Modelling of Buoyancy Driven Turbulent...
Transcript of Modelling of Buoyancy Driven Turbulent...
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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
2KNOO Workshop, Manchester, July 16, 2008
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
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3KNOO Workshop, Manchester, July 16, 2008
4KNOO Workshop, Manchester, July 16, 2008
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
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6KNOO Workshop, Manchester, July 16, 2008
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.
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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
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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
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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,…
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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
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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
…
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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
=
=
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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 !
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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
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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
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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.
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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.
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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!
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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
=
=
=
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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)
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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)
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Some Experience in Modelling of
Buoyancy Driven Turbulent Flows
at TU Delft
Contribution by:
S. Kenjeres, H. Dol, S. Gunarjo, M. Reeuwijk
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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
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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.
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( )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, , −=∏−=∏
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( )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
28KNOO Workshop, Manchester, July 16, 2008
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 *
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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):
30KNOO Workshop, Manchester, July 16, 2008
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:
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31KNOO Workshop, Manchester, July 16, 2008
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).
32KNOO Workshop, Manchester, July 16, 2008
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
33KNOO Workshop, Manchester, July 16, 2008
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:
34KNOO Workshop, Manchester, July 16, 2008
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
37KNOO Workshop, Manchester, July 16, 2008
Examples of ComputationsExamples of Computations
Heat transfer and friction factor on a free-hanging
20
39KNOO Workshop, Manchester, July 16, 2008
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)
40KNOO Workshop, Manchester, July 16, 2008
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
41KNOO Workshop, Manchester, July 16, 2008
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)
42KNOO Workshop, Manchester, July 16, 2008
Internally heated horizontal annulusInternally heated horizontal annulus
2θε −−k RAFM model, Hanjalic et al. 1996temperature streamlines
Ret k
22
43KNOO Workshop, Manchester, July 16, 2008
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:
44KNOO Workshop, Manchester, July 16, 2008
__
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
45KNOO Workshop, Manchester, July 16, 2008
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
46KNOO Workshop, Manchester, July 16, 2008
Illustration of the AFM / ASMIllustration of the AFM / ASM 22k fυ ε θ− − − −
5:1 side heated vertical cavity, Ra=5x108
(Kenjeres, Gunarjo & Hanjalic, IJHFF 2005)
24
47KNOO Workshop, Manchester, July 16, 2008
Illustration of the AFM / ASMIllustration of the AFM / ASM 22k fυ ε θ− − − −
Rayleigh-Bérnad convection, Ra=4x105- 2x109
48KNOO Workshop, Manchester, July 16, 2008
Illustration of the AFM / ASMIllustration of the AFM / ASM 22k fυ ε θ− − − −
Side-heated cubic enclosure, Ra=5x1010
(Kenjeres, Gunarjo & Hanjalic, IJHFF 2005)
25
49KNOO Workshop, Manchester, July 16, 2008
Illustration of the AFM / ASMIllustration of the AFM / ASM 22k fυ ε θ− − − −
2D enclosure with supply and exhaust under stable stratification(Kenjeres, Gunarjo & Hanjalic, IJHFF 2005)
50KNOO Workshop, Manchester, July 16, 2008
Illustration of the AFM / ASMIllustration of the AFM / ASM 22k fυ ε θ− − − −
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
(Kenjeres, Gunarjo & Hanjalic, IJHFF 2005)
26
51KNOO Workshop, Manchester, July 16, 2008
Illustration of the AFM / ASMIllustration of the AFM / ASM 22k fυ ε θ− − − −
Indoor climate under summer cooling conditions; Experiments: Fossdal (1990)
(Kenjeres, Gunarjo & Hanjalic, IJHFF 2005)
52KNOO Workshop, Manchester, July 16, 2008
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
53KNOO Workshop, Manchester, July 16, 2008
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
54KNOO Workshop, Manchester, July 16, 2008
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.
Hanjalić, K. 1994, Achievements and limitations in modelling natural convection, Proc. 10th Int. Heat Transfer Conf., Brighton, UK, IChemE/Taylor & Francis, Vol. 1, 1-18, 1994.
Hanjalic´, K., 2002. One-point closure models for buoyancy-driven turbulent flows. Ann. Rev. Fluid Mech. 34, 321–347.
Hanjalic´, K., Kenjeresˇ, S., Durst, F., 1996. Natural convection in partitioned two-dimensional enclosures at higher Rayleigh numbers.
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Kenjeresˇ, S., 1998. Numerical modelling of complex buoyancy-driven flows. Ph.D. Thesis, Delft University of Technology, Delft, Nl.
Kenjeresˇ, S., Hanjalic´, K., 1995. On the prediction of thermal convection in concentric and eccentric annuli. Int. J. Heat Fluid
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