A meshless finite difference method for fluid flow and heat transfer
Mesh generation consumes a substantial portion of human time in computational fluid dynamics (CFD) simulations of complex industrial geometries. Despite the progress in developing solvers and mesh generation techniques for unstructured meshes, the task remains onerous. Therefore, there has been a great deal of interest in recent years to develop computational techniques that eliminate the mesh generation task altogether, through the use of meshless methods. A number of successful schemes have been published, most commonly for structural analysis, but also for fluid flow. Nevertheless, it is fair to say that the field is still in its infancy, and many needs exist for improving discretization accuracy and solution speed. ^ In this thesis, a meshless finite difference scheme is developed for steady incompressible flows of Newtonian fluids using a weighted least-squares method. The weighted least-squares method is used to fit a polynomial which is then compared to the Taylor series in order to compute approximations to the derivatives appearing in the governing equations. The method is applied in sequence to the heat diffusion equation, the convection-diffusion equation, and finally to the incompressible steady Navier-Stokes equations, and its accuracy and convergence properties are evaluated. ^ Heat conduction in a constant conductivity domain is first computed using structured and unstructured distributions of points in order to establish the order of accuracy of the method. Conjugate heat conduction problems are addressed subsequently, with conductivity ratios of up to 1000. The solutions obtained using the meshless finite difference method are compared to those obtained using the commercial software, FLUENT. Good comparisons with published analytical and numerical solutions are obtained. The scheme is shown to be free of spurious spatial oscillations that plague many published meshless schemes for conjugate heat transfer, especially at high conductivity ratios. ^ Scalar transport in the presence of a given velocity field is simulated next. The focus here is to develop analogues to convection schemes used in traditional finite differences. These include the first-order upwind scheme, the second-order central difference scheme and a new technique called the minimum gradient method, inspired by the essentially non-oscillatory (ENO) scheme. The stability of the solution procedure is demonstrated for a range of Peclet numbers. The order of accuracy is established by comparing computed solutions to available exact solutions. ^ Finally, the most important element of CFD, namely the fluid flow solution method, is tackled. A non-staggered velocity-pressure formulation is the most convenient option for a meshless method. Therefore, the scheme stores pressure and velocity at all computational points. Our focus is on the solution of incompressible flows using sequential and iterative schemes. Hence, a modified explicit fractional-step technique is developed for the meshless method. At each time step, the momentum equations are first solved without a pressure gradient term using an explicit time step, and yield an auxiliary velocity field. This auxiliary velocity is not continuity satisfying, and therefore must be corrected; the pressure is computed in such as way as to ensure that the resulting velocity field is divergence-free. In this work, the auxiliary velocity field is decomposed into curl-free and divergence-free components. The curl-free component is cast as the gradient of a scalar field, and this field is solved for, using boundary conditions derived from those imposed on pressure. The pressure is then computed posteriori from the scalar field through a simple algebraic relationship. ^ The explicit fractional time-stepping algorithm for fluid flow is tested on three fluid flow problems: 2D channel flow, the driven cavity problem, and a vortical flow problem based on the method of manufactured solutions. In all three cases, stable and accurate solutions free of pressure and velocity checkerboarding are obtained. Two different convective schemes, the first order upwind scheme and the central difference scheme, are tested for each of the three problems. The order of accuracy of the solution is established, and is found to be limited by that of the convective operator. Furthermore, the computational effort and CPU time for the computations are also found. ^ The thesis establishes that a viable meshless finite difference method may be developed for incompressible flows. Future work includes the extension of the work to implicit fractional time-stepping schemes, alternative u-v-p coupling algorithms, application to complex geometries in porous media, particle beds and foams, and in fluid-structure interaction problems combining meshless methods for both fluid and structure.^
Jayathi Y. Murthy, Purdue University.
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