Transient RANS and Hybrid RANS/LES

Title
1/4
Transient RANS and Hybrid RANS/LES
by
K. Hanjalić
Hanjali
Department of Multi-scale Physics, Delft University of Technology
Delft, The Netherlands
CONTENT:
¾ Limitations of LES and needs for combined (hybrid) LES/RANS
¾ Hybrid RANS/LES (HRL)
¾ Rationale and a priori tests; zonal and seamless coupling; interface issues
¾ One-, two- and multi-equation RANS models
¾ Examples of application of HRL in attached and separated flows
¾ Plane channel at high Re
¾ Hill flow
¾ Transient -RANS based VLES
¾ Rationale, justification, validation and limitations
¾ Application to thermal R-B convection at extreme Ra’s
(“ultra-turbulent” regime)
¾ Examples of practical relevance:
¾ Diurnal dynamics over a mezzo-scale town valley
¾ Diurnal wind over Arctic ice sheet
1
Motivation: Predictions of complex wallwall-bounded turbulent flows
and heat transfer at very high Reynolds and Rayleigh numbers
¾ The mainstay of the contemporary industrial CFD are the RANS
turbulence closures: affordable, economical,…but:
¾ too much empiricism, lack of universality, difficulties in predicting
complex unsteady and nonequilibrium flows, ..
¾ LES: less empirical, captures better the turbulence physics,
considered as the future industrial standard,…but:
¾ expensive and time consuming, especially for high Re and Ra
number wall-bounded flows in complex geometries:
Time- or ensemble-averaged (RANS) or filtered (LES)
momentum and energy Equations:
⎞
D Ui
1∂ P
∂ ⎛⎜ ∂ U i
ν
= Fi −
+
− τ ij ⎟⎟
⎜
Dt
ρ ∂xi ∂x j ⎝ ∂x j
⎠
DT
Dt
=
q
ρc p
+
∂
∂x j
⎛ν ∂T
⎞
⎜
− τθ i ⎟
⎜ σ T ∂x j
⎟
⎝
⎠
Common practice in RANS approach: Linear Eddy Viscosity/Diffusivity models:
2
3
⎛ ∂U i
τ ij = uiu j = kδ ij − ν t ⎜⎜
τθ i = θ ui = −
⎝ ∂x j
+
ν t ∂T
σ Tt ∂xi
∂U j ⎞
⎟
∂xi ⎟⎠
2
Grids issues for LES and RANS for wallwall-bounded turbulent flows
¾ LES of wall-bounded flows require high resolution grid in all directions
for resolving near-wall processes (∆x+ O(50), ∆y+ O(1), ∆z+ O(20))
¾ For resolving viscous near-wall boundary layer: No of grid cells ∝ Reτ1.8
as compared to Reτ0.4 for outer layer (Chapman, 1979).
¾ For R-B conv.: ∆/H ≈ O(Pr2/NuRa)1/4 ⇒ Total No of grid cells ∝ Ra!
¾ In contrast, for near-wall RANS N∝ ln Reτ , for R-B N∝ Ra1/3
¾ Hence, for high Reynolds and Raleigh numbers LES still too expensive
¾ Options for very high Re and Ra numbers:
• Hybrid LES/RANS (Balaras, Davidson, Spalart, Hamba, Piomelli,…
• RANS-based VLES
Wall functions versus nearnear-wall RANS (Hybrid)
¾ Wall functions
¾ No universal character
¾ Standard log-law adequate in simple wall-attached flows
¾ Inadequate for separated flows
¾ Hybrid LES-RANS (HRL) strategies (including DES)
¾ Substantial part of turbulence is modelled by RANS
¾ Significantly smaller number of cells (large aspect ratio)
¾ Criteria for location the RANS-LES interface:
¾ Decided by user or
¾ Controlled by cell dimensions –comparison
between length scale and typical mesh size
(critical in some separating flows)
3
Key questions regarding HRL
¾ Where to locate the interface?
¾ Which matching conditions are to be used at the interface?
¾ How does a RANS model react to unsteadiness (“receptivity”)?
¾ Will the dynamics be rightly returned?
¾ What is the impact of RANS layer on the LES region?
¾ Will the modelled contribution correctly compensate the
reduction in the resolved contribution?
¾ Which models are suitable?
DES approach of Spalart et al.
• One-equation transport model (Spalart-Allmaras, 1993, Nikitin et al. 2000)
used as RANS model in the near-wall region, and as an ssg model for LES
in the outer region
2
2
⎛ ∂ν ⎞ ⎤
⎛ ν ⎞
Dν
1 ∂ ⎡⎢
∂ν
i
⎥,
ν +ν
= cb1 Sν − cw1 f w ⎜ i ⎟ +
+ cb 2 ⎜
⎜ ∂x ⎟⎟ ⎥
Dt
σ
x
x
∂
∂
d
⎢
j
j
j
⎝ ⎠
⎝
⎠ ⎦
⎣
(
ν
ν t = ν fυ1 , χ = ,
ν
fυ1 =
g = r + cw 2 ( r 6 − r ) ,
χ3
χ 3 + cυ31
r=
)
χ
fυ 2 = 1 −
,
1 + χ fυ1
,
ν
2
Si di κ 2
,
1
⎡ 1 + cw6 3 ⎤ 6
fw = g ⎢
,
6 ⎥
⎣ g + cw3 ⎦
ν
Si = S +
f
2 υ2
κ 2 di
• Switching from RANS to LES: d = min(d , CDES ∆ ), where ∆ = max(∆ x, ∆ y, ∆ z )
and d is the distance from the nearest wall
4
DES: RANS/LES interface location in a channel flow
Grid: 96x64x64
Grid: 64x64x32
DES: RANS/LES interface location in a channel flow
Grid: 64x64x32
5
A priori test of the response of a RANS model to external
LES perturbations
¾ Prescribed RANS layer by reference to the distance to the wall, or separate
RANS and SGS models (ideally same type of model)
¾ A-priori test – overlap of the two domains
¾ A-posteriori test – two separate domains
A priori study
¾ Rationale:
• Identifying / quantifying the response of the RANS layer to LES
¾ Methodology
• LES provides information to RANS
• RANS does not provide information to LES
• LES is solved down to the wall
¾ Case Description
• Periodic channel flow
• Reb = 10935 – DNS of Moser, Kim and Mansour (1999)
• Computational domain: 2πh x 2h x πh
• Grid: 96 x 64 x 64 with
• Interface location:
• SGS model: Smagorinsky
6
A-priori test results
Instantaneous streamwise velocity profiles for the a-priori RANS and equivalent LES;
Time history for the velocity U (y+ = 30) and Uτ for a-priori RANS and equivalent LES
LES
RANS
A-priori test of Wall Function approach for LES
Wall-normal variations of the correlations for the a-priori RANS and LES,
(Temmerman, Leschziner & Hanjalic, 2002)
7
A-priori test results
Time-averaged kinetic energy and eddy viscosity for the reference
DNS, the a-priori RANS and the equivalent LES
A posteriori study: coupled RANS and LES
¾ RANS and LES regions are solved using the same solver
¾ Coupling strategy:
• Switch from RANS to LES at an imposed location and blending
of RANS and LES viscosity on the LES-RANS interface;
LES→RANS
data transfer
RAN
S
LES
° °
° °
•
•
•
•
• Switch from RANS to LES
controlled by wall distance
d and cell side ∆
Prescribed
y+int
8
Case Description: Fully developed channel flow
• Reb = 10935 (Reτ=590) (DNS by Moser, Kim and Mansour (1999)
• Reb = 40000 (Reτ=2000) (Experiments by Wei & Willmarth, 1998
• Computational domain: 2πh x 2h x πh
Hybrid RANS/LES and Coarse LES
• Grid: Nx × Ny × Nz= 64 x 32 x 64 with y+(1) = 0.5
• Interface locations: y+int=65 (Reτ=590); y+int=135 (Reτ=2000);
Fine-resolved LES
• Reτ=590 : Nx × Ny × Nz= 64 x 64 x 128 and 96x64x64, y+(1) = 0.5
• Reτ=2000: Nx × Ny × Nz= 512x128x128, y+(1) = 0.75
Channel Flow – Results, Reτ=2000
(L. Temmerman)
Time-averaged velocity and shear stress profiles for the LES computations.
9
HRL Modelling Practice: One-eqn. RANS
(L.Temmerman,
2001)
L.
¾ RANS model: one-equation transport model for turbulence
energy (Wolfshtein, 1969);
¾ SGS model: One-equation transport model for SGS energy
(Yoshizawa and Horiuti, 1985)
¾ Assumption: RANS and LES grids are identical at the interface;
¾ Target:
• Viscosity:
ν tRANS
= ν tLES
,int
, int
• Velocity:
RANS
U int
= U int
LES
• Modelled energy:
RANS
.
kmod,
int = k mod,int
LES
A two-layer hybrid scheme: Matching criteria
¾ Matching criteria: continuity of total eddy viscosity at the interface
ν SGS + ν
res
LES
= ν t +ν
ν
res
RANS
res
LES
(ui'u j' − uk' uk' δ ij / 3) S ij
=
S ij S ij
with overbar denoting filtered, and <> some local smoothing.
¾ Resolved stresses continuous across the interface ⇒ ν SGS
=νt
Cµ at the interface:
One-eqn model: ν t = Cµ lµ k
Cµ =
ν SGS
0.5
lµ k RANS ,int
0. 5
k-ε model: ν t = Cµ f µ
f µ ( k / ε )ν SGS
2
Cµ =
( f ( k / ε ))
µ
2
k2
ε
2
10
Adjustment of Cµ
Variant 1
⎡1 − exp ( − y ∆ ) ⎤⎦
Cµ = 0.09 + ( Cµ ,int − 0.09 ) ⎣
⎡⎣1 − exp ( − yint ∆ int ) ⎤⎦
Variant 2
⎧
y+
+
⎪ C µ = 0.09 27 for y ≤ 27
⎪
⎨
⎡C
− 0.09 ⎦⎤ ⎣⎡1 − exp( − ( y − y ( y + = 34)) / ∆ ) ⎦⎤
⎪ C = 0.09 + ⎣ µ ,int
for y + > 27
+
⎪ µ
⎣⎡1 − exp( − yint − y ( y = 34) / ∆ int ) ⎤⎦
⎩
Channel Flow – Results, Reτ=2000
One-equation RANS, (L. Temmerman)
Time-averaged velocity profiles for the hybrid RANS-LES computations.
11
HRL: Channel Flow, Reτ=2000, One-equation RANS,
(L. Temmerman)
Interface issues in Two-equation RANS model
¾ Interface B.C. for 2-eqn RANS, kint and εint - options for k:
c: Scale similarity
a: k = k res
int
LES
kint = k SGS = 0.5( Û i − U i )2
b: Isotropic spectrum distrib.
kint = k SGS =
3 Cκ −2 / 3 ⎛ vSGS ⎞
π
⎜
⎟
2 CS8 / 3
⎝ ∆ ⎠
where Û i - test-filtered velocity
U i - filtered velocity
2
¾ Interface B.C. for 2-eqn RANS, kint and εint - options for εint :
ε int =
k3/ 2
2.5 yn
Or, from least-square error between the
total viscosity on both sides of interface.
12
Channel Flow – Results, Reτ=2000
Two-equation RANS (M. Hadziabdic)
Streamwise vorticity, Reτ=590
fine-resolved LES (96x64x64)
coarse LES (64x64x32)
∆ z+
hybrid RANS/LES (64x64x32)
RANS/LES interface
13
Zonal k-v2-f RANS/LES: Equations
(Hadziabdic & Hanjalic 2003)
• RANS model:
k − ε −υ 2 − f
∂
∂k ∂U j k
=
+
∂x j
∂x j
∂t
⎛ LRANS
⎝ LLES
α = max⎜⎜1,
⎞
⎟⎟ ,
⎠
⎛
⎞
⎜ (ν + ν t ) ∂k ⎟ + P − α ⋅ ε
⎜
∂x j ⎟⎠
⎝
LRANS = Cl
α ≤1
RANS region
1 < α ≤ 1.5
Buffer region
α >1
LES region
1.5
ktot
ε
,
LLES = 0.8 ⋅ (∆X ⋅ ∆Y ⋅ ∆Z )1 / 3
ktot = k res + k mod
• LES model: Dynamic Smagorinsky model
Hybrid RANS (k-v2-f) / LES (dynamic): Velocity profiles
14
Hybrid RANS (k-v2-f) / LES (dynamic):
Shear stress and kinetic energy
Reτ=2000
Reτ=20000
Hill Flow – Case Description
¾ Periodic channel flow with constriction at both ends
¾ Re number based on channel height and bulk velocity is 21560
¾ Data from highly resolved LES computations (5 x 106 nodes) by
Temmerman et al (2003)
¾ Domain size: 9h × 3.036h × 4.5h (h=hill height)
¾ Grid details:
• Discretisation for HRL 112 × 64 × 56
+
• Near-wall resolution: yc (1) ≈ 1
• Spanwise and streamwise resolution: ∆x = ∆z
15
Hill Flow - Results
(x/h)sep. = 0.22
(x/h)reat. = 4.72
(x/h)sep. = 0.21
(x/h)reat. = 5.30
(x/h)sep. = 0.23
(x/h)reat. = 4.64
(x/h)sep. = 0.23
(x/h)reat. = 5.76
Averaged streamlines for the reference simulation, LES, DES and RANS-LES
cases.
Hill Flow - Results
Streamwise velocity profiles at x/h = 2.0.
16
Some observations
¾ Resolved motion in the URANS region is as strong as the
resolved motion in the equivalent LES region.
¾ Hence, in the RANS region, both resolved and modelled
contributions to the motion are substantial.
¾ The sum of both contributions is too high, hinting at the
need of an ad hoc modification to reduce the total motion.
¾ Channel Flow:
Flow Encouraging results.
¾ The response to the parameters change is small.
¾ Response to the location of the interface: in proportion of
modelled motion;
¾ Hill Flow:
Flow agreement with the reference data reasonable;
¾ Compared to the channel case, Cµ has a similar behaviour.
¾ Difficult to draw definitive conclusions because of the low Re.
Concluding Remarks on HRL
¾ New hybrid RANS-LES (HLR) method allowing:
• Freedom in locating the interface;
• Dynamic adjustment of the RANS model to ensure
continuity across the interface.
¾ For identical grids, the HRL results are significantly better
than those obtained with LES for the same (coarse) mesh.
¾ Application to a recirculating flow:
• Results are non-conclusive due to low Reynolds number;
• The hybrid RANS-LES approach overestimates the
recirculation zone length.
¾ Fundamental inconsistency in on the LES side next to RANS
¾ (unrealistic streaks structure, insufficient stress);
¾ Needs for further adjustment (smoothing, extra forcing,
artificial backscatter, …) irrespective of RANS model
17
Schematics of TRANS –VLES rationale
Semi-deterministic Modelling (SDM), (Ha Minh et al.)
T-RANS Niche: HighHigh-Ra challenge in thermal RB convection
¾ Nu∝ Ra1/3 for Ra<1012 (Pr O(1))
¾ Nu∝ Ra1/2 for Ra→∞
¾ λv /H ∝ Ra-1/7
¾ λθ/H ∝ Ra-1/3
18
T-RANS EQUATIONS AND SUBSCALE MODELS:
⎞ 1 ∂( P − Pref )
∂ Ui
∂ Ui
∂ ⎛⎜ ∂ Ui
+ βgi ( T − Tref )
+ Uj
=
ν
− τij ⎟ −
⎟ ρ
∂xi
∂t
∂x j
∂x j ⎜⎝ ∂x j
⎠
∂T
∂t
∂C
∂t
+ Uj
+ Uj
∂T
∂x j
=
⎞
∂ ⎛⎜ ν ∂ T
− τ θj ⎟
⎜
⎟
∂x j ⎝ Pr ∂x j
⎠
⎞
∂ ⎛⎜ ν ∂ C
=
− τcj ⎟
⎟
∂x j ∂x j ⎜⎝ Sc ∂x j
⎠
∂C
+final closure: 3eqn. model
Dk
= Dk + Pk + Gk − ε
Dt
Dε
Dt
D θ2
Dt
= Dε + Pε1 + Pε 2 + G ε − Y
= Dθ + Pθ − ε θ
assuming weak equilibrium
(D / Dt − Diff)ϕui = 0
Subscale ASM/AFM/ACM
⎤
⎡ ∂T
∂ Ui
+ ξτθj
+ ηβ g i θ 2 ⎥
⎢τij
∂x j
⎥⎦
⎣⎢ ∂x j
τθi = −Cφ
k
ε
τ ci = −C φ
∂ Ui ⎤
k ⎡ ∂C
+ ξτ cj
⎢ τ ij
⎥
ε ⎣⎢ ∂x j
∂x j ⎥⎦
⎛ ∂ Ui
∂ Uj
τ ij = − ν t ⎜
+
⎜ ∂x j
∂x i
⎝
⎞ 2
⎟ + k δ + C k β g θu
ij
i
j
⎟ 3
ε
⎠
Verification: Long-term averaged temperature profiles and heat flux
in R-B convection for different Ra numbers; DNS and TRANS
(Kenjereš and Hanjalić, 1999-2003)
^^
~~
Φ Ψ − ΦΨ = Φ Ψ + φψ
α
∂T ~ ~
q
− T W − θw = w = const.
∂z
ρcp
Wall scaling with heat-fluxbased buoyancy velocity
19
MON1(z/D=0.5,x/L=0.5, y/L=0.5)
MON2(z/D=0.01,x/L=0.5,y/L=0.5)
T-RANS
LES
Time spectra of <U>, <V>, <W> and <T> signals at
characteristic monitoring points, Ra=109
Mean vertical profiles of
temperature for different Ra
20
High-Ra number challenge in thermal RB convection
¾ Nu∝ Ra1/3 for Ra<1012 (Pr O(1))
¾ Nu∝ Ra1/2 for Ra→∞
Ra1/2
Ra1/3
Ra-1/7
¾ λv /H ∝ Ra-1/7
Ra-2/9
¾ λθ/H ∝ Ra-1/3
Ra-1/3
z
z
CASE (I): weak CASE (II): strong
∆Τ=1
T=T(x,y,z,τ)
z/H=2/3
1600m
800m
∆Τ=2
∆T=4
z/H=1/3
C=C(x,y,z,τ)
Residential Industrial
INITIAL STRATIFICATIONS
y
TEMPERATURE CONCENTRATION
y
T-RANS of pollutant dispersion in a town valley
21
Q0>0
Q0<0
TIME
INITIAL STRATIFICATION
CASE (I): weak
CASE (II): strong
Instantaneous trajectories in vertical plane over hilly terrain
T-RANS of pollutant dispersion in a town valley
Passive pollutant dispersion visualized by concentration isosurface
Evolution of the pollutant front (C=0.05 Cmax): strong stratification
22
Diurnal winds over Arctic ice sheet
380 km
Solution domain:
380x45x3 km
Mesh:
180x40x40
45km
3 km
Ra~1010 , Pr~ 1
Assumed near-ground
temperature
V. van Huijen, S. Kenjeres and K.Hanjalic
Some instantaneous streamline
patterns over an ice sheet
Wind velocity profiles:
comparison with measurements
23
Some Conclusions on T-RANS/VLES:
¾ Both RANS and LES will long be in use, each in its niche, but industry
cannot count on LES for large-scale problems in the foreseeable future
¾ There will be an increasing research on merging RANS and LES
strategies for very high Re and Ra numbers and complex flows
¾ T-RANS based VLES captures well main flow features in flows dominated
by large-scale (pseudo)deterministic structures
¾ T-RANS can be used to predict natural convection at very high Ra and in
complex domains, which are inaccessible to LES, DNS or other methods.
References:
1.
2.
3.
4.
5.
6.
7.
Balaras, E., Benocci, C., Piomelli, U., Two-layer approximate boundary conditions for large-eddy simulations,
AIAA Journal 34 (1996), 1111-1119.
Cabot, W., Moin, P., Approximate wall boundary conditions in the large eddy simulation of high Reynolds
number flow, Flow, Turbulence and Combustion, 63 (1999), 269-291.
Hanjalic, K., Hadziabdic, M., Temmerman, L. and Leschziner M., Merging RANS and LES strategies: zonal or
seamless coupling, Invited lecture DLES V, Munchen Aug. 27-29, 2003 (to appear in R. Friedrich, B. Geurs
and O. Metais, (eds) Durect and Large-Eddy Simulations V, Kluwer Acad. Publ. 2004
L.Temmerman, M.A.Leschziner, K.Hanjalic, A priori studies of a near-wall RANS model within a hybrid
LES/RANS scheme , 5th Internacional Symposium on Engineering Turbulence Modelling and Measurements,
Mallorca, Spain, 16-18 September, 2002
Spalart, P.R., Jou, W-H., Strelets, M., Allmaras, S.R., Comments on the feasibility of LES for wings and on the
hybrid RANS/LES approach, in Advances in DNS/LES, 1st AFOSR Int. Conf. On DNS/LES (Greden Press)
(1997).
Spalart P.R. and Allmaras, S.R., A one-equation turbulence model for aerodynamic flows. AIAA Paper
92-0439. (1992).
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constrictions, Int. J. Heat Fluid Flow (to appear).
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