A Calculation Procedure for Solution of Incompressible Navier-Stokes Equations on Curvilinear Non Staggered Grids

SUMMARYThe convergence rate of a methodology for solving incompressible flow in general curvilinear co-ordinates is analyzed. Double-staggered grids (DSGs), each defined by the same boundaries as the physical domain, are used for discretization. Both grids are MAC quadrilateral meshes with scalar variables (pressure, temperature, etc.) arranged at the center and the Cartesian velocity components at the middle of the sides of the mesh cells. The problem was checked against benchmark solutions of natural convection in a squeezed cavity, heat transfer in concentric horizontal cylindrical annuli, and a hot cylinder in a duct.Poisson's pressure-correction equations that arise from the SIMPLE-like procedure are solved by several methods: successive overrelaxation, symmetric overrelaxation, modified incomplete factorization preconditioner, conjugate gradient (CG), and CG with preconditioner. A genetic algorithm was developed to solve problems of numerical optimization of SIMPLE-like calculation time in a space of iteration numbers and relaxation parameters. The application provides a means of making an unbiased comparison between the DSGs method and the widely used interpolation method. Furthermore, the convergence rate was demonstrated by application to the calculation of natural convection heat transfer in concentric horizontal cylindrical annuli. Calculation times when DSGs were used were 2-10 times shorter than those achieved by interpolation. With the DSGs method, calculation time increases slightly with increasing non-orthogonality of the grids, whereas an interpolation method calls for very small iteration parameters that lead to unacceptable calculation times.

Accelerated convergence of the numerical simulation of incompressible flow in general curvilinear co‐ordinates by discretizations on the double‐staggered grids

Shklyar

Arbel

2007

A novel Cartesian cut cell solver for incompressible viscous flow in irregular domains

Luo

Lei

et al. 2011

SUMMARYA Cartesian cut cell solver with solution-based adaptive mesh refinement is developed for simulating viscous, incompressible flows with arbitrary complex geometries. The cut cells are automatically generated using Volume CAD (VCAD), a framework for storing geometric and material attribute data. Unlike earlier cut cell methods, this solver organizes the cutting patterns into only six categories and further subdivides the resulting pentagon into two quadrilaterals, such that mesh data can be stored by uniform data structure and the post-processing of flow data can be handled conveniently. A novel method is proposed to treat minuscule cut cells without the process of cell merging. A collocated finite volume method, which can be used even when multiple cell shapes and orthogonal and non-orthogonal grids exist in the decomposition, is employed to discretize the Navier-Stokes equations. A modified SIMPLE-based smoothing pressure correction scheme is applied in this cut cell method to suppress checkerboard pressure oscillations caused by collocated arrangement. The solver is first used to simulate a channel flow to demonstrate its calculation accuracy expressed with L 1 and L ∞ norm errors and then the method is utilized to solve three benchmark problems of flow and heat transfer within irregular domains to verify its feasibility, efficiency, accuracy and potential in engineering applications.

A three‐dimensional Cartesian cut cell method for incompressible viscous flow with irregular domains

Luo

Lei

et al. 2011

SUMMARY A three‐dimensional Cartesion cut cell method is presented for the simulations of incompressible viscous flows with irregular domains. A new model (referred to as ‘6+N’ model) is proposed to describe arbitrarily shaped cut cells and treat all the cells as polyhedrons with 6+N faces. The finite volume discretization of the Navier–Stokes equation is then implemented by using the ‘6+N’ model to separate the surface flux integrals into two parts, that is, the fluxes through the basic face of the hexahedron and those through the cutting surfaces. The previously proposed Kitta Cube algorithm and volume computer‐aided design platform (J. Comput. Aided. Des. 2005; 37(4): 1509–1520. Doi:10.1016/j.cad.2005.03.006) are adopted to generate cut cells and provide shape data and physical attributes for the numerical analysis. A modified SIMPLE‐based smoothing pressure correction scheme is applied to suppress checkerboard pressure oscillations caused by the collocated arrangement of velocities and pressure. The calculation accuracy of the numerical method expressed by L1 and L ∞ norm errors is first demonstrated by the simulation of a pipe flow. Then its feasibility, efficiency, and potential in engineering applications are verified by applying it to solve natural convections between concentric spheres and between eccentric spheres. The heat transfer patterns in eccentric spheres are also obtained by using the numerical method. Copyright © 2011 John Wiley & Sons, Ltd.