M. Perić

Solution of the Navier-Stokes Equations

In Chaps. 3, 4 and 6 we dealt with the discretization of a generic conservation equation. The discretization principles described there apply to the momentum and continuity equations (which we shall collectively call the Navier-Stokes equations) . In this chapter, we shall describe how the terms in the momentum equations which differ from those in the generic conservation equation are treated.

FURTHER DISCUSSION OF NUMERICAL ERRORS IN CFD

The methods of estimating numerical errors given in an earlier paper are extended in directions that make them useful in actual CFD applications. In particular, the method of estimating convergence error (the error due to insufficient iteration) is extended to allow the possibility of complex eigenvalues; an ad hoc method that can be applied to any case is also given. For the discretization error, which arises from the numerical approximation of the differential equation(s), methods that can be used on non-uniform drids are presented; they can be extended to unstructured grids as well. The utility of these methods is demonstrated for linear problems as well as solutions of the Navier-Stokes equations. The examples show that the estimation of errors is neither difficult nor expensive.

A fourth-order finite volume method with colocated variable arrangement

Abstract A finite volume solution method for the two-dimensional Navier-Stokes equations and temperature equation with 4th order discretization on cartesian grids is presented. The method uses colocated variable arrangement and the SIMPLE-kind of velocity-pressure coupling. The surface integrals (convection and diffusion fluxes through control volume faces) are approximated by Simpsons rule and polynomial interpolation, and the volume integrals (source terms) are approximated by fitting a fourth-order polynomial through nine points and integrating it analytically. Applications to the solution of a scalar transport on a known velocity field and to the lid-driven and buoyancy-driven cavity flows show superior accuracy as compared to the first and second order schemes. The approach can be readily extended to control volumes of arbitrary shape and unstructured grids.

AN IMPLICIT FINITE-VOLUME METHOD USING NONMATCHING BLOCKS OF STRUCTURED GRID

Abstract This article presents a finite-volume method for computing flow problems using block-structured grids. The grids may move in some blocks, and they do not have to match at block interfaces. However, in the resulting linear equation systems, all blocks are implicitly coupled and the method is fully conservative. The discretization is of second order in both space and time, and the solution algorithm is based on the SIMPLE method. The approach of treating the nonmatching block interfaces can be applied to other types of solution methods as well.

BENCHMARK SOLUTIONS OF SOME STRUCTURAL ANALYSIS PROBLEMS USING FINITE-VOLUME METHOD AND MULTIGRID ACCELERATION

In this paper a set of benchmark test cases for solid-body stress analysis and their solutions are presented. The results are obtained using finite-volume discretization and segregated solution procedure. Sets of progressively finer grids are used in a full multigrid algorithm based on V cycles and a correction scheme, ensuring high computational efficiency. Solutions obtained on systematically refined grids are used to estimate the solution error, which was found to be less than 1 per cent on the finest grids. In addition to graphical presentation of the solutions, tabular data for some characteristic profiles is included to make future comparisons easier. Some details about the convergence properties of the method as well as an outline of the methodology are also presented. It is hoped that the test problems and the solutions presented in this paper will be used in the future for assessing the accuracy and efficiency of new solution methods for solid-body stress analysis.

EFFICIENCY AND ACCURACY ASPECTS OF A FULL-MULTIGRID SIMPLE ALGORITHM FOR THREE-DIMENSIONAL FLOWS

This article reports on the analysis of efficiency and accuracy of a full-multigrid SIMPLE algorithm for three-dimensional flows using co-located grids and central difference discretization for both convective and diffusive fluxes. It is shown that the central differencing scheme--contrary to common belief--offers both good convergence properties and high accuracy, even at large Peclet numbers. Accurate solutions, with discretization errors below 0.5%, were obtained for lid-driven flows in a cubic cavity using grids with up to 128{sup 3} (nearly 2.1 million) control volumes. For a grid with 64{sup 3} (262,114) control volumes, solution can be obtained on a personal computer in 5--10 min. The computer code used for the calculations reported here is available from authors on request (free of charge).

COMPUTATION OF UNSTEADY FLOWS USING NONMATCHING BLOCKS OF STRUCTURED GRID

Abstract This article presents application of a finite-volume method for computing unsteady flow problems using nonmatching blocks of structured grid. The discretization is of second order in both space and time, and the solution algorithm is based on the SIMPLE method. Results of computations for three unsteady-flow problems are presented to demonstrate the capabilities of the method: flow around a free cylinder, flow around a cylinder confined by two parallel walls, and flow between co-rotating square cylinders. The emphasis is on accuracy and efficiency of the method, so computations are done on systematically refined grids and using several time steps.

Parallel DNS with Local Grid Refinement

A parallel finite volume method for unstructured grids is used for a direct numerical simulation of the flow around a sphere at Re = 5000 (based on the sphere diameter and undisturbed velocity). The observed flow structures are confirmed by visualization experiments. A quantitative analysis of the Reynolds averaged flow provides a data base for future model evaluations.

Computation of Flows with Free Surfaces

The paper presents two methods for comuting flows with free surfaces: an interface-tracking and an interface-capturing method. The former computes the liquid flow only, using a numerical grid which adapts itself to the shape and position of the free surface; the kinematic and dynamic boundary conditions are applied there. The second method considers both fluids as a single effective fluid with variable properties; the interface is captured as a region of a sudden change in fluid properties. An additional transport equation is solved to determine the volume fraction of one of the fluids. Advantages and disadvantages of the two methods are discussed and several application examples are presented.

Computation of Complex Turbulent Flows

This paper presents the results of a joint effort by two research groups to improve the understanding and the predictability of complex turbulent flows. On one hand, the effort was made to reliably evaluate modelling errors of several popular low-Reynoldsnumber, two-equation turbulence models by making sure that all other errors in the computations were negligibly small. On the other hand, methods suitable for direct numerical simulation of turbulent flows in complex geometries have been thoroughly evaluated by performing a systematic grid and time step refinement, evaluating discretization errors, and comparing the results with established solutions of other groups for flows in simple geometries. Finally, some complex flows have been analysed using both direct numerical simulation and turbulence modelling approaches.