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LARGE EDDY SIMULATION (LES)
Lars Davidson, www.tfd.chalmers.se/˜lada
Chalmers University of Technology
Gothenburg, Sweden
THREE-DAY CFD COURSE AT CHALMERS
◮This lecture is a condensed version of the course
Unsteady Simulations for Industrial Flows: LES, DES, hybrid
LES-RANS and URANS
5-7 November 2012 at Chalmers, Gothenburg, Sweden
Max 16 participants
50% lectures and 50% workshops in front of a PC
For info, see http://www.tfd.chalmers.se/˜lada/cfdkurs/cfdkurs.html
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 2 / 1
LECTURE NOTES
The slides are partly based on the course material at
(click here)
http://www.tfd.chalmers.se/˜lada/
comp turb model/lecture notes.html
This course is part of the MSc programme Applied Mechanics at
Chalmers. For Fluid courses, click here
http://www.tfd.chalmers.se/˜lada/
msc/msc-programme.html
The MSc programme is presented here
http://www.chalmers.se/en/education/programmes/maste
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 3 / 1
http://www.tfd.chalmers.se/~lada/comp_turb_model/lecture_notes.htmlhttp://www.tfd.chalmers.se/~lada/msc/msc-programme.htmlhttp://www.chalmers.se/en/education/programmes/masters-info/Pages/Applied-Mechanics.aspx
LARGE EDDY SIMULATIONS
GS
SGS
SGS
In LES, large (Grid) Scales (GS) are resolved and the small
(Sub-Grid) Scales (SGS) are modelled.
LES is suitable for bluff body flows where the flow is governed by
large turbulent scales
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 4 / 1
BLUFF-BODY FLOW: SURFACE-MOUNTED CUBE[14]Krajnović & Davidson (AIAA J., 2002)
Snapshots of large turbulent scales illustrated by Q = −∂ūi∂xj
∂ūj∂xi
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 5 / 1
BLUFF-BODY FLOW: FLOW AROUND A BUS[15]
Krajnović & Davidson (JFE, 2003)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 6 / 1
BLUFF-BODY FLOW: FLOW AROUND A CAR[16]Krajnović & Davidson (JFE, 2005)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 7 / 1
BLUFF-BODY FLOW: FLOW AROUND A TRAIN[12]
Hem
ida
&K
rajn
ović
,2006
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 8 / 1
SEPARATING FLOWS
Wall
TIME-AVERAGED flow and INSTANTANEOUS flow
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 9 / 1
SEPARATING FLOWS
Wall
TIME-AVERAGED flow and INSTANTANEOUS flow
In average there is backflow (negative velocities). Instantaneous,
the negative velocities are often positive.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 9 / 1
SEPARATING FLOWS
Wall
TIME-AVERAGED flow and INSTANTANEOUS flow
In average there is backflow (negative velocities). Instantaneous,
the negative velocities are often positive.
How easy is it to model fluctuations that are as large as the mean
flow?
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 9 / 1
SEPARATING FLOWS
Wall
TIME-AVERAGED flow and INSTANTANEOUS flow
In average there is backflow (negative velocities). Instantaneous,
the negative velocities are often positive.
How easy is it to model fluctuations that are as large as the mean
flow?
Is it reasonable to require a turbulence model to fix this?
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 9 / 1
SEPARATING FLOWS
Wall
TIME-AVERAGED flow and INSTANTANEOUS flow
In average there is backflow (negative velocities). Instantaneous,
the negative velocities are often positive.
How easy is it to model fluctuations that are as large as the mean
flow?
Is it reasonable to require a turbulence model to fix this?
Isn’t it better to RESOLVE the large fluctuations?
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 9 / 1
TIME AVERAGING AND FILTERING
RANS: time average. This is called Reynolds time averaging:
〈Φ〉 =1
2T
∫ T
−T
Φ(t)dt , Φ = 〈Φ〉+Φ′
In LES we filter (volume average) the equations. In 1D we get:
Φ̄(x , t) =
1
∆x
∫ x+0.5∆x
x−0.5∆xΦ(ξ, t)dξ
Φ = Φ̄ + Φ′′
(1)
no filter one filter
2 2.5 3 3.5 40.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
u,
ū
x
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 10 / 1
EQUATIONS
The filtering is defined by the discretization (nothing is done)
The filtered Navier-Stokes (N-S) eqns, i.e. the LES eqns, read
∂ūi∂t
+∂
∂xj
(
ūi ūj)
= −1
ρ
∂p̄
∂xi+ ν
∂2ūi∂xj∂xj
−∂τij∂xj
,∂ūi∂xi
= 0 (2)
where the subgrid stresses are given by
τij = uiuj − ūi ūj
Contrary to Reynolds time averaging where 〈u′i 〉 = 0, we have here
u′′i 6= 0 ū i 6= ūi
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 11 / 1
FILTERING: HOW IS EQ. 2 OBTAINED?
The N-S eqns are filtered (=discretized) using Eq. 1
The pressure gradient term, for example, reads
∂p
∂xi=
1
V
∫
V
∂p
∂xidV
Now we want to move the derivative out of the integral. It is
allowed if V is constant.
The filtering volume, V=grid cell which is not constant
Fortunately, the error is proportional to V 2, i.e. it is 2nd-order error
∂p
∂xi=
∂
∂xi
(
1
V
∫
V
pdV
)
+O(
V 2)
=∂
∂xi(p̄) +O
(
V 2)
All linear terms are treated in the same way.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 12 / 1
NON-LINEAR TERM
First we filter the term and move the derivative out of the integral
∂uiuj∂xj
=∂
∂xj
(
1
V
∫
V
uiujdV
)
+O(
V 2)
=∂
∂xj(uiuj) +O
(
V 2)
We have∂
∂xjuiuj ; we want
∂
∂xjūi ūj
Let’s add want we want (on both LHS ans RHS) and subtract want
we don’t want
This is how we end up with the convective term and the SGS term
in Eq. 2, i.e. −∂τij∂xj
= −∂
∂xj(uiuj − ūi ūj)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 13 / 1
LARGE EDDY SIMULATIONS
GS
SGS
SGS
Large scales (GS) are resolved; small scales (SGS) are modelled.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 14 / 1
ENERGY SPECTRUM
The limit (cut-off) between GS and SGS is supposed to take place in
the inertial subrange (II)
I
II
III
κ
E(κ) cut-off
I: large scales
II: inertial subrange, −5/3-range
III: dissipation subrange
GS SGS
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 15 / 1
SUBGRID MODEL
We need a subgrid model for the SGS turbulent scales
The simplest model is the Smagorinsky model [23]:
τij −1
3δijτkk = −2νsgss̄ij
νsgs = (CS∆)2√
2s̄ij s̄ij ≡ (CS∆)2 |s̄|
s̄ij =1
2
(
∂ūi∂xj
+∂ūj∂xi
)
, ∆ = (∆VIJK )1/3
(3)
A damping function fµ is added to ensure that νsgs ⇒ 0 as y ⇒ 0
fµ = 1 − exp(−y+/26)
A more convenient way to dampen the SGS viscosity near the wall
is
∆ = min{
(∆VIJK )1/3 , κy
}
where y is the distance to the nearest wall.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 16 / 1
SMAGORINSKY MODEL VS. MIXING-LENGTH MODEL
• The eddy viscosity in the mixing length model reads inboundary-layer flow [13, 22]
νt = ℓ2
∣
∣
∣
∣
∂U
∂y
∣
∣
∣
∣
.
• Generalized to three dimensions, we have
νt = ℓ2
[(
∂Ui∂xj
+∂Uj∂xi
)
∂Ui∂xj
]1/2
= ℓ2(
2SijSij)1/2
≡ ℓ2|S|.
• In the Smagorinsky model the SGS length scale ℓ = CS∆ i.e.
νsgs = (CS∆)2|s̄|
which is the same as Eq. 3
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 17 / 1
ENERGY PATHE(κ)
E(κ) ∝ κ−5/3
κ
inertial rangeεsgs ≃ 〈
νsgs ∂u ′
i∂xj
∂u ′i∂x
j
〉
ε
dissipating range
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 18 / 1
LES VS. RANS
LES can handle many flows which RANS (Reynolds Averaged Navier
Stokes) cannot; the reason is that in LES large, turbulent scales are
resolved. Examples are:
o Flows with large separation
o Bluff-body flows (e.g. flow around a car); the wake often includes
large, unsteady, turbulent structures
o Transition
• In RANS all turbulent scales are modelled ⇒ inaccurate
• In LES only small, isotropic turbulent scales are modelled ⇒ accurateLES is very much more expensive than RANS.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 19 / 1
FINITE VOLUME RANS AND LES CODES.
RANS LES
Domain 2D or 3D always 3D
Time domain steady or unsteady always unsteady
Space discretization 2nd order upwind central differencing
Time discretization 1st order 2nd order (e.g. C-N)
Turbulence model ≥ two-equations zero- or one-eq
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 20 / 1
TIME AVERAGING IN LES
t1: Start time averaging
t2: Stop time averaging
tt1: start t2: end
v̄1
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 21 / 1
NEAR-WALL RESOLUTION
Biggest problem with LES: near walls, it requires very fine mesh in
all directions, not only in the near-wall direction.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 22 / 1
NEAR-WALL RESOLUTION
Biggest problem with LES: near walls, it requires very fine mesh in
all directions, not only in the near-wall direction.
The reason: violent violent low-speed outward ejections and
high-speed in-rushes must be resolved (often called streaks).
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 22 / 1
NEAR-WALL RESOLUTION
Biggest problem with LES: near walls, it requires very fine mesh in
all directions, not only in the near-wall direction.
The reason: violent violent low-speed outward ejections and
high-speed in-rushes must be resolved (often called streaks).
A resolved these structures in LES requires ∆x+ ≃ 100,∆y+min ≃ 1 and ∆z+ ≃ 30
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 22 / 1
NEAR-WALL RESOLUTION
Biggest problem with LES: near walls, it requires very fine mesh in
all directions, not only in the near-wall direction.
The reason: violent violent low-speed outward ejections and
high-speed in-rushes must be resolved (often called streaks).
A resolved these structures in LES requires ∆x+ ≃ 100,∆y+min ≃ 1 and ∆z+ ≃ 30
The object is to develop a near-wall treatment which models the
streaks (URANS) ⇒ much larger ∆x and ∆z
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 22 / 1
NEAR-WALL RESOLUTION
Biggest problem with LES: near walls, it requires very fine mesh in
all directions, not only in the near-wall direction.
The reason: violent violent low-speed outward ejections and
high-speed in-rushes must be resolved (often called streaks).
A resolved these structures in LES requires ∆x+ ≃ 100,∆y+min ≃ 1 and ∆z+ ≃ 30
The object is to develop a near-wall treatment which models the
streaks (URANS) ⇒ much larger ∆x and ∆z
In the presentation we use Hybrid LES-RANS for which the grid
requirements are much smaller than for LES
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 22 / 1
NEAR-WALL RESOLUTION CONT’D
In RANS when using
wall-functions,
30 < y+ < 100 for thewall-adjacent cells
x
y wall
y+
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 23 / 1
NEAR-WALL RESOLUTION CONT’D
In RANS when using
wall-functions,
30 < y+ < 100 for thewall-adjacent cells
In LES, ∆z+ ≃ 30
x
y wall
y+
x
z
∆z+
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 23 / 1
NEAR-WALL RESOLUTION CONT’D
In RANS when using
wall-functions,
30 < y+ < 100 for thewall-adjacent cells
In LES, ∆z+ ≃ 30EVERYWHERE
x
y wall
y+
x
z
∆z+
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 23 / 1
NEAR-WALL RESOLUTION CONT’D
In RANS when using
wall-functions,
30 < y+ < 100 for thewall-adjacent cells
In LES, ∆z+ ≃ 30EVERYWHERE
AND ∆x+ ≃ 100,∆y+min ≃ 1
x
y wall
y+
x
z
∆z+
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 23 / 1
NEAR-WALL TREATMENT
from Hinze (1975)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 24 / 1
NEAR-WALL TREATMENT
0 1 2 3 4 5 6
Z
0
0.5
1
1.5
x
z
Fluctuating streamwise velocity at y+ = 5. DNS of channel flow.
We find that the structures in the spanwise direction are very
small which requires a very fine mesh in z direction.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 25 / 1
ZONAL PANS MODEL
L. Davidson
A New Approach of Zonal Hybrid RANS-LES Based on a
Two-equation k − ε Model [7]ETMM9, Thessaloniki, 7-9 June 2012
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
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 26 / 1
PANS LOW REYNOLDS NUMBER MODEL [17]
∂k
∂t+
∂(kUj )
∂xj=
∂
∂xj
[(
ν +νtσku
)
∂k
∂xj
]
+ (Pk − ε)
∂ε
∂t+
∂(εUj)
∂xj=
∂
∂xj
[(
ν +νtσεu
)
∂ε
∂xj
]
+ Cε1Pkε
k− C∗ε2
ε2
k
νt = Cµfµk2
ε,C∗ε2 = Cε1 +
fkfε(Cε2f2 − Cε1), σku ≡ σk
f 2kfε, σεu ≡ σε
f 2kfε
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
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 27 / 1
PANS LOW REYNOLDS NUMBER MODEL [17]
∂k
∂t+
∂(kUj )
∂xj=
∂
∂xj
[(
ν +νtσku
)
∂k
∂xj
]
+ (Pk − ε)
∂ε
∂t+
∂(εUj)
∂xj=
∂
∂xj
[(
ν +νtσεu
)
∂ε
∂xj
]
+ Cε1Pkε
k− C∗ε2
ε2
k
νt = Cµfµk2
ε,C∗ε2 = Cε1 +
fkfε(Cε2f2 − Cε1), σku ≡ σk
f 2kfε, σεu ≡ σε
f 2kfε
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
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 27 / 1
PANS LOW REYNOLDS NUMBER MODEL [17]
∂k
∂t+
∂(kUj)
∂xj=
∂
∂xj
[(
ν +νtσku
)
∂k
∂xj
]
+ (Pk − ε)
∂ε
∂t+
∂(εUj)
∂xj=
∂
∂xj
[(
ν +νtσεu
)
∂ε
∂xj
]
+ Cε1Pkε
k− C∗ε2
ε2
k
νt = Cµfµk2
ε,C∗ε2 = 1.5 +
fkfε(1.9 − 1.5), σku ≡ σk
f 2kfε, σεu ≡ σε
f 2kfε
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
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 27 / 1
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 bereduced 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 = fkkRANSNothing 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 Helsinki 4 October 2012 28 / 1
(Nx × Nz) = (64 × 64). y+int = 500
1 100 1000 300000
5
10
15
20
25
30
y+
U+
0 0.05 0.1 0.15 0.2-1
-0.8
-0.6
-0.4
-0.2
0
y〈u
′v′〉+
Reτ = 4 000 Reτ = 8 000 Reτ = 16 000;Reτ = 32 000.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 29 / 1
INTERFACE LOCATION. Reτ = 8 000.
1 100 1000 80000
5
10
15
20
25
30
y+
U+
0 500 1000 1500 2000-1
-0.8
-0.6
-0.4
-0.2
0
y+〈u
′v′〉+
y+ = 130 y+ = 500 y+ = 980
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 30 / 1
EFFECT OF fk . Reτ = 16 000. y+int = 500
1 100 1000 100000
5
10
15
20
25
30
y+
U+
0 0.05 0.1 0.15 0.2-1
-0.8
-0.6
-0.4
-0.2
0
y〈u
′v′〉+
fk = 0.2 fk = 0.3 fk = 0.5 fk = 0.6
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 31 / 1
EFFECT OF RESOLUTION: VELOCITY
1 100 1000 300000
5
10
15
20
25
30
(Nx × Nz) = (32 × 32)
y+
U+
1 100 1000 300000
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 Helsinki 4 October 2012 32 / 1
EFFECT OF RESOLUTION: RESOLVED SHEAR STRESS
0 0.05 0.1 0.15 0.2-1
-0.8
-0.6
-0.4
-0.2
0
(Nx × Nz) = (32 × 32)
y
〈u′v′〉+
0 0.05 0.1 0.15 0.2-1
-0.8
-0.6
-0.4
-0.2
0
(Nx × Nz) = (128 × 128)
y〈u
′v′〉+
Reτ = 4 000 Reτ = 8 000 Reτ = 16 000;Reτ = 32 000.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 33 / 1
EFFECT OF RESOLUTION: TURBULENT VISCOSITY
0 0.05 0.1 0.15 0.2 0.25 0.30
0.005
0.01
0.015Reτ = 4000
ν t/(
uτδ)
0 0.05 0.1 0.15 0.2 0.25 0.30
0.005
0.01
0.015Reτ = 8000
0 0.05 0.1 0.15 0.2 0.25 0.30
0.005
0.01
0.015Reτ = 16 000
y
ν t/(
uτδ)
0 0.05 0.1 0.15 0.2 0.25 0.30
0.005
0.01
0.015Reτ = 32 000
y(Nx × Nz) = 64 × 64 32 × 32 128 × 128
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 34 / 1
EFFECT OF RESOLUTION: TURBULENT VISCOSITY
0 0.05 0.1 0.15 0.2 0.25 0.30
0.005
0.01
0.015Reτ = 4000
ν t/(
uτδ)
0 0.05 0.1 0.15 0.2 0.25 0.30
0.005
0.01
0.015Reτ = 8000
0 0.05 0.1 0.15 0.2 0.25 0.30
0.005
0.01
0.015Reτ = 16 000
y
ν t/(
uτδ)
0 0.05 0.1 0.15 0.2 0.25 0.30
0.005
0.01
0.015Reτ = 32 000
y(Nx × Nz) = 64 × 64 32 × 32 128 × 128
Grid
indep
ende
nttur
bulen
t visc
osity
!
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 34 / 1
SGS MODELS BASED ON GRID SIZE
When the grid is refined, νt gets smaller
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 35 / 1
SGS MODELS BASED ON GRID SIZE
When the grid is refined, νt gets smallerE
εsgs
κ
Pk ,res
κc
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 35 / 1
SGS MODELS BASED ON GRID SIZE
When the grid is refined, νt gets smallerE
εsgs,∆εsgs,0.5∆
κ
Pk ,res
κc 2κc
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 35 / 1
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 = 2〈νt s̄ij s̄ij〉 − 〈τ12,t〉∂〈ū〉
∂y
Grid refinement ⇒ must beaccompanied with larger s̄ij s̄ij
⇒ s̄ij s̄ij must take place athigher wavenumbers
if not ⇒ grid dependent
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 35 / 1
POWER DENSITY SPECTRA OF ν0.5t∂w̄ ′
∂z
0 20 40 60 80 1000
0.005
0.01
0.015
0.02
E(
ν t(∂
w̄′/∂
z)2)
κz
One-eq ksgs model
0 20 40 60 80 1000
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 Helsinki 4 October 2012 36 / 1
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).
κcκ
E(κ)
εsgs
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 37 / 1
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).
κcκ
E(κ)
εsgs
εsgs,κ
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 37 / 1
SGS DISSIPATION, Reτ = 8000
SGS dissipation in the ū′i ū′
i/2 eq, εsgs = 2〈νt s̄ij s̄ij〉 − 〈τ12,t〉∂〈ū〉
∂y
0.1 0.15 0.2 0.25 0.30
5
10
15
y
ε sg
s
One-eq ksgs model
0.1 0.15 0.2 0.25 0.30
5
10
15
y
ε sg
s
Zonal PANS
(Nx × Nz) = 64 × 64 32 × 32 128 × 128
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 38 / 1
LOCAL EQUILIBRIUM. Reτ = 4000, Nx ×Nz = 64 × 64.
0 0.1 0.2 0.3 0.4 0.5 0.60
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4 0.5 0.60
2E-3
4E-3
0 0.1 0.2 0.3 0.4 0.5 0.60
2E-3
4E-3
y
k equation
0 0.1 0.2 0.3 0.4 0.5 0.60
0.01
0.02
0.03
0 0.1 0.2 0.3 0.4 0.5 0.60
2E-4
4E-4
0 0.1 0.2 0.3 0.4 0.5 0.60
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 Helsinki 4 October 2012 39 / 1
LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium??
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 40 / 1
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 Helsinki 4 October 2012 40 / 1
LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium??
If
〈Pk 〉 = 〈ε〉
then
C1〈ε〉
〈k〉〈Pk 〉6=C
∗
2
〈ε〉2
〈k〉,because C1 6= C
∗
2
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 40 / 1
LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium??
If
〈Pk 〉 = 〈ε〉
then
C1〈ε〉
〈k〉〈Pk 〉6=C
∗
2
〈ε〉2
〈k〉,because C1 6= C
∗
2
However, the previous slide shows
C1
〈 ε
kPk
〉
= C∗2
〈
ε2
k
〉
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 40 / 1
LOCAL EQUILIBRIUM IN ε EQUATION.
How can both the k eq. and ε be in local equilibrium??
If
〈Pk 〉 = 〈ε〉
then
C1〈ε〉
〈k〉〈Pk 〉6=C
∗
2
〈ε〉2
〈k〉,because C1 6= C
∗
2
However, the previous slide shows
C1
〈 ε
kPk
〉
= C∗2
〈
ε2
k
〉
Answer: when time-averaging 〈ab〉 6= 〈a〉〈b〉
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 40 / 1
LOCAL EQUILIBRIUM IN ε EQUATION.
The answer is because of time averaging (〈ab〉 < 〈a〉〈b〉, (seebelow)
0 0.1 0.2 0.3 0.4 0.5 0.61
1.05
1.1
1.15
1.2
y
〈εPk/k〉
〈ε〉〈Pk 〉/〈k〉
〈ε2/k〉
〈ε2〉/〈k〉
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 41 / 1
RESOLVED AND MODELLED TURBULENT KINETIC
ENERGY.
0 0.1 0.2 0.3 0.40
1
2
3
4
5
6
7
8Resolved
y
〈kre
s〉
0 0.1 0.2 0.3 0.40
1
2
3
4
5
6
7
8
Modelled: bottom; total: top
y〈k〉,
〈kre
s+
k〉
Reτ = 4 000 Reτ = 8 000 Reτ = 16 000;Reτ = 32 000.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 42 / 1
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 Helsinki 4 October 2012 43 / 1
HYBRID LES-RANS
Near walls: a RANS one-eq. k or a k − ω model.In core region: a LES one-eq. kSGS model.
y
x
Interface
wall
wall
URANS
URANS
LES
y+ml
• Location of interface either pre-defined or automatically computed
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 44 / 1
MOMENTUM EQUATIONS
• The Navier-Stokes, time-averaged in the near-wall regions andfiltered in the core region, reads
∂ūi∂t
+∂
∂xj
(
ūi ūj)
= −1
ρ
∂p̄
∂xi+
∂
∂xj
[
(ν + νT )∂ūi∂xj
]
νT = νt , y ≤ yml
νT = νsgs, y ≥ yml
• The equation above: URANS or LES? Same boundary conditions ⇒same solution!
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 45 / 1
TURBULENCE MODEL
• Use one-equation model in both URANS region and LES region.
∂kT∂t
+∂
∂xj(ūjkT ) =
∂
∂xj
[
(ν + νT )∂kT∂xj
]
+ PkT − Cεk
3/2T
ℓ
PkT = 2νT S̄ij S̄ij , νT = Ckℓk1/2T
LES-region: kT = ksgs, νT = νsgs, ℓ = ∆ = (δV )1/3
URANS-region: kT = k , νT = νt ,ℓ ≡ ℓRANS = 2.5n[1 − exp(−Ak
1/2y/ν)], Chen-Patel model (AIAAJ. 1988)
Location of interface can be defined by min(0.65∆, y),∆ = max(∆x ,∆y ,∆z)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 46 / 1
DIFFUSER[9]
Instantaneous inlet data from channel DNS used.
Domain: −8 ≤ x ≤ 48, 0 ≤ yinlet ≤ 1, 0 ≤ z ≤ 4.
xmax = 40 gave return flow at the outlet
Grid: 258 × 66 × 32.
Re = UinH/ν = 18 000, angle 10o
The grid is much too coarse for LES (in the inlet region
∆z+ ≃ 170)
Matching plane fixed at yml at the inlet. In the diffuser it is located
along the 2D instantaneous streamline corresponding to yml .
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 47 / 1
DIFFUSER GEOMETRY. Re = 18 000, ANGLE 10oH = 2δ
7.9H
21H
29H
4H
4.7H
periodicb.c.
convective outlet b.c.
no-slip b.c.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 48 / 1
DIFFUSER: RESULTS WITH LES• Velocities. Markers: experiments by Buice & Eaton (1997)
x = 3H 6 14 17 20 24H
x/H = 27 30 34 40 47Hwww.tfd.chalmers.se/˜lada Helsinki 4 October 2012 49 / 1
DIFFUSER: RESULTS WITH RANS-LESx = 3H 6 14 17 20 24H
x/H = 27 30 34 40 47H
forcing; no forcingwww.tfd.chalmers.se/˜lada Helsinki 4 October 2012 50 / 1
SHEAR STRESSES (×2 IN LOWER HALF)x = 3H 6 13 19 23H
x/H = 26 33 40 47H
resolved; modelledwww.tfd.chalmers.se/˜lada Helsinki 4 October 2012 51 / 1
RANS-LES: νt/νx = 3H 6 14 17 20 24H
x/H = 27 30 34 40 47H
forcing; no forcing
At x = 24H, νT ,max/ν ≃ 450
At x = −7H νT ,max/ν ≃ 11
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 52 / 1
k − ω MODEL SST-DES
DES [24]: Detached Eddy Simulation
SST [18, 19]: A combination of the k − ε and the k − ω model
∂k
∂t+
∂
∂xj(ūjk) =
∂
∂xj
[(
ν +νtσk
)
∂k
∂xj
]
+ Pk − β∗kω
∂ω
∂t+
∂
∂xj(ūjω) =
∂
∂xj
[(
ν +νtσω
)
∂ω
∂xj
]
+ αPkνt
− βω2+ . . .
The dissipation term in the k equation is modified as [19, 25]
β∗kω → β∗kωFDES , FDES = max
{
Lt
CDES∆,1
}
∆ = max {∆x1,∆x2,∆x3} , Lt =k1/2
β∗ω
⇒ RANS near walls and LES away from walls
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 53 / 1
RESOLUTION
For the near-wall region, we know how fine the mesh should be in
terms of viscous units (see Slide 22)
An appropriate resolution for the fully turbulent part of the
boundary layer is δ/∆x ≃ 10 − 20 and δ/∆z ≃ 20 − 40
This may be relevant also for jets and shear layers
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 54 / 1
HOW TO ESTIMATE RESOLUTION IN GENERAL? [4, 5]
Energy spectra (both in spanwise direction and time)
Two-point correlations
Ratio of SGS turbulent kinetic energy 〈ksgs〉 to resolved0.5〈u′u′ + v ′v ′ + w ′w ′〉
Ratio of SGS shear stress 〈τsgs,12〉 to resolved 〈u′v ′〉
Ratio of SGS viscosity, 〈νsgs〉 to molecular, ν
Energy spectra of SGS dissipation
Comparison of SGS dissipation due to ∂u′i/∂xj and ∂〈ūi〉/∂xj
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 55 / 1
CHANNEL FLOW, Reτ = 4000, y+ = 440
100
101
102
10-4
10-3
10-2
10-1
Ew
w(k
z)
κz = 2π(kz − 1)/zmax
Energy spectra
0 0.1 0.2 0.3 0.4-0.2
0
0.2
0.4
0.6
0.8
1
Bw
w(ẑ)/
w2 rm
s
ẑ = z − z0
Two-point correlations
(∆x ,∆z) 0.5∆x 0.5∆z ◦ 2∆x ; +: 2∆z
The (∆x ,∆z) mesh is (δ/∆x , δ/∆z) = (10,20)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 56 / 1
CHANNEL FLOW, Reτ = 4000, y+ = 440
100
101
102
10-4
10-3
10-2
10-1
Ew
w(k
z)
κz = 2π(kz − 1)/zmax
Energy spectra
0 0.1 0.2 0.3 0.4-0.2
0
0.2
0.4
0.6
0.8
1
Bw
w(ẑ)/
w2 rm
s
ẑ = z − z0
Two-point correlations
(∆x ,∆z) 0.5∆x 0.5∆z ◦ 2∆x ; +: 2∆z
The (∆x ,∆z) mesh is (δ/∆x , δ/∆z) = (10,20)
Two-point correlation is better
Shows that 2∆z and 2∆x (two-point corr in x) are too coarse.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 56 / 1
CHANNEL FLOW, Reτ = 4000, y+ = 440
0 1000 2000 3000 40000
1
2
3
4
5
6
kre
s
y+
kres = (u′2 + v ′2 + w ′2)/2
0 1000 2000 3000 40000.4
0.5
0.6
0.7
0.8
0.9
1
γ
y+
γ = kres+kres
Pope [20] suggests γ > 0.8 indicates well resolved flow
(∆x ,∆z) 0.5∆x 0.5∆z ◦ 2∆x ; +: 2∆z
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 57 / 1
CHANNEL FLOW, Reτ = 4000, y+ = 440
0 1000 2000 3000 40000
1
2
3
4
5
6
kre
s
y+
kres = (u′2 + v ′2 + w ′2)/2
0 1000 2000 3000 40000.4
0.5
0.6
0.7
0.8
0.9
1
γ
y+
γ = kres+kres
Pope [20] suggests γ > 0.8 indicates well resolved flow
(∆x ,∆z) 0.5∆x 0.5∆z ◦ 2∆x ; +: 2∆z
Pope criterion does not work here
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 57 / 1
SGS VS. MOLECULAR VISCOSITY [5]
0 1 2 3 40
0.2
0.4
0.6
0.8
1
y/H
〈νsgs〉/ν0 2 4 6 8 10 12
-4
-3
-2
-1
0
1
y/H
〈νsgs〉/ν
Nz = 32; Nz = 64; Nz = 128.
x = −H
x = 20H
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 58 / 1
SGS VS. RESOLVED SHEAR STRESSES
0 0.05 0.1 0.15 0.2 0.25 0.30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
y/H
〈τsgs,12〉/〈u′v ′〉
0 0.01 0.02 0.03 0.04 0.05
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
y/H
〈τsgs,12〉/〈u′v ′〉
Nz = 32; Nz = 64; Nz = 128.
x = −H
x = 20H
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 59 / 1
THE PANS MODEL
The PANS model is a modified k − ε model
It can operate both in RANS mode and LES mode
In the present work a low-Reynolds turbulence version of the
PANS is used
A method how to implement embedded LES is proposed
It is evaluated for channel flow and hump flow
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 60 / 1
Embedded LES Using PANS [10, 11]
Lars Davidson1 and Shia-Hui Peng1,2
Davidson& Peng
1Department of Applied Mechanics
Chalmers University of Technology, SE-412 96 Gothenburg, SWEDEN2FOI, Swedish Defence Research Agency, SE-164 90, Stockholm,
SWEDEN
PANS LOW REYNOLDS NUMBER MODEL [17]
∂ku∂t
+∂(kuUj)
∂xj=
∂
∂xj
[(
ν +νuσku
)
∂ku∂xj
]
+ (Pu − εu)
∂εu∂t
+∂(εuUj)
∂xj=
∂
∂xj
[(
ν +νuσεu
)
∂εu∂xj
]
+ Cε1Puεuku
− C∗ε2ε2uku
νu = Cµfµk2uεu
,C∗ε2 = Cε1 +fkfε(Cε2f2 − Cε1), σku ≡ σk
f 2kfε, σεu ≡ σε
f 2kfε
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
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 62 / 1
CHANNEL FLOW: DOMAIN
d
x
y
δ 2.2δ
LES, fk < 1RANS, fk = 1.0
Interface
Interface: Synthetic turbulent fluctuations are introduced as
additional convective fluxes in the momentum equations and the
continuity equation
fk = 0.4 is the baseline value for LES [17]
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 63 / 1
INLET FLUCTUATIONS
0 0.5 1 1.5 20
0.5
1
1.5
2
y
〈u′v ′〉, v2rms, w2rms, u
2rms/u
2τ
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-p
oin
tco
rr
ẑ
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)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 64 / 1
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 are
set to zero
Next, new convective fluxes are added. Which “inlet” valuesshould be used at the interface?
◮ ku,int = fk kRANS(x = 0.5δ), εu,int = C3/4µ k
3/2u,int/ℓsgs, ℓsgs = Cs∆,
∆ = V 1/3
◮
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 65 / 1
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 are
set to zero
Next, new convective fluxes are added. Which “inlet” valuesshould be used at the interface?
◮ ku,int = fk kRANS(x = 0.5δ), εu,int = C3/4µ k
3/2u,int/ℓsgs, ℓsgs = Cs∆,
∆ = V 1/3
◮ Baseline Cs = 0.07; different Cs values are tested
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 65 / 1
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
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 66 / 1
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
lve
dstr
esse
s
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
〉 ma
x
〈u′u′〉+ 〈u′u′〉+max (left)〈v ′v ′〉+ 〈νu〉
+max (right)
〈w ′w ′〉+
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 67 / 1
CHANNEL FLOW: DIFFERENT Cs VALUE FOR εinterface
ku,int = fkkRANS
εu,int = C3/4µ k
3/2u,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
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 68 / 1
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/δuτ
Cs = 0.07 Cs = 0.1 Cs = 0.2
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 69 / 1
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
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 70 / 1
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/δuτ
fk = 0.4 fk = 0.2 fk = 0.6
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 71 / 1
HUMP FLOWxI/c = 0.6 RS
NTS 2D RANS PANS
Inlet, Separation xS/c = 0.65; reattachment xR/c = 1.1
Rec = 936 000Uijc
ν (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)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 72 / 1
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, w2rms, u
2rms/u
2τ
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-p
oin
tco
rr
ẑ
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 by
fbl = max {0.5 [1 − tanh(y − ybl − ywall)/b] ,0.02}, ybl = 0.2,b = 0.01.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 73 / 1
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)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 74 / 1
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)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 75 / 1
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)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 76 / 1
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)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 77 / 1
RESOLVED AND MODELLED (< 0) SHEAR STRESSES
-5 0 5 10 15
x 10-3
0.1
0.15
0.2
0.25
x/c = 0.65y
0 0.01 0.02 0.030
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)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 78 / 1
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)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 79 / 1
TURB VISCOSITY: AMPLITUDES OF INLET FLUCT
0 5 10 15 200.1
0.12
0.14
0.16
0.18
0.2x/c = 0.65
y
0 20 40 60 80 100 1200
0.05
0.1
0.15
0.2x/c = 0.80
0 20 40 60 80 100 1200
0.05
0.1
0.15
0.2x/c = 1.00
〈νt〉/ν
y
0 20 40 60 80 100 1200
0.05
0.1
0.15
0.2x/c = 1.10
〈νt〉/νbaseline 1.5× (baseline) 0.5× (baseline)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 80 / 1
TURB VISCOSITY: AMPLITUDES OF INLET FLUCT
0 20 40 60 80 100 1200
0.05
0.1
0.15
0.2x/c = 1.20
〈νt〉/ν
y
0 20 40 60 80 100 1200
0.05
0.1
0.15
0.2x/c = 1.30
〈νt〉/ν
baseline 1.5× (baseline) 0.5× (baseline)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 81 / 1
PRESSURE: fk = 0.5; NO INLET FLUCT; Nk = 128
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
Nk = 128 no inlet fluct fk = 0.5
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 82 / 1
SKIN FRICTION: fk = 0.5; NO INLET FLUCT; Nk = 128
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
Nk = 128 no inlet fluct fk = 0.5
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 83 / 1
VELOCITIES: fk = 0.5; NO INLET FLUCT; Nk = 128
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/UbNk = 128 no inlet fluct fk = 0.5
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 84 / 1
VELOCITIES: fk = 0.5; NO INLET FLUCT; Nk = 128
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
Nk = 128 no inlet fluct fk = 0.5
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 85 / 1
RESOLVED AND MODELLED (< 0) SHEAR STRESSES
-5 0 5
x 10-3
0.1
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
Nk = 128 no inlet fluct fk = 0.5www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 86 / 1
SHEAR STRESSES: fk = 0.5; NO INLET FLUCT; Nk = 128
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
Nk = 128 no inlet fluct fk = 0.5
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 87 / 1
TURB VISCOSITY: fk = 0.5; NO INLET FLUCT; Nk = 128
0 5 10 15 200.1
0.12
0.14
0.16
0.18
0.2x/c = 0.65
y
0 50 100 150 200 2500
0.05
0.1
0.15
0.2x/c = 0.80
0 50 100 150 200 2500
0.05
0.1
0.15
0.2x/c = 1.00
〈νt〉/ν
y
0 50 100 150 200 2500
0.05
0.1
0.15
0.2x/c = 1.10
〈νt〉/νNk = 128 no inlet fluct fk = 0.5
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 88 / 1
TURB VISCOSITY: fk = 0.5; NO INLET FLUCT; Nk = 128
0 50 100 150 200 2500
0.05
0.1
0.15
0.2x/c = 1.20
〈νt〉/ν
y
0 50 100 150 200 2500
0.05
0.1
0.15
0.2x/c = 1.30
〈νt〉/ν
Nk = 128 no inlet fluct fk = 0.5
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 89 / 1
CONCLUDING REMARKS
LRN PANS has been shown to work well as an embedded LES
method
Channel flow: At two δ downstream the interface, the resolvedturbulence in good agreement with DNS data and the wall friction
velocity 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 but the recover rate sligtly
too slow
Hump flow: large (small) inlet fluctuations gives a smaller (larger)
recirculation
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 90 / 1
PANS: CONCLUDING REMARKS
Embedded LES with k − ε PANS and Synthetic b.c.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 91 / 1
PANS: CONCLUDING REMARKS
Embedded LES with k − ε PANS and Synthetic b.c.
Channel flow
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 91 / 1
PANS: CONCLUDING REMARKS
Embedded LES with k − ε PANS and Synthetic b.c.
Channel flow◮ Isotropic fluctuations work well for channel flow
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 91 / 1
PANS: CONCLUDING REMARKS
Embedded LES with k − ε PANS and Synthetic b.c.
Channel flow◮ Isotropic fluctuations work well for channel flow◮ Strong dependence on interface ku & εu values
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 91 / 1
PANS: CONCLUDING REMARKS
Embedded LES with k − ε PANS and Synthetic b.c.
Channel flow◮ Isotropic fluctuations work well for channel flow◮ Strong dependence on interface ku & εu values◮ No strong dependence on amplitude, L or T of fluctuations
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 91 / 1
PANS: CONCLUDING REMARKS CONT’D
Hump flow
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 92 / 1
PANS: CONCLUDING REMARKS CONT’D
Hump flow◮ PANS & synthetic inlet b.c. with fk everywhere gives good results
except Cf (error > 50%)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 92 / 1
PANS: CONCLUDING REMARKS CONT’D
Hump flow◮ PANS & synthetic inlet b.c. with fk everywhere gives good results
except Cf (error > 50%)◮ With embedded isotropic fluctuations, interface must be located far
upstream
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 92 / 1
PANS: CONCLUDING REMARKS CONT’D
Hump flow◮ PANS & synthetic inlet b.c. with fk everywhere gives good results
except Cf (error > 50%)◮ With embedded isotropic fluctuations, interface must be located far
upstream◮ With embedded anisotropic fluctuations, good results are obtained,
still poor Cf
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 92 / 1
PANS: CONCLUDING REMARKS CONT’D
Hump flow◮ PANS & synthetic inlet b.c. with fk everywhere gives good results
except Cf (error > 50%)◮ With embedded isotropic fluctuations, interface must be located far
upstream◮ With embedded anisotropic fluctuations, good results are obtained,
still poor Cf◮ On-going work . . .
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 92 / 1
Large Eddy Simulation of Heat Transfer in Boundary layer and
Backstep Flow Using PANS [6]
Lars Davidson
THMT-12, Palermo, Sept 2012
PANS LOW REYNOLDS NUMBER MODEL [17]
∂ku∂t
+∂(kuUj)
∂xj=
∂
∂xj
[(
ν +νuσku
)
∂ku∂xj
]
+ (Pu − εu)
∂εu∂t
+∂(εuUj)
∂xj=
∂
∂xj
[(
ν +νuσεu
)
∂εu∂xj
]
+ Cε1Puεuku
− C∗ε2ε2uku
νu = Cµfµk2uεu
,C∗ε2 = Cε1 +fkfε(Cε2f2 − Cε1), σku ≡ σk
f 2kfε, σεu ≡ σε
f 2kfε
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.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 94 / 1
NUMERICAL METHOD
Incompressible finite volume method
Pressure-velocity coupling treated with fractional step
Differencing scheme for momentum eqns:◮ 95% 2nd order central and 5% 2nd order upwind differencing
scheme (baseline) OR◮ 100% 2nd order central differencing
Hybrid 1st order upwind/2nd order central scheme k & ε eqns.
2nd -order Crank-Nicholson for time discretization
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 95 / 1
BOUNDARY LAYER FLOW: DOMAIN
x
y
L
H
δinlet
Inlet: δinlet = 1 (covered by 45 cells), Reθ = 3 600, Uin = ρ = 1.Stretching 1.12 up to y/δ ≃ 1.
Domain: L/δin = 3.2, H/δin = 15.6, Zmax = 1.5δinResolution: ∆z+in ≃ 27, ∆x
+
in ≃ 54
Grid: 66 × 96 × 64 (x , y , z)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 96 / 1
ANISOTROPIC SYNTHETIC FLUCTUATIONS: I [3, 2, 8]
Prescribe an homogeneous Reynolds tensor, uiuj (here from
DNS)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 97 / 1
ANISOTROPIC SYNTHETIC FLUCTUATIONS: I [3, 2, 8]
(u′
1u′
1)λ
x1,λ
(u′2 u
′2 )λ
x2,λu′1,λu
′
2,λ = 0
Prescribe an homogeneous Reynolds tensor, uiuj (here from
DNS)
isotropic fluctuations in principal directions, (u′1u′
1)λ = (u′
2u′
2)λ,u′1,λu
′
2,λ = 0
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 97 / 1
ANISOTROPIC SYNTHETIC FLUCTUATIONS: I [3, 2, 8]
(u′
1u′
1〉λ
x1,λ
(u′2 u
′2 〉λ
x2,λu′1,λu
′
2,λ = 0
Prescribe an homogeneous Reynolds tensor, uiuj (here from
DNS)
isotropic fluctuations in principal directions, (u′1u′
1)λ = (u′
2u′
2)λ,u′1,λu
′
2,λ = 0
re-scale the normal components, (u′1u′
1)λ > (u′
2u′
2)λ,u′1,λu
′
2,λ = 0
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 97 / 1
ANISOTROPIC SYNTHETIC FLUCTUATIONS: II
u′1u′
1 > u′
2u′
2
x1
u′2 u
′2
x2
u′1u′
2 6= 0
Transform from (x1,λ, x2,λ) to (x1, x2)
u′21u′22
= 23,u′21u′23
= 5 from (u′1u′
1)peak in DNS channel flow, Reτ = 500
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 98 / 1
INLET CONDITIONS FOR ku AND εu AS IN [10]
A pre-cursor RANS simulation using the AKN model (i.e. PANS
with fk = 1) is carried out. At Reθ = 3 600, URANS, VRANS, kRANSare taken.
ūin = URANS + u′
synt , v̄in = VRANS + v′
synt , w̄in = w′
synt
Anisotropic synthetic fluctuations are used. The fluctuations are
scaled with ku/ku,max .
ku,in = fkkRANS , εu,in = C3/4µ k
3/2u,in/ℓsgs, ℓsgs = Cs∆, ∆ = V
1/3,
Cs = 0.05
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 99 / 1
INLET TURB. FLUCTUATION, TWO-POINT CORRELATIONS
-2 0 2 4 60
200
400
600
800
1000
stresses
y/H
0 0.1 0.2 0.3 0.4 0.5
0
0.2
0.4
0.6
0.8
1
ẑ/δ, ẑ/HB
ww(ẑ)
Two-point correlation
u+2rms, v+2rms,
w+2rms〈u′v ′〉+ ◦: inlet; x = 3δin
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 100 / 1
BOUNDARY LAYER: VELOCITY AND SKIN FRICTION
1 10 50 10000
5
10
15
20
25
y+
U+
100%CDS
0 0.5 1 1.5 2 2.5 3
2.6
2.8
3
3.2
3.4
3.6
x 10-3
x
Cf
x = δin; x = 2δin;x = 3δin; �: DNS [21]
100% CDS; 100%CDS, Uin from AKN; 25%larger inlet fluct.; 25% largerinlet fluct., Cs = 0.07; markers:0.37 (log10Rex)
−2.584 (+: AKN; ◦:DNS)
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 101 / 1
REYNOLDS STRESSES
-1 0 1 2 30
500
1000
1500
y+
〈u′v ′〉 〈u′u′〉-1 0 1 2 3
0
500
1000
1500
y+
〈u′v ′〉 〈v ′v ′〉, 〈w ′w ′〉, 〈u′u′〉
x = δin; x = 2δin;x = 3δin.
x = 3δin; Markers: DNS [21]
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 102 / 1
BACKWARD FACING STEP: DOMAIN
4.05H 21H
4H
H x
y
qw
ReH = 28 000 Experiments by Vogel & Eaton [26]
Mean inlet profiles from RANS (same as in boundary layer)
Grid: 336 × 120 in x × y plane. Zmax = 1.6H, Nk = 64, ∆z+
in = 31.
Anisotropic synthetic fluctuations, u′, v ′,w ′ (same as for boundarylayer flow); no fluctuations for t ′
Constant heat flux, qw , on lower wall.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 103 / 1
BACKSTEP FLOW. SKIN FRICTION AND STANTON
NUMBER
-5 0 5 10 15 20-2
-1
0
1
2
3
4x 10
-3
x/H
Cf
0 5 10 151
1.5
2
2.5
3
3.5
4x 10
-3
x/HS
t
PANS; PANS, 50% smaller inlet fluctuations;WALE; •: PANS, no inlet fluctuations; : 2D RANS; ◦,•:
experiments [26].
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 104 / 1
BACKSTEP FLOW: VELOCITIES.
0 0.2 0.4 0.6 0.8 11
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
〈ū〉/Uin
x = −1.13H
-0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
〈ū〉/Uin
x = 3.2H
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
〈ū〉/Uin
x = 14.86H
PANS; PANS, 50% smaller inlet fluctuations;WALE;
•: PANS, no inlet fluctuations; : 2D RANS; ◦: experiments [26].
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 105 / 1
BACKSTEP FLOW: RESOLVED STREAMWISE STRESS.
0 0.05 0.1 0.151
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
urms/Uin
y/H
x = −1.13H
0 0.05 0.1 0.15 0.20
0.5
1
1.5
2
2.5
urms/Uin
x = 3.2H
0 0.05 0.1 0.150
0.5
1
1.5
2
2.5
urms/Uin
x = 14.86H
PANS; PANS, 50% smaller inlet fluctuations;WALE;
•: PANS, no inlet fluctuations; : 2D RANS; ◦: experiments [26].
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 106 / 1
BACKSTEP FLOW: TURBULENT VISCOSITIES.
0 2 4 6 81
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
νu/ν
y/H
x = −1.13H
0 5 10 15 200
0.5
1
1.5
2
2.5
νu/ν
x = 3.2H
0 5 10 150
0.5
1
1.5
2
2.5
νu/ν
x = 14.86H
PANS; PANS, 50% smaller inlet fluctuations;WALE;
•: PANS, no inlet fluctuations; : 2D RANS/10;
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 107 / 1
FORWARD/BACKWARD FLOW
Fraction of time, γ, when the flow along the bottom wall is in thedownstream direction.
0 2 4 6 8 10 12 140
0.2
0.4
0.6
0.8
1
x/H
γ
PANS; PANS, 50% smaller inlet fluctuations;WALE;
•: PANS, no inlet fluctuations; ◦: experiments [26].www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 108 / 1
SHEAR STRESSES. x = 3.2H
-5 0 5 10 15
x 10-4
0
0.02
0.04
0.06
0.08
0.1
y/H
PANS
-5 0 5 10 15
x 10-4
0
0.02
0.04
0.06
0.08
0.1RANS
2〈νt s̄12〉; ν∂〈ū〉
∂y; −〈〈u′v ′〉〉; ◦: 2〈νt s̄12〉− 〈〈u
′v ′〉〉.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 109 / 1
SHEAR STRESSES. x = 14.86
0 0.2 0.4 0.6 0.8 1
x 10-3
0
0.02
0.04
0.06
0.08
0.1
y/H
PANS
0 0.2 0.4 0.6 0.8 1
x 10-3
0
0.02
0.04
0.06
0.08
0.1RANS
2〈νt s̄12〉; ν∂〈ū〉
∂y; −〈〈u′v ′〉〉; ◦: 2〈νt s̄12〉− 〈〈u
′v ′〉〉.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 110 / 1
TERMS IN THE 〈ū〉 EQUATION. x = 3.2H
-0.06 -0.04 -0.02 0 0.02 0.04 0.060
0.02
0.04
0.06
0.08
0.1
y/H
PANS
-0.06 -0.04 -0.02 0 0.02 0.04 0.060
0.02
0.04
0.06
0.08
0.1RANS
∂
∂y(2〈νt s̄12〉); ν
∂2〈ū〉
∂y2; −
∂〈ū〉〈ū〉
∂x; +: −
∂〈ū〉〈v̄〉
∂y;
⋆: −∂〈p̄〉
∂x, △: −
∂〈〈u′v ′〉〉
∂y.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 111 / 1
TERMS IN THE 〈ū〉 EQUATION. x = 14.86H
-0.06 -0.04 -0.02 0 0.02 0.04 0.060
0.02
0.04
0.06
0.08
0.1
y/H
PANS
-0.06 -0.04 -0.02 0 0.02 0.04 0.060
0.02
0.04
0.06
0.08
0.1RANS
∂
∂y(2〈νt s̄12〉); ν
∂2〈ū〉
∂y2; −
∂〈ū〉〈ū〉
∂x; +: −
∂〈ū〉〈v̄〉
∂y;
⋆: −∂〈p̄〉
∂x, △: −
∂〈〈u′v ′〉〉
∂y.
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HEAT FLUXES. x = 3.2H
-1 -0.8 -0.6 -0.4 -0.2 00
0.02
0.04
0.06
0.08
0.1
y/H
PANS
-1 -0.8 -0.6 -0.4 -0.2 00
0.02
0.04
0.06
0.08
0.1RANS
〈
νtσt
∂T̄
∂y
〉
;ν
σℓ
∂〈T̄ 〉
∂y; −〈vθ〉. ◦:
〈
νtσt
∂T̄
∂y
〉
− 〈vθ〉.
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HEAT FLUXES. x = 14.86H
-1 -0.8 -0.6 -0.4 -0.2 00
0.02
0.04
0.06
0.08
0.1
y/H
PANS
-1 -0.8 -0.6 -0.4 -0.2 00
0.02
0.04
0.06
0.08
0.1RANS
〈
νtσt
∂T̄
∂y
〉
;ν
σℓ
∂〈T̄ 〉
∂y; −〈vθ〉. ◦:
〈
νtσt
∂T̄
∂y
〉
− 〈vθ〉.
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TERMS IN THE 〈T̄ 〉 EQUATION. x = 3.2H
-100 -50 0 50 1000
0.02
0.04
0.06
0.08
0.1
y
PANS
-100 -50 0 50 1000
0.02
0.04
0.06
0.08
0.1RANS
∂
∂y
(
νtσt
∂〈T̄ 〉
∂y
)
;ν
σℓ
∂2〈T̄ 〉
∂y2; −
∂〈ū〉〈T̄ 〉
∂x; +:
−∂〈v̄〉〈T̄ 〉
∂y; △: −
∂〈vθ〉
∂y.
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TERMS IN THE 〈T̄ 〉 EQUATION. x = 14.86H
-50 0 500
0.02
0.04
0.06
0.08
0.1
y/H
PANS
-50 0 500
0.02
0.04
0.06
0.08
0.1RANS
∂
∂y
(
νtσt
∂〈T̄ 〉
∂y
)
;ν
σℓ
∂2〈T̄ 〉
∂y2; −
∂〈ū〉〈T̄ 〉
∂x; +:
−∂〈v̄〉〈T̄ 〉
∂y; △: −
∂〈vθ〉
∂y.
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CONCLUDING REMARKS
Developing boundary layer◮ Synthetic fluctuations give fully developed conditions after a couple
of boundary layer thicknesses◮ 5% upwinding dampens resolved fluctuations; can be compensated
by 25% larger inlet fluctuations
Backstep flow◮ Very good agreement with experiments◮ 2D RANS predicts turbulent diffusion surprisingly well◮ Synthetic inlet fluctuations give an improved Stanton number;
otherwise small effect in the reciculation region◮ LRN PANS and WALE equally good◮ 5% upwinding has a negligble effect in the recirculation region
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REFERENCES I
[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] BILLSON, M.
Computational Techniques for Turbulence Generated Noise.
PhD thesis, Dept. of Thermo and Fluid Dynamics, Chalmers
University of Technology, Göteborg, Sweden, 2004.
[3] BILLSON, M., ERIKSSON, L.-E., AND DAVIDSON, L.
Modeling of synthetic anisotropic turbulence and its sound
emission.
The 10th AIAA/CEAS Aeroacoustics Conference, AIAA
2004-2857, Manchester, United Kindom, 2004.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 118 / 1
REFERENCES II
[4] DAVIDSON, L.
Large eddy simulations: how to evaluate resolution.
International Journal of Heat and Fluid Flow 30, 5 (2009),
1016–1025.
[5] DAVIDSON, L.
How to estimate the resolution of an LES of recirculating flow.
In ERCOFTAC (2010), M. V. Salvetti, B. Geurts, J. Meyers, and
P. Sagaut, Eds., vol. 16 of Quality and Reliability of Large-Eddy
Simulations II, Springer, pp. 269–286.
[6] DAVIDSON, L.
Large eddy simulation of heat transfer in boundary layer and
backstep flow using pans (to be presented).
In Turbulence, Heat and Mass Transfer, THMT-12 (Palermo,
Sicily/Italy, 2012).
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 119 / 1
REFERENCES III
[7] 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).
[8] DAVIDSON, L., AND BILLSON, M.
Hybrid LES/RANS using synthesized turbulence for forcing at the
interface.
International Journal of Heat and Fluid Flow 27, 6 (2006),
1028–1042.
[9] DAVIDSON, L., AND DAHLSTRÖM, S.
Hybrid LES-RANS: An approach to make LES applicable at high
Reynolds number.
International Journal of Computational Fluid Dynamics 19, 6
(2005), 415–427.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 120 / 1
REFERENCES IV
[10] DAVIDSON, L., AND PENG, S.-H.
Emdedded LES with PANS.
In 6th AIAA Theoretical Fluid Mechanics Conference, AIAA paper
2011-3108 (27-30 June, Honolulu, Hawaii, 2011).
[11] DAVIDSON, L., AND PENG, S.-H.
Embedded LES using PANS applied to channel flow and hump
flow (to appear).
AIAA Journal (2013).
[12] HEMIDA, H., AND KRAJNOVIĆ, S.
LES study of the impact of the wake structures on the
aerodynamics of a simplified ICE2 train subjected to a side wind.
In Fourth International Conference on Computational Fluid
Dynamics (ICCFD4) (10-14 July, Ghent, Belgium, 2006).
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 121 / 1
REFERENCES V
[13] HINZE, J.
Turbulence, 2nd ed.
McGraw-Hill, New York, 1975.
[14] KRAJNOVIĆ, S., AND DAVIDSON, L.
Large eddy simulation of the flow around a bluff body.
AIAA Journal 40, 5 (2002), 927–936.
[15] KRAJNOVIĆ, S., AND DAVIDSON, L.
Numerical study of the flow around the bus-shaped body.
Journal of Fluids Engineering 125 (2003), 500–509.
[16] KRAJNOVIĆ, S., AND DAVIDSON, L.
Flow around a simplified car. part II: Understanding the flow.
Journal of Fluids Engineering 127, 5 (2005), 919–928.
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REFERENCES VI
[17] 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.
[18] MENTER, F.
Two-equation eddy-viscosity turbulence models for engineering
applications.
AIAA Journal 32 (1994), 1598–1605.
[19] MENTER, F., KUNTZ, M., AND LANGTRY, R.
Ten years of industrial experience of the SST turbulence model.
In Turbulence Heat and Mass Transfer 4 (New York, Wallingford
(UK), 2003), K. Hanjalić, Y. Nagano, and M. Tummers, Eds.,
begell house, inc., pp. 624–632.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 123 / 1
REFERENCES VII
[20] POPE, S.
Turbulent Flow.
Cambridge University Press, Cambridge, UK, 2001.
[21] SCHLATTER, P., AND ORLU, R.
Assessment of direct numerical simulation data of turbulent
boundary layers.
Journal of Fluid Mechanics 659 (2010), 116–126.
[22] SCHLICHTING, H.
Boundary-Layer Theory, 7 ed.
McGraw-Hill, New York, 1979.
[23] SMAGORINSKY, J.
General circulation experiments with the primitive equations.
Monthly Weather Review 91 (1963), 99–165.
www.tfd.chalmers.se/˜lada Helsinki 4 October 2012 124 / 1
REFERENCES VIII
[24] SPALART, P., JOU, W.-H., STRELETS, M., AND ALLMARAS, S.
Comments on the feasability of LES for wings and on a hybrid
RANS/LES approach.
In Advances in LES/DNS, First Int. conf. on DNS/LES (Louisiana
Tech University, 1997), C. Liu and Z. Liu, Eds., Greyden Press.
[25] STRELETS, M.
Detached eddy simulation of massively separated flows.
AIAA paper 2001–0879, Reno, NV, 2001.
[26] VOGEL, J., AND EATON, J.
Combined heat transfer and fluid dynamic measurements
downstream a backward-facing step.
Journal of Heat Transfer 107 (1985), 922–929.
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