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The effect of sound on natural convection from a vertical flat plate
Journal of Sound and Vibration (1992) 158(2), 213-218
THE EFFECT OF S O U N D O N N A T U R A L C O N V E C T I O N F R O M A VERTICAL FLAT PLATE H. ENGELBRECHT AND L. PRETORIUS
Systems Laboratory, Rand Afrikaans University, P.O. Box 524, Johannesburg 2000, Republic of South Africa (Received 5 July 1990, and in final form 26 April 1991) An experimental investigation has been carried out to study the influence of sound waves on the transition from laminar to turbulent flow in the boundary layer associated with natural convection from a vertical flat plate with uniform surface heat flux. A relationship between frequency and Grashof number at transition has been found. Buckling flow theory provides a means of explaining this relationship. 1. INTRODUCTION The effect of sound on both natural and forced convection from cylinders has been studied to quite an extent. References [ 1-4] provide an overview of the work done in this field. Parker and Welsh [5] and Cooper et al. [6] conducted experiments to determine the influence of sound on forced convection from horizontal flat plates. In both cases the reattachment length of the boundary layer was shown to be shortened when sound was applied. Parker and Welsh also showed that for flow across a plate with a square leading edge the reattachment point oscillates at the applied frequency. No effect of sound was observed for a plate with a round leading edge. Cooper et al. measured the maximum heat transfer coefficient at the reattachment point and observed that the value increases when sound is applied. In contrast to the above-mentioned work at sound frequencies between 200 and 1200 Hz, Kimura [7] studied the effect of low frequency sound (10-20 Hz) on buckling flow. He concluded that the point of transition from laminar to turbulent flow in an axisymmetric plume can be controlled by the application of an external sound field. Some of the results shown by Kimura indicate transition from laminar to turbulent flow in a narrow range of frequencies relatively insensitive to sound level (amplitude). This led to the investigation described in what follows. The effect of sound on natural convection for a vertical flat plate has not been studied extensively. It is the purpose of the work described in this paper to investigate experimentally the effect of sound on the transition from laminar to turbulent flow in the boundary layer associated with natural convection from a flat vertical plate. 2. EXPERIMENTAL APPARATUS AND INSTRUMENTATION The following experimental apparatus were used to obtain results quoted in this paper. 2.1. PLATE The vertical plate consists of a piece of hardboard with dimensions 130 x 200 x 15 m m . Tightly stretched across the length of the plate are I I strips of I0 m m wide copper-zinc 213 0022--460X/92/200213 +06 $08.00/0 © 1992 AcademicPress Limited
214
lq. E N G E L B R E C H T A N D L. P R E T O R I U S
shimstock of thickness 50 pm. The strips are 1 mm apart, and two springs are fitted at the back of the hardboard to ensure that the strips remain tight when they expand as a result of heating. The metal strips are soldered together to form a continuous conductor. The plate resistance is connected to a direct current power source capable of supplying 0-50 amp6re for 0 10 volts. 2.2. S P E A K E R A S S E M B L Y A 60 watt 8 12 cone-type loudspeaker is connected through a music amplifier to a signal generator, generating sinusoidal signals. The signals were measured to contain a single frequency, pointing to an undistorted sinusoidal signal. 2.3. HOT-WInE ANEMOMETER The characteristics of the flow in the flow region near the vertical plate are determined by means of a TSI, IFA 1000 hot-wire anemometer. A cylindrical wire probe with a resistance of 6-66 -('2 is used. The operating resistance of the hot-wire anemometer is set at 12.08 12. The output from the hot-wire anemometer is connected to an Iwatsu 5S-5212 oscilloscope. 3. EXPERIMENTAL PROCEDURE The loudspeaker and hot-wire probe are placed in positions relative to the plate as indicated in Figure 1. The hot-wire probe is at a vertical position of 0-16 m from the lower edge of the plate. For set voltages between 0 and 10 volts, the frequency of the frequency generator is slowly adjusted and the signal on the oscilloscope observed. Frequencies are observed at which the signal on the screen of the oscilloscope is disturbed form the continuous signal existing without sound applied. The disturbed signal is more random in nature and does not show definite periodicity. The frequency, voltage and current are noted. The ambient temperature is also recorded.
Imm i
J
Probe
position
i
|
I
.J Front 50
view
mm
Side view Figure 1. Positions of speaker and probe relative to plate.
f
,
4 0 mrn
i Plate
EFFECT
OF
SOUND
ON
NATURAL
215
CONVECTION
The experiment is conducted with the vertical plate in air only. The goal of the experiment is not to characterize the full extent of the flow field, but only to show the onset of turbulence at a certain point.
4. ANALYSIS Sparrow and Gregg [ 14] have described an exact solution of the laminar boundary-layer equations for free convection from a vertical plate having uniform surface heat flux. They obtained dimensionless velocity distributions for Prandtl numbers of 0-1, 1, I0 and 100. The equations provided by Sparrow and Gregg were reworked and solved on a HP 41C computer to obtain the dimensionless velocity distribution for Pr = 0-708 shown in Figure 2. Because current experiments by the authors have been carried out in air, a value of
03[ 0-2
01
O0
I I
I
2
I
3
i
4
5
Figure 2. Dimensionlessvelocitydistributionfor Pr=0.708. 0.708 for the Prandtl number is assumed. (A list of nomenclature is given in the Appendix.) From Figure 2 it can be seen that the maximum velocity occurs at a value of F ' = 0 . 2 6 for Pr--0" 708. At this point 2/5 = 0" 2 6 .
(1)
U,,ox = (0" 26)( 5)( Gr.* / 5)2/5( v / x ) .
(2)
F' = ( u x / v ) / 5 ( G r * / 5 )
Therefore
Also from Figure 2 it can be seen that the boundary-layer thickness is given by 1/.... = 5" rI = ( 6 / x ) ( G r * / 5 ) '/5 = 5.
(3)
= 5x(5/Gr*)'/5.
(4)
Therefore
Bejan [9-12] has described the theory of buckling flows. He showed that under certain conditions a fluid can show a tendency to buckle in a fashion analogous to the way that a slender column axially loaded at the ends may buckle. Bejan reasoned that this buckling of a fluid can be the onset of turbulence. The buckling wavelength of a two-dimensional
216
H. ENGELBRECHT
AND
L. P R E T O R I U S
jet is found to scale with the transverse dimension of the jet as =
,rDl~.
(5)
Kimura [7] used this wavelength to obtain a frequency at which the flow buckles during flow in a thermal plume by considering an average velocity U,,ax/2:
f = ( U,,ax/2)/2..
(6)
Substituting from equation (5) for Z yields
T= x/~U,,~x/Zrt D.
(7)
With D = S and substituting from equations (2) and (4), one has f = 0.02729(Gr*x)3/5v /x 2.
(8)
In Figure,,) is shown the relationship between f and Gr* for a value of x = 0. 16 and v = 16.84 × 10 -6 m2/s, which corresponds to Pr=0-708. In this case f = 1"7952 x lO-5(Gr*x)3/5.
(9)
50
4o
"~ 30
g .= ~ 20
tO
10"~
10 9
10 e Grashof
1010
number, Gr~
Figure 3. Relationshipbetween Gr~ and frequencyat which flowis disturbed, x, Experimentalvalues; - theoretical prediction. 5. RESULTS With the temperature of the surrounding, T o , measured, the values of fl, k and v can be obtained. By measuring the voltage and current the value of q can be calculated and x, the position where all measurements are made, is known. The modified Grashof number Gr* can now be calculated as
Gr* = gflqx" / k v 2.
(10)
The experimental frequency at which the flow is disturbed in a random fashion was obtained for various values of G ~ and is plotted on Figure 3. Also shown in Figure 3 is
EFFECT OF SOUND ON NATURAL CONVECTION
217
the analytical expression (equation (9)) for disturbing frequency as a function of G r a s h o f number. 6. CONCLUSION An experimental investigation has been carried out to study the influence o f sound waves on the transition from laminar to turbulent flow in the boundary layer associated with natural convection from a vertical flat plate. The following conclusions are drawn. (1) A relationship exists between the frequency at which transition occurs and the G r a s h o f number, in terms of heat flux, at a specific vertical position o f a heated plate. (2) Transition from laminar to turbulent flow has been shown to occur at a G r a s h o f number lower than the normally cited value o f Gr*,~ l0 II [13]. (3) Buckling flow theory provides a means of explaining the influence of sound on transition in the boundary layer associated with natural convection from a vertical flat plate. N o t only could these findings be applied in industrial natural convection plate heat exchangers, but also a better understanding of the process of natural convection which is of fundamental importance to the functioning of the human body and the explanation of global weather patterns has been gained. REFERENCES 1. J. A. PETERKA and P. D. RXCHARDSON1969 Journal of Fluid Mechanics 37(2), 265-287. Effects of sound on separated flows. 2. J. ADACHI, S. OKAMATO and M. ADACm 1979 Bulletin of the Japan Society of Mechanical Engineers 122(172). The effect of sound on the rate of heat transfer from a cylinder placed normal to an air stream. 3. P. D. RICHARDSON 1969 Journal of Sound Vibration 10(1), 32-41. Local effects of horizontal and vertical sound fields on natural convection from a horizontal cylinder. 4. H. KIMOTO 1986 Bulletin of the Japan Society of Mechanical Engineers 29(258). A study on heat transfer from a horizontal circular cylinder in a progressive sound field. 5. R. PARKER and M. C. WELSH 1983 International Journal of Heat and Fluid Flow 4, ! 13-127. Effects of sound on flow separation from blunt flat plates. 6. P. J. COOPER,J. C. SHERIDANand C. J. FLOOD 1986 International Journal of Heat and Fluid Flow 7, 61-68. The effect of sound on forced convection over a flat plate. 7. S. KIMURA 1983 Ph.D. Thesis, University of Colorado, Boulder, Colorado, U.S.A. Buckling flow and transition to turbulence in axisymmetric plumes. 8. H. SCHUCHTING 1979 Boundary Layer Theory (7th edition). New York: McGraw Hill. See pp. 316-317. 9. A. BEJAN 1982 Entropy Generation Through Heat and Fluid Flow. New York: John Wiley. 10. A. BEJAN 1984 Convection Heat Transfer. New York: John Wiley. 11. A. BEGAN1986 American Journal of Heat and Mass Transfer 19, 83-103. Buckling flows: A new frontier in convection heat transfer. 12. A. BEJAN 1987 in Annual Review of Numerical Fluid Mechanics and Heat Transfer (T. C. Chawla, editor), 262-304. Buckling flows: a new frontier in fluid mechanics. Washington D.C. : Hemisphere. 13. J. P. HOLMAN 1976 Heat transfer. Tokyo: McGraw-Hill. 14. E. M. SPARROW and J. L. GREGG 1956 Transactions of the American Society of Mechanical Engineers, 435-440. Laminar free convection from a vertical plate with uniform surface heat flux. APPENDIX: NOMENCLATURE D
F'
transverse dimension of a two-dimensional jet, m = (ux/v)/5(GrUS) 2"
218
Gr,* Pr Too g q U
U, nax x
y Cf
P
d~
V
H. E N G E L B R E C H T
A N D L. P R E T O R I U S
=gflqx4/kv 2, modified Grashof number for constant heat flux = v/a, Prandtl number temperature of surrounding, K frequency, Hz =9-81 ms -2, gravitational acceleration heat flux, w a t t / m 2 velocity in x direction, ms -~ maximum velocity in x direction, ms -~ vertical position from lower edge of plate, m horizontal position from surface of plate thermal diffusivity, m 2 s -~ coefficient of thermal expansion (/3 = I/T~ for ideal gas), K -~ boundary-layer thickness, m ( y / x)( Gr* / 5) ~/5, similarity variable buckling wavelength, m kinematic viscosity, m z s -~
THE EFFECT OF S O U N D O N N A T U R A L C O N V E C T I O N F R O M A VERTICAL FLAT PLATE H. ENGELBRECHT AND L. PRETORIUS
Systems Laboratory, Rand Afrikaans University, P.O. Box 524, Johannesburg 2000, Republic of South Africa (Received 5 July 1990, and in final form 26 April 1991) An experimental investigation has been carried out to study the influence of sound waves on the transition from laminar to turbulent flow in the boundary layer associated with natural convection from a vertical flat plate with uniform surface heat flux. A relationship between frequency and Grashof number at transition has been found. Buckling flow theory provides a means of explaining this relationship. 1. INTRODUCTION The effect of sound on both natural and forced convection from cylinders has been studied to quite an extent. References [ 1-4] provide an overview of the work done in this field. Parker and Welsh [5] and Cooper et al. [6] conducted experiments to determine the influence of sound on forced convection from horizontal flat plates. In both cases the reattachment length of the boundary layer was shown to be shortened when sound was applied. Parker and Welsh also showed that for flow across a plate with a square leading edge the reattachment point oscillates at the applied frequency. No effect of sound was observed for a plate with a round leading edge. Cooper et al. measured the maximum heat transfer coefficient at the reattachment point and observed that the value increases when sound is applied. In contrast to the above-mentioned work at sound frequencies between 200 and 1200 Hz, Kimura [7] studied the effect of low frequency sound (10-20 Hz) on buckling flow. He concluded that the point of transition from laminar to turbulent flow in an axisymmetric plume can be controlled by the application of an external sound field. Some of the results shown by Kimura indicate transition from laminar to turbulent flow in a narrow range of frequencies relatively insensitive to sound level (amplitude). This led to the investigation described in what follows. The effect of sound on natural convection for a vertical flat plate has not been studied extensively. It is the purpose of the work described in this paper to investigate experimentally the effect of sound on the transition from laminar to turbulent flow in the boundary layer associated with natural convection from a flat vertical plate. 2. EXPERIMENTAL APPARATUS AND INSTRUMENTATION The following experimental apparatus were used to obtain results quoted in this paper. 2.1. PLATE The vertical plate consists of a piece of hardboard with dimensions 130 x 200 x 15 m m . Tightly stretched across the length of the plate are I I strips of I0 m m wide copper-zinc 213 0022--460X/92/200213 +06 $08.00/0 © 1992 AcademicPress Limited
214
lq. E N G E L B R E C H T A N D L. P R E T O R I U S
shimstock of thickness 50 pm. The strips are 1 mm apart, and two springs are fitted at the back of the hardboard to ensure that the strips remain tight when they expand as a result of heating. The metal strips are soldered together to form a continuous conductor. The plate resistance is connected to a direct current power source capable of supplying 0-50 amp6re for 0 10 volts. 2.2. S P E A K E R A S S E M B L Y A 60 watt 8 12 cone-type loudspeaker is connected through a music amplifier to a signal generator, generating sinusoidal signals. The signals were measured to contain a single frequency, pointing to an undistorted sinusoidal signal. 2.3. HOT-WInE ANEMOMETER The characteristics of the flow in the flow region near the vertical plate are determined by means of a TSI, IFA 1000 hot-wire anemometer. A cylindrical wire probe with a resistance of 6-66 -('2 is used. The operating resistance of the hot-wire anemometer is set at 12.08 12. The output from the hot-wire anemometer is connected to an Iwatsu 5S-5212 oscilloscope. 3. EXPERIMENTAL PROCEDURE The loudspeaker and hot-wire probe are placed in positions relative to the plate as indicated in Figure 1. The hot-wire probe is at a vertical position of 0-16 m from the lower edge of the plate. For set voltages between 0 and 10 volts, the frequency of the frequency generator is slowly adjusted and the signal on the oscilloscope observed. Frequencies are observed at which the signal on the screen of the oscilloscope is disturbed form the continuous signal existing without sound applied. The disturbed signal is more random in nature and does not show definite periodicity. The frequency, voltage and current are noted. The ambient temperature is also recorded.
Imm i
J
Probe
position
i
|
I
.J Front 50
view
mm
Side view Figure 1. Positions of speaker and probe relative to plate.
f
,
4 0 mrn
i Plate
EFFECT
OF
SOUND
ON
NATURAL
215
CONVECTION
The experiment is conducted with the vertical plate in air only. The goal of the experiment is not to characterize the full extent of the flow field, but only to show the onset of turbulence at a certain point.
4. ANALYSIS Sparrow and Gregg [ 14] have described an exact solution of the laminar boundary-layer equations for free convection from a vertical plate having uniform surface heat flux. They obtained dimensionless velocity distributions for Prandtl numbers of 0-1, 1, I0 and 100. The equations provided by Sparrow and Gregg were reworked and solved on a HP 41C computer to obtain the dimensionless velocity distribution for Pr = 0-708 shown in Figure 2. Because current experiments by the authors have been carried out in air, a value of
03[ 0-2
01
O0
I I
I
2
I
3
i
4
5
Figure 2. Dimensionlessvelocitydistributionfor Pr=0.708. 0.708 for the Prandtl number is assumed. (A list of nomenclature is given in the Appendix.) From Figure 2 it can be seen that the maximum velocity occurs at a value of F ' = 0 . 2 6 for Pr--0" 708. At this point 2/5 = 0" 2 6 .
(1)
U,,ox = (0" 26)( 5)( Gr.* / 5)2/5( v / x ) .
(2)
F' = ( u x / v ) / 5 ( G r * / 5 )
Therefore
Also from Figure 2 it can be seen that the boundary-layer thickness is given by 1/.... = 5" rI = ( 6 / x ) ( G r * / 5 ) '/5 = 5.
(3)
= 5x(5/Gr*)'/5.
(4)
Therefore
Bejan [9-12] has described the theory of buckling flows. He showed that under certain conditions a fluid can show a tendency to buckle in a fashion analogous to the way that a slender column axially loaded at the ends may buckle. Bejan reasoned that this buckling of a fluid can be the onset of turbulence. The buckling wavelength of a two-dimensional
216
H. ENGELBRECHT
AND
L. P R E T O R I U S
jet is found to scale with the transverse dimension of the jet as =
,rDl~.
(5)
Kimura [7] used this wavelength to obtain a frequency at which the flow buckles during flow in a thermal plume by considering an average velocity U,,ax/2:
f = ( U,,ax/2)/2..
(6)
Substituting from equation (5) for Z yields
T= x/~U,,~x/Zrt D.
(7)
With D = S and substituting from equations (2) and (4), one has f = 0.02729(Gr*x)3/5v /x 2.
(8)
In Figure,,) is shown the relationship between f and Gr* for a value of x = 0. 16 and v = 16.84 × 10 -6 m2/s, which corresponds to Pr=0-708. In this case f = 1"7952 x lO-5(Gr*x)3/5.
(9)
50
4o
"~ 30
g .= ~ 20
tO
10"~
10 9
10 e Grashof
1010
number, Gr~
Figure 3. Relationshipbetween Gr~ and frequencyat which flowis disturbed, x, Experimentalvalues; - theoretical prediction. 5. RESULTS With the temperature of the surrounding, T o , measured, the values of fl, k and v can be obtained. By measuring the voltage and current the value of q can be calculated and x, the position where all measurements are made, is known. The modified Grashof number Gr* can now be calculated as
Gr* = gflqx" / k v 2.
(10)
The experimental frequency at which the flow is disturbed in a random fashion was obtained for various values of G ~ and is plotted on Figure 3. Also shown in Figure 3 is
EFFECT OF SOUND ON NATURAL CONVECTION
217
the analytical expression (equation (9)) for disturbing frequency as a function of G r a s h o f number. 6. CONCLUSION An experimental investigation has been carried out to study the influence o f sound waves on the transition from laminar to turbulent flow in the boundary layer associated with natural convection from a vertical flat plate. The following conclusions are drawn. (1) A relationship exists between the frequency at which transition occurs and the G r a s h o f number, in terms of heat flux, at a specific vertical position o f a heated plate. (2) Transition from laminar to turbulent flow has been shown to occur at a G r a s h o f number lower than the normally cited value o f Gr*,~ l0 II [13]. (3) Buckling flow theory provides a means of explaining the influence of sound on transition in the boundary layer associated with natural convection from a vertical flat plate. N o t only could these findings be applied in industrial natural convection plate heat exchangers, but also a better understanding of the process of natural convection which is of fundamental importance to the functioning of the human body and the explanation of global weather patterns has been gained. REFERENCES 1. J. A. PETERKA and P. D. RXCHARDSON1969 Journal of Fluid Mechanics 37(2), 265-287. Effects of sound on separated flows. 2. J. ADACHI, S. OKAMATO and M. ADACm 1979 Bulletin of the Japan Society of Mechanical Engineers 122(172). The effect of sound on the rate of heat transfer from a cylinder placed normal to an air stream. 3. P. D. RICHARDSON 1969 Journal of Sound Vibration 10(1), 32-41. Local effects of horizontal and vertical sound fields on natural convection from a horizontal cylinder. 4. H. KIMOTO 1986 Bulletin of the Japan Society of Mechanical Engineers 29(258). A study on heat transfer from a horizontal circular cylinder in a progressive sound field. 5. R. PARKER and M. C. WELSH 1983 International Journal of Heat and Fluid Flow 4, ! 13-127. Effects of sound on flow separation from blunt flat plates. 6. P. J. COOPER,J. C. SHERIDANand C. J. FLOOD 1986 International Journal of Heat and Fluid Flow 7, 61-68. The effect of sound on forced convection over a flat plate. 7. S. KIMURA 1983 Ph.D. Thesis, University of Colorado, Boulder, Colorado, U.S.A. Buckling flow and transition to turbulence in axisymmetric plumes. 8. H. SCHUCHTING 1979 Boundary Layer Theory (7th edition). New York: McGraw Hill. See pp. 316-317. 9. A. BEJAN 1982 Entropy Generation Through Heat and Fluid Flow. New York: John Wiley. 10. A. BEJAN 1984 Convection Heat Transfer. New York: John Wiley. 11. A. BEGAN1986 American Journal of Heat and Mass Transfer 19, 83-103. Buckling flows: A new frontier in convection heat transfer. 12. A. BEJAN 1987 in Annual Review of Numerical Fluid Mechanics and Heat Transfer (T. C. Chawla, editor), 262-304. Buckling flows: a new frontier in fluid mechanics. Washington D.C. : Hemisphere. 13. J. P. HOLMAN 1976 Heat transfer. Tokyo: McGraw-Hill. 14. E. M. SPARROW and J. L. GREGG 1956 Transactions of the American Society of Mechanical Engineers, 435-440. Laminar free convection from a vertical plate with uniform surface heat flux. APPENDIX: NOMENCLATURE D
F'
transverse dimension of a two-dimensional jet, m = (ux/v)/5(GrUS) 2"
218
Gr,* Pr Too g q U
U, nax x
y Cf
P
d~
V
H. E N G E L B R E C H T
A N D L. P R E T O R I U S
=gflqx4/kv 2, modified Grashof number for constant heat flux = v/a, Prandtl number temperature of surrounding, K frequency, Hz =9-81 ms -2, gravitational acceleration heat flux, w a t t / m 2 velocity in x direction, ms -~ maximum velocity in x direction, ms -~ vertical position from lower edge of plate, m horizontal position from surface of plate thermal diffusivity, m 2 s -~ coefficient of thermal expansion (/3 = I/T~ for ideal gas), K -~ boundary-layer thickness, m ( y / x)( Gr* / 5) ~/5, similarity variable buckling wavelength, m kinematic viscosity, m z s -~