Fluid mechanics of turbulent flows - Markus Uhlmann

Fluid mechanics of turbulent flows - Markus Uhlmann

Evolution of a round jet
Boundary layer approximation
Energy budget
Fluid mechanics of turbulent flows
Fluidmechanik turbulenter Strömungen
Markus Uhlmann
Institut für Hydromechanik
Karlsruher Institut für Technologie
www.ifh.kit.edu
WS 2010/2011
Lecture 4
1/27
Evolution of a round jet
Boundary layer approximation
Energy budget
LECTURE 4
Free shear flows
2/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Questions to be answered in the present lecture
How does a turbulent flow develop away from solid boundaries?
How can the equations be simplified for slow spatial evolution?
What is the evolution in the self-similar region?
What is the turbulence structure in a plane jet?
3/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Types of free shear flows
Flows developing far from solid boundaries
I
wake
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jet
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mixing layer
4/27
Evolution of a round jet
Boundary layer approximation
Energy budget
The round jet
De
He
200
| C.Braun | Institut für Hydromechanik | Experimente in der Ström
Re ≈ 5000 (Deutsch & Haneka)
Einleitung – Eigenschaften der Turbulenz
Re = 3000 (van Dyke)
Re = 13000 (van Dyke)
(movie)
5/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Configuration and coordinate system
r, V
θ, W
Steady round jet
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ambient surroundings
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nozzle diameter d
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exit velocity UJ
UJ
d
x, U
→ Reynolds: Re = UJ d/ν
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statistically stationary,
stat. axisymmetric flow
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initial development (20d)
then self-similar
6/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Mean velocity and self-similarity
Observations from experiments
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mean axial velocity hu(x, r)i
→ velocity decays on axis
& jet spreads radially
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profile shape remains similar
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define centerline velocity:
U0 (x) ≡ hu(x, 0)i
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define jet half width r1/2 (x):
hu(x, r1/2 )i = U0 (x)/2
(Re = 95500, Hussein et al. 1994)
7/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Mean velocity and self-similarity (2)
Observations from experiments
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scaling of radial coordinate:
ξ ≡ r/r1/2
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scaling of velocity:
f (ξ) ≡ hui(x, r)/U0 (x)
→ profiles collapse
I
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spreading rate S ≈ 0.094:
r1/2 (x) = S(x − x0 )
velocity decay (B ≈ 5.8):
U0 (x) = B · d · UJ /(x − x0 )
→ local Reynolds is constant!
Re0 = r1/2 (x)U0 (x)/ν
(Re = 105 , Wygnanski & Fiedler 1969)
8/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Reynolds stresses
Observations from experiments
I
coordinate system:
x = (x, r, θ), u = (u, v, w)
I
due to symmetry:
hwi = hu0 w0 i = hv 0 w0 i = 0
 0 0

hu u i hu0 v 0 i
0
 hu0 v 0 i hv 0 v 0 i

0
0
0
0
0
hw w i
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self-similarity of hu0i u0j i/U02
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significant anisotropy
(Hussein et al. 1994)
9/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Reynolds stresses
CHAPTER 5: FREE SHEAR FLOWS
Turbulent Flows
Observations from experiments
Stephen B. Pope
Cambridge University Press, 2000
I
c
°Stephen
B. Pope 2000
supposing stress-strain
relation, shear stress can be
written:
hu0 v 0 i = −νT
∂hui
∂r
0.03
ν^ T
0.02
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positive eddy viscosity νT
0.01
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self-similarity:
νT (x, r)
ν̂T (r) ≡
U0 (x) r1/2 (x)
0.00
0.0
⇒ plateau across jet
1.0
r/r1/2
0.0 (Hussein et al.
0.11994)
η
2.0
0.2
Figure 5.10: Normalized turbulent diffusivity ν̂ T (Eq. (5.34)) in the
self-similar round jet. From the curve fit to the experimental data
10/27
of Hussein et al. (1994).
Evolution of a round jet
Boundary layer approximation
Energy budget
Navier-Stokes equations in cylindrical coordinates
Instantaneous equations
1
1
∂x u + ∂r (rv) + ∂θ w
r
r
w
∂t u + u∂x u + v∂r u + ∂θ u
r
w
w2
∂t v + u∂x v + v∂r v + ∂θ v −
r
r
w
vw
∂t w + u∂x w + v∂r w + ∂θ w +
r
r
= 0
1
= − ∂x p + ν∇2 u
ρ
1
v
2
2
= − ∂r p + ν ∇ v − 2 − 2 ∂θ w
ρ
r
r
1
w
2
2
= − ∂θ p + ν ∇ w − 2 + 2 ∂θ v
rρ
r
r
with:
1
1
∇2 = ∂xx + ∂r r∂r + 2 ∂θθ
r
r
11/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Reynolds-averaged Navier-Stokes in cylindrical coordinates
Averaged equations
I
non-swirling, statistically stationary & axisymmetric
1
0 = ∂x hui + ∂r (rhvi)
r
D̄hui
1
1
= − ∂x hpi − ∂x hu0 u0 i − ∂r (rhu0 v 0 i) + ν∇2 hui
ρ
r
D̄t
D̄hvi
1
1
hw0 w0 i
= − ∂r hpi − ∂x hu0 v 0 i − ∂r (rhv 0 v 0 i) +
ρ
r
r
D̄t
hvi
+ν ∇2 hvi − 2
r
with:
D̄
= ∂t + hui∂x + hvi∂r
D̄t
12/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Boundary layer approximation
Applicable to flows with slow spatial evolution
I
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small streamwise gradients: ∂x ∼ 1/L
large cross-stream variation: ∂r ∼ 1/δ
with L δ
⇒ simplification of Navier-Stokes equations
Examples:
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free shear flows (jet, wake, mixing layer)
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wall-bounded flows (boundary layer)
13/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Boundary layer equations for round jet
Perform order-of-magnitude analysis
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U , V are reference velocities
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continuity equation:
1
∂ hui + ∂ (rhvi) = 0
| x{z } |r r {z }
O( U
L)
O( V )
δ
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balance of the terms:
V ∼
Uδ
L
⇒ reference velocity in radial direction:
Uδ
L
14/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Boundary layer equations for round jet (2)
Radial momentum equation
term
∂t hvi
+hui∂x hvi
+hvi∂r hvi
= − ρ1 ∂r hpi
−∂x hu0 v 0 i
− 1r ∂r (rhv 0 v 0 i)
+ 1r hw0 w0 i
+ν∂xx hvi
+ν 1r ∂r (r∂r hvi)
+ν r12 ∂θθ hvi
−ν hvi
r2
⇒
1
ρ ∂r hpi
order O(·)
0 (stationary)
U V /L = U 2 δ/L2
V 2 /δ = U 2 δ/L2
∆pr /(ρδ)
R12 /L
R22 /δ
R33 /δ
νV /L2 = νU δ/L3
νV /δ 2 = νU/(δL)
0 (axisymmetric)
νV /δ 2 = νU/(Lδ)
O(·), divided by U 2 δ/L2
0
1
1
∆pr L2 /(ρU 2 δ 2 )
R12 L/(U 2 δ)
R22 L2 /(U 2 δ 2 )
R33 L2 /(U 2 δ 2 )
1/Re
dominant
•
•
•
1 L2
Re δ 2
0
1 L2
Re δ 2
+ 1r ∂r (rhv 0 v 0 i) − 1r hw0 w0 i = 0
15/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Boundary layer equations for round jet (3)
Obtaining the streamwise pressure gradient
I
integrate the approximate radial momentum equation:
Z ∞ 0 0
hpi
p0 (x)
hv v i hw0 w0 i
=
− hv 0 v 0 i +
−
dr0
0
0
ρ
ρ
r
r
r
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1
1 dp0
∂x hpi =
− ∂x hv 0 v 0 i + ∂x
ρ
ρ dx
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Z
r
∞
hv 0 v 0 i hw0 w0 i
−
r0
r0
dr0
neglecting streamwise gradients of Reynolds stresses:
∂x hpi ≈
dp0
dx
16/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Boundary layer equations for round jet (4)
Streamwise momentum equation
term
∂t hui
+hui∂x hui
+hvi∂r hui
= − ρ1 ∂x hpi
−∂x hu0 u0 i
− 1r ∂r (rhu0 v 0 i)
+ν∂xx hui
+ν 1r ∂r (r∂r hui)
+ν r12 ∂θθ hui
⇒
order O(·)
0 (stationary)
U 2 /L
V U/δ = U 2 /L
∆px /(ρL)
R11 /L
R12 /δ
νU/L2
νU/δ 2
0 (axisymmetric)
O(·), divided by U 2 /L
0
1
1
∆px /(ρU 2 )
R11 /U 2
R12 L/(U 2 δ)
1/Re
dominant
•
•
•
•
1
Reδ 2
0
hui∂x hui + hvi∂r hui = − ρ1 ∂x hpi − 1r ∂r (rhu0 v 0 i)
17/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Boundary layer equations for round jet – final result
1
∂x hui + ∂r (rhvi) = 0
r
hui∂x hui + hvi∂r hui = −
1
1 dp
0 − ∂r (rhu0 v 0 i)
ρ dx
r
Using the BL approximation:
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it can be shown that self-similarity implies:
1
(x − x0 )
∼ x − x0
U0 ∼
r1/2
→ consistent with experimental observations (cf. above)
18/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Closure of equations in BL approximation
Assuming a uniform turbulent viscosity
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define −hu0 v 0 i = νT ∂r hui
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experiments show similarity:
νT (x, r) = r1/2 (x)U0 (x)ν̂T (r)
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ν̂T ≈ 0.028 across jet
⇒ assume ν̂T = cst.
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BL equations closed
⇒ reasonable prediction of hui
(disagreement near edge)
νT = cst approx. vs. Hussein et al. 1994
19/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Kinetic energy equation
Instantaneous kinetic energy
I
definition: Ek (x, t) ≡ 12 u · u
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transport equation (cf. lecture 2):
DEk
= − (uj p/ρ),j + (2νui Sij ),j − 2νSij Sij
Dt
Mean kinetic energy
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decomposition:
1
hui · hui
hEi =
|2 {z }
≡ Ē
(due to mean flow)
+
1 0 0
hu · u i
|2 {z }
≡k
(due to turbulence)
20/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Kinetic energy of mean flow and turbulence
Transport equations
∂t Ē + huj iĒ + hui ihu0i u0j i + huj ihpi/ρ − 2νhui iS̄ij ,j
1 0 0 0
0 0
0 0
∂t k + huj ik + hui ui uj i + huj p i/ρ − 2νhui Sij i
2
,j
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= −P − ε̄
= +P − ε
production term: P ≡ −hu0i u0j i hui i,j
⇒ sink for Ē, source for k → exchange term!
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dissipation due to mean flow: ε̄ ≡ 2ν S̄ij S̄ij
0 S0 i
dissipation due to turbulence: ε ≡ 2νhSij
ij
21/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Scaling of dissipation
Round jet flow
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considering self-similarity of hui, hu0i u0j i
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production term (BL approximation) scales as:
P/(U03 /r1/2 ) ≈ −
hu0 v 0 i r1/2
∂r hui
U02 U0
→ turbulent dissipation will also scale as:
ε̂ = ε/(U03 /r1/2 )
⇒ dissipation defined as 2νhs0ij s0ij i, but found independent of ν!
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finer scales at higher Re → higher gradients s0ij
22/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Finer scales with increasing Reynolds
Re = 105
Re = 2 · 105
(Brown & Roshko 1974)
23/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Kolmogorov scales
Why is dissipation independent of the value for viscosity?
I
define characteristic scales of smallest turbulent motion:
1/4
η ≡ ν 3 /ε
, τη ≡ (ν/ε)1/2 , uη ≡ (νε)1/4
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Kolmogorov-scale Reynolds number: Reη ≡ η uη /ν = 1
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length scale decreases with Reynolds: η/r1/2 = Re0
→ compensates changes in viscosity
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more details on scaling in lecture 5
−3/4 −1/4
ε̂
24/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Turbulent kinetic energy budget
Observations from experiments
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mean-flow convection:
− (huj ik),j
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turbulent transport:
− 21 hu0i u0i u0j i + hu0j p0 i/ρ
0 i
−2νhu0i Sij
,j
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production: P
dissipation: −ε
(Panchapakesan & Lumley 1993)
25/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Comparison of time scales in turbulent free shear flow
Round jet data
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reference scale:
τ0 ≡ r1/2 /U0
mean shear:
τS ≡ 1/∂r hU i
τ0 /τ0 = 1
τS /τ0 ≈ 1.7
turbulence production:
τP ≡ k/P
τP /τ0 ≈ 6
turbulence dissipation:
τε ≡ k/ε
τε /τ0 ≈ 4.5
mean “flight time” from virtual origin:
τJ ≡ 12 x/U0
τJ /τ0 ≈ 5.3
→ turbulence is long-lived
26/27
Evolution of a round jet
Boundary layer approximation
Energy budget
Other canonical free shear flows
Plane jet
Mixing layer
Plane or axisymmetric wake
Homogeneous shear flow
Homogeneous-isotropic flow
27/27
Conclusion
Summary
Main questions of the present lecture
I
How does a turbulent flow develop away from solid
boundaries?
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How can the equations be simplified for slow spatial
evolution?
I
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What is the evolution in the self-similar region?
I
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boundary layer approximation
round jet: linear spreading, mean velocities ∼ 1/x
Turbulence structure in the round jet:
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turbulent kinetic energy budget
crude approximation with uniform turbulent viscosity
1/4
Conclusion
Problem
Derive the formula for the mean “flight time” of a fluid particle on
the round jet axis, measured from the virtual origin x0 to a
position x. (Answer: τJ = 12 x/U0 )
2/4
Conclusion
Outlook on next lecture: The scales of turbulence
How are energy and anisotropy distributed among scales?
Which physical processes occur on each scale?
3/4
Conclusion
Further reading
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S. Pope, Turbulent flows, 2000
→ chapter 5
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H. Tennekes and J.L. Lumley, First Course in Turbulence, 1972
→ chapter 5
4/4