Refined turbulence modelling for reactor thermal-hydraulicsPart of UK’s “Keeping Nuclear Options Open” project

School of Mechanical, Aerospace & Civil Engineering (MACE)The University of Manchester

Dr. Marc Cotton, Dr. Yacine Addad, Amir Keshmiri, Stefano Rolfo,

+ F. Billard, D. Laurence

Code Saturne

User’s Annual Meeting, EDF R&D Chatou

FR, Nov. 2007

http://www.knoo.org/

- to maintain and develop skills relevant to nuclear power generation.- 4 years, 6M £, 50 PhDs and Post-docs, - largest commitment to fission reactor research in UK for over 30 years, - collaboration with industrial and governmental stakeholders and international partners.

KNOO WORKPACKAGES

Ex: Molecular dynamics of radiation enhanced helium re-solution

Droplets on hot solid surfaces: Modelling and Experimentation

Ballooning fuel pins

Heat transfer through crud

LES with adaptive FE meshing

CFD @ Imperial College

CFX

STAR

FE (Fetch

Test Case 1 :Test Case 1 : Mixed convection in vertically flowing heated pipeMixed convection in vertically flowing heated pipe

(buoyancy aiding or opposing)(buoyancy aiding or opposing)

Problem specifications:Re=2650

Pr=0.71

Wall constant heat flux

Boussinesq approximation

Heat transfer Regimes:Gr/Re2=0.000 Forced Convection

Gr/Re2=0.063 Forced/Mixed Convection

Gr/Re2=0.087 Re-Laminarization

Gr/Re2=0.241 Recovery

KNOO CFD @ Manchester: Heat transfer test cases

AGR working schemeAGR working schemeRelevance to AGR and VHTR

DNS Relaminarization

point, Impairment of heat transfer

Buoyancy aided heated pipe flow

(Buoyancy number)

K-omega

or SST worst !

Range of RANS models

Buoyancy aided heated pipe flow

LES Addad Manch.

best models: Launder Sharma, V2F

GrGr/Re**2 = 0.000/Re**2 = 0.000

Buoyancy aided heated pipe flow

GrGr/Re**2 = 0.063/Re**2 = 0.063Buoyancy aided heated pipe flow

GrGr/Re**2 = 0.087 (/Re**2 = 0.087 (relaminarizationrelaminarization))

Buoyancy aided heated pipe flow

GrGr/Re**2 = 0.241 (recovery)/Re**2 = 0.241 (recovery)Buoyancy aided heated pipe flow

The model and Code_Saturne• A low-Reynolds (near-wall integration) eddy viscosity

model derived from second moment closure models• No damping functions, no wall functions, less empirical

assumptions• Best results on range of test cases, heat transfer and

natural convection in particular. • The original model is stiff (requires coupled solver or very

small time-step)• Degraded version available in StarCD, Fluent, NUMECA..• Long collaboration Stanford, Delft, Chatou, Manchester

(Durbin, Parneix, Hanjalic, Manceau, Uribe) => “several code friendly” versions since 1995.

• Present: Reconsider all historical choices with numerical stability and known asymptotic states as principal objectives

v2− fAccuracy

Robustness

Stanford 1991

TU-Delft 2004

Manchester 2004

Stanford 1996(Fluent, STAR-CD)

Revisiting the V2F model

Durbin's original model

the walls affect the whole domain through an elliptic operator : near walleffects (mainly wall echo and wallblocking effect)

Classic near wall modelν t = fμ Cμk T ; T = k /ε

fμ =ν t you wantν t you have

fμ = f (y +,ν t /ν )

Elliptic Relaxation

ν t = Cμ v 2 T

Dv 2

Dt= φ22 − ε22 +∇((ν +ν t )∇v 2 )

φ22 = −2ρ

v ∂p' /∂y

Revisiting the V2F model

The usual second moment closure• Usual closure for the source term of : v2

Dv 2

Dt= φ22 − ε22 −

23

ε⎛ ⎝ ⎜

⎞ ⎠ ⎟ −

23

ε +Diffν +ν tv 2

Term to be modelled

ϕ 22

Launder Reece Rodi (LRR) :

ϕ22h = −

1T

C1v2

k−

23

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ +C2

Pk

Speziale, Sarkar, Gatski

(SSG) :

ϕ22h = −

1T

C1 +C'2Pε

⎛ ⎝ ⎜

⎞ ⎠ ⎟

v2

k−

23

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ −

C4

3− C5

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Pk

εh = 23

εModel for the dissipation :

wall

2u

2u

2v

2w

2w

2v

2u

2v2w

2w

2v

Wall blocking effect:

2u

The redistribution is anisotropic near the wall

Revisiting the V2F model : original

The asymptotic behaviour

• Near wall limit of the k

–eps model

DkDt

= (ν + ν t ) Δk + P − ε ≈ νΔk − ε ≈ νΔk n +1 −εk

⎛ ⎝ ⎜

⎞ ⎠ ⎟

n

k n +1

( O 1( )+O y 3( ) ).ΔO y 2( )

( )3yO

)1(O

( )2yO=k ( )1O=ε

No stability problem

Negative but implicit

source

Numerical stability

The asymptotic behaviour

• Taylor series expansion : ( )42 yO=v

Dv 2

Dt= ν ∂2 v 2

∂xk2 −

∂v 2ui

∂x i

+ φ22 + D22p − ε22 −

εk

v 2⎛ ⎝ ⎜

⎞ ⎠ ⎟ −

εk

v 2

( )2yO ( )5yO ( )2yO ( )2yO( )2yO

( )22 yO=u ( )22 yO=w

two-component limit of the turbulence

term to be modelled

The balancebetween

quadratic terms must be ensured

( )yO ( )yO

2Ay fkBy =22Cy

0)1()5()6(0

=−+−++=++ wallthenearCBA

2 42 2 2

hom 4 40 0

( )20lim lim( )

!y y

v O yL f f = f with f = =y O y

νε→ →

⎛ ⎞∇ − − => −⎜ ⎟⎜ ⎟

⎝ ⎠

Revisiting the V2F model

Stanford “code friendly” model (1)

It involves a change of variable so that gf=f + 0lim0

=fy→

The new equation for f reads :

L2∇2 f − f = C1 −1T

v 2

k−

23

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ − C2

Pε

+g − L2∇2gneglected term

Original Durbin : 2220 CyByAy ++=

fk

Stanford

: 222220 DyCyDyByAy ++−+=gkfk − gk

0=− DB

Lien and Durbin (1996)

Numerical stability vs. accuracy

Stanford “code friendly” model (2)Numerical stability vs. accuracy

Delft / UMIST approach

• A new change of variable to reduce the stiffness of the B.C. : kv=

2

ϕ

UMIST (Laurence et al. (2004)) and TU-Delft (Hanjalic et al. (2004))

DϕDt

= f − P ϕk

+2k

ν + ν t

σ k

⎛

⎝ ⎜

⎞

⎠ ⎟ ∇ϕ∇k+∇((ν + ν t

σϕ

)∇ϕ)

A B C

wallthenearCBA )( +−=

fw = −2ε ϕy 2

UMIST

modelϕϕνϕν 22

∇+∇∇+= kk

ff

0=fw

Delft

modelς

fw = −2ε ςy 2

Numerical stability vs. accuracy

Numerical instabilities of the model

2 options :•

f is forced to be equal to 0 at the wall : excellent robustness,

similar to the one

of Stanford's model, but bad prediction of the turbulent viscosity near the wall.• the turbulent viscosity is removed : numerical instabilities

kv=

2

ϕ

Numerical stability vs. accuracy

Natural convection in a heated cavity (non convergence in C_S V 1.2 !)

HOTCOLD

Very low value of k

Very low value of k

ϕνϕν 22∇+∇∇+= k

kff

ϕ model

Code_Saturne

Numerical stability vs. accuracy

The model• Using the elliptic blending of Manceau and Hanjalic 2002 (with Re stress model)

ϕ −α

122 −−∇ =L ααDϕDt

= α 3 fhom + 1−α 3( )fw − P ϕk

+ 2ν t

k∇ϕ∇k +∇((ν + ν t

σϕ

)∇ϕ)

Successfully tested on channel flows for many Re numbers, flow around airfoil trailing edge, heated pipe, heated channel flow, heated cavityNormal time-step (external flow CFL values as for k-omega)Unlike, code friendly Stanford model, no term has been neglected hereUnlike UMIST and Delft model, the correct asymptotic behaviour of and

is accurately predicted without impairing the numerical robustnessv2

tν

0=wα

2yfw

ϕε−= )1(2 ofw =∇− ϕν

Final model

Results (Channel flow, Re*=395

tν

v2

Results (Channel flow, Re*=395) (2)

V2v2

tν

y+

Conclusions and further work

• 4 different versions of V2F revisited => numerical stability improved while respecting known asymptotic states,

• Relaminarisation OK, next: prediction of laminar-turbulent transition,

input some ingredients of Launder and Sharma model (extra viscous source terms in epsilon equation)

• Recalibration of source terms in the ε

equation (all literature focuses only on near wall layer but prediction of the core region can be improved)

• An accurate and robust near-wall low-Reynolds RANS model also suitable for RANS/LES coupling

[1] Launder, B.E. and Sharma, B.I., 1974, “Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc”, Lett. Heat Mass Transfer, 1, pp. 131-138.[2] Cotton, M.A., Ismael, J.O., 1998, “A strain parameter turbulence model and its application to homogeneous and thin shear flows”,

Int. J. Heat Fluid Flow 19, pp.

326–337.[3] Chen, Y.S., and Kim, S.W. 1987, “Computation of turbulent flows using an extended k-ε

turbulence closure model”,

NASA CR-179204.

[4] Wilcox, D.C. 1998, “Turbulence Modelling for CFD”,

2nd edition, DCW Industries, Inc.[5] Durbin, P.A. 1995, “Separated flow computations with the k-ε−v2 model”

, AIAA

Journal, 33(4), pp. 659–664.[6] Laurence, D., Uribe, J.C. and Utyuzhnikov, S., 2004, “

A robust formulation of the

v2-f model”, Flow, Turbulence and Combustion, 73, pp. 169-185[7] You, J., Yoo, J.Y. and Choi. H., 2003, “Direct Numerical Simulation of Heated Vertical Air Flows in Fully Developed Turbulent Mixed Convection”, Int. J. Heat Mass Transfer, 46, pp.1613-1627