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UNCORRECTED PROOF
CNM 954pp: 1–16 (col.fig.: Nil)
PROD. TYPE: COMED: Senthil
PAGN: PR -- SCAN: NitShree
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng (in press)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.954
1
Extension of weakly compressible approximationsto incompressible thermal flows3
Mofdi El-Amrani1,∗,† and Mohammed Seaıd2
1Departamento Matematicas ESCET, Universitat Rey Juan Carlos, C/Tulipan, s/n, Mostoles-Madrid 28933, Spain52AG Technomathematik, FB Mathematik, Universitat Kaiserslautern, Kaiserslautern 67663, Germany
SUMMARY7
Weakly compressible and advection approximations of incompressible isothermal flows were developedand tested in (Commun. Numer. Methods Eng. 2006; 22:831–847). In this paper, we extend the method to9solve equations governing incompressible thermal flows. The emphasis is again on the reconstruction ofunconditionally stable numerical scheme such that, restriction on time steps, projection procedures, solution11of linear system of algebraic equations and staggered grids are completely avoided in its implementation.These features are achieved by combining a low-Mach asymptotic in compressible flow equations with13a semi-Lagrangian method for the weakly compressible approach. The time integration is carried outusing an explicit Runge–Kutta with variable stages. The method is applied to the natural convection flows15in a squared cavity for both steady and transient computations. The numerical results demonstrate highresolution of the proposed method and confirm its capability to provide accurate and efficient simulations17for thermal flow problems. Copyright q 2006 John Wiley & Sons, Ltd.
Received 12 May 2006; Revised 15 September 2006; Accepted 29 September 2006
KEY WORDS: weakly compressible flows; low-Mach asymptotic; semi-Lagrangian method; incom-19pressible Navier–Stokes equations; natural convection
21
1. INTRODUCTION
Many thermal incompressible flows use the incompressible Navier–Stokes/Boussinesq equations in23
their mathematical modelling. For flow problems in two space dimensions, the governing equations
can be formulated in dimensionless form as25
∇ ·u = 0 (1a)
∗Correspondence to: Mofdi El-Amrani, Departamento Matematicas ESCET, Universitat Rey Juan Carlos, C/Tulipan,s/n, Mostoles-Madrid 28933, Spain.
†E-mail: mofdi.elamrani@urjc.es
Copyright q 2006 John Wiley & Sons, Ltd.
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CNM 954
M. EL-AMRANI AND M. SEAID
�u
�t+ u · ∇u + ∇ p −
1
Re∇2u=
Gr
Re2�j (1b)
��
�t+ u · ∇� −
1
Pr Re∇2
� = 0 (1c)
where u= (u, v)T is the velocity field, p the pressure, � the temperature, Pr the Prandlt number,1
Gr the Grashof number, and j the unit vector in direction of gravity. The Rayleigh number Ra,
which is a parameter of interest, is the product of Pr and Gr. Equations (1) have to be solved in an3
open bounded domain � ⊂ R2 with smooth boundary � and a time interval [0, T ]. We assume that
appropriate initial and boundary conditions are given in such a way the problem is well defined5
and has a unique solution.
Most of the numerical methods for solving (1) suffer from two major computational prob-7
lems namely, the velocity–pressure coupling and the zero divergence restriction. Furthermore,
when convection dominates, spurious spatial oscillations may occur if standard Eulerian-based9
discretizations are used. These numerical difficulties have been treated by combining a low-Mach
number asymptotically in compressible flows with a semi-Lagrangian discretization for advection.11
The low-Mach analysis replaces Equations (1) by a weakly compressible flow problem which,
for small Mach numbers, asymptotically approximates the solution of (1). This approach does not13
need staggered grids and avoids Poisson or Stokes solvers for its numerical solution. On the other
hand, the semi-Lagrangian procedure is not limited by a stability-based CFL condition. This is15
due to the fact that each gridpoint is treated in Lagrangian manner. Since high-order difference
schemes have the potential to reduce numerical diffusion, the spatial discretization is performed17
using a fourth-order centred scheme.
The time integration is carried out using the modified Runge–Kutta Chebyshev method imple-19
mented in [1] for the weakly compressible approximation of the incompressible Navier–Stokes
equations for isothermal flows. The idea of this scheme is to stabilize the Runge–Kutta method21
by using extended number of stages in its formulation. The proposed method is efficient, easy
to implement and uniformly accurate such that the selection of time steps in the computational23
process is independent of the Mach number and is based only on the required numerical accuracy.
We demonstrate these claims by presenting results on two benchmarks for steady and transient25
natural convection problems.
This work is an extension of Reference [1] to incompressible thermal flows. Therefore, the dif-27
ferent formulations will be simply described and then the numerical results presented. In Section 2,
we formulate the weakly compressible approximation of incompressible thermal flows. A brief29
description of the solution procedure is presented in Section 3. Section 4 is devoted to numerical
results for natural convection flows. Conclusions are drawn in Section 5.31
2. WEAKLY COMPRESSIBLE MODEL FOR INCOMPRESSIBLE THERMAL FLOWS
Following the same formalism as in [1], the weakly compressible problem we consider in this33
paper to approximate Equations (1) reads
�p
�t+ u · ∇ p + p∇ ·u = 0 (2a)35
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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EXTENSION OF WEAKLY COMPRESSIBLE APPROXIMATIONS 3
�u
�t+ u · ∇u +
1
M2∇ p −
1
Re∇2u=
Gr
Re2�j (2b)
��
�t+ u · ∇� −
1
Pr Re∇2
� = 0 (2c)
where M is the Mach number, namely the ratio of a characteristic velocity in the flow with the1
sound speed in the fluid. Integrating (1a) over the space domain, we obtain the incompressibility
condition as follows:3
∮
�
u ·n ds = 0 (3)
where n denotes the unit outward normal vector on boundary �. This condition has to be satisfied by5
Equations (2) also. Formally, for small Mach number M , the solution of the weakly compressible
model (2) asymptotically converges to the solution of incompressible equations (1) with an error7
of O(M2), we refer to [2–4] for more details and further references are cited therein. Thus, for
small Mach number and appropriate numerical methods, the solution of thermal incompressible9
Navier–Stokes equations (1) can be recovered by solving the weakly compressible problem (2).
Assume M�1, the asymptotic equations are obtained by expanding the pressure, temperature11
and velocity variables in terms of the Mach number
u= u(0) + M2u(2) + · · ·
p = p(0) + M2 p(2) + · · ·
� = �(0) + M2
�(2) + · · ·
(4)
13
where the terms u(0) and u(2) satisfy the incompressibility condition (3). The variables p(0) and
p(2) can be interpreted as a thermodynamic pressure and a hydrodynamic pressure, respectively.15
Introducing expansions (4) into Equations (2) and balancing all terms of the same power of M ,
the asymptotic equations of leading, first and second order are obtained, see [2–4] among others.17
Thus, the leading order equations of (2) are
�p(0)
�t+ u(0) · ∇ p(0) + p(0)∇ ·u(0) = 0 (5a)
∇ p(0) = 0 (5b)
�u(0)
�t+ u(0) · ∇u(0) + ∇ p(2) −
1
Re∇2u(0) =
Gr
Re2�
(0)j (5c)
��(0)
�t+ u(0) · ∇�
(0) −1
Pr Re∇2
�(0) = 0 (5d)
From (5b), it follows that p(0) depends on time only, i.e.19
p(0) = p(0)(t) (6)
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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CNM 954
M. EL-AMRANI AND M. SEAID
Consequently, the term ∇ p(0) cancels out in (5a). Integrating Equation (5a) over the spatial domain1
� and applying the Gauss theorem we obtain the following:
�p(0)(t)
�t
∫
�
d� + p(0)(t)
∮
�
u(0) ·n ds = 0 (7)3
Since the velocity u(0) satisfies (3), Equation (7) gives that p(0) is constant in time, i.e.
p(0)(t) = p0 (8)5
with p0 is a constant. Back-substituting (8) in Equation (5a) implies that the velocity field
is divergence free. All together lead to the following thermal incompressible Navier–Stokes7
equations:
∇ ·u(0) = 0 (9a)
�u(0)
�t+ u · ∇u(0) + ∇ p(2) −
1
Re∇2u(0) =
Gr
Re2�
(0)j (9b)
��(0)
�t+ u(0) · ∇�
(0) −1
Pr Re∇2
�(0) = 0 (9c)
The role of the pressure in the above asymptotic analysis can be understood as follows. At small9
Mach number limit, two pressure terms p(0) and p(2) appear in the asymptotic equations. The
leading order pressure, p(0), becomes constant in both space and time. Whereas, the second pressure11
term, p(2), is used to establish the divergence-free condition for the velocity field. This can be
formally obtained from the fact that13
1
M2∇ p −→ ∇ p(2) for M −→ 0 (10)
In order to show the connection between a compressible fluid dynamics problem and the weakly15
compressible model (2), we consider the compressible Navier–Stokes equations written in conser-
vative form as17
��
�t+ ∇ · (�u) = 0 (11a)
�(�u)
�t+ ∇ · (�u ⊗ u) + ∇ p = ∇ · � + �g (11b)
�E
�t+ ∇ · (u(E + p)) = ∇ · (�u) + ∇ · (k∇�) (11c)
where �, E and � denote, respectively, the density, total energy and viscous stress; whereas k is
the thermal conduction coefficient and g is the gravitational force. The stress tensor � is given by19
the following:
� = 2�D − 23�∇u with D= 1
2(∇u + (∇u)T)21
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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EXTENSION OF WEAKLY COMPRESSIBLE APPROXIMATIONS 5
An equation of state is required to close system (11). For simplicity, we consider perfect gases1
such that
p= R��, E = cv� (12)3
with cv and cp are, respectively, the specific heat at constant volume and pressure; R = cp − cv
is the universal gas constant, and � = cp/cv is the ratio of specific heat. We assume that fluid5
properties are constant, viscous dissipation and fluid density variations are neglected except in the
buoyancy term, i.e. the Boussinesq approximation holds. Using these assumptions the governing7
equations (11) can be reformulated in primitive variables (p,u, �) instead of the conservative
variables (�, �u, E) as9
�p
�t+ u · ∇ p + p∇ · u= k∇2
� (13a)
�
(
�u
�t+ u · ∇u
)
+ ∇ p = �∇2u + �(1 − �(� − �ref))g (13b)
�cv
(
��
�t+ u · ∇�
)
+ p∇ ·u= k∇2� (13c)
where �ref is a reference temperature and � is the thermal expansion coefficient. To rewrite
Equations (13) in non-dimensional form we introduce the following scales:11
x∗ =x
xref, y∗ =
y
xref, t∗ =
ureft
xref, u∗ =
u
uref, v∗ =
v
vref, p∗ =
p − �ref
�refu2ref
g∗ =xrefg
u2ref
, �∗ =
� − �ref
��, �∗ = ���
with the subscript ‘ref’ presents the reference quantities, and�� denotes the temperature difference.
We also define the following non-dimensional numbers:13
� =�
�ref, M =
uref√
�R�ref
, Pr =�
k, Gr =
|g|�x3ref��
�2, Re=
urefxref
�(14)
Hence, in dimensionless form Equations (13) become15
�p
�t+ u · ∇ p + �p∇ ·u= (� − 1)∇2
� (15a)
�u
�t+ u · ∇u +
1
M2∇ p =
1
Re∇2u +
Gr
Re2�j (15b)
��
�t+ u · ∇� + p∇ ·u=
1
Pr Re∇2
� (15c)
where we have dropped the asterisk of the dimensionless variables for ease of notation. The formal
asymptotic analysis reported in [2] shows that, for small values of Mach number, the divergence-17
free condition (1a) is almost satisfied. Thus, by neglecting the effect of heat flux in (15a) and setting
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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CNM 954
M. EL-AMRANI AND M. SEAID
∇ ·u= 0, Equations (13) can be rearranged as those in the weakly compressible equations (2). We1
should mention that the pressure p in Equations (2) plays the same role as the one appears in the
compressible equations (13).3
3. SOLUTION PROCEDURE
In this paper, we adapt the same solution procedure as the one used in [1] for the isothermal flow5
problems. The key idea is to rewrite Equations (2) in transport form as
Dp
Dt+ p∇ · u= 0 (16a)
Du
Dt+
1
M2∇ p −
1
Re∇2u=
Gr
Re2�j (16b)
D�
Dt−
1
Pr Re∇2
� = 0 (16c)
where Dw/Dt measures the rate of change of a generic function w following trajectories of the7
flow particles with the velocity u
Dw
Dt=
�w
�t+ u · ∇w (17)9
Next, we consider a semi-Lagrangian method for the numerical solution of (16). The semi-
Lagrangian method imposes a regular grid at the new time level, and backtracks the flow trajectories11
to the previous time level. At the old time level, the quantities that are needed are evaluated by
interpolation from their known values on a regular grid.13
Let us discretize the time interval into subintervals [tn, tn+1] of length �t with tn := n�t . For the
spatial discretization, we assume without loss of generality a rectangular domain�=[a, b]× [c, d],15
covered with a uniform numerical mesh defined as
�h ={xi, j := (xi , y j )T, xi = a + i�x, y j = c + j�y, i = 0, 1 . . . , I, j = 0, 1 . . . , J }17
where �x = (b−a)/I ,�y = (d−c)/J and h denotes the maximum cell size, h := max(�x,�y). We
use the notations wi, j (t) :=w(xi , y j , t) and wni, j :=w(xi , y j , tn). Hence, the characteristic curves19
Xi, j := (X i, j , Yi, j )T associated with the material derivative (17) are solutions of the following
initial value problem:21
dXi, j (t)
dt= u(t,Xi, j (t)), t ∈ [tn, tn+1]
Xi, j (tn+1) = xi, j
(18)
The solution of the ordinary differential equation (18) is a continuous curve Xi, j (t), known as23
departure point at time t of a fluid particle passing through the gridpoint xi, j at time t = tn+1. The
semi-Lagrangian method does not follow the flow particles forward in time, as the Lagrangian25
schemes do, instead it traces backwards the position at time tn of particles that will reach the
points of a fixed mesh at time tn+1. By so doing, the semi-Lagrangian method avoids the grid27
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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EXTENSION OF WEAKLY COMPRESSIBLE APPROXIMATIONS 7
distortion difficulties that the conventional Lagrangian schemes have. In the current work, we solve1
Equations (18) using a Runge–Kutta scheme along with an extrapolation method and an iterative
procedure to decouple the equations. Details on these techniques are given in [1, 5] and are not3
repeated here.
Let the departure points Xi, j (tn) be accurately calculated and let the mesh elements where5
Xi, j (tn) reside is correctly identified. Then, the material derivative (17) is approximated by
Dwi, j
Dt=
wn+1i, j − wn
i, j
�twith wn
i, j := w(tn,Xi, j (tn)) (19)7
The solutions wni, j are computed by bicubic Lagrangian interpolation from gridpoints of the
mesh element where Xi, j (tn) belong. Hence, a spatial discretization of Equations (16) can be9
formulated as
Dpi, j
Dt+ pi, j (Dxui, j + Dyvi, j ) = 0 (20a)
Dui, j
Dt+
1
M2Dx pi, j −
1
Re�hui, j = 0 (20b)
Dvi, j
Dt+
1
M2Dy pi, j −
1
Re�hvi, j =
Gr
Re2�i, j (20c)
D�i, j
Dt−
1
Pr Re�h�i, j = 0 (20d)
whereDx ,Dy and �h =D2x +D
2y are difference approximations of �/�x , �/�y and ∇2 = �
2/�x2+11
�2/�y2. A second-order centred finite difference discretization of these operators has been used
in [1]. In the present work, we consider the following fourth-order approximation13
Dxwi, j =wi−2, j − 8wi−1, j + 8wi+1, j − wi+2, j
12�x, 1�i�I − 1, 1� j�J − 1
D2xwi, j =
−wi−2, j + 16wi−1, j − 30wi, j + 16wi+1, j − wi+2, j
12(�x)2
(21)
The operators Dy and D2y are approximated in similar manner. At boundary points we use the15
notion of ‘ghost points’. For example, at the left boundary (x = a) and the right boundary (x = b),
the values w−1, j and wI+1, j are respectively needed for Dx and D2x in (21). To calculate these17
values we use a fourth-order interpolation formula obtained by Taylor expansion as follows:
1. For Dirichlet boundary conditions w(a, y)=wa(y); w(b, y)=wb(y), we use19
w0, j = wa(y j )
w−1, j = 5w0, j − 10w1, j + 10w2, j − 5w3, j + w4, j
wI, j = wb(y j )
wI+1, j = 5wI, j − 10wI−1, j + 10wI−2, j − 5wI−3, j + wI−4, j
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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CNM 954
M. EL-AMRANI AND M. SEAID
2. For Neumann boundary conditions (�w/�x)(a, y)= ga(y), (�w/�x)(b, y)= gb(y), we use1
w0, j = −12�x
25ga(y j ) +
48
25w1, j −
36
25w2, j +
16
25w3, j −
3
25w4, j
w−1, j = −12�x
3ga(y j ) −
10
3w0, j +
18
3w1, j −
6
3w2, j +
1
3w3, j
wI, j = −12�x
25gb(y j ) +
48
25wI−1, j −
36
25wI−2, j +
16
25wI−3, j −
3
25wI−4, j
wI+1, j = −12�x
3gb(y j ) −
10
3wI, j +
18
3wI−1, j −
6
3wI−2, j +
1
3wI−3, j
Similar work has to be done for other boundary values wi,−1 and wi,J+1 for the boundary
reconstruction of D2y and Dy .3
The time integration of (20) can be carried out using any robust ordinary differential equa-
tions solver. However, the presence of the term 1/M2 in the pressure variable makes the ex-5
plicit schemes unstable. A way to stabilize explicit schemes without solving linear systems
is to treat the pressure term in (20) implicitly. In our computations presented in Section 4,7
we have used a modified Runge–Kutta Chebyshev scheme. This method offers an uncondi-
tionally stable explicit time stepping where stability is maintained by extending the number9
of stages in the Runge–Kutta scheme. Its implementation for Equations (20) is carried out as
follows:
11
1. Compute
P(0)i, j = pni, j , U
(0)i, j = uni, j , V
(0)i, j = vni, j , T
(0)i, j = �
ni, j13
2. Solve
P(1)i, j =P
(0)i, j − �1�tP
(1)i, j (Dx u
ni, j + Dy v
ni, j )15
3. Compute
T(1)i, j =T
(0)i, j + �1
�t
Pr Re�
ni, j
U(1)i, j =U
(0)i, j − �1�t
(
1
M2DxP
(1)i, j −
1
Re�h u
ni, j
)
(22a)
V(1)i, j =V
(0)i, j − �1�t
(
1
M2DyP
(1)i, j −
1
Re�h v
ni, j −
Gr
Re2T
(1)i, j
)
(22b)17
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
UNCORRECTED PROOF
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EXTENSION OF WEAKLY COMPRESSIBLE APPROXIMATIONS 9
4. For k = 2, 3, . . . , s:1
(i) Solve
P(k)i, j = (1 − �k − �k)P
(0)i, j + �kP
(k−1)i, j + �kP
(k−2)i, j
− �k�tP(k)i, j (DxU
(k−1)i, j + DyV
(k−1)i, j )
− �k�tP(0)i, j (DxU
(0)i, j + DyV
(0)i, j )
(ii) Compute3
T(k)i, j = (1 − �k − �k)T
(0)i, j + �kT
(k−1)i, j + �kT
(k−2)i, j
+ �k�t
Pr Re�hT
(k−1)i, j + �k
�t
Pr Re�hT
(0)i, j
U(k)i, j = (1 − �k − �k)U
(0)i, j + �kU
(k−1)i, j + �kU
(k−2)i, j
− �k�t
(
1
M2DxP
(k)i, j −
1
Re�hU
(k−1)i, j
)
− �k�t
(
1
M2DxP
(1)i, j −
1
Re�hU
(0)i, j
)
(22c)
V(k)i, j = (1 − �k − �k)V
(0)i, j + �kV
(k−1)i, j + �kV
(k−2)i, j
− �k�t
(
1
M2DyP
(k)i, j −
1
Re�hV
(k−1)i, j −
Gr
Re2T
(k)i, j
)
− �k�t
(
1
M2DyP
(1)i, j −
1
Re�hV
(0)i, j −
Gr
Re2T
(1)i, j
)
(22d)
6. Update the new solution
pn+1i, j =P
(s)i, j , un+1
i, j =U(s)i, j , vn+1
i, j =V(s)i, j , �
n+1i, j =T
(s)i, j5
All the coefficients �k , �k , �k and �k , appearing in the above equations, are given in the analytical
form for k = 1, 2, . . . , s with arbitrary s>0. Their expressions are the same as for conventional7
Runge–Kutta Chebyshev scheme, compare [6] among others. The starting point is the Chebyshev
polynomial of the first kind of degree k9
Tk(z) = cos(k arcos z), −1�z�1
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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CNM 954
M. EL-AMRANI AND M. SEAID
Then, the parameters used by the Runge–Kutta Chebyshev scheme are defined as1
� =2
13, q0 = 1 +
�
s2, q1 =
T ′s (q0)
T ′′s (q0)
b0 = b2, b1 = b2, bk =T ′′k (q0)
(T ′k(q0))
2(2�k�s)
�1 = b1q1, �k =2bkq0
bk−1, �k =
−bk
bk−2(2�k�s)
�k =2bkq1
bk−1, �k = − (1 − bk−1Tk−1(q0))�k (2�k�s)
Note that the pressure and temperature terms in Equations (22) are treated implicitly. This comes
from the fact that the semi-Lagrangian method decouples the energy equation in (20d) from the3
momentum equations (20b) and (20c), which can be solved separately once the advective step is
performed.5
4. NUMERICAL RESULTS
To demonstrate the performance of the weakly compressible approximation we consider two7
test examples on thermal incompressible flows. Both examples were used in the literature as
benchmarks to verify and validate numerical methods. Therefore, our results will be compared9
with those available from the literature for the considered test examples. In all the computations
we used a Mach number M = 0.1. Other smaller values of M have also been tested and from11
our comparisons we have observed that M = 0.1 produces reasonable results. The time step is
fixed to �t = 0.01 and the number of stages required for this time-step selection in the modified13
Runge–Kutta Chebyshev scheme is adjusted as in [1].
4.1. Steady natural convection in a squared cavity15
As a first example we consider the well-established problem of natural convection in a squared
cavity. The flow domain is a squared cavity with dimension L as shown in Figure 1 (left plot).17
The left and right vertical walls are maintained at dimensionless hot temperature �H = 1 and
cold temperature �c = 0, respectively. The bottom and top horizontal walls are insulated, whereas19
no-slip boundary conditions are imposed on all cavity walls. The literature is abundant for this
flow configuration which shows many flow features depending on the Rayleigh number. Authors21
in [7–10] presented numerical solutions up to Ra = 108 using standard Eulerian-based methods.
Their works represent a comprehensive study of natural convection in squared cavity and therefore,23
our results will be compared with those reported therein.
We present results for Prandtl number Pr = 0.71 with Ra ranging from 103 to 108 and the25
number of gridpoints is set to 128× 128. In Figure 2, we present the isotherms, velocity vectors
and streamlines for different Rayleigh numbers at time t = 200. At this time, the flow has all the27
characteristic features of the steady state. As can be seen, the flow is symmetric with respect to the
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
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L
L /43L /8
L /4
LL
L
L /4
Figure 1. Flow domain for the steady natural convection (left) and for transient buoyancy flow (right).
Figure 2. Isotherms (first row), velocity fields (second row), streamlines (third row) for Rayleigh numbers
(from left to right) Ra = 103, 104, 105, 106, 107 and 108.
cavity centre for all the Rayleigh numbers. At low Ra numbers, the flow exhibits a central vortex,1
and the heat transfer is dominated by conduction regime. At Ra = 105, the vortex breaks into two
vortices moving towards the vertical walls. At high Rayleigh numbers, the flow becomes fully3
convection dominated, the cold fluid is entrained right to the hot wall where high-temperature
gradients are created. There is excellent agreement between these results and those published5
in [7–10]. Note that the performance of our approach is very attractive since the computed
solutions remain stable and highly accurate without solving linear systems of algebraic equations7
or requiring special projection methods or relying on pressure correction procedures.
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
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Table I. Comparison between published results for maximum horizontal velocity u at the cavity mid-widthx = 0.5. The number in parenthesis corresponds to the y co-ordinate.
Ra Reference [7] Reference [8] Reference [9] Reference [10] Present work
103 3.634 (0.813) 3.68 (0.817) 3.6493 (0.8125) 3.6434 (0.8167) 3.6471 (0.815)
104 16.2 (0.823) 16.1 (0.817) 16.1798 (0.8235) 15.967 (0.8167) 16.1063 (0.819)
105 34.81 (0.855) 34.0 (0.857) 34.7741 (0.8535) 33.51 (0.85) 34.0128 (0.853)
106 65.33 (0.851) 65.4 (0.875) 64.6912 (0.8460) 65.55 (0.86) 65.2551 (0.860)
107 — 139.7 (0.919) 145.2666 (0.8845) 145.06 (0.92) 144.9833 (0.898)
108 — — 283.689 (0.9455) 295.67 (0.94) 291.6950 (0.946)
In order to quantitatively assess the accuracy of weakly compressible approximation, we dis-1
play in Table I a comparison between the published results for maximum horizontal velocity
at the cavity mid-width and results obtained with weakly compressible approach. The results3
mostly agree with all the model results in References [7–10]. As is obvious from the table,
the small difference to other methods can be attributed to the mesh size used in the present5
investigation.
4.2. Transient incompressible buoyancy flow7
Our second example solves the pure advection of a hot squared plume in a cold fluid placed in
a squared cavity. This test example has been proposed by Harvat et al. [11] as a benchmark for9
incompressible buoyancy flows. Here, a squared thermal plume is placed in the lower half of a
squared cavity as shown in the right plot in Figure 1. Both the plume and the surrounding medium11
are composed of the same fluid and initially maintained at rest. The plume has a dimensionless hot
temperature �H = 1, while the surrounding medium has a dimensionless cold temperature �c = 0.13
For the fluid temperature, all the wall boundaries are considered adiabatic and no-slip boundary
conditions are used for the fluid velocity. Following the formulation in [11], the problem statement15
consists of solving Equations (2) with
1
Pr Re= 0,
Gr
Re2= 1 and
1
Re= �17
Hence, the dimensionless viscosity � is the only free parameter in the problem considered. In
contrast to the previous test example, the considered energy equation is solved with zero diffusion19
term. As a consequence, the later flow is more difficult to handle; the results shown here illustrate
the robustness of the semi-Lagrangian method. Furthermore, the considered advective heat is a21
problem unsteady in nature and therefore, good numerical accuracy is required in order to capture
the different phenomena present in its evolving solution. The fourth-order centred differencing23
and the modified Runge–Kutta Chebyshev scheme show high accuracy and good stability for this
transient flow problem.25
In all our computations the spatial domain is discretized into 128× 128 gridpoints and the
results are displayed at the non-dimensional time T when the plume reaches the upper cavity wall.27
The dimensionless time T depends on the values taken by the viscosity coefficient �. Here, we use
T = 50, 5, 4, 3.5 and 3 for �= 10−1, 10−2, 10−3, 10−4 and 0, respectively. In Figure 3 we show29
the evolution in time of the isotherms. The time evolution of velocity field is presented in Figure 4.
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
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EXTENSION OF WEAKLY COMPRESSIBLE APPROXIMATIONS 13
0
0
Figure 3. Isotherms at times from bottom to top: T/4 (first row), T/2 (second row), 3T/4 (third row), T
(fourth row) for viscosity coefficients from left to right: �= 10−1, 10−2, 10−3, 10−4 and 0.
In these figures, we plot the results at four non-dimensional instants t = T/4, T/2, 3T/4 and T1
for each viscosity coefficient. As expected, for small values of viscosity the thermal plume travels
faster towards the upper cavity wall. For � = 0, the flow is purely convection such that no physical3
diffusion is present in the equations. This last case presents a challenge for many Eulerian-based
methods for which numerical diffusion is consequence of their stabilization. For our approach, no5
excessive numerical diffusion has been detected for the case with � = 0.
The plume temperature and the fluid velocity in Figures 3 and 4 demonstrate the effect of7
buoyancy force on the fluid dynamics. For large viscosity values the buoyancy force strongly
increases the shear stresses on the cavity walls. These wall effects slow the advection of the9
thermal plume towards the upper cavity wall allowing the plume to preserve its compact shape
during the whole simulation period. Small viscosity values decrease the wall shear stresses leading11
to an increase in the velocity field. The thermal plume also exhibits Rayleigh–Taylor instabilities
which cause its deformation and rolling up along the advection process. At zero viscosity, the13
wall stresses disappear and the fluid motion is purely driven by convection. The presented results
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
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00
Figure 4. Velocity vectors at times from bottom to top: T/4 (first row), T/2 (second row), 3T/4 (third
row), T (fourth row) for viscosity coefficients from left to right: � = 10−1, 10−2, 10−3, 10−4 and 0.
also indicate that as the viscosity decreases, the size of the recirculation zones increases with the1
flow exhibiting eddies with different magnitudes and separating shear layers. We can see the small
complex patterns of the flow being captured by the weakly compressible approximation.3
A quantitative comparison between our results and those presented in [11] is also performed in
the present example. To do so, we compute the averaged kinetic energy Q(t) at each time step as5
Q(t) =1
2
∑
i, j
(u2i, j + v2i, j )�x�y (23)
In Figure 5, we plot the averaged kinetic energy Q(t) versus the time t for the selected viscosity7
coefficients � = 10−1, 10−3 and 0. We also plot the results from [11] for comparison. It is clear that
the viscosity has direct influence on the kinetic energy of the fluid. For instance, decreasing the9
viscosity coefficients leads to an increase in the averaged kinetic energy within the simulation time.
This influence is more pronounced for zero viscosity test which indicates that the buoyancy force11
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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EXTENSION OF WEAKLY COMPRESSIBLE APPROXIMATIONS 15
Figure 5. Evolution in time of the averaged kinetic energy for three values of viscosity coefficients (from
left to right) � = 10−1, 10−3 and 0.
has dominant effect on the flow field. In most cases, our results agree well with those reported by1
Harvat et al. [11]. The small discrepancies observed in Figure 5 may be due to numerical diffusion
introduced by the solution procedure used.3
5. SUMMARY
In this paper we have extended the weakly compressible and advection approximations proposed5
by El-Amrani and Seaıd [1] to solve equations governing incompressible thermal flows. The
emphasis was again on the reconstruction of unconditionally stable numerical method such that (i)7
iterative solvers required for Poisson or Stokes problems, (ii) CFL restrictions on the time steps,
(iii) staggered grids, and (iv) projection procedures are completely avoided in its implementation.9
These properties have been achieved in our approach by combining a low-Mach asymptotic in
the compressible flow equations with a semi-Lagrangian method. The low-Mach analysis avoids11
projection procedures and staggered grids while the semi-Lagrangian method eliminates the CFL
condition and difficulties related to the treatment of convection terms.13
Numerical results have been presented for two benchmark problems on thermal flows in steady
and transient regimes. Good agreements have been obtained between the computed results and15
those published in the literature for the benchmark problems. The proposed method has shown its
capability to provide accurate and efficient simulations for thermal flow problems.17
REFERENCES
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3. Lin CK. On the incompressible limit of the compressible Navier–Stokes equations. Communications in Partial23Differential Equations 1995; 20:677–707.
4. Klein R. Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics. I.25One-dimensional flow. Journal of Computational Physics 1995; 121:213–237.
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm
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5. Staniforth A, Cote J. Semi-Lagrangian integration schemes for the atmospheric models: a review. Weather Review11991; 119:2206–2223.
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9. Mayne DA, Usmani AS, Carpper M. h-adaptive finite element solution of high Rayleigh number thermally driven9cavity problem. International Journal of Numerical Methods for Heat and Fluid Flow 2000; 10:598–615.
10. Wan DC, Patnaik SV, Wei GW. A new Benchmark quality solution for the Buoyancy-driven cavity by discrete11singular convolution. Numerical Heat Transfer, Part B 2001; 40:199–228.
11. Horvat A, Kljenak I, Marn J. On incompressible Buoyancy flow Benchmarking. Numerical Heat Transfer,13Part B 2001; 39:61–78.
Copyright q 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng (in press)
DOI: 10.1002/cnm