Modelling of Buoyancy Driven Turbulent Flowscfd.mace.manchester.ac.uk/.../pub/Main/...16July08.pdf · Buoyancy Driven Turbulent Flows Motivations: Thermal and concentration buoyancy

1

1KNOO Workshop, Manchester, July 16, 2008

K. HanjalićMarie Curie Chair, University of Rome, ‘La Sapienza’, Rome Italy(Professor Emeritus, Delft University of Technology, The Netherlands)

CFD Workshop on Test Cases, Databases & BPG for Nuclear Power Plants Applications

The University of Manchester, July 16, 2008

• ERCOFTAC SIG 15 on Refined Turbulence Modelling

Modelling of Buoyancy Driven Turbulent Flows

• Some experience from Delft University of Technology, Nl

• Issues, progress and pertaining challenges


the Pilot Centres, that coordinate the research in Flow, Turbulence, and

Combustion on a regional or national scale,

http://www.ercoftac.org/

ERCOFTAC is a scientific association of

research, education and industry

groups in the technology of flow,

turbulence and combustion. It is

organised around 2 pillars :

the Special Interest Groups, that stimulate European-wide research

efforts on specific topics in flow, turbulence and combustion.

ERCOFTAC members benefit from specific products and services,

specialized publications and targeted workshops and conferences

2



1st SIG-15 Workshop in Lyon, France,1992

2nd SIG-15 Workshop in Lyon, France, 1993 (Natural convection)

3rd SIG-15 Workshop in Lisabon, Portugal, 1994

4th SIG-15 Workshop in Karlsruhe, Germany, 1995

5th SIG-15 Workshop in Chatou/Paris, France, 1996

6th SIG-15 Workshop in Delft, The Netherlands,1997

7th SIG-15 Workshop in Manchester, UK, 1998 (Natural convection)

8th SIG-15 Workshop in Helsinki, Finland1999

9th SIG-15 Workshop in Darmstadt, Germany, 2001

10th SIG-15 Workshop in Poitiers, France, 2002

11th SIG-15 Workshop in Gothenburg, Sweden, 2005

12th ERCOFTAC/IAHR in Berlin, Germany, October 12-13, 2006

13th ERCOFTAC/IAHR in Graz, Austria, September 22-23, 2008

ERCOFTAC/IAHR Workshop on Refined Turbulence Modelling

3



5th workshop at EdF Chatou (25-26 April 1996)

Case 5.1 2D Plane Wall Jet

Case 5.2 Natural Convection Boundary Layer

Case 5.3 Natural Convection in a Tall Cavity

Case 4.5 (Repeated) Developing Flow in a Curved Rectangular Duct

7th workshop at UMIST (28-29 May 1998)

Case 7.1 Fully developed flow in a plane channel in orthogonal mode rotation with rotation numbers up to 0.5

Case 7.2 Two-dimensional flow and heat transfer over a smooth wall "roughened" with square-sectioned ribs

Case 7.3 Fully developed flow and heat transfer in an orthogonally rotating square sectioned rib-roughened duct

Case 6.2 Fully developed flow and heat transfer in a matrix of surface mounted cubes

Case 5.3 (Repeated) Natural convection in an infinite cavity formed by a heated and cooled wall.

4


Case 5.3: Natural convection of air in a tall cavity;

Experiments by P.L. Betts and I.H. Bokhari,

DNS by R. Boudjemadi, V. Maupu, D. Laurence and P. Le Quere

Flow configuration

Experiments

The experiment was conducted with a cavity of internal dimensions of H=2.18m high, D= 0.0762m wide

and 0.52rn deep. One vertical wall was at 15 degrees C and the other at 35 degrees C for Ra= 8.5

×105, and 15 degrees C and 55 degrees C for Ra =15.3 ×105. Results include mean and turbulent

(rms) vertical velocity and temperature.

Direct Numerical Simulation

On the other hand, the incompressible Navier-Stokes equations using the Boussinesq approximation of

density have been solved by a pseudo-spectral/finite difference DNS code for the flow between two

infinite vertical walls and for Ra= 1.0 ×105 & Ra= 5.4 ×105. All first and second moments, and their

budgets, are provided. Mean and rms fluctuating values normalized by the buoyancy velocity agree

well with the experiment. A still open question is the slight disagreement concerning the friction

velocity with an other DNS performed by Niewstadt and Versteegh.

Flow parameters

Directions 1,2,3 correspond to x,y,z respectively, where x is the stream wise ascending co-ordinate, y is

the spanwise co-ordinate, z is the wall normal co-ordinate (z=0 for the hot wall and z=1 for the cold

one). Rayleigh no, Dgap widthD T temperature difference between walls Pr=v/kPrandtl n….etc


explicit ASMFu et al, Tsinghua Univ. TS-SF-EARSM

RSM + algebraic heat fluxFu et al, Tsinghua Univ. TS-SF-RSAHF

RSM + eddy diffusivityFu et al, Tsinghua Univ. TS-SF-RSED

LES Peng & Davidson, Chalmers UTCUT-SP-LES

A

low-Re RSM + flux transport Craft, UMISTUM-TC-RSM

cubic RSM + flux transport Armitage, UMISTUM-CA-RSMC

linear RSM + flux transportArmitage, UMISTUM-CA-RSML

RSM + flux transport Fu et al, Tsinghua Univ. TS-SF-RSMF

A

Turbulence Model Contributor Identifier Group

Example of Cross plots:

Case 5.3 Flow in a infinite side heated/cooled cavity, Ra=5x10

5


Issues, Progress and Pertaining Challenges

in Modelling of

Buoyancy Driven Turbulent Flows

Motivations:

Thermal and concentration buoyancy in building structures, space

cooling and heating, nuclear engineering – reactor containments,

radioactive waste storage, electronics equipment, solar collectors

and ponds, crystal growth, atmosphere, oceans, lakes,…


RANS Equations for ThermallyRANS Equations for Thermally-- buoyant Flowsbuoyant Flows

D

Dj

p j j p

T q Tu

t c x x c

λα θ α

ρ ρ

∂ ∂= + − = ∂ ∂

where

−

∂

∂

∂

∂+

∂

∂−=∑ ji

j

i

jin

n

i uux

U

xx

PF

t

Uν

ρ

1

D

D i

∑n

n

iF

i Mean momentum and energy equations (in terms of temperature T ) :

kk xUtDtD ∂∂+∂∂= ///where : is the material derivative,

are the mean body forces acting on the fluid,

q is the internal energy source

i To close the equations, we need to provide stress and flux,

(Note: for a more general case, we can replace T by Φ, denoting any scalar,

e.g. species concentration)

, ;i j iu u uθ

6


ti t

T i

Tu C

xθ

νθ

σ

∂= −

∂...iD u

Dt

θ=

Approaches to RANS Approaches to RANS modellingmodelling of turbulent heat fluxof turbulent heat flux

(Isotropic) eddy diffusivity (EDM, SGDH) Differential (Re-) flux Models (DFM)

Modified linear EVM/EDMs

Nonisotropic EDMs (GGDH)

“Realisable” EVM/EDM

Elliptic relaxation EVM/EDMs

….

Implicit

Explicit

Reduced

…

Nonlinear EVM/EDMs

Algebraic flux models (AFM)

Quasi-linear

Quadratic

Cubic

…


Basic EDM for Buoyant FlowsBasic EDM for Buoyant Flows

0,k i i i iG f u g uβ θ= = ≠

Tx

Tu

t

T

t

j

t

T

t

j ∇−≡∂

∂−=

σ

ν

σ

νθ

• Eddy-diffusivity model is the standard approach in most commercial CFD codes

2

t

kCµν

ε=

• 2nd Boussinesq hypothesis: turbulent density fluctuations accounted only in body

force terms and expressed in term of temperature and/or concentration

• The standard Isotropic Eddy Diffusivity model (EDM) for heat (and other scalars):

0 00 , ,

'1

p s p T

dT dST S

ρ ρ ρρ ρ ρ

ρ

∂ ∂= + = + + ∂ ∂

T T dT T θ≈ + ≈ +

0

0

'/

(1 )T S s

ρ ρ

ρ ρ β θ β= + +

' ii

gf

ρ

ρ=

0

1T

T

ρβ

ρ

∂ = −

∂ 0

1S

S

ρβ

ρ

∂ = +

∂

k k k

DkP G D

Dtε= + − + 1 3 2k kC P C G CD

DDt

ε ε εε

εε

τ

+ −= +

S S dS S s≈ + ≈ +

Assume:

where:

;

.tT

C const

const

µ

σ

=

=

7


Principal deficiencies of the Basic EDM for buoyant flowsPrincipal deficiencies of the Basic EDM for buoyant flows

0!i i

q uθ= − ≠

0,i i

G g uβ θ= ≠

Tx

Tu

t

T

t

j

t

T

t

j ∇−≡∂

∂−=

σ

ν

σ

νθ

i Isotropic eddy-diffusivity model (EDM) for heat flux (“Simple Gradient Diffusion

Hypothesis”, SSGD) :

Consider two generic situations:

1. A fluid layer heated from below, gi || ∇∇∇∇T

Outside the thin wall layers, ∇∇∇∇T ≈≈≈≈ 0 (or > 0!),

yet, the vertical heat transport

2. Vertical heated walls, gi ⊥⊥⊥⊥ ∇∇∇∇T

Buoyancy source of k (and ε)

yet, the vertical ∇∇∇∇T ≈≈≈≈ 0 !


1

21

2

21

dx

dUu

dx

dTuu

t

T

θ

σ =

ε

εθ

τ

τ θθ

/2

/2

kR th ==

Ra

Ra

DNS scrutiny: infinite, vertical side-heated plane channel

DNS of Versteegh 1998, TUD

Time scale ratio:

Turbulent Prandtl-

Schmidt number:

Further deficiencies of the Basic EDM for buoyant flowsFurther deficiencies of the Basic EDM for buoyant flows

8


ijijijji

j

i

jiij

ji

fij

ij

GP

ufufx

Uuu

t

uuε−Π+++

∂

∂−=

DD

D

,D

D;

D

D;

D

D2

ttt

θθεθε

iii

j

i

j

j

jii

i

imi

thi

GPP

gx

Uu

x

Tuu

t

uθθθ εθβθ

θ

θ

θθ

Π+−−∂

∂−

∂

∂−=

−−

2DD

D

The alternative:lternative: Second-moment (Re stress/flux) closures)

Plus Total 17 differential equations!

For double-diffusion fields (thermal + concentration): 24 equations


Differential Flux Model for Buoyant FlowsDifferential Flux Model for Buoyant Flows

:iuθ

2

i ju u θand

( )

ii

Li

Tim

ithi

i

ti

i

k

i

k

r

i

L

ii

k

i

k

k

ki

i

ki

k

i

i

kk

i

x

u

xG

f

G

f

G

g

P

x

Uu

P

x

Tuu

x

p

D

uu

D

x

uu

xxt

u

iD

θθθθ

θθ

θ

θνθ

ε

θναθθθβθ

ρ

θθθν

θα

θ

θ

∂

∂

∂

∂+−++−

∂

∂−

∂

∂−

∂

∂−

−∂

∂+

∂

∂

∂

∂=

Ω

∏

Production

2

D

D

i The starting point: the exact equation for turbulent heat flux

Note 1: Terms in boxes need to be modelled (including molecular diffusion,

if Pr ≠1) assuming that are provided.

9


Body Forces in EquationBody Forces in Equationjiuu

( )

( )

ijD

pij

D

uup

tij

D

uuu

ijD

x

uu

x

ij

x

u

x

u

ij

x

u

x

u

p

p

nij

G

ufuf

ijP

x

Uuu

x

Uuu

ijC

x

uuU

ijL

t

uu

t

uu

ikjjkikji

k

ji

k

k

j

k

i

i

j

j

i

n

i

n

jj

n

i

k

i

kj

k

j

ki

k

ji

k

jiji

+−−∂

∂

∂

∂+

∂

∂

∂

∂−

Φ

∂

∂+

∂

∂+++

∂

∂+

∂

∂−=

∂

∂+

∂

∂=

∑

δδρ

ν

ν

ε

ν2

D

D

Note: Terms in boxes must be modelled.


Body Forces in Equation (cont.)Body Forces in Equation (cont.)jiuu

jiuu

( )∑ +=n i

n

jj

n

i

n

ij ufufG

where: ρ = fluid density, T = mean temperature, S = mean species concentration

θβ iT

T

i gf −=

∂

∂−=

PS

TT ,

1 ρ

ρβ- thermal buoyancy force, where

S

i S if g sβ=

,

1S

T PS

ρβ

ρ

∂ =

∂ - concentration buoyancy force, where

The term in the double box: source/sink of due to fluctuating body forces,

Hence, contributions by various body forces are:n

ijG

( )T

ij T i j j iG g u g uβ θ θ= − + - thermal buoyancy

( )S

ij S i j j iG g su g suβ= + - concentration buoyancy

etcx

Tuk

x

U

x

Uuu

i

t

T

t

jiij

i

j

j

i

tji ,,3

2

∂

∂=−−

∂

∂+

∂

∂=−

σ

νθδν

Note that simple EVM/Note that simple EVM/EDMsEDMs ignore effects of body force ignore effects of body force fluctuation! fluctuation!

10


MiddleMiddle--ofof--the road: Algebraic Flux Models, AFMthe road: Algebraic Flux Models, AFM

02

=

−

θ

θ

k

uD

Dt

D i

−+

−=

−

=−

k

k

DDt

Dk

kD

Dt

Du

DkDt

D

k

uD

Dt

uD

i

i

i

i

11

2

12

2

2

22/1

22/1

22/1

θ

θ

θθ

θθ

θθ

θθ

i Differential flux model (DFM) can be truncated to algebraic form (AFM)

by applying the weak equilibrium hypothesis (Rodi 1972)

which leads to (Gibson and Launder 1976)

1

2

2

1 12

22

ii l i i

j j

i

ii i l i i

j j

T Uu u u g

x xu

T UC u u u g u

k x k xθ θθ

ξθ ηβ θ

θε

θ ε β θ εθ

∂ ∂+ +

∂ ∂=

∂ ∂− + + + + +

∂ ∂

+

2

/ ..;

/ ..;

/ ..

Dk Dt

D Dt

D Dt

ε

θ

=

=

=


Another route: (Semi)Another route: (Semi)--Explicit Explicit AFMsAFMs (EAFM)(EAFM)

from Tensor Representation methodfrom Tensor Representation method

i where are the integrity base vectors (characteristic polynomials) in terms

of and (passive + buoyant): ,,,, igijijSjuiu ΩixT ∂∂ /

)( j

iT

ix

T

∂

∂=(1)

iTi

jix

Tuu

∂

∂=(2)

iTi

ijx

TS

∂

∂=(3)

iTi

ijx

T

∂

∂Ω=(4)

iT

j

kjikx

T

∂

∂ΩΩ=(6)

iTi

kjikx

TSS

∂

∂=(5)

iT

ig=(11)

iT jji guu=(12)

iTjij gS=(13)

iT jij gΩ=(14)

iT

jkjik gSS=(15)

iT jkjik gΩΩ=(16)

iT

+ other nonlinear terms

+ other nonlinear terms

∑=

==20

1

(n)

i

(n)model)(

n

ii Tuu ζθθ

i The Representation Theory can be used to derive explicit and semi-explicit

expression (usually in a nondimensional form)

11


i No extensive validation of EAFM or SEAFM has been reported

EAFM from the Representation TheoryEAFM from the Representation Theory

i Coefficients evaluated from equalization of corresponding terms on

LHS (model) and RHS (vector expansion) of the equation for

(n)ζ

iuθ

i The Representation Theory satisfies only the mathematical formalism: the

final expression for can at best be only as good as the parent model itself

(if earlier derived (linear) AFM expressions are applied, all nonlinear integrity

base vectors disappear)

iuθ

(n)ζ

i The only advantage is that the expression is explicit (or, more often, semi-explicit,

particularly for 3D), hence numerical convenience.

i EAFM and SEAFM were proposed by So and Sommer (1996) Taulbee (1997),

Younis et al (1997), Girimaji and Balachandar (1998), Girimaji and Hanjalic (2000)


12


Some Experience in Modelling of

Buoyancy Driven Turbulent Flows

at TU Delft

Contribution by:

S. Kenjeres, H. Dol, S. Gunarjo, M. Reeuwijk


A Comprehensive Set of Test Cases A Comprehensive Set of Test Cases (1994)(1994)

ThTc

Side heated cubes, Opstelten 1996

From Hanjalić, Proc. 10th Int. Heat Transfer Conf., Brighton, UK, IChemE/Taylor & Francis, Vol. 1, 1-18, 1996.

Many more new results (especially DNS and LES) appeared in the meantime!

e.g. DNS of Rayleigh Bernard convection, Kerr, 1996, 2000 (Ra=107), Reeuwijk, 2007 (Ra=106-108), and others

13


Differential Flux Model for Buoyant FlowsDifferential Flux Model for Buoyant Flows

:iuθ

2

i ju u θand

( )

ii

Li

Tim

ithi

i

ti

i

k

i

k

r

i

L

ii

k

i

k

k

ki

i

ki

k

i

i

kk

i

x

u

xG

f

G

f

G

g

P

x

Uu

P

x

Tuu

x

p

D

uu

D

x

uu

xxt

u

iD

θθθθ

θθ

θ

θνθ

ε

θναθθθβθ

ρ

θθθν

θα

θ

θ

∂

∂

∂

∂+−++−

∂

∂−

∂

∂−

∂

∂−

−∂

∂+

∂

∂

∂

∂=

Ω

∏

Production

2

D

D

i The starting point: the exact equation for turbulent heat flux

Note 1: Terms in boxes need to be modelled (including molecular diffusion,

if Pr ≠1) assuming that are provided.


( )θθθε

θθθθ kjkijjijii uaaCuaCuCk

1

'

111, ++−=Π

DNS

DNS

linear

linear

quadratic

quadratic

cubic

cubic

Pressure scrambling:

First order (linear ) models (Launder ’76)

Higher order (nonlinear) models,

e.g. the ‘slow term’:

DNS scrutiny of DNS scrutiny of ,iθ∏

,1 ,2 ,2 ,2 ..m th g w

i i i i i iθ θ θ θ θ θ∏ = ∏ + ∏ + ∏ + ∏ + + ∏

mi

mi

i

iPC

uC θθθθθ τ

θ22,11,

, −=∏−=∏

i

g

i

th

ii

th

i GCPC θθθθθθ 32,

'

2, , −=∏−=∏

14


( )kjkijjijii uaaCuaCuCk

θθθε

θθθθ''

1

'

111,ˆ ++−=∏

A New Model of A New Model of (Dol et al. 1999),iθ∏

:)][1:( 328

9AAA −−=Note

[ ] [ ]'

1 1

''

1

6.4 1 exp( 4 ) 8.1 1 exp( 5.5 )

1 exp( 20 ) 1 4.5exp( 28 )

0

A AC C

A A

C

θ θ

θ

− − − − −= =

+ − + −

=

i

th

i

m

ii GCPCPC θθθθθθθ 3

'

223/2,ˆ −−−=∏

2 ' 2 3 4

2 2

3

1.25 , 6.15 19.3 15 ,

0.45

C A C A A A

C

θ θ

θ

= = − +

=

( ),1 ,2 /3

1/ 2

| |

max(0, 0.58 0.69 )

ˆ ˆ ˆw w

i ij j j

w

C a

C A

θ θ θ θ

θ

= +

= −

∏ ∏ ∏

where

• The “slow” term:

• The “rapid” term:

• The wall-effect term:

Dol.H., Hanjalic, K, Versteegh T..M., JFM 391, 1999


Models of Models of ((DolDol et alet al. 1999). 1999)andi iDθ θε

i Molecular diffusion (models are marked by the ‘hat’ symbol):

iThe last two terms can be neglected because of large difference in scales

of fluctuations and their second derivatives.

iTurbulent diffusion:

i where Cθ =0.11; the last term is usually neglected.

iDissipation rate * 3/ 23

4

exp[ ] )i

f Aθε = −( where

i

kk

i

i

k

i

i uxx

uD

x

uD

2

2

2

2

2

2

)()()(ˆ2

1

2

1

2

1

∂

∂−−

∂

∂−+=

∂

∂+=

θναθνα

θνα ν

θνθ

∂

∂+

∂

∂+

∂

∂

∂

∂=

l

ki

l

l

k

li

l

i

lk

k

t

ix

uuu

x

uuu

x

uuu

kC

xD θ

θθ

εθθ

ˆ

i

i

iii uk

DDu

kf θθ

εε

νθν

θθ

+−+

+=

Pr

11ˆ

2

1

Pr

11ˆ

4

1

2

1 *

15


Models of equation (Summary) Models of equation (Summary) iuθ

iuθ

2 21 0.5 1 0.5C Cθ θξ η= − ≈ = − ≈and

ii

m

i

th

i

w

iii

t

ii

i GPPDDDt

uDθθθθθθθθ

νθ ε

θˆˆˆˆˆˆ

3/2,1, −+++∏+∏+∏++=

iii

j

i

j

j

ji

k

k

kj

k

i

ki

j

i

i

kuCg

x

Uu

x

Tuu

x

uuu

x

uuu

kC

xD

Dt

uD

θθ

θνθ

εε

θθηβθξ

θθ

ε

θ

−+−∂

∂−

∂

∂−

∂

∂+

∂

∂

∂

∂−=

1

2

ˆ

i Simplified equation ( with Gibson-Launder model for ):iθ∏

i The full modelled transport equation for (Dol et al. 1999):


Modeling equation (Modeling equation (DolDol et alet al. 1999). 1999)2θ

θθθθ

θθ

θθνθθ

ε

θθαθθ

θα

θ

kkk

kk

kk xx

P

x

Tu

D

D

u

D

xxDt

D

t∂

∂

∂

∂−

∂

∂−

−∂

∂

∂

∂= 222

22

∂

∂+

∂

∂+

∂

∂

∂

∂=

l

lk

l

k

l

l

lk

k

t

x

Tuu

x

uu

xuu

kC

xD θ

θθ

θ

εθθθθ 22ˆ

2

5.0ˆ2

≈==m

thRkR τ

τθεεθθ where

:2θi Exact equation for

i Model of turbulent diffusion:

i Model of scalar dissipation rate:

16


Example of application of DFMExample of application of DFM

3-D flow in a side-heated near-cubic cavity, Ra = 5 x 1010 (Dol 1998).

Streamlines in the cavity corner with —– adiabatic and – – isothermal

Horizontal walls : (a) EDM, (b) DFM (PH).


A DNS scrutiny of A DNS scrutiny of AFMsAFMs for Buoyant Flowsfor Buoyant Flows

εθβ −−∂

∂−=− ii

j

i

jik ugx

UuuD

Dt

Dk

1

2

2

1 12

22

ii l i i

j j

i

ii i l i i

j j

T Uu u u g

x xu

T UC u u u g u

k x k xθ θθ

ξθ ηβ θ

θε

θ ε β θ εθ

∂ ∂+ +

∂ ∂=

∂ ∂− + + + + +

∂ ∂

θθθεθ

θ−

∂

∂=− −

j

jx

TuD

Dt

D2

2

2

+−−

∂

∂−

∂

∂−=−

2

2

1

θ

εεεθθηβθξ

θ θεθθθ

k

fk

Cugx

Uu

x

TuuD

Dt

uD

iii

j

i

j

j

ji

i

iu

iuθlead to the following Full Implicit Algebraic Expressions for

i Equations for k and with weak-equilibrium assumption2θ

NOTE: the AFM depends on the adopted modes of Πθ i and εθ i in the parent

Differential model equation for iuθ

17


DNS Validation of DFM / AFM TruncationDNS Validation of DFM / AFM Truncation

+=

kiD

kDuD i

11

2

1

2 θθθθ

θ

I Model DNS

1 o –—

2 ∆∆∆∆ – –

i Comparison of the a-priori evaluated diffusion terms for the two flux components

with the DNS data show only qualitative, but rather poor agreement, questioning

the basic (weak-equilibrium) hypothesis behind the AFM:

i In a fully developed flow between two differentially heated vertical plates

D/Dt = 0, hence the weak equilibrium hypothesis (DFM / AFM truncation)

reduces to:


Note:

- RAFM is regarded as the minimum modelling level at which all major

effects (all sources of ) are accounted for!

- Note the importance of ati third term and the need ao obtain from ias

transport equation (in tomogeneous flow regions, e. g. R-B conv.

- This further simplification requires a slight returning of coefficienas.

The recommended values are :

Further Simplification: Further Simplification: ‘‘ReducedReduced’’ AFM (AFM (‘‘RAFMRAFM’’))

iuθ

2 / /m

kτ υ ε ζ ε= =

thτ .thmττ

iuθ

2θ

)0=∇=∇ UT

6.0,51 === ηξθC

/m

kτ ε=

+

∂

∂+

∂

∂−= 2

1

1θηβθξ

εθ

θi

j

i

j

j

jii gx

Uu

x

Tuu

k

Cu

i Full neglect of the transport terms in equation leads to a simpler (‘reduced’)

algebraic flux model (RAFM

i Note: can be replaced by or mixed scale

i In the framework of v2-f or ζ-f ER models:

19


Examples of ComputationsExamples of Computations

Heat transfer and friction factor on a free-hanging

20


Example: SideExample: Side--heated 5:1 cavityheated 5:1 cavity

i Velocity and vertical heat flux profiles at y/H=0.5. Contribution of the terms

in the RAFM expression (Kenjeres and Hanjalic 1996)


Velocity and turbulence Re number Ret=νt /ν

in a side-heated partitioned enclosure with

Hob / H = 0.5, RaL =5.65 x 1010

( +RAFM, Hanjalic et al. 1996)

Example: Partitioned EnclosuresExample: Partitioned Enclosures

2θε −−k

21


Example: Penetrative convective layerExample: Penetrative convective layer

Time evolution of the temperature and the normalized vertical heat flux in a

mixed layer heated from below (Kenjeres 1999)


Internally heated horizontal annulusInternally heated horizontal annulus

2θε −−k RAFM model, Hanjalic et al. 1996temperature streamlines

Ret k

22


2

2 2Dkf D

Dt k υ

υ υε= − +

An Elliptic Relaxation AFM / ASMAn Elliptic Relaxation AFM / ASM 22k fυ ε θ− − − −

k

DkP G D

Dtε= + − +

2

2 2D

P DDt

θ θ θθ

ε= − +

1 2( )k kC P G CD

DDt

ε

εε ερε

τ

+ −= +

1/ 22

2

2

1; max ,0.6

1k

AR A

ARθ

θ

θε ε=

= +

(Kenjeres, Gunarjo & Hanjalic, IJHFF 2005)

2 1.5 ; 0.3ii i j j i ij j

j j

T Uu C u u u g a u C

x xθ θθ τ θ β θ θ ∂ ∂

= − + + + = ∂ ∂

2 221

2 2

1 20

3j

C P G fC f L

k l x

υ

τ

− + ∂= − + − +

∂

i j

j

UP u u

x

∂= −

∂j

j

TP u

xθ θ

∂= −

∂i iG g uβ θ= −

t

t

j j

Dx x

φ

φ

ν φν

σ

∂ ∂= +

∂ ∂

1 / 41 / 23/ 2 3

max , ; max ,L

k kC L C Cµ η

ν ντ

ε ε ε ε

= =

2

tCµν τυ=

2 2

3 3

ji

i j t i j j i k k ij

j i

UUu u k C g u g u g u

x xθν τ θ θ θ δ ∂∂

= − + + + − ∂ ∂

where:

A2θ , A2, A3, A = flux/stres invariants

A more recent development:


__

T g

∇

A prioriA priori test of some models in generic flowstest of some models in generic flows

2 1.5i

i i j j i ij j

j j

UTu C u u u g a u

x xθθ τ θ β θ θ

∂∂= − + + +

∂ ∂

The AFM of Kenjeres et al(2004) in conjunction with the v2-f -θθθθ

2 model reproduces

best the heat flux components in both generic cases of natural convection: vertical and

horizontal plane channels with | and respectively.

Wall normal heat flux in a side heated 6, Pr=0.71

(Symbols: DNS, Versteegh 1998)

Wall normal heat flux in a heated-from-below 5,

Pr=0.71 (Symbols: DNS Woerner1994)

||

T g

∇

2

AFM-new

AFM GGDH SGDH

23


1/ 22

2

2

1; max ,0.6

1k

AR A

ARθ

θ

θε ε=

= + A2θ , A2, A3, A = flux/stres invariants

Equation for scalar dissipation Equation for scalar dissipation εεεεεεεεθθθθθθθθ

i Several proposals for a model equation for Dεεθθ //DtDt (source in (source in eqn,

thermal time scale (e.g. Nagano, 200, 2006, Groetzbach 2007)

i To much uncertainty (too many new coefficients, too little benefits,..)

i More plausible alterative: algebraic expression for time-scale ratio

2θ


Illustration of the AFM / ASMIllustration of the AFM / ASM 22k fυ ε θ− − − −

5:1 side heated vertical cavity, Ra=5x108


24



Rayleigh-Bérnad convection, Ra=4x105- 2x109



Side-heated cubic enclosure, Ra=5x1010


25



2D enclosure with supply and exhaust under stable stratification(Kenjeres, Gunarjo & Hanjalic, IJHFF 2005)



Indoor climate under summer cooling conditions; Experiments: Fossdal (1990)

Scenario E1: Q=0.0158 m3/sTa=15C, Tw=30C

Scenario E2:

Q=0.0315 m3/sTa=10C, Tw=30C


26



Indoor climate under summer cooling conditions; Experiments: Fossdal (1990)



LargeLarge--scale realscale real--life applications: life applications:

Use of AFM/ASM as Use of AFM/ASM as subscalesubscale model in Transient RANSmodel in Transient RANS

(Kenjeres, & Hanjalic, J. of Turbulence 2000)

27


Summary and ConclusionsSummary and Conclusions

i Several levels of truncation are possible: satisfactory results for a variety of

buoyancy-driven flows were obtained by the Reduced Algebraic Flux Model

(RAFM), closed by k, ε and equations.2θ

i Differential second-moment closures (DSM/DFM) should be the optimum

framework for RANS modelling effects of body forces in turbulent flows

i Despite their appeal, DSM/DFM are impractical: 17 differential transport

equations for 3-D buoyancy driven flows and 25 for 3-D double diffusion flows!

i In contrast, linear isotropic eddy-diffusivity models are simple, but have

serious deficiencies, particularly in flow driven by body forces.

i Algebraic flux/stress models (ASM/AFM) offer a good compromise, despite

the fact that the basic assumption (weak-equilibrium) is not justified.

i No wall functions exist for flows driven by gravitational force, development

of WF and their combination with ItW is an urging challenge for high Re/Ra.

i Notable improvements have been achieved by implementing elliptic relaxation

concept into AFM/ASM


Dol, H.S., Hanjalic´, K., 2001. Computational study of turbulent natural convection in a side-heated near-cubic enclosure at a high

Rayleigh number. Int. J. Heat Mass Transfer 44, 2323–2344.

Dol, H.S., Hanjalic´, K., Kenjeresˇ, S., 1997. A comparative assessment of the second-moment differential and algebraic

models in turbulent natural convection. Int. J. Heat Fluid Flow 18 (1), 4–14.

Dol, H.S., Hanjalic´, K., Versteegh, T.A.M., 1999. A DNS-based thermal second-moment closure for buoyant convection at vertical

walls. J. Fluid Mech. 391, 211–247.

Gunarjo, S.B., 2003. Contribution to advanced modelling of turbulent natural and mixed convection. Ph.D. thesis, Delft University of

Technology, Delft, The Netherlands.

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Documents

Modelling of Buoyancy Driven Turbulent Flowscfd.mace.manchester.ac.uk/.../pub/Main/...16July08.pdf · Buoyancy Driven Turbulent Flows Motivations: Thermal and concentration buoyancy