NUMERICAL STUDIES OF SOME CONVECTIVE FLOW IN POROUS MEDIA: STABILITY IN VERTICAL PIPE

Abstract

The concept of transport through porous media has received considerable atten-tion in recent decades owing to its wide range of applications in the engineering and applied sciences; ranging from contaminant transport [39], paper manufactur-ing [52], geophysics and petroleum engineering [51], to marine science [25]. Un-derstanding the dynamic behavior of fluid flow through porous media, especially flow-transition, is still a challenge in fluid dynamics research. Most of the stud-ies relevant to free convection in a porous media are induced by uniform bound-ary conditions. Whereas, convection due to non-uniform boundary condition is largely overlooked. Apart from this, convection due to external pressure gradient and buoyancy forces (i.e mixed convection) in a porous media is a fundamental problem in fluid dynamics and still a field of active and ongoing research. For example, convec-tion in permeable sediment due to hydrothermal vent is a combination of forced and natural convection. Because of the high pressure difference the hot and highly dense mixture of mineral water used to move in the upward direction through 11 permeable sediment layer and leads to a forced convection, whereas, the temper-ature gradient of the sediment layer leads to density gradients in the fluid and in the presence of gravitational field, these variation in density induce varying body forces in the fluid, which in turn, causes natural convection. In the present thesis, an attempt has been taken to understand the flow dy-namics induced by non-uniform boundary condition in porous media as well as hydrodynamic stability of mixed convective flow in a vertical pipe. Present thesis is concerned with two types of-problems: (i) free convective heat and mass transfer in porous cavity due to non-uniform boundary conditions, (ii) mixed convection in a vertical pipe filled with a porous medium and its stability. The former type of problems have been solved completely by using the Spectral Element method [79], whereas, the Spectral Collocation Method [19] has been used to study numerically the mixed convection in vertical porous-pipe and its stability. The outline of the thesis is as follows. Chapter 1, is introductory and it contains a brief outline of the fundamentals of flow in porous media, hydrodynamics stability theory and a review of the litera-ture, mainly related to (i) single diffusive natural convection in a porous enclosure, (ii) double-diffusive natural convection in a porous enclosure, and (iii) mixed con-vection in vertical system and its stability. In chapter 2, introduction to four numerical methods (Finite-Difference, Finite 111 Element, Spectral Methods, and Spectral Element Method) and mathematical for-mulations of Pseudo Spectral Method and Spectral Element method are given. In Chapter 3, a comprehensive numerical investigation on the natural convec-tion in a hydro-dynamically anisotropic porous enclosure is presented. The flow is due to non-uniformly heated bottom wall and maintenance of constant tempera-ture at cold vertical walls along with adiabatic top wall. It is an extension work of Basak et al. [4] by incorporating the medium as hydrodynamically anisotropic. Brinkman extended non-Darcy model, including material derivative, ist consid-ered. The principal direction of the permeability tensor has been taken oblique to the gravity vector. The Spectral Element method has been adopted to solve numer-ically the governing conservative equations of mass, momentum and energy, by using a stream-function vorticity formulation. Special attention is given to under-stand the effect of anisotropic parameters on the heat transfer rate as well as flow configurations. The numerical experiments show that in case of isotropic porous enclosure, the maximum rate of bottom as well as side heat transfer (Nub and Nus) take place at the aspect ratio, A, of the enclosure equal to 1, which is, in general, not true in the case of anisotropic porous enclosures. The flow in the enclosure is governed by two different types of convective cells: rotating (i) clockwise, (ii) anti-clockwise. Based on the value of media permeability as well as orientation angle, in anisotropic case, one of the cell will dominant on the other one. In contrast to isotropic porous media, enhancement of flow convection in the anisotropic porous iv enclosure does not mean to increase the side heat transfer rate always. Further-more, the results show that anisotropy causes significant changes in the bottom as well as side average Nusselt numbers. In particular, the present analysis shows that permeability orientation angle has a significant effect on the flow dynamics and temperature profile; and consequently on the heat transfer rates. In chapter 4, we have extended the work of chapter 3 by incorporating mass transfer into the system. Here, flow is induced by equal and non-uniform (sinu-soidal) temperature and concentration at the bottom wall, and equal and uniform temperature and concentration at vertical walls. The top wall is adiabatic and im-permeable. Rigorous numerical investigation are made to understand the effect of anisotropic parameters i.e. Orientation angle of media permeability tensor (0), per-meability of the media through Darcy number (Da) and permeability ratio (K*) on the fluid flow as well as heat and mass transfer mechanisms. The heat and mass transfer rate have been investigated via Nusselt and Sherwood number respec-tively. Here also two types of convective cells: (i) rotating clockwise (ii) rotating anticlockwise, are observed in the entire flow mechanism and depending on the media permeability one cell dominants on the other. Apart from this, average bot-tom as well as side heat and mass transfer rates are symmetric about the line 4 = 45° and =135° in the domain [0°,901 and [90°,180°] respectively. It is also found that due to anisotropic permeability the flow dynamic becomes complex. In chapter 5, mixed convective flow in vertical pipe filled with a porous medium V is considered. The flow is induced by an external pressure gradient and buoyancy force. The wall temperature varies linearly with vertical coordinate. The non Darcy Brinkman-Forchheimer-Wooding extended model has been used. Here, the flow is governed by controlling parameters: (i) heat source intensity i.e. Rayleigh number (Ra), (ii) media permeability in terms of Darcy number, Da, (iii) Forchheimer num-ber, F. Rigorous parametric studies have been considered for both buoyancy as-sisted as well as buoyancy opposed cases. For F equal to zero, the governing equa-tions of the fully developed mixed convective flow is solved analytically, whereas, for F 0, the same is solved by Chebyshev Spectral Collocation method. It has been found that when buoyancy is in favour of flow, depending on the values of other parameters the velocity profile posses point of inflection beyond a threshold value of Ra. In case of buoyancy opposed flow, the velocity profile may Contains point of inflection in the center zone and point of separation at the vicinity of the wall. The appearance of point of separation cause back flow near the wall. In con-trast to buoyancy assisted case, where enhancement of Ra increases heat transfer as well as flow strength, in buoyancy opposed case enhancement of Ra always does not increase the flow strength as well as heat transfer. The point of separation as well as inflection are died out on reducing media permeability. It has been found that based on other controlling parameters, there exist a threshold value of Ra be-yond that the temperature profile posses point of inflection. A kind of distortion of velocity as well as temperature is found on the enhancement of Ra further. vi In chapter 6, the linear stability analysis of the above problem has been pre-sented for both buoyancy assisted as well as opposed cases. It has been found that in contrast to buoyancy opposed case, where first azimuthal mode is always least stable, in buoyancy assisted case depending on the value of media permeability as well as Reynolds number, the flow will be least stable under either first azimuthal mode or zero azimuthal mode. The stability of the basic flow is controlled by gov-erning parameters Reynolds number, Rayleigh number, Prandtl number, Darcy number and porosity, and specific heat capacity ratio. In this study, porosity and specific heat capacity ratio. and Forchheimer number are fixed at 0.9, 1 and 0 re-spectively. The stability analysis indicates that for the same Reynolds number (Re), the fully developed base flow is highly unstable for fluid of high Prandtl number for both cases buoyancy assisted as well as opposed. In contrast to a pure viscous fluid, where the effect of Pr is not significant, in isotropic porous medium Prandtl number takes a significant role in characterizing the flow stability in buoyancy op-posed case. Both pattern as well as magnitude of the secondary isotherm profile varies significantly on changing media permeability as well as Prandtl number. In chapter 7, the conclusions and future scope are