Evaluation of Navier-Stokes solutions using the integrated effect ofnumerical dissipation

Abstract

A method for evaluating the quality of solutions to the Navier-Stokes equations is developed and illustrated with representative examples. In solutions to the Navier-Stokes equations it is important that added numerical dissipation does not overwhelm the real viscous dissipation. To verify this, it is necessary to be able to estimate quantitatively the effect of numerical dissipation. A method for estimating the integrated effect of numerical dissipation on solutions to the Navier-Stokes equations is developed in this paper. The method is based on integration of the momentum equations, and the computation of corrections due to numerical dissipation to the drag integral. These corrections can then be considered as estimates of the error due to dissipation. Solutions to the Navier-Stokes equations for laminar and turbulent flows over airfoils are used to illustrate the method. The errors due to numerical dissipation are compared with the total numerical errors in the solutions. The effect of Mach number scaling of the numerical dissipation terms is discussed. HE evaluation of the quality of any numerical solution of the Navier-Stokes equations and the validation of the computer code that yielded the solution necessarily require an estimation of the errors in the solution. As Hoist 1 has pointed out, these errors fall under two broad categories—physical modeling errors and numerical errors. The physical modeling errors include, among others, those arising from the approximations involved in the NavierStokes equations themselves, or their thin-layer approximation, as well as those introduced by any model for the effects of turbulence. The numerical errors include those due to the basic discretization scheme, including any implicit or explicit numerical dissipation, and are dependent upon the fineness and distribution of the grid. Physical modeling errors can be quantified only by comparison with the results of experiments or with the results of direct numerical simulations in which the corresponding approximations are not made. Before these comparisons can be meaningful, however, it is important to understand the level of numerical error, and this can be done without recourse to comparison with experiments. It is with these numerical errors and, in particular, with the effects of numerical dissipation, that the present article is concerned. The calculation of fluxes in several widely used finite volume schemes used to solve the Euler and Navier-Stokes equations can be shown to be equivalent to central differencing. Such schemes, applied to the Euler equations, do not contain any inherent dissipation. To prevent odd-even point decoupling and oscillations near shock waves or stagnation points, numerical dissipation terms must be added when solving the Euler equations. The NavierStokes equations, on the other hand, possess dissipative properties due to the presence of the viscous terms, but the physical dissipation provided by these terms in regions far away from the surface is usually small, and the addition of numerical dissipation terms is still necessary to ensure the stability and robustness of the schemes. While the added dissipation terms must be large enough for this purpose, they must also be small enough not to overwhelm