STABILITY OF NON-ISOTHERMAL PARALLEL FLOW IN VERTICAL ANNULUS

Abstract

This thesis reports a linear as well as weakly nonlinear stability analysis of stably stratified non-isothermal flow in a vertical annulus formed by two coaxial circular cylinders. The stably stratified flow (i.e., when buoyant force is in the direction of forced flow) is induced due to external pressure gradient and main tenance of linear variation of the temperature of the inner cylinder as well as the adiabatic nature of the outer cylinder. The major part of the thesis is con cerned with problems in purely viscous media, whereas, one chapter is concerned with porous medium to understand the instability mechanism of flow through porous medium in an annular domain. The spectral collocation method has been used to solve the governing equations of the problems. Following the pre vious efforts of some researchers [122, 129, 155], the weakly nonlinear stability of non-isothermal Poiseuille/parallel flow in vertical annulus filled with purely viscous/porous medium is performed. We have mainly emphasized the impact of the gap between cylinders in terms of curvature parameter and Prandtl number on the bifurcation and instability mechanism of the considered flow. The entire study is split into six chapters and the chapter wise description is given below. Chapter 1 contains some basic definitions and preliminaries of the flow in porous media, a brief description of hydrodynamic stability theory, a brief state of the art in the direction of linear and nonlinear stability analysis of duct flow, and justification regarding the model considered for the flow through porous medium. In Chapter 2, the linear stability of stably stratified non-isothermal annular Poiseuille flow (here onwards referred as NAPF) is studied using normal mode analysis. The flow is governed by four non-dimensional parameters: curvature parameter (C), Reynolds number (Re), Rayleigh number (Ra) and Prandtl number (Pr). The curvature parameter is defined as the ratio of inner cylinder radius (ri) to the difference of radii of inner and outer cylinders (i.e., ro − ri, where ro is the radius of the outer cylinder). It is found that the basic flow velocity profile possesses a point of inflection, which shifts from the inner cylinder to the outer cylinder on increasing the value of C and decreasing the value of Ra. As it has been pointed out in the literature [153], the point of inflection in the velocity profile could give rise to the instability of the basic flow, thus the stability analysis of NAPF is taken into account in Chapter 1. The analysis is carried out after a partial re-investigation of Rogers & Yao’s [112] numerical study. The impact of curvature parameter and Prandtl number on the instability of NAPF is examined under the superposition of axisymmetric as well as non-axisymmetric disturbances. The numerical experiments show that the flow of low Pr fluids under axisymmetric disturbance is more stable in comparison with the flow under non-axisymmetric disturbance. When the gap between cylinders is relatively small (i.e., C is large), the flow under non-axisymmetric disturbance is most stable for the fluids having high Pr. It is known that the Newtonian pipe flow is linearly stable for all values of Reynolds number. However, in the present study, it is found that the pipe Poiseuille flow with a thin rod placed at the center of the pipe is linearly unstable even for a very small value of heat source intensity (in terms of Rayleigh number) at Re = 2000. In this situation, the disturbance velocity is concentrated in the vicinity of the inner cylinder and the instability is due to the transfer of kinetic energy from the basic flow through Reynold’s stress. Depending on the value of controlling parameters, three types of instability: thermal-shear, interactive, thermal-buoyant have been observed. In general, for a fixed value of Pr, the type of instability is not affected by the value of C and Re. The NAPF of low Pr fluids (including mercury and gases) is governed by thermal-shear instability and the same of high Pr fluids (including liquid and oil) is governed by thermal-buoyant instability. In Chapter 3, the bifurcation and instability of NAPF of air as well as water is studied by using weakly nonlinear theory. To identify the type of bifurcation, the finite-amplitude analysis based on the cubic Landau equation [69,122,155] is used. The influence of curvature parameter is observed for axisymmetric as well as non-axisymmetric disturbance. The results from the linear stability analysis reveal that the first azimuthal mode acts as the least stable mode of NAPF of air for relatively small values of C. In this situation, even though for some values of C the NAPF has supercritical bifurcation but the same flow may experience subcritical bifurcation under zero azimuthal mode. It is also observed that for relatively larger values of Reynolds number and lower values of C the NAPF under axisymmetric disturbance always persists subcritical bifurcation. However, for small values of Re, the flow persists only supercritical bifurcation. Furthermore, the finite-amplitude analysis suggests only supercritical bifurcation of NAPF of water. The influence of nonlinear interaction of different harmonics on the amplitude profile as well as kinetic energy spectrum is also investigated in Chapter 3. The amplitude profile possesses a jump in the vicinity of a point where the type of bifurcation is changed. The nonlinear kinetic energy spectrum shows that in the subcritical regime the induced shear production due to modification of the gradient production acts as a main destabilizing factor for NAPF, and it is balanced by gradient production of kinetic energy. Chapter 4 is an extension of Chapter 3 by considering the relative influence of momentum diffusivity and thermal diffusivity, in terms of the Prandtl number (Pr), on the finite-amplitude instability of NAPF as well as pattern variation of secondary flow under linear and weakly nonlinear theories. The limiting value of the growth of instabilities under nonlinear effects is studied by the cubic Lan dau equation derived in Chapter 3. An especial attention is given on the impact of low Prandtl number and the curvature parameter (C) on the bifurcation and the pattern variation of the secondary flow for both axisymmetric as well as non axisymmetric disturbances. The finite-amplitude analysis manifests that in con trast to NAPF of water or fluid with Pr ≥ O(1) where the flow is supercritically unstable, the NAPF of low Pr fluids, in particular liquid metals, has shown both supercritical and subcritical bifurcation in the vicinity as well as away from the critical point. The nonlinear interaction of different harmonics for the liquid metal predicts a lower heat transfer rate than those by the laminar flow model, whereas for a fluid with Pr > 2 it is the other way. The maximum heat transfer takes place for the considered minimum value (0.01) of C. For fluids with low Pr, a probable lower critical Rayleigh number is also reported in Chapter 4. The corre sponding variation of neutral stability curves as a function of wavenumber reveals that the instability that is supercritical for some wavenumber may be subcritical or vice versa at other nearby wavenumbers. The structural feature of the pattern of the secondary flow under the linear theory differs significantly from those of the secondary flow under nonlinear theory away from the bifurcation point. This is a consequence of the intrinsic interaction of different harmonics that are re sponsible for the stabilizing or destabilizing nature of different components in the disturbance kinetic energy balance. In Chapter 5, the linear and weakly nonlinear stability analysis of stably strat ified non-isothermal parallel flow in a vertical annulus filled with a high permeable porous medium is studied. The Darcy extended Brinkman-Forchheimer model is considered to describe the flow stability in porous medium. The physical prob lem is mainly governed by the porosity of porous medium ( ), permeability of porous medium (defined in terms of Darcy number, Da), form drag coefficient (cF), Reynolds number (Re), Rayleigh number (Ra), curvature parameter (C) and Prandtl number (Pr). The linear stability analysis of the physical problem considered in Chapter 5 is first studied by Bhowmik et al. [16], where the consid ered fluid was water. In the present study, we have considered fluid as air (i.e., Pr = 0.7). Thus, a brief discussion of linear stability results of the flow of air is made with the help of normal mode analysis. According to the linear stability the ory, the least stable disturbance can be either axisymmetric or non-axisymmetric. Wefind that the linear instability boundary curve rapidly decreases as the value of Reynolds number increases from a small value to a threshold value. This thresh old value of Reynolds number is determined by the form drag coefficient and the permeability of the porous medium. The linear theory also demonstrates that the curvature parameter (C) has a stabilizing impact. The finite-amplitude analysis based on the cubic Landau equation is used to conduct the weakly nonlinear sta bility analysis. The effect of the different controlling parameters (Re, C, Da and cF) and the type of disturbance on the bifurcation and instability of the consid ered flow is investigated. The finite-amplitude analysis predicts the supercritical as well as a subcritical bifurcation in the vicinity as well as away from the insta bility boundary. In comparison to the non-axisymmetric disturbance, the range of Reynolds number in which subcritical bifurcation exists enhanced when the axisymmetric disturbance is considered. For some sets of values of controlling pa rameters, the change of bifurcation from subcritical to supercritical or vice-versa occurs due to the change of least stable mode. The equilibrium (threshold) am plitude is analyzed in the regime of supercritical (subcritical) as a function of different controlling parameters. The threshold amplitude below which the flow remains nonlinearly stable increases on decreasing the value of C. The presence of subcritical/supercritical bifurcation at and away from the instability boundary is also validated by studying nonlinear kinetic energy balance. Apart from this, the influence of nonlinear interaction of different harmonics on the heat transfer across the concentric cylinders (in terms of Nusselt number), friction coefficient (skin friction) on both the walls of the annulus, as well as steady secondary flow is also investigated in Chapter 5. It has been found that the Nusselt number is larger for smaller values of Re and smaller for the larger values of Re. It increases on decreasing the value of curvature parameter and Darcy number. The friction coefficient at the outer wall is always less than the same at the inner wall, and in general, both are decreasing functions of C and Re. On increasing the media permeability, the skin friction at the outer wall decreases while the skin friction at the inner wall increases. The typical increasing (at the inner wall) or decreasing (at the outer wall) characteristic of skin friction as a function of Ra is a consequence of the fact: the flow near the inner wall is buoyancy assisted and it will retard the flow velocity near the outer wall which in turn an increase in the skin friction at the inner wall and a decrease in the same at the outer wall. The analysis of secondary flow pattern reveals that from subcritical to the supercritical regime, the radius of the primary vortex cell reduced appreciably with increasing the value of Ra, although the strength of the vortices inside it enhanced greatly. The temperature contour pattern illustrates the azimuthal dependency of the solution and proves the temperature diffusion in the azimuthal direction. On varying the value of C, the multicellular isotherms change into the uni-cellular structure. In the end, Chapter 6 presents the summary and concluding remarks of the thesis and some possible directions for future work.