1. science
2. mathematics
3. algebra

Proceedings of the
4th JSME-KSME Thermal Engineering Conference
October 1-6, 2000, Kobe, Japan
NATURAL CONVECTION IN ENCLOSED CAVITY
UNDER PERIODIC GRAVITY VARIATION
Keisuke MATSUNAGA 1 , Ichiro UENO2 , Hiroshi KAWAMURA 2
1
Graduate Student, Dept. of Mech. Eng., School of Science and Technology, Science University of Tokyo
2
Dept. of Mech. Eng., Fac. of Science and Technology, Science University of Tokyo
2641 Yamazaki, Noda-shi, Chiba 278-8510, Japan
ABSTRACT A numerical analysis is conducted to investigate the effect of periodic variation of acceleration
upon the natural convection of fluid in a closed cavity with heated end walls. Acceleration or g-jitter is applied with
inclination against the temperature gradient direction. Even if the acceleration is periodic, a net circulation remains
by integrating the velocity and temperature distribution over one period. With the increase of the frequency, the
direction of convection is inversed, because, the phase shift of horizontal velocity and temperature fields against the
g-jitter becomes prominent. The numerical results are examined in comparison with simplified analytical solutions.
Keywords: Natural Convection, Closed Cavity, Micro Gravity, G-jitter
1. INTRODUCTION
The perfect null-gravity condition cannot be achieved in
an orbiting spacecraft or on the space station. Nelson et al.[1]
categorized the residual accelerations as quasisteady, oscillatory, and transient ones. The quasisteady acceleration is caused
by atmospheric drag and Keplerian effects. The oscillatory
one arises from various natural frequencies of the spacecraft
structure. Internal disturbances are raised primarily by crew
activities, and external disturbances are thruster firings. The
transient acceleration is caused by orbital control, docking,
and others. The residual acceleration levels are ranging from
10-6g 0 to 10-3g 0, where g 0 refers to the Earth’s gravity. Among
them, the time dependent component of residual acceleration
is called “g-jitter”.
Now the International Space Station ( ISS hereafter ) is
under construction, the g-jitter is unavoidable in use of the
space environment. Periodic accelerations typical of the gjitter can cause fatal effects on a large class of microgravity
experiments involving diffusion measurements, crystal growth
from the melt or the vapor, and directional solidification of
alloys. Particularly, in hydrodynamic fields, fluid flow with
heat and mass transport is sensitive to the gravity disturbances.
However, the effects of such a periodic environment on heat
and mass transport are not well understood and there has been
no general theory to predict such effects. It is thus necessary
to examine the fluid behavior influenced by the g-jitter.
R.Savino and R.Monti[2] investigated the effect of convection by residual-g and g-jitter on the on-orbit diffusion experiment. Farooq and Homsy[ 3 ] made an analytical study on
the streaming flows induced by the g-jitter up to a fairly high
Raireigh number region. Grassia and Homsy[4][5] made a series of analytical studies or thermocapillary and buoyant flow
in an open cavity imposed with low frequency and investigated a nontrivial flow originated from the spanwise buoyancy forces. As for the steady state natural convection related
to the present work, Cormack et al.[6] derived an analytical
solution of the natural convection in a shallow rectangular
cavity with differentially heated end walls. Later Bejan and
Tien[7] extended it to the natural convection in a cylindrical
cavity. An on-orbit experiment of natural convection in a closed
cavity was proposed by Naumann[8] and was conducted in the
Space shuttle flight STS-95 as a joint Japan-US venture called
as Japanese-United States Thermal Science Acceleration
Project (JUSTSAP).
[8]
In his preliminary theoretical analysis, Naumann predicted an induced flow field as follows. The essence of periodic acceleration can be understood by referring to the sketch
below (Fig. 1). The fluid is contained in a differentially-heated
cavity, whose walls are heated on one side and cooled on the
other one. The connecting walls are assumed to be conducy
hot
A
cold
B
x
acceleration during
second half cycle
acceleration during
first half cycle
Fig. 1 Natural convection in a cavity imposed with
periodic gravity acceleration
tive or adiabatic. If an oscillating acceleration is applied vertically ( Case A in Fig. 1 ), the fluid will circulate clockwise
during the first half cycle, ( g direction downwards ), while
counter-clockwise during the second one when the g upwards.
The magnitude of the induced flow is equal for the both cycles,
thus no net flow is resulted. The case of an oscillating acceleration inclined to the vertical axis is shown as Case B in Fig.
1. In the first half cycle, the acceleration component in the xaxis is directed the hot wall. It enhances the clockwise flow in
the first half cycle. In the second half cycle, on the other hand,
the x-component is towards the cold wall, thus the counterclockwise circulation is suppressed. As the result a clockwise
net circulation flow remains over one period. Accordingly the
resulting net temperature perturbation becomes positive in the
upper half and negative in the lower half. In the present study,
a numerical analysis is conducted to investigate the effect of
periodic variation of acceleration upon the natural convection
in a closed cavity.
Spatial derivation
Grid
Coupling Algorithm
Convection term
Finite Difference Method
Staggered Grid
Fractional Step Method
2nd-order Central Scheme
2nd-order Central Scheme
(Consistent Scheme)
SOR Method
Euler Method Scheme
80 × 40
Viscous term
Solution of Poisson equation
Time Advancement
Mesh number ( X × Y )
Two non-dimensional parameters, the Prandtl number Pr and
the Grashof number Gr
Gr,, are also introduced;
Pr =
ν
g β∆TH 3
, Gr =
κ
ν2
(2)
where κ is the thermal diffusivity; β the volume expansion
coefficient, g the gravitational acceleration. The Boussinesq
approximation is employed for the buoyancy effect. The periodic change of acceleration, i.e., the g-jitter is proposed as
follows;
2. NUMERICAL CALCULATION
∆Τ
Adiabatic wall
g jitter = g sin(2π ft )
θ = 45 °
Hot
(65℃)
Table 1 Computational conditions
g jitter= g sin(2πft)
Cold
(25 ℃)
(3)
where g = 10-6 g 0, with g 0 = 9.8 m/s 2. With the above equation,
the continuity, momentum and energy equations are solved
numerically. The non dimensionalized fundamental equations
are given below.
Continuity equation
Adiabatic wall
Monitoring line
Fig. 2 Calculation model
A numerical analysis of the natural convection in a 2-D
closed rectangular cavity is performed. Following the
JUSTSAP experiment, a cavity of 0.10m in horizontal length
L and of 0.05m in vertical height H is employed. Water of
Pr=7.0
Pr
=7.0 is assumed as a test fluid. The end walls are held at
uniform and different temperatures TH and TC with a difference of ∆T (= TH - TC ) = 40 K. The top and bottom walls are
of adiabatic condition. All surfaces are rigid non-slip boundaries. Periodic acceleration is applied with an inclination o f
45°° against the temperature gradient direction as shown in
45
Fig. 2. The velocity components u , v are in the horizontal and
vertical directions, respectively. Fundamental equations are
non-dimensionalized by introducing the non-dimensional variables as follows;
x
y
H
H
 * ν
*
*
*
*
 t = H 2 t , x = H , y = H , u = ν u, v = ν v,
(1)

2
2
 P* = ν P , T * = T − Tcold , F * = fH

ρH 2
∆T
ν
where t is the time, x and y the horizontal, and vertical positions, u , v are the horizontal, and vertical velocities, T temperature, P pressure, H and L cavity height, and length, ν the
kinematic viscosity, ρ the density,, f the frequency. The above
quantities denoted by an asterisk represent the non-dimensional
variables, however, the asterisk is omitted for the simplicity
in the followings.
∂u ∂v
+ =0
∂x ∂y
Momentum equations
(4)
∂u
∂ u ∂ u ∂ P ∂ 2 u ∂ 2 u 
+
+
 +u +v = −

∂x
∂y
∂ x ∂ x2 ∂ y2 
 ∂t

+ Gr ⋅ T ⋅ cosθ ⋅ sin(2π Ft )

2
2
 ∂ v + u ∂v + v ∂v = − ∂P +  ∂v + ∂ v 
 2

 ∂t
∂x
∂y
∂y ∂x
∂y 2 

+ Gr ⋅ T ⋅ sin θ ⋅ sin(2π Ft)

(5)
Energy equation
∂T
∂T
∂T
1  ∂2 T ∂ 2 T 
+u
+v
=
+


∂t
∂x
∂y Pr  ∂x 2 ∂y 2 
(6)
Computational conditions are summarized in Table 1. The
finite difference method with the second order accuracy is
Table 2 Calculation Parameter
Case
Frequency f [Hz]
F=fH2 /ν [-]
1
1.0×10-6
2.5×10-3
2
1.0×10-5
2.5×10-2
3
1.0×10
-4
2.5×10-1
4
1.0×10-3
5
1.0×10
-2
2.5×10
6
1.0×10-1
2.5×102
2.5
employed. The staggered grid arrangement is adopted. The
frequency f and corresponding non-dimensional frequency F
= f H2 //ν
ν are given in Table 2. Temperature and velocity fields
are monitored on the longitudinal center line of the cavity.
3. CODE VERIFICATION
Before discussing the results, numerical calculation is compared with analytical solution to verify the accuracy of numerical code. The numerical calculation has been made in
reference to the configuration and condition of the JUSTSAP
experiment. The steady acceleration is added vertically against
the temperature gradient direction.
direction. The gravity level is assumed
to be 0.5µ
0.5µg 0 , 1.5µ
1.5µg 0 , 10.0µ
10.0µg 0. Cormack[6] obtained an analytical solution of velocity and temperature distributions along
the longitudinal center line for a long rectangular cavity with
adiabatic side wall.
Horizontal velocity
u (η )
1
= − (η 3 −η )
Gr ⋅ A
6
(7)
Temperature distribution
T (η)
1
=
(3η5 − 10η3 + 15η)
Gr ⋅ Pr ⋅ A2 360
(8)
where A is the aspect ratio H/L and η is the non-dimensional
vertical position scaled by H/2, u and T the non-dimensional
horizontal velocity and temperature, respectively The numerical results are given in Figs. 3 (a) and (b) in comparison with
the above analytical equations. The present numerical result
of horizontal velocity is in good agreement with the analysis
when the gravity level is small. As for the temperature solution, however, the numerical solution for the smaller gravity
level differs appreciably from the analytical ones. Thus a
smaller aspect ratio of A = 1/8 is also calculated. It is in good
agreement with the Cormack’s analysis so that the validity of
the present numerical method is confirmed. Thus, the deviation in the case of A=1/4 can be deduced from the effect of
the short rectangular cavity length.
4. RESULTS AND DISCUSSIONS
The net circulation flow and temperature disturbance averaged over one period of the fully developed convection are
illustrated in Figs. 4 (a) and (b), respectively. Induced convection under the smaller g-jitter frequencies (F
(F<1.0 or f =
10-6 to 10-4 Hz) indicates a clockwise net circulation flow. The
direction of the induced net flow is the same as predicted
theoretically by Naumann[7]. A counter-clockwise net circulation flow appears, on the other hand, under higher frequencies (F
(F >1.0 or f = 10-3 to 10-1 Hz). Such the reverse of net
circulation flow direction can be explained by considering the
phase shifts of horizontal velocity u and temperature against
the g-jitter frequencies. Figs. 5 (a)-(f) indicate the phase shifts
among those three quantities over one period of g-jitter at y =
4/5 H on the monitoring line in the case of g-jitter frequency f
from 10-6 to 10-1 Hz, respectively. Horizontal velocity and
temperature fields respond immediately to the imposed periodic acceleration in case of the lower frequencies. With in-
creasing frequency, the phase shifts become more prominent.
In the case of frequency of 10-1Hz (Fig. 5 (f) ), the phase shifts
in velocity and temperature fields against the imposed g-jitter
reach almost π/2 and π, respectively. This can be explained as
follows.
The phase shifts of horizontal velocity and temperature against
the g-jitter component are considered from momentum and
energy equations. A long vertical channel with sinusoidal g is
assumed for simplicity. The equations are non-dimensionalized
using periodic acceleration frequency f. New non-dimensional
variables denoted by tilde are introduced.
x
y
u
v
%
*
*
t = ft , x = H , y = H , u% = Hf , v% = Hf ,


P
T − Tcold
fH 2
 P% =
, T* =
, F* =
2

ρ ( Hf )
∆T
ν
(9)
With the above variables, the horizontal Navier-Stokes equation is rewritten as
 ∂u
∂u
∂u 
∂P ∂ 2u ∂ 2u 
F  +u
+v = −
+
+

∂x
∂y 
∂ x ∂ x 2 ∂ y 2 
 ∂t
(10)
Gr
+
⋅ sin(2π t )
F
Note that the asterisk and tilde is omitted for the simplicity.
Because of the microgravity condition, the convection term is
negligible. If non-dimensional frequency F is much smaller
than 1.0, the equation (10) can be approximated as
∂ 2 u ∂ 2 u Gr
+
+
⋅ sin(2π t ) = 0
∂x2 ∂y 2 F
(11)
Thus the above equation gives the following solution.
u∝
Gr
⋅ sin(2π t )
F
(12)
The solution (12) indicates that the horizontal velocity u is in
proportion to the buoyant term. Thus the no phase shifts exists between the two quantities.
In case of F >> 1.0, on the other hand, the rate of change
term in the axial momentum equation balances primarily with
the buoyancy term. Thus, the equation (10) becomes
∂u Gr
=
⋅ sin(2π t )
∂t F
(13)
Integration of equation (13) gives the horizontal velocity u as
u∝−
Gr
⋅ cos(2π t )
2π F
(14)
This indicates that the phase shift of horizontal velocity u
against buoyant term or g-jitter component becomes π/2 in
the case of higher frequencies ( F >> 1.0 ). This tendency has
already been pointed out by Lamb [9]
In the same way, the energy equation is nondimensionalized as follows;
∂T
∂T
∂T 1 1  ∂ 2T ∂ 2T 
+u
+v
=
+


∂t
∂x
∂y F Pr  ∂x2 ∂y 2 
(15)
θ% = − x + T ( x , y )
(16)
The temperature perturbation θ% from the linear temperature
distribution is introduced as
With use of the temperature perturbation θ% , equation (15) is
replaced by the following form;

∂θ%
∂θ%  ∂θ% 1 1  ∂ 2θ% ∂ 2θ%  (17)
+ u  −1 +
=
+
 +v


∂t
∂x 
∂y F Pr  ∂x2 ∂y2 

The temperature variation is governed by the convective effect in conjunction with the linear steady temperature gradient. In addition u , v, and T% are very small , then the rest of the
convective term is negligible. When F is much higher than
1.0, The following solution is obtained.
∂θ%
Gr


∝ u = −
⋅ cos2π t 
(18)
∂t
2
π
F


Once the above solution is integrated, the temperature perturbation is obtained;
θ% ∝ −
Gr
(2π )2 F
⋅ sin2π t
n
cos(nπ )sin ( nπη )
n 3π 3
Θ+net = −Pr 2 ⋅ Grx ⋅ Gr y∑ An
n
cos(nπ )sin ( nπη )
n 5π 5
(20)
(21)
where
n π − Ω Pr
( n 4π 4 + Ω 2 )( n 4π 4 + Ω 2 Pr)
2y
H
Θ
+
η=
, U net
= U net , Θ+net =
H
2ν
H  dT 
(22)
2  dx 
4
g y β  dT  H 4
g β  dT  H 
Grx = x 2 
,
Gr
=

y

ν  dx 
ν 2  dx 
 2 
 2 
An =
4
4
5. CONCLUSION
(19)
The above expression indicates the further phase shift of π/2
against
again
st the velocity. So the phase shift of temperature perturbation against g-jitter becomes π when F is higher than 1.0.
[10]
Recently, Naumann obtained an analytical solution for
the net horizontal velocity with an assumption of an infinitely
long channel with a constant axial temperature dT/dx imposed.
[10]
According to Naumann , the resultant net flow ( Unet ) and
the net temperature deviation ( Θnet ) from the linear profile
can be expressed as
+
U net
= − Pr ⋅ Grx ⋅ Gr y∑ An
For a small aspect ratio ( A = 1/8 ) and for a low frequency
( Ω+ = 0.0039 ), the numerical results agree well with the above
simplified solution. In Fig. 6 (a), however they deviates from
Eq. (23) with the increase of the aspect ratio and the frequency.
For the high frequencies, comparison is made in Figs.6 (c)
and 7 (c). The numerical result tends to Eqs. (25) and (26).
Note that the direction of circulation is indeed reversed with
the increase of the frequency. Accordingly the net flow must
become zero between these two limits. The obtained velocity
and temperature in the intermediate frequency is given in Figs.
6 (b) and 7 (b). These figures indicate that the reversal of the
net flow does not take place uniformly over the channel section but starts from the central region of the channel.
2
2π f  H 
ν  2 
The conductive condition is assumed for the side walls. For a
+
very small and large Ω , Eqs. (20) and (21) can be simplified
respectively as
sin ( πη )
+
(23)
U net
= Pr ⋅ Grx ⋅ Gry
( Ω → 0)
π7
sin (πη )
+
U net
= −Grx ⋅ Gry
(Ω → ∞ )
(24)
Ω 2π 3
The effects of periodic accelerations, g-jitter upon the natural convection in an enclosed cavity has been numerically studied. The net circulation flow and temperature disturbance averaged over one period remains when the g-jitter is inclined
against temperature gradient direction.
direction. The flow direction of
the net circulation depends upon the periodic acceleration
frequencies. Low frequencies (F
(F<1.0) gives no phase shifts in
the horizontal velocity u and temperature against gravitational
component. On the other hand, High frequencies (F
(F>1.0) results in the phase shifts among the gravity, velocity and temperature profiles. The calculated net flow agrees well with the
simplified analytical solutions.
ACKNOWLEDGEMENT
The authors appreciate the collaboration of Japan Space
Utilization Promotion Center (JSUP) for providing useful information regarding JUSTSAP. They also wish to thank Prof.
R. J. Naumann of University of Alabama in Huntsville for the
helpful advices and supports, especially for offering the analytical solution of the net circulation. A part of this work was
supported by Japan Space Forum.
REFERENCES
2
Ω+ =
sin (πη )
π9
sin ( πη )
= −Pr ⋅ Grx ⋅ Gry
Ω 2π 5
Θ+net = Pr 2 ⋅ Grx ⋅ Gry
( Ω → 0)
(25)
Θ+net
(Ω → ∞ )
(26)
The present numerical results are compared with these analytical solutions in Figs. 6 (a)-(c) and Figs. 7 (a)-(c).
[1]Nelson, E.S., NASA TM-103775 (1991).
[2] Savino, R., and Monti,R., Convection induced by residualg and g-jitters In diffusion experiments, Int. J. Heat Mass
Transfer 42, pp111-126, (1999)
[3] Farooq, A., and Homsy, G. M., Streaming flows due to gjitter-induced natural convection, J. Fluid Mech. vol.271,
vol.271,
pp.351-378, (1994)
[4] Grassia, P., and Homsy, G. M., Thermocapillary and buoyant flows with low frequency jitter. I Jitter confined to the
plane, Phys. Fluids 10 (6), pp.1273-1290, (1998)
[5] Grassia, P., and Homsy, G. M., Buoyant flows with low
frequency jitter, Phys. Fluids 10 (8), pp.1903-1923, (1998)
[6] Cormack, D. E., Leal, L. G., and Imberger. J., Natural
convection in a shallow cavity with differentially heated end
walls. Part 1. Asymptotic theory, J. Fluid Mech.,
Mech., 65 (2),
pp.209-229, (1974)
[7] Bejan, A., and Tien, C. L., Fully developed natural counter-
flow in a long horizontal pipe with different end temperatures,
Int. J. Heat Mass Transfer 21, pp701-708, (1978)
[8] Naumann, R. J., An Analytical Model for Transport from
Quasi-Steady and Periodic Accelerations on Spacecraft, Int.
J. Heat Mass Transfer.,
Transfer. , (in printing).
[9] Lamb, H., Hydrodynamics, Cambridge University Press
(1975).
[10] Naumann, R. J., Private Communication (2000).
1
η
η
1
0
0
Analytical ( Cormack 1974 )
Numerical ( 0.5 ×1 0-6 g 0 , A=1/4 )
Numerical ( 1.5 ×1 0-6 g 0 , A=1/4 )
-6
Numerical ( 10.0 ×1 0 g 0 , A=1/4 )
-1
-0.05
0
u (η )
Gr ⋅ A
Analytical ( Cormack
1974 )
-6
Numerical ( 0.5 ×1 0 g0 , A=1/8 )
-6
Numerical ( 0.5 ×1 0 g0 , A=1/4 )
-6
Numerical ( 1.5 ×1 0 g0 , A=1/4 )
-6
Numerical ( 10.0 × 1 0 g 0 , A=1/4 )
-1
0.05
-0.02
(a) Horizontal velocity
-0.01
0
T (η )
Gr ⋅ Pr ⋅ A 2
0.01
0.02
(b) Temperature
Fig. 3 (a)-(b) Comparison of numerical calculation with analytical solution along the longitudinal center line
1
η [-]
η [-]
1
0
0
Case 1 ( F =0.0025 )
Case 2 ( F =0.025
Case 3 ( F =0.25
-1
C a s e 1 ( F= 0 . 0 0 2 5 )
)
)
C a s e 2 ( F= 0 . 0 2 5 )
C a s e 3 ( F= 0 . 2 5 )
-1
-1
0
1
-8
Net horizontal velocity u [m/s] [× 1 0 ]
-0.004 -0.002
0
0.002 0.004
Temperature disturbance [ ℃]
1
1
C a s e 4 ( F=2.5 )
C a s e 5 ( F= 2 5 . 0 )
0
-1
C a s e 6 ( F= 2 5 0 . 0 )
η [-]
η [-]
C a s e 4 ( F=2.5 )
C a s e 5 ( F= 2 5 . 0 )
C a s e 6 ( F= 2 5 0 . 0 )
-2
-1
0
1
Net horizontal velocity u [m/s] [ ×1 0
(a) Net horizontal velocity
2
-10
]
0
-1
-0.0001
0
Temperature disturbance [
(b) Averaged temperature profile
Fig 4 (a)-(b) Horizontal net velocities and averaged temperatures over one period
℃]
0.0001
Time over one period
Case 1 F =0.0025 [-]
(b)
Time over one period
Case 2 F=0.025 [-]
(c)
Time over one period
(d)
Time over one period
Case 4 F=2.5 [-]
(e)
Time over one period
Case 5 F=25.0 [-]
Amplitude
Amplitude
Amplitude
Amplitude
(a)
Amplitude
Amplitude
Acceleration
Horizontal velocity [-]
Temperature [-]
Case 3 F=0.25 [-]
(f)
Time over one period
Case 6
Fig. 5 (a)-(f) Phase shifts among acceleration, velocity and temperature at a fixed point in a cavity
1
1
F=250.0 [-]
1
-3
-π sin(πη )
+
Ω =390.0, A=1/4
+
Ω =39.0, A=1/4
+
η
0
0
-7
η
η
Ω =390.0, A=1/8
0
π sin(πη )
+
Ω =0.0039, A=1/4
+
Ω =0.039,
+
Ω =0.39,
+
Ω =2.0 ,
-1
A=1/4
A=1/4
-0.0002
0
+
Ω =3.9, A=1/4
A=1/4
+
Ω
0 . =0 00.0020 3 9 , A = 1 / 8
-1
-2
0
+
U net
+
U net
Pr ・Gr x ・Gr y
-1
2
-6
[ ×1 0 ]
Pr ・Grx ・Gr y
-0.02
0
+2
0.02
+
Ω ・U n e t
Gr x・Gr y
(b)
(a)
(c)
Fig. 6 (a)-(c) Comparison of net horizontal velocity between numerical calculation and analytical solution
along the longitudinal center line ( conductive wall )
1
1
1
-5
+
-π sin(π η)
Ω =3.9, A=1/4
+
Ω =390.0, A=1/4
+
Ω =39.0,
A=1/4
+
-9
π sin(πη )
η
0
η
η
Ω =390.0, A=1/4
0
0
+
Ω =0.0039, A=1/4
+
Ω =0.039,
+
Ω =0.39,
+
Ω =2.0,
-1
+
-0.00002
2
Pr
A=1/4
A=1/4
Ω =0.0039, A=1/8
0
Θ
A=1/4
0.00002
-1
-1
0
+
net
・Gr x ・Gr y
Θ
2
-1
1
+
net
Pr ・Gr x ・Gr y
-7
[× 1 0 ]
-0.002
0
0.002
・Θ +n e t
Pr ・G rx ・G ry
Ω
+2
(b)
(a)
(c)
Fig. 7 (a)-(c) Comparison of net temperature disturbance between numerical calculation and analytical solution
along the longitudinal center line ( conductive wall )