10 000 processors or more

UNSTEADY FLOW MODELLING AND COMPUTATION

Franck Nicoud

University Montpellier II – I3M CNRS UMR 5149 and

CERFACS

MOTIVATION - 1

• CFD of steady flow is now mature

• Many industrial codes available, robust and accurate

• Complex flow physics included: moving geometries, turbulence, combustion, mixing, fluid-X coupling

• But only the averages are computed

MOTIVATION - 2

• Averages are not always enough

(instabilities, growth rate, vortex shedding)

• Averages are even not always meaningful

T1 T2>T1

T

time

Oscillating flame ^T2

T1

Prob(T=T_mean)=0 !!

MOTIVATION - 3

• Some phenomena are unsteady in nature.

Ex: turbulence in a channel with ablation

O. Cabrit – CERFACS & UM2

MOTIVATION - 4

• Some phenomena are unsteady in nature.

Ex: ignition of an helicopter engine

Y. Sommerer & M. Boileau CERFACS

MOTIVATION - 5

• Some phenomena are unsteady in nature.

Ex: thermoacoustic instability - Experiment

D. Durox, T. Schuller, S. Candel – EM2C

MOTIVATION - 6

• Some phenomena are unsteady in nature.

Ex: thermoacoustic instability - CFD

P. Schmitt – CERFACS

PARALLEL COMPUTING

• www.top500.org – june 2007

# Site Computer

1 DOE/NNSA/LLNL United States

BlueGene/L - eServer Blue Gene Solution IBM

2 Oak Ridge National Laboratory United States

Jaguar - Cray XT4/XT3 Cray Inc.

3 NNSA/Sandia National Laboratories United States

Red Storm - Sandia/ Cray Red Storm, Opteron 2.4 GHz dual core

Cray Inc.

4 IBM Thomas J. Watson Research Center United States

BGW - eServer Blue Gene Solution IBM

5

Stony Brook/BNL, New York Center for Computional Sciences

United States

New York Blue - eServer Blue Gene Solution IBM

6 DOE/NNSA/LLNL United States

ASC Purple - eServer pSeries p5 575 1.9 GHz

IBM

7

Rensselaer Polytechnic Institute, Computional Center for Nanotechnology Innovations

United States

eServer Blue Gene Solution IBM

8 NCSA

United States

Abe - PowerEdge 1955, 2.33 GHz, Infiniband Dell

9 Barcelona Supercomputing Center Spain

MareNostrum - BladeCenter JS21 Cluster, PPC 970, 2.3 GHz, Myrinet

IBM

10 000 processors or more

PARALLEL COMPUTING

• Large scale unsteady computations require huge computing resources, an efficient codes …

processors

Speed up

SOME KEY INGREDIENTS

• Flow physics:

– turbulence, combustion modeling, heat loss, radiative transfer, wall treatment, chemistry, two-phase flow, acoustics/flame

coupling, mode interaction, …

• Numerics:

– non-dissipative, low dispersion scheme, robustness, linear stability, non-linear stability, conservativity, high order,

unstructured environment, parallel computing, error analysis, …

• Boundary conditions:

– characteristic decomposition, turbulence injection, non-reflecting, pulsating conditions, complex impedance, …

BASIC EQUATIONS

reacting, multi-species gaseous mixture

SIMPLER BASIC EQUATIONS

• Navier-Stokes equations for a compressible fluid

• Finite speed of propagation of pressure waves

= 0

∂ + ∂

∂

∂

i i

x u t

ρ ρ

( )

j ij i

j i j

i

x x

u P x u

t u

∂

∂τ

∂ ρ ∂

∂

ρ ₊ ∂ _{= − +}

∂

∂( )

k k ij

i j j

i

ij x

u x

u x

u

∂ µδ ∂

∂

∂

∂ µ ∂

τ 3

− 2











 +

=

P rT

ρ

( ) ( )

j j j

ij i j

ij i

i

j x

q x

u x

P E u

x u t

E

∂

∂

∂ τ

∂

∂ δ ρ ∂

∂∂

ρ _{+ = − + −}

∂

∂( )

i

i x

q T

∂ λ ∂

−

=

SIMPLER BASIC EQUATIONS

• Navier-Stokes equations for an incompressible fluid

• Infinite speed of propagation of pressure waves

• Contain turbulence

= 0

∂

∂

i i

x u

( )

j ij i

j i j

i

x x

u P x u

t u

∂

∂τ

∂ ρ ∂

∂

∂

ρ _{+ = − +}

∂

∂( )











 +

=

i j j

i

ij x

u x

u

∂

∂

∂ µ ∂ τ

ρ

ρ

Turbulence

Turbulence

TURBULENCE IS

EVERYWHERE

TURBULENCE

Leonardo da Vinci

TURBULENCE

The flow regime (laminar vs turbulent) depends on the Reynolds number :

Re = Ud

ν

Velocity Length scale

viscosity

Re

laminar turbulent

TURBULENCE

•Turbulence is increasing mixing

–Most often favorable (flames can stay in combustors)

–Some drawbacks (drag, relaminarization techniques…)

TURBULENCE AND CHAOS

ONERA WIND TUNNEL

CHAOS OF THE POOR

] 1 , 1 [ ,

2

1

1

= − ∈ −

+

v v

v

66667 .

0 v

6667 .

0

0

= ersus v =

v

TURBULENCE

• The flow is very sensitive to the initial/boundary conditions. This leads to the impression of chaos,

because the tiny details of the operating conditions are never known

• This sensitivity is related to the non-linear (convective) terms

• Need for an academic turbulent case

HOMOGENEOUS ISOTROPIC TURBULENCE

• No boundary

• L-periodic 3D domain

• being any physical quantity

L L

L

( ) ^{, ˆ , 2 Z}

e L

t φ

π

φ ^x = ∑

^k =

k

x k k

Ω ∫∫∫ ( )

Ω

⋅

= 1

,

ˆ t e dx dx dx

L

i k

x

x

φ φ

( ) ^{x , t}

φ

BUDGETS IN HIT

• Spatial averaging

• Momentum

• Turbulent Kinetic Energy (TKE)

= 0 dt

u d _i

4 4 4 3 4

4 4 2 1

rate n dissipatio

TKE : 2 2

2 2

/

ε

ν 











∂ + ∂

∂

− ∂

=

i j j

i i

x u x

u dt

u d

∫∫∫ ( )

Ω

= 1

)

( dx dx dx

t L φ x

φ

FLUX OF TKE

• For any K>0 and any

• KE of scales larger than 2

π

• Non linear terms are responsible for the energy transfer from the largest to the smallest scales

( )

:

, 2

, ˆ Z

e L t

K

i

π

φ

φ = ∑ =

<

⋅

<

x k

k k

x k k

j i j i i

x u u

dt u u d

∂

− ∂

−

= ^{< < >}

<

2 ε

2 /

( ) ^{x , t}

φ

TURBULENCE: A SCENARIO

1. The largest scales of the flow are fed in energy (l₀)

• Must be done at rate

2. The energy is transferred to smaller and smaller scales (l)

• Must be done at rate

3. When scales become small enough (

η

• Must occur when:

0 3 0

l

≡ u ε

l u³ ε ≡

≡1 νη_K uK

SCALES IN TURBULENCE

• Kolmogorov scales (the smallest ones)

• Large-to-small scales ratio

• Remark: in CFD, the number of grid points in each direction must follow the same scaling

4 / 1

4 / 3

ε

η

≡ ν u

≡ ν

ε

4 / 3 0

0

R

l

K

η ≡

TURBULENCE SPECTRUM

( )

R_ij r = _i x _j x+ r

• Two-point correlation tensor:

• Velocity spectrum tensor

• Energy spectrum: E(k)dk is the kinetic energy contained in the scales whose wave number are between k and k+dk

( )

∫∫∫

3

3 2

3 ( ) 1

) 2 (

1

R

i ij

ij k R r e ^krdr dr dr

φ π

( )

∫∫∫

3

3 2

) 1

(

R

i ij

ij e dk dk dk

R r φ k ^{k r}

( )

k k

S

k dS k

E _ii

radius of

sphere :

) (

) ( ) 2 (

1

∫

= φ k

i iu u dk

k

E 2

) 1 ( :

that shows

one

0

∫

∞

KOLMOGOROV ASSUMPTION

• After Kolmogorov (1941), there is an inertial zone

(l₀<<l<<

η

ε

It follows that:

E ( k ) = C

ε

k

( ^{( )} )

ln E k k

k

ln

TURBULENCE SPECTRUM

CHANNEL

ONERA WIND TUNNEL

JET

3 / 5 3

/

)

( k ∝ k

E ε

Well supported by the experiments

TURBULENCE IN CFD

• Turbulence is contained in the Navier-Stokes equations

• So it is deterministic

• BUT: because the flow is so sensitive to the (unknown) details of IC and BC, only averages can be predicted, or used for comparison purposes ex: numerical/experimental comparisons

• “simply” resolve the Navier-Stokes equations to obtain turbulence: Direct Numerical Simulation

TURBULENCE IN CFD

TURBULENCE IN CFD

TURBULENCE IN CFD

TURBULENCE IN CFD

DNS OF ISOTROPIC

TURBULENCE

DIRECT NUMERICAL

SIMULATION OF TURBULENCE

• Solve the Navier-Stokes equations and represent all scales in space and time

• Compute the average, variance, of the unsteady solution

• The main limitation comes from the computer resources required:

– the number of points required scales like – The CPU time scales like

• In many practical applications, the Reynolds number is large, of order 10⁶ or more

( )

3

R0

THE RANS APPROACH

• Ensemble average the (incompressible) Navier-Stokes equations

• Reynolds decomposition

• Reynolds equations:

• A model for the Reynolds stress is required

= 0

∂

∂

i i

x u

j ij i

j j i i

x x

P x

u u t

u

∂ τ

∂

∂

∂

∂ ρ ∂

ρ + = − +

∂

∂

i'

i

i u u

u = +

= 0

∂

∂

i i

x

u

( )

j

j i ij

i j

j i i

x

u u x

P x

u u t

u

∂ ρ τ

∂

∂

∂

∂ ρ ∂

ρ + = − + ^{− ' '}

∂

∂

' 'u ρu

−

DNS vs RANS in a combustor

Modern combustion chamber - TURBOMECA Multiperforated plate

Solid plate Dilution hole

DNS vs RANS in a combustor

• RANS: used routinely during the design process.

• Approx. 10⁶ nodes for the whole geometry

•DNS: used to know more about the flow physics in the multi-perforated plate region

•Approx. 10⁶ nodes for only one perforation

THE RANS APPROACH

• Intense scientific activity in the ’70, ’80 and ’90 to derive the ultimate turbulence model.

Ex: k-ε, RSM, k-ε-v², …

• No general model

• No flow dynamics

• Thanks to increasing available computer resources, unsteady calculations become affordable

Jet of hot gas

DNS RANS

RANS – LES - DNS

• Reynolds-Averaged Navier-Stokes:

– Relies on a model to account for the turbulence effects – efficient but not predictive

• Direct Numerical Simulation:

– The only model is Navier-Stokes

– predictive but not tractable for practical applications

• Large Eddy Simulation:

– Relies on a model for the smallest scales, more universal – Resolves the largest scales

– Predictive and tractable

Large-Eddy Simulation

RANS – LES - DNS

time

Streamwise velocity

RANS

DNS LES

Large Eddy Simulation

Wave number

Energy spectrum

« Navier-Stokes » Model

• Formally: replace the average operator by a spatial filtering to obtain the LES equations

( )

( ) ( )

φ ^t =

∫∫∫

R

ξ ξ

x ξ

x ³

3

, ,

LES equations

• Assumes small commutation errors

• Filter the incompressible equations to form :

• Sub-grid scale stress tensor to be modeled

= 0

∂

∂

i i

x

u

( )

j sgs ij ij

i j

j i i

x x

P x

u u t

u

∂ τ τ

∂

∂

∂

∂ ρ ∂

ρ + = − + ⁺

∂

∂

j i j

i sgs

ij

ρ u u ρ u u

τ = −

NS

The Smagorinsky model

• From dimensional consideration, simply assume:

• The Smagorinsky constant is fixed so that the proper dissipation rate is produced, C_s =0.18

( )

sgs = C ∆ ² 2S S

ν













∂ + ∂

∂

= ∂

=

−

i j j

i ij

ij sgs ij

sgs kk sgs

ij x

u x

S u

S 2

with 1 ,

3 2

1τ δ ρν

τ

The Smagorinsky model

• The sgs dissipation is always positive

• Very simple to implement, no extra CPU time

• Any mean gradient induces sub-grid scale activity and dissipation, even in 2D !!

• Strong limitation due to its lack of universality.

Eg.: in a channel flow, Cs=0.1 should be used

ij ij sgs ij

sgs ij

m τ S ρν S S

ε = ≈ ²

Solid wall

0

and ) (

because

0 but

0

12 ≠

=

≠

=

=

S y

U U

W

V ν ^sgs

No laminar-to-turbulent transition possible

The Germano identity

• By performing , the following sgs contribution appears

• Let’s apply another filter to these equations

• By performing , one obtains the following equations

• From A and B one obtains

j i j

i sgs

ij u u u u

T = ρ − ρ

j i j

i sgs

ij ρu u ρuu

τ ^{= −}

NS

NS

j i j

i sgs

ij ρu u ρu u

τ = −

( )

j sgs ij ij i

j j i i

x x

P x

u u t

u

∂ τ τ

∂

∂

∂

∂ ρ ∂

ρ + = − + ⁺

∂

∂

j sgs ij ij

i j

j i i

x x

P x

u u t

u

∂ τ τ

∂

∂

∂

∂ ρ ∂

ρ ^

 



 +

+

−

=

∂ +

∂

j sgs ij ij

i j

j i i

x T x

P x

u u t

u

∂ τ

∂

∂

∂

∂ ρ ∂

ρ ^

 



 +

+

−

=

∂ +

∂

A

B

j i j

i sgs

ij sgs

ij u u u u

T =τ −ρ + ρ

The dynamic Smagorinsky model

• Assume the Smagorinsky model is applied twice

• Assume the same constant can be used and write the Germano identity

( )

ij sgs kk sgs

ij 2 C 2S S S

3

1 ²

∆

=

− τ δ ρ

τ^{ijsgs Tkksgs ij}

( )

T 2 2

3

1 ²

∆

=

− δ ρ

j i j

i sgs

ij sgs

ij u u u u

T =τ −ρ + ρ

( )

^{( )}

ρ ∆ + = ∆ + − +

3 2 1

3 2 2 1

2 ²

2

ij sgs kk sgs

kk ij

ij

s M N T

C = + ^_^τ − ^_^δ 3

2 1 C_s dynamically obtained

from the solution itself

The dynamic Smagorinsky model

• The model constant is obtained in the least mean square sense

• No guaranty that . Good news for the backscattering of energy; Bad news from a numerical point of view.

• In practice on takes

• Must be stabilized by some ad hoc procedure.

E.g.: plan, Lagragian or local averaging

• Proper wall behavior

ij ij

ij ij

sgs kk sgs

kk ij

s M M

M T

N C



 



 



 



 −

+

= τ δ

3 1

2

2 > 0 Cs

∆

≈

∆ 2

• The dynamic procedure is one of the reason for the development of LES in the last 15 years

• It allows to overcome the lack of universality of the Smagorinsky model

• The dynamic procedure can be (has been) applied to other models E.g.: dynamic determination of the sgs Prandtl number when

computing the heat fluxes

• BUT:

– it requires some ad hoc procedure to stabilize the computation – Defining a filter is not an easy task in complex geometries

The dynamic Smagorinsky model

∆

≈

∆ 2

• From the eddy-viscosity assumption, modeling means finding a proper expression for

• The Smagorinsky model reads

• More generally, from dimensional argument

An improved static model

sgs

τij

νsgs

( )

sgs = C ∆ ² 2S S

ν

( ) [ ( ) ]

2 , , with , = ⁻

∆

= C_m OP t OP t T

sgs x x

ν

Dynamic procedure Should depend on the filtered velocity.

No necessarily equal to 2S_ij S_ij

• A good candidate appears to be based on the traceless symmetric part of the square of the velocity gradient tensor

• Considering its second invariant

1. Involves both the strain and rotation rates 2. Exactly zero for any 2D field

3. Near solid walls, goes asymptotically to zero

Choice of the frequency scale OP

(

)

d

ij g² g² g² δ

3 1 2

1 + −

= Σ

(

)

= Σ

Σ^d_{ij ij}^d S S S 2IV_S 2

3 6

1 _{2 2 2 2 2 2}

li jl kj ik S

ij ij ij

ijS IV S S

S

S² = Ω² = Ω Ω _Ω = Ω Ω

• The WALE model makes use of the previous invariant to define the sgs viscosity:

1. Proper asymptotic behavior 2. No extra filtering required 3. Simple implementation

The Wall Adapting Local Eddy viscosity model

( ) ( )

( )

( )

2 / 3

2 =

Σ Σ + Σ

∆ Σ

= _m

d ij d ij ij

ij

d ij d ij m

sgs C

S S ν C

43

( )

42 1

0 3

→

=

y

sgs O y

ν

Periodic cylindrical tube at bulk Reynolds number 10000

Simple geometry with complex mesh

An academic case

• Close to solid walls, the largest scales are small …

About solid walls

Boundary layer along the x-direction: Vorticity ω_x

- steep velocity profile, L_t ~ κy

- Resolution requirement: ∆y⁺ = O(1), ∆z⁺ and ∆x⁺ = O(10) !!

- Number of grid points: O(R_τ²) for wall resolved LES

z

y

x

u

• In the near wall region, the total shear stress is

constant. Thus the proper velocity and length scales are based on the wall shear stress τ_w:

• In the case of attached boundary layers, there is an inertial zone where the following universal velocity law is followed

About solid walls

τ ντ

ρ τ

l u

u = ^w =

2 5

, constant"

Universal

"

:

0.4

constant, Karman

Von :

, ,

1 ln

. C

C

y yu u

u u C

y u

≈

≈

=

= +

= ^{+ + +}

+

κ κ

ν

κ _τ ^τ

y

u

• A specific wall treatment is required to avoid huge mesh refinement or large errors,

• Use a coarse grid and the log law to impose the proper fluxes at the wall

Wall modeling

u

v ττττ₁₂^model

ττττ₃₂^model

y

u

Exact velocity gradient at wall

Velocity gradient at wall assessed from a coarse grid

• Coarse grid LES is not LES !!

• Numerical errors are necessary large, even for the mean quantities

• No reliable model available yet

Wall modeling in LES

Log. zone => L = κ.y

L < ∆y !!

for j=1, 2

∆y⁺ = O(100)

+ +

+

+ +

+ +

∆

= + +

≠ ∆

≈ ∆

∆

∆

−

≈ ∆

+

+ y y y

y u

y u

y

y κ κ

1 3

ln )

2 / (

) 2 / 3

( dy

du

2

1 /

1 y y

u → = ∆

2 /

2 3

2 y y

u → = ∆

Numerical schemes for

unsteady flows

Classical numerical methods

• Finite elements

• Finite volumes

• Finite differences

• Finite differences are only adapted to Cartesian meshes

• The most intuitive approach

• In 1D, the three methods are equivalent

• Thus the FD are well suited to understand basic phenomena shared by the three methods

A few words about Finite

Differences

• Instead of seeking for f(x), only the values of f at the nodes x_i are considered. Thus the unknowns are f_i=f(x_i)

• The basic idea is to replace each partial derivative in the PDE by expressions obtained from Taylor expansions

written at node i

i i +1 i + 2

−1 i

− 2 i

−3

i x

xi x

+1

xi x_i+2

−1

xi

−2

xi

−3

xi

A few words about Finite

Differences

First spatial derivative

• Taylor expansion at node i:

• Only derivatives at node i are involved

i i +1 i + 2

−1 i

− 2 i

−3

i x

(

) (

)

( (

)

)

1 2 _x ^{i i}

i i

x i

i i

i O x x

dx f d x

x dx

x df x

f f

i i

−

− + +

− +

= ₋ ⁻ ₋

−

(

) (

)

( (

)

)

1 2 ^{i i}

x i

i x

i i

i

i O x x

dx f x d

x dx

x df x

f f

i i

−

− + +

− +

= ₊ ⁺ ₊

+

• Defining and

i i +1 i + 2

−1 i

−2 i

−3

i x

( )

2 2 2

1 1

1 2 ⁻

− −

− = −∆ + ∆ + ∆_i

x i

x i

i

i O

dx f d dx

f df f

i i

( )

2 2 2

1 2 ⁱ

x i

x i

i

i O

dx f d dx

f df f

i i

∆

∆ + +

∆ +

+ =

×

∆_i−1²

×

∆_i²

(

) (

) (

)

1 2 1

2

1 i i

x i

i i

i i

i i i

i i

i O

dx f df

f f

i

∆

∆ +

∆

∆ +

∆

∆ +

∆

−

∆

=

∆

−

∆ _{− + − − − − −}

( )

(

)

( )

1 2

1 − ∆ −∆ −∆ + ∆

= ∆ ⁻ f ⁺ f ⁻ f ⁻ O

df _{i i i i i i i}

1

1 −

−

= −

∆

x

x

∆

= x

− x

First spatial derivative

• If the mesh is uniform then and one recovers the classical FD formula :

• The truncation error is

• Second order centered scheme

x f f f

dx D

df

i

x_i

∆

= −

≈

2

1 0 1

1

) ( x

O ∆

Uniform mesh

i

x

i

= ∆ = ∆

∆