TURBULENCE HEAT-FLUX MODELING OF NATURAL
CONVECTION IN TWO-DIMENSIONAL THIN-ENCLOSURE
Zarko M. Stevanovic
VINCA Institute for Nuclear Sciences
Laboratory for Thermal Engineering and Energy
1001 Belgrade, P.O. Box 522, Yugoslavia
zare@rt270.vin.bg.ac.yu
Computer: Pentium III
Operating System: Microsoft Windows 2000 Professional / NT
PHOENICS: Version 3.3
Abstract
This paper deals with the thermal buoyancy-driven flow in two-dimensional thin enclosure with
an aspect ratio of 28.6:1 that can be regarded as thermal buoyancy-driven flow between two- heated
infinite vertical plates. Results of direct numerical simulation (DNS) of the natural-convection flow
between differentially heated infinite vertical plates of Versteegh, [18] and as well as experimental
data of Betts and Bokhari [1] were used for validation of the proposed algebraic heat-flux model for
computing turbulent thermal convection in two-dimensional thin-enclosure. Despite the large
differences in the turbulence parameter-by-parameter comparison, the complete second-moment
closure should be replaced by low-Re- k--  2 turbulence model with the algebraic heat-flux terms.
KYEWORDS:
model,
thermal buoyancy-driven flow, turbulence modelling, algebraic heat-flux
PHOENICS CFD Code
Contents List:
1. Introduction
2. Objective of Work
3. Turbulence Modelling
3.1. The Full Differential Stress - Flux Second-Moment Closure
3.2. Differential - to - Algebraic Truncation
4. PHOENICS Settings
5. Computational Details
6. Presentation of Results
7. Discussion of Results
8. Conclusions
9. Literature References
1. Introduction
Thermal buoyancy-driven turbulent flow is interesting from a fundamental point of view,
because the production of turbulence by shear and thermal buoyancy takes place in the same
direction. This flow is also interesting from an applied point of view because it is representative for
many practical heat transfer applications.
On the subject of natural convection in vertical slots and channels many studies have been
published. However, these studies were mainly concerning laminar and transitional flow. Only during
the last decade, experimental data on the turbulent regime of this flow were produced by the group of
Betts [1], [4]. All experiments were carried out for airflow in a thin vertical slot with aspect ratio of
28.6. Besides the distribution of the time-averaged velocity and temperature, also distributions of
variances were measured.
Only a few DNS-studies on this geometry have been carried out. The first has been reported by
Phillips [13] for turbulent flows just above the critical Rayleigh number. DNS-data for higher
Rayleigh-number turbulent flows were published by Boudjemadi [2], Boudjemadi et al. [3], Versteegh
and Nieuwstadt [17], Nieuwstadt and Versteegh [11] and Versteegh [18]. It gave an impulse to
turbulence modeling for this type of flow and improvement was obtained by Boudjemadi et al. [3],
Dol et al. [6], Versteegh [18] and Kenjeres [8].
2. Objective of Work
Application of the second-moment closure, even in its simplest form, to the solution of thermal
buoyancy-driven flows represents still a formidable task. This is not so much because of a large of
differential equations involved, but more because of still too high uncertainties in modeling various
terms in the equations that obscure real physics. It should be kept in mind that in buoyant convection,
irrespective of how large the bulk Rayleigh number may be, the molecular effects will remain
important in a significant portion of the flow domain, requiring the implementation of low-Reynolds
number modifications. This makes the use of standard wall function inapplicable and requires the
integration straight up to the wall, which in turns calls for a very fine numerical mesh. There is,
therefore, much to be gained if the differential model can be truncated to an algebraic form in which
transport equations will be solved only for major scalar quantities, primarily k, , 2 .
Therefore, the application of simpler models (at most at the algebraic level) should be served as
a convenient alternative for handling complex industrial situations. These prospects have motivated
the present work, aimed at verifying the current practice in modeling of the transport equations for the
turbulent heat-flux and temperature-variance and at exploring the limits of simpler turbulence models
at the algebraic level. Results of direct numerical simulation and experimental data of turbulent
natural convection between two differentially heated vertical plates for Ra = 8.227x105 [1, 18] and for
Ra = 5.4x105 [2, 3] have been used to validate the algebraic heat flux model.
3. Turbulence Modelling
3.1. The Full Differential Stress - Flux Second-Moment Closure
The full differential stress / flux second-moment closure for buoyancy-driven flows can be
summarized as following [8]:
D( ui u j )
Dt
 Dij  Pij  Gij  ij   ij
(1)
D( ui )
U i
T

 Di   ui u j
 u j
 gi  2  C1 ui   i
Dt
x j
x j
k
(2)
D( k )
 Dk  P  G  
Dt
(3)
2
D( )

 D    C1 f 1 P  C1 G  C 2 f  2   S 
Dt
k
(4)
D(  2 )
 D  2P  2 
Dt
(5)
2
D(   )
 
 
 
  



 D  C1 P 2  C 3 P
 C 4  2  C5 f 
Dt
k
k


(6)

U j
U i 
Pij  ui u k
 u j uk
,
x k
x k 

(7)



Gij    g i u j  g j ui




ij   C1


ui u j  23 kij  C2 Pij  23 Pij  C3 Gij  23 Gij
k
k
P  ui u j
U i
,
x j
D 
G  gi ui ,
P  u j

(8)
T
x j
(9)
  
 
 C ui u j


x j  x j
xi 
(10)
2


   
2  
S   max  0.83   1  , 0 ,
k  
  


Table 1. Specification of coefficients
C1
C1


5
0.6
0.6
1.44
 
k 3/ 2
,   C xn

(11)
C2
C3
C1
C3
C4
C5
1.92
1.44
1.3
0.72
2.2
0.8
C1
C2
C3
Cs
Ck
C
C 
C
C
f 
1.8
0.6
0.6
0.22
0.09
0.07
0.09
0.09
0.09
1.0
Pij and Gij represent Reynolds stress production by strain and buoyancy, respectively. The
pressure strain term ij is modeled as the sum of slow-term [10], rapid-term (the simple isotropic
model [12, 10]) and buoyancy-pressure term [7]. P and G represent the turbulence energy production
by strain and buoyancy, respectively. P is the production of temperature variance. D denotes the
total general diffusion term, in which the turbulent part is modeled by the simple gradient hypothesis,
2
which the tensorial exchange coefficient kuk ui  replaced by the scalar form k  . For
homogeneous turbulence, the dissipation rate i is negligible (isotropic model). In that case, the
pressure scrambling balances the production. A consistent approach would require that dissipation
rate i be obtained from its differential transport equation. However, in non-homogeneous turbulence
(specially near walls), an appropriate algebraic model can be obtained. This model should have the
correct limiting value at the wall and should reduce to isotropic model for homogeneous turbulence. A
very simple model that satisfied these requirements is given by [14]:

i   a   ui  uk nk ni
 x1
(12)
2
n
3
In addition to using the wall-normal unit vector, this model contains the normal distance to the nearest
wall xn.
3.2. Differential - to - Algebraic Truncation
Presented full differential stress/flux model for three-dimensional flows consists of 17 transport
equations (for velocity components, pressure, components of Reynolds stresses, temperature,
dissipation of turbulent kinetic energy, components of turbulent heat fluxes, temperature variance and
its dissipation rate). Furthermore, in the buoyancy-driven flows major interactions occur very close to
the wall where molecular effects are important and resolving the flow details and heat transfer in nearwall regions requires the low-Re number modification as well as integration up to the wall for which a
very fine numerical grid is needed. Much can be gained if the full differential model can be truncated
to a simpler form in which differential transport equations will be solved only for major scalar
quantities, e.g. for the turbulence kinetic energy, temperature variance and their dissipation rates. This
implies the elimination of the convective and diffusive transport terms in the equations for turbulent
stresses (Eq.1) and turbulent heat fluxes (Eq.2), which are the only terms containing the time and
spatial derivatives of ui u j and ui .
The first step in the analysis of possible simpler mathematical form that can describe
sufficiently accurately defined phenomena is to recognize the dominant mechanisms that determine
the character of the complete problem under consideration. In the present case it is obvious that the
effects of external force in the form of a buoyancy term must be included in the basic equations for
the fluid flow and the heat transfer. Besides this direct influence, the effect of the gravitational field
must be taken into account in the analysis of pressure fluctuations. The effect of the gravitational field
on pressure fluctuations will be expressed in pressure-velocity gradient and pressure-temperature
gradient correlations via additional terms. These two features of buoyancy-driven flows must be taken
into account in any kind of proposed mathematical form. The simplest form of turbulence differential
model can be derived using the Boussinesq isotropic eddy viscosity/diffusivity formulation for
Reynolds stresses and turbulent flux components, knowing as the simple gradient transport models
(SGDH):
 U
U j 
ui u j  23 kij  t  i 
,
xi 
 x j
ui  
t T
,
 xi
t  C f 
k 3/ 2

(13)
As shown, the first two expressions need to be closed by eddy viscosity ( t ). The problem is
that due to its scalar nature, turbulent viscosity will be identical for all stress components as well as
for turbulent heat flux components. It is obvious that such a closure is not appropriate for buoyancydriven flows where the turbulent heat flux components will be basically a function of the orientation
between the gravitational vector and the temperature gradient imposed by boundary conditions. In the
vertical thin-enclosure, where the one wall is heating from the side, gravitational vector is normal to
the temperature gradient. The temperature gradient in the vertical direction will produce a very small
value and consequently, the vertical turbulent heat flux component and the integral Nusselt number
will be heavily underestimated.
The next step towards the improvement of SGDH is obviously to replace the scalar eddy
diffusivity / conductivity formulations with more general form. The direction that we followed is to
truncate the transport differential equation for turbulent heat fluxes (Eq.2) in the appropriate algebraic
form. The development of algebraic models for turbulent heat flux components usually followed the
approach outlined by Rodi [15], for development of an algebraic model for Reynolds stresses. Two
different levels in proposed modeling strategy can be distinguished - reduced and more refined
4
formulation. Assuming that we can neglect the transport entirely, D ui / D t  Di  0, the reduced
algebraic expression of ui can be written in the following form:
ui  

U i
1 k
T
 1  C2  uk
 1  C3  g i 2 
 ui u k
C1  
x k
x k

(14)
The refined formulation can be derived assuming proportionality between transport of turbulent
heat flux and transports of temperature variance and turbulent kinetic energy:
1
D ui
1
 Di  ui  2
Dt
2
 
 D 2
 1 D k


 D   
 Dk 


D
t
k
D
t


(15)
This corresponds also to the assumption of weak equilibrium of the thermal field, defined by the
condition: D(ui / 2 k ) / D t  0 . In the case of a fully developed flow between the two infinite
parallel plates, the convection is zero, so that the hypothesis relates only the diffusive transport,
Di  21 ui (D / 2  Dk / k ) . Replacing the transport of ui
by the modeled right-hand side of the
equation for turbulent heat flux (Eq.2) and the transport of k (Eq.3) and  2 (Eq.5) by their sources,
yields the algebraic expression for the turbulent heat-flux vector:
U i
T
 u k
 g i  2
x k
x k
ui 
 1 

U i

1 
T
 C1  2  2u k
    
 g i ui  
 ui u k
k 2 
x k
x k
 2k 

ui u k
(16)
where   (1  C2 ) and   (1  C3 ). The closure of the algebraic expressions, irrespective of the
modeling level (reduced or more complete), requires that the basic scalar variables, k,  ,  and   be
supplied from separate modeled transport equations (four-equation model). The modeling of the  
equation is very difficult. A further simplification can be achieved by expressing the dissipative ratio
2
of temperature variance, in terms of , k,  and , from the assumed ratio of the thermal to mechanical
turbulence time scales, R    /  m , with either its constant value, or expressed by an algebraic
2
2
function in terms of available variables,    ( / R ) ( / 2 k ). This will reduce the model to a
three-equation model, e.g. k--  . The constant time-scale ratio assumption is valid only when the
turbulence is in local equilibrium. Fortunately, the numerical simulation for buoyancy-driven flows
are not so sensitive to the choice of calculation  , so the thermal to mechanical turbulence time
scales ratio can be taken constant (R = 0.5).
In buoyancy-driven flows, both laminar and turbulent regions may exist in the same flow
domain. The model is required to reproduce a gradual transition from one to another regime, not only
across the molecular wall sublayer, but also at the edge of turbulence zones away from the wall. In the
latter case, the turbulent fluctuations decay freely under the action of molecular forces. This process is
physically different from the turbulence damping in the near-wall region, where the molecular effects
are mixed with the eddy splitting due to the wall blockage and pressure reflection. A consistent
approach would require separate modeling of these effects, as practiced in some recent secondmoment closure models. In the present model, we have followed the low-Re-number modifications of
2
5
the k and  equations appeared to suffice and no direct modifications of the  2 equation were found
necessary. We applied Lam-Bremhorst [9] extension of k- model, with the advantage that the model
requires no additional source terms. The dumping functions, appearing in the equations (Eq.4) and
(Eq.13), are expressed in the following form:
f   1  e
Rek 
0. 0165 Rek

 0.05 
 20.5 
 Re 2
 ; f 2  1  e t
1 
 ; f 1  1  
Ret 

 f 
3
2
k
xn k
; Ret 
lam
lam
(17)
(18)
4. PHOENICS Settings
Besides the embodied low-Re- k- Lam-Bremhorst turbulence model, additional GROUND
coding is needed. Additional source terms G  gi ui that represent energy production by
buoyancy has to be coded, in both equations of k and . Implicit form of algebraic terms for ui
(Eq.14) can be expressed in full explicit form in 2D case; therefore, a simple formulas for thermal
fluxes have to be coded. Finally, a new transport equation of  2 (Eq. 5) must be coded. Fortunately,
this equation is very similar to the equation of turbulent kinetic energy (Eq. 3); therefore, the
appropriate analogy can be applied to resolve this equation.
5. Computational Details
In order to test the accuracy of the proposed algebraic heat-flux (AHF) model, we performed a
simulation of a two-dimensional square thin-enclosure. The left wall is heated and the right wall is
cooled. The configuration is shown on Fig. 1. Since we where only be comparing normalized
quantities, the dimensions and fluid property values can be freely chosen. We took h = 0.5 and T =
Th - Tc = 1, then suitable values should be assigned to, g,  and  to give the required Rayleigh
numbers Ra  gT (2h) 3 / va . The results are normalized using the friction velocity and
temperature, defined by:
 dW 
u   
 ;
 dx  wall
12
*
 
a dT
u  dx
(19)
wall
T /x = 0; W = 0
Th = 1
U=0
W=0
Ra = 5.4 x 105
Ra = 8.227 x 105
Pr = 0.71
Tc = 0
U=0
W=0
2h = 1
z
T /x =0; W = 0
x
Figure 1. The test problem configuration.
6
6. Presentation of Results
Figures 2a-f and 3a-d show dynamics and thermal field profiles for Ra = 8.227x105,
respectively. Also, figures 4a-d show thermal field profiles for Ra = 5.4x105 [3], only, because the
dynamic field shows the similar behavior as the previous one. For comparison with experimental data
we used the measurements performed by Betts and Bokhari [1].
6
W / u*
5
4
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(a)
8
uu / ( u* ) 2
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(b)
10
8
k / ( u* ) 2
6
4
2
0
0.0
0.1
0.2
0.3
x / (2h)
7
0.4
0.5
(c)
8
7
vv / ( u* ) 2
6
5
4
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(d)
2.5
2.0
uw / ( u* ) 2
1.5
1.0
0.5
0.0
-0.5
0.0
0.1
0.2
0.3
x / (2h)
0.4
0.5
(e)
(f)
Figure 2. Dynamic field profiles for Ra = 8.227 x 105 , scaled by friction velocity u * .
(  - Betts and Bokhari, [1];  - Versteegh, [18]);  AHF Model )
8
12
10
T / *
8
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(a)
2.0
* *
u / (  u )
1.5
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(b)
6
5
2
* 2
 /( )
4
3
2
1
0
0.0
0.1
0.2
0.3
x / (2h)
9
0.4
0.5
(c)
5
* *
w / (  u )
4
3
2
1
0
-1
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(d)
Figure 3. Thermal field profiles for Ra = 8.227x10 , scaled by friction velocity u * and
5
temperature  * . (  - Betts and Bokhari, [1];  - Versteegh, [18];  AHF Model )
12
T / *
10
8
6
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(a)
2.0
* *
u / (  u )
1.5
1.0
0.5
0.0
0.0
0.1
0.2
0.3
x / (2h)
10
0.4
0.5
(b)
6
2
* 2
 /( )
5
4
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(c)
5
* *
w / (  u )
4
3
2
1
0
-1
0.0
0.1
0.2
0.3
0.4
0.5
x / (2h)
(d)
Figure 4. Thermal field profiles for Ra = 5.4x10 , scaled by friction velocity u and temperature  * .
(  - Betts and Bokhari, [1];  - Versteegh, [18];  AHF Model )
5
*
7. Discussion of Results
For the average profiles the main discrepancy is found in the average velocity, which lies about
25% higher in the measurements. Also the average temperature is somewhat higher in the
measurements than in the DNS data [18]. The discrepancies in the average velocity and temperature
profiles of DNS and measured data may have been caused by the fact that the experimental enclosure
is finite. Due to the side and top walls, the average flow may have been influenced by threedimensional effects. Given the higher temperature gradient in the center region of the experimental
tank, the mixing properties of turbulence may not be as much developed as in an infinite channel. As a
result, the buoyant force is larger and the momentum exchange is less in the thank. Thus allowing a
higher average velocity in the tank. In the central region, the velocity variance ww , given in Fig. 2f,
lie for the measurements somewhat higher than for the numerical data. Nevertheless, the turbulence
intensities relative to the average velocities are nearly the same. The experimental temperature
variances  2 , given in Fig. 3c, are much lower than in the DNS as well as present calculations.
Furthermore, a near-wall peak that is always found in horizontal and vertical turbulent thermal
boundary layers is only visible in the numerical data. Such a near wall peak in the measurements of
Betts and Bokhari [1] was not found. The reason for that may be that response time of the
11
thermocouple used in the measurements was lower than required. Therefore, it was reasonable that the
DNS data of Versteegh [18] have been used for comparing, because they seem to be the most
accurate.
The figures 2a, 3a, 4a and 2c show that AHF results and the results of Versteegh [18] and
Boudjemadi [3] agree quite well for the average velocity, average temperature and turbulent kinetic
energy profiles, respectively. Discrepancies in Reynolds-stress components (figures 2b, 2e, 2f) have
been expected due to the eddy-viscosity concept of k- turbulence model. The largest shortcoming of
the AHF model is the large deviation of the vertical heat-flux component w from the DNS data (see
Fig.3d and 4d). This is probably caused by the turbulent-diffusion model [6] in the modeled equations
(Eq.10) and the reduced temperature flux model (Eq.14), which are too dissipative. Predictions of
 2 by AHF model (Fig.3c and 4c) show the same deviation in shape, which is caused manly by the
turbulent-diffusion model. The wall-normal heat-flux component u is predicted quite well (see
Fig.3b and 4b).
8. Conclusions
Results of direct numerical simulation (DNS) of the natural-convection flow between
differentially heated infinite vertical plates of Versteegh, [18] and as well as experimental data of
Betts and Bokhari [1] were used for validation of the algebraic heat-flux model for computing
turbulent thermal convection in two-dimensional thin-enclosure. Despite the large differences in the
turbulence parameter-by-parameter comparison, the complete second-moment closure should be
replaced by low-Re- k--  2 turbulence model with the algebraic heat-flux terms (Eq.14).
AHF model reproduces well the mean flow (average velocity and temperature) and major
turbulence property (turbulence kinetic energy) in the considered thin-cavity case. No gain is achieved
by employing the refined, computationally more inconvenient, algebraic expression (Eq.16), as
compared with the reduced expression (Eq.14). However, even the reduced expressions (Eq.14) show
important advantages as compared with simple isotropic eddy-diffusivity model SGDH, (Eq.13),
because the heat-flux includes all major generation terms (mean temperature, and velocity gradients,
and thermal turbulence self-amplification through the temperature variance).
Finally, we can conclude that the minimum level of modeling which can reproduce the major
flow features in more complex geometry is the algebraic model which accounts for all major sources
of turbulent heat flux and thus far expresses the turbulent heat flux vector in terms of the mean
temperature gradient, mean flow deformation and temperature variance interacting with the
gravitational vector.
Also, in the complex geometry of natural convection fluid flow problems, one of the most
convenience turbulence models is LVEL model, proposed by professor D.B. Spalding [19].
9. Literature References
[1]
[2]
[3]
[4]
[5]
Betts, P.L. and Boghari, I.H., (1996), “Experiments on turbulent natural convection of air in
tall cavity”, Proc. of the 5th ERCOFTAC workshop on Refined Flow Modeling, Chatou,
France.
Boudjemadi, R., (1996), “Simulation numerique directe et modelisation de la vonvection
naturalle dans un canal vertical differentiellement chauffe”, Ph.D. Thesis, Universite de
Paris-Sud U.F.R. Scientifique D’Orsay.
Boudjemadi, R., Maupu, V., Laurence, D., Le Quere, P., (1997), “Budgets of turbulent stress
and fluxes in a vertical slot natural convection flow at Rayleigh Ra = 10 5 and 5.4x105”, Int. J.
Heat and Fluid Flow, Vol. 18, pp. 70-79.
D’Afa’alla, A.A., (1988), “Turbulent natural convection in a tall closed cavity”, Ph.D.
Thesis, University of Manchester, UK.
Daley, B.J. and Harlow, F.H., (1970), “Transport equations of turbulence”, Phys. Fluids, Vol.
13, p 2634.
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[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Dol, H.S., Hanjalic, K., and Kenjeres, S., (1997), “A comparative assessment of the secondmoment differential and algebraic models in turbulent natural convection”, Int. J. Heat and
Fluid Flow, Vol. 18, pp. 4-14.
Gibson M.M. and Launder B.E., (1978), “Ground Effects on Pressure Fluctuations in the
Atmospheric Boundary Layer”, J. Fluid Mechanic, Vol. 86, p 491
Kenjeres S., (1999). “Numerical Modeling of Complex Buoyancy-Driven Flows”, Ph.D.
Thesis, Technische Universiteit Delft, The Netherlands.
Lam, C.G.K. and Bremhorst, K., (1981), “A modified form of the k-epsilon model for
predicting wall turbulence”, ASME J. Fluids Eng., Vol.103, p 456.
Launder B.E., Reece, G.J. and Rodi, W., (1975), “Progress in the Development of a Reynolds
Stress Turbulence Closure”, J. Fluid Mechanic, Vol. 68, pp. 537-566.
Nieuwstadt, F.T.M., Versteegh, T.A.M., (1997), “DNS of natural convection between two
vertical, differentially heated walls”, Proc. 11th Turbulent Shear Flows Symposium,
Grenoble, France.
Noat, D., Shavit, A., and Wolfshtein, M., (1970), “Interaction between Components of the
Turbulent Velocity Correlation Tensor due to Pressure Fluctuation”, Israel J. Technology,
Vol. 8, p 259.
Phillips, J.R., (1996), “Direct simulations of turbulent unstratified natural convection in a
vertical slot for Pr = 0.71”, Int. J. Heat and Fluid Flow, Vol. 39, pp. 2485-2494.
Peeters, T.W.J. and Henkes,R.A.W.M., (1992), "The Reynolds-stress model of turbulence
applied to the natural-convection boundary layer along a heated vertical plate", Int. J. Heat
Mass Transfer, Vol. 35, pp. 403-420.
Rodi,W., (1976), "A new algebraic relation for calculating the Reynolds stresses", ZAMM,
Vol. 56, pp. 219-221.
Stevanovic, M.Z., (1996), “A point of view to the standard two-equations turbulence model and
possibilities of its improvement to the fluid flow with low-Re-number”, Proc. 2nd Int.
Symposium - Contemporary Problems of Fluid Mechanics, Belgrade, Yugoslavia.
Versteegh, T.A.M., Nieuwstadt, F.T.M., (1997), “Coherent structures in natural convection
between two vertical, differentially heated walls”, Proc. 2nd Int. Symposium on
Turbulence, Heat and Mass Transfer, Delft, The Netherlands.
Versteegh, T.A.M., (1998), “Numerical simulation of natural convection in a differentially
heated, vertical channel”, Ph.D. Thesis, Printed by Ponsen & Looijen, The Netherlands.
Spalding, D.B., (2002), PHOENICS encyclopedia - TURBULENCE MODELS IN
PHOENICS (www.cham.co.uk).
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