Collocated Grid System Research Articles

Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow

Conservation properties of the mass, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations. Existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out. In particular, it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass, momentum, and kinetic energy. This deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems. Finite difference schemes in a collocated grid system are also analyzed, and a violation of kinetic energy conservation is revealed. The predicted conservation properties are demonstrated numerically in simulations of inviscid white noise, performed in a two-dimensional periodic domain. The proposed fourth order schemes in a staggered grid system are generalized for the case of a non-uniform mesh, and the resulting scheme is used to perform large eddy simulations of turbulent channel flow.

Three-dimensional numerical modelling of mass transport in curved channels

A three-dimensional hydrodynamic model has been developed to study the flow and mass transport in curved channels. A curvilinear coordinate system is used in the horizontal xy-plane. In the vertical z direction, the σ-transformation is introduced to fit the free surface and variable bed bathymetry. The collocated grid system is adopted. To reduce the numerical diffusion, the second-order upwind scheme of Roe is incorporated to discretize the convection terms. The k-e model is used to estimate the turbulent eddy viscosity and diffusivity. The governing equations are solved by a fractional three-step implicit algorithm. Two curved channels are considered: a 180° curved channel and a meandering channel. Some results are compared with the available data, which show that the secondary currents have a strong influence on the mixing of mass in curved channels.

Depth-Averaged Hydrodynamic Model in Curvilinear Collocated Grid

A two-dimensional (2D) depth-averaged model of turbulent flows has been developed in a boundary-fitted curvilinear coordinate system. Based on the finite-volume method (FVM), the convection terms are discretized by the Roe's scheme of second-order accuracy, as well as the Power-law scheme. The governing equations are solved in a collocated grid system by the efficient fractional two-step implicit algorithm that is regarded superior to the SIMPLEC algorithm. The “time marching” technique has been used to obtain a steady-state solution as a limiting case. The new model has been applied to several problems including pure convection of a sharp step profile, side discharge into an open channel, flow in a meandering channel, and flow in a Parshall flume with supercritical outflow. Comparison with the available data shows that the model is efficient and robust.

MODIFIED SIMPLE FORMULATION ON A COLLOCATED GRID WITH AN ASSESSMENT OF THE SIMPLIFIED QUICK SCHEME

The simplified QUICK scheme (transverse curvature terms are neglected) is extended to a nonuniform, rectangular, collocated grid system for the solution of two-dimensional fluid flow problems using a vertex-based finite-volume approximation. The influence of the non-pressure gradient source term is added to the Rhie-Chow interpolation method [5], and a local-mode Fourier analysis of the modified scheme demonstrates that characteristically, it is strongly elliptic and has high-frequency damping capability, which effectively eliminates the grid-scale pressure oscillations. Within this framework, the SIMPLE iteration procedure is constructed. A comparison between the present method and the control-volume-based finite-element method (CVFEM) with vorticity-stream function formulation for free convection in a cavity indicates that the proposed scheme can be applied successfully to fluid flow and heat transfer problems.

FOURIER ANALYSIS OF THE SIMPLE ALGORITHM FORMULATED ON A COLLOCATED GRID

Abstract The Fourier error amplification matrix has been developed for the SIMPLE algorithm formulated on a collocated grid using the Rhie-Chow approximation. The cause of and remedy for the checkerboard pressure effect associated with collocated grid systems is shown. Comparisons with observations from the technical literature suggest that Fourier analysis is capable of describing algorithm operation. This analysis also suggests that there is a conditional relationship between the momentum underrelaxation factor and the pressure-correction damping factor for convergence of the collocated form, and indicates that the collocated form should serve as a superior multigrid smoother.

Conservative Properties of Finite Difference Schemes for Incompressible Flow. 2nd Report. Schemes in Staggered and Collocated Grid Systems.

In this report, finite difference schemes in staggered and collocated grid systems are considered in light of the analytical requirements that were discussed in the first report. The standard second-order accurate finite difference scheme with divergence form in a staggered grid system is proper. Proper second-order accurate advective and skew-symmetric forms have been proposed in the staggered grid system. Existing fourth-order accurate convective schemes in a staggered grid system are not proper. Proper higher-order accurate finite difference schemes in a staggered grid system are proposed. This is novel because higher-order staggered grid schemes that conserve momentum and kinetic energy simultaneously do not previously appear in the literature. The pressure term is not conservative in the kinetic energy equation in the collocated grid system, although we can construct proper-second and fourth-order accurate convective schemes in this system.

Conservative Properties of Finite Difference Schemes for Incompressible Flow. 3rd Report. Numerical Tests.

In this report, numerical tests of the conservation properties of finite difference schemes are performed. Inviscid flow simulations in a two-dimensional periodic domain are performed using several finite difference schemes that are discussed in the first and second reports. The momentum and kinetic energy are conserved when the proper schemes in regular and staggered grid systems are used. The defect of the finite difference schemes in a collocated grid system is emphasized in the simulation. In addition, plane channel flows at Re=180 and 2000 are simulated using the proper fourth-order accurate finite difference scheme in a staggered grid system and the results are better than those of the second-order accurate algorithm and an existing fourth-order accurate scheme.

Finite volume computation of the turbulent flow over a hill employing 2D or 3D non‐orthogonal collocated grid systems

AbstractA general numerical method for the solution of the complete Reynolds‐averaged Navier‐Stokes equations for 2D or 3D flows is described. The method uses non‐orthogonal co‐ordinates, Cartesian velocity components and a pressure‐velocity‐coupling algorithm adequate for non‐staggered grid systems. The capability of the method and the overall performance of the κ–ϵ eddy viscosity model are demonstrated by calculations of 2D and 3D flow over a hill. Solution error estimations based on fine grids, e.g. 320 × 192 control volumes, together with comparisons with standard turbulence model modifications, low‐Reynoldsnumber or streamline curvature effects, have allowed the investigation of model drawbacks in predicting turbulent flows over surface‐mounted hills.

Finite Volume Computation Of Internal Turbulent Flows Promoted By Wind Action And Heat Release Employing A 3D Nonorthogonal Collocated Grid System

A general numerical method for the solution of the complete 3D averaged Navier-Stokes equations is presented. This method uses nonorthogonal co-ordinates, Cartesian velocity components and a pressure-velocity-coupling algorithm suitable for nonstaggered grid systems. A k-e two-equation turbulence model is used with this method to calculate the turbulent flow inside a heated space in communication with the outside environment through two small ventilation openings.