48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2010, Orlando, Florida
AIAA 2010-1480
Scaling Parameters in Buoyancy-Driven Flows
John Kizito1, Richard Opoku
North Carolina A&T State University, Greensboro, NC 27411
Buoyancy driven flow over a vertically heated flat plate ensues due to fluid density
gradient in a gravitational field caused by density gradients due to temperature
effects. In this configuration when the density gradient is orthogonal to the
gravitational vector, any density gradient results into immediate flow. Buoyancydriven convection over a vertically heated flat plate can be modeled and solved by
Grashof and/or Rayleigh scaling parameters. The solution to the natural convection
over a heated flat plate was first solved by S. Ostrach in 1952 using Grashof scaling
parameters. The analytical solution obtained in such flows depends on the scaling
parameters used in the derivation of the similarity equations to be solved. We have
solved the similarity problem obtained by a variety of scaling parameters to
describe the development of boundary layer and the magnitude of the velocity that
ensues. The solution obtained using Grashof and/or Rayleigh scaling parameters
have been compared with numerical results using Boussinesq approximation by
comparing the thermal and velocity magnitudes obtained. We observed that the
system of ordinary differential equations which describe the natural convection over
a vertical flat plate become stiff with Rayleigh scaling when considering low Prandtl
number fluids. The converse is true for high Prandtl numbers. Our results show
that when these different scaling methods are compared in a dimensional form, both
methods are identical.
Nomenclature
Gr
Pr
Ra
ν
α
β
Re
η
ƒ
uo
L
u
v
=
=
=
=
=
=
=
=
=
=
=
=
=
Grashof number
Prandtl number
Rayleigh number
kinematic viscosity
thermal diffusivity
volumetric expansion coefficient
Reynolds number
similarity length scale
similarity velocity
reference velocity
characteristic length scale
velocity in x-direction
velocity in y-direction
1
Assistant Professor, Mechanical Engineering, 1601 East Market St, North Carolina A&T State University,
Greensboro, NC 27411, AIAA Member.
Copyright © 2010 by John Kizito. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
I.
Introduction
Buoyancy-driven flows are important in systems where the fluid has variable density in a gravitational
field. The parameter which describes the importance of these flows is proportional to the cubic
characteristic length especially when the length is oriented parallel to the gravitational vector. These
flows are still significant in microgravity (or lunar gravitational field) where the gravitational vector
magnitude is smaller than that of normal Earth’s gravity but the length scale is significant. In
microgravity and lunar gravitational field the cryogenic thermal management systems are still in need of
development. For example, buoyancy plays a significant role is in fuel storage of hydrogen/oxygen in
space vehicle and/or satellites. The fuel in these space systems is kept at cryogenic temperatures and heat
leakages into the storage system require deployment of heat removal assets. The heat comes from a
variety of sources, solar being the main source which causes the temperatures to be as high as 250oF
(121oC) at the side exposed to the solar radiation while on the dark side the temperatures may plunge to
minus 250oF (-157oC) if no thermal control hardware were present. A leading design strategy to control
temperature fields in the fuel storage tanks is the Zero Boil-Off (ZBO) pressure control concepts [8, 9].
Buoyancy effects can be significant in issues like long-term storage of fuel, pressure control i.e. ZBO,
mass gauging of propellant quantities [10]. In summary, for lunar and other planetary missions the
thermal control options are limited by the size and materials which can cost effectively be carried
onboard.
In this paper, we have examined the coupling between the thermal and momentum fields using scaling
and numerical methods first suggested by Ostrach [2] who used Grashof number coordinate stretching in
a similarity problem. We have compared these results with subsequent researchers (i.e. Bejan [5]) who
claimed better scaling intuition. Bejan uses Rayleigh number coordinate stretching in this similarity
problem. Whereas the similarity results are different, the results in this paper show that when converted
back to dimensional coordinates, Ostrach results still holds.
The similarity solutions have been obtained using both Grashof and Rayleigh scaling parameters to
describe the development of boundary layer and the magnitude of the velocity that ensues. T he similarity
equations are also solved numerically and the results are compared with the analytical solutions. The
existence of a thermal boundary layer thickness describes the fundamental length scale because it is over
this length that the buoyancy force acts. The magnitude of the reference velocity can be obtained from
equation 1 and 2, derived by Ostrach, for small and large Prandtl numbers respectively [1]. The Grashof
number is described as:
3
g (Ts  T ) L
GrL 
2

For low Pr number ( Gr  1 and
u 0  g T L 
1/ 2
 Gr 
1/ 2

L
Pr  1 )
(1)
For large Pr number ( Gr  1 and
 g T L 
u0  

 Pr 
1/ 2
1/ 2
 Gr 
 
 Pr 

L
Pr  1 )
(2)
The estimated reference velocity for air, water and mercury has been computed using equations 1 and
2 and is reported in Table 1. The reference velocities have been computed for a characteristic length of
0.5 meters and a temperature difference of 10 K.
Table 1: Estimates of reference velocities for Air, Mercury and Water
Low Prandtl number (Pr < 1)
Air [Pr = 0.71]
[β = 0.00343(l/K)]
u0 = 0.410 [m/s]
High Prandtl number (Pr >1)
Water [Pr = 7.5]
[β = 207 x10-6 (l/K) ]
u0 = 0.038 [m/s]
Mercury [Pr = 0.025]
[β = 18 x10-6(l/K)]
u0 = 0.094 [m/s]
The estimated reference velocities shown in table 1 are higher than the actual velocities but are of the
same order of magnitude as those obtained in the similarity or numerical solutions.
II.
Analytical Methods
Theoretical models based on Grashof and Rayleigh scaling parameters are analyzed for the buoyancy
driven convection problem. The laminar free convection flow equations with heat transfer and their
boundary conditions to be solved in non-dimensional form are presented as equations 3 [3]. Following
these previous researcher, a steady-state, two-dimensional, constant properties, and the validity of the
boundary layer approximations have been assumed. The corresponding similarity equations to be solved
which were obtained with both Grashof and Rayleigh scaling parameters are presented below.
Scaling parameters:
x* 
x
L
y* 
y
L
u* 
u
u0
v* 
v
u0
T* 
T  T
Ts  T
Boundary Conditions
{ y*  0,
u*  v*  0,
T *  1}
{ y*  ,
u*  0,
T *  0}
Scaled Momentum and Energy Equations
u *
u * g (Ts  T ) L *
1  2 u*
 v*

T

2
Re L y*2
u0
x*
y*
T *
T *
1  2T *
u*
 v*

2
x*
y* Re L Pr y*
u*
(3a)
(3b)
A. Scaling Using Grashof Number
In buoyancy driven flows, the causality force in these flows can be balanced by either viscous forces
or inertia forces which are resisting the flow depending on the Prandtl number describing the thermal
physical properties of the fluid. The causality characteristic thermal length scale must be the fundamental
length because it is over that length that the buoyancy force acts. Therefore, a coordinate stretching is
required in equation 3a to make the viscous term of order unity and comparable to the buoyancy term.
This results in characteristic reference velocity u0  g T L .
1/ 2
When we require coordinate
stretching in equation 3b, the conduction term will be of the same order of magnitude as the convection
term. Similar relationship can be obtained.
The non-dimensional partial differential equations described in equation 3 can be transformed to equation
4 using similarity parameters below with Grashof number coordinate stretching [see 3 and 11 for details]
Stream function
Similarity parameter
y  Gr 
  x
x 4 
x-velocity component
  Grx 1 / 4 
 ( x, y )  f ( ) 4 
 
  4  
1/ 4
u
2
1/ 2
Grx f ' ( )
x
The partial differential equations (3) can be shown to reduce to a system of ordinary differential equations
(4) as follows:
f ' ' '  3 f f ' '  2( f ' ) 2  T *  0
''
T *  3 Pr fT *'  0
(4a)
(4b)
To get proper physical meaning in equation 4 we redefine i.e. conduction is balanced by convection
terms.
f  F
to get
F ' ' '3FF' '2 F '2 T *  0
(4c)
T *' '3 Pr FT*'  0
( 4d )
Equation 4 is subjected to the following boundary conditions.
Boundary conditions:
  0,
f  f '  0,
  ,
f '  0,
T* 1
(4 e)
T*  0
(4 f )
The system of ordinary differential equations can be solved in a variety of ways. These equations are
reduced to system of first order equations and the ordinary differential equations are discretized using
finite difference techniques of first order accuracy. The results are refined further using finite difference
schemes that are fourth order accurate. This method is akin to predictor-corrector method. The results
are compared with MATLAB in build codes which solve boundary value problems. In order to compare
numerical and analytical scaling methods, we convert the solution back to dimensional form. The results
are discussed in this paper.
B. Scaling Using Rayleigh Number
The Rayleigh number is defined as the product of the Grashof number and the Prandtl number. The
Rayleigh number is presented as equation 5 below.
Rax  Grx . Pr 
g (Ts  T ) x
3
(5)

The similarity momentum and energy equations which describe buoyancy driven flow due to thermal
effects using Rayleigh scaling parameters are indicated as equations 6.

y
Rax 1/ 4
x
u
 Ra x1 / 2
x
 ( x, y)   Rax1/ 4 f ( )
f ' ( )
4 Pr( f ' ' '  T * )  2 ( f ' 2 3 f f " )  0
*"
4 T  3 f T *'  0
(6a)
(6b)
To get proper physical meaning in equation 6 we redefine
f  F
to get
4 Pr F ' ' 'T *  2 F '2 3FF ' '  0
(6c)
4T *' '3FT*'  0
(6 d )
Table 2 shows typical non dimensional parameters for a one-half local length of the plate.
Table 2: Typical Grashof and Rayleigh numbers for common working fluids
Fluid Type
Prandtl number
β (1/K)
ν (m2/s)
Gr x
Air
Water
Mercury
0.72
7.50
0.025
0.003200
0.000207
0.000182
0.0000151
0.0000014
0.0000001
1.72E+08
1.30E+09
2.23E+11
Ra x
1.24E+08
9.75E+09
5.58E+09
III.
Numerical Methods
The underlying problem has also been solved using numerical techniques. Boussinesq approximation
[3] was used in obtaining the numerical solution. To compare numerical solution with analytical
solutions based on Grashof and Rayleigh scaling parameters, three fluids with different Prandtl numbers
were simulated in ANSYS Fluent Software. The data used in the numerical simulation are presented as
Table 3 below. The computational domain was meshed and indicated as Figure 1. The node spacing was
based on exponential meshing scheme to obtain finer mesh near the heated plate. One meter long plate
was selected for analysis. The one meter long plate is maintained at isothermal conditions and an outflow
boundary condition is set at the upper part of the computational domain. A zero-velocity inlet boundary
condition was used at the leading edge (bottom part) of the domain. Symmetry boundary condition was
used for the side far and opposite to the heated plate.
Figure 1: Meshed domain for numerical simulation with exponential placed nodes
Table 3 below presents the data and boundary conditions used for the modeling process.
Table 3: Data for numerical simulation
DIMENSIONS [meters]
Domain height
1.0
Domain width
Minimum node space
Leading edge
Upstream
Plate
Far plate
0.5
0.0005
BOUNDARY CONDITIONS
Velocity Inlet
Outflow
Wall
Zero Stress
[0 m/s]
The numerical solutions were obtained by solving the continuity, momentum and energy equations
which describe natural convection over an isothermal plate. The differential equations are presented as
equation 7.
Continuity equation
u v

0
x y
(7 a )
X-Momentum Equation
  2u  2u 
u
u
u
p
u
v
     2  2 
t
x
y
x
y 
 x
Y-Momentum Equation
(7b )
  2v  2v 
v
v
v
p
 u  v    g (T  T )    2  2 
t
x
y
y
y 
 x
(7c)
Energy equation
  2T  2T 
T
T
T
u
v
   2  2 
t
x
y
y 
 x
(7 d )
The no slip condition at the plate wall and zero velocity of fluid far from the plate were used to
describe the boundary conditions needed to solve equation 7.
Thus,
y  0,
u  v  0,
T  Ts
u  0,
T  T
and
y  ,
Before the numerical studies were carried out, we performed a grid independent study using a stream
function as an indicator of results not depending on the node spacing or grid. Two different mesh sizes
were simulated. The mesh sizes are presented in Table 4 below.
Table 4: Grid Independence Study for Numerical Simulation (Air)
Mesh
No of cells No of faces
No of nodes
Stream function (kg/s)
1
56000
112480
56481
0.00217
2
39600
79600
40001
0.00199
IV.
Results and Discussion
The results obtained with the analytical and numerical methods are presented below. The stream
function plot obtained with the numerical simulation for air is indicated in Figure 2. The stream function
plot was obtained for a temperature difference of 10 K.
Table 5: Maximum velocity using different methods (m/s)
Air (Pr = 0.72)
Water (Pr = 7.5)
Mercury (Pr = 0.025)
Gr Scaling
0.22706
0.02576
0.09513
Ra Scaling
Numerical methods
0.22713
0.21620
0.02555
0.09478
Table 5 shows the maximum velocity magnitudes obtained using different solution techniques with
different working fluids. The maximum velocity obtained using direct numerical simulation differs
slightly from the analytical solutions due to a variety of factors. Lack of ability to simulate an infinite
domain for the direct numerical method might have contributed to this discrepancy. We observed return
flow in the far field. Lack of control of the outflow boundary condition in the commercial software also
could have contributed to this difference. Accumulation of hotter fluids at the domain exit results into a
stabilizing mode, thus, a relatively lower maximum velocity for the direct numerical method when
compared to the analytical solution. Table 5 shows that there is no significant difference between the
Rayleigh and Grashof solution techniques in determining the maximum velocity magnitudes.
Figure 2 shows the stream function plot for the direct numerical solution for air. At the mid plane of
the domain the stream functions are consistent with boundary layer flows. Results obtained at the mid
plane give a good approximation of an infinite domain. We obtained the velocity and the temperature
profiles at the mid plane.
Figure 2: Stream Function plot for Air Natural Convection over a heated plate
The velocity profile for the buoyancy driven convection on a vertically heated flat plate for a
temperature difference of 10 K with air as the working fluid is shown in Figure 3. This figure shows a
comparison between the two analytical methods described above and the direct numerical technique. The
maximum velocity obtained for both Grashof and Rayleigh scaling is of the same order of magnitude as
shown in Table 5. A small difference between the analytical solution and the direct numerical technique
is observed. The difference might be due to the explanations already given above. Figure 3 shows that
all methods results in a boundary layer thickness which is about 7% of the domain size.
Figure 4 shows the temperature profile for the conditions described in figure 3 where the temperature
profile is compared for the analytical and numerical techniques. It is observed that the direct numerical
method gives a larger temperature gradient than the analytical methods.
Results shown in figures 5 and 6 were obtained with water as the working fluid and those shown in
figure 7 and 8 are for mercury as the working fluid. This was done to investigate the effect of Prandtl
number on the difference between the two analytical methods using Rayleigh and Grashof scaling
parameters. Figures 5 and 6 show the results for high Prandtl number fluid (Pr =7.5 for water) in terms of
a velocity and a temperature profile respectively. Figures 7 and 8 show results for low Prandtl number
fluid (Pr = 0.025 for mercury), similarly, for velocity and temperature profiles respectively. From the
results shown in figures 5-8, there is no significant difference between the two analytical methods. For
low Prandtl numbers the Grashof scaling results in less stiff systems of differential equations whereas the
Rayleigh systems of equations become stiffer. This translates into fewer number of equations to be
solved with Grashof scaling and hence the accuracy. The converse is true at high Prandtl numbers.
0.25
Ra Scaling (Air Pr=0.72)
Gr Scaling (Air Pr=0.72)
Numerical (Air Pr=0.72)
Velocity (m/s)
0.2
0.15
0.1
0.05
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Distance (m)
Figure 3: Velocity Profile Comparison for Air
320
319
Ra Scaling (Air Pr=0.72)
Gr Scaling (Air Pr=0.72)
Numerical (Air Pr =0.72)
318
317
Temperature (K)
316
315
314
313
312
311
310
0
0.005
0.01
0.015
0.02
Distance (m)
Figure 4. Temperature Profile Comparison for Air
0.025
0.03
0.035
0.04
0.03
0.025
Ra Scaling (H20 Pr=7.5)
Gr Scaling (H20 Pr=7.5)
Velocity (m/s)
0.02
0.015
0.01
0.005
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Distance (m)
Figure 5. Velocity Profile Comparison for Water
320
319
Ra Scaling (H2O Pr=7.5)
Gr Scaling (H2O Pr=7.5)
318
317
Temperature (K)
316
315
314
313
312
311
310
0
0.002
0.004
0.006
0.008
Distance (m)
Figure 6. Temperature Profile Comparison for Water
0.01
0.012
0.014
0.1
0.09
0.08
Ra Scaling (Hg Pr=0.025)
Velocity (m/s)
0.07
Gr Scaling (Hg Pr=0.025)
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.002
0.004
0.006
0.008
0.01
0.012
Distance (m)
Figure 7. Velocity Profile Comparison for Mercury
320
319
Ra Scaling (Hg Pr=0.025)
318
Gr Scaling (Hg Pr=0.025)
317
Temperature (K)
316
315
314
313
312
311
310
0
0.002
0.004
0.006
Distance (m)
Figure 8. Temperature Profile Comparison for Mercury
0.008
0.01
0.012
At high Pr number, the velocity boundary layer is much larger than the thermal boundary layer as
observed in figures 5 and 6. This can be attributed to momentum diffusivity being larger than the thermal
diffusivity. At low Pr numbers, we observed that the velocity and thermal boundary layers have the same
order of magnitude as shown in figures 7 and 8. Figures 3 and 4 show the results for Pr number close to
unity; both the velocity and thermal boundary layers are of the same size.
V.
Conclusion
This paper has revisited the scaling analysis first introduced by S. Ostrach in 1952 and compared it to
subsequent researchers who suggested different scaling parameters in a natural convection problem over a
vertical flat plate. The coupled partial differential equations are reduced to system of ordinary differential
equations via similarity transformation methods. These are further reduced to a system of first order
differential equations which are discretized using finite difference methods. The solution proceeds using
a first order accurate solver and refined using a fourth order accuracy scheme. This method is akin to
predictor-corrector method. We observed that the system of ordinary differential equations which
describe the natural convection over a vertical flat plate become stiff with Rayleigh scaling when
considering lower Prandtl number fluids. The converse is true for high Prandtl numbers. Our results
show that when these different scaling methods are compared in a dimensional form, both methods are
identical.
Acknowledgement
The authors thank the North Carolina Space Grant for financial assistance.
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