Typical contours of secondary flow patterns and temperature profiles are also obtained, and it is found that the unsteady flow consists of asymmetric single-, two-, three- and four-vortex solutions.
It is found that the steady-state flow turns into chaotic flow through periodic and multi-periodic flows, if Dn is increased. Time evolution calculations as well as their phase spaces show that the unsteady flow is steady-state for and this region increases as Gr becomes large. The main concern of the present study is to investigate the nonlinear behavior of the unsteady solutions such as whether the unsteady flow is steady-state, periodic, multi-periodic or chaotic, if Dn or Gr is increased. Numerical calculations are carried out by using the spectral method, and covering a wide range. The outer wall of the duct is heated while the inner wall cooled, the top and bottom walls being adiabatic. In this paper, a numerical study is presented for the fully developed two-dimensional flow of viscous incompressible fluid through a curved rectangular duct of aspect ratio 0.5 and curvature 0.1. In particular we consider some periodic regular structures and some cases of random media. The aim of the present paper is to extend the analysis of to a more physically realistic three dimensions. A suitable com-parison between the microscopic and the macro-scopic solutions then yields the value of h, which is a function of the conductivity ratio and the geom-etry of the porous matrix. Rees also determined various ana-lytical expressions and numerical correlations for h in both one–dimensional and two–dimensional me-dia, by using the simple device that heat is gener-ated uniformly within one phase. Various correlations for h have appeared in the published literature, but Rees showed that these give unphysical expressions in the limit of no flow.
The ease with which heat is transferred between the phases is measured by h, the coefficient of the source/sink terms. microscopic transfer of heat between the phases due to differ-ences in their local intrinsic values. In the absence of fluid flow, these equations take the form of Fourier's equation with additional source/sink terms which allow for the. Two heat transport equations are used which gov-ern the evolution of the temperatures of each phase. In many nonisothermal flows in porous media it is often necessary to treat the heat transfer char-acteristics of the fluid and solid phases separately. We obtain new equitightness and \begin\times (0,\infty)$. In addition, using the discrete maximum principle, we prove the third order convergence rate of the scheme for the general elliptic cross-interface problem. In the general case, we derive a compact third-order finite difference scheme, also yielding a linear system with an $M$-matrix. Our numerical experiments demonstrate the fourth (for the general case) and sixth (for the special case) accuracy orders of the proposed schemes. Furthermore, we show that the resulting linear system for the special case has an $M$-matrix, and prove the theoretical sixth order convergence rate using the discrete maximum principle. Moreover, for the special case when the intersecting point of the two lines of coefficient jumps is a grid point, such a compact scheme, involving relatively simple formulas for computation of the stencil coefficients, can even reach sixth order of accuracy. In the present paper we propose a fourth-order $9$-point finite difference scheme on uniform Cartesian meshes for an elliptic problem whose coefficient is piecewise constant in four rectangular subdomains of the overall two-dimensional rectangular domain. It is well known that if the values of the permeability in the four subregions around a point of intersection of two such internal interfaces are all different, the solution has a point singularity that significantly affects the accuracy of the approximation in the vicinity of the intersection point. In contrast with such problems involving one smooth non-intersecting interface, that have been extensively studied, there are very few papers addressing elliptic interface problems with intersecting interfaces of coefficient jumps. In this paper we develop finite difference schemes for elliptic problems with piecewise continuous coefficients that have (possibly huge) jumps across fixed internal interfaces.
0 kommentar(er)
