi & 16. The LDR–HDR
transition occurs at around DR% ≈ 13−15% under different β and b, which we expect
to be a function of Re. The discovery of the LDR–HDR transition at the current
151
low Re and especially, at a relatively low DR%, indicates that this is a qualitative
transition between two stages of viscoelastic turbulent flows and not a quantitative
effect of the amount of drag reduction. Drag reduction reaches its upper limit at
DR% ≈ 26% in the asym-DR stage, where DR% converges upon increasing Wi. This
upper limit is universal with respect to different β and b, and it is to our knowledge the
first time the universality of MDR with respect to polymer parameters is examined
in numerical simulations. After the asym-DR stage, which persists for a finite range
of Wi at given β, b and Re, the flow returns to the laminar state.
The LDR–HDR transition is associated with a change in the underlying dynamics
of the self-sustaining process of turbulence. At the LDR stage, the essential coherent
structure to sustain turbulence is similar to that of Newtonian turbulence, which consists of one undulating low-speed streak and its surrounding counter-rotating streamwise vortices. At the HDR stage, the essential structure is more complicated and
involves more than one streak; inter-streak interactions may be important. Nevertheless, the streamwise streaks and vortices are still the major components of the
self-sustaining process in all turbulent stages in our MFU solutions. This change of
the basic structure is reflected in the length scale of the MFU, resulting in a sudden
change in the slope of the L+
z ∼ DR% curve: the minimal box size increases more
sharply with DR% at the HDR stage compared with the LDR stage. Several qualitative changes in flow statistics are observed during this transition, including: (1)
change of the log-law slope in the mean velocity profile, from the Newtonian log-law
to a larger slope; (2) disappearance of the concavity in the root-mean-square spanwise velocity fluctuation profile; (3) change in the location of the suppression of the
Reynolds shear stress profile, which is suppressed locally (in the buffer layer) at LDR
152
while globally (in most of the channel) at HDR and asym-DR. These changes cannot
be correlated with any observed qualitative transitions in the statistics of the polymer
conformation tensor.
At the asym-DR stage, the mean velocity profiles converge onto a single curve
at the given Re. The Reynold stresses either converge to a limit or at least lose
their dependence on Wi, β and b, and fluctuate within certain ranges. In contrast,
polymer is increasingly stretched by the flow with increasing Wi despite the converged
flow rate, and the polymer conformation tensor continues to dependent on Wi, β
and b. In the asym-DR stage, the spatiotemporal flow structure seems similar as
that of the HDR stage; the self-sustaining process also shows complex dynamics
involving multiple streaks. The minimal length scale in z to sustain turbulence keeps
on increasing with Wi in the asym-DR stage; however, the length scale of the MFU
solutions in the asym-DR stage under different β and b all approximately fall in the
range of 200 6 L+
z 6 260.
This study shows that the drag reduction process with varying parameters is composed of several key stages of transition, which are present in both fully developed
turbulence (according to other studies) and the laminar-turbulence-transition regime.
The mechanism of these transitions, especially the LDR–HDR transition and the existence of a universal asym-DR, is as yet unclear. Spatiotemporal images of turbulent
coherent structures suggest that a shift of the underlying self-sustaining mechanism
occurs at the LDR–HDR transition. Further study of this change will be important
in understanding drag reduction behaviors in HDR and MDR regimes. In addition,
the capability of isolating the minimal transient solutions, and the knowledge that
these transitions can all be studied in the near-transition regime, will greatly facilitate
153
future insight into the polymer drag reduction phenomenon.
Dynamics of turbulence in MFU, both Newtonian and viscoelastic, show intermittent occurrence of relatively-long periods when substantially-lower magnitudes of
wall shear rate are observed. These “hibernating” periods display many features of
experimentally-observed MDR in polymer solutions, including weak streamwise vortices, nearly nonexistent streamwise variations, strongly suppressed Reynolds shear
stress, and most importantly, a mean velocity gradient that quantitatively matches
experiments. Frequency of hibernation increases significantly in the high-Wi regime,
where polymer compresses the lifetime of an active turbulence interval, and facilitates the occurrence of these periods. Once inside hibernation, polymer is weakly
stretched by the flow, and has little effect on hibernation itself. These results point
toward a fundamentally-new direction of understanding turbulence in the high-Wi
regime, especially the maximum drag reduction.
154
Chapter 13
Future work: dynamics of
viscoelastic turbulence and drag
reduction in turbulent flows
Our study on viscoelastic turbulence raises more questions than it answers. Although
it does not provide a complete mechanism for MDR, nor does it offer a clear explanation of the LDR–HDR transition, a new direction has be pointed to understand the
regime of high–Wi viscoelastic turbulence. In particular, the resemblance between
the intermittent turbulence hibernation and experimentally-observed MDR provides
an important clue that might lead to the eventual revelation of the nature of this
long-lasting mystery. Further knowledge is needed about this newly-recognized hibernating period: understanding its nature and its connection with MDR is the most
important task in the future.
155
13.1
Hibernation statistics: effect on the LDR–
HDR transition
With the current numerical method and data, the short-term plan is to quantify
the differences between active and hibernating turbulence. Results on turbulence
behaviors during hibernation discussed in Chanpter 11 is based on a typical example, although other instances have been examined to ensure that those observations
are not specific to the selected set of data, more general analysis on the flow and
polymer conformation statistics in both active and hibernating turbulence should be
performed. As reviewed in Chapter 7, the LDR–HDR transition is marked by a series
of qualitative transitions in turbulence statistics and flow structures: e.g. increased
log-law slope in the mean velocity profile, reduced Reynolds shear stress across the
channel and dramatically weakened three-dimensionality in the streak-vortex structures (Warholic, Massah & Hanratty 1999, Ptasinski et al. 2003, White et al. 2004, Li,
Sureshkumar & Khomami 2006). Many of these observations consist with the characteristics of hibernating turbulence. A natural conjecture is that the LDR–HDR
transition is caused by an event leading to the frequent occurrence of hibernation.
Before this transition, hibernation is rare and active turbulence dominates the statistics. In HDR and MDR regimes, hibernation makes a substantial contribution to the
overall statistics, which causes all the changes reported in previous studies; meanwhile
active turbulence remains qualitatively the same, although quantitatively it should
have a smooth dependence on Wi as well (as seen from the dependence of LDR statistics on Wi, where hibernation frequency is almost constant). To test this hypothesis,
one needs to effectively divide data from each time series into the two categories,
156
such that statistics for either of them can be computed separately, and the difference
between these two regimes can be compared with statistical certainty.
13.2
A hypothetical dynamical-scenario
From a nonlinear dynamics point of view, the distinct separation between hibernating
and active turbulence in the solution state space usually suggests the existence of certain solution objects governing the hibernation dynamics, which are located far away
from the major TWs that construct the latter (Guckenheimer & Holmes 1983). In the
simplest case, hibernation is caused by intermittent visits of the proximity of some
saddle point: TW solution with both stable and unstable dimensions (Figure 13.1).
The system is pulled toward the saddle point via trajectories going along the stable
manifold; it turns near the saddle and is ejected away toward active turbulence along
the direction of the unstable manifold. In Newtonian turbulence, these excursions
are rare: the system is trapped in active turbulence for long time periods before it
hits the orbit toward the saddle. At high Wi, active turbulence can sustain for much
shorter time, and these orbits are visited more frequently. Differences in duration
and “depth” of hibernation among individual instances are accounted for by different closeness between the incoming orbits (orbits entering hibernation from active
turbulence) and the stable manifold. Although an one-saddle scenario is shown in
Figure 13.1, it may also involve more than one saddles, or even more complex solution
objects such as periodic orbits.
Characteristics of hibernating turbulence observed in Chapter 11 recall us to the
concept of “lower-branch” TWs. In the regime near the critical Re for the laminar-
157
Edge
Stru
cture
Active Turbulence
ld
n?
Hibernatio
ab
ifo
St
an
M
le
le
M
an
b
ta
ifo
ld
s
Un
Edge Structure
TW (Saddle)
Laminar Flow
Figure 13.1: Schematic of near-transition turbulent dynamics: intermittent excursions
toward certain saddle points and the laminar-turbulence edge structure.
158
turbulence transition, TWs typically appear in pairs through saddle-node bifurcations (Waleffe 1998, 2001, 2003). By lower-branch solutions we refer to the ones in
each pair that are relatively closer to the laminar state: i.e. those have lower turbulence intensity. Jiménez et al. (2005) summarized TW solutions obtained by various
groups in Newtonian plane Couette flow and concluded that these solutions can all be
categorized as either lower-branch or upper-branch solutions. TWs of both categories
are in the form of a sinuous low-speed streak straddled by a pair of staggered counterrotating streamwise vortices, but the lower-branches generally have smaller transverse
velocity fluctuations, much weaker vortical motions and less streamwise waviness in
the streak. Upper-branch TWs are widely believed to be the building blocks of the
chaotic structure of active turbulence (Waleffe 2001, 2003, Jiménez et al. 2005, Gibson et al. 2008), while characteristics of lower-branch TWs are remarkably similar to
the hibernating turbulence discovered in this study. In the context of the scenario
illustrated in Figure 13.1, it is likely that the saddle(s) dominating hibernation dynamics belong to the category of lower-branch TWs. Previous studies on viscoelastic
ECS (Stone et al. 2002, Stone & Graham 2003, Stone, Roy, Larson, Waleffe & Graham 2004, Li, Stone & Graham 2005, Li, Xi & Graham 2006, Li & Graham 2007)
showed that upper-branch ECSs are strongly suppressed by polymer stress, and at Wi
sufficiently high, they are completely eliminated. This is consistent with the current
observation that the average life time of active turbulence is significantly shortened
at high Wi. Meanwhile, since lower-branch TWs have much weaker vortical structures and less three-dimensionality, they might not stretch polymer substantially to
generate enough polymer stress, and these solutions may be largely unaffected with
increasing Wi. This would explain the invariant duration time scale of hibernation
159
for different Wi, and the similarity between Newtonian and viscoelastic hibernation
observed in this study.
Recent studies in Newtonian turbulence suggested that lower-branch TWs play an
important role in the laminar-turbulence transition. For ECS in plane Couette flow,
Wang et al. (2007) showed that the stable manifold of lower-branch ECS forms part
of the separating boundary between basins of attraction of laminar and turbulence
states. This solution has only one unstable dimension (Waleffe 2003). One side of
the unstable manifold points to the laminar state and the other leads to turbulence
(see Figure 13.1). Initial states “above” the stable manifold will become turbulent
and those below will laminarize. This separatrix is commonly known as the “edge
structure”, and has been widely studied recently (Skufca et al. 2006, Schneider et al.
2007, Duguet et al. 2008, Viswanath & Cvitanović 2009). Although the lower-branch
ECS and its stable manifold form part of the edge structure, they probably are not the
only contribution. Skufca et al. (2006) observed that the stable manifold of a periodic
orbit coincides with the edge at low Re, and at higher Re a higher-dimensional chaotic
object is involved. Study of Duguet et al. (2008) in pipe flow suggested that a few TWs
and the heteroclinic connections between them are the key structures that organized
the edge. Although highly fractal in shape (Schneider et al. 2007), this laminarturbulence edge is presumed to be a surface insulating states in the turbulence side
from the laminar attractor. Based on the above discussion on lower-branch TWs,
we may assume that the edge is also hardly affected by polymer, and would prevent
turbulence from laminarization even at very high Wi. In this case, turbulence would
eventually be stuck near the edge structure as it is moved toward the laminar side
with increasing viscoelasticity; the dynamics of the edge would persist as Wi further
160
increases. This should echo the puzzle of experimentally observed MDR upper-bound.
After connecting all these threads, a picture would emerge presenting the dynamics underlying the transitions of viscoelastic turbulence at moderate Re. In Newtonian
flows, a number of upper-branch TWs form a chaotic saddle of active turbulence. Turbulence stays active for long time, while occasionally embarks on excursions toward
the laminar state. These trajectories can extend no further than the edge surface,
and would be reflected back near certain saddle structures on the edge (lower-branch
TWs or others). These excursions are observed as hibernating turbulence. In dilute polymer solutions, at low Wi (pre-onset stage), polymer has little effect on any
components of the turbulence dynamics. When Wi exceeds Wionset , polymer is significantly stretched in the active turbulence regime. Upper-branch TWs are modified:
polymer stress weakens the streamwise vortices, and the friction factor of these solutions is reduced (Stone et al. 2002, Stone & Graham 2003, Stone, Roy, Larson,
Waleffe & Graham 2004, Li, Stone & Graham 2005, Li, Xi & Graham 2006, Li &
Graham 2007). Changes in these TWs collectively cause drag reduction in active
turbulence. At LDR, hibernation still occurs on a occasional basis; its frequency
starts to increase at the LDR–HDR transition. The cause of this transition is unclear. One straightforward possibility is: as Wi increases, upper-branch TWs and
active turbulence is moved in the state space; when they are close enough to the edge
structure, formation of certain dynamical objects, such as heteroclinic orbits between
certain upper- and lower-branch solutions, greatly facilitates the visits of edge structure and thus hibernation. Another possibility is with sufficient viscoelasticity, some
of the TWs are eliminated (Stone et al. 2002, Li, Xi & Graham 2006, Li & Graham
2007), and more exits are created in the domain of active turbulence. This change
161
in hibernation frequency is reflected in various experimentally-measurable quantities (Warholic, Massah & Hanratty 1999). At sufficiently high Wi, turbulence stay in
hibernation for the majority of time. Since polymer is only mildly deformed during
hibernating turbulence, it is not able to keep turbulence stay in hibernation. Active
turbulence occurs intermittently, which is quickly quenched by polymer stress. Hibernating turbulence dominates experimental measurements due to the large fraction of
time it occupies. Since polymer is largely ineffective in changing flow structure during turbulence hibernation, this would be the upper-limit of polymer-induced drag
reduction.
The notion of “edge state” is mentioned by Benzi et al. (2005, 2006) and Procaccia
et al. (2008) in their phenomenological model of MDR. However, we need to clarify
that their “edge state” is fundamentally different from the edge structure in our
scenario. In their work, “edge” refers to the limit of turbulent kinetic energy (TKE)
reducing to zero, where their model for mean velocity profile approaches the Virk
MDR profile. The edge structure in our discussion is an actual object in the solution
state space, and there is no indication so far about its quantitative features. Studies
on Newtonian turbulence show that flow structures on the edge closely resemble many
TWs (Schneider et al. 2007, Duguet et al. 2008), which clearly have finite TKE. Also,
according their model, Reynolds shear stress is proportional to TKE, which in our
simulation although reduces, does not approach zero during hibernation. What is in
common between their model and our simulation, however, is that the universality of
MDR is rooted in Newtonian turbulence. Our observation that hibernating turbulence
exists in Newtonian flows, and is unaffected by polymer, is the key element that could
potentially explain the universality of MDR.
162
All current results are obtained at a relatively low Re, close to the critical Re
(≈ 1000) of laminar-turbulence transition. Turbulent structures in this regime are
relatively simple. In addition, at Re this low, the active turbulence regime is close
to the laminar-turbulence edge in the state space; if the above scenario is true, this
might be the reason the intermittent hibernation dynamics is easier to observe in our
study. At higher Re where most experiments are performed, hibernation in Newtonian
turbulence might be very rare, which would only become frequent at very high Wi.
Nevertheless, after the dynamics at near-transition-Re is further understood in the
future, effect of increasing Re should also be investigated; the whole physical picture
should be verified in the high-Re regime.
13.3
Development of methodology
The scenario above consists of two hypotheses: first, hibernating turbulence is built
around certain remote (w.r.t. active turbulence) TW solution(s), which might belong
to the category of lower-branch TWs; second, these TWs form at least part of the
laminar-turbulence edge structure. To verify this picture, the first step to take is
to find the corresponding TWs responsible for hibernating dynamics, analyze their
linear stability, and study their connection with the turbulence trajectory. During a
hibernation period, there has to be certain amount of time when the system passes
through the vicinity of these solution objects; if these are indeed TWs (steady-states
in traveling reference frames), using properly selected snapshots during hibernation
as initial guesses, Newton iteration should converge to these solutions. Therefore, developing an algorithm of finding steady-state solutions in viscoelastic turbulent flows
163
is the first task we propose in this section. Although our past work has been successful in numerically finding one class of viscoelastic TWs (ECS), the algorithm used in
that study is restricted to solutions with certain imposed symmetry-conditions (Stone,
Roy, Larson, Waleffe & Graham 2004, Li & Graham 2007). All other previous studies on viscoelastic turbulence were based on transient solutions (see Chapter 7). A
general algorithm of solving for viscoelastic TW solutions is thus due. Given the viscoelastic DNS (time-integration) code we have already developed, the Newton-Krylov
method is the more preferable algorithm (Sánchez et al. 2004, Viswanath 2007, 2009,
Gibson et al. 2008), which, instead of computing the Jacobian matrix directly, estimates its product with the state vector using a time-integration algorithm. This
method has been successfully applied in Newtonian turbulence problems in finding
steady states, traveling waves, periodic orbits and relative periodic-orbits (which allows phase shifts) (Viswanath 2007, 2009, Gibson et al. 2008).
Initial conditions for the Newton iteration should be taken at several important
instances during typical hibernation periods, including turning points of significant
signals such as Ubulk , h∂vx /∂yi, and the mean velocity profile slope. Solution objects
in control of the hibernating orbits could then be identified; although the case of
TW is discussed in Section 13.2, the Newton-Krylov method mentioned above is not
limited to TWs. In comparison, solutions dominating the active turbulence regime
should also be studied. Two aspects of these solutions are of interest. The first is how
do these solutions quantitatively recover the flow structures and statistics observed
in either hibernating or active turbulence. In particular, we are interested in if there
is one of them that can quantitatively match with the experimental observations
during MDR. The second problem to investigate is the effect of viscoelasticity on
164
these solutions: much weaker dependence on the viscoelasticity is expected for those
governing the hibernating regime.
Beyond the study of these solutions themselves, their connection with the dynamical trajectory should also be inspected. The relevant importance of each solution
to different stages of the trajectory can be determined by the frequency at which
these solutions are visited, and the distance between them and the trajectory during
each visit. The closeness of these solutions to a given instant on the trajectory, and
in general the distance between any two states in the state space, can be measured
by a form of inner product with the translational (and also rotational for the pipe
geometry) symmetry taken account of (Kerswell & Tutty 2007). A clearer view of
how these solutions determine the state-movement on the trajectory would require an
effective projection of the high-dimensional state space onto a 2D or 3D coordinate
system. Gibson et al. (2008) projected the transient trajectory of a Newtonian plane
Couette flow onto a set of orthonormal basis-states constructed with the upper-branch
ECS (Waleffe 2003) and its symmetric copies; and the geometry of the state space
was clearly visualized (Figure 7.4).
Computation of unstable eigenvalues and eigenvectors can be achieved through
time integration as well using Arnoldi iteration (Viswanath 2007, Gibson et al. 2008).
This would enable us to include unstable manifolds into the visualization discussed
above, along which the trajectory moves away from each solution. With these information, heteroclinic orbits, trajectories connecting the unstable manifold of one
solution to the stable manifold of another, could be numerically found (Duguet et al.
2008, Halcrow et al. 2009). These orbits determine the transitions from one TW to
another, thus would be important in the understanding of transitions between active
165
and hibernating turbulence, as well as the movement of state within the latter.
As to the second hypothesis, dynamical trajectories embedded on the edge structure can be computed directly with a time integration algorithm using a bisectionbased edge-tracking method (Skufca et al. 2006, Schneider et al. 2007, Duguet et al.
2008). This method is illustrated in Figure 13.2. For a given form of perturbation on
the laminar state with one adjustable parameter measuring its amplitude, through
bisection, one can find two amplitudes that are close to one other to the required numerical precision, while one of the corresponding states is below the edge, the other is
above. Using these two states as initial conditions, time integration will generate two
trajectories stay close to the edge for a fairly long amount of time; both trajectories
are good approximations to the actual edge they move along. As time proceeds, these
trajectories start to diverge, both moving away from the edge; once the distance between them grow larger than the required precision, a new pair of initial conditions
should be obtained by another bisection process, starting from which the tracking
would continue. This process is repeated for the desired length of the edge trajectory.
Using this method, we can readily find trajectories on the edge for viscoelastic
flows with our current DNS code. The proposed edge-tracking study should address
two questions: first, how close is the edge to hibernating turbulence; second, how
does viscoelasticity affect the edge. For both questions, beyond analyzing the edge
structure statistically, better understanding could be achieved when edge-tracking is
carried out in conjunction with the search and stability analysis of TWs discussed
above. On one hand, same as above, TWs and other solution objects forming the
edge can be found by carefully selecting the initial conditions (Duguet et al. 2008),
which can be compared with those found governing hibernating turbulence. On the
166
Figure 13.2: Schematic of the edge-tracking method based on repeated bisection (Skufca et al. 2006).
other, given a TW found relevant to hibernating turbulence, if this particular solution
is indeed on the edge, one can also find its stable manifold on the edge using the
method proposed by Viswanath & Cvitanović (2009).
In summary, many methods have been developed by the Newtonian turbulence
community to analyze the nonlinear dynamics governing the laminar-turbulence transition; most of them are readily adaptable to viscoelastic problems. The overall goal
is to obtain a clear view of the dynamical structure in the state space that causes all
these transitions in viscoelastic turbulence, especially the unique turbulence structure
and statistics in HDR and MDR, and the university of the latter.
167
13.4
Further extensions: other drag-reduced turbulent flow systems
Polymer is not the only type of drag-reducing agent, drag reduction in turbulent flows
are observed in many other fluid systems, such as fiber suspensions (Metzner 1977),
worm-like-micelle-forming surfactant solutions (Shenoy 1984, Zakin et al. 1998), and
even liquids with injected micro-bubbles (Madavan et al. 1984). Surfactant-induced
turbulent drag reduction is a particularly interesting extension to the current study.
Practically speaking, unlike polymer molecules which gradually degrade upon strong
deformations, and lose their drag-reducing capability (Culter et al. 1975, Vanapalli
et al. 2005), scission of worm-like micelles is non-permanent. Micelles broken in
a strong flow can regain their formation after the deformation is released. This
is particularly desirable in closed circulation flow systems, such as district heating
and cooling systems. In addition, solutions of worm-like micelles are fundamentally
viscoelastic fluids with more complicated interactions between the microscopic micelle
structures and macroscopic flow behaviors than dilute polymer solutions (Cates &
Candau 1990, Butler 1999, Raghavan & Kaler 2001, Qi & Zakin 2002); one would
thus expect both similarities and disparities between the drag-reducing mechanisms of
these two systems. With all the understanding we are about to acquire of turbulence
in dilute polymer solutions, the mechanism of surfactant-induced drag reduction could
be more accessible.
Perhaps the most interesting difference between these two drag-reducing systems
is that the maximum drag reduction limit in surfactant solutions can be higher than
in polymeric fluids (Bewersdorff & Ohlendorf 1988, Chara et al. 1993, Zakin et al.
168
1996, 1998). This clearly indicates a different drag-reducing mechanism, at least in
the high-extent of drag reduction limit, from the polymer system. If the hypothetical
picture in Section 13.2 would be verified, this difference would become even more
intriguing: the understanding of how surfactant might change the dynamics near the
edge would bring further insight into the problem of laminar-turbulence transition.
Mean velocity profiles of surfactant solutions appear similar to polymeric fluids at
low-extent of drag reduction (Bewersdorff & Ohlendorf 1988, Zakin et al. 1998), while
in high-extent of drag reduction, they can be qualitatively different. For cases close
to or even above the Virk MDR limit, the “S-shape” profile is often reported: the
profile is lower than the Virk MDR in the buffer layer, at y + ≈ 30 it starts to raise
with a much larger slope, and later crosses the Virk MDR profile (Chara et al. 1993,
Myska & Zakin 1997). Other shapes have been observed in different experimental
conditions (Warholic, Schmidt & Hanratty 1999, Itoh et al. 2005, Tamano et al. 2009).
Li, Kawaguchi, Segawa & Hishida (2005) even reported that for a fixed experimental
setup, the mean velocity profile and other turbulence statistics quantities can be
qualitatively different at the same level of drag reduction, in different regimes of
transition (note that in surfactant solutions, along an experimental path, DR% can
change non-monotonically with increasing Re; see e.g.
Qi & Zakin (2002) and Li,
Kawaguchi, Segawa & Hishida (2005)). Since in near-wall turbulence, different types
of coherent structures dominate different layers away from the wall (Robinson 1991),
these complexities observed in mean velocity profiles suggest a variety of complicated
interactions between worm-like micelles and different turbulent coherent structures.
As to the mechanism of the micro-structure-flow interaction, many studies (Bewersdorff et al. 1989, Myska & Zakin 1997, Myska & Stern 1998, Warholic, Schmidt &
169
Hanratty 1999) suggested that drag reduction is closely linked with the formation of
“shear-induced structures” (SIS) (super-molecular aggregates of worm-like micelles
formed under flow (Cates & Candau 1990, Liu & Pine 1996, Butler 1999, Förster
et al. 2005)); while some other studies indicated that the ability of forming SIS is not
a necessity for drag reduction (Myska & Zakin 1997, Qi & Zakin 2002). The role of
SIS in surfactant drag reduction is another major unsolved problem in this area.
Understanding of surfactant drag reduction is very limited even compared with
that of polymer drag reduction. Dynamics of worm-like micelles under flow are so
complicated that a satisfactory micro-mechanical model (like the bead-spring model
for flexible linear polymer (Bird, Curtis, Armstrong & Hassager 1987) for computer
simulation is still missing. Even if there is one, it is unlikely to be simple enough that
a constitutive equation can be derived analytically, which is necessary for DNS studies
with the state-of-the-art computation capacities. Among the very few computational
studies on turbulence of surfactant solutions, e.g. Yu et al. (2004) and Yu & Kawaguchi
(2005), constitutive equations for polymer were used; only the parameters were fitted
with rheological data of drag-reducing surfactant solutions. These simulations were
not able to capture the qualitatively-different dynamics in surfactant solutions.
Since surfactant solutions are also viscoelastic fluids, semi-empirical constitutive
equations can be built based on polymer models, with qualitative features of wormlike-micelle dynamics included. Bautista et al. (1999), Manero et al. (2002) and Boek
et al. (2005) proposed a constitutive equation, which based on the Oldroyd-B equation for polymer solutions (Bird, Armstrong & Hassager 1987), included an additional
partial-differential equation taking account of the dynamical destruction and reformation of micellar structures. Change of structure is parameterized as a varying micellar
170
contribution to the shear viscosity, the destruction rate is proportional to the rate of
work done by the flow on micellar structures, and the reformation rate is determined
by the distance from the current state to equilibrium. Simple as it is, this model
demonstrates how different conceptual elements of surfactant-solution dynamics can
be incorporated by modifying a viscoelastic constitutive equation for polymer solutions. More features can be included in the model in a similar manner; in particular,
the effect of SIS can be modeled by a shear-rate dependent term contributing to the
structure change. A good starting point to study surfactant turbulent flow is to perform DNS with the Boek-improved Bautista-Manero model (Boek et al. 2005). By
comparing the results with experimental observations, the constitutive model can be
further improved by including more features of surfactant dynamics, or better parameterization of the micellar structure. In addition, DNS results with and without
the SIS contribution should be compared to determine the importance of SIS relative
to other surfactant-specific features such as the dynamical destruction-reformation
process.
Another extension to the study of polymer drag reduction is the active control of
turbulence. Besides these intrusive drag-reducing agents like polymer and surfactant,
drag reduction can also be achieved with more controllable engineering techniques,
such as mechanical actuators (Rathnasingham & Breuer 2003), temperature variation (Yoon et al. 2006), blowing and suction of fluids through the wall (Choi et al.
1994), and electrical forces (Du & Karniadakis 2000). Previous feedback control
strategies focus on empirically-determined objective functions (e.g. Lee et al. (1998)).
With the recent understandings of the nonlinear dynamics in the regime of laminarturbulence transition, more rational schemes can be developed: these schemes can
171
either aim at restricting the flow state to the region close to the edge, or bringing
the state over the edge to laminar flow (Kawahara 2005, Wang et al. 2007). Generally speaking, further knowledge of the dynamical structures in the state space has
a two-fold impact on improving the design of active turbulence control strategies:
first, knowing the characteristics of important solution objects, measurements can be
better designed to provide a good estimation of the system state; second, given information about the current state, a clearer objective of control can be generated with
the knowledge of the state space geometry: e.g. to move the system to the closest
relatively-stable low-drag state.
Our proposed study on polymer drag reduction would also benefit the design of
more rational control strategies. For example, since hibernating turbulence is believed
to be a Newtonian structure, with a better knowledge of how polymer increases its
frequency of appearance, we can design a control strategy to maximize the probability of hibernating turbulence without adding polymer into the fluid. Furthermore,
polymer drag-reducing agents can be applied in conjunction with active control techniques to achieve larger drag reduction: the former can bring the turbulence close to
MDR whereas the latter can potentially break the MDR limit.
172
Appendix A
Numerical algorithm for the direct
numerical simulation of viscoelastic
channel flow
This appendix provides the detailed algorithm for the direct numerical simulation
(DNS) of viscoelastic flows in the plane Poiseuille geometry, used in Part II of this
dissertation. This algorithm is an extension of that of the Newtonian DNS code ChannelFlow, developed and maintained by Gibson (2009) (see also Canuto et al. (1988)),
to the viscoelastic system. A summary of the numerical method and parameters used
in this study is provided in Section 8.2; listed here is the corresponding formulation
for the method. For convenience, the equation system to be solved (Equations (8.1),
173
(8.2), (8.3) & (8.4)) is relisted below:
∂v
β 2
2 (1 − β)
+ v · ∇v = −∇p +
∇ v+
∇ · τ p,
∂t
Re
ReWi
∇ · v = 0.
α
Wi ∂α
T
+
+ v · ∇α − α · ∇v − (α · ∇v)
1 − tr(α)/b
2
∂t
b
=
δ,
b+2
b+5
α
2
τp =
− 1−
δ .
b
1 − tr(α)/b
b+2
(A.1)
(A.2)
(A.3)
(A.4)
We start by discussing the numerical algorithm of solving the Navier-Stokes equation: (A.1) & (A.2). The velocity and pressure fields are decomposed into the base
and perturbation components:
v = U ex + v † ,
(A.5)
p = Πx + p† .
(A.6)
Hereinafter, † indicates the perturbation component. Constant Π is the mean pressure
gradient; for plane Poiseuille flow with the fixed-pressure-drop constraint, its value is
−2/Re. The base flow velocity profile U = U (y) is chosen to be that of the laminar
plane Poiseuille flow, i.e. U (y) = 1 − y 2 (note: walls locate at y = ±1); ex is the unit
vector in the streamwise direction (similarly, ey and ez , which will appear below, are
unit vectors in wall-normal and spanwise directions, respectively). Plugging (A.5) &
(A.6) into (A.1) & (A.2), we obtain the partial differential equations for perturbation
174
variables v † and p† :
∂v †
β ∂ 2U
β 2 † 2 (1 − β)
= −v · ∇v − ∇p† − Πex +
∇v +
∇ · τ p,
ex +
2
∂t
Re ∂y
Re
ReWi
(A.7)
∇ · v † = 0.
(A.8)
We introduce simplified notations for the terms on the right-hand side of (A.7):
N ≡ v · ∇v,
(A.9)
β 2 †
∇v,
Re (A.10)
Lv † ≡
β ∂ 2U
− Π ex ,
Re ∂y 2
2 (1 − β)
S≡
∇ · τ p.
ReWi
C≡
(A.11)
(A.12)
Here, N is the inertia term (nonlinear); Lv † is the viscosity term (linear); C is the
constant term; S is the contribution of the divergence of polymer stress (nonlinear).
Equation (A.7) is then simplified as:
∂v †
= −N − ∇p† + Lv † + C + S.
∂t
(A.13)
Taking Fourier transform in x and z directions on both sides of the equation, we
obtain:
∂ ṽ †
˜ † + L̃ṽ † + C̃ + S̃,
= −Ñ − ∇p̃
∂t
(A.14)
where ∼ denotes variables in Fourier space in x and z dimensions, and in physical
175
space in the y dimension. Differential operators in (A.14) are defined as:
˜ =∇
˜ kx ,kz ≡ 2πi kx ex + ∂ ey + 2πi kz ez ,
∇
Lx
∂y
Lz
2
2
2
˜2 = ∇
˜ 2k ,k ≡ ∂ − 4π 2 ( kx + kz ),
∇
x z
∂y 2
L2x L2z
β ˜2
L̃ = L̃kx ,kz ≡
.
∇
Re kx ,kz
(A.15)
(A.16)
(A.17)
As introduced earlier in Section 8.2, the semi-implicit Adams-Bashforth/backwarddifferentiation scheme is used for temporal discretization. Linear terms, L̃ṽ † and
˜ † , are discretized with the implicit backward-differentiation method; nonlin−∇p̃
ear terms, −Ñ and S̃, are discretized with the explicit Adams-Bashforth method.
Detailed discussion of this scheme is given in Peyret (2002) (pages 131-132, Section 4.5.1(b)); for (A.14), the discretized time-stepping equation is:
1
∆t
k−1
X
bj −Ñ
n−j
+ S̃
n−j
ηṽ †,n+1 +
k−1
X
!
aj ṽ †,n−j
=
j=0
(A.18)
˜ †,n+1 ,
+ L̃ṽ †,n+1 + C̃ − ∇p̃
j=0
which, after rearrangement, becomes
η †,n+1
˜ †,n+1
ṽ
− L̃ṽ †,n+1 + ∇p̃
∆t
k−1 n−j
X
aj
n−j
=
− ṽ †,n−j − bj Ñ
+ C̃
− S̃
∆t
j=0
(A.19)
n
≡ R̃ .
Here, n is the index of the current step; n + 1 is the index of the next step, i.e. that of
176
Order
1
2
3
4
η
1
3/2
11/6
25/12
a0
−1
−2
−3
−4
a1
1/2
3/2
3
a2
a3
b0
1
2
3
4
−1/3
−4/3 1/4
b1
b2
b3
−1
−3
−6
1
4
−1
Table A.1: Numerical coefficients for the Adams-Bashforth/backwarddifferentiation temporal discretization scheme with different orders-ofaccuracy (Peyret 2002).
quantities to be solved. For an algorithm with k-th order accuracy in time, solutions
at k previous steps, including that at the n-th step, are needed at each time step.
These known solutions are indexed with n − j (0 6 j < k). Numerical coefficients η,
n
aj and bj are listed in Table A.1. Hereinafter, R̃ denotes the summation of terms
known at the n-th time step: i.e. terms do not involve quantities at the to-be-solved
(n + 1)-th step.
Expanding (A.19) with (A.17), we obtain:
2
kz2
kx
η
β ∂ 2 †,n+1
2 β
˜ †,n+1 = −R̃n .
ṽ
− 4π
+ 2 +
ṽ †,n+1 − ∇p̃
2
2
Re ∂y
Re Lx Lz
∆t
(A.20)
For each (kx , kz ) pair, Equation (A.20) is a differential equation with derivatives in y
only. The following quantities are constant for a given wavenumber pair:
λ = λkx ,kz
β
ν≡
,
Re
2
kx
kz2
η
2 β
+ 2 +
.
≡ 4π
2
Re Lx Lz
∆t
(A.21)
(A.22)
With the simplified notation above, (A.20) is rewritten below together with the
continuity equation in Fourier space (take Fourier transform in x and z dimensions
177
on both sides of (A.8)) and the no-slip boundary conditions at both walls:
ν
∂ 2 ṽ †
˜ † = −R̃,
− λṽ † − ∇p̃
∂y 2
˜ · ṽ † = 0,
∇
ṽ † |y=±1 = 0.
(A.23)
(A.24)
(A.25)
The above equation is referred to as the tau-equation. For each time step, the tauequation is solved for each wavenumber pair (kx , kz ). Note that time step indices n
and n + 1 are omited from these equations; at each time step, ṽ † and p̃† are unknown
quantities to be solved, and R̃ is known with information from solutions at previous
time steps.
Kleiser & Schumann (1980) proposed an elegant way, the influence matrix method,
of solving the tau-equation with both the divergence-free and boundary conditions satisfied analytically, which is discussed in detail in Canuto et al. (1988) (pages 216-221,
Section 7.3.1). Below we summarize the basic ideas of this method. To separate equations for ṽ † and p̃† , we take divergence of (A.23) and apply (A.24) to obtain (A.26);
then we take the y-component of (A.23) to get (A.28). Boundary conditions, (A.27)
and (A.29), are obtained by evaluating (A.24) at the no-slip walls, and taking the
y-component of (A.25), respectively. Here are the equations and boundary conditions
178
for p̃† and ṽy† :
∂ 2 p̃†
˜ · R̃,
− κ2 p̃† = −∇
∂y 2
∂ṽy†
|y=±1 = 0,
∂y
∂ 2 ṽy†
∂ p̃†
ν 2 − λṽy† −
= −R̃y ,
∂y
∂y
(A.26)
(A.27)
(A.28)
ṽy† |y=±1 = 0,
(A.29)
where κ is a constant for a given wavenumber pair:
2
κ =
κ2kx ,kz
β
≡ 4π
Re
2
kz2
kx2
+
L2x L2z
.
(A.30)
Equations (A.26), (A.27), (A.28) & (A.29) are called the A-problem. This problem
is not ready to solve since there are no boundary conditions for p̃† , while there are
two boundary conditions for ṽy† at each wall. If we could replace boundary conditions
(A.27) with Dirichlet boundary conditions for p̃† (see (A.32)), these equations would
be much easier to solve. This hypothetical problem equivalent to the original Aproblem, called the B-problem, is listed below:
∂ 2 p̃†
˜ · R̃,
− κ2 p̃† = −∇
∂y 2
(A.31)
p̃† |y=±1 = P± ,
(A.32)
∂ 2 ṽy†
ν 2
∂y
− λṽy† −
†
∂ p̃
= −R̃y ,
∂y
(A.33)
ṽy† |y=±1 = 0.
(A.34)
Of course boundary values for the pressure field P± are unknown. However, it can
179
be shown that a general solution to the B-problem can be constructed with a particular solution from the inhomogeneous version of the B-problem with homogeneous
boundary conditions (the B’-problem):
∂ 2 p̃†p
˜ · R̃,
− κ2 p̃†p = −∇
∂y 2
†
∂ 2 ṽy,p
ν
∂y 2
(A.35)
p̃†p |y=±1 = 0,
∂ p̃†p
†
− λṽy,p
= −R̃y ,
−
∂y
(A.36)
†
ṽy,p
|y=±1 = 0,
(A.38)
(A.37)
and basis solutions from two corresponding homogeneous problems, the B+ -problem:
†
∂ 2 ṽy,+
ν
∂y 2
∂ 2 p̃†+
− κ2 p̃†+ = 0,
2
∂y
(A.39)
p̃†+ |y=−1 = 0,
(A.40)
p̃†+ |y=+1 = 1,
(A.41)
∂ p̃†+
= 0,
∂y
(A.42)
†
ṽy,+
|y=±1 = 0;
(A.43)
∂ 2 p̃†−
− κ2 p̃†− = 0,
2
∂y
(A.44)
p̃†− |y=−1 = 1,
(A.45)
p̃†− |y=+1 = 0,
(A.46)
†
− λṽy,+
−
and the B− -problem:
†
∂ 2 ṽy,−
ν
∂y 2
†
− λṽy,−
−
∂ p̃†−
= 0,
(A.47)
†
ṽy,−
|y=±1 = 0.
(A.48)
∂y
180
All these equations are readily solvable with a standard numerical scheme. Take
the B’-problem for example, (A.35) is a complex Helmholtz equation with Dirichlet
boundary conditions (A.36), which can be solved with the Chebyshev-tau method
(see Canuto et al. (1988), pages 129-133, Section 5.1.2; this method is included in
the ChannelFlow code by Gibson (2009)). Once p̃†p is known, (A.37) and (A.38) is
another comlex Helmholtz equation system solvable with the same method. Same
procedures are performed for B+ - and B− -problems; note that these two problems do
not vary from one time step to the next, and only need to be solved once in the whole
simulation run. The general solution to the B-problem is thus:

†


p̃†p


p̃†+


p̃†−






 p̃  
 + δ− 
.
 + δ+ 
=

†
†
†
ṽy,+
ṽy,−
ṽy,p
ṽy†
(A.49)
There are two undetermined coefficients in the solution, δ+ and δ− , because the
boundary values of pressure in the B-problem are unknown. These coefficients can be
determined by matching the solution to the yet-unused boundary condition (A.27) in
the A-problem (which is equivalent to the B-problem):



†
∂ṽy,+
/∂y |y=+1
†
∂ṽy,−
/∂y |y=+1
†
†
/∂y |y=−1
∂ṽy,+
/∂y |y=−1 ∂ṽy,−




†
  δ+ 
 ∂ṽy,p /∂y |y=+1 
=
−



 . (A.50)
†
δ−
∂ṽy,p /∂y |y=−1
Equation (A.50) is known as the influence matrix equation. After we obtain the
numerical solutions of ṽy† and p̃† for the A-problem, an additional step called taucorrection is performed. This step corrects the discretization error in the above procedure, and is found necessary to maintain numerical stability. Detailed discussion of
181
tau-correction is found in Canuto et al. (1988) (Section 7.3.2), and not repeated here;
the tau-correction procedure is also included in the ChannelFlow code. Plugging p̃†
into the x and z components of (A.23), we get the complex Helmholtz equations for
ṽx† and ṽz† , both solvable with the Chebyshev-tau method. This completes the solution
for the Navier-Stokes equation ((A.1) & (A.2)).
The FENE-P equation for the polymer conformation tensor field (A.3) is easier
to solve. The equation, with the artificial diffusivity term 1/(ScRe)∇2 α added, is
rearranged as:
∂α
= − v · ∇α + α · ∇v + (α · ∇v)T
∂t
2
2 b
1
α
−
+
δ+
∇2 α.
Wi 1 − tr (α) /b Wi b + 2
ScRe
(A.51)
We again simplify the notation by defining:
α
2
,
Wi 1 − tr (α) /b
2 b
Cp ≡
δ,
Wi b + 2
1
Lp α ≡
∇2 α.
ScRe
N p ≡ −v · ∇α + α · ∇v + (α · ∇v)T −
(A.52)
(A.53)
(A.54)
Here N p denotes all nonlinear terms; Lp α is the linear term; and C p is the constant
term. The simplified convection-diffusion equation is:
∂α
= N p + C p + Lp α.
∂t
(A.55)
182
Taking Fourier transform in x and z directions, we obtain:
∂ α̃
= Ñ p + C̃ p + L̃p α̃,
∂t
(A.56)
where,
L̃p = L̃p |kx ,kz ≡
1 ˜2
∇
.
ScRe kx ,kz
(A.57)
Same as above, a semi-implicit scheme is used for temporal discretization: the
nonlinear term Ñ p is discretized with the explicit Adams-Bashforth method; the
linear term L̃p α̃ is discretized with the implicit backward-differentiation method.
The resulting time-stepping equation is:
1
∆t
n+1
η α̃
+
k−1
X
!
aj α̃
n−j
=
j=0
k−1
X
n−j
bj Ñ p
+ L̃p α̃n+1 + C̃ p .
(A.58)
j=0
After rearrangement, it becomes:
k−1
X aj
η n+1
n−j
− α̃n−j + bj Ñ p
α̃
− L̃p α̃n+1 =
+ C̃ p
∆t
∆t
j=0
(A.59)
n
≡ R̃p .
n
Here R̃p denotes terms that can be calculated with information known at the n-th
step. Numerical coefficients are the same as those given in Table A.1. Expanding
(A.59) with:
1
L̃p =
ScRe
∂2
− 4π 2
∂y 2
kx2
kz2
+
L2x L2z
,
(A.60)
183
we obtain:
1 ∂ 2 n+1
α̃
−
ScRe ∂y 2
4π 2
ScRe
kx2
kz2
+
L2x L2z
η
+
∆t
n
α̃n+1 = −R̃p .
(A.61)
Once boundary values of the α̃n+1 tensor are known, each component of (A.61) is
a complex Helmholtz equation that can be solved with the Chebyshev-tau method.
The boundary values are obtained by updating (A.58) without the artificial diffusivity
term L̃p α̃n+1 :
1
∆t
n+1
η α̃
+
k−1
X
!
aj α̃
n−j
=
k−1
X
n−j
bj Ñ p
+ C̃ p ,
(A.62)
j=0
j=0
which, after rearrangement, gives:
α̃n+1
∆t
=
η
k−1 X
j=0
aj
n−j
− α̃n−j + bj Ñ p
+ C̃ p
∆t
!
.
(A.63)
This equation can be explicitly computed.
The overall procedure is as follows. At the beginning of each time step, inverse
Fourier transform is performed for all fields, and the nonlinear terms, N , S and N p ,
are computed directly at each grid point. Note that for the computation of N , the
alternating form is used:


 ∇ · (vv) divergence form, when n is odd;
n
N =

 v · ∇v
convection form, when n is even.
(A.64)
Among several forms for evaluating this term discussed in Zang (1991), the alternating form offers the best combination of efficiency, accuracy and numerical stability.
184
Forward Fourier transform is then performed for all fields including results of these
n
nonlinear terms. A loop over every (kx , kz ) is started. In each step of the loop, R̃ ,
n
R̃p and boundary conditions for α̃n+1 are computed, after which the tau-equation
((A.23), (A.24) & (A.25)) for velocity and pressure fields, and Helmholtz equations
for the polymer conformation tensor field (A.61) are constructed and solved. At each
time-step and for each wavenumber pair (kx , kz ), a total of 10 complex Helmholtz
equations are solved: 4 for velocity and pressure fields (2 in the B’-problem, 2 more
for ṽx† and ṽz† ), and 6 for the FENE-P constitutive equation (only 3 out of the 6 offdiagonal components need to be computed because of the symmetry of the tensor).
Other than the 1st-order algorithm, all higher-order algorithms require initial
conditions at more than one consecutive time steps. For a typical situation where
initial condition at only one instant is available, initialization of the algorithm is
required. We use lower-order algorithms to initialize higher-order ones. For example,
if the 3rd-order algorithm is used as the main algorithm (as in all simulations presented
in Part II), and if we denote the initial condition as step n = 0, we compute the n = 1
solution with the 1st-order algorithm; then we use the n = 0 and n = 1 solutions to
compute the n = 2 solution with the 2nd-order algorithm. After these steps, sufficient
solutions at previous steps are available for the 3rd-order algorithm. Increasing the
order-of-accuracy in time slightly increases the computation time, it however requires
substantially larger memory space to store additional previous steps. In general, 2ndor higher-order algorithms are recommended for reasons of numerical stability and
accuracy.
185
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Xi, L. - FCF Research Group - University of Wisconsin–Madison

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