Differential Quadrature Method For Viscous/Viscoelastic Fluid Flow And Heat Transfer Problems

In general, it is impossible to obtain the exact solutions of problems in fluid mechanics, due to their governing equations are a very complicated system of nonlinear equations. Hence various numerical methods are presented for solving these ones. The finite difference (FD) method and finite element (FE) method are two kinds of usual methods. In many cases it is enough to obtain moderately accurate solutions at a few points. But one has to use a large quantity of grid points in order to obtain moderately accurate solutions at a few points as the FD method and FE method are adopted. Thus, a large quantity of the computational workloads and storages are required when these methods are used. However, for the differential quadrature (DQ) method proposed by R. Bellman in 1970, it only needs applying a few grid points in order to get high-precise solutions. Furthermore the DQ method possesses advantages including easy to use and arbitrary to choose the grid spacing. Hence the DQ method has attracted many researchers attention in recent years.But the traditional DQ method is only suitable for solving problems with regular domains and there is lack of upwind mechanism to treat with the convection in the fluid flow. In order to make the DQ method to be appropriate for solving problems in the fluid mechanics with irregular domains, a localized DQ method having upwind mechanism (ULDQ) is proposed in this dissertation. By using the ULDQ method, some numerical results for incompressible two-dimensional flow problems of viscous or viscoelastic fluid coupled with heat transfer are obtained. The main results contain as follows:1. Because there is a lack of upwind mechanism to characterize the convection of the fluid flow in the traditional differential quadrature method, so the numerical experiments for fluid flow are usually became defeat as the Reynolds number are larger. In the Chapter 3, at the every temporal iterative step, a prediction-adjustment method in which at first the traditional upwind difference scheme is applied to predict for convective terms, then the DQ method are used to adjust all terms in spatial variables. This method is called the mixed differential quadrature method. By using this mixed method the numerical experiments for the coupled problems of two-dimensional incompressible Navier-Stokesequations with heat equation are made successfully. The results obtained show that the mixed differential quadrature method is appropriate to solve the fluid flow with higher Reynolds numbers. These results also point out that the mixed differential quadrature method owns advantages including the good convergence, high accuracy and less workloads comparing with the conventional differential quadrature method.2 . Although the differential quadrature method has been applied to successfully solve various problems in fluid mechanics, it is limited with regular domains and an absence upwind mechanism to characterize the convection of the fluid flow. The upwind mechanism is directly introduced into the traditional DQ method and a localization technique is applied to deal with the irregularity of flow regions in Chapter 4. The DQ method improved above is called the upwind local differential quadrature method. By using this method, the numerical simulations for the coupled problem of incompressible laminar flow with heat transfer in an irregular region are made successfully. Comparing with the low-order finite difference method, the upwind local differential quadrature method is more accurate and it only requires less computational workload.3. A problem of two-dimensional steady flow for an incompressible second-order viscoelastic fluid between two parallel plates is discussed by using the perturbation technique in Chapter 5. By expanding the governing equations with respect to a small parameter, the zero and first order approximation equations are obtained. By using the differential quadrature method and an iterative technique presented in the thesis the numerical solution is successfully obtained.The numerical r...