Thermal Modulation of Rayleigh-Benard Convection

Transcription

1 Thermal Moulation of Rayleigh-Benar Convection B. S. Bhaauria Department of Mathematics an Statistics, Jai Narain Vyas University, Johpur, Inia-3400 Reprint requests to Dr. B. S.; Z. Naturforsch. 57 a, (00); receive June 8, 00 Thermal convection in a flui layer confine between two horizontal rigi bounaries has been stuie with the help of the Floquet theory. The temperature istribution consists of a steay part an an oscillatory time-epenent part. Disturbances are assume to be infinitesimal. Numerical results for the critical Rayleigh numbers an wave numbers are obtaine. It is foun that the isturbances are either synchronous with the primary temperature fiel or have half its frequency. Some comparisons have also been mae. Key wors: Moulation; Stability; Rayleigh Number; Galerkin Metho; Thermal Convection.. Introuction Thermal convective instability often arises when a flui is heate from below. This instability can occur in the atmosphere an the ocean. Thermal convection in a horizontal flui layer heate from below has been stuie extensively uring the last fifty years. Chanrasekhar [] has given a comprehensive review of this stability problem. Drazin an Rei [], Koschmieer [3] an Getling [4] have also iscusse in etail about Rayleigh-Benar convection. The present paper eals with the thermal convection in a flui layer confine between two horizontal rigi planes. The flui layer is heate from below an above perioically with time. In 969 Venezian [5] investigate this problem as the thermal analogue of Donnelly s experiment [6]. Venezian's theory oes not fin any such finite frequency, as obtaine by Donnelly, but fins that for the case of moulation only at the lower surface, the moulation woul be stabilizing with maximum stabilization occurring as the frequency goes to zero. Consiering low-frequency moulation, Rosenblat an Herbert [7] have investigate the linear stability problem an foun an asymptotic solution. They consiere free-free surfaces an employe perioicity an amplitue criteria to calculate the critical Rayleigh number. Rosenblat an Tanaka [8] have investigate the problem by using the Galerkin proceure. However, in their analysis they consiere rigi-rigi bounaries. Yih an Li [9] have investigate the formation of convective cells in a horizontal flui layer, confine between two rigi bounaries. They foun that the isturbances (or convection cells) oscillate either synchronously or with half frequency. Gresho an Sani [0] have treate the linear stability problem with rigi bounaries. They foun that gravitational moulation coul significantly affect the stability limits of the system. Finucane an Kelly [] have carrie out an analytical-experimental investigation to confirm the results of Rosenblat an Herbert. Besies investigating the linear stability, Roppo et al. [] have also carrie out the weakly non-linear analysis of the problem. Weimin an Charles [3] have obtaine a numerical solution of the linear Rayleigh- Benar convection an compare the results with the analytic solution. Aniss et al. [4] have carrie out an investigation on linear problem of the convection parametric instability in the case of a Newtonian flui confine in a Hele-Shaw cell an subjecte to a vertical perioic motion. In their asymptotic analysis they have investigate the influence of the gravitational moulation on the instability threshol. Bhatia an Bhaauria [5, 6], Bhaauria an Bhatia [7] have stuie the linear stability problem for more general temperature profiles. Recently Bhaauria et. al. [8] have stuie the thermal convection in a Hele-Shaw cell with parametric excitation uner saw-tooth an step-function oscillations. The objective of the present stuy is to fin the critical conitions uner which thermal convection / 0 / $ c Verlag er Zeitschrift für Naturforschung, Tübingen

2 B. S. Bhaauria Thermal Moulation of Rayleigh-Benar Convection 78 with the bounary conitions (.5) amit an equilibrium solution in which Fig.. Benar Configuration. starts. A sinusoial profile is consiere to moulate the wall temperatures, an the effect of moulation on the thermal convective instability is investigate. Galerkin s metho is use to solve the problem. The results have their relevance with convective flows in the terrestrial atmosphere.. Formulation We consier a layer of a viscous, incompressible flui, confine between two parallel horizontal walls, one at an the other at. The walls are infinitelyexteneanrigi. The configurationis shown in the Figure. The governing equations in the Boussinesq approximation are ( 0)ˆ + (.) 0 (.) (.3) where 0 an 0 are the constant reference ensity an temperature, respectively, is the acceleration ueto gravity, the kinematicviscosity, the thermal iffusivity, the coefficient of volume expansion, an ( )theflui velocity. The relation between 0 an 0 is given by 0 [ ( 0)] (.4) Here we consier the moulation of the temperature of both the lower an upper bounary. The externally impose wall temperatures are: 0 cos( ) at 3 (.5) cos( ) at 3 where is the moulating frequency an / is the perio of oscillation. The equations (. -.4) along ( )0 ( ) ( ) (.6) An equation for the pressure ( ), which balances the buoyancy force, is not require explicitly, however the temperature ( ) can be given by the iffusion equation (.7) The ifferential equation (.7) can be solve with the help of the bounary conitions (.5). Here my objective is to examine the behaviour of infinitesimal isturbances to the basic solution (.6). With this in view, substitute ( ) + + (.8) into (. -.3) an linearize the equations with respect to the perturbation quantities,. Now, to simplify these equations we procee as follows: (i) The isturbance equations are reuce to the equations one for an the other one for the velocity component. (ii) These quantities are Fourier analyze as ( 3 ) ( 3 )exp[ ( + )] (.9) ( 3 ) ( 3 )exp[ ( + )] where an are the wave numbers in the an irection, respectively. (iii) All quantities are non-imensionalize, we put ( ) ( 3 ) + ( 0 ) 0 (.0) where is the non-imensionalize horizontal wave number. Then the bounary conitions (.5) reuce to cos at cos at where ( 0 ) (.)

3 78 B. S. Bhaauria Thermal Moulation of Rayleigh-Benar Convection Then the solution of (.7) can be written as sinh( ) + Re (.) sinh( ) where (.3) Also, the perturbation equations obtaine after the processes (i), (ii) an (iii) are foun to be + (.4) (.5) an ( ) cosh cosh sinh sinh cos cos if is o, sin sin if is even. (3.4) The functions ( )an ( ) above are efine in a manner so that they vanish at. For their erivatives to vanish at these bounaries, it is require that are to be the roots of the characteristic equation tanh ( ) tan 0 (3.5) where ( 0 ) 3 / is the Rayleigh number, / is the Prantl number an is the temperature graient which can be obtaine irectly from the equation (.). The bounary conitions on an are 3. Metho 0 at 0 at (.6) (.7) The presence of the time-epenent coefficients in (.5) prohibits a separation of variables solution to the system (.4,.5). Therefore we use Galerkin s technique, which transforms the two partial ifferential equations (.4) an (.5) into a system of orinary ifferential equations. The latter are then solve numerically. Here we have a set of linearly inepenent functions, which are calle the basis functions. The epenent variables of the problem are represente by the finitesum of these basisfunctions. We put ( ) ( ) where ( ) sin + ( ) ( ) (3.) ( ) ( ) (3.) (3.3) The roots of (3.5) are given in the book of Chanrasekhar ([], p. 636). It is clear that the functions ( )an ( ) each form an orthonormal set in the interval ( /, /). Alsotheyareevenin( /, /) when is o, an vice versa. Now we substitute (3.) an (3.) into (.4) an (.5). Then multiply (.4) by ( ) an (.5) by ( ),,, 3,...,. The resulting equations are then integrate with respect to in the interval ( /, /). The outcome is a system of, orinary ifferential equations for the unknown coefficients ( )an ( ). + ( ) + 4 ( ) (3.6) ( + ) ( ) + Re ( ) ( ) (3.7) where is the Kronecker elta. The other coefficients, which occur in (3.6) an (3.7) are ( ) ( ) (3.8)

4 B. S. Bhaauria Thermal Moulation of Rayleigh-Benar Convection 783 an ( ) ( ) (3.9) cosh( ) ( ) ( ) sinh( ) (3.0) All these coefficients can be easily evaluate in their close form, but here we have calculate them numerically using Simpson s (/3)r rule (Sastry [9], p. 5). It is convenient for computational purpose to take, i. e. total N equations an then rearrange them. For this, first multiply the equations (3.6) by the inverse of the matrix ( ), an then introuce the notations 3 4 (3.) Now combine (3.6) an (3.7) to the form (3.) ( 3 an ) where ( )is the matrix of the coefficients in (3.6) an (3.7). 4. Analysis It is clear that the coefficients ( ) of (3.) are either constant or perioic in with perio 0 /, therefore the stability of the solution of (3.) can be iscusse by the Floquet theory (Cesari [0], p. 55). We know that the solution in the Floquet theory is of the form ( ), where ( ) is a perioic function of with perio 0. The vanishing of the real part r of gives the stability bounary. Here our aim is to etermine the conition uner which 0. If the imaginary part of is zero, the isturbance is synchronous with the unsteay part of the mean temperature fiel. However if 0 is equal to or, then the isturbance is subharmonic, having a frequency half that of the unsteay mean temperature fiel. Let ( ) ( )col ( ) ( ) ( ) ( 3 ) (4.) be the solutions of (3.), which satisfy the initial conitions (0) (4.) The solutions (4.) with the conitions (4.) form linearly inepenent solutions of (3.). Once these solutions are foun, one can get the values of ( ), an then arrange them in the constant matrix [ ( )] (4.3) The eigenvalues,, 3,..., of the matrix are also calle the characteristic multipliers of the system (3.), an the numbers,efine by the relations exp( ) 3 (4.4) are the characteristic exponents. The values of the characteristic exponents etermine the stability of the system. We assume that the are orere so that Re( ) Re( ) Re( ) (4.5) Then the system is stable if Re( ) < 0, while Re( ) 0 correspons to one perioic solution an represents a stability bounary. This perioic isturbance is the only isturbance which will manifest itself at marginal stability. To obtain the matrix C we have integrate the system (3.), using the Runge-Kutta-Gill Proceure (Sastry [9], p. 7). The eigenvalues,, 3,..., of the matrix are foun with the help of Rutishauser metho (Jain, Iyengar, an Jain [], p. 6). 5. Results an Discussion As usual in the Galerkin proceure, we etermine the value of in the process of numerical solution. It is foun that for the parameter ranges of our interest it is sufficient to take (two Galerkin terms-one even an one o). Therefore all the following results are relate to. In this case the critical value of the Rayleigh number an the wave number in the absence of moulation ( 0) are foun as c (5.)

5 784 B. S. Bhaauria Thermal Moulation of Rayleigh-Benar Convection Fig.. Variation of c with respect to. 073, 5. Fig. 4. Variation of neut with respect to. 073, 5. Fig. 3. Variation of c with respect to. 073, 5. Fig. 5. Variation of neut with respect to. 073, 0.5.

6 B. S. Bhaauria Thermal Moulation of Rayleigh-Benar Convection 785 Fig. 6. Variation of neut with respect to. 0 73, 0.5, 00. When 0an 0, we calculate the moifie value of c with variation in other parameters. We also check the critical value of the wavenumber. Here the results have been obtaine by solving (3.) for,, 3 an 4, The results are calculate at moerate values of as we are intereste only in the moulating effect of the oscillation; there seems to be no reason why this theory can not be applie for large values of the parameters. Here it is possible to obtain a relationship between the critical Rayleigh number c an the corresponing critical wave number c in terms of the imensionless frequency of the oscillating primary temperature fiel. Since the critical Rayleigh number is the minimum value of as a function of the wave number for a fixe value of, the critical curve consists of the ifferent curves, corresponing to synchronous solutions (S-curve) an subharmonic solutions (Hcurve). In Fig., we epict the variation of the critical Rayleigh number c with. The graph consists S- curves an H-curves, which are alternative. From the figure one can see that each cusp in the c- curve is really the intersection of an S-curve an an H-curve, both of which can be continue beyon their intersections. Thus in the area above the H-curve there are also synchronous isturbances, but isturbances with half-frequency can be expecte to be more unstable. Similarly above an S-curve there are isturbances with half frequency, but synchronous isturbances are more unstable. In the above figure we note that when is small, the effect of moulation on the stability of the system is stabilizing with the convection occurring at a later Fig. 7. Variation of neut with respect to. 3., 0.5, 5. point than in the unmoulate case ( 0). However as increases the effect of the unsteay part of the primary temperature fiel becomes estabilizing. The results agree very much with the finings of Yih an Li [9]. They also agree with the results of Rosenblat an Tanaka [8]) an Bhaauria an Bhatia [7], who foun that the effect of the out of phase moulation of the temperature fiel is stabilizing. Comparing the results of Rosenblat et. al. [8] an Bhaauria an Bhatia [7] one can see that their curves are smooth while here we have an alternative S-curve an H- curve. Their unmoulate case ( 0)correspons to 0 of this stuy. Initially, as increases we have a smooth curve upto 06 an then an alternative H- curve an S-curve. As we go beyon a certain value of we fin a estabilizing effect, however in their cases the stabilizing effect is continuously increasing as increases. This ifference may be because here the bounary conitions are slightly ifferent from theirs. The above results also agree with the finings of Venezian [5] an Bhatia an Bhaauria [5], who foun that for out of phase moulation, initially the effect of the unsteay part is stabilizing but becomes estabilizing at a particular as increases. The results agree with that of Yih an Li [9], who foun whilestuyingtheinstabilityof an unsteayflow that the effect of moulation is estabilizing. They also foun that the critical curve is compose of the two curves, one corresponing to the synchronous solution an the other correspons to subharmonic one. Figure 3 shows the variation of the critical wavenumber corresponing to the critical Rayleigh number c in Figure. The critical value appears to be iscontinuous from the c - curves. This is because

7 786 B. S. Bhaauria Thermal Moulation of Rayleigh-Benar Convection in Fig., the S-curves an H-curves are not continue beyon their intersections. The c - curves are also compose of S-curves an H-curves. The existence of synchronous an subharmonic isturbances has alreay been inicate by Gresho an Sani [0], Yih an Li [9], Clever et al. [] an Aniss et al. [4] in their investigations. Figure 4 epicts the value of the Rayleigh number neut (at neutral stability) with respect to at 3.. Therefore the value of neut is slightly greater than the value of c in Figure. However the values of in both the figures o not iffer much, as the value of in Fig. 3 is never very much ifferent from 3.. In Fig. 5 we epict the value of neut with at 0 73 an 0 5. We fin that the effect of moulation is stabilizing, ecreasing with increasing frequency.as increases to infinity, the effect of the unsteay part of the primary temperature fiel isappears altogether. These results are in agreement with Bhatia an Bhaauria [5], Bhaauria an Bhatia [7], Rosenblat an Tanaka [8] an Venezian [5]. Figures 6 an 7 correspon to the variations of neut with an, respectively. From Fig. 6 it is very clear that the critical value of is very close to 3.. Figure 7 shows thattheenhancementof neut is greatest when 06. This means that the effect of moulation is highest stabilizing when 06. Acknowlegement The author is thankful to his wife Preeti Singh Bhaauria for typing the paper an rawing all the figures by computer. [] S. Chanrasekhar, Hyroynamic an Hyromagnetic Stability, Oxfor University Press, Lonon, 96. [] P. G. Drazin an W. H. Rei, Hyroynamic Stability. Cambrige University Press (98). [3] E. L. Koschmieer, Benar Cells, an Taylor Vortices. Cambrige University Press (993). [4] A. V. Getling, Rayleigh-Benar Convection: Structures an Dynamics. Worl Scientific Publishing Company (998). [5] G. Venezian, J. Flui Mech. 35(), 43 (969). [6] R. J. Donnelly, Proc. Roy. Soc. A 8, 30 (964). [7] S. Rosenblat an D. M. Herbert, J. Flui Mech. 43(), 385 (970). [8] S. Rosenblat an G. A. Tanaka, Phys. Fluis 4(7), 39 (97). 97 (963). [9] C.-S. Yih an C.-H. Li, J. Flui Mech. 54(), 43 (97). [0] P. M. Gresho an R. L. Sani, J. Flui Mech. 40(4), 783 (970). [] R. G. Finucane an R. E. Kelly, Int. J. Heat Mass Transfer 9, 7 (976). [] M. N. Roppo, S. H. Davis, an S. Rosenblat, Phys. Fluis 7(4), 796 (984). 663 (993). [3] Xu Weimin an Charles A. Lin, Tellus 45A, 93 (993). [4] S. Aniss, M. Souhar, an M. Belhaq, Phys. Fluis (), 6 (000). [5] P. K. Bhatia an B. S. Bhaauria, Z. Naturforsch. 55a, 957 (000). [6] P. K Bhatia an B. S. Bhaauria, Z. Naturforsch. 56a, 509 (00). [7] B. S. Bhaauria an P. K. Bhatia, Physica Scripta, 66(), 59 (00). [8] B. S. Bhaauria, P. K. Bhatia, an Lokenath Debnath, Int. J. Math. Math. Sci., in press (00). [9] S. S. Sastry, Introuctory Methos of Numerical Analysis, n E. Prentice-Hall of Inia Private Limite, New Delhi, 993. [0] L. Cesari, Asymptotic Behavior an Stability Problems, Springer Verlag, Berlin 963. [] M.K.Jain,S.R.K.Iyengar,anR.K.Jain, Numerical Methos for Scientific an Engineering Computation, Wiley Eastern Limite, New Delhi, 99. [] R. Clever, G. Schubert, an F. H. Busse, J. Flui Mech. 53, 663 (993). [3] G. Z. Gershuni an E. M. Zhukhovitskii, J. Appl. Math. Mech. 7, 97 (963).