Large Eddy Simulation (LES) - Chalmerslada/slides/slides_les_helsinki.pdf · LES VS. RANS LES can...

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


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


BLUFF-BODY FLOW: SURFACE-MOUNTED CUBE[14]Krajnović & Davidson (AIAA J., 2002)

Snapshots of large turbulent scales illustrated by Q = −∂ūi∂xj

∂ūj∂xi


BLUFF-BODY FLOW: FLOW AROUND A BUS[15]

Krajnović & Davidson (JFE, 2003)


BLUFF-BODY FLOW: FLOW AROUND A CAR[16]Krajnović & Davidson (JFE, 2005)


BLUFF-BODY FLOW: FLOW AROUND A TRAIN[12]

Hem

ida

&K

rajn

ović

,2006


SEPARATING FLOWS

Wall

TIME-AVERAGED flow and INSTANTANEOUS flow


SEPARATING FLOWS

Wall


In average there is backflow (negative velocities). Instantaneous,

the negative velocities are often positive.


SEPARATING FLOWS

Wall




How easy is it to model fluctuations that are as large as the mean

flow?


SEPARATING FLOWS

Wall





flow?

Is it reasonable to require a turbulence model to fix this?


SEPARATING FLOWS

Wall





flow?

Is it reasonable to require a turbulence model to fix this?

Isn’t it better to RESOLVE the large fluctuations?


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


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


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.


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)


LARGE EDDY SIMULATIONS

GS

SGS

SGS

Large scales (GS) are resolved; small scales (SGS) are modelled.


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


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.


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


ENERGY PATHE(κ)

E(κ) ∝ κ−5/3

κ

inertial rangeεsgs ≃ 〈

νsgs ∂u ′

i∂xj

∂u ′i∂x

j

〉

ε

dissipating range


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.


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


TIME AVERAGING IN LES

t1: Start time averaging

t2: Stop time averaging

tt1: start t2: end

v̄1


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








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


NEAR-WALL RESOLUTION CONT’D

In RANS when using

wall-functions,

30 < y+ < 100 for thewall-adjacent cells

x

y wall

y+



In RANS when using

wall-functions,


In LES, ∆z+ ≃ 30

x

y wall

y+

x

z

∆z+



In RANS when using

wall-functions,


In LES, ∆z+ ≃ 30EVERYWHERE

x

y wall

y+

x

z

∆z+



In RANS when using

wall-functions,


In LES, ∆z+ ≃ 30EVERYWHERE

AND ∆x+ ≃ 100,∆y+min ≃ 1

x

y wall

y+

x

z

∆z+


NEAR-WALL TREATMENT

from Hinze (1975)


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.


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


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



∂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


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]

}



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


(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.


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


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


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.


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.


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


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

!


SGS MODELS BASED ON GRID SIZE

When the grid is refined, νt gets smaller



When the grid is refined, νt gets smallerE

εsgs

κ

Pk ,res

κc



When the grid is refined, νt gets smallerE

εsgs,∆εsgs,0.5∆

κ

Pk ,res

κc 2κc



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


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


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


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


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


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.


LOCAL EQUILIBRIUM IN ε EQUATION.

How can both the k eq. and ε be in local equilibrium??




If

〈Pk 〉 = 〈ε〉




If

〈Pk 〉 = 〈ε〉

then

C1〈ε〉

〈k〉〈Pk 〉6=C

∗

2

〈ε〉2

〈k〉,because C1 6= C

∗

2




If

〈Pk 〉 = 〈ε〉

then

C1〈ε〉

〈k〉〈Pk 〉6=C

∗

2

〈ε〉2


∗

2

However, the previous slide shows

C1

〈 ε

kPk

〉

= C∗2

〈

ε2

k

〉




If

〈Pk 〉 = 〈ε〉

then

C1〈ε〉

〈k〉〈Pk 〉6=C

∗

2

〈ε〉2


∗

2

However, the previous slide shows

C1

〈 ε

kPk

〉

= C∗2

〈

ε2

k

〉

Answer: when time-averaging 〈ab〉 6= 〈a〉〈b〉



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〉


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.


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


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


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!


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)


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 .


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.


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


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


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


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


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


(∆x ,∆z) 0.5∆x 0.5∆z ◦ 2∆x ; +: 2∆z

The (∆x ,∆z) mesh is (δ/∆x , δ/∆z) = (10,20)



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


(∆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.



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



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


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


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


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


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


∂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


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]

}



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]


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)


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

◮


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


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


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 ′〉+


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


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


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


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


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)


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.


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)


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



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)


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)


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



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



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)


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



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


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



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


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



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



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


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



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


PANS: CONCLUDING REMARKS

Embedded LES with k − ε PANS and Synthetic b.c.




Channel flow




Channel flow◮ Isotropic fluctuations work well for channel flow




Channel flow◮ Isotropic fluctuations work well for channel flow◮ Strong dependence on interface ku & εu values




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


PANS: CONCLUDING REMARKS CONT’D

Hump flow



Hump flow◮ PANS & synthetic inlet b.c. with fk everywhere gives good results

except Cf (error > 50%)




except Cf (error > 50%)◮ With embedded isotropic fluctuations, interface must be located far

upstream





upstream◮ With embedded anisotropic fluctuations, good results are obtained,

still poor Cf





upstream◮ With embedded anisotropic fluctuations, good results are obtained,

still poor Cf◮ On-going work . . .


Large Eddy Simulation of Heat Transfer in Boundary layer and

Backstep Flow Using PANS [6]

Lars Davidson

THMT-12, Palermo, Sept 2012


∂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


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.


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


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)


ANISOTROPIC SYNTHETIC FLUCTUATIONS: I [3, 2, 8]

Prescribe an homogeneous Reynolds tensor, uiuj (here from

DNS)



(u′

1u′

1)λ

x1,λ

(u′2 u

′2 )λ

x2,λu′1,λu

′

2,λ = 0


DNS)

isotropic fluctuations in principal directions, (u′1u′

1)λ = (u′

2u′

2)λ,u′1,λu

′

2,λ = 0



(u′

1u′

1〉λ

x1,λ

(u′2 u

′2 〉λ

x2,λu′1,λu

′

2,λ = 0


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


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


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


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


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)


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]


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.


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].


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].


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, no inlet fluctuations; : 2D RANS; ◦: experiments [26].


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, no inlet fluctuations; : 2D RANS/10;


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, 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 ′〉〉.


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 ′〉〉.


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.


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.


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〉.


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〉.


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.


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.


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


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.


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).


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

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[8] DAVIDSON, L., AND BILLSON, M.

Hybrid LES/RANS using synthesized turbulence for forcing at the

interface.

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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.


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).


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.


Numerical study of the flow around the bus-shaped body.

Journal of Fluids Engineering 125 (2003), 500–509.


Flow around a simplified car. part II: Understanding the flow.

Journal of Fluids Engineering 127, 5 (2005), 919–928.


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.

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[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.


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.

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[22] SCHLICHTING, H.

Boundary-Layer Theory, 7 ed.

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[23] SMAGORINSKY, J.

General circulation experiments with the primitive equations.

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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.

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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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