Post on 12-Aug-2021
EMBEDDED LES USING PANS [2, 3]LARS DAVIDSON
Department of Applied MechanicsChalmers University of Technology, SE-412 96 Gothenburg,
SWEDEN
PANS LOW REYNOLDS NUMBER MODEL [4]∂ku
∂t+
∂(kuUj)
∂xj=
∂
∂xj
[(
ν +νu
σku
)
∂ku
∂xj
]
+ (Pu − εu)
∂εu
∂t+
∂(εuUj)
∂xj=
∂
∂xj
[(
ν +νu
σεu
)
∂εu
∂xj
]
+ Cε1Puεu
ku− C∗
ε2ε2
u
ku
νu = Cµfµk2
u
εu, C∗
ε2 = Cε1 +fkfε
(Cε2f2 − Cε1), σku ≡ σkf 2k
fε, σεu ≡ σε
f 2k
fε
Cε1, Cε2, σk , σε and Cµ same values as [1]. fε = 1. f2 and fµ read
f2 =
[
1 − exp(
−y∗
3.1
)
]2 {
1 − 0.3exp[
−( Rt
6.5
)2]}
fµ =
[
1 − exp(
−y∗
14
)
]2{
1 +5
R3/4t
exp[
−( Rt
200
)2]
}
Baseline model: fk = 0.4. Range of 0.2 < fk < 0.6 is evaluated
Davidson ( ) www.tfd.chalmers.se/˜lada 2 / 40
CHANNEL FLOW: DOMAIN
d
x
y
δ 2.2δ
LES, fk < 1RANS, fk = 1.0
Interface
Interface: Synthetic turbulent fluctuations are introduced asadditional convective fluxes in the momentum equations and thecontinuity equation
fk = 0.4 is the baseline value for LES [4]
Davidson ( ) www.tfd.chalmers.se/˜lada 3 / 40
INLET FLUCTUATIONS
0 0.5 1 1.5 20
0.5
1
1.5
2
y
〈u′v ′〉, v2rms, w2
rms, u2rms/u2
τ
0 0.1 0.2 0.3 0.4 0.5
0
0.2
0.4
0.6
0.8
1
w′w
′tw
o-po
intc
orr
z
Anisotropic synthetic fluctuations, u′, v ′, w ′,
Integral length scale L ≃ 0.13 (see 2-p point correlation)
Asymmetric time filter (U ′)m = a(U ′)m−1 + b(u′)m witha = 0.954, b = (1 − a2)1/2 gives a time integral scale T = 0.015(∆t = 0.00063)
Davidson ( ) www.tfd.chalmers.se/˜lada 4 / 40
INTERFACE CONDITIONS FOR ku AND εu
For ku & εu we prescribe “inlet” boundary conditions at theinterface.
First, the usual convective and diffusive fluxes at the interface areset to zeroNext, new convective fluxes are added. Which “inlet” valuesshould be used at the interface?
◮ ku,int = fk kRANS(x = 0.5δ), εu,int = C3/4µ k3/2
u,int/ℓsgs, ℓsgs = Cs∆,
∆ = V 1/3
◮
Davidson ( ) www.tfd.chalmers.se/˜lada 5 / 40
INTERFACE CONDITIONS FOR ku AND εu
For ku & εu we prescribe “inlet” boundary conditions at theinterface.
First, the usual convective and diffusive fluxes at the interface areset to zeroNext, new convective fluxes are added. Which “inlet” valuesshould be used at the interface?
◮ ku,int = fk kRANS(x = 0.5δ), εu,int = C3/4µ k3/2
u,int/ℓsgs, ℓsgs = Cs∆,
∆ = V 1/3
◮ Baseline Cs = 0.07; different Cs values are tested
Davidson ( ) www.tfd.chalmers.se/˜lada 5 / 40
NUMERICAL METHOD
Incompressible finite volume method
Pressure-velocity coupling treated with fractional step
2nd order central differencing scheme for momentum eqns in LESregion.
2nd order upwind differencing scheme (van Leer) for momentumeqns in RANS region.
Hybrid 1st order upwind/2nd order central scheme k & ε eqns.
2nd -order Crank-Nicholson for time discretization
Davidson ( ) www.tfd.chalmers.se/˜lada 6 / 40
CHANNEL FLOW: VELOCITY AND SHEAR STRESSES
100
101
102
0
5
10
15
20
25
30
y+
U+
0 0.5 1 1.5 2
−1
−0.5
0
0.5
1
y+〈u
′v′〉+
x/δ = 0.19 x/δ = 1.25 x/δ = 3
Davidson ( ) www.tfd.chalmers.se/˜lada 7 / 40
CHANNEL FLOW: STRESSES AND PEAK VALUES VS. x
0 200 400 600 8000
0.5
1
1.5
2
2.5
3
3.5
y/δ
reso
lved
stre
sses
x/δ = 3
0 0.5 1 1.5 2 2.5 3 3.50
2
4
0 0.5 1 1.5 2 2.5 3 3.50
50
100
x
〈u′u′〉+ m
ax
〈νu/ν
〉 max
〈u′u′〉+ 〈u′u′〉+max (left)〈v ′v ′〉+ 〈νu〉
+max (right)
〈w ′w ′〉+
Davidson ( ) www.tfd.chalmers.se/˜lada 8 / 40
CHANNEL FLOW: DIFFERENT Cs VALUE FOR εinterface
ku,int = fkkRANS
εu,int = C3/4µ k3/2
u,int/ℓsgs, ℓsgs = Cs∆
100
101
102
0
5
10
15
20
25
30
y+
U+
x/δ = 3
0 0.5 1 1.5 2
−1
−0.5
0
0.5
1
y+
〈u′v′〉+
Cs = 0.07 Cs = 0.1 Cs = 0.2
Davidson ( ) www.tfd.chalmers.se/˜lada 9 / 40
CHANNEL FLOW: DIFFERENT Cs VALUE FOR εinterface
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
y+
〈νu〉/
ν
x/δ = 3
0 1 2 3 40.85
0.9
0.95
1
1.05
x/δu τ
Cs = 0.07 Cs = 0.1 Cs = 0.2
Davidson ( ) www.tfd.chalmers.se/˜lada 10 / 40
CHANNEL FLOW: DIFFERENT fk VALUES
100
101
102
0
5
10
15
20
y+
U+
x/δ = 3
0 0.5 1 1.5 2
−1
−0.5
0
0.5
1
y+〈u
′v′〉+
fk = 0.4 fk = 0.2 fk = 0.6
Davidson ( ) www.tfd.chalmers.se/˜lada 11 / 40
CHANNEL FLOW: DIFFERENT fk VALUES
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
3.5
4
y+
〈νu〉/
ν
x/δ = 3
0 1 2 3 40.85
0.9
0.95
1
1.05
x/δu τ
fk = 0.4 fk = 0.2 fk = 0.6
Davidson ( ) www.tfd.chalmers.se/˜lada 12 / 40
HUMP FLOWxI/c = 0.6 RS
NTS 2D RANS PANS
Inlet, Separation xS/c = 0.65; reattachment xR/c = 1.1
Rec = 936 000Uijcν (Uin = c = ρ = 1, ν = 1/Rec
H/c = 0.91, h/c = 0.128, x/c = [0.6, 4.2]
Mesh: 312 × 120 × 64, Zmax = 0.2c (baseline)
Davidson ( ) www.tfd.chalmers.se/˜lada 13 / 40
BASELINE INLET FLUCTUATIONS
−1 0 1 2 3 4 5 6
0.15
0.2
0.25
0.3
0.35
0.4
y/c
〈u′v ′〉, v2rms, w2
rms, u2rms/u2
τ
0 0.02 0.04 0.06 0.08 0.1
0
0.2
0.4
0.6
0.8
1
w′w
′tw
o-po
intc
orr
z
Integral length scale L ≃ 0.04 (see 2-p point correlation)Asymmetric time filter (U ′)m = a(U ′)m−1 + b(u′)m witha = 0.954, b = (1 − a2)1/2 gives a time integral scale T = 0.038∆t = 0.002. 7500 + 7500 time steps (100 hours one core)Fluctuations multiplied byfbl = max {0.5 [1 − tanh(y − ybl − ywall)/b] , 0.02}, ybl = 0.2,b = 0.01.
Davidson ( ) www.tfd.chalmers.se/˜lada 14 / 40
NUMERICAL METHOD
Incompressible finite volume method
Pressure-velocity coupling treated with fractional step
95% 2nd order central and 5% 2nd order upwind differencingscheme for momentum eqns.
Hybrid 1st order upwind/2nd order central scheme k & ε eqns.
2nd -order Crank-Nicholson for time discretization
Davidson ( ) www.tfd.chalmers.se/˜lada 15 / 40
PRESSURE: AMPLITUDES OF INLET FLUCT
0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x/c
−C
p
baseline inlet fluct 1.5× (baseline inlet fluct)0.5× (baseline inlet fluct)
Davidson ( ) www.tfd.chalmers.se/˜lada 16 / 40
SKIN FRICTION: AMPLITUDES OF INLET FLUCT
0 0.5 1 1.5−2
0
2
4
6
8
10x 10
−3
x/c
Cf
0.6 0.8 1 1.2 1.4 1.6
−2
−1
0
1
2
3x 10
−3
x/c
zoom
baseline inlet fluct 1.5× (baseline inlet fluct)0.5× (baseline inlet fluct)
Davidson ( ) www.tfd.chalmers.se/˜lada 17 / 40
VELOCITIES: AMPLITUDES OF INLET FLUCT
0 0.2 0.4 0.6 0.8 1 1.20.1
0.15
0.2
0.25
x/c = 65y
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 80
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 100
U/Ub
y
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 110
U/Ubbaseline 1.5× (baseline) 0.5× (baseline)
Davidson ( ) www.tfd.chalmers.se/˜lada 18 / 40
VELOCITIES: AMPLITUDES OF INLET FLUCT
0 0.2 0.4 0.6 0.8 1 1.20
0.05
0.1
0.15
0.2
0.25
x/c = 120
U/Ub
y
0 0.2 0.4 0.6 0.8 1 1.20
0.05
0.1
0.15
0.2
0.25
x/c = 130
U/Ub
baseline 1.5× (baseline) 0.5× (baseline)
Davidson ( ) www.tfd.chalmers.se/˜lada 19 / 40
RESOLVED AND MODELLED (< 0) SHEAR STRESSES
−0.005 0.005 0.010.1
0.15
0.2
0.25
x/c = 0.65y
0 0.01 0.02 0.03 0.040
0.05
0.1
0.15
0.2
0.25
x/c = 0.80
0 0.01 0.02 0.03 0.040
0.05
0.1
0.15
0.2x/c = 1.00
〈τ12,u〉, 〈−u′v ′〉/U2b
y
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.10
〈τ12,u〉, 〈−u′v ′〉/U2b
baseline 1.5× (baseline) 0.5× (baseline)
Davidson ( ) www.tfd.chalmers.se/˜lada 20 / 40
SHEAR STRESSES: AMPLITUDES OF INLET FLUCT
Resolved and Modelled (< 0) Shear stresses
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.20
〈τ12,u〉, 〈−u′v ′〉/U2b
y
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.30
〈τ12,u〉, 〈−u′v ′〉/U2b
baseline inlet fluct 1.5× (baseline inlet fluct)0.5× (baseline inlet fluct)
Davidson ( ) www.tfd.chalmers.se/˜lada 21 / 40
TURB VISCOSITY: AMPLITUDES OF INLET FLUCT
0 2 4 6 8 100.1
0.12
0.14
0.16
0.18
0.2x/c = 0.65
y
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 0.80
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 1.00
〈νt〉/ν
y
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 1.10
〈νt〉/νbaseline 1.5× (baseline) 0.5× (baseline)
Davidson ( ) www.tfd.chalmers.se/˜lada 22 / 40
TURB VISCOSITY: AMPLITUDES OF INLET FLUCT
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 1.20
〈νt〉/ν
y
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 1.30
〈νt〉/ν
baseline 1.5× (baseline) 0.5× (baseline)
Davidson ( ) www.tfd.chalmers.se/˜lada 23 / 40
PRESSURE: fk = 0.3; NO INLET FLUCT; fk = 0.5
0.5 1 1.5 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x/c
−C
p
fk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 24 / 40
SKIN FRICTION: fk = 0.3; NO INLET FLUCT; fk = 0.5
0 0.5 1 1.5−2
0
2
4
6
8
10x 10
−3
x/c
Cf
0.6 0.8 1 1.2 1.4 1.6
−2
−1
0
1
2
3x 10
−3
x/c
zoom
fk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 25 / 40
VELOCITIES: fk = 0.3; NO INLET FLUCT; fk = 0.5
0 0.2 0.4 0.6 0.8 1 1.20.1
0.15
0.2
0.25
x/c = 65y
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 80
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 100
U/Ub
y
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 110
U/Ubfk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 26 / 40
VELOCITIES: fk = 0.3; NO INLET FLUCT; fk = 0.5
0 0.2 0.4 0.6 0.8 1 1.20
0.05
0.1
0.15
0.2
0.25
x/c = 120
U/Ub
y
0 0.2 0.4 0.6 0.8 1 1.20
0.05
0.1
0.15
0.2
0.25
x/c = 130
U/Ub
fk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 27 / 40
RESOLVED AND MODELLED (< 0) SHEAR STRESSES
−0.0025 0.0025 0.0050.1
0.12
0.14
0.16
0.18
0.2x/c = 0.65
y
0 0.01 0.02 0.03 0.040
0.05
0.1
0.15
0.2
0.25
x/c = 0.80
0 0.01 0.02 0.03 0.040
0.05
0.1
0.15
0.2x/c = 1.00
〈τ12,u〉, 〈−u′v ′〉/U2b
y
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.10
〈τ12,u〉, 〈−u′v ′〉/U2b
fk = 0.3 no inlet fluct fk = 0.5Davidson ( ) www.tfd.chalmers.se/˜lada 28 / 40
SHEAR STRESSES: fk = 0.3; NO INLET FLUCT; fk = 0.5
Resolved and Modelled (< 0) Shear stresses
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.20
〈τ12,u〉, 〈−u′v ′〉/U2b
y
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.30
〈τ12,u〉, 〈−u′v ′〉/U2b
fk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 29 / 40
TURB VISCOSITY: fk = 0.3; NO INLET FLUCT; fk = 0.5
0 2 4 6 8 100.1
0.12
0.14
0.16
0.18
0.2x/c = 0.65
y
0 50 100 1500
0.05
0.1
0.15
0.2x/c = 0.80
0 50 100 1500
0.05
0.1
0.15
0.2x/c = 1.00
〈νt〉/ν
y
0 50 100 1500
0.05
0.1
0.15
0.2x/c = 1.10
〈νt〉/νfk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 30 / 40
TURB VISCOSITY: fk = 0.3; NO INLET FLUCT; fk = 0.5
0 50 100 1500
0.05
0.1
0.15
0.2x/c = 1.20
〈νt〉/ν
y
0 50 100 1500
0.05
0.1
0.15
0.2x/c = 1.30
〈νt〉/ν
fk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 31 / 40
PRESSURE: WALE, Nk = 128, Zmax = 0.4, CDS
0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x/c
−C
p
fk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 32 / 40
SKIN FRICTION: WALE, Nk = 128, Zmax = 0.4, CDS
0 0.5 1 1.5−2
0
2
4
6
8
10x 10
−3
x/c
Cf
0.6 0.8 1 1.2 1.4 1.6
−2
−1
0
1
2
3x 10
−3
x/c
zoom
fk = 0.3 no inlet fluct fk = 0.5
Davidson ( ) www.tfd.chalmers.se/˜lada 33 / 40
VELOCITIES: WALE, Nk = 128, Zmax = 0.4, CDS
0 0.2 0.4 0.6 0.8 1 1.20.1
0.15
0.2
0.25
x/c = 65y
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 80
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 100
U/Ub
y
0 0.5 10
0.05
0.1
0.15
0.2
0.25
x/c = 110
U/UbWALE Nk = 128, Zmax = 0.4 CDS
Davidson ( ) www.tfd.chalmers.se/˜lada 34 / 40
VELOCITIES: WALE, Nk = 128, Zmax = 0.4, CDS
0 0.2 0.4 0.6 0.8 1 1.20
0.05
0.1
0.15
0.2
0.25
x/c = 120
U/Ub
y
0 0.2 0.4 0.6 0.8 1 1.20
0.05
0.1
0.15
0.2
0.25
x/c = 130
U/Ub
WALE Nk = 128, Zmax = 0.4 CDS
Davidson ( ) www.tfd.chalmers.se/˜lada 35 / 40
RESOLVED AND MODELLED (< 0) SHEAR STRESSES
−0.02 −0.01 0 0.01 0.020.1
0.15
0.2
0.25
x/c = 0.65y
0 0.01 0.02 0.03 0.040
0.05
0.1
0.15
0.2
0.25
x/c = 0.80
0 0.01 0.02 0.03 0.040
0.05
0.1
0.15
0.2x/c = 1.00
〈τ12,u〉, 〈−u′v ′〉/U2b
y
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.10
〈τ12,u〉, 〈−u′v ′〉/U2b
WALE Nk = 128, Zmax = 0.4 CDS
Davidson ( ) www.tfd.chalmers.se/˜lada 36 / 40
SHEAR STRESSES: WALE, Nk = 128, Zmax = 0.4, CDS
Resolved and Modelled (< 0) Shear stresses
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.20
〈τ12,u〉, 〈−u′v ′〉/U2b
y
0 0.01 0.02 0.030
0.05
0.1
0.15
0.2x/c = 1.30
〈τ12,u〉, 〈−u′v ′〉/U2b
WALE Nk = 128, Zmax = 0.4 CDS
Davidson ( ) www.tfd.chalmers.se/˜lada 37 / 40
TURB VISCOSITY: WALE, Nk = 128, Zmax = 0.4, CDS
0 2 4 6 8 100.1
0.12
0.14
0.16
0.18
0.2x/c = 0.65
y
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 0.80
0 20 40 60 800
0.05
0.1
0.15
0.2x/c = 1.00
〈νt〉/ν
y
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 1.10
〈νt〉/νWALE Nk = 128, Zmax = 0.4 CDS
Davidson ( ) www.tfd.chalmers.se/˜lada 38 / 40
TURB VISCOSITY: WALE, Nk = 128, Zmax = 0.4, CDS
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 1.20
〈νt〉/ν
y
0 10 20 30 40 50 60 700
0.05
0.1
0.15
0.2x/c = 1.30
〈νt〉/ν
WALE Nk = 128, Zmax = 0.4 CDS
Davidson ( ) www.tfd.chalmers.se/˜lada 39 / 40
CONCLUDING REMARKS
LRN PANS has been shown to work well as an embedded LESmethod
Channel flow: At two δ downstream the interface, the resolvedturbulence in good agreement with DNS data and the wall frictionvelocity has reached 99% of its fully developed value.
Channel flow: The treatment of the modelled ku and εu across theinterface is important.
LRN PANS predicts the hump flow well. LRN PANS better thanWALE.
Hump flow: CDS gives unphysical fluctuations near the inlet. Thebecause larger the smaller the inlet fluctuations
Davidson ( ) www.tfd.chalmers.se/˜lada 40 / 40
[1] ABE, K., KONDOH, T., AND NAGANO, Y.A new turbulence model for predicting fluid flow and heat transfer inseparating and reattaching flows - 1. Flow field calculations.Int. J. Heat Mass Transfer 37 (1994), 139–151.
[2] DAVIDSON, L., AND PENG, S.-H.Emdedded LES with PANS.In 6th AIAA Theoretical Fluid Mechanics Conference, AIAA paper2011-3108 (27-30 June, Honolulu, Hawaii, 2011).
[3] DAVIDSON, L., AND PENG, S.-H.Embedded LES using PANS applied to flow in a channel and overa hump (submitted).AIAA Journal (2012).
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Davidson ( ) www.tfd.chalmers.se/˜lada 40 / 40