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International Journal of Numerical Methods for Heat & Fluid Flow
Development and validation of an incompressible Navier-Stokes solver including
convective heat transfer
Konstantinos Stokos Socrates Vrahliotis Theodora Pappou Sokrates Tsangaris
Article information:
To cite this document:
Konstantinos Stokos Socrates Vrahliotis Theodora Pappou Sokrates Tsangaris , (2015),"Development
and validation of an incompressible Navier-Stokes solver including convective heat transfer",
International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 25 Iss 4 pp. 861 - 886
Permanent link to this document:
http://dx.doi.org/10.1108/HFF-01-2014-0023
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Development and validation of an
incompressible Navier-Stokes
solver including convective
heat transfer
Konstantinos Stokos and Socrates Vrahliotis
Department of Mechanical Engineering,
National Technical University of Athens, Athens, Greece
Theodora Pappou
FiDES DV-Partner GmbH, Munich, Germany, and
Sokrates Tsangaris
Department of Mechanical Engineering,
National Technical University of Athens, Athens, Greece
Abstract
Purpose –The purpose of this paper is to present a numerical method for the simulation of steady
and unsteady incompressible laminar flows, including convective heat transfer.
Design/methodology/approach –A node centered, finite volume discretization technique is applied
on hybrid meshes. The developed solver, is based on the artificial compressibility approach.
Findings –A sufficient number of representative test cases have been examined for the validation of
this numerical solver. A wide range of the various dimensionless parameters were applied for different
working fluids, in order to estimate the general applicability of our solver. The obtained results agree
well with those published by other researchers. The strongly coupled solution of the governing
equations showed superiority compared to the loosely coupled solution as inviscid effects increase.
Practical implications –Convective heat transfer is dominant in a wide variety of practical
engineering problems, such as cooling of electronic chips, design of heat exchangers and fire
simulation and suspension in tunnels.
Originality/value –A comparison between the strongly coupled solution and the loosely coupled
solution of the Navier-Stokes and energy equations is presented. A robust upwind scheme based on
Roe’s approximate Riemann solver is proposed.
Keywords Heat transfer, Boussinesq approximation, Hybrid mesh, Incompressible flow,
Roe’s approximate Riemann solver, Strongly coupled solution
Paper type Research paper
Nomenclature
x, y, z Cartesian coordinates
Q
!dependent variables
vector
p pressure
u, v, w Cartesian velocity
components
International Journal of Numerical
Methods for Heat & Fluid Flow
Vol. 25 No. 4, 2015
pp. 861-886
© Emerald Group Publishing Limited
0961-5539
DOI 10.1108/HFF-01-2014-0023
Received 30 January 2014
Revised 8 July 2014
Accepted 6 August 2014
The current issue and full text archive of this journal is available on Emerald Insight at:
www.emeraldinsight.com/0961-5539.htm
Most of the computations were performed on the high-performance platform “VELOS”of the
Laboratory of Thermal Turbomachines, Parallel CFD and Optimization Unit, School of Mechanical
Engineering of the National Technical University of Athens. This work was funded under the
EUROSTARS Project (E!5292) “Structural and Aerodynamic Design of TUNnels under Fire
Emergency Conditions”from: the Greek General Secretariat for Research and Technology,
Bundesministerium fur Bildung and Forschung (BMBF), FiDES DV-Partner GmbH, Germany and
ELXIS Engineering Consultants S.A. It was also funded from the Greek State Scholarships Foundation.
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heat transfer
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T dimensionless
temperature
t dimensionless
physical time
L
o
characteristic length
V
o
characteristic
velocity
F
!inv inviscid fluxes vector
F
!vis viscous fluxes
vector
S
!source term vector
b
n¼ðnx;ny;nzÞunit normal vector
τ
x
i
x
j
shear stresses
q
x
i
heat fluxes
k thermal conductivity
C
p
specific heat at
constant pressure
g gravitational
acceleration
_
qcheat release rate per
unit volume
b
tij unit vector along
edge “ij”
‘ij length of edge “ij”
f
vortex shedding
frequency
CFL
Courant-Friedrichs-
Lewy number
a
o
thermal diffusivity
nedge(i) number of edges
connected to node i
Dimensionless numbers
Re ¼V
o
L
o
/v
o
Reynolds number
Pr ¼μ
o
C
p
/k Prandtl number
Gr ¼L3
ogbΤDΤo=n2
z:omicr; Grashof number
S_
q¼Lo
_
qc=VoDΤoroCppower of the heat
source
Ra ¼GrPr Rayleigh number
Ri ¼Gr/Re
2
Richardson number
Str ¼L
o
f/V
o
Strouhal number
Nu Nusselt number
C
D
drag coefficient
C
L
lift coefficient
Greek symbols
τpseudo-time
βartificial
compressibility
parameter
β
Τ
thermal expansion
coefficient
κconstant (2 for 2D, 3
for 3D simulations)
δKronecker’s delta
ρdensity
Ωfinite volume
Θ¼un
x
+vn
y
+wn
z
normal velocity
νkinematic viscosity
ΔΤtemperature
difference
ΔS
j
contribution of edge
j to the control
volume surface
subscripts
d dimensional
parameters
o reference values
used for the
non-
dimensionalization
rms root mean square
values
h hot wall
c cold wall
1. Introduction
Convective heat transfer is dominant in a wide variety of practical engineering
problems, such as cooling of electronic chips , design of heat exchangers and
fire simulation and suspension in tunnels. Nowadays, with the great progress
in computer science, Computational Fluid Dynamics (CFD) has become a
valuable tool for the simulation of such cases (Aliabadi et al., 2014; Xie et al.,
2014; Stokos et al., 2013b).
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The difficulty with the incompressible flows is the coupling of the pressure and
velocity fields during the numerical solution. For this coupling many techniques have
been developed (Lakshminarayana, 1991). The most popular approaches are the
pressure correction method originally introduced by Harlow and Welch (1965) and the
artificial compressibility method originally introduced by Chorin (1967). A thorough
comparison of these two methods has been done by Tamamidis et al. (1996). According
to this study the most attractive feature of the artificial compressibility method is the
faster convergence as continuity and momentum equations are fully coupled.
Today, hybrid meshes are widely used (Anderson et al., 1995; Haselbacher and
Blazek, 2000; Kallinderis and Ahn, 2005; Vrahliotis et al., 2012; Stokos et al., 2012, 2013a).
The hybrid meshes can combine good viscous layer resolving capability obtained from
their structured elements, with the geometric flexibility of unstructured meshes
(Kallinderis and Ahn, 2005). Therefore, they lead to significant savings in memory and
computational time with satisfactory results.
Inviscid fluxes evaluation has received considerable attention in the CFD
community. Developed schemes are distinguished into central and upwind schemes
(Hirsch, 1988). Hybrid schemes combining central and upwind schemes are also found
(Fereidoon et al., 2013). Many works have been found in the literature based on central
differencing schemes (Swanson and Turkel, 1992; Lin et al., 2006; Malan et al., 2002a, b;
Kallinderis and Ahn, 2005; Lin and Sotiropoulos, 1997). They gain in their simplicity
but are susceptible for the odd-even mode decoupling, making artificial dissipation
necessary. The most dominant in the upwind schemes is the scheme that is based on
the Roe’s approximate Riemann solver (Kallinderis and Ahn, 2005; Liu et al., 1998;
Yuan, 2002; Anderson et al., 1995; Shin, 2001; Azhdarzadeh and Razavi, 2008). This
approach is more complicated, but relieves us from artificial dissipation, since upwind
schemes introduce artificial dissipation implicitly.
In this paper we present a numerical solver for the simulation of steady and
unsteady incompressible laminar flows including convective heat transfer. Density
variations due to temperature differences are simulated by means of Boussinesq
approximation. A node centered, finite volume discretization technique is applied.
The developed solver, based on the artificial compressibility approach, utilizes hybrid
meshes containing triangles and quadrilaterals in the 2-D version and hexahedra,
prisms, tetrahedra and pyramids in the 3-D version.
Attention was paid to the strong coupling of the equations and the development of
an efficient upwind scheme for the calculation of the inviscid fluxes. The Navier-Stokes
and energy equations are solved simultaneously leading to the concurrent convergence
of flow and temperature fields. A Roe’s approximate Riemann solver was developed for
the evaluation of the inviscid fluxes. For the discretization of the viscous fluxes a CPU-
time efficient central scheme is used. Temporal accuracy is achieved by an implicit,
dual time stepping scheme. The algorithms for spatial and temporal discretization are
mesh transparent. A sufficient number of benchmark test cases have been solved for
the validation of the present numerical method.
2. Governing equations
The augmented with artificial compressibility incompressible equations in
dimensionless integral form are presented below:
ZO
@Q
!
@tdOþΕZO
@Q
!
@tdOþZOrF
!invdOZOrF
!visdO¼ZO
S
!dO(1)
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Applying the divergence theorem, the volume integrals of inviscid and viscous terms
turn into surface integrals:
ZO
@Q
!
@tdOþΕZO
@Q
!
@tdOþI@O
F
!invUb
ndSI@O
F
!visUb
ndS ¼ZO
S
!dO(2)
where:
Q
!¼
p
u
v
w
T
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
;F
!invUb
n¼
bY
uYþnxp
vYþnyp
wYþnzp
YT
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
;F
!visUb
n
¼
0
txxnxþtxy nyþtxznz
tyxnxþtyy nyþtyz nz
tzxnxþtzy nyþtzznz
qxnxþqynyþqznz
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
;S
!¼
0
0
Gr
Re2Tdk2
Gr
Re2Tdk3
S_
q
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
(3)
Ε¼diag 0;1;1;1ðÞ
txixj¼1
Re
@ui
@xjþ@uj
@xi
;qxi¼1
RePr
@T
@xi
;(4)
i;j¼1;2;3;u1;u2;u3
ðÞ¼u;v;wðÞ;x1;x2;x3
ðÞ¼x;y;zðÞ
This system of non-linear Equations (2) expresses the conservation of mass,
momentum and energy for non-isothermal incompressible laminar flows, of Newtonian
fluids. The viscous dissipation term in the energy equation is neglected, because the
thermal energy due to viscous shear in incompressible flows at low velocities is small
(Malan et al., 2002a).
The dimensionless variables and numbers used are:
t¼tdVo
Lo
;xi¼xd;i
Lo
;p¼pd
roV2
o
;ui¼ud;i
Vo
;T¼TdTo
DTo
Re ¼VoLo=no;Pr ¼moCp=k;Gr ¼L3ogbTDTo=v2o;Sq_¼Lo
_
qc=VoDToroCp(5)
where the subscripts “d”and “o”mean the dimensional dependent variables and the
reference values used for the non-dimensionalization respectively.
The source terms in the “y”and “z”direction momentum equations model fluid
buoyancy due to density differences. It is necessary to include this extra source term,
as incompressible equations cannot predict density differences. Buoyancy forces by
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means of Boussinesq approximation become ( ρ−ρ
o
)g¼ρ
o
gβ
T
(T−T
o
), where β
T
is the
thermal expansion coefficient.
3. Spatial discretization
Integrating the above conservation laws (2) on the finite control volume “Ω
i
”, we have:
@Q
!iOi
@tþΕ
@Q
!iOi
@t¼S
!iOiI@O
F
!invUb
ndSþI@O
F
!visUb
ndS
¼STiFINViþFVISi¼RH Si(6)
The dependent variables vector “Q
!i”is calculated at each node “i”of the grid. Every
node “i”is associated to a median dual volume “Ω
i
”, being its control volume. For the
construction of this dual volume in two dimensions, we connect edge midpoints and
centroids of the cells sharing a common node “i”, and in three dimensions we connect
faces defined by edge midpoints, cell centers and face centers sharing a common
node “i”, as it is shown in Figure 1. An efficient edge-wise algorithm is used for the
computation of the flow fluxes. Edges are visited only once giving their contribution to
the associated nodes. Moreover mesh transparency is achieved.
3.1 Inviscid fluxes
The system of conservative laws is transformed from parabolic to hyperbolic with the
introduction of artificial compressibility. Therefore, numerical treatment similar to
compressible flow equations can be applied. For the calculation of the inviscid fluxes
two upwind schemes were developed.
primary grid
QLQRdual grid
QLQR
ii
iiii
Figure 1.
Dual volumes
in two and three
dimensions
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The inviscid flux term is numerically approximated as:
I@O
F
!invUb
ndS X
nedge iðÞ
j¼1
F
!invUb
njDSj¼FINVi(7)
where “F
!inv
ij
”is the numerical inviscid flux vector evaluated at the mid-point of
edge “ij”,“b
nj”is the control volume boundary surface outward unit normal vector
associated with edge “ij”.
The first alternative for the calculation of the inviscid fluxes is the characteristics’
based method of Roe’s approximate Riemann solver (Roe, 1981). An earlier similar
attempt was made by Azhdarzadeh and Razavi (2008) in two dimensions. Inviscid
fluxes are formed from data on either side of the face as it is presented below:
F
!invUb
n¼1
2F
!inv Q
!L
Ub
nþF
!inv Q
!R
Ub
n
þ1
2AQ
!L;Q
!R;b
n
Q
!LQ
!R
(8)
where “F
!inv Q
!L
”and “F
!inv Q
!R
”are the inviscid flux vectors calculated from the
values of the dependent variables on the left and right side (“Q
!L”and “Q
!R”
respectively) of the control volume surface associated with edge “ij”. The flux Jacobian,
known as Roe’s matrix is defined as A¼TL
jj
T1, where “Λ”is a diagonal matrix
with the eigenvalues of Roe’s matrix, “T”is the eigenvectors matrix of the flux Jacobian
and “T
−1
”is the inverse matrix of “T”. All quantities are evaluated with algebraic
averaging of “Q
!L”and “Q
!R”, in order that the properties of Roe’s matrix are satisfied
(Taylor and Whitfield, 1991). The eigenvalues and eigenvectors of the Jacobian matrix
are presented in the next two paragraphs.
3.1.1 Two-dimensional version. The eigenvalues are: λ
1
¼Θ−c, λ
2
¼Θ+c, λ
3
¼Θ,
λ
4
¼Θ, where c¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Y2þb
q.
Matrices “T”and “T
−1
”are the following:
Τ¼
cl2cl100
cnxþnyj
cnxnyj
ny0
cnynxj
cnyþnxj
nx0
TT01
2
6
6
6
43
7
7
7
5;
Τ1¼
1
2c2
l1nx
2c2
l1ny
2c20
1
2c2
l2nx
2c2
l2ny
2c20
j
c2
c2nyþjYnx
ðÞ
c2c2nxþjYny
ðÞ
c20
T
c2YnxT
c2YnyT
c21
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
;(9)
where φ¼vn
x
−un
y
.
3.1.2 Three-dimensional version. The eigenvalues are: λ
1
¼Θ−c,λ
2
¼Θ+c,λ
3
¼Θ,
λ
4
¼Θ,λ
5
¼Θ, where c¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Y2þb
q.
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Matrices “T”and “T
−1
”are the following:
Τ¼
cl2cl1000
cnxþnyjzþnzjy
cnxnyjznzjy
a1a20
cnynxjzþnzjx
cnyþnxjznzjx
b1b20
cnznxjynyjx
cnzþnxjyþnyjx
c1c20
TT001
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
;
Τ1¼
1
2c2
l1nx
2c2
l1ny
2c2
l1nz
2c20
1
2c2
l2nx
2c2
l2ny
2c2
l2nz
2c20
s1
2c2s3
2c2s5
2c2s7
2c20
s2
2c2s4
2c2s6
2c2s8
2c20
T
c2TYnx
c2TYny
c2TYnz
c21
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
;(10)
where:
jx¼wnyvnz;jy¼wnxunz;jz¼vnxuny;
X
!1X
!2¼a1;b1;c1
ðÞa2;b2;c2
ðÞ¼
b
n¼nx;ny;nz
and:
s1¼2a2jxn2
yþa2jxn2
zb2jyn2
xb2jyn2
zþc2jzn2
xþc2jzn2
yþa2jynxnya2jznxnz
b2jznynzb2jxnxnyþc2jynynzc2jxnxnz;
s2¼2a1jxn2
yþa1jxn2
zb1jyn2
xb1jyn2
zþc1jzn2
xþc1jzn2
yþa1jynxnya1jznxnz
b1jznynzb1jxnxnyþc1jynynzc1jxnxnz;
s3¼cc2l1nycc2l2nyþb2jyl1nxþb2jyl2nxþb2jxl1nyþb2jxl2nyc2jzl1nx
c2jzl2nxþc2jxl1nzþc2jxl2nzb2cl1nzþb2cl2nz;
s4¼cc1l1nycc1l2nyþb1jyl1nxþb1jyl2nxþb1jxl1nyþb1jxl2ny
c1jzl1nxc1jzl2nxþc1jxl1nzþc1jxl2nzb1cl1nzþb1cl2nz;
s5¼cc2l1nxcc2l2nxþa2jyl1nxþa2jyl2nxþa2jxl1nyþa2jxl2ny
þc2jzl1nyþc2jzl2nyþc2jyl1nzþc2jyl2nza2cl1nzþa2cl2nz;
s6¼cc1l1nxcc1l2nxþa1jyl1nxþa1jyl2nxþa1jxl1nyþa1jxl2nyþc1jzl1ny
þc1jzl2nyþc1jyl1nzþc1jyl2nza1cl1nzþa1cl2nz;
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s7¼a2jzl1nxþa2jzl2nxa2jxl1nza2jxl2nzþb2jzl1nyþb2jzl2ny
þb2jyl1nzþb2jyl2nza2cl1nyþa2cl2nyþb2cl1nxb2cl2nx;
s8¼a1jzl1nxþa1jzl2nxa1jxl1nza1jxl2nzþb1jzl1nyþb1jzl2ny
þb1jyl1nzþb1jyl2nza1cl1nyþa1cl2nyþb1cl1nxb1cl2nx
Vectors “X
!1”and “X
!2”are tangential to the surface where the flux is calculated,
as can be seen from their definition. Vector “X
!1”is selected; we find the largest in
magnitude component of normal vector “b
n”and we set equal to zero the next in cyclic
order component of vector “X
!1”. For the calculation of the other two components we
know that the scalar product of vector “X
!1”with the normal vector “b
n”is equal to zero
and we impose that the length of “X
!1”is equal to one. Vector “X
!2”is calculated by
using the cross product properties as: X
!2¼b
nX
!1.
The second inviscid fluxes upwind scheme (IFUS) that was developed is presented
below:
X
nedge iðÞ
j¼1
F
!inv
ij
Ub
njDSjX
nedge iðÞ
j¼1
Vþ
ne Q
!LþV
ne Q
!R
ijDSj(11)
where:
Vþ
ne ¼max Vne;0ðÞ;V
ne ¼min Vne;0ðÞ
Vne ¼uenx
ðÞ
eþveny
eþwenz
ðÞ
e(12)
ue¼uLþuR
ðÞ
2;ve¼vLþvR
ðÞ
2;we¼wLþwR
ðÞ
2
A third-order approach, proposed by Tai et al. (2005) was applied for the reconstruction of
the dependent variables vectors “Q
!L”and “Q
!R”for both upwind schemes:
Q
!L¼Q
!iþ1
21kðÞij
!UrQ
!iþkDþ
i
!
and Q
!R
¼Q
!j1
21kðÞij
!UrQ
!jþkD
j
!
hi (13)
where:
Dþ
i
!¼D
i
!¼Q
!jQ
!i;k¼1
3;(14)
For the test cases presented in the fifth section Roe’s approximate Riemann
solver was only used, apart from the case of the differentially heated square cavity in
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Section 5.2.1, where a comparison of the proposed schemes and coupling methods
is conducted.
3.2 Viscous fluxes
The viscous flux term is numerically approximated as:
I@O
F
!visUb
ndS X
nedge iðÞ
j¼1
F
!vis
ij
Ub
njDSj¼VISi(15)
where “F
!vis
ij
”is the numerical viscous flux evaluated at the midpoint of edge “ij”.
As can be seen from Equations (3) and (4), for the calculation of the viscous fluxes
through the control volume boundary it is necessary to calculate the gradients of the
velocities and temperature at the mid-point of every edge. A CPU-time efficient and
mesh transparent method is used for the calculation of these gradients. This method
was proposed by Vrahliotis et al. (2012) for the Navier-Stokes equations and is
implemented for the energy equation successfully. Cell gradients are first evaluated
using Green’s theorem. A face-wise loop is performed for the calculation of the
contribution of every face to the surface integral and this contribution is distributed
with the appropriate sign to the cells that share that face. After this face-wise loop
every cell has gathered the sum of the surface integrals from the faces that it is
consisted of. By dividing this sum with the volume of the corresponding cell we find the
gradients of the variables for that particular cell:
rF
ðÞ
cell ¼1
Vcell I@Vcell
Fb
ndS (16)
Afterwards, a cell-wise loop is performed for the calculation of nodal gradients via
volume averaging (the cell to nodes information is needed):
rFðÞ
i¼Pncell ðiÞ
k¼1VkrFðÞ
k
Pncell ðiÞ
k¼1Vk
(17)
Having found the nodal gradients it is easy to compute the gradients at edge midpoints
with the formula proposed by Weiss et al. (1997):
rFðÞ
ij ¼ rF
ij rF
ij
U
b
tij@F
@‘
ij
"#
U
b
tij (18)
where:
rF
ij ¼1
2rFðÞ
iþrFðÞ
j
;@F
@‘
ij ¼FjFi
‘ij
;(19)
“
b
tij”is the unit vector along edge “ij”,Φ¼u,v,w,T and “‘ij ”denotes the length of
edge “ij”.
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4. Temporal discretization
For the temporal discretization a dual-time stepping scheme is used, which constitutes
a fully implicit time integration procedure. Specifically, for the physical time marching
we apply an implicit second order backward difference scheme and for the pseudo-time
marching an implicit first order backward Euler scheme. This way we avoid the severe
CFL number restrictions associated with explicit time integration schemes. After the
spatial and temporal discretization and applying Newton’s method for the necessary
linearization, Equation (6) becomes:
A
½
nþ1;kdq
!¼b
!nþ1;k
(20)
where:
A
½
nþ1;k¼Oi
DtiþE3Oi
2Dt@RHSnþ1;k
i
@q(21)
b
!nþ1;k
¼RHSnþ1;k
iOi
qk
iQnþ1;m
i
Dti
!
þEOi3qk
iþ4Qn
iQn1
i
2Dt(22)
dq
!¼qkþ1qk(23)
“dq
!”is the variation vector of the dependent variables between two successive
Newton iterations “k”and “k+1”.
To pass from the pseudo-time step “m”to the next pseudo-time step “m+1”, one or
two Newton iterations are applied. To pass from one physical time step to another,
either a predefined number of pseudo-time steps is completed or convergence in
pseudo-time is achieved. The calculation of the local pseudo-time step for every node is
the same with the one applied by Kallinderis and Ahn (2005).
5. Numerical method validation
In this section we present some representative test cases solved for the validation of
the developed solver. These cases are the steady flow in a differentially heated cubic
cavity, the steady flow in a differentially heated square cavity, the steady flow in an
internally heated square cavity and the unsteady flow past a square cylinder, under the
influence of aiding and opposing buoyancy. For all test cases mesh independency of
our solution was tested.
For the prediction of steady flows two approaches were applied. In the first
approach we set an extremely large physical-time step (physical-time term in
Equation (21) becomes negligible), CFL number between 0.1 and 100 and many
pseudo-time steps in order to achieve convergence of all dependent variables in the
first physical-time step. In the second approach we set a smaller physical-time
step (10
−2
is a typical value), CFL number between 1 and 100 and approximately 100
pseudo-time steps for each physical-time step. After some physical-time steps
convergence for all dependent variables is achieved. The value of 1 was chosen for the
artificial compressibility parameter “β”for all simulations.
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5.1 Differentially heated cubic cavity
The case of the differentially heated cubic cavity is a widely investigated case (Tric
et al., 2000; Lo and Leu, 2009; Ravnik et al., 2008). A closed cube of size W (Figure 2) is
assumed, with a fluid subjected to a temperature difference between two opposite walls
and four adiabatic walls. The fluid near the hot wall is heated, and due to density
difference, goes up, while the reverse phenomenon takes place near the cold wall, with
the fluid going down.
Specifically, temperature T
h
¼0.5 is prescribed at x ¼1 (hot wall) and T
c
¼−0.5 at
x¼0 (cold wall). Air is the working fluid with Prandtl number equal to Pr ¼0.71. The
reference value used for the non-dimensionalization of the velocity is Vo¼ao=W,
where ao¼k=roCpis the thermal diffusivity of air. Thus Reynolds number is
Re ¼1=Pr . There is not any heat source or sink in the cube (S_
q¼0). Grashof number
is calculated for the Rayleigh values of 10
3
,10
4
and 10
5
(Gr ¼Ra/Pr).
The three Rayleigh varied cases were run successively for increasing Rayleigh
number. For every case the converged flow field of the previous case was used as initial
condition, except for the first case (Ra ¼10
3
) where all variables were set equal to zero.
The finer numerical hybrid mesh (for which solution was independent from the
mesh) consisted of 357,911 nodes and 343,000 cells. For the prismatic region of the
hybrid meshes used, the first layer thickness was equal to 0.01 and the growing factor
equal to 1.2. The prismatic region was extended from the wall about 10 percent of the
size of the cube.
Figure 3 presents temperature profiles and Figure 4 velocities profiles. The
comparison with the results of the other researchers is satisfactory. Global Nusselt
number (Nu ¼R1
0R1
0
@T
@xdydz) values at the hot wall and maxima values of velocities
are presented at Tables I and II respectively. Nusselt number expresses the heat
flux through a wall. Differences between Nusselt numbers at hot and cold walls
were approximately equal to 10
−7
percent for all Rayleigh varied cases, indicating
conservation of energy.
5.2 Differentially heated square cavity
The case of the differentially heated square cavity is a widely studied case (de Vahl
Davis, 1983; Barakos and Mitsoulis, 1994; Markatos and Pericleous, 1984; Fusegi et al.,
Z
Y
X
Z
Y
X
Figure 2.
Sketch of the cubic
cavity (left) and
a numerical
mesh (right)
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0.50 0.50
0.25
0.00
–0.25
–0.500.00 0.25 0.50 0.75 1.00
x0.00 0.25 0.50 0.75 1.00
x0.00 0.25 0.50 0.75 1.00
x
T
0.25
0.00
–0.25
–0.50
T
0.50
0.25
0.00
–0.25
–0.50
T
Notes: Comparison of temperature profiles on y=0.5 and z=0.5, Ra =103 (left), Ra =104 (middle), Ra =105 (right) (o: Ravnik et al.
(2008); continuous line: present solver)
Figure 3.
Differentially heated
cubic cavity
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1.00
0.75
0.50
0.25
0.00–30 –20
30
20
10
w
–10
–20
–30 0 0.25 0.5 0.75 1
x0 0.25 0.5 0.75 1
x0 0.25 0.5 0.75 1
x
0
30 80
70
60
50
40
30
20
10
0
–10
–20
–30
–40
–50
–60
–70
–80
20
10
ww
–10
–20
–30
0
–10 u
0 102030 –30 –20 –10 u
0102030 –30–40–50 –20 –10 u
01020304050
z
1.00
0.75
0.50
0.25
0.00
z
1.00
0.75
0.50
0.25
0.00
z
Notes: Comparison of velocities profiles. u velocity profile on x=0.5 and y=0.5 (top row). w velocity profile on y= 0.5 and z=0.5
(bottom row). Ra=103 (left), Ra =104 (middle), Ra =105 (right) (o: Ravnik et al. (2008); continuous line: Present solver)
Figure 4.
Differentially heated
cubic cavity
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1992). A 2D square cavity of size H is assumed, with a fluid subjected to a temperature
difference at the two vertical walls and the horizontal walls being adiabatic (Figure 5).
Density differences, produced by temperature differences, results in a recirculating
flow in the interior of the cavity.
More specifically, temperature T
h
¼1 is prescribed at x ¼0 (hot wall) and T
c
¼0at
x¼1 (cold wall). At all walls no-slip boundary conditions are applied (u ¼v¼0). Air is
the working fluid with Prandtl number equal to Pr ¼0.71 . The reference value used for
the non-dimensionalization of the velocity is Vo¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gbΤDΤoH
p. Thus Reynolds number
is Re ¼ffiffiffiffiffiffi
Gr
p. There is not any heat source or sink in the cube (S_
q¼0). Rayleigh
number values are equal to 10
3
,10
4
,10
5
and 10
6
.
The four Rayleigh number varied cases were run successively for increasing
Rayleigh number. For faster convergence the converged flow field of the previous case
Ra ¼10
3
Ra ¼10
4
Ra ¼10
5
Velocities
Tric
et al.
(2000)
Lo and
Leu
(2009)
Present
solver
Tric
et al.
(2000)
Lo and
Leu
(2009)
Present
solver
Tric
et al.
(2000)
Lo and
Leu
(2009)
Present
solver
u
max
3.5435 3.5227 3.5255 16.7198 16.5312 16.7108 43.9037 43.6877 43.7705
v
max
0.1733 0.1726 0.1694 2.15657 2.1092 2.1143 9.6973 9.3720 9.4812
w
max
3.5446 3.5163 3.5312 18.9835 18.6971 18.8822 71.0680 70.6267 71.2915
Table I.
Maxima values of
velocities compared
to other researchers
(differentially heated
cubic cavity)
Ra Tric et al. (2000) Lo and Leu (2009) Ravnik et al. (2008) Present solver
10
3
1.0700 1.0710 1.0713 1.0713
10
4
2.0542 2.0537 2.0591 2.0659
10
5
4.3370 4.3329 4.3570 4.3932
Table II.
Nusselt number
values compared to
other researchers
results (differentially
heated cubic cavity)
1
0.8
0.6
Y
0.4
0.2
00 0.2 0.4 X0.6 0.8 1
Notes: Typical hybrid numerical mesh
Figure 5.
Differentially heated
square cavity
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was used as initial condition of the next one, except for the first case (Ra ¼10
3
) where
velocities and pressure were equal to zero and temperature equal to 0.5.
The finer numerical mesh consisted of 7,626 nodes and 10,962 cells. For its
structured region, consisted of quadrilaterals, the first layer thickness was equal to
0.001 and the growing factor equal to 1.2. The prismatic region was extended from the
wall 5 to 10 percent of the size of the square cavity.
Our results are presented and compared to those presented by other researchers.
Figure 6 presents temperature and velocities profiles. Average Nusselt number
1.00
0.75
T0.50
0.25
0.00
1.00
0.75
T0.50
0.25
0.00
1.00
0.75
T0.50
0.25
0.00
1.00
0.75
T0.50
0.25
0.00
1.00
1.00
0.75
0.75
0.50
0.50
yv
0.25
0.25
0.00
1.00
0.75
0.50
y
0.25
0.00
1.00
0.75
0.50
y
0.25
0.00
1.00
0.75
0.50
y
0.25
0.00
0.000.00 0.25 0.50
x0.75 1.00
0.00 0.25 0.50
x0.75 1.00
0.00 0.25 0.50
x0.75 1.00
0.00 0.25 0.50
x0.75 1.00
–0.40 –0.40
–0.20
–0.20
0.00
0.00
u x
1.000.750.500.250.00 x
1.000.750.500.250.00 x
1.000.750.500.250.00 x
0.20
0.20
0.40
–0.40 –0.20 0.00
u0.20 0.40
–0.40 –0.20 0.00
u0.20 0.40
–0.40 –0.20 0.00
u0.20 0.40
0.40
v
–0.40
–0.20
0.00
0.20
0.40
v
–0.40
–0.20
0.00
0.20
0.40
v
–0.40
–0.20
0.00
0.20
0.40
Notes: Comparison of temperature and velocities profiles. Temperature profile (left). u
velocity profile on x=0.5 (middle). v velocity profile on y=0.5 (right). Ra=103 (first row),
Ra=104 (second row), Ra=105 (third row) , Ra=106 (fourth row) (o: Barakos and Mitsoulis
(1994); continuous line: present solver)
Figure 6.
Differentially heated
square cavity
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Nu ¼R1
0ð@T=@xÞdy
values at the hot wall and maxima values of velocities are
presented at Tables III and IV, respectively. Our results match very well with the results
of the other researchers. Figure 7 shows velocity vectors for the cases of Ra ¼10
3
and
Ra ¼10
6
. A huge stagnation region exists at the center of the cavity for the Ra ¼10
6
case.
Generally, stagnation region increases with increasing Rayleigh number and a highly
convective region is created near side walls. The satisfactory simulation
of this case shows the general applicability of our solver, as highly diffusive and
highly convective regions coexist in the same cavity.
5.2.1 Coupling methods comparison. In this subsection we present a comparison
of the convergence behavior of both coupling methods. When Roe scheme is used
equations are strongly coupled (S-C) (simultaneous solution of Navier-Stokes and
Ra
Markatos and Pericleous
(1984)
de Vahl Davis
(1983)
Fusegi et al.
(1992)
Barakos and Mitsoulis
(1994)
Present
solver
10
3
1.108 1.118 1.105 1.114 1.1175
10
4
2.201 2.243 2.302 2.245 2.2459
10
5
4.430 4.519 4.646 4.510 4.5085
10
6
8.754 8.799 9.012 8.806 8.8194
Table III.
Nusselt number
values compared to
other researchers
results (differentially
heated square
cavity)
Ra ¼10
3
Ra ¼10
4
Ra ¼10
5
Ra ¼10
6
Authors u
max
v
max
u
max
v
max
u
max
v
max
u
max
v
max
de Vahl Davis (1983) 0.136 0.138 0.192 0.234 0.153 0.261 0.079 0.262
Fusegi et al. (1992) 0.132 0.131 0.201 0.225 0.147 0.247 0.084 0.259
Barakos and Mitsoulis (1994) 0.153 0.155 0.193 0.234 0.132 0.258 0.077 0.262
Present solver 0.137 0.139 0.192 0.233 0.130 0.256 0.076 0.262
Table IV.
Maxima values of
“u”and “v”velocities
on the vertical and
horizontal mid-
planes respectively
(differentially heated
square cavity)
1
0.8
0.6
Y
0.4
0.2
0
1
0.8
0.6
Y
0.4
0.2
0
0 0.2 0.4 X0.6 0.8 1 0 0.2 0.4 X0.6 0.8 1
Notes: Velocity vectors for the cases of Ra= 103 (left) and Ra= 106 (right)
Figure 7.
Differentially heated
square cavity
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energy equations), while when IFUS scheme is used equations are loosely coupled (L-C).
For the L-C method the flow equations are first solved with the temperature field fixed.
Then, the energy equation is solved with the velocity field fixed. This procedure is
repeated until convergence in pseudo-time is achieved.
The two aforementioned methods were compared at the case of the differentially
heated square cavity and for “Ra”number ranging from 10
3
to 10
6
. This range is chosen
in order to evaluate the performance of both methods as inviscid fluxes increase. All
dependent variables were set equal to zero as initial condition field for each simulation.
Runs were conducted as being steady, setting an extremely large physical time step
and CFL number was equal to 100 from the beginning till the end of the simulations.
Figure 8 shows the convergence historiesof both methods for the three different Rayleigh
cases. For the case of Ra ¼10
3
both methods produce approximately equal convergence
rates. However, as inviscid fluxes dominate over viscous fluxes (with increasing Rayleigh
0
–5
–10
log10(δp)
–15
S-C (Ra=103)
L-C (Ra=103)
S-C (Ra=103)
L-C (Ra=103)
S-C (Ra=103)
L-C (Ra=103)
S-C (Ra=103)
L-C (Ra=103)
S-C (Ra=105)
L-C (Ra=105)
S-C (Ra=105)
L-C (Ra=105)
S-C (Ra=106)
L-C (Ra=106)
S-C (Ra=106)
L-C (Ra=106)
S-C (Ra=105)
L-C (Ra=105)
S-C (Ra=106)
L-C (Ra=106)
S-C (Ra=105)
L-C (Ra=105)
S-C (Ra=106)
L-C (Ra=106)
–20
0
–5
–10
log10(δu)
–15
–20
0
–5
–10
log10(δv)
–15
–20
0
–5
–10
log10(δT)
–15
–20
0
–5
–10
log10(δv)
–15
–20
0
–5
–10
log10(δT)
–15
–20
0
–5
–10
log10(δu)
–15
–20
0
–5
–10
log10(δp)
–15
–20
0
–5
–10
log10(δp)
–15
–20
0
–5
–10
log10(δu)
–15
–20
0
–5
–10
log10(δv)
–15
–20
0
–5
–10
log10(δT)
–15
–20
01,000 2,000 3,000
iterations 4,000 5,000
01,000 2,000 3,000
iterations
4,000 5,000
0 1,000 2,000 3,000
iterations 4,000 5,000
0 1,000 2,000 3,000
iterations 4,000 5,000
01,000 2,000 3,000
iterations 4,000
0 1,000 2,000 3,000
iterations
4,000
01,000 2,000 3,000
iterations 4,000
0 1,000 2,000 3,000
iterations 4,000
0 2,000 4,000 6,000
iterations
8,000
0 2,000 4,000 6,000
iterations 8,000
0 2,000 4,000 6,000
iterations 8,000
0 2,000 4,000 6,000
iterations 8,000
Notes: Convergence histories for the cases of Ra=103 (left), Ra=105 (middle) and Ra=106
(right) through pseudo-time
Figure 8.
Differentially heated
square cavity
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number) the S-C method requires less pseudo-time steps compared to the L-C one. This
benefit is attributed to the fact that all dependent variables of the inviscid fluxes are
“alive”in the S-C approach and not fixed as it happens in the L-C one. For the converged
flow field, differences between the maximum and minimum u and v velocities, predicted
by the two methods were less than 0.3 percent.
The L-C method requires lower numerical effort and hence less CPU time. This
is because of the less numerically demanding calculation of the convective terms
and Jacobi iterations. Specifically, for each node in each Jacobi iteration a system
of 4 equations needs to be solved, when the S-C method in two dimensions is
implemented. On the contrary when the L-C method is implemented a system of three
equations needs to be solved followed by the solution of one equation. Since it takes
longer to solve a 4 ×4 system than a 3 ×3 system followed by a 1 ×1 system, the CPU
time for each Jacobi iteration using the S-C method is augmented. After some numerical
experiments with the serial version of the code (it is difficult to assess the time needed
for message passing), the CPU time per pseudo-time step for the strongly coupled
method and the loosely coupled method were approximately determined as 0.057 and
0.035 s respectively. Runs were conducted on an Intel Xeon E5530 processor at 2.4 GHz.
Table V shows the total CPU time needed for each simulation. The convergence
criterion was that corrections between two successive pseudo-time steps of all
dependent variables reach machine zero. For the case of Ra ¼10
6
the S-C method gives
a speed-up approximately equal to 1.255 compared to the L-C method.
Apart from the CPU time, we compared the coupling methods as to the CFL
numbers and the density of the numerical mesh needed for their convergence. The
strongly coupled method can lead to convergence for higher CFL numbers and coarser
numerical meshes.
5.3 Internally heated cavity
The case of natural convection in a cavity due to uniform heat generation throughout
the fluid was also studied. A 2D square cavity of size H internally heated is assumed,
leading to a circulating flow of the fluid.
No slip boundary conditions are applied at all walls which are maintained
isothermal with the temperature equal to zero (T
c
¼0). Water is the working fluid with
Pr ¼7. The reference value used for the non-dimensionalization of the velocity is
Vo¼n=Η, where vis the kinematic viscosity of the fluid (thus Re ¼1). The reference
value used for the non-dimensionalization of the temperature is DTo¼_
qcH2=8k. The
simulation performed corresponds to Ra ¼8.10
4
. The dimensionless heat source is
S_
q¼8=Pr. As initial conditions all variables were set equal to zero.
Ra L-C S-C
10
3
106.75 s 174.99 s
10
5
78.75 s 109.44 s
10
6
211.75 s 168.72 s
Table V.
Total CPU time
needed in seconds
for each simulation
using the loosely
coupled (L-C) and
strongly coupled
(S-C) methods
(differentially heated
square cavity)
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For this case six structured, uniform numerical meshes were used, in order to achieve
independent results from the numerical mesh (Figure 9). Figure 10 presents
temperature on some points of the symmetry axis x ¼0.5, compared with that of
Deshmukh et al. (2011). Isolines of temperature, inside the square cavity are also given.
5.4 Unsteady mixed convection past a square cylinder
To evaluate the transient capabilities of the proposed method, the 2D case of mixed
convection past a square cylinder is simulated. This problem is challenging as it
contains a stagnation point in front of the cylinder and a recirculation region in the
wake. Sharma and Eswaran have published a series of studies on heat transfer across a
0.4
11×11
21×21 51×51
71×71 101×101
121×121
0.3
0.2
T
0.1
00 0.2 0.4 0.6 0.8
y1
Notes: Independence of solution from the
numerical mesh
Figure 9.
Internally heated
cavity
0.4 0.0416 0.0828
0.33
0.33
0.2064
0.1652
0.124
0.124
0.0828
0.0416
0.2888
0.2888
0.1652 0.124 0.0828 0.0416
0.2064
0.2476
0.0416
0.0828
0.2064
0.124
0.1652
0.2476
0.3
0.2
T
0.1
0
0.00 0.25 0.50 0.75 1.00
y
Notes: Comparison of temperature profile along the symmetry axis (left) (o: Deshmukh et al.
(2011); )
)
Present solver). Isolines of temperature (right)
Figure 10.
Internally heated
cavity
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square cylinder for unconfined and channel-confined flows either in the vertical or in
the horizontal direction (Sharma and Eswaran, 2004a, b, 2005a, b).
In our case the square cylinder is heated or cooled to a constant temperature T
h
¼1
and is exposed to a constant and uniform free-stream upward velocity (u ¼0, v ¼1)
and temperature (T
c
¼0). Free-slip boundary conditions are implemented on left and
right computational boundaries. Pressure is constant at the outlet. This flow is
periodical with the Strouhal number “St”being dependent on the Richardson number
“Ri”, a dimensionless number that provides a measure of the influence of free
convection compared to forced convection.
The governing flow parameters are: Re ¼100, Pr ¼0.7, S_
q¼0. Richardson number
is defined as: Ri ¼Gr=Re2, from which Gr number is calculated. CFL number was
equal to 100 for all simulations conducted.
The first studied case was the case of Ri ¼0. As initial conditions for all other cases
the results predicted from the solved case with the closer “Ri”number was used.
For the simulation of the flow three different numerical meshes were used, in order
to find a solution independent from the mesh. The finer numerical mesh consisted of
36,395 nodes and 71,524 cells. For its prismatic region, consisted of quadrilaterals,
the first layer thickness was equal to 0.001 and the growing factor equal to 1.2. The
prismatic region was extended from the wall about 20 percent of the size of the square
cylinder. Each edge of the cylinder was divided in 40 boundary edges. Figure 11
presents a typical mesh that was used and the mesh in the vicinity of the cylinder,
where the prismatic region is obvious. The numerical mesh was denser in the region
around the cylinder that extended three units upstream and sideways and till the outlet
of the computational domain.
Strouhal number for Reynolds number ranging from 70 to 150 and without taking
into account buoyancy effects (Ri ¼0) is presented in Figure 12, compared to the
numerical results of Robichaux et al. (1999) and Sharma and Eswaran (2004b).
Figure 12 also shows Strouhal number (Str ¼L
o
f/V
o
, where “f”is the vortex
shedding frequency) for various Richardson numbers ranging from −1 to 0.1,
Notes: Typical mesh that was used (left). Mesh near the cylinder (right)
Figure 11.
Mixed convection
past a square
cylinder
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compared to results of Sharma and Eswaran (2004b). Differences are less than
5 percent. It is evident that increasing buoyancy leads to an increase in the shedding
frequency. For Ri ¼0.15 breakdown of vortex shedding was observed. Generally, for
Ri ⩾0.15 the flow turns to steady as it happens and for the circular cylinder (Chang and
Sa, 1990). Increasing buoyancy leads to an increase in the fluid’s velocity in the wake
region. For Ri ¼0.15 this velocity becomes big enough to stop the vortex shedding.
In Figure 13 streamlines for various “Ri”are presented. It is obvious that cooling
the cylinder, which means decreasing Richardson number, the wake region increases
0.17 0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.16
0.15
Str
Str
0.14
0.13
0.1270 90 110
Re 130 150 –1.00 –0.80 –0.60 –0.40
Ri –0.20 0.00 0.20
Notes: Comparison of “Str” for various “Re”, when Ri=0 (left) (♦:Robichaux et al. (1999);
Sharma and Eswaran (2004b); • Present solver). Comparison of “Str” for various “Ri”
(right) (o: Sharma and Eswaran et al. (2004b); )
)
Present solver)
Figure 12.
Mixed convection
past a square
cylinder
Ri=–1
Ri=–0.15 Ri=0 Ri=0.1 Ri=0.15
Ri=–0.75 Ri=–0.5 Ri=–0.25
Notes: Streamlines for various “Ri” numbers
Figure 13.
Mixed convection
past a square
cylinder
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in width. Heating the cylinder the wake region decreases and vortices gradually
fade away. For Ri ¼0.15 vortex shedding is vanished. Similar conclusions were
conducted by Sharma and Eswaran (2004b). In Figure 14 variation of coefficient
of total drag “C
D
”and rms (root mean square) values of the fluctuations of drag and
lift coefficients with Richardson number at Re ¼100, are given.
6. Conclusions
A numerical method for the simulation of 2D and 3D steady and unsteady,
incompressible laminar flows including convective heat transfer, was presented.
Three natural convection and a mixed convection heat transfer problems were
simulated. A wide range of input parameters producing highly diffusive and/or
highly convective cases, were applied. The computed results are quite promising. The
proposed upwind scheme, based on Roe’s approximate Riemann solver, showed
accuracy and robustness.
2
1.5
1
0.5
0
0.2
0.15
0.1
0.05
0
–1 –0.5 0
Ri 0.5 1 –1 –0.5 0
Ri 0.5 1
CLrms
CDrms
2.5
2
1.5
1–1 –0.5 0 0.5 1
Ri
CD
Notes: Variation of coefficient of total drag “CD” and rms values of the drag and lift
coefficients fluctuations with Richardson number at Re=100 (o: Sharma and Eswaran
(2004b); Present solver)
)
)
Figure 14.
Mixed convection
past a square
cylinder
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A comparison of loosely and strongly coupled methods was conducted, for the case of
the differentially heated square cavity and for a wide range of Ra number. Comparison
showed superiority of the strongly coupled method as inviscid effects dominate over
viscous effects with the increase of the Ra number.
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About the authors
Konstantinos Stokos, Mechanical Engineer MScEng., currently a PhD student in the National
Technical University of Athens. Stokos research and working areas are Heat Transfer (finite
volume methods development for convective heat transfer, finite element methods for heat
conduction, finite volume and analytical methods for radiative heat transfer). Computational
Fluid Dynamics in Biofluids (development of 1-D hyperbolic type methods for compressible and
incompressible flows, applications in cerebral circulatory system and circulatory system
in general). Distinctions: funded PhD studies from the Greek State Scholarships Foundation.
Stokos publications include five presentations in International conferences. Konstantinos Stokos
is the corresponding author and can be contacted at: kstokos@mail.ntua.gr
Socrates Vrahliotis, Mechanical Engineer PhD, currently a Software Engineer in Sofistik
Hellas S.A. Company. Research and working areas are Computational Fluid Dynamics
(Aerodynamics, fluid-structure interaction, mesh-generation). Socrates has one publication in
scientific journals and three presentations in International conferences.
Dr Theodora Pappou, Mechanical Engineer PhD, currently a Research Engineer in FiDES
DV-Partner GmbH Company. Research and working areas are Computational Fluid Dynamics
in Aerodynamics (upwind schemes, unsteady flows, moving boundaries, aeroelasticity effects).
Computational Fluid Dynamics in Biofluids (unsteady flows, moving boundaries).
Adjoint Techniques in Optimization and Inverse Design Problems in aeronautics.
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Mesh generation algorithms. Finite Element methods for the solution of structural problems.
Fluid-Structure interaction. Theodora Pappou has six publications in scientific journals and more
than 20 presentations in International conferences.
Sokrates Tsangaris, Dr Techn. currently a Professor in the Mechanical Engineering Department
of the N.T.U.A (Director of Lab. of Biofluid –Mechanics and Biomedical Engineering). Research and
working areas and teaching experience include Fluid Mechanics in general, Biofluid-Mechanics,
Biomedical Engineering, Theoretical Aerodynamics. Sokrates Tsangaris has 90 publications in
scientific journals.
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