Спектр краевой задачи двумерной тепловой конвекции

We study the problem of two-dimensional fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall. Two-dimensional ideal fluid flow past a gas bubble on whose boundary surface-tension forces act (or a gas bubble bounded by an elastic film) has been studied by several authors. Zhukovskii, who first studied jet flows with consideration of the capillary forces, constructed an exact solution of the problem of symmetric flow past a gas bubble in a rectilinear channel [1]. However, Zhukovskii's solution is not the general solution of the problem; in particular, we cannot obtain the flow past an isolated bubble from his solution. Slezkin [2] reduced the problem of symmetric flow of an infinite fluid stream past a bubble to the study of a nonlinear integral equation. The numerical solution of this problem has recently been found by Petrova [3]. McLeod [4] obtained an exact solution under the assumption that the gas pressure p1 in the bubble equals the flow stagnation pressure p0. Beyer [5] proved the existence of a solution to the problem of flow of a stream having a given velocity circulation provided p1≥p0. We examine the problem of two-dimensional ideal fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall. The solution depends on the value of the contact angle βπ. The existence of a solution is proved in some range of variation of the parameters, and a technique for finding this solution is given. The situation in which β=1/2 is studied in detail.

This paper deals with the solution of two-dimensional fluid flow problems using the truly meshless Local Petrov-Galerkin (MLPG) method. The present method is a truly meshless method based only on a number of randomly located nodes. Radial basis functions (RBF) are employed for constructing trial functions in the local weighted meshless local Petrov-Galerkin method for two-dimensional transient viscous fluid flow problems. No boundary integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions due to satisfaction of kronecker delta property in RBFs. Three different radial basis functions (RBFs), i.e. Multiquadrics (MQ), Gaussian (EXP) and Thin Plate Splines (TPS) are examined and the selection of their shape parameters is studied based on closed-form solutions. The effect of quadrature domain size is also studied. The variational method is used for the development of discrete equations. The results are obtained for a two-dimensional model problem using three RBFs and compared with the results of finite element and exact methods. Results show that the proposed method is highly accurate and possesses no numerical difficulties.

Flow characteristics and flow-induced forces of a stationary and rotating triangular cylinder with different incidence angles at low Reynolds numbers

In this study, the problem of two-dimensional non-Newtonian Oldroyd-B fluid flow over the upper horizontal paraboloid surface (UHPS) is investigated. The shape of a submarine, the bonnet of a car, the shape of the jet plane, and the shape of the upper pointed bullet are some daily life examples of the paraboloid surface. At a free stream, the Oldroyd-B fluid flow over UHPS is created by the reaction of the catalytic surface and the stretching between fluid layers. With the help of appropriate similarity parameters, the determining nonlinear coupled partial differential equations (PDE’s) are reduced into nonlinear coupled ordinary differential equations (ODE’s). The numerical solution of governing ODE’s is obtained by using the Runge–Kutta Fehlberg method associated with the shooting technique through MATLAB software. The numerical results are achieved for the velocity field, concentration, and temperature fields by varying different physical parameters like thickness parameter, reaction consumption parameter, fluid’s parameters, and velocity power index parameter. The obtained results are investigated numerically and graphically. The velocity field of fluid flow is a rising function of the thickness parameter. The temperature field is increasing with an extension in the quantity of the velocity index parameter. The concentration field is a growing function of the thickness parameter. The coefficient of local skin friction decreases due to the increment of the thickness parameter of the paraboloid surface.

An effective technique is proposed for obtaining exact formulas for estimating the area of flow regions in two-dimensional fluid flow problems with free boundaries, that allow an exact solution in terms of elliptic functions. The effectiveness of the technique is demonstrated using a specific example of the problem of capillary waves on the surface of a liquid of finite depth. This example is characterized by mirror symmetry of the flow region, but the technique can be generalized to the case of other symmetry of the flow region.

To simulate three-dimensional flow problems in this thesis, the vector potential formulations of three-dimensional incompressible Navier-Stokes are chosen to govern the motion of fluid flow. The vector potential formulation belongs to one of the pressure-free algorithms which are obtained by taking curl to the momentum equations. By replacing the vorticity with Laplacian vector potential to the vorticity transport equations, the Navier-Stokes equations are transformed to fourth-order partial differential equations (PDEs) with one variable---vector potential. Comparing with other pressure-free algorithms, vector potential formulations are simpler and more accurate, and, moreover, the computation is more efficient. To the boundary conditions of vector potential, except the presented defined boundary conditions for confined flow, we further improved the algorithm to through-flow problem by introducing the concept of Stokes' theorem. To author's best knowledge, this improvement is groundbreaking. To accurately approximate these fourth-order governing equations, fourth-order-accuracy localized differential quadrature (LDQ) methods are employed. Through adopting the non-uniform mesh grids, the solutions can be obtained efficiently. To examine the ability of the proposed scheme to fourth-order governing equations, two benchmark problems are considered, including two-dimensional cavity flow problems and backward-facing step flow problems. The results show the accuracy and feasibility of the proposed scheme. By the successful implementation of the present scheme to two-dimensional flow problems, the proposed scheme is further employed to solve three-dimensional benchmark problems, including three-dimensional driven cavity flow problems and backward-facing step flow problems. The good performance not only demonstrates that the proposed scheme is able to be employed to solve the vector potential formulation, but also validates the correctness of the presented formulation. Furthermore, we specifically visualized the contour of vector potential by numerical simulation. The comparison between vector potential and stream functions show the difference of these two algorithms. Conclusively, the vector potential formulations of Navier-Stokes equations are successfully used to simulate the three-dimensional fluid motion, especially the fluid flow problems with through-flow. Through the application of the fourth-order-accuracy of localized differential quadrature method, the solutions can be accurately obtained, and the vector potential can be specifically visualized. From the previous literatures, no literature has ever presented the similar idea of this research. It is convinced that the groundbreaking findings in this thesis can provide a feasible way to simulate three-dimensional fluid motion.

In the previous researches about the benchmark solutions in the irregular regions, there are some obvious shortcomings, such as small number of computational grids, simple computational domain, small characteristic number, and lack of mixed convection benchmark solution. In order to improve this situation, this article deeply studied the benchmark solutions for two-dimensional fluid flow and heat transfer problems in the irregular regions under the body-fitted coordinate system. Taking the lid-driven flow, natural convection, and mixed convection problems as the research objects and considering the influence of boundary configurations of the computational domains, Reynolds number, Prandtl number, and Grashof number, this article designed three groups of test cases. The program code is first verified through employing the nonlinear multigrid method based on the collocated finite volume method and then the numerical solutions of three test cases with 1024 × 1024 grids and convergence criterion of 10−14 are obtained to get the benchmark solutions estimated by the Richardson extrapolation method.

The simplified QUICK scheme (transverse curvature terms are neglected) is extended to a nonuniform, rectangular, collocated grid system for the solution of two-dimensional fluid flow problems using a vertex-based finite-volume approximation. The influence of the non-pressure gradient source term is added to the Rhie-Chow interpolation method [5], and a local-mode Fourier analysis of the modified scheme demonstrates that characteristically, it is strongly elliptic and has high-frequency damping capability, which effectively eliminates the grid-scale pressure oscillations. Within this framework, the SIMPLE iteration procedure is constructed. A comparison between the present method and the control-volume-based finite-element method (CVFEM) with vorticity-stream function formulation for free convection in a cavity indicates that the proposed scheme can be applied successfully to fluid flow and heat transfer problems.

This thesis deals with the formulation of a computationally efficient adaptive grid system for two-dimensional elliptic flow and heat transfer problems. The formulation is in a curvilinear coordinate system so that flow in irregular geometries can be easily handled. An equal order pressure-velocity scheme is formulated in this thesis to solve the flow equations. An adaptive grid solution procedure is developed in which the grid is automatically refined in regions of high errors and consecutive calculations are performed between the coarse grid and adapted grid regions in the same spirit as that of a Multi-Grid method. In orthogonal coordinate systems, checkerboard pressure and velocity fields are avoided by using staggered grids. In curvilinear coordinates however, the geometric complications associated with staggered grids are overwhelming and therefore a non-staggered grid arrangement is desirable. To this end, an equal order pressure-velocity interpolation scheme is developed in this thesis. This scheme is termed as the SIMPLEM algorithm and is shown to have good convergence characteristics, and to suppress checkerboard pressure and velocity fields. The adaptive grid technique developed flags the important regions in the calculation domain from an initial coarse grid calculation. Then, adaptation is performed by generating a nonuniform mesh in the flagged region using Poisson's equations in which the nonhomogeneous terms are chosen so that a denser clustering of grid points is obtained where needed most in the flagged region. Coarse grid calculations in the whole domain, and fine grid calculations in the flagged region are consecutively performed until convergence, with correction terms from the fine grid solution added to the coarse grid equations in the flagged region in every cycle of calculation. Thus, the solution in the non-refined regions improves due to the influence of the correction terms added to the coarse grid equations. The effectiveness of the method is demonstrated by solving a variety of test problems and comparing the results with those obtained on a uniform or fixed grid. The adaptive grid solutions are shown to be more accurate than the fixed uniform grid solutions for the same level of computational effort.

Mobility changes from region to region can occur in a reservoir due to inhomogeneities in permeability and due to change in saturating fluid. In either case interfaces are present across which mobility changes abruptly. The problem of fluid flow in porous media in the presence of such interfaces is solved in this paper by using complex velocity potentials. On the interface tangential component of efflux vector is discontinuous. A vortex sheet appears on the interface and it affects the flow. Complex velocity potential in such a case can be expressed as a linear superposition of the potentials due to sources (injection wells), sinks (production wells) and due to vortex sheets along the interfaces of mobility discontinuity. Vorticity distribution on interfaces and the velocity field are inter-related. Therefore we get an integral equation for the vorticity distribution. The solution of the integral equation enables us to define the complex velocity potential completely and hence the velocity field in the flow region. It is shown that in the general case of several wells and interfaces the integral equation has a unique solution and leads to a unique flow pattern. An exact solution of the integral equation for a circular interface is obtained. An impervious no-flow boundary is an extreme case of mobility discontinuity. It is shown that its effect on the flow can be considered by an appropriate vortex sheet along it. Thus a formal solution in the form of an integral equation is obtained for instantaneous velocity distribution for arbitrary well pattern and mobility distribution.

The initial-boundary value problem of two-dimensional incompressible fluid flow in stream function form is considered. A fully discrete Legendre spectral scheme is proposed. By a series of a priori estimations and a compactness argument, it is proved that the numerical solution converges to the weak solution of the original problem. If the genuine solution is suitably smooth, then this approach provides higher accuracy. The numerical results show the advantages of this method. The techniques used in this paper are also applicable to other related problems with derivatives of high order in space.

An efficient segregated algorithm for two-dimensional incompressible fluid flow and heat transfer problems with unstructured grids

On the stability analysis of the PISO algorithm on collocated grids

In this paper, we investigated the effects of magnetic field and thermal in Stokes' second problem for unsteady second grade fluid flow through a porous medium.The expressions for the velocity field and the temperature field are obtained analytically.The effects of various pertinent parameters on the velocity field and temperature field are studied through graphs in detail.

In this paper, we studied the effect of thermal in Stokes' second problem for unsteady second grade fluid flow through porous medium. The expressions for the velocity field and the temperature field are obtained analytically. The effects of various emerging parameters on the velocity field and temperature field are studied through graphs in detail.