Generalized SIMPLE-Based Pressure Correction Method for Unstructured Colocated Grids

A full multigrid/full approximation storage (FMG/FAS) method is developed along with a unified pressure-based algorithm to handle the complex fluid flow problems encompassing both compressible and incompressible regimes. Besides highlighting the key elements of the present algorithm on a curvilinear staggered grid system, several flow problems, ranging from incompressible to hypersonic speeds, are computed, in conjunction with an adaptive grid method, to demonstrate its computational capability. With wide variations in geometry, fluid physics, and grid distribution, speedup between the single- and multigrid procedures can be obtained. Because different pressure boundary conditions are needed for incompressible, subsonic, and supersonic inlet and outlet, the number of grid levels that can be effectively used by the present multigrid method appears flow dependent; the implications of this observation are discussed.

This paper describes the solution of a steady-state natural convection problem in porous media by the radial basis function collocation method (RBFCM). This mesh-free (polygon-free) numerical method is for a coupled set of mass, momentum, and energy equations in two dimensions structured by the Hardy's multiquadrics with different shape parameter and different order of polynomial augmentation. The solution is formulated in primitive variables and involves iterative treatment of coupled pressure, velocity, pressure correction, velocity correction, and energy equations. Numerical examples include convergence studies with different collocation point density and arrangements for a two-dimensional differentially heated rectangular cavity problem at filtration Rayleigh numbers Ra*=25, 50 and 100, and aspect ratios A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine-mesh finite volume method in terms of mid-plane velocity components, mid-plane and insulated surface temperatures, streamfunction minimum, and Nusselt number.

A multiple-step pressure correction algorithm, similar in spirit to the PISO algorithm, has been developed for the calculation of viscous flows in nonorthogonal curvilinear coordinates. The resulting pressure correction equations arc solved using a multigrid correction scheme. These developments were pursued because the pressure correction equation is the most time-consuming part of a Navier-Stokes flow calculation and because of the effect of the coupling between the pressure and velocity variables on the convergence rate. The new algorithm does improve the convergence rate for laminar flows or flows calculated on nearly orthogonal meshes. However, for turbulent or reacting flows, or flows computed on highly nonorthogonal meshes, there is little or no improvement in the convergence rate, and the CPU time is generally higher than for a single-step algorithm. A suitable balance between updating the velocity and static pressure variables is important, since a tightly converged pressure field can ex...

A coupled pressure-based computational method for incompressible/compressible flows

Adaptive finite element simulations of the surface currents in the North Sea

The SIMPLE method and its various modified versions have been widely used over the past two decades in the fluid flow modeling. One important feature in these SIMPLE-like algorithms is the use of a pressure correction in the pressure-correction equation, which is known to be effective in correcting the velocity field but not so in correcting the pressure field. This paper presents a modified SIMPLE algorithm to overcome this problem. This modified SIMPLE method, while still using the pressure correction for correcting the velocity field like the SIMPLE method, introduces a secondary pressure correction for correcting the pressure field. Numerical calculations of gas-solid flow in a fluidized bed suggest that this method can accelerate convergence, improve accuracy and save CPU time, compared to the SIMPLE method.

An efficient strategy for the solution of N-S Equations using collocated, non-orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co-ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non-orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non-orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid-driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti-symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.

Towards the Parallelisation of Pressure Correction Method on Unstructured Grids

A pressure-correction method is presented to solve incompressible viscous flows. The development of this method is aimed at dealing with unstructured grids, which are made of control volumes with arbitrary topology. To enhance the robustness of the method, all variables are collocated on the cell centers. The divergence theorem of Gauss is employed for discretization, and vector forms are used throughout the formulation. In this way the method is equally applicable to two- and three-dimensional problems. An overrelaxed approach is adopted for the approximation of the cross-diffusion flux to deal with “skew” grids. It can be seen that this approach is equivalent to some other approximations available in the literature. However, the present approach is more suitable for three-dimensional calculations without causing complication. This overrelaxed approach is also employed in the pressure-correction equation derived from the continuity constraint. Most prevailing methods simply ignore the cross-derivative term of the pressure-correction equation, which not only causes instability but also slows down convergence rate if the grid is skew. This cross-derivative term is taken into account in the present calculations by using a successive correction procedure. The application of the methodology to flows in lid-driven cavities and diffusers shows that no more than two pressure-correction steps are enough to obtain fast and stable convergence. The method is also applied to a three-dimensional flow in an impeller-stirred tank.

A computational study of unsteady, separated, low-Reynolds number fluid flow is made using a second-order accurate, cell-centered finite-volume method on unstructured grids. The three-dimensional simulations include large- eddy simulation turbulence modeling using a fixed-coefficient Smagorinsky subgrid-scale model. The flow about a three-dimensional circular cylinder at Reynolds number 3900 and Mach number 0.2 was studied using a standard Navier-Stokes formulation with and without the subgrid-scale modeling. A third case was also tested using a modified form of the Navier-Stokes equations with subgrid-scale modeling. Comparisons are made with experiment and selected three-dimensional numerical calculations performed on structured grids. The frequency of the unsteady vortex shedding was in good agreement with experiment, however, the numerical dissipation inherent in the upwind flux calculations affected the details of the time-averaged flowfield. The results showed an increased base pressure, shortened recirculation zones in the cylinder wake, and excessive drag compared to experiment for all cases.

A depth-averaged elliptic computational model in curvilinear coordinates is presented for velocity and depth computations in shallow water river confluences of any geometry. The pressure correction equation has been used with SIMPLE or SIMPLE-like procedures in depth-averaged models to date; here this is replaced by a depth correction equation to improve the convergence. This is a significant enhancement to the SIMPLE procedure when applied to depth-averaged flow computations of river elements with irregular bed topography. First, the model is validated using laboratory measurements from a flat-bed confluence with flow recirculation. Second, it is validated using laboratory measurements from a confluence with irregular bed topography derived at the dynamic equilibrium of a movable bed experiment. Next the model is employed to simulate the complex confluence flow at a bend of a river with a floodplain. The model predictions give satisfactory agreement with field observations. Though the flow in the scour hole region of a river confluence is three-dimensional, if proper parameters are used to represent bottom friction and mixing, a depth-averaged model can be used to obtain sufficiently accurate velocity and water surface elevation information for the design of flood control measures in shallow water river confluences.

Preconditioning methods for compressible flow CFD codes: Revisited

In this paper, we consider a multilithology diffusion model used in the field of stratigraphic basin simulations to simulate large scale depositional transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main variables of the system are the sediment thickness h, the L surface concentrations cis in lithology i of the sediments at the top of the basin, and the L concentrations ci in lithology i in the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for cis with a linear advection equation for ci for which cis appears as an input boundary condition. For this coupled system, a weak formulation is introduced. The system is discretized by an implicit time integration and a cell centered finite volume method. This numerical scheme is shown to satisfy stability estimates and to converge, up to a subsequence, to a weak solution of the problem.

This paper deals with a numerical solution of laminar incompressible steady flows of Newtonian and non–Newtonian fluids through bypass of a restricted vessel. Blood flow is considered to be Newtonian in the case of vessels of large diameters as aorta. On the other hand, with decreasing diameter of a vessel the non–Newtonian behavior of blood can play a significant role. One could describe these problems using Navier–Stokes equations and continuity equation as a model. In the case of Newtonian fluids one considers constant viscosity compared to non–Newtonian fluids where viscosity varies and can depend on the tensor of deformation. In order to find numerical solution, the system of equations is completed using an artificial compressibility method. The space derivatives are discretised using a cell centered finite volume method and arising system of ordinary differential equations is solved using an explicit multistage Runge–Kutta method with given steady boundary conditions. The steady solution is achieved for time t→∞ and steady boundary conditions. The results can be used in the field of cardiovascular research. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The convergence and accuracy characteristics of the preconditioned incompressible Euler and Navier–Stokes equations are studied. An object-oriented C++ numerical code has been developed for solving the inviscid and viscous, steady, incompressible flows problems. The code is based on the cell-centred finite volume method. In this scheme, two-dimensional incompressible Euler and Navier–Stokes equations are modified by a robust artificial compressibility (AC) and a local preconditioning matrix of pressure-sensor type. The preconditioned equations are solved with the Jameson's numerical approach, i.e. artificial dissipation and artificial viscosity terms under the form of a fourth- and second-order derivative, respectively. An explicit four-stage Runge–Kutta integration algorithm is applied to obtain the steady-state condition. The computed results include the steady-state solution of flow past the NACA-hydrofoils and a circular cylinder in free stream, for which the numerical results are compared with numerical works of other researchers. Good agreement is observed. The effects of AC parameter, artificial viscosity and dissipation factor, and local preconditioning coefficient on convergence rate and solution accuracy are tested by computing flow over the NACA0012 hydrofoil. In addition, some important design criteria of a preconditioner, such as stiffness reduction, hyperbolicity, symmetrisability, accuracy preservation for M → 0, and M-property have been examined analytically.