PRESSURE WORK EFFECTS IN UNSTEADY CONVECTIVELY DRIVEN FLOW ALONG A VERTICAL PLATE

Transcription

1 Proceeding of 3ICCHMT 3 rd International Conference on Computational Heat and Ma Tranfer May 6 3, 3, Banff, CANADA Paper Number 87 PRESSURE WORK EFFECTS IN UNSTEADY CONVECTIVELY DRIVEN FLOW ALONG A VERTICAL PLATE Alan Shapiro and Evgeni Fedorovich School of Meteorology, Univerity of Oklahoma, Norman, OK, USA ABSTRACT Thi paper reviit the claical problem of convectively driven one-dimenional (parallel) flow along an infinite vertical plate. We conider flow induced by an impulive (tep) change in plate temperature and by a udden application of a plate heat flux. Proviion i made for preure work and vertical temperature advection in the thermodynamic energy equation, procee that are generally neglected in previou onedimenional tudie of thi problem. In a tatically table environment thee additional procee provide a imple negative feedback mechanim: warm air rie, expand and cool relative to the environment, wherea cool air ubide, compree and warm relative to the environment. Exact olution of the vicou equation of motion are obtained by the method of Laplace tranform for the cae where the Prandtl number i unity. The preure work and vertical temperature advection are found to have a ignificant impact on the tructure of the olution at later time. NOMENCLATURE c pecific heat at contant preure (=cont); p - g gravitational acceleration ( 9.8 m ); p preure t time T temperature (T ' i it deviation from the environmental value) w velocity component along the plate; W non-dimenional velocity along the plate; x direction normal to the plate; z direction along the plate; buoyancy parameter; molecular thermal diffuivity (=cont); non-dimenional temperature perturbation; kinematic vicoity coefficient (=cont); non-dimenional time; non-dimenional x coordinate. INTRODUCTION The tranient natural convection flow of a vicou fluid adjacent to vertical urface i a fundamental problem in fluid mechanic and heat tranfer, with ignificance for a variety of engineering application []. The implet form of thi problem i one-dimenional tranient convective flow adjacent to an infinite vertical plate, firt conidered in [] for an impulive change in plate temperature. Author of [3]- [6] have obtained analytic olution to thi problem for a variety of temporal variation in plate

2 temperature and plate heat flux. In thee tudie preure work i neglected and ambient thermal tratification i not conidered. Accordingly, the thermodynamic energy equation reduce to the tandard one-dimenional heat conduction equation. After olving thi equation for the temperature field, the vertical velocity i recovered from the vertical equation of motion which ha the form of a diffuion equation with inhomogeneou buoyancy forcing term. Thee exact unteady olution of the Bouineq equation are potentially valuable a imple conceptual/pedagogical model of natural convection a well a tool for validating numerical model of convection. The preent tudy refine the claical theory of one-dimenional tranient convectively driven flow along a vertical plate by including the preure work term in the thermodynamic energy equation. It alo extend the claical theory by making proviion for a linearly varying ambient temperature. A we will ee, in the context of the one-dimenional model, the preure work and vertical temperature advection term are of the ame form o the refinement and extenion can be taken into account imultaneouly by combining both procee into a ingle advection term. With attention retricted to a perfect ga with a Prandtl number (Pr) of unity, analytical olution are readily obtained by the method of Laplace tranform. Proviion for temperature tratification or preure work allow the unteady olution to approach a teady tate at large time, wherea if there i no temperature tratification or preure work (i.e., claical olution), the olution grow without bound. A in the claical cae, the new olution will only be appropriate for time prior to the arrival of the leading edge effect and prior to the onet of any flow intabilitie. The teady-tate olution for the tratified cae ha already been obtained in [7], and it linear tability ha been analyzed in [8] and [9]. In thoe tudie the tability wa found to decreae with increaing plate perturbation temperature and increae with increaing tratification. In light of thoe tudie and the experiment decribed in [] and [], we anticipate that the main interet in our olution will be in cae where the temperature tratification i large enough to delay (or prevent) flow intability. For weak temperature tratification, the deviation of our olution from the correponding claical olution may not become apparent before the flow become untable. The outline of the paper i a follow. We firt formulate the problem of one-dimenional natural convection for a fluid, whoe thermal expanion coefficient i that of a perfect ga. The governing equation are introduced and reduced to a ingle fourth order linear partial differential equation for the perturbation temperature. For Pr=, thi equation i olved analytically for the cae of an impulively changed plate perturbation temperature (hereafter referred to a ST cae) and for the cae of a uddenly applied plate heat flux (hereafter referred to a SF cae) by the method of Laplace tranform. The new olution are then compared to the claical olution, in which preure work i neglected and the environment i conidered to be iothermal. Latly, we briefly decribe numerical method for computing velocity and temperature olution for the cae of arbitrary Pr value and dicu temperature olution for different Pr in the cae of a uddenly applied plate heat flux. GOVERNING EQUATIONS We conider a Carteian coordinate ytem in which the z-axi oppoe the gravity vector, the y-z plane coincide with an infinite vertical plate, the x-axi i directed perpendicular to the plate, and fluid fill the region x. The fluid i quiecent with zero horizontal temperature gradient until thermal condition at the plate are abruptly changed at t=. The enuing motion i one-dimenional with the only non-zero velocity component, the vertical velocity w, varying only in the x direction. Accordingly, the ma conervation equation (incompreibility condition) i trivially atified. In order for the horizontal equation of motion to be atified (albeit trivially), the horizontal preure gradient force mut be zero everywhere. Thu the local preure p( xzt,, ) mut equal p, the environmental preure at x, which atifie the hydrotatic equation p / z g.

3 3 With the denity ( x, zt, ) and temperature T( x, z, t ) decompoed into it environmental ( ) and perturbation (') component, ( x, zt, ) = ( z) + '( x, t), T( x, z, t ) =T ( z) +T'( x, t), and with linearized equation of tate, '/ r T'/ Tr, the Bouineq form of the vertical equation of motion i w w T ', () t x where the ubcript r denote a contant reference value, and the term g( T'/ Tr ) T' i the buoyancy force per unit ma of the fluid. (a) (b) Figure. Contour of (a) (, ) and (b) W (, ) for the ST cae. The contour increment i. in W (, ) and.5 in (,). Negative contour are dahed. The thermodynamic energy equation in the cae under conideration ha a form []: T T p p T cp w w cp, () t z t z x where the firt two term on the right-hand ide repreent the preure work effect. Decompoing the temperature in the above equation, making ue of p( xzt,, ) = p, retricting attention to linearly varying T ( z), and auming we come to / T' T' w, (3) t x where dt / dz g / cp i a contant parameter. The w term in (3) ariing from the combined effect of preure work and vertical temperature advection introduce a coupling between w and T ' beyond the appearance of the buoyancy force in the vertical equation of motion (). Since the temperature gradient in a tatically neutral adiabatic environment i dt dz g/ c [3], we can interpret a the difference / p between the environmental temperature gradient and the temperature gradient in a tatically neutral adiabatic environment. Under tatically table condition (> the w term provide a imple negative feedback in () and (3): warm fluid rie, expand and cool relative to the environment, wherea cool

4 4 fluid ubide, compree and warm relative to the environment. Proviion for thi feedback add a new level of realim to the claical problem. The plate boundary condition for t> are the no-lip condition for velocity, w(,t)=, with either a pecified temperature perturbation T ', a in ST cae, or a pecified kinematic heat flux Q( T'/ x ), a in SF cae, are applied at the plate urface, at x=. In both cae w and T ' are aumed to vanih far from the plate. We introduce dimenionle independent variable and a / /4 / x ( ) and t( ), and cale temperature perturbation T ' and velocity w depending on the thermal condition pecified at the plate. In the ST cae, the dependent variable w and T ' are caled / / a T'( T' ) and W w ( T' ). In the SF cae, the caling of T ' and w ha the form: T' ( ) /4 / Q 3/4 /4 Q and W w. In both cae, the ytem (), (3) become: W W, W, (4) Pr with Pr / being the Prandtl number. The following boundary condition for the dimenionle variable W (, ), W (, ), (, ), and either (, ) (ST cae) or / (, ) (SF cae). Next, we ubtitute W from the econd equation of (4) into the firt one and obtain the fourth order equation for dimenionle temperature 4 3 ( / Pr), (5) 4 Pr which we will olve analytically in the following ection for the pecial cae of Pr=. IMPULSIVE CHANGE IN PLATE TEMPERATURE (ST CASE) Multiplying (5) by e and integrating (by part, where poible) from = to = yield the ordinary differential equation 4 d ˆ d ˆ ( / Pr) ( ) ˆ, (6) 4 Pr d d where ˆ e d i the Laplace tranform of. Retricting attention to Pr=, equation (6) reduce to 4 d ˆ d ˆ ( ) ˆ, (7) 4 d d A general olution of (7) for ˆ vanihing at i ˆ aexp( i) bexp( i ). (8) Evaluating coefficient a and b from the boundary condition at the plate (which provide a=b=.5 ), and taking the invere tranform of ˆ, we arrive at exp( ) exp( ) L i L i, (9) where L denote the invere Laplace tranform operator. The invere tranform in (9) can be expreed a [4]: L exp( i) exp i ' d ' 3/, () ' 4 ' and o (9) become

5 5 co ' d 3/ ' 4 '. () (, ) exp ' The velocity field W i calculated from the econd equation of (4) with upplied from (). After integration by part and rearrangement of term, we obtain in ' W(, ) exp ' 3/ ' 4 ' d. () Although it i not obviou, careful analyi of () how that it provide the deired temperature value at the plate urface: lim (, ). Differentiation of () with repect to yield d co (, ) d. (3) Thu, the plate heat flux i infinite at = and undergoe a decaying ocillation a it approache / in the limit. Since the non-dimenional period correpond to a dimenional period of / ( ), tronger tratification, or larger buoyancy parameter value, are aociated with higher ocillation frequencie. Contour plot of (, ) and W (, ) are preented in Fig.. Thee plot how the boundary-layer character of the olution and the ocillatory approach to teady-tate condition. The plot were contructed by numerically evaluating the integral in () and (). (a).5 (b) W Figure. Cro-ection of (a) (, ) and (b) W (, ) at dimenionle time. Solid line preent claical olution and heavy olid line preent new olution for the cae of an impulive (tep) change in plate perturbation temperature. Dahed line preent claical olution and heavy dahed line preent new olution for the cae of uddenly applied plate heat flux. Claical olution for the ST cae when the preure work i neglected, the environment i conidered to be iothermal, and Pr=, have the form [5]:

6 6 d 3/, (4) ' 4 ' (, ) erfc exp ' 3 W (, ) ierfc exp ' exp 3/ 4 ' 4 ' d, (5) 4 where erfc( x) exp( x' ) dx' i the complementary error function and ierfc( x) erfc( x') dx', ee x x [5]. To facilitate the comparion, we have non-dimenionalized the claical olution in the ame manner a our new olution. New and claical olution for the horizontal ditribution of (, ) and W (, ) at in the ST cae are compared in Fig.. It i clearly een that the claical model predict larger temperature perturbation and larger velocitie than predicted by the new model that account for negative feedback mechanim aociated with tratification and preure work effect. The teady-tate olution for dimenionle temperature ( ), correponding to arbitrary value of Pr, i obtained from (5) with time derivative neglected. In thi cae, (5) reduce to a linear fourth-order ordinary differential equation, whoe olution in our cae i given by /4 /4 ( ) co( Pr / )exp( Pr / ). (6) The teady-tate velocity W ( ) i readily obtained from the teady verion of the econd equation of (4): / /4 /4 W ( ) Pr in( Pr / )exp( Pr / ). (7) Equation (6) and (7) were firt obtained in [7]. It may be hown, ee [6], that our unteady olution () and () approach the teady-tate olution (6) and (7) with Pr=. SUDDEN APPLICATION OF PLATE HEAT FLUX (SF CASE) Now we conider the cae when a plate heat flux i uddenly applied at = (SF cae). A in the ST cae, we firt come to (5) and, after the aumption i made that Pr=, obtain the general expreion (8) for the tranformed temperature ˆ. Applying boundary condition for the SF cae, we come to ˆ i i exp( i ) exp( i i ). (8) The invere tranform of (6) given by L ˆ i evaluated with the convolution theorem ued in conjunction with (): ' in( ') co '' (, ) exp d'' d 3/ 3/ ( ') '. (9) '' 4 '' Comparing the olution (9) for the temperature in the SF cae with the temperature olution () for the ST cae (we will denote it by ), we have ST 3/ ST d, () ( ') in( ') (, ) (, ') ' which mean that at any location the olution for the SF cae i a weighted average over time of the olution for the ST cae. To obtain the temperature perturbation at the plate in the SF cae, we take a limit of (). Integration by part yield x in 3/ (, ) C F ( / ), () where C F ( x) co( x' /) dx' i a Frenel coine integral [5], which tend to / a. Thu, the

7 7 dimenionle plate temperature approache a. The velocity W (, ) i again obtained a reidual of the econd equation of (4): in( ') in '' W(, ) 3/ exp '' ' 3/ d d. () ( ') '' 4 '' Comparing () and (), one may ee that W (, ) for the SF cae can be expreed a weighted average of the W (, ) olution for the ST cae (we will denote it by W ): in( ') W(, ) W 3/ ST(, ') d'. (3) ( ') Contour plot of (, ) and W (, ) for the SF cae obtained by numerical evaluation of the integral in (9) and () are preented in Fig. 3. Equation () wa ued to evaluate (, ). The olution for the SF cae are qualitatively imilar to the olution for the ST cae (Fig. ). ' ST (a) (b) Figure 3. Contour of (a) (, ) and (b) W (, ) for the SF cae. Contour increment are a in Fig.. Claical olution for the SF cae when the preure work i neglected, the environment i conidered to be iothermal, and Pr=, have the form [5]: (, ) ierfc exp d ' exp 3/, (4) ' 4 ' 4 W (, ) i erfc exp ' exp 3/ 4 ' 4 ' d, (5) 4 where i erfc( x) ierfc( x ') dx '. Careful analyi of (4) and (5) in comparion with new olution (9) x and () indicate that, analogouly to the ST cae, the new and claical olution for the SF cae are in cloe agreement only for. Already at, the new olution for the SF cae ignificantly diverge

8 8 from their claical counterpart, ee Fig.. The teady-tate olution ( ) and W ( ) for the SF cae, /4 /4 /4 ( ) Pr co( Pr / )exp( Pr / ), (6) 3/4 /4 /4 W ( ) Pr in( Pr / )exp( Pr / ), (7) are obtained in the ame manner a for the ST cae (ee previou ection). The olution (6) and (7) differ from their counterpart for the ST cae, equation (6) and (7), by a multiplicative factor of /4 Pr. A hown in [6], the unteady olution (9) and () at approach the teady-tate olution (6), (7) with Pr=. NUMERICAL INVESTIGATION OF CONVECTION WITH ARBITRARY Pr NUMBER The analytical olution for temperature and vertical velocity dicued in the previou ection of the preent paper have been obtained for the particular cae of Pr=. In order to invetigate Prandtl-number variability of dynamic and thermal regime in the conidered convectively driven flow, we addreed the problem numerically. The following ytem of equation in the Bouineq approximation wa ued: ui ui p ' u j T' i3 ui t x x x x, (8) 3 j i j ' x, (9) T' T' T uj u3 t xj xj j i j ui =, (3) x where p' ( p p )/ r i the normalized preure deviation from it hydrotatic value, i=,,3; j=,,3 u ( u, u, u ) ( u, v, w) i the three-dimenional flow velocity vector with the component along the coordinate axe x x,, y x and z x3, ij i the Kronecker delta, and the Eintein convention of ummation over repeated indice i applied. Other notation correpond to the one introduced in the "Governing Equation" ection. The above ytem can be ued for invetigation of both the regime of laminar and turbulent convection along a heated vertical plate. For the preent tudy, the plate thermal forcing (which i either temperature perturbation or heat flux) and phyical parameter of the problem (,,, and ) have been choen in a way to enure the laminar flow regime. We dicretized equation (8)-(3) in a rectangular domain tretched along the x axi. Equation for the prognotic variable (u, v, w, T ') were integrated over time by the leapfrog cheme with a weak filter, and the patial derivative were approximated by the econd-order finite-difference expreion, a decribed in [7]. The Poion equation for the preure wa olved by the Fat Fourier Tranform technique over the y-z plane, and by the tridiagonal factorization method in the x direction. The no-lip boundary condition for the velocity component were applied at the urface of the plate, where either the temperature perturbation value wa precribed (in the ST cae) or the temperature gradient related to the urface kinematic heat flux Q a T'/ xq/ (in the SF cae). The normalized preure at the urface of the plate wa calculated from the truncated verion of the third (i=3) equation of motion. At the oppoite ide of the domain, at large x, the normal gradient of all prognotic variable were et to zero. Over the x-y and x-z computational boundarie of the domain (four in total), the periodic boundary condition for all computed variable were applied. The output velocity and temperature perturbation value were averaged over the y-z plane. However, variation of thee quantitie in the y-z plane (i.e., parallel to the wall) were found to be negligibly mall.

9 Figure 4. Pr-number dependence of dimenionle temperature (, ) at τ= for the SF cae. Solution for Pr= are given by ymbol: the croe preent analytical olution and the circle preent the numerical olution. Numerical olution for Pr=,, and.5 are depicted by olid, dahed-dotted, and dahed line, repectively. Reult of numerical imulation of perturbation temperature in the SF cae for four different Pr number are preented in Fig. 4. The computed (, ) ditribution for Pr= almot perfectly overlap with the correponding analytical curve (ee alo Fig. ), which prove the adequacy of the employed numerical procedure for imulation of the conidered flow type. Within the range of invetigated Pr number variation (from.5 to, which cover mot of it variability in natural fluid), the dimenionle plate temperature turn out to be rather enitive to the value of Pr number. Smaller Pr lead to noticeably higher temperature value at the plate and thicker thermal boundary layer. CONCLUSIONS Thi tudy reviit one of the implet cenario of natural convection, the one-dimenional (parallel) convectively driven flow of a vicou fluid along an infinite vertical plate. Our model refine the claical theory by including the preure work term in the thermodynamic energy equation, and extend the theory by making proviion for a linearly varying ambient temperature. With attention retricted to a Prandtl number of unity, exact analytic olution of the vicou equation of motion are obtained for flow driven by an impulively changed plate perturbation temperature, and a uddenly impoed plate heat flux. The conidered thermodynamic procee introduce a negative feedback mechanim whereby warm fluid rie, expand and cool relative to the environment, while cool fluid ubide, compree and warm relative to the environment. Thi negative feedback mechanim reult in a convective flow that approache a teady tate at large time. In contrat, in the claical olution where preure work i neglected and there i no temperature tratification, the diturbance continue to pread outward from the plate and no teady tate i approached. In thee latter flow, the fluid experience a peritent buoyancy-

10 induced acceleration, and the vertical velocity grow without bound. It i traightforward to how, ee [6], how the method of Laplace tranform can be ued to obtain olution for arbitrary temporal variation of plate perturbation temperature or plate heat flux, again with a Prandtl number of unity. However, analytic olution for Prandtl number different from unity appear to be more challenging to obtain, and numerical analyi may be the mot convenient way to proceed. In thi tudy we have preented preliminary numerical olution for the cae of a uddenly impoed plate heat flux for Prandtl number in the range.5-. The numerical reult indicate that thicker boundary layer are obtained at maller Prandtl number. ACKNOWLEDGEMENTS The author gratefully acknowledge the aitance of Robert Conzemiu in redeigning the computer code ued in the numerical part of thi tudy. REFERENCES. Gebhart, B., Jaluria, Y., Mahajan, R. L., and Sammakia, B., 988, Buoyancy-Induced Flow and Tranport. Hemiphere Publihing.. Illingworth, C. R., 95, Unteady Laminar Flow of a Ga Near an Infinite Flat Plate. Proc. Cambridge Philo. Soc., 46, pp Siegel, R., 958, Tranient Free Convection From a Vertical Flat Plate. Tran. ASME, 8, pp Menold, E. R., and Yang, K.-T., 96, Aymptotic Solution for Unteady Laminar Free Convection on a Vertical Plate. Tran. ASME: J. Appl. Mech., 9, pp Goldtein, R. J., and Brigg, D. G., 964, Tranient Free Convection About Vertical Plate and Circular Cylinder. Tran. ASME: J. Heat Tranfer, 86, pp Da, U. N., Deka, R. K., and Soundalgekar, V. M., 999, Tranient Free Convection Flow Pat an Infinite Vertical Plate with Periodic Temperature Variation. J. Heat Tranfer,, pp Gill, A. E., 966, The Boundary Layer Regime for Convection in a Rectangular Cavity. J. Fluid Mech., 6, pp Gill, A. E., and Davey, A., 969, Intabilitie of a Buoyancy-Driven Sytem. J. Fluid Mech., 35, pp Bergholz, R. F., 978, Intability of Steady Natural Convection in a Vertical Fluid Layer. J. Fluid Mech., 84, pp Brooker, A. M. H., Patteron, J. C., Graham, T., and Schöpf, W.,, Convective Intability in a Time-Dependent Buoyancy Driven Boundary Layer. Intl J. Heat Ma Tranfer, 43, pp Patteron, J. C., Graham, T., Schöpf, W., and Armfield, S. W.,, Boundary Layer Development on a Semi-Infinite Suddenly Heated Vertical Plate. J. Fluid Mech., 453, pp Schlichting, H., 979, Boundary-Layer Theory. 7th edition. McGraw Hill. 3. Holton, J. R., 99, An Introduction to Dynamic Meteorology. Academic Pre. 4. Doetch, G., 96, Guide to the Application of Laplace Tranform. Van Notrand. 5. Abramowitz, M., and Stegun, I. A., 964, Handbook of Mathematical Function with Formula, Graph, and Mathematical Table. National Bureau of Standard, Wahington, D.C. 6. Shapiro, A., and Fedorovich, E., 3, Unteady Convectively Driven Flow Along a Vertical Plate Immered in a Stably Stratified Fluid. J. Fluid Mech. (in review). 7. Fedorovich, E., Nieuwtadt, F. T. M., and Kaier, R.,, Numerical and Laboratory Study of Horizontally Evolving Convective Boundary Layer. Part I: Tranition Regime and Development of the Mixed Layer. J. Atmo. Sci., 58, pp