SLIDE 1
A NEW APPROACH OF ZONAL HYBRID RANS-LES BASED ON A TWO-EQUATION k − ε MODEL [2] LARS DAVIDSON
ETMM9, Thessaloniki, 7-9 June 2012 Lars Davidson, www.tfd.chalmers.se/˜lada Financed by the EU project ATAAC (Advanced Turbulence Simulation for Aerodynamic Application Challenges) DLR, Airbus UK, Alenia, ANSYS, Beijing Tsinghua University, CFS Engineering, Chalmers, Dassault Aviation, EADS, Eurocopter Deutschland, FOI, Imperial College, IMFT, LFK, NLR, NTS, Numeca, ONERA, Rolls-Royce Deutschland, TU Berlin, TU Darmstadt, UniMAN
SLIDE 2 PANS LOW REYNOLDS NUMBER MODEL [3]
∂k ∂t + ∂(kUj) ∂xj = ∂ ∂xj
σku ∂k ∂xj
∂ε ∂t + ∂(εUj) ∂xj = ∂ ∂xj
σεu ∂ε ∂xj
ε k − C∗
ε2
ε2 k νt = Cµfµ k2 ε , C∗
ε2 = Cε1 + fk
fε (Cε2f2 − Cε1), σku ≡ σk f 2
k
fε , σεu ≡ σε f 2
k
fε Cε1, Cε2, σk, σε and Cµ same values as [1]. fε = 1. f2 and fµ read f2 =
3.1 2 1 − 0.3exp
Rt 6.5 2 fµ =
14 2 1 + 5 R3/4
t
exp
Rt 200 2 Baseline model: fk = 0.4. Range of 0.2 < fk < 0.6 is evaluated
www.tfd.chalmers.se/˜lada Zonal PANS 2 / 18
SLIDE 3 PANS LOW REYNOLDS NUMBER MODEL [3]
∂k ∂t + ∂(kUj) ∂xj = ∂ ∂xj
σku ∂k ∂xj
∂ε ∂t + ∂(εUj) ∂xj = ∂ ∂xj
σεu ∂ε ∂xj
ε k − C∗
ε2
ε2 k νt = Cµfµ k2 ε , C∗
ε2 = Cε1 + fk
fε (Cε2f2 − Cε1), σku ≡ σk f 2
k
fε , σεu ≡ σε f 2
k
fε Cε1, Cε2, σk, σε and Cµ same values as [1]. fε = 1. f2 and fµ read f2 =
3.1 2 1 − 0.3exp
Rt 6.5 2 fµ =
14 2 1 + 5 R3/4
t
exp
Rt 200 2 Baseline model: fk = 0.4. Range of 0.2 < fk < 0.6 is evaluated
www.tfd.chalmers.se/˜lada Zonal PANS 2 / 18
SLIDE 4 PANS LOW REYNOLDS NUMBER MODEL [3]
∂k ∂t + ∂(kUj) ∂xj = ∂ ∂xj
σku ∂k ∂xj
∂ε ∂t + ∂(εUj) ∂xj = ∂ ∂xj
σεu ∂ε ∂xj
ε k − C∗
ε2
ε2 k νt = Cµfµ k2 ε , C∗
ε2 = 1.5 + fk
fε (1.9 − 1.5), σku ≡ σk f 2
k
fε , σεu ≡ σε f 2
k
fε Cε1, Cε2, σk, σε and Cµ same values as [1]. fε = 1. f2 and fµ read f2 =
3.1 2 1 − 0.3exp
Rt 6.5 2 fµ =
14 2 1 + 5 R3/4
t
exp
Rt 200 2 Baseline model: fk = 0.4. Range of 0.2 < fk < 0.6 is evaluated
www.tfd.chalmers.se/˜lada Zonal PANS 2 / 18
SLIDE 5 CHANNEL FLOW: ZONAL RANS-LES
x y ku,int, εu,int wall yint LES, fk < 1 RANS, fk = 1.0 Interface: how to treat k and ε over the interface? They should be reduced from their RANS values to suitable LES values The usual convection and diffusion across the interface is cut off, and new “interface boundary” conditions are prescribed ku,int = fkkRANS Nothing is done for ε xmax = 3.2 (64 cells), zmax = 1.6 (64 cells), y dir: 80 − 128 cells CDS in entire region
www.tfd.chalmers.se/˜lada Zonal PANS 3 / 18
SLIDE 6 (Nx × Nz) = (64 × 64). y+
int = 500
1 100 1000 30000 5 10 15 20 25 30
y+ U+
0.05 0.1 0.15 0.2 −1 −0.8 −0.6 −0.4 −0.2
y uv+ Reτ = 4 000 Reτ = 8 000 Reτ = 16 000; Reτ = 32 000.
www.tfd.chalmers.se/˜lada Zonal PANS 4 / 18
SLIDE 7 INTERFACE LOCATION. Reτ = 8 000.
1 100 1000 8000 5 10 15 20 25 30
y+ U+
500 1000 1500 2000 −1 −0.8 −0.6 −0.4 −0.2
y+ uv+ y+ = 130 y+ = 500 y+ = 980
www.tfd.chalmers.se/˜lada Zonal PANS 5 / 18
SLIDE 8 EFFECT OF fk. Reτ = 16 000. y+
int = 500
1 100 1000 10000 5 10 15 20 25 30
y+ U+
0.05 0.1 0.15 0.2 −1 −0.8 −0.6 −0.4 −0.2
y uv+ fk = 0.2 fk = 0.3 fk = 0.5 fk = 0.6
www.tfd.chalmers.se/˜lada Zonal PANS 6 / 18
SLIDE 9 EFFECT OF RESOLUTION: VELOCITY
1 100 1000 30000 5 10 15 20 25 30
(Nx × Nz) = (32 × 32) y+ U+
1 100 1000 30000 5 10 15 20 25 30
(Nx × Nz) = (128 × 128) y+ U+ Reτ = 4 000 Reτ = 8 000 Reτ = 16 000; Reτ = 32 000.
www.tfd.chalmers.se/˜lada Zonal PANS 7 / 18
SLIDE 10 EFFECT OF RESOLUTION: RESOLVED SHEAR STRESS
0.05 0.1 0.15 0.2 −1 −0.8 −0.6 −0.4 −0.2
(Nx × Nz) = (32 × 32) y uv+
0.05 0.1 0.15 0.2 −1 −0.8 −0.6 −0.4 −0.2
(Nx × Nz) = (128 × 128) y uv+ Reτ = 4 000 Reτ = 8 000 Reτ = 16 000; Reτ = 32 000.
www.tfd.chalmers.se/˜lada Zonal PANS 8 / 18
SLIDE 11 EFFECT OF RESOLUTION: TURBULENT VISCOSITY
0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015
Reτ = 4000 νt/(uτδ)
0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015
Reτ = 8000
0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015
Reτ = 16 000 y νt/(uτδ)
0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015
Reτ = 32 000 y (Nx × Nz) = 64 × 64 32 × 32 128 × 128
www.tfd.chalmers.se/˜lada Zonal PANS 9 / 18
SLIDE 12 EFFECT OF RESOLUTION: TURBULENT VISCOSITY
0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015
Reτ = 4000 νt/(uτδ)
0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015
Reτ = 8000
0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015
Reτ = 16 000 y νt/(uτδ)
0.05 0.1 0.15 0.2 0.25 0.3 0.005 0.01 0.015
Reτ = 32 000 y (Nx × Nz) = 64 × 64 32 × 32 128 × 128
G r i d i n d e p e n d e n t t u r b u l e n t v i s c
i t y !
www.tfd.chalmers.se/˜lada Zonal PANS 9 / 18
SLIDE 13 SGS MODELS BASED ON GRID SIZE
When the grid is refined, νt gets smaller
www.tfd.chalmers.se/˜lada Zonal PANS 10 / 18
SLIDE 14 SGS MODELS BASED ON GRID SIZE
When the grid is refined, νt gets smaller E εsgs κ Pk,res κc
www.tfd.chalmers.se/˜lada Zonal PANS 10 / 18
SLIDE 15 SGS MODELS BASED ON GRID SIZE
When the grid is refined, νt gets smaller E εsgs,∆ εsgs,0.5∆ κ Pk,res κc 2κc
www.tfd.chalmers.se/˜lada Zonal PANS 10 / 18
SLIDE 16 SGS MODELS BASED ON GRID SIZE
When the grid is refined, νt gets smaller E εsgs,∆ εsgs,0.5∆ κ Pk,res κc 2κc εsgs,∆ = εsgs,0.5∆ εsgs = 2νt¯ sij¯ sij − τ12,t∂¯ u ∂y Grid refinement ⇒ must be accompanied with larger ¯ sij¯ sij ⇒ ¯ sij¯ sij must take place at higher wavenumbers if not ⇒ grid dependent
www.tfd.chalmers.se/˜lada Zonal PANS 10 / 18
SLIDE 17 POWER DENSITY SPECTRA OF ν0.5
t
∂ ¯ w′ ∂z
20 40 60 80 100 0.005 0.01 0.015 0.02
E
w′/∂z)2 κz One-eq ksgs model
20 40 60 80 100 0.005 0.01 0.015 0.02 0.025 0.03
κz Zonal PANS (Nx × Nz) = 64 × 64 32 × 32 128 × 128
www.tfd.chalmers.se/˜lada Zonal PANS 11 / 18
SLIDE 18 SGS DISSIPATION VS. WAVENUMBER
Energy spectra of the SGS dissipation show that the peak takes place at surprisingly low wavenumber (length scale corresponding to 10 cells or more). κc κ E(κ) εsgs
www.tfd.chalmers.se/˜lada Zonal PANS 12 / 18
SLIDE 19 SGS DISSIPATION VS. WAVENUMBER
Energy spectra of the SGS dissipation show that the peak takes place at surprisingly low wavenumber (length scale corresponding to 10 cells or more). κc κ E(κ) εsgs εsgs,κ
www.tfd.chalmers.se/˜lada Zonal PANS 12 / 18
SLIDE 20 SGS DISSIPATION, Reτ = 8000
SGS dissipation in the ¯ u′
i ¯
u′
i/2 eq, εsgs = 2νt¯
sij¯ sij − τ12,t∂¯ u ∂y
0.1 0.15 0.2 0.25 0.3 5 10 15
y εsgs One-eq ksgs model
0.1 0.15 0.2 0.25 0.3 5 10 15
y εsgs Zonal PANS (Nx × Nz) = 64 × 64 32 × 32 128 × 128
www.tfd.chalmers.se/˜lada Zonal PANS 13 / 18
SLIDE 21 LOCAL EQUILIBRIUM. Reτ = 4000, Nx × Nz = 64 × 64.
0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 0.6 2E−3 4E−3 0.1 0.2 0.3 0.4 0.5 0.6 2E−3 4E−3
y k equation
0.1 0.2 0.3 0.4 0.5 0.6 0.01 0.02 0.03 0.1 0.2 0.3 0.4 0.5 0.6 2E−4 4E−4 0.1 0.2 0.3 0.4 0.5 0.6 2E−4 4E−4
ε equation y Pk+ ε+ Cε1Pk/ε+ Cε2ε2/k+ Left vertical axes: URANS region; right vertical axes: LES region.
www.tfd.chalmers.se/˜lada Zonal PANS 14 / 18
SLIDE 22 LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium??
www.tfd.chalmers.se/˜lada Zonal PANS 15 / 18
SLIDE 23 LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium?? If Pk = ε
www.tfd.chalmers.se/˜lada Zonal PANS 15 / 18
SLIDE 24 LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium?? If Pk = ε then C1 ε kPk=C∗
2
ε2 k , because C1 = C∗
2
www.tfd.chalmers.se/˜lada Zonal PANS 15 / 18
SLIDE 25 LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium?? If Pk = ε then C1 ε kPk=C∗
2
ε2 k , because C1 = C∗
2
However, the figure on previous slide shows C1 ε k Pk
2
ε2 k
- www.tfd.chalmers.se/˜lada
Zonal PANS 15 / 18
SLIDE 26 LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium?? If Pk = ε then C1 ε kPk=C∗
2
ε2 k , because C1 = C∗
2
However, the figure on previous slide shows C1 ε k Pk
2
ε2 k
- Answer: when time-averaging ab = ab
www.tfd.chalmers.se/˜lada Zonal PANS 15 / 18
SLIDE 27 LOCAL EQUILIBRIUM IN ε EQUATION.
The answer is because of time averaging (ab < ab, (see below)
0.1 0.2 0.3 0.4 0.5 0.6 1 1.05 1.1 1.15 1.2
y εPk/k εPk/k ε2/k ε2/k
www.tfd.chalmers.se/˜lada Zonal PANS 16 / 18
SLIDE 28 RESOLVED AND MODELLED TURBULENT KINETIC ENERGY.
0.1 0.2 0.3 0.4 1 2 3 4 5 6 7 8
Resolved y kres
0.1 0.2 0.3 0.4 1 2 3 4 5 6 7 8
Modelled: bottom; total: top y k, kres + k Reτ = 4 000 Reτ = 8 000 Reτ = 16 000; Reτ = 32 000.
www.tfd.chalmers.se/˜lada Zonal PANS 17 / 18
SLIDE 29 CONCLUDING REMARKS
LRN PANS works well as zonal LES-RANS model for very high Reτ (> 32 000) The model gives grid independent results The location of the interface is not important (it should not be too close to the wall) Values of 0.2 < fk < 0.5 have little impact on the results
www.tfd.chalmers.se/˜lada Zonal PANS 18 / 18
SLIDE 30 [1] ABE, K., KONDOH, T., AND NAGANO, Y. A new turbulence model for predicting fluid flow and heat transfer in separating and reattaching flows - 1. Flow field calculations.
- Int. J. Heat Mass Transfer 37 (1994), 139–151.
[2] DAVIDSON, L. A new approach of zonal hybrid RANS-LES based on a two-equation k − ε model. In ETMM9: International ERCOFTAC Symposium on Turbulence Modelling and Measurements (Thessaloniki, Greece, 2012). [3] MA, J., PENG, S.-H., DAVIDSON, L., AND WANG, F. A low Reynolds number variant of Partially-Averaged Navier-Stokes model for turbulence. International Journal of Heat and Fluid Flow 32 (2011), 652–669.
www.tfd.chalmers.se/˜lada Zonal PANS 18 / 18
