STABILITY OF NON-ISOTHERMAL POISEUILLE FLOW IN VERTICAL ANNULUS FILLED WITH POROUS MEDIUM

Abstract

The non-isothermal Poiseuille flow in porous media has been a subject of intense research for over four decades. This type of flow in pipe/channel/annulus is used in many industrial situations such as extraction of bio-fuel [51], packed-bed chemical reactors [2], oil recovery process [63], solid-matrix heat exchangers and cooling of nuclear plants [54], etc. However, most of the available studies are done in vertical channel or pipe. The results of channel • or pipe can not be used to predict the flow mechanism in annular geometry. Therefore, to • understand the flow configuration in annular geometry formed by two concentric cylinders a step has been taken in the present thesis. Both linear and nonlinear theories have been used to examine the stability mechanism of the flow. The objective of this study is to investigate the effect of gap between the two concentric cylinders on the above flow for different permeable medium as well as non-isothermal resources. The annulus is filled with a homogeneous and isotropic porous medium. An external pressure gradient and a buoyancy force (due to temperature difference) drive the fully developed water flow in the annular region. The inner wall temperature of the annulus increases linearly with the axial coordinate from an upstream reference temperature and the outer wall is adiabatic. In the limit of fully developed flow, this simulates a constant heat flux condition on the inner cylinder. Note that depending on the sign of Rayleigh number, the fully developed flow may be stably stratified (i.e., the buoyancy force acts in the direction of forced flow) or unstably stratified (i.e., buoyancy force in the negative direction of forced flow). The linear stability of the above flow for both stably stratified and unstably stratified cases are analyzed in this thesis. Following the previous efforts of Yao & Rogers [123], the weakly 11 nonlinear stability of non-isothermal Poiseuille flow in vertical annulus filled with porous medium is developed. The present thesis is compiled in six chapters and the chapter wise description is given below. Chapter 1 is an introductory and contains some basic definitions, preliminaries of the flow in porous medium, brief description of hydrodynamic stability theory, work done by various authors in the field of linear and nonlinear stability analysis of Poiseuille flow, and justification regarding the model, which has been adopted for this problem. Chapter 2 addresses the basic flow characteristic of the non-isothermal Poiseuille flow in vertical annulus filled with porous medium. Both stably stratified and unstably stratified situations are considered for this study. The non-Darcy-Brinkman-Forchheimer model is used. The governing equations are solved analytically for a special case: form drag equal to zero and numerically by Chebyshev spectral collocation method. Along with the other controlling parameters, a special attention is given to understand the effect of curvature parameter (C) of the annulus on the flow configuration as well as heat transfer rate. The numerical experiments show that reducing the value of C enhances the maximum magnitude of the velocity along with heat transfer rate in the system. The impact of C (C> 10) on the flow profile as well as heat transfer rate is negligible. Furthermore, the analysis shows that the tendency of appearance of back flow, point of inflection and flow separation (in case of unstably stratified flow) in the flow profile is highly sensitive to C. Apart from this, for a small increase in Ra, a drastic change (up side down) in the flow profile can also be seen. The appearance of flow separation shifted from the vicinity of the inner wall to the outer wall. Hence, to shed more light on this phenomenon and to find the appropriate non-isothermal parameter space as a function of gap between the two concentric cylinders, in which the flow will remain as fully developed, stability analysis is needed. Chapter 3 contains the linear stability of the above Poiseuille flow for stably stratified case. For a given annulus, the stability of the basic flow is controlled by different parameters such as Reynolds number (Re), Rayleigh number (Ra), Darcy number (Da), Prandtl number (Pr), heat capacity ratio (a), viscosity ratio (A), porosity (E), and modified Forchheimer 111 number (F'). Since curvature parameter (C) plays a vital role to describe the size of the annulus, therefore impact of C on the transition mechanism of basic flow for relatively high permeable medium is considered in this chapter. To avoid numerous parametric study we have fixed the value of some of the parameters such as A, Pr, and a. at 1, 7 and 1, respectively. The disturbance momentum and energy equations are numerically solved by spectral collocation method. We have also analyzed the energy budget spectrum at critical point. The linear stability results show that increasing C as well as decreasing Da stabilizes the basic flow. 1-lowever, beyond C = 10 the impact of curvature parameter on the stabilization of the basic flow is almost negligible. From the energy analysis at critical level it is observed that the thermal-buoyant instability is the only mode of instability. Furthermore, the analysis of linear stability shows that although the impact of form drag upto a threshold value is negligible on instability but its contribution in energy dissipation - is significant. In Chapter 4, we have investigated the stability of stably stratified non-isothermal Poiseuille flow of water in vertical porous-medium annulus using weakly nonlinear stability theory, with particular emphasis on the impact of gap between the two vertical axisymmetric cylinders. For a comparative study, we have considered three different values (1 0, 0.6, 10) of C for three different values (10_I, 10_2, 10) of Da. The flow in the annulus is governed by the volume-averaged forms of the Naiver-Stokes and continuity equations derived by [117]. To carry out the weakly nonlinear analysis, we started by analyzing the range of Ra, beyond the critical point, in which the growth rate varies linearly using perturbation series solution approach. From this analysis it has been found that for high permeable medium the linear relationship between growth rate and Ra holds good for very small neighborhood of critical (bifurcation) point, however for low permeable medium it is relatively large. This gives an impression that the nonlinear interaction is not effective for low permeable medium, which is also supported by finite amplitude analysis. The finite amplitude analysis predicts both the supercritical as well as subcritical bifurcation at and in the vicinity of bifurcation point, which are also investigated by nonlinear energy spectrum. The analysis lv of the nonlinear energy spectrum for the disturbance reveals that in case of Da = IO 2 or C = 10 an instability that is supercritical for some wavenumber may be supercritical or subcritical at other nearby wavenumber. The equilibrium amplitude increases on decreasing the media permeability as well as reducing the gap between inner and outer cylinders. In the limiting case (i.e., at C = 10) the fundamental disturbance of stably stratified nonisothermal Poiseuille flow (SSNPF) of water in vertical channel filled with porous medium will have minimum amplitude. The influence of nonlinear interaction of different superimposed waves on some physical aspects: heat transfer, friction coefficient, nonlinear energy spectrum, and steady secondary flow is also investigated. Investigation related to impact of superimposed waves on the pattern of secondary flow, based on linear stability theory gives an impression that cells of flow pattern are just shifted. This is the consequence of negligible modification in the buoyant production of disturbance kinetic energy and significant modification in the rate of the viscous dissipation of disturbance energy for the considered set of parameters. In Chapter 5, the instability mechanism of the above flow is analyzed for unstably stratified case. Linear stability analysis predicts first azimuthal mode as the least stable mode in the entire range of C for Da = 10_I and 10. For Da = 10-2 first azimuthal mode is also the least stable mode except for 0.02 < C < 0.1 where zero azimuthal mode is the least stable mode. However, for Da = 10-2 (except for 0.02 <C < 0.1) and 10 the least stable mode at n = 1 is under R-T (Rayleigh-Taylor) mode. Energy analysis at critical level shows the change in the characteristic: stabilizing to destabilizing, of disturbed kinetic energy due to shear factor (Es) on changing C for Da = 10_I and 10-2, which is the cause of changing the shape of secondary flow from uni-cellular to bi-cellular. Moreover, depending on the media permeability as well as curvature parameter three types of instability namely, thermal-buoyant, interactive and Rayleigh-Taylor are observed. This Rayleigh-Taylor type instability is independent of Re and becomes the least stable mode in (I Ra , Re)-plane on decreasing Da. C takes a significant role on the appearance of Rayleigh-Taylor instability. Although for stably stratified case no relation between the appearance of point of inflection V and instability of the flow is observed but for unstably stratified water flow, the appearance of flow separation is the sufficient condition for instability. Furthermore, to analyze the nature of the Rayleigh-Taylor instability and the finite amplitude behavior of unstable disturbance that occurs beyond the linear stability, especially when the permeability of the medium is relatively low we have used weakly nonlinear stability theory in terms of finite amplitude analysis. Our analysis on Landau constant and amplitude as a function of Ra reveals two important facts. First, for both Da = 10-2 and 10 depending on C as well as Ra, Rayleigh-Taylor instability shifts from supercritical to subcritical (reverse) at and beyond Rae. Second, the amplitude profile experiences a sudden jump whenever the type of instability changes away from the critical point. Finally, Chapter 6 presents the summary and concluding remarks of this thesis and the possible directions of the future scope.