A Numerical Study Of Semi-Implicit And Fast Algorithm For Fluid Flow Using Stagger, Collocated And Non-Orthogonal Grids

At the heart of computational fluid dynamics (CFD) is the velocity-pressure coupling algorithm that drives the fluid flow simulations to convergence. Over the past decades efforts to develop more robust and efficient velocity-pressure algorithms have resulted in a better understanding of the numerical issues affecting the performance of these algorithms, such as the choice of primitive variables, the type of variable arrangement, and the kind of solution approach (coupled versus segregated), and so on. While consensus regarding best practices of many issues have been reached within the CFD community, for pressure-based algorithms the coupled versus segregated approach dichotomy has not been completely resolved, this was clearly indicated in a recent review of pressure-based algorithms for single and multiphase flow conducted, it was mentioned that the situation is currently in favor of the segregated approach. There is the convergence problem experienced by segregated solvers when used with dense computational meshes, but the convergence issue has been addressed successfully through multigrid, parallel processing, and domain decomposition. Although the coupled versus segregated issue is not directly related to the method used, traditionally it has been the case that pressure-based methods follow a segregated approach. This state of affairs owes more to the development history of pressure-based algorithms than to any algorithmic limitation.Pressure-based algorithms originated from the work of Harlow and Welch and Chorin. However, the real thrust to this group of algorithms was generated in the early 1970s by the CFD group at Imperial College through the development of the well-known segregated SIMPLE algorithm (semi-implicit method for pressure linked equations) for incompressible flows. The CFD research community widely adopted the SIMPLE algorithm which led to the development of a number of SIMPLE-like algorithms. Furthermore, the work of Rhie and Chow provided a solution to the checkerboard problem and expanded the application area of the SIMPLE-like algorithms by enabling the use of a collocated variable arrangement and setting the ground for a geometric flexibility similar to that of the finite element method (FEM). Additional developments resulted in extending the applicability of the SIMPLE-like algorithms to a wide range of fluid physics such as compressible, free-surface, and multiphase flows. The recent article reported on a pressure-based segregated algorithm for the solution of incompressible flow problems over several kinds of structured grid systems. The objective of this work is comparing the performance of the new algorithm with the SIMPLE-like algorithm for structured grid by solving a series of test problems showing the effects of grid size, grid non-uniformity, and mesh skewness on the convergence rate. At the same time, a brief description of the Finite Volume discretization process is presented, followed by a short review of the new algorithm. The main work includes:First, a new method-SIMPLEXT algorithm for heat transfer and flow is proposed to improve calculation speed and accuracy on the premise of full consideration of pressure-correction . In the SIMPLEXT method, the pressure field is determined by the way adopted in the SIMPLER algorithm; the way adopted in the PISO algorithm achieves that pressure field and velocity field meet continuity equation and momentum conservation equation, the influence of velocity correction on neighbors for velocity correction on numbering nodes and the synchrony between source term and velocity field are considered. The reliability and validity of the way and procedure are verified by the numerical simulation of the lid-driven square cavity flow.Second, SIMPLEXT algorithm has been carried out based on collocated grids. The process of derivation and the numerical simulation of the lid-driven square cavity flow are given. A collocated arrangement of variables is introduced on numerical grids, and the cell face velocities are calculated by momentum interpolation on the premise of considering the influence of convection-diffusion term for velocity correction on numbering nodes and the synchrony between source term and velocity field. Numerical results include the relationship between outer iteration numbers and residuals of different conditions and the convergence of different algorithm. Results show great satisfaction.Third, the SIMPLEXT algorithm is applied to nonorthogonal collocated grids. The algorithm treats the influence of adjacent velocity, source term and cross-derivatives terms for velocity corrections explicitly. The process of derivation and the numerical simulation of the inclined lid-driven square cavity flow are given. By comparison, the number of the eddy is increasing with increasing Reynolds number and Inertia Force is enhancing the flow, so SIMPLEXT algorithm can simulate physical phenomenon effectively and accurately. Also, it obtains the convergent solution under relatively wide underrelaxation and haves good robustness and convergence, which can be used for calculation of flow fields with complex boundaries.Fourth, the section has combined multigrid method with SIMPLEXT algorithm to solve the problems of large-scale computational complexity of computing flow field governing equation based on application of grids. The numerical simulation of the lid-driven square cavity flow is given. The results show that the multigrid method could decrease the number of iterations and computation time notably with finer grid for outer iteration, which makes great difference from single-grid versions of the algorithm. The efficiency is also much less sensitive to the choice of underrelaxation. Meanwhile, the paper makes relative error analysis with the result, which obtains high efficiency of calculation and fast convergence.At last, we give a summary and comments on simple series of algorithms.