Proceedings of 4 th ICCHMT May 7 0, 005, Paris-Cachan, FRANCE ICCHMT 05-53 NATURAL CONVECTION IN INCLINED RECTANGULAR POROUS CAVITY SUBJECT TO HEAT FLUXES ON THE LONG SIDE WALLS L. Storesletten*, D.A.S. Rees**, and I. Pop*** *Department of Mathematics, Agder University College, Servicebos 4, 4604 Kristiansand, Norway **Department of Mech. Engineering, University of Bath, Bath, UK ***Department of Mathematics, University of Cluj, Cluj, Romania *Correspondence author: Fa: + 38407 Email: leiv.storesletten@hia.no ABSTRACT In this paper we consider natural convection in inclined rectangular cavities filled with a fluid-saturated porous medium. Convection is driven by uniform, but generally different, heat flues at the long side walls, the remaining walls being perfectly insulated. An analytical parallel flow solution is found in the core region. For the case of horizontal cavity an eplicit analytical solution is derived. Finally, numerical solutions are presented which are valid in the whole cavity. NOMENCLATURE A aspect ratio K permeability pq, parameters S, S constants (stratification) m thermal diffusivity heating parameter, heat flues, non-dimensional heat flues heat capacity ratio non-dimensional temperature non-dimensional stream function INTRODUCTION Natural convection in cavities filled with a porous medium plays a vital role in various geophysical and engineering problems. A rich variety of eperimenttal, numerical and analytical results has been published over recent decades. However, most of the studies have been related to either a vertically or horizontally imposed temperature or to heat flu differences. There is relatively little wor on natural convection in inclined cavities with prescribed but generally different, heat flues at two opposing sidewalls, the remaining walls being perfectly insulated. In the present paper a class of natural convection flows inside an inclined, rectangular porous cavity is studied using both analytical and numerical means. Sen et al.[] considered this problem for the special case of equal heat flues. In the same year Vasseur et al [] presented a detailed analysis of convection in long cavities of various orientations with sidewall heating on either the long or short walls. An etension to include Brinman effects was also undertaen by Vasseur et al [3]. In the present paper we revisit this general problem, but generalize the analysis to cases where there are unequal heat flues on the boundaries. To date such cases have only been considered by Sundström and Kimura [4]. MATHEMATICAL FORMULATION We consider natural convection in an inclined rectangular cavity filled with a fluid-saturated porous medium. The cavity is two-dimensional with height H and length L, H L, the inclination angle is, 0 /, and a Cartesian frame is chosen with the origin located in the centre of the cavity and the -ais parallel with the long side walls as shown in Fig.. Furthermore, the long sidewalls are subject to the different but uniform heat flues and, while the short endwalls are perfectly insulated. The governing equations are based on the equation of continuity, Darcy s law and the energy equation, where the Boussinesq approimation applies.
F( y)sin, 0 /, (8a) f ( y) t S( sin ycos ), (8b) Figure Depicting the cavity under investigation. The -ais is aligned with the long walls. In the diagram, the cavity is inclined at an angle of =5 degrees. Also shown are typical streamlines (continuous) and isotherms (dashed). On introducing non-dimensional variables by setting * * * t ( / H ) t, / H, y y / H, m * ( / gk H ), ( T T0 )/( H), A L/ H, / /, /, ( ) the governing equations and boundary conditions become sin cos () y Ra t y y 0, on y y 0, on y y () (3) (4) 0, 0 on A/. (5) y The starred terms denote dimensional quantities and Ra is the Rayleigh number given by gk H Ra. (6) m The two heat flues may be described using only one parameter : cos, sin. (7) Therefore the full problem has four non-dimensional parameters: the Rayleigh number Ra, the aspect ratio A, the inclination angle and the heating parameter. ANALYTICAL SOLUTIONS Following Sundström and Kimura [4], who studied the corresponding clear fluid problem, we assume that a parallel flow solution of equations ()-(4) is valid in the core region, (i.e. sufficiently far from the end walls at A/), and it is of the form, where and S are unnown constants. It is evident that this solution does not satisfy the boundary conditions (5) at the end walls A/. However, we assume that matching end-region solutions eist, which is confirmed by our full numerical simulations. On substituting (8) into equations () and (), we obtain the following ordinary differential equations F f, f S( Rasin ) F (9) subject to the boundary conditions F( ) 0, f ( ) S cos, f ( ) S cos. (0) where primes denote differentiation with respect to y. On integrating second equation in (9) from -½ to ½ and using the boundary equations, it follows that. () The stratification parameter S is obtained by applying the energy flu condition of Bejan [5], / ( Ra ) dy 0 () y / which means that the heat transport through a cross section constant should be zero. From equations (9) to () it follows that (3) F S( Rasin ) F with the boundary conditions F( ) 0 (4) F( ) S cos, F( ) S cos. There are two different cases depending on whether S is positive (natural stratification) or negative (unnatural stratification) when 0 /. There is a third case, 0, corresponding to a horizontal cavity. Case : S 0, 0 /. We introduce the parameter 0 ( Rasin ) S., defined by / The solution of equations (3) and (4) is given by
cosh y sinh y F( y) p q y cosh( ) sinh( ) (5) where q (6) By applying the energy flu condition () and integration by parts, we obtain the following nonlinear equation for the stratification parameter S p q cosh ( ) sinh ( ) sinh sinh q p sinh( ) S 4 cos 4 S cosh( ) Ra (7) Figs. and 3 show stratification curves as a function of log(ra) for various inclination angles. When Ra is small, the stratification is also small, and may be shown to be proportional to Ra. This situation corresponds essentially to slow flows where the isotherms are parallel to the heated or cooled surfaces. At present the situation when Ra is large is unclear, and will require an asymptotic analysis of equation (7). However, since S>0, the stratification of the cavity is such that warmer fluid lies over cooler fluid, and therefore the flow is thermoconvectively stable, even for Rayleigh numbers as high as a million. Figure 3 Depicting how the stratification, S, varies with Ra for various inclinations when =45 degrees. Case : S 0, 0 /. The parameter 0 is now defined by ( Rasin ) S / of equations (3) and (4) is given by cos y sin y F( y) p q y cos( ) sin( ). The solution (8) where n, n,, 3,... On following the same procedure as in Case, the nonlinear equation for S in the present case becomes p q cos ( ) sin ( ) sin sin q p sin( ) S 4 cos 4 S cos( ) Ra (9) Numerical calculations of the full elliptic equations indicate that solutions (8) for negative S are unstable for all Rayleigh numbers, and therefore we do not present such solutions. Figure Depicting how the stratification, S, varies with Ra for various inclinations when =0 degrees. Case 3: An eplicit solution for the horizontal case When the cavity is horizontal the physics of the problem is entirely different. The solution given by (8) breas down when 0 : the stream function vanishes and the temperature is no longer allowed to depend on. However, an alternative assumption would be to introduce a linear stratification S in the -direction rather than in the direction of gravity.
Following Sundström and Kimura [4], we tae a solution of the form F( y), f ( y) t S (0) where and S are constants. On substituting (0) into () and (), we get the following equations F S, f ( S Ra) F () subject to the boundary conditions F( ) 0, f ( ), f ( ), () where is again found to be given by equation (). The solution of equations ()-() is given by F y S y (3) ( ) ( ) 4 f ( y) RaS ( y ) ( ) y ( ).(4) 4 On applying the energy flu condition () we obtain a nonlinear equation for the stratification S S [ ( Ra) S 4 ( ) Ra ] 0. (5) 30 6 Thus, S 0 is always one solution, corresponding to pure conduction. For bottom heating, 0, and for Ra 4/( ) there are also two other solutions 5 S / {4 ( ) Ra. (6) Ra These solutions are both stable, whereas the solution S 0 is only stable when the other two do not eist, i.e. for sub-critical Rayleigh numbers Ra Rac. These nonzero solutions may be related to the stability properties of the S 0 state. Here the critical Rayleigh number is 4 Ra c, (7) which corresponds to the triple solution S 0. It turns out that the corresponding critical wave number is equal to zero. For the important special case with equal negative 0 heat flues, / (i.e. 5 ), the critical Rayleigh number, Rac. If we define Ra in accordance with Sen et al.[], we have to multiply it by /, which gives the critical value. This value agrees with the result found by Sen et al.[]. The same critical value was also reported by Nield [6] as the limit for linear stability of the conductive state with these boundary conditions. We do not pursue the aspect of stability in the present paper. RESULTS AND DISCUSSION Equations () and () were solved for the cavity displayed in Fig. using a straightforward finite difference method. Equation () used a combination of the DuFort Franel method and Araawa s [7] discretisation of the nonlinear terms to update the temperature at each timestep. The corresponding Poisson s equation for the streamfunction used a standard pointwise Gauss-Seidel solver with Correction Scheme multigrid. A cavity of aspect ratio A=6 was chosen and a 93 grid used. We concentrated on cases where the cavity is heated from above, and therefore the resulting solutions are unique and stable for all choices of the parameters. In practice this means that the resulting solutions are independent of the initial conditions. Figure 4 shows how the flow and temperature fields vary with Ra when the layer is inclined at =45 degrees and when =0 degrees (i.e. the upper surface is insulated and the lower surface cooled). For low Rayleigh numbers the flow proceeds in a clocwise direction, with the isotherms displaying an increasing deformation as Ra increases. It is clear from these subfigures that A=6 is a sufficiently large aspect ratio within which there is a clearly identifiable core region with parallel flow. When the Rayleigh number rises to 00 a second anticlocwise circulation near the upper surface begins to grow. If one traces out the temperature along a horizontal line from the upper surface to the lower surface, it is apparent that the temperature rises to a maimum within the cavity, and this causes an upflow in the middle of the cavity with downflow along the surfaces. At even higher Rayleigh numbers this effect still eists, there remains two long contrarotating cells. However, the isotherms align to be horizontal, ecept within the boundary layers along the longer walls. Finally, it is important to note that the -derivative of the temperature field along y=0, the centre of the cavity, yields Ssin (see equation (8b)); these numerically obtained values match well the analytical solutions given by (7), although the error increases gradually with Ra as a fied grid is being used.
Figure 5 shows how the structure of the flow varies as changes from 45 degrees to 5 degrees, when =45 degrees and Ra=000. The top frame corresponds to cooling from below and heating from above for which the flow is clocwise. On the other hand, the bottom frame corresponds to heating from below and cooling from above, and therefore the flow is in the opposite direction. At intermediate values of there is a gradual transformation from one etreme case to the other. At first the rotational symmetry about =y=0 is lost. And then a narrow region of anticlocwise circulation is formed which grows with increasing and eventually supplants the clocwise motion. In all cases, the isotherms are close to being horizontal at y=0. Solutions for values of outside of our chosen range are found to be identical to those already presented but correspond to a 80 degree rotation in the depicted patterns. Figure 4 Showing the effect of increasing Ra on the flow and temperature fields for =45 degrees and =0 degrees. Figure 5 Showing the effect of varying on the flow and temperature fields for =45 degrees and Ra=000.
cases the isotherms are horizontal in the main bul of the cavity. A full analysis of this problem would require a greater range of parameter selection including those cases where heating from below taes place in a horizontal or nearly horizontal channel, for in these instances instability may be epected. It is highly liely that the instabilities will tae the form of longitudinal vortices with aes in the -direction. REFERENCES Figure 6 Showing the effect of varying on the flow and temperature fields for =0 degrees and Ra=000. Finally we display the effect of different inclination angles on the Ra=000 and =0 o case. When =45 0 we recover a case shown in Figure 4 with two cells, where the lower cell is stronger than the upper cell. For all cavity inclinations isotherms attempt to remain horizontal, but the insulating condition at the upper surface ensures that there remains a circulation in the opposite direction to that induced by the lower cooled surface. Indeed, this upper recirculation increases in size as the cavity tends towards the horizontal. In the opposite limit of a vertical cavity a single cell circulation is obtained. The strength of the flow (as may be monitored by the thicness of the momentum boundary layers) increases as increases due to the direct action of buoyancy forces on the cooling surface. CONCLUSIONS To conclude, we have studied a cavity flow problem in porous media using both numerical and analytical means. The analytical techniques assume that the flow is fully developed, while the numerical solutions tae place within a finite but sufficiently large aspect ratio cavity. The two approaches agree sufficiently well when the core region of the numerical solution is compared with the analytical solutions. At sufficiently large values of Ra distinct boundary layers form, which are equivalent to having large values of in the analytical study. In these. Sen, K., Vasseur, P. and Robillard, L., 987, Multiple steady states for unicellular natural convection in an inclined porous layer, Int. J. Heat Mass Transfer 30, pp. 097-3.. Vasseur, P., Satish, M.G. and Robillard, L., 987, Natural convection in a thin, inclined, porous layer eposed to a constant heat flu, Int. J. Heat Mass Transfer 30, pp. 537-550. 3. Vasseur, P., Wang, C.H. and Sen, M., 990, Natural convection in an inclined rectangular porous slot: the brinman-etended Darcy model, Trans. A.S.M.E. J. Heat Transfer, pp507-5. 4. Sundström, L-G. and Kimura, S., 996, On laminar convection in inclined rectangular enclosures, J.Fluid Mech. 33, pp. 343-366. 5. Bejan, A., 983, The boundary layer regime in a porous layer with uniform heat flu from the side, Int.J.Heat Mass Transfer 6, pp. 339-446. 6. Nield, D.A., 968, Onset of thermohaline convection in a porous medium, Water Resour.Res. 4, pp. 553-560. 7. Araawa, A., 966, Computational design of long-term numerical integration of the equations of fluid motion. I. Twodimensional incompressible flow, J. Comp. Phys., pp. 9-43.