STABILITY OF MIXED CONVECTION FLOW IN VERTICAL CHANNEL FILLED WITH POROUS MEDIUM

Abstract

The thesis reports a linear as well as weakly non-linear stability analysis of non-isothermal parallel flow in a vertical channel filled with porous medium. The flow is induced by external pressure gradient and buoyancy force due to maintenance of non-isothermal walls of the channel. Two different boundary conditions are considered: (i) when both walls of the channel are kept at constant but different temperatures (called as differentially heated), and (ii) when the temperature of both walls varies linearly with respect to the vertical coordinate. The non-Darcy model which gives rise to the volume averaged Navier-Stokes (VANS) equation is used except for some comparative study where Darcy model is also used. In porous medium, two different situations: local thermal equilibrium state (LTE) and local thermal non-equilibrium state (LTNE) are possible to explain the mechanism of heat transfer. Both these situations are considered in the thesis. The spectral method has been adopted to solve the governing equations of the problem. The weakly nonlinear analysis of parallel mixed convective flow in vertical channel filled with porous medium is developed on the basis of pioneering work of Stuart [99, 100], Stuart and Stewartson [101] and Yao & Rogers [130]. Motivation for the present study is based on the following three facts: (i) mixed convection through wall bounded domain filled with porous medium has numerous applications, (ii) stability analysis has not been extended to parallel mixed convective flow in a differentially heated channel, (iii) there are subtle differences between parallel mixed convective flow in a linearly heated channel and differentially heated channel. The entire study is split into six chapters and a brief description of each chapter is given below. i ii The first chapter contains some basic concepts related to porous medium as well as hydrodynamic stability, a brief state of the art in this direction, and a note on considered models in this thesis is presented. In Chapter 2, the linear stability of parallel mixed convective flow (PMCF) in a differentially heated vertical channel filled with incompressible fluid saturated porous medium is analyzed . Here, it is assumed that the two phases of porous medium: fluid and the solid matrix are in LTE state. The flow is controlled by six governing parameters: Darcy number (Da), ratio of Grashof number to Reynolds number (Gr0 = Gr=Re), Reynolds number (Re), product of Forchheimer number and Reynolds number (F0 = FRe), Prandtl number (Pr), and porosity (e). The stability analysis of PMCF is carried out after a partial reinvestigation of Kwok and Chen’s [62] numerical study which confirms that Gill’s [41] result on stability of parallel natural convection flow (PNCF) is not valid when no-slip condition and inertia impact are taken into account. Similar to the parallel flow due to linearly varying wall temperature [5], in the present problem also the basic flow possesses three different types of instability, namely thermal-shear, interactive, and thermal-buoyant, which depends on controlling parameters, mainly Pr. The regimes of above three instabilities over the domain of Pr are functions of Da as well as Re. Stability analysis also reveals that when Re is fixed at 1000, the appearance of point of inflection in the basic velocity for fluid with Pr less than 30 acts as a necessary condition for instability for all considered values of Da. For a very small range of Pr, in the vicinity of Pr = 0:01, velocity disturbances are more responsible for the instability and kinetic energy production due to shear force is most dominant in balancing the dissipation of disturbance kinetic energy (KE). So, for these values of Pr the type of instability of the basic flow is thermal-shear. On the other hand, for relatively larger values of Pr (i.e., water or heavy oil), the instability is primarily due to the thermal disturbances and KE production due to buoyant term is most dominant in the energy balance and results in thermal-buoyant instability of the flow. In contrast to the flow in purely fluid medium, where the production of KE is solely suppressed by viscous dissipation (Ed), in porous medium the KE can be suppressed by surface drag (ED), iii form drag (EF) and viscous dissipation (Ed). For relatively low permeable medium (i.e., for Da = 10􀀀4) dissipation through ED dominates in the entire range of Pr for Re = 1000. From the influence of modified Forchheimer number it has been found that for Pr equal to 7 or 100 there exist a minimum value (Fo) of F0 below which the effect of form drag on the critical Gr0 as well as corresponding wave number is negligible and above this minimum value F0 stabilizes the flow. It is also found that form drag may destabilize the flow for very less viscous (i.e., Pr <1) fluid. For the range of parameters considered in this study, the scale analysis reveals that when the permeability of the medium is less than or equal to 2:5 10􀀀6m2 and half width of the channel is 5cm, then the critical value of DT (i.e, temperature difference between the channel walls) for PMCF of water is higher than the same for PNCF (e.g., DT for PNCF and PMCF when K = 2:5 10􀀀6m2 are 8.58oC and 13.3oC, respectively), which may be the consequence of thermal-buoyant instability of the flow. In the case of PMCF there exists an optimum value of Re, depending on Da and F0, at which the value of DT corresponding to critical Gr0 is least. Thus, the flow will remain stable for all Re if the value of DT is less than this least value. Also it is expected that for a given media permeability there will be a least value of Pr above which the instability of the flow will take place for DT less than 20oC, i.e., results from linear stability analysis using Boussinesq approximation will be more realistic. Chapter 3 is an extension of the previous chapter by considering the two phases of porous medium in LTNE state. This study is carried out to get a better perspective of how two different modes, namely local thermal equilibrium and local thermal non-equilibrium, of transfer of energy between solid and fluid phases inside the porous medium affect the instability of the considered flow. The interphase heat transfer coefficient (H) and porosity scaled thermal conductivity ratio of fluid and solid phases (g) are new parameters in LTNE state whose effects are analyzed. The linear instability boundary shows that for a given value of Da, the interphase heat transfer coefficient has a stabilizing effect on the instability of the flow. The relative change in critical Gr0 as a function of H for different values of Da shows that the impact of local thermal non-equilibrium state is relatively higher in the case iv of high permeable porous medium. In comparison to the LTE state, the disturbance kinetic energy balance at the critical level in LTNE state shows that shear instability is dominant in larger range of Pr for all considered values of Da. For Pr = 7, the disturbance kinetic energy balance shows that the contribution of Eb decreases whereas the contribution of Es increases by almost 10% as compared to LTE state. The interphase heat transfer coefficient affects the instability of the flow when quadratic form drag is relatively low, i.e., up to F0 = 100. The finite amplitude instablity of stably stratified parallel mixed convective flow due to linearly varying wall temperature in vertical channel filled with porous medium has not been carried out yet. Thus, before studying the finite amplitude instability of the flow considered in Chapter 2, we consider the finite amplitude instability of stably stratified parallel mixed convective flow of air as well as water in a vertical channel filled with porous medium in Chapter 4. The objective of this study is to analyze the nature of bifurcation and the finite amplitude behavior of unstable disturbances that occur beyond the linear instability boundary, specially when the permeability of the medium and strength of the flow are reasonably high. This is accomplished by reviewing the linear stability results, and then a weakly nonlinear analysis is made to trace the evolution of finite amplitude perturbation. From the review it has been checked through dimensional analysis that the non-isothermal PMCF becomes unstable under mild heating conditions. For example, when the channel is filled with water saturated porous medium with permeability equal to 10􀀀7m2, PMCF becomes unstable even when the temperature gradient, C, is equal to 3.7. In the case when channel is filled with air saturated porous medium with permeability equal to 9 10􀀀7m2, PMCF becomes unstable when C = 13:9. The results obtained using Boussinesq approximation remains valid for a wide range of input parameters. To study the evolution of finite amplitude perturbation, we have analyzed the variation of real part of Landau constant ((a1)r) and amplitude in the vicinity of the least linearly stable point as a function of Re for air as well as water. Depending on the flow strength as well as media permeability, the weakly nonlinear analysis predicts both supercritical and subcritical bifurcations for air v and only supercritical bifurcation for water. In the case of air, an increase in Forchheimer number or decrease in media permeability delays the shift of bifurcation from supercritical to subcritical or vice versa in Reynolds number space. In general, compared to subcritical bifurcation, the supercritical bifurcation occurs at relatively lower values of Re. The amplitude profile shows a peak, due to change in sign of (a1)r, at the Reynolds number where the shift in bifurcation from supercritical to subcritical takes place. The similar characteristic is also observed in physical quantities such as Nusselt number and friction coefficient, which is a consequence of the distortion in basic flow velocity and temperature. For air, when Da = 10􀀀3, the nonlinear spectrum of kinetic energy in supercritical regime of Re shows that due to change in the shape of fundamental wave, modification in the buoyant production of KE (T11) becomes main destabilizing factor, however modification in the gradient production (P110) as well as modification in the surface drag dissipation (K11) become major stabilizing factors. On the other hand in the subcritical regime of Re, P110 is a destabilizing factor along with T11. Furthermore, based on very small value of imaginary part of complex linear eigenvalue (ci), we have analyzed the bifurcation away from the critical point for particular choice of Re in super as well as subcritical regimes. It has been found that for all the considered values of Da the supercritical bifurcation as a function of Rayleigh number (Ra) remains supercritical whereas for Re in subcritical regime it may change to supercritical one. Also, the heat transfer rate (skin friction) increases (decreases) significantly and experiences jump in the neighbourhood of Ra where the change of bifurcation takes place. The nonlinear balance of kinetic energy for the finite disturbances also supports the results obtained through Landau constant. It is important to mention here that while studying mixed convection flow in vertical annulus, Rogers et. al. [88] have found that the buoyant instability is supercritical at all wavenumbers whereas with the shear and interactive (or mixed) instabilities, both subcritical and supercritical branches appear on the neutral curves. In contrast to the above results it has been observed in the present study that with the buoyant and mixed instabilities both subcritical and supercritical branches appear vi on the neutral curves. Finally, for values of Re in supercritical regime, the disturbance temperature contours maintain the same shape but due to nonlinear interaction of waves they move towards the center of the channel. In the case of Re in subcritical regime, the shape of disturbance velocity as well as temperature contours gets changed drastically which in turn enhances the destabilizing characteristic of T11. In Chapter 5, we have analyzed the finite amplitude instability of fully developed mixed convection flow in differentialy heated vertical channel filled with porous medium. The results are presented with respect to two different fluids with Prandtl number equal to 0:7 and 7 using non-Darcy model. In contrast to the non-isothermal flow in linearly heated channel where subcritical bifurcations were observed, in this case the finite amplitude analysis predicts only supercritical bifurcation of the flow at and beyond the critical points for both fluids as the sign of real part of Landau constant (a1)r is found to be negative for all the considered values of Da. The magnitude of equilibrium amplitude experiences sharp fall when the Reynolds number is increased form 0 to 1000 and remains almost constant beyond Re = 1000 due to non-linear saturation. Also, only supercritical bifurcation is predicted in the neighborhood of critical wavenumbers. Due to the interaction of different harmonics, increased heat transfer rate is obtained for distorted flow as compared to the same for basic flow. Considering the importance of results in practical situations, higher order weakly nonlinear stability analysis and direct numerical simulation could be used to further explore the stability behavior of the present flow. In the end, Chapter 6 presents the summary and concluding remarks of the thesis and some possible directions for future work.