J.
Fluid Me&.
(1974),
vol.
66,
part
2,
pp.
209-229
Printed
in
Cheat Britain
209
Natural convection in a shallow
cavity with differentially heated end walls.
Part
1.
Asymptotic
theory
By
D.
E.
CORMACK, L. G. LEAL
Chemical Engineering, California Institute
of
Technology, Pasadena
AND
J.
IMBERGER
Department
of
Mathematics and Mechanical Engineering, University
of
Western Australia, Nedlands
(Received
23
March
1973
and in revised form
15
February
1974)
The problem of natural convection in
a
cavity of small aspect ratio with dif-
ferentially heated end walls
is
considered.
It
is
shown by use of matched asymp-
totic expansions that the flow consists of two distinct regimes
:
a parallel flow in
the core region and a second, non-parallel flow near the ends of the cavity.
A
solution valid
at
all orders in the aspect ratio
A
is
found for the core region, while
the
first
several terms of the appropriate asymptotic expansion are obtained
for the end regions. Parametric limits of validity for the parallel flow structure
are discussed. Asymptotic expressions for the Nusselt number and the single
free parameter
of
the parallel flow solution, valid in the limit as
A
-+
0,
are
derived.
1.
Introduction
Convection due to buoyancy forces is an important and often dominant
mode of heat and mass transport. Of particular significance to the dispersion of
pollutants and heat waste in estuaries are the buoyancy-driven convective
motions induced by gradients in
salt
concentration or temperature.
Unfortunately the direct modelling of these natural systems
is
very complex,
mainly because the flow is turbulent. However, the idealized problem of laminar
flow in an enclosed rectangular cavity with differentially heated ends does
provide some insight into these more difficult problems, and has been studied
extensively in other contexts by prior investigators. The majority of these studies
have used finite-difference numerical solutions
of
the full equations of motion,
subject to the Boussinesq approximation, to consider cavities which were either
square
or
had height hlarger than their length
I
(cf. Quon
1972;
Wilkes
&
Churchill
1966;
Newel1
&
Schmidt
1970;
Szekely
&
Todd
(1971);
De Vahl Davis
1968).
However, Batchelor
(1954),
EIder
(1965)
and Gill
(1966)
have shown that ana-
lytical progress is possible when the cavity aspect ratio h/l is large.
Batchelor
(1954)
considered both large and small Grashof numbers
Gr.
In
the latter case, he obtained an asymptotic solution about the pure conduction
I4
FLhf
65
210
D.
E.
Cormack,
L.
G.
Leal and
J.
Imberger
-1-
.l,
FIGURE
1.
Schematic diagram of system.
mode of heat transfer. For large
Gr,
Batchelor envisaged
a
flow with thin boun-
dary layers on all solid surfaces and a closed-streamline isothermal core of con-
stant vorticity. Motivated by the experimental measurements of Elder
(1965),
Gill
(1966)
proposed an alternative structure for the case
h/l
$
1
and
Gr
9
1.
In Gill's model the flow is decomposed into boundary layers adjacent to the end
walls in which the horizontal temperature gradients are large, and a core region
in which the temperature is assumed to be
a
function only
of
the vertical
CO-
ordinate. In spite of the approximations necessary to solve the resulting equations
Gill reported moderate agreement with the experimental measurements of
Elder
(1965).
A
key feature of the case
h/Z
$
1,
which is implicit in Gill's model,
is that the core dynamics play only
a
secondary role in establishing the overall
flow structure, which is dominated by the buoyancy-driven boundary layers.
A
natural question is whether this qualitative feature persists as the aspect ratio
h/Z
is varied. In particular, in the limit as
h/l
-+
0,
which is most relevant for the
naturally occurring flows of interest in the present investigation, one might
anticipate that viscous effects in the core would play an increasingly important
role in establishing the flow structure for all fixed (though large) values of
Gr.
In the present paper, we use the standard methods of matched asymptotic
expansions to consider the cavity flow problem in this limiting case
h/l
<
1,
Gr
fixed. We shall show that the flow structure consists of two parts: a parallel-
flow core region in which essentially all of the horizontal temperature drop occurs
and which
is
dominated by viscous effects; and end regions which serve primarily
to turn the core flow through
180"
as required by the solid end walls. The numeri-
cal and experimental results reported in parts
2
and
3
of the present study show
excellent agreement with this asymptotic theory for large, though finite values of
of
(h/l)-l.
2.
Mathematical formulation
of
the problem
We consider a closed rectangular two-dimensional cavity of length
1
and height
h
which contains a Newtonian fluid, and
is
shown schematically in figure
1.
The
end walls
are
held at different but uniform temperatures and
qb,
with
T,
<
Th.
Convection
in
a shallow
cavity
with heated walls. Part
1
21
1
The top and bottom are insulated, and all surfaces are rigid no-slip boundaries.
Actually, the upper boundary of the environmental systems mentioned in the
introduction is more closely approximated as
a
zero-shear surface. However,
it
was found that the experimental measurements, to be presented in part
3,
could
be
obtained only in
a
cavity with
a
no-slip lid. Hence, the present analysis was
undertaken to provide a solution that could be compared directly with the ex-
perimental results.
A
systematic investigation of the influence
of
the upper
surface conditions on flow structure may be found in Cormack, Stone
&
Leal
(1974).
The appropriate governing equations, subject to the usual Boussinesq ap-
proximations, are
with corresponding boundary conditions
u’
=
v’
=
0
on all solid boundaries,
aT/ay‘
=
0
on
y’
=
o,h,
T
=
T,,T,
on
2‘
=
0,l.
(5)
Here,
u’
and
vf
are the horizontal and vertical velocity components;
v,
po,
Cp,
E
and
p
are the kinematic viscosity, density, heat capacity, thermal conductivity
and coefficient
of
thermal expansion, all referred to some mean temperature of
the fluid.
Non-dimensionalizing, using the definitions
0
=
(T-c)/(Th-q),
t
=
t‘g/3h2(T,-(P,)/d,
and introducing a stream function
$
such that
one can reduce
(1)-(4)
to
u
=
a$/ay,
v
=
-a$/ax,
02$
=
--w.
with boundary conditions
9
=
a$/a~
=
0,
e
=
AX
at
x
=
0,
A-I
and
$
=
a$/ay
=
ae/ay
=
o
at
y
=
0,
1.
212
D.
E.
Cormaclc,
L.
G.
Leal
and
J.
Imberger
Although the characteristic velocity scaling may at first appear an arbitrary
choice,
it
is consistent with the physical picture of a buoyancy-driven parallel
flow which is moderated by viscous effects over
a
length
I,
and may in fact be
justified
a
posteriori
by the theory which is presented in this paper. The dimen-
sionless parameters are
Gr
=
gp(Th
-
T,)
h3/v2
Pr
EE
C,,uu/k
(Prandtl number)
(Grashof number),
and
A
=
h/b
(aspect ratio).
In what follows, we consider the asymptotic problem in which
A
-+
0
with
Pr
and
Gr
held fixed.
3.
The
core
flow
The key to a proper asymptotic solution, in the present case, is a proper
resolution of the central
or
core region of the cavity. Fortunately, the flow struc-
ture
in
this region is surprisingly simple and amenable to direct analytical solu-
tion of the governing equations. Both the numerical and experimental evidence
which we shall present in parts
2
and
3
in fact indicate that the streamlines in the
core region become more nearly parallel as the aspect ratio is decreased, with
substantial deviations from this structure only occurring in the immediate
vicinity of the end walls. Acceptance
of
a parallel flow structure as a first ap-
proximation in the core would imply that the appropriate characteristic scale
length in the
x
direction must be
O(A-l).
With introduction of the characteristic horizontal scale
x
=
O(A-l),
equa-
tions
(6)-(
8)
become
where
B
=
Ax.
Using
(lo)-(
12),
one may now obtain the full asymptotic solution for the core
temperature and velocity fields, as a regular expansion in the small parameter
A.
Although the precise forms of the gauge functions in this expansion are strictly
obtainable only from the requirements for a proper asymptotic match with the
corresponding solutions in the end regions, we anticipate the simple form (which
will be verified
a
posteriori)
e=
o,+Ao,+A~~,+
...,
$
=
$0+A$1+A2$2+...
,
w
=
w0+A~,+A%,+
....
The systematic solution, valid
for
A
<
1,
which results on substituting these
Convection
in
a
shallow cavity with heated walls.
Part
1
213
expansions into
(lo)-(
12)
and equating terms of like order in
A
has the same form
at all orders in
A,
i.e.
+
=
K1(&Y"i%Y3+AY2),
(14)
O
=
K12
+
K2,Gr
Pr
A2(&y5
-
&y4
+
&y3)
+
K,,
(15)
where
K,
=
c1+c2A+c3A2+
...,
K,
=
cT+cZA+C:A~...
and
cl,
c,,
...,
c:, ct,
.
..,
c,*
are constants which depend on
Gr
and
Pr.
The velocity field corresponding to
(14)
is strictly parallel to the top and
bottom walls of the cavity, and cannot, therefore, satisfy the boundary con-
ditions
(9a)
at the end walls. These conditions must be satisfied by solutions valid
in the end regions, and in general, the two parameters
Kl
and
K,
are evaluated
by matching the core solution with these two end-region solutions. In the present
case, however, the problem simplifies somewhat owing to the centro-symmetry
property of the equations and boundary conditions (discussed by Gill
1966).
This property imposes the requirement on the solutions that
WAY)
=
$(1-~,1-Y),
@,Y)
=
41-2,1-Y)
and
B(2,y)
=
1-O(1-2,l-y).
Hence, one half of the cavity is
an
inverted mirror image
of
the other. Moreover,
it
is
apparent that
so
that, according to
(15),
O(jT,*)
=
4,
&Kl+&K2,GrPrA2+K2
=
4.
This relationship allows the constants
c:
(and hence
K,)
to be entirely eliminated
in favour
of
the single set
(ci>,
i
=
1,2,
.
.
.,
00,
e.g.
2
3
1440c2,GrPr. (16a,b,c)
c,
*
=
-gc2, c;
=
--lc
-1
c;
=
Q-'c
With the constant
K,
thus eliminated,
it
is possible to evaluate
K,
completely
(and hence the
ci,
i
=
1,2,
...,
co,
which depend on
Gr
and
Pr)
by matching
the core solution with a proper solution that is valid in either
of
the two end
regions. This matching process is, of course, considerably simplified by the fact
that the basic form of the core solution is preserved at all orders in the small
parameter
A.
Before proceeding to
a
resolution of the flow in the end regions,
it
is useful
to note the key structural features of the basic core solution for
Gr
fixed,
A
.+
0
[equations
(14)
and
(15)]
and to contrast these with the structure in the pre-
viously noted conduction and boundary-layer limits
A
fixed,
Gr
-+
0
and
A
fixed,
Gr
-+
co
of Batchelor and Gill. The solution
(14)
and
(15)
exhibits two key
features. First, the velocity field in the core is parallel to all orders in the small
parameter
A.
Second, to
a
first approximation,
B
is independent of vertical
position, and varies linearly between the end walls. The primary driving force for
motion is the horizontal temperature gradient in the core. In fact, we shall show
in the next section that
c,
=
1,
so
that effectively all of the temperature drop