A NUMERICAL MODEL OF A CONVECTIVE CELL
DRIVEN BY NON-UNIFORM
HORIZONTAL HEATING
by
John F. Festa
B.S., The City College of New York
(1968)
SUBMITTED IN
PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
January 1971
Signature of Author:
epartment of Meteorology, October 30, 1970
Certified by:
Thesis Supervisj
Accepted by:
ate Students
ES
IL____I___
_l______tl
I-~UI-YII~ LLX
I-U~.II
~-I -^--~-LI
-Ln
-
2 -
A NUMERICAL MODEL OF A CONVECTIVE CELL
DRIVEN BY NON-UNIFORM HORIZONTAL HEATING
John F. Festa
Submitted to the Department of Meteorology on October 30
1970 in partial fulfillment of the requirements for the
degree of Master of Science.
Abstract
A numerical model of thermal convection for a two-dimensional
laminar, non-linear, non-rotating, incompressible, viscous fluid contained in a rectangle and heated non-uniformly from below is investigated. Two studies are made, one for a stress free bottom boundary and
the other for a constant applied stress along the bottom. The Boussinesq
equations are integrated numerically by means of the explicit scheme of
DuFort and Frankel (1953) and in each case the top and side boundaries
are both insulating and rigid.
In the first study the dependence of the cellular convective
motion and isotherms upon the Prandtl number, F, and the horizontal Rayleigh number, Ra, is examined. A single asymmetric convective cell develops as the Rayleigh number is increased. The asymmetry and intensity
of the circulation increase markedly with increasing horizontal Rayleigh
number but respond slightly to changes in the Prandtl number. The bottom
boundary layer thickness is also examined and is found to decrease as the
Rayleigh number increases.
In the second study, calculations are presented for a single
Rayleigh number and Prandtl number and a moderate range in applied stress.
The stress is applied in opposition to the thermal driving. As the stress
increases, the thermal circulation weakens and the opposing circulation
becomes stronger. The increase in stress results in a change from a thermal to a dual, containing both thermal and stress driven circulations of
the same magnitude, to a stress driven circulation.
Thesis Supervisor:
Title:
Robert C. Beardsley
Assistant Professor of Oceanography
DEDICATED TO
MOM & DAD
-3-
Table of Contents
Page
1.
Introduction ...............................................
2.
Governing Equations .......................................
3.
Dimensional Analysis ........................................
4.
Boundary Conditions ..........................
5.
Numerical Formulation ......................... ............
6.
(a)
(b)
Grid Geometry ........
Numerical Approximatioi
(c)
(d)
(e)
(f)
Stability and Accuracy
Computational Procedur
Initialization .......
Steady State Requiremei
.... .........
.................
Approximations .
............ . .
..............0 ...
00. 0.00......0...0..
eoooeoeeeeleeeeee
ooeeoooooeoeooooo
ooooeooooooeeoooo
Previous Investigations ... .................................. 18
(a)
Theoretical ...
(b)
(c)
Experimental ..
Numerical.....
7.
Results (Part 1) ....
8.
Results
(Part 2) ....
.....
......................... .......
23
. . oeee..
. e.....
.. e.....
.......
57
Summary ............. ........................................ 69
Bibliography ........ ........................................ 72
Acknowledgements .... ........................................ 73
Appendix A .......... ........................................ 74
Appendix B .......... ........................................ 76
Appendix C .......... ......................................
o. 80
-
1.
4 -
Introduction
The general circulation of the oceans is primarily a result of the
thermal driving of the sun.
Circulations can be induced by the unequal
heating between the equator and poles, the conversion of the sun's energy
into a wind stress or, more likely, a coupling of these two effects.
A study by Stommel (1962) initiated a series of investigations of
thermally driven circulations induced by horizontal heating.
In a simple
pipe model, he found a strikingly asymmetrical convective cell.
Stommel
suggested that this phenomenon might have some relationship to the scarcity and small size of oceanic sinking regions.
Since then experiments
by Rossby (1965), and Miller (1968), as well as numerical studies by Barcilon and Veronis (1965) and Somerville (1967) have exhibited this asymmetric flow.
The purpose of this paper is to obtain a numerical simulation of
thermal convection induced by non-uniform horizontal heating and to examine the effects of various parameters on the asymmetry, mean temperature and boundary layers of the flow.
In addition, an attempt is made
to study the effects of applying a "wind stress" boundary condition to
the model.
Since the range of parameters is limited by the size and
speed of our computers, the initial investigations will be in a range
significantly below the actual experimental values used by Rossby and
Miller.
-
2.
5 -
Governing Equations
The problem considered is a two dimensional, laminar, non-linear,
non-rotating, incompressible, viscous fluid contained in a rectangular
The coordinate system is Cartesian in
heated non-uniformly from below.
x and z (Fig. 1).
The Boussinesq approximation (Spiegel and Veronis, 1960), in which
the equation of state is expressed linearly as
/
=o(l-T)
and where
density changes are neglected except when coupled with gravity, is employed.
The Boussinesq equations are
+
Cx-momentum)
(continuity)
+D_
+
-
(z-momentum)
UY
tUf-
X
(energy)
(la)
V'
+Cd
T
I
(lb)
(Ic)
O
T
(d)
where u and w are the horizontal and vertical components of velocity respectively, p is the reduced pressure, T is the temperature deviation from
an arbitrary temperature field To,
at temperature T,
/0 is the mean density of the fluid
o( is the coefficient of thermal expansion, X
is the
thermal diffusivity, > is the kinematic viscosity, and g is the acceleration of gravity.
Here D/Dt and
V represent the two dimensional sub-
stantial derivative and Laplacian operator respectively, i.e. Du/Dt =
e
~IIY--I~-L~-~Y~I
-
ut
+ uu x + wu Z and
6 -
V2u = uxx + uzz .
Incompressibility (ic) allows introduction of the stream function,
such that
U
The vorticity q
I
X V(2)
is defined as
The governing vorticity equation, obtained by taking the curl of the
momentum equation thus eliminating the pressure field, is
7It
T-0)±+
1cj Tx .
(4a)
The governing temperature equation, expressed in terms of the stream
function and vorticity, is
Tt4 --J(C ,T)
X V17T.
Here the Jacobian is defined by
Jc(A,e) = Ax Be - 4e , .
(4b)
-7-
H
zL
x
FIG. I
ZSO
X O
Ol)
J
(2 ))
(O,J) (I, J
iJ(
(1,J
le
I
II
FIG. 2
(01)
or
1)
(I
I
0iO ) 1, 0) (20)
.
aIl
.j
isx
•0
*i.
I
FIG. 3
9i * I,j
i - ,Ij
AX
ij-
-
3.
8 -
Dimensional Analysis
We non-dimensionalize equations 4a and 4b by introducing the non-
dimensional variables
WEC'
. wkW-)
L
T4TT,
where L is the horizontal and H is the vertical extent of the fluid cell,
r=
H2/)( is the characteristic diffusive period, W =X/H is a typical
diffusive vertical velocity, E = H/L is the aspect ratio, and 4T is the
characteristic horizontal temperature difference imposed by the boundary
conditions.
After dropping the primes, the non-dimensional equations become
=
I(P,T) + 72T,
(5a)
and
e 7(5b)
where Ra =
/X
*£
dTP
7
is the "horizontal" Rayleigh number, d=
is the Prandtl number and
--
>
(5c)
where
"--
it%
0d'
-
4.
9 -
Boundary Conditions
In each case considered, the following boundary conditions are im-
posed.
(i)
Normal components of velocity are zero on all boundaries.
= 0 for x = 0, L and z = 0, H.
(ii)
Top and side boundaries of the fluid cell are both rigid
(no-slip condition) and adiabatic (thermally insulated).
Tx =k
T
(iii)
=
= 0 for x = 0, L.
, = 0 for z = H.
Thermal driving is maintained by a linear temperature distribution along the bottom.
T = T(x) = x for z = 0.
Boundary condition (i) enables us to express the stress along the bottom
boundary,
in terms of the non-dimensional vorticity,
, as
Thus, if we set a value for the vorticity along the bottom boundary, we
are in fact applying a certain stress to the fluid.
the effects of a stress free bottom boundary.
(iv)
1
=
0 along z = 0 (free slip condition).
In part I, we study
-
10 -
In part II, we study the effects of applying a constant "wind" stress
at the bottom boundary,
(v)
7
= constant along z = 0.
-
5.
11 -
Numerical Formulation
(a) Grid Geometry
Letting Ax, d z and At be increments of x, z and t, we define a
finite different grid such that
I,
x
= iAx,
i =0,
z
= jAz,
j = 0, 1,
t
= nAt,
n = 0, 1, 2, ..., N,
1, 2,
2,
... ,
...
, J,
Ax = 1/I,
Az = 1/J,
where I4land J41represent the number of horizontal and vertical grid points
respectively (Fig. 2).
(b) Numerical Approximations
In keeping with the requirement that a finite difference scheme be
both consistent and stable, the following approximations are used.
Laplacian operator,
,
The
, is given by the usual 5 point difference
scheme (Fig. 3), that is
Conservative approximations (Arakawa, 1966; Lilly, 1965) for the advection of heat, J (
,T) and of vorticity, J (Br
) are given as
-
12 -
.T( ,T '
addj I (g I -)
and
J-
C 1)
)
/* ud
J-3 ( 01 1)
where
-T ,T-) =
TL,.
-
5., (q
TL-4-,
(qk,-
I.,,P_
,p,) +
-
)
(7a)
and
F~
~(4{
L1
C + +- 1 ., .,
,T) conserves both temperature and temperature squared while J3
J2 (
(
(7b)
I,
) conserves vorticity and kinetic energy.
The time integration of equations 5a and 5b is carried out by using
the explicit scheme of DuFort and Frankel (1953).
In finite difference
I~IILI__YI____IPI__II1IX~I_
-
I~.----XII--*UWI ~^X
13 -
form the non-dimensional equations become
t- I
av-I
7-..
m+
C2At
M
IWh
+
I
tI
M1%.
lit
(8a)
and
in I
At-I
0CY')/,/
+
C-
-
, (T,:OK,
j,
(8b)
M+1
Once the "new" vorticity field,
, is predicted, the values of the
LI,
m4I
"new" stream functions,
j
, are obtained by means of an alternating
direction-implicit solution (Ralston, 1960) to the finite difference
Poisson equation,
Sisj
T)jL
.
(8c)
(c) Stability and Accuracy of the Approximations
The non-linearity of our system requires that solutions be both dif-
.
-
14 -
The diffusive stability criterion
fusively and advectively stable.
(Richtmyer, 1967), normally associated with most explicit difference
schemes, is eliminated by using the DuFort-Frankel method of time integration.
The advective stability condition, sometimes called the
Courant-Friedrick-Lewy criterion, requires that for a given grid spacing,
Ax, there exists a limit as to the size of the time increment, A t,
that may be taken.
According to Deardorff (1967) this condition is given
as
(9)
where u
max
represents the maximum velocity of the fluid.
We can satisfy
this condition by making Ax large, but in so doing, we must consider the
truncation errors that will result.
As in all numerical models, the ac-
curacy of the solutions must be weighted against the computational time
required to reach steady state.
While allowing us to avoid satisfying a
diffusive stability criterion, the DuFort-Frankel scheme will produce
spurious transient results if the time increment is much larger than
that permitted by diffusive requirements.
Initial comparisons between
the DuFort-Frankel and the standard time (Leap-Frog) and spacially centered difference approximations (Richtmyer) suggest that our scheme does
produce accurate steady state solutions.
(d)
Computational Procedure
In order to generate the steady state solutions for our model we
follow this procedure:
-
15 -
(1)
All fields are initialized.
(2)
Interior temperature values are obtained from 8a.
(3)
Temperature boundary conditions are set.
(4)
Interior vorticity values are obtained from 8b.
(5)
Interior stream functions are obtained from 8c.
(6)
Vorticity boundary conditions are set.
(7)
Kinetic energy is calculated periodically.
(8) Adjacent time steps are averaged periodically.
(9)
Solutions are checked for steady state conditions at specified intervals.
(10)
Repeat above procedures starting from (2).
A first order Taylor series approximation (Bryan, 1963) is used to
calculate the vortical distribution on the boundaries, i.e.
)
(10)
and the kinetic energy (KE) is calculated using the stream functionvorticity relationship (Appendix A),
I'KE -AAAP
(e)
(11)
Initialization
Steady state solutions are obtained for a 26 x 26 equally spaced
finite difference grid by choosing a constant temperature field (T = .5),
a symmetrical stream function,
-
16 -
and its corresponding vorticity field,
1T
as initial values.
%;o-
)
(13)
These results are linearly extrapolated to a 41 x 41
grid and the new fields are then used to initialize solutions to the finer
grid mesh.
Such a method was found to have saved them one-third to one-
half the computer time required for initial zero fields.
(f)
Steady State Requirements
Successive time steps of the temperature, stream function, and vorticity fields are averaged periodically in order to speed up convergence
and maintain computational stability.
A steady state check, requiring
the time derivatives of adjacent kinetic energy, temperature and stream
function values to be less than 1%, is performed periodically.
When each
of the selected requirements is satisfied, steady state is achieved and
the computation is terminated.
In addition to these requirements, contours of the temperature and
stream function and vorticity fields are obtained as well as plots of
the vertical temperature structure, and heat flux along the bottom boundary.
Also a complete analysis of the steady state temperature equation,
L
and normalized vorticity equation,
(14)
-
17 -
1747- =
m/I - / e
m(15)
is performed.
This includes contour plots of each component of the tem-
perature and vorticity equations as well as their x and z position plots.
All calculations were performed in the NCAR CDC-6600 computer.
For
each case considered approximately 45 minutes of computer time and between 2 and 3 non-dimensional time scales were necessary to reach steady
state.
This would be roughly equivalent to 6 hours of computer time, for
each case, on the M.I.T. IBM-360-65 computer.
IIL____IYCYLYI_~__^_~__--111-~-1.11.
-
-
18 -
6. Previous Investigations
(a) Theoretical
In 1962, H. Stommel considered a system of a number of vertical
tubes to which a non-uniform temperature distribution is applied at the
top and whose bottoms are suspended in a fluid reservoir.
Upon examina-
tion of the equations governing this system he was able to predict that
the motion is always upward in all but the coldest tube (Fig. 4).
Though
this model hardly describes an oceanic phenomenon, Stommel suggested that
this asymmetry might help explain the smallness of sinking regions in
the ocean.
(b) Experimental
In 1965, T. Rossby conducted an experiment in which a fluid cell
was heated non-uniformly from below while thermally insulated boundaries
were maintained elsewhere.
tric cell (Fig. 5).
He too observed the appearance of an asymme-
In examining the bottom boundary layer, Rossby as-
sumed that the stream function and boundary layer thickness were functions
of the applied temperature gradient.
He found them to be proportional to
Ral / 5 and Ra-1 / 5 respectively, by assuming a temperature balance between
vertical diffusion and advection, and a vortical balance between vertical
diffusion and buoyancy.
His data, based on experiments in which the Ray-
leigh number was varied only by changing the viscosity of the fluid, would
appear to agree with these results.
In 1968 a similar experiment was performed by R.C. Miller; however
he considered a saw-tooth temperature distribution along the bottom
9Li
ll~--~
-
boundary (Fig. 6).
19 -
Unlike Rossby, Miller examined a variety of cases
in which the applied temperature gradient, as well as the viscosity of
the fluid was varied.
Once again the asymmetry of the cell was evident.
Miller defined a bottom boundary layer thickness, a
, as "the distance
measured parallel to the z axis, directly below a vortex eye, from the
base of the cell to the point where )T/
z is equal to an arbitrary
His
constant" and measures it by using a modified Schlieren technique.
results suggest that S
is proportional to Ra-1/8; however, he does ad-
mit that the resolution of his Schlieren photographs is inadequate to
completely resolve a thermal boundary layer.
He also finds that for a
constant applied temperature gradient, the circulation appears to be
37
proportional to Ra' 3 7 .
A couple of Miller's streak photographs, showing
the asymmetric cell, as well as his Schlieren photographs, showing lines
of equal vertical temperature gradient, can be seen in Fig. 7.
The homo-
geneous interiors and asymmetric convective cells are clearly evident.
(c)
Numerical
Barcilon and Veronis (1965) considered numerically the same problem
as Rossby, making the applied temperature distribution cosinusoidal and
the boundaries free rather than rigid.
For Rayleigh numbers of 103,
5 x 103, and 10 , and several orders of magnitude in Prandtl number, they
found a weak asymmetric circulation, which appeared to be independent of
the Prandtl number.
Somerville (1967) considered essentially this same problem; however
he employed a different numerical technique.
He maintained the top of
-
20 -
the cell at a constant temperature rather than adiabatic, and considered a cell with an aspect ratio of .1.
His results were qualita-
tively similar to those of Barcilon and Veronis.
He too observed the
appearance of a weak asymmetric cell and also indicated that the solutions were independent of the Prandtl number.
Both achieved similar
results in suggesting that the intensity of the circulation appeared
to be proportional to Ra3/5; however, neither study could adequately
resolve a bottom boundary layer.
Since both numerical investigations
did not consider rigid boundaries, quantitative comparisons between
their results and those of Rossby and Miller cannot be made.
-
<
T
g
L
21 -
T2
i-o|m
<
R
T3 ..... Tn
•m-m
FIG. 4
KI
NhI-i
t
COLDH
/' i '1
IA
)
J
COLD
HOT
L-T T(x)
FIG. 5
COLD
FIG. 6
-
Fig. 7
o, b
Ro
22 -
3
= 1.757 x 10
= 8.274 x 1051
Exposure time
c, d
Ra = 9.580 x 105,
Exposure time
= 50.02 sec
o
= 9.077 x 10 2
= 45.09 sec
-
7.
23 -
Results (Part 1)
Steady state solutions are obtained for Rayleigh numbers of 103
104, 3 x 104, 105, 3 x 105, an aspect ratio of 1i,and a 41 x 41 point
finite difference grid.
Boundary conditions, in which a horizontal li-
near temperature gradient, T = x, and stress free (
= 0) bottom boundary
with remaining boundaries both rigid and insulated, are employed.
Pre-
liminary calculations are made for a Rayleigh number of 104 and Prandtl
numbers of 1, 10 and 100, and slightly different boundary conditions, no
slip bottom and free slip right wall.
Deviations of less than 1% in
max are found for this range in Prandtl number (Table 1).
Table 1.
1!ax_
100
10
1
2.50
2.51
2.52
(Ra = 104 )
% deviation
.4%
.8%
These results as well as the previous investigations of Rossby, Barcilon
and Veronis, and Somerville suggest that the solutions are virtually independent of Prandtl number, for 6Z 1; thus, a constant Prandtl number,
4f= 10 (approximately that of water), is chosen.
The major effects of varying the Rayleigh number can be seen in the
contours of the stream function (Fig. 8) and temperature (Fig. 9) fields.
As the Rayleigh number is increased the geometry of the cell becomes successively more asymmetrical.
The motion of the fluid is counter-clockwise,
with the formation of a jet of warm buoyant fluid near the right boundary,
accompanied by gradual sinking throughout the remainder of the cell.
Upon
-
24 -
investigating further the asymmetry of the cell, namely the x and z
movements of the vortex eye with increasing Rayleigh number (Fig. 10)
we see the linear movement of the eye towards the lower right region of
the cell (Fig. 11) with the intersection occurring at z = .15.
The upper portion of the fluid becomes nearly isothermal as the
Rayleigh number is increased.
Since the flow is counter-clockwise and
the isotherms are "tied" to the bottom boundary, we find the isotherms
becoming closely packed over the coolest portion of the bottom and showing little movement over the warmest portion of the bottom boundary.
This can easily be seen in the plots of the movement of isotherms along
the x = 0 (Fig. 12) and x = 1 (Fig. 13) boundary.
As the Rayleigh number
increases, the temperature at the center of the vortex eye approaches a
value of .7 (Fig. 14) and the mean temperature of the fluid approaches a
value between .7 and .8 (Fig. 15).
It is remarkable that these same re-
sults were observed experimentally by both Rossby and Miller, and numerically by both Somerville and Barcilon and Veronis.
This would seem to
indicate that for sufficiently large Rayleigh numbers, the temperature of
the upper layer of fluid reaches an asymptotic value which is independent
of the boundary conditions considered.
For low Rayleigh numbers the convection of heat is of little importance; this is readily seen in the nearly symmetric temperature and
stream function fields for Ra = 103.
However, for higher Rayleigh number
(>104), the convective processes become increasingly important.
Heat
is convected upward in the jet, outward and down in a weakly stratified
interior, and conducted out
as the fluid particles are being accelera-
-
25 -
ted to the right over the cooler portion of the bottom boundary.
If we
examine the quantitative growth of the stream function at the center of
the vortex eye (Fig. 16), we see a functional change in its dependence
upon the Rayleigh number for those between 10
3
4
4
and 10 , and those>10
For the larger Rayleigh numbers, the intensity of the circulation appears
to be proportional to Ra' 3 6 + .01
As mentioned before, Miller's calcu-
lations include a Ra.37 dependency, and although Rossby's results suggest
an Ra
1/5
37
dependence, the scatter in his data will include an Ra. 3 7 as
well.
Examination of the vertical temperature structure at various horizontal positions (Fig. 17A-D), enables us to calculate the bottom boundary layer thickness,
C
.
Specifically,
J
is defined as the distance
from the bottom boundary to the point where JT/
z = 0.
Plots of bottom
boundary layer thickness as a function of Rayleigh number (Fig. 18) show
that the thickness decreases with increasing Rayleigh numbers, and appears to fit a curve proportional to Raagreement with Rossby's Ra
-1/5
.
This seems to be in better
dependency than with Miller's Ra
-1/8
depen-
dency; however one must remember that the boundary conditions are slightly
different.
Also Miller's bottom boundary layer thickness is defined be-
neath a moving vortex eye, while ours is defined at fixed horizontal positions.
For each case considered we verify the attainment of steady state
by calculating the net heat flux through the bottom boundary.
Since a
non-uniform temperature distribution is maintained at the bottom, and
all other boundaries are insulated, the net heat flux must be zero at
-
steady state.
26 -
Vertical temperature gradients along z = 0 are obtained
and the net heat flux is calculated by integrating these results along
z = 0.
B).
The error in all cases considered is found to be small (Appendix
Plots of the vertical temperature gradient along the stress free
bottom (Fig. 19) indicate an increase in the heat flux entering and
leaving the system as the Rayleigh number is increased.
The nearly sym-
metric plot for the lowest Rayleigh number indicates that the heat transfer process in the fluid is primarily conductive.
However, for the lar-
ger Rayleigh numbers, we find that the transfer of heat is increasingly
convective.
As a particle slowly moves across the cooler portion of the
bottom, heat is conducted out of the system.
The particle is then ac-
celerated over the warmer portion of the bottom boundary, causing an increase in its horizontal velocity.
Concurrently, the heat flux into the
system will increase, since the particle appears to be cooler with respect to the bottom.
peak in the 3T/
city.
We find that the region of maximum heat input, the
z plot, occurs at the region of maximum horizontal velo-
Since the velocity is required to be zero at the right wall, we
see that as the particle slows down there is a decrease of heat flux into
the system.
However in the area close to the wall, due to no-slip, there
is an upswing of the curves.
This signifies a stagnant region where the
advection of heat is negligible and conduction dominates, accounting for
the similarity between the curve for pure conduction and those for the
various Rayleigh numbers.
It is interesting to note that this does not
occur near the cooler side where the same stagnant region exists.
As the
Rayleigh number increases, we find that the isotherms are compressed, the
~~X~I~__
-
__^^___il~l1_r~___
__I .LL
27 -
fluid particles near the bottom become warmer, and thus the amount of
heat leaving the system increases.
The different function relationship
for low (103) and high (>104) Rayleigh numbers is once again evident in
the plots of the magnitude
of the heat flux, IH.F.1, as a function of
Rayleigh number (Fig. 20) and ?ax (Fig.
21).
H.F1
is defined as the
average of the absolute values of heat entering and leaving the system
along the bottom boundary.
Here we find that the magnitude of heat flux
appears to be proportional to Rd
1/4 and o( = .67 + .01 ~
and
max
where
= .25 + .01
2/3.
If we define the Nusselt number, Nu, as the ratio of the total heat
flow through the system to the theoretical heat transferred by conduction only, we can express the Nusselt number in terms of the non-dimensional magnitude of the heat flux, IH.F.1, and the non-dimensional heat
transfer due to conduction, (H.F.(
o
.
Since
H.F.I c
will be inde-
pendent of the Rayleigh number, we see that Nu is directly proportional
to IH.F. 1
which in turn is proportional to Ra.
1/5
3
Nu - Ra 1/5 however Miller suggests Nu C Ra .
Rossby indicates that
Our results show a
slightly higher Rayleigh number dependency than Rossby's; however, this
is reasonable since our bottom boundary condition allows the existence
of a horizontal velocity along the bottom which will tend to increase
the magnitude of the heat flux.
number.
This in turn will increase the Nusselt
Since both Rossby and Miller had essentially the same boundary
conditions, it is hard to believe that Miller's results would be considerably different from Rossby's.
__ ___
-
28 -
In order to better examine the dynamics and non-linearity of the
system, steady state contours of the three components of the temperature
equation (Fig. 22A-C) and the four components of the vorticity equation
(Fig. 23A-D) are obtained.
The temperature calculations indicate a
highly non-linear system, in which, for increasing Rayleigh numbers, the
interior of the fluid becomes essentially homogeneous and the region near
the bottom boundary is the most active.
The steady state temperature
balances near the bottom (Fig. 24) show the increasing importance of the
advective terms while those near the right boundary (Fig. 25) show the
presence of the homogeneous upper layer of the fluid.
Unlike the temperature balances, we find that the vorticity system
is highly linear.
For all Rayleigh numbers considered, the major balance
exists between viscous dissipation and buoyancy.
The non-linear terms
are very weak for all Rayleigh numbers; however, they do make some slight
contribution (>5%) for higher Rayleigh numbers (105 ).
Their signifi-
cance is confined to the lower region of the jet through which the fluid
is accelerated and where the horizontal and vertical components of velocity are at a maximum.
Plots of the vorticity near the bottom (Fig.
26) and right wall (Fig. 27) exhibit this characteristic.
Here we can
see the growth in relative importance of the two non-linear vorticity
terms (C,D) as the Rayleigh number increases.
-
29 -
COLD
HOT
COLD
HOT
COLD
COLD
HOT
FIG. 8
Ra
COLD
Contour
From
To
Contours
Scaled by
a
10
0
b
104
0
2.7
102
c
3x10 4
0
4.2
10
d
10 5
0
7.0
10
e
3x10 5
0
9.0
.48
2
2
__l_~_l~---(~^.L~.I..-iil
~~.I-.-I---XIIX~--.
--- La~_.lt
-~O~~--tll^l-i~-P-2~.-LrrYL
--
-
30 -
COLD
HOT
COLD
HOT
COLD
HOT
COLD
HOT
FIG. 9
T
Ra
b
c
COLD
All contour plots
10
3 x 10
4
d
105
e
3 x 10 5
are from .0 to 1.0
2
scaled by 10
100
1
~
1
I I I I II
I
_ I I___
II I
I
I I I
I
I I , , II
_
1 _ ______
I
I
I
111II
I
I
I
I
_
I
I I I _I __
II
I
I
III
I
I
I I I I
A
B
1 I
,I I I
I
I
10
I I II
I
10 5
10 4
Ro
FIG. 10
A = X POSITION OF
B = Z POSITION OF
4
4
max AS A FUNCTION OF RAYLEIGH NUMBER
max AS A FUNCTION OF RAYLEIGH NUMBER
10
I
I
I
i
I
I
I
I
I
I
I
I
.4
Ro = 10
.3 5
3 x 10'
10O
.25 3x 10
I
I
I
.55
FIG. II
VARIATION OF
X-Z PLANE
I.
I
I
I
I
I
.65
X
Position
I
I
I
I
.7
max AS A FUNCTION OF RAYLEIGH NUMBER IN THE
I
II II II q__I_
I
I
I 1 11111
I
11 1 1
III__ _W
II II I I II
I
fl
YV
II
I
I I I I II I
I
I
I
I
I
I
I
I
I
I
I
I
I II
I11
z
'0-'
I I I
.
.
I
0
1O s
4
Ra
FIG. 12
I I I II
,I,
I
I I
I I
106
POSITION OF THE ISOTHERMS, T = .35 (A) AND T: .30 (B),
ALONG X=O AS A FUNCTION OF RAYLEIGH NUMBER
I I
I I
I lii
II
I
I
I
I IIIJ
I I I I f il
I
I
I
1lII
II I 1 1111
_
I
I
I
I
I
I
I
11
IIll
I0-I
,,B,
I I I I11
103
I
I
I
I I I I Il
1 1 11
1
I
I II II II iI il i'll
I
I
I
II II II II II II II
105
104
106
Ro
FIG. 13
POSITION OF THE ISOTHERMS,
T= .85 (A) AND T = .90
ALONG X= I AS A FUNCTION OF RAYLEIGH NUMBER
(B)
100
_ _ _____
I
I
I
I
I
II
I
I
_ _ ____
I lI
I
II
I
I
_ _____
I I I I
I
I
I
_ ____
I
I I
I
i-
lim
iiiiiiiiiii0
xww
IO'
I I
. .
I
I
10
FIG. 14
I
II
..
I
I
I
1 11111
I.
.
I
I
I
3
Ro
I
I II
1111
. . . ...
S
.I
. I. I.I ..I I
I
IO5
TEMPERATURE AT THE CENTER OF THE VORTEX EYE AS A
FUNCTION OF RAYLEIGH NUMBER
106
~~~1~
_ Iill1 1 IIJ-~--~
"1
I I I 11
-I
1
I1 I1 I
I
I
11111
11 1
I 111111
104
FIG. 15
1
1 I~~J uhf
I
I
I
1 11111
I11111
I
I
I
I
I
I I liii
I I I II I
I
I I l i t
6
Ra
MEAN TEMPERATURE AS A FUNCTION OF RAYLEIGH NUMBER
10
- 37 -
10I
mox
100
10'
IO"'
I
I0
, , , , ,,
I
1
I10
11111
IOs
I
, 11,,,
Ro
FIG. 16
MAXIMUM VALUE OF STREAM FUNCTION (I max) AS A FUNCTION
OF RAYLEIGH NUMBER ( Ro )
o10
-
t.00
.co
.00
....
.
.
.
....
....
....
.
38 -
. ....
0
.,
b
.90
.00
.50
.60
.40.
.30
.70
0.0
.50
.20
.10
I.,.
.
l"O.49
.0
.
.g
.9
..
wO
.6
."
..
.6
.62
UI
.04
a
..
4
.0
.
6
.40
.70
.20
.50
..00
.
.00
.
.
.. 6
.5
.65
.76
.
FIG. 17A
Plot of Vertical Tem-
perature Structure at
.90
x = .50.
abscissa = T
ordinate = z
.80
.70
.60
Ra
.50
103
10
.40
.36
3 x 10
.20
105
.00
.45
.50
.
.
.00
.65
.70
.
.
5
3 x 10
.
- 39
I. an
a
b
.90
.00
.70
.60
.50
.40.30
.20
.20
I
.0O
)((
.00
0.O06
60
.
'
"'
.
.53
.6
;
....
-
.
.
.
.
O.00|
.
,555
.550
.345
.540
.0
.
.
.
2
,
.04
.62
.6
.m
.7
,,vv
C
.90
i
.80
.70
.60
.50
.40
.30
.20
.10
52
.94
.5
.8
.6
.62
.64
.66
.68
.70
I
I
0.00
.,
1-a
.90
o
.95
FIG. 17 B
e
Ra
..
l
/
.00
.50
.10E
-.
L-
.55
-
i i l l
.60
....
.
.0
..
..
.70
.70
.
.75
.75
perature Structure at
x = .55.
abscissa = T
ordinate = z
2
. .
.0
Plot of Vertical Tem-
.70
.
.60
.0
a
103
b
104
c
3 x 104
d
105
e
3 x 105
-
40 -
~~.~~.-
......-'--~'~'~'
''''''''''''-'''--
I.
0
.90
I .
. .
.
, , ,
.
. . . .
. . . .
.I
I r "
.60
.70
.60
. . . . .
. . ..
. . . ..
. . . .
. . . .
.50
.40
.a
.21
.60
.
..
.570 .60
. . .i. . . .. . . . . ., . . . . J. . .
; .. .
............ ...l i ---l ....I ....
.640
.A0
.650
.620
.600 .610
.699
I"
.6M
.6O1 .420
.
.63
.0
.670
.660
. 0
d
.
.O0
.30
.
.-
.20
.
.10
16
.76
.64.6
a.62
.72
6
.60
FIG. 17 C
e
I
/
-
.62
.64
|
.66
I
I
.
m
.60
I
I-7-.....
.62
.
.64
.
.66
.
.
.61
.
I
.60
.
.70
.
.
.72
.
.
.74
.
I
.70
,
.72
Plot of Vertical Temperature Structure at
x = .65.
abscissa = T
ordinate = z
Ra
10
.1O
O.
- - ' --
103
10
104
4
3 x 10
.
-
.20
.-
5
5
3 x 10
.6
-
41 1.00
.90
b
.80
.70
.60
.50o
.40
.30
.20
.oo0
4
C
.670
.MI
.6"
.740
.70
.20
.710
.700
,
,
00.
.70
.90
.80
.70
.60
.50
.40
.50
.20
66
.66
.4
.72
.70
.A
.
.00
.O0
.60
I
|
I
.64
FIG. 17 D
.90
.70
0.00 .62
e
i
.66
i
I
.60
l
i
abscissa = T
ordinate = z
Ra
3
.40
10
.30
10
.20
3 x 10
.10
10
4
4
5
.66
-6--
6
.
.
- -,[
.
-.
.74
Plot of Vertical Temperature Structure at
x = .75.
.50
0.000.0.62
.
.72
.70
5
3 x 10
'p1
.U
I
I11111
I'
-
Z.
1
I
1 II
1
I
I
I
I
I
I
5
10-
II
_ _
I
I
I
I
1111
_ ___
10 '4
FIG. 18
BOUNDARY LAYER THICKNESS (8)
AT VARIOUS X POSITIONS
I
I
I II _111
_ ___
10
AS A FUNCTION OF RAYLEIGH NUMBER ( Ro)
$new x
0..
*
C-
I
I
I I l
j
I
I1 11
I
I
I
I I
I
I I I1 1111
I
.
I
I
I .
.
...
I I I I II
I
I
I
I
.I.1
1
I I I II
100
IH.F.I
I
P
rP
I
I
I I III
I
103
I
I
I
I11
I
.II
..
I I
.
I
I
I
. . .I
I
I
I
I
II
I
106
Ro
FIG. 20
I
MAGNITUDE OF HEAT FLUX,
OF RAYLEIGH NUMBER
IH.F.
,
AS A FUNCTION
I
I
I
I
I
I
I
I I I
I
I
I
I
I
I I I
I
I
II
to
.3
4)
100
I
II
I I I I
I
•I
100
L
I
i Ii
I
I
I
I
I I I
I
I
I
I
.10'
max
FIG. 21
MAGNITUDE OF HEAT FLUX, IH.F.I, AS A FUNCTION
THE STRENGTH OF THE CIRCULATION, *fmox
OF
-
46 -
COLD
HOT
COLD
FIG. 22A
Ra
COLD
2
V
Contour
From
To
T
Contours
Scaled by
2
a
103
-
b
104
-6.0
15.0
101
c
3x10 4
-12.0
28.0
101
d
105
-25.0
50.0
101
e
3x10 5
-40.0
-- 88.0
10
.6
2.4
10
1
-
COLD
47 -
HOT
COLD
COLD
FIG. 22 8
- uTK
Contour
From
To
Ra
COLD
HOT
COLD
Contours
Scaled by
2
10 3
- 2.5
104
-15.0
0.0
101
3x10l
-29.0
1.0
101
105
-54.0
3.0
101
-96.0
6.0
101
3x10
5
.1
10
_
-
COLD
HOT
COLD
HOT
~I~
48 -
HOT
COLD
FIG. 22 C
Ra
COLD
-wTz
Contour
From
To
Contours
Scaled by
.85
103
7.0
102
a
103
b
104
- 1.0
c
3x10 4
- 2.0
14.0
101
d
105
- 6.0
26.0
101
e
3x10 5
-12.0
48.0
101
~_~_
LYII/~~_I~_______l_*ri~-~li~iY-l-~
-
49 -
COLD
COLD
HOT
COLD
HOT
COLD
HOT
FIG. 23A
Ra
COLD
V 7 /Ra
Contour
From
To
Contours
Scaled by
.96
a
103
b
104
0
1.02
102
c
3x10 4
0
1.12
102
d
105
0
1.44
e
3x10 5
-. 1
1.90
102
-
COLD
50 -
HOT
COLD
COLD
HOT
COLD
HOT
FIG.23B
Ra
COLD
Contour
From
To
Contours
Scaled by
a
103
-
.96
0
103
b
104
-1.02
0
102
c
3x104
-1.12
0
102
d
105
-1.44
0
102
e
3x10 5
-1.80
.1
102
-
51 -
COLD
COLD
HOT
COLD
HOT
COLD
HOT
FIG. 23 C
Ra
COLD
-u
x / Roo-
Contour
From
To
Contours
Scaled by
a
103
-.0048
.0042
105
b
104
-.028
.012
104
c
3x10 4
-.052
.024
104
d
105
-.099
.045
e
3x10 5
-.170
.080
II
-
__
-l~--. L_~-1LL--_II
52 -
COLD
COLD
COLD
COLD
HOT
FIG. 23 D
-W77z / RocContour
Ra
103
From
To
Contours
Scaled by
-. 0027
.0(027
105
-.020
16
.0:
10
-.036
.0:28
10
-. 070
.04 49
10
-.120
.0180
103
104
4
3x10
5
10
3x105
COLD
-
53 -
FIG. 24
Temperature Balance at z = .05
abscissa = x
Ra
a
103
A = T
b
104
B = T
c
4
3 x 10
d
105
e
3 x 10 5
xx
zz
C = -wT
z
D = -uT
x
-
54 -
d
20
-10
-20
-50
'8 00
.,0
.20
.50
e
60
.40
.50
.60
.70
.80
.90
,.00
FIG. 25
Temperature Balance at x = .90
abscissa = z
Ra
10 3
A = T
104
B = T
xx
-20
-40
3 x 10
-60
105
-80
0
S.00
.
1..
.10
. .
.20
..
.30
0
.40
.0. . .
.50
.60
.
.
.70
80
1
.80
.
.90
1 .0
1.00
3 x 105
zz
C = -wT
z
D = -uT
x
-
55 -
...........
....
... I
I
-
I
a',
7
.6
-.
"
.8...........
.
. o
.20
.50
.5o0
.40
. 70
80
90
1.00
....
.0o
.20
.3o
.0
.50
.60
.70
.80
.90
.00
SA.
..",
""
....... .. . . .......
::1"
~-.,.... ............... E... .... ...
... ....
..
-1.0 F
t.oo
>-
D
SI
.to
.20
.0'
.
.
~ .0
o0 .70
.80
.90
1.00
I2
e
FIG. 26
.
Vorticity Balance at z = .05
abscissa = x
.
.......
Ra
a
......... ..... .....
10
A = xx /Ra
l
4
10
-1.0
3 x 10 4
-0.5
10
B =T
/Ra
zz
C = -wl
.o10.20o.3
.0
.
6o .6
.7o .e0 .90 1.00
3 x 10
/RaG
D = -unx /Raa
-' n|
*oo
z
E = -T
x
-
56 -
d
'.1oo
.o
.
o
.
. 0
50.
.
60
70
. 0
.90 t.00
FIG. 27
Vorticity Balance at x = .90
abscissa = z
Ra
a
10
A =
xx/Ra
b
10
B =
zz/Ra
c
3 x 10 4
C = -wz/Ra
d
105
D = -unx/Rau
e
3 x 10 5
E = -T
x
-
8.
57 -
Results (Part 2)
In the next series of experiments we employ a second set of bound-
ary conditions in which a uniform "wind"stress, I = constant, is maintained along the bottom in opposition to the thermal circulation.
The
Rayleigh number is held constant at 104 as vorticities of 0, 100, 200,
400, and 600 are applied at the bottom boundary.
A Rayleigh number of
104 enables us to examine the effects of the stress on a convective cell,
while minimizing the error in the heat flux calculation.
Since the stress opposes the prevailing circulation, we might expect the motion in the cell to simply slow down, stop and eventually reverse itself as the stress is increased.
this does not occur.
Contrary to our expectations,
Examination of the stream function contours (Fig.
28) reveals the evolution of a stress driven bottom circulation.
the I
= 200, we see that the opposing circulation is weak.
For
The warm
fluid, that is slowly being advected over the bottom toward the left hand
side, cools rapidly and is unable to rise any appreciable distance.
the stress increases, the opposing circulation becomes stronger.
As
The
strength of the thermal circulation decreases (Fig. 29) accompanied by
an upward movement of the vortex eye (Fig. 30).
With the continued
growth of the stress there is a change from a dual system, containing
both thermal and stress driven circulations of the same magnitude, to a
system where the thermal circulation becomes non-existent.
Plots of the vertical temperature structure at x = .5 (Fig. 31) as
well as the plot of the mean temperature (Fig. 32) indicate the existence
of a dual state for 200<)
400, and a stress dominant state for I
400.
-
58 -
As the stress increases, the mean temperature of the fluid decreases
and approaches an asymptotic value between .3 and .4.
The clockwise
motion of the stress driven circulation is now carrying cooler fluid
into the interior.
If we examine the contours of the temperature field
(Fig. 33) we see a gradual bending back of the isotherms for the lower
stresses, ) : 200.
For the higher stresses studied,
-)r 400, we see
that the isotherms are "whipped" around and exhibit close packing over
the warmer portion of the bottom boundary.
The movement of isotherms
along x = 1 (Fig. 34) also indicates this packing over the warmer portion of the bottom, while the movement of the isotherms along x = 0
(Fig. 35) shows little change over the coolest section of fluid.
This
is a complete reversal from the effects of the counter-clockwise thermal
circulation in the previous section.
At steady state, the vertical temperature gradient along the bottom
is calculated and the net heat flux is obtained by integrating these results along z = 0 (Fig. 36).
By increasing the stress, we find that the
magnitude of the heat flux first decreases and then increases.
This is
also true for the kinetic energy calculations (Table 2).
Table 2.
SNetR.F.
HF.
Relative Error
K.E.
0
100
200
400
.0078
-.0085
-.0208
-.0556
.640
.531
.456
.726
1.2%
- 1.6%
- 4.5%
- 7.6%
600
17.64
10.92
8.43
22.42
t.1423
1.360
-10.5%
128&.01
59 -
-
For the lower stresses, work is being done against the system by slowing down the thermal circulation.
However for the higher stresses, in
which the circulation is stress dominant, the work no longer opposes
the circulation.
Thus the kinetic energy increases, the isotherms are
"whipped" around and the magnitude of the heat flux increases.
For the
higher stresses, warm fluid particles are being advected along the bottom towards the cooler portion of the tank.
As they move across the
cooler bottom, the heat leaving the system increases.
The maximum hori-
zontal velocity, just above the boundary, is now associated with the
maximum heat output.
The no-slip condition at x = 0 produces a stagnant
region in which advection is negligible and conduction dominates.
This
accounts for the similarity between the pure conduction and high stress
curves in this region.
However, this does not occur near x = 1.
As the
stress increases, the clockwise motion of the cell compresses the isotherms over this warmer region.
The fluid next to the bottom becomes
cooler thus producing an increase in the heat entering the system.
Again,
this is a complete reversal from the case in which the Rayleigh number is
varied.
Contours of the steady state temperature and vorticity components,
similar to those presented in the previous section, can be found in Appendix C.
These plots again reflect the change from a thermal, to a dual,
to a stress driven circulation, with the non-linear vorticitv terms becoming increasingly important near the bottom boundary.
-
60 -
HOT
COLD
HOT
COLD
COLD
FIG. 28
n
COLD
HOT
Contour
From
To
Contours
Scaled by
a
0
0.0
2.7
102
b
100
- .3
1.8
10 2
c
200
- .2
1.6
10
d
400
-2.1
.3
10
e
600
-6.4
0.0
10
2
2
2
-
61 -
3.0
2.0
Tmnox -
1.0-
0
FIG.29
100
200
300
400
Stress (liz.o)
STRENGTH OF THERMAL CIRCULATION
FUNCTION OF APPLIED STRESS
500
('Tmax)
600
AS A
- 62 -
.8
.7
.6
.5
4
.3
.2
0
I
I
100
200
300
400
500
Stress (lz,o)
FIG. 30
VERTICAL MOVEMENT OF THERMAL CIRCULATION VORTEX
EYE AS A FUNCTION OF STRESS
600
- 63 -
.7
.6
M
e
0 .5
n
t
e
m
p
e
r .4a
t
u
e
.3
.2
0
I
100
I
200
I
300
I
400
I
500
Stress ('7z.o)
FIG. 31
MEAN TEMPERATURE AS A FUNCTION OF THE
APPLIED STRESS
- 64 -
0
I
" ' ' ' ' ' ' ' ' ' ~ ~~
A0
.5
.5
.56
.56A
.
6
.6
.*66
FIG. 32
Plot of Vertical Temperature Structure at
x = .50.
abscissa = T
ordinate = z
1r
0
100
200
400
600
-
65 -
b
COLD
COLQ
HOT
d
COLD
HOT
HOT
COLD
FIG. 33
T
a
0
b
100
All contour
c
200
plots are from
.0 to 1.0 scaled
COLD
d
400
e
600
by 102.
- 66 -
.6
.5
.4
Z
.3
.2
.1
0
Stress ( r z =o)
FIG.34
MOVEMENT OF ISOTHERMS AS A FUNCTION OF STRESS
A = Plot of T= .70 isotherm along X = 1.0
B
=
Plot of T= .75 isotherm along X= 1.0
-
67 -
.5
.4
Z
.3
.2
0
0
100
200
Stress
FIG. 35
300
400
500
(/z, o )
MOVEMENT OF ISOTHERMS AS A FUNCTION OF STRESS
A = PLOT OF T = .25 ISOTHERM ALONG X = 0.0
B=PLOT OF T=.20 ISOTHERM ALONG X=O.0
600
...... __.I___1~~II___L
_I___
I__I ~_~1
I
I
ST
8Z
X Axis
SS
PLoTS or TMs VOTCL RTMPEATURE UANT AT TE
OF HORIONTA POSITION
SOMu
#WroM . (
).
, A PUTomN
-
69 -
Summary
A numerical simulation of thermal convection in which the circulation is driven by a non-uniform temperature distribution along the bottom
boundary was presented.
The first case studied involved a cell with a
stress free bottom boundary.
several Rayleigh numbers, 103
6
=10.
Steady state solutions were obtained for
Ra
3 x 105 and a single Prandtl number,
Temperature and stream function contours and heat flux calcu-
lations along the bottom boundary were examined.
It was found that with
increasing Rayleigh number the cell became increasingly asymmetrical with
the formation of a warm buoyant jet close to the lower right boundary.
The top was isothermal and the cool bottom exhibited closely packed isotherms.
For large Rayleigh numbers the intensity of the circulation
36 + .01
seemed to be proportional to Ra'3 6
.0
This appears to be in better
agreement with portions of Miller's data than with Rossby's results,
however the scatter in Rossby's data will include this result as well.
Upon examination of the bottom boundary layer, it was found that the
boundary layer thickness appeared to be proportional to Ra
This is
in better agreement with Rossby's results than with Miller's, however
Miller defines his boundary layer thickness differently than we do.
Heat
flux calculations made along the bottom boundary indicated a stagnant
conductive region near the warmest end.
The magnitude of the heat flux
increased with increasing Rayleigh number and appeared to be proportional
to Ral /4
-
70 -
In the second case studied, the Rayleigh number was held constant,
Ra = 10
,
and a stress was applied along the bottom boundary in opposi-
tion to the thermal circulation.
Stream function contours showed the ap-
pearance of a weak stress driven bottom circulation for the smaller stress.
As the stress was increased the thermal circulation weakened, the vortex
eye rose, and finally the stress driven circulation dominated.
With the
highest stress the thermal circulation disappeared and the mean temperature decreased as the isotherms became packed over the warm portion of
the bottom boundary.
A complete reversal of the counter-clockwise circu-
lation seen in the previous study resulted.
Calculations of the magni-
tude of heat flux along the bottom first showed a decrease then an
crease as the applied stress increased.
in-
For low stresses, work is being
done against the system by slowing down the thermal circulation.
The
isotherms slowly reverse themselves and the magnitude of the vertical
temperature gradients along the bottom decreases.
However for the higher
stresses, in which the circulation is stress dominant, the work no longer
opposes the circulation.
The isotherms are "whipped" around and there is
a net increase in the magnitude of the heat flux.
A stagnant conductive
region, which appeared over the warmest end in the first study, appears
near the coolest portion of the bottom boundary for the higher stresses.
Future Work
Development of finite difference models of these problems using a
non-uniform grid might be the next step in this investigation.
Use of
such a model could possibly enable the exploration of higher Rayleigh
- 71 -
numbers and applied stresses.
An experiment simulating a convective cell
in which a stress is applied at the boundary of the thermal driving might
also aid in further understanding the problem of thermal-wind oceanic
circulations.
-
72 -
BIBLIOGRAPHY
Arakawa, A. (1966) Computational Design for Long-Term Numerical Integratipn of the Equations of Fluid Motion: Two-Dimensional Incompressible Flow. Part 1., J. Computational Physics, 1, 119-143.
Bryan, K. (1963) A Numerical Investigation of a Nonlinear Model of a
Wind Driven Ocean, J. Atm. Sci., 20 (6), 594-606.
Deardorff, J.W. (1967)
215-228.
Thermal Convection:
A Colloquium, NCAR-TN-24,
DuFort, E.C., and Frankel, S.P. (1953) Stability Conditions in the
Numerical Treatment of Parabolic Differential Equations, Math.
Tables and other Aids to Computation, 7, 135.
Lilly, D.K. (1965) On the Computational Stability of Numerical Solutions of Time-Dependent Non-Linear Geophysical Fluid Dynamics
Problems, Monthly Wea. Rev., 93, 11-26.
Miller, R.C. (1968) A Thermally Convecting Fluid Heated Non-Uniformly
from Below, Unpublished Phd. Thesis, M.I.T.
Ralston, A. (1960) Mathematical Methods for Digital Computers, Wiley &
Sons, New York.
Richtmyer, R.D., and Morton, K.W. (1967) Difference Methods for Initial
Value Problems, Wiley & Sons, New York.
Rossby, H.T. (1965) On Thermal Convection Driven by Non-Uniform Heating
from Below: An Experimental Study, Deep Sea Research, 12, 9-16.
Somerville, R.C.J. (1967) A Non-Linear Spectral Model of Convection in
a Fluid Unevenly Heated From Below, J. Atm. Sci., 24 (6), 665-676.
Spiegel, E.A., and Veronis, G. (1960) On the Boussinesq Approximation
for a Compressible Fluid, Astrophys. J., 131, 442-447.
Stommel, H. (1962) On the Smallness of Sinking Regions in the Ocean,
Proc. Nat. Acad. Sci., 48, 766-772.
Veronis, G., and Barcilon, V. (1965) Thermal Convection Generated by
Differential Heating along a Horizontal Surface, Unpublished paper,
M.I.T.
-
73 -
Acknowledgements
I wish to thank Professor Robert Beardsley for his guidance and
encouragement during the course of this study.
I am especailly grateful
for his patience and understanding in the many discussions we had.
I would also like to thank Dr.'s J. Deardorff, A. Kasahara, R.
Miller, N. Phillips, P. Saunders, and W. Washington for their discussions
and helpful suggestions related to this investigation.
I am indebted to the National Center for Atmospheric Research,
sponsored by N.S.F., for allowing me to use their CDC-6600 computer.
special thanks go
My
to Mrs. Jeanne Adams and David Robertson for their
help when programming difficulties arose, and to all others at N.C.A.R.
who introduced me to scientific programming a few years ago.
I also thank Mr. S. Ricci who helped with the drafting and Miss D.
Lippincott who did the typing.
The research reported in this paper was supported by the Office of
Naval Research (Grant number NONR-1841(74)), and the Atmospheric Science
Program, N.S.F. (Grant number GP5053).
74 -
-
Appendix A
Kinetic energy calculations are based on the stream function-vorticity relationship
kE
-AX'
i4
-
I
qkj C'
(Al)
!to dW*
Here we will show, for the boundary condition employed, that this is
equivalent to the familiar expression
kE =
( U.'+ LIc)dxde .
'I/
(A2)
By transforming equation Al into its integral form, we have
Ih dA .
A
Since
we have
k E - Y,SY 0 V'dA
=- -
V.
dA.
Now
V
kE'
9 .W7
'bPi4
dA
A.
7
V;ptr
(A3)
-
75 -
By transforming the area integral,
into the line integral,
vq .
and requiring
4
,Ads
= 0 on all boundaries, we find
r
SS~ i;. $P ~dA -
pV.,
Therefore,
k
ker
/E
l S 1 '* P dA
/ .
U +te) 'Ix<j e
s-=o
-
76 -
Appendix B
Calculations of the heat flow through the bottom boundary are made
by line integrating the heat equation around a small box near the boundary.
This is done by transforming the area integral of the steady state
temperature equation
( VT-
5
dA = o,
into the line integral
o
' T- a ).A
Since
A
A
we have
Now
and
as
therefore,
T. ,^ds9
Td
=
o.
-
77 -
For a typical box
3
S,
we have
VTo
S, + Ta St -Tx 5s -T
s
and
T' Tdr -
)+
T(
- )V}.
Since
d
-
and
Sic Sz.= S
= S=
= AX =AZ >
we find
Thus, we have an expression for the vertical temperature gradient along
the bottom boundary.
By placing the corners of the box at grid points
-
78 -
and then between grid points, we are able to obtain a large number of
values of
T/3 z and plot a smooth curve.
The values of
3T/I z at
each wall are calculated by using the first point above the boundary;
i.e.)
( T/t j
%-
(Toll -7bO)6VbR,
.T
and
(JT/j)~F
Listed in Table B-l are the relative errors associated with each
case considered.
The relative error,
is proportional to the net heat flux divided by the magnitude of heat
entering or leaving the system.
Table B-l.
Ra
Net H.F.
IH.F.1
Relative Error
10 3
.0027
.41
.67%
10
.0078
.64
1.22%
.0250
.87
2.9%
3 x 10
10 5
3 x 10
5
.0750
1.18
.1708
1.55
6.4%
10.9%
Here we see that an increase in Rayleigh number is associated with
an increase in relative error.
This seems reasonable since the boundary
- 79 -
layer thickness decreases and thus our finite difference approximations
for
)T/
z are not as accurate.
This is particularly true in the re-
gion of change between heat input and output, where the major errors will
occur.
In this region the temperature gradients are small and the dyna-
mics of the system will not appreciably be changed.
-
80 -
Appendix C
Contours of the steady state temperature and vorticity components.
V2T
Fig. C'l
Fig. C-2
- QTX/
Fig. C-3
- WTe
Fig. C-4
Fig. C-5
-
Fig. C-6
-
Fig. C-7
-
4
40 dr
t
d
-
81 -
COLD
HOT
COLD
HOT
COLD
HOT
COLD
HOT
V2 T
FIG. C-I
Contour
From
To
COLD
Contours
Scaled by
a
0
b
100
-3.6
6.6
c
200
-6.3
4.9
d
400
-30.0
18.0
101
e
600
-70.0
49.0
101
-
.6
15.0
101
102
102
-
COLD
82 -
HOT
COLD
HOT
==
COLD
HOT
2
HOT
COLD
- uT
FIG. C-2
n
a
0
Contour
To
From
-15.0
0.0
Contours
Scaled by
101
2
b
100
-7.0
1.0
10
2
10
COLD
6.3
c
200
-4.9
d
400
-2.0
32.0
e
600
-4.0
68.0
101
101
-
83 -
COLD
COLD
COLD
HOT
COLD
HOT
FIG. C-3
COLD
-wTz
Contours
Scaled by
T1
Contour
From
To
a
0
-1.0
7.0
102
b
100
-1.2
3.9
102
c
200
-3.0
2.4
102
d
400
-20.0
1.0
101
e
600
-51.0
6.0
101
-
84 -
COLD
COLD
COLD
HOT
COLD
V2 77/Ra
FIG. C-4
nI
COLD
Contour
From
To
a
0
0.0
1.02
b
100
0.0
.96
c
200
0.0
.96
d
400
0.0
.96
e
600
-0.3
1.30
Contours
Scaled by
102
102
-
85 -
COLD
HOT
COLD
COLD
HOT
COLD
HOT
FIG. C-5
Contour
From
To
a
0
-1.02
0.0
b
100
- .96
0.0
c
200
- .96
0.0
d
400
-1.02
0.0
e
600
-1.30
0.3
Contours
Scaled by
102
103
103
102
COLD
102
-
COLD
HOT
COLD
HOT
86 -
HOT
COLD
COLD
FIG. C-6
-u r7x/ Rao,
Contour
From
To
COLD
a
0
-.028
.012
b
100
-.016
.009
c
200
-.027
.030
d
400
-.180
.160
e
600
-.450
.400
Contours
Scaled by
(I--Lr.-r.^llrxl
l- 1~--~111-.
.--ii-sl1lll~
- 1111
-
COLD
HOT
COLD
HOT
87 -
COLD
HOT
HOT
COLD
FIG. C-7
-w 71z / Ro c
Contour
To
Contours
Scaled by
1
From
0
-.020
.016
10
100
-.011
.019
10
200
-.021
.030
10
400
-.200
.180
10
600
-.560
.490
4
4
4
3
3
COLD
10