A second-order projection method for the incompressible navier-stokes equations

TL;DR: In this paper, a second-order projection method for the Navier-Stokes equations is proposed, which uses a specialized higher-order Godunov method for differencing the nonlinear convective terms.

About: This article is published in Journal of Computational Physics.The article was published on 1989-12-01 and is currently open access. It has received 1287 citations till now. The article focuses on the topics: Projection method & Godunov's scheme.

Summary

1. INTRODUCTION

• Lly).the authors.
• The authors will also assume that the mesh spacing is uniform in the x and y directions.
• The method provides a second-order discretization for smooth flow and selectively introduces dissipation near discontinuities by means of a "monotonized" slope computation.
• The description of the method is completed in Section 4 where the authors discuss the numerical approximation of the projection P.

At

• Unfortunately, this analysis does not extend to the case in which boundaries are present.
• The potential difficulties associated with boundary conditions appear to be an artifact of the analysis.
• Before discussing the spatial discretizations used in the algorithm, it will be useful to summarize the steps in the algorithm.
• This introduces a perturbation term to (2.10) but does not substantially affect convergence or the analysis.
• The bulk of the computational work associated with the method is spent on the linear algebra problems associated with the parabolic equations (2.5) and the projection.

3. ApPROXIMATION OF DIFFUSION AND NONLINEAR CONVECTION TERMS

• There are essentially two pieces to the spatial approximation of (2.5): discretization of the Laplacian used to model the diffusion terms and the second-order Godunov procedure that is used to compute [( U . V) uy + 1/2. The discretization of the Laplacian is done using standard, cell-centered finite difference approximations.
• It was observed by Russell and Wheeler [17J that for self-adjoint elliptic problems, cell-centered differences were equivalent to a mixed finite-emement method using the lowest order Raviart-Thomas space and special quadrature rules.
• Weiser and Wheeler [18J exploit this relationship to show that the cell-centered difference approximation is second-order accurate.
• For the uniform grids considered in this paper the cell-centered approximation is equivalent to the standard five-point discretization of the Laplacian at interior cells; the only modifications occur for cells for which some edge lies on the boundary.
• For cells bordering the left boundary (corresponding to index I,}) the authors approximate Un by 2(U I,J-U I/2,)).

Ax

• The reader is referred to the above papers for the construction of the approximation for more general grids and for details of the analysis.
• Unlike standard upwind differencing methods, these types of schemes couple the spatial and temporal discretization by propagating information along characteristics.
• More precisely, if vii ~ 0, or, if vii < 0, This asymmetry in the treatment of the derivatives implies that there are actually two separate evaluations of the first-order derivatives in (3.5) corresponding to left-right edges and top-bottom edges, respectively.
• The time-step restriction of the Godunov method is used to set the time step for the overall algorithm.
• For slope computations and for the predictor, the authors have used first-order, one-sided difference approximations for cells adjacent to the boundary.

1 (2U l2Vl1)

• When appropriately scaled, Eq. (4.7) has an interesting interpretation.
• If the authors use the definition of the 'I"s to re-express Eq. (4.7) in terms of the rPi+ 1/2.}+ 1/2 and then sum by parts, they find that the coefficients of the ex's form a discretization of -L1 and that the right-hand side is an approximation to the vorticity (the curl of V).
• Thus, the ex's computed during the projection define a discrete stream function for the velocity field.

5. NUMERICAL RESULTS

• At Re = 20,000, higher harmonics of the initial perturbation appear on the fine grid results which are not resolved on the coarse grid.
• As a final comment concerning the performance of the method, the authors will briefly discuss some representative timings for the method on 128 x 128 grids on a singleprocessor Cray XMP.
• For the examples with boundary conditions, the projection was done using an MILU(O) preconditioned conjugate gradient algorithm.
• Addition of the viscosity increased the time to 23.7 J1S per zone.

6. COI"CLUSIONS

• This will allow us to model more complex geometries and to use boundary-layer zoning near solid walls to obtain better resolution of boundary-layer phenomena at high Reynolds number.
• The quadrilateral-grid version of the algorithm will then form the basis for the development of a local mesh refinement algorithm that will allow us to focus computational effort where it is required to resolve complex flow features.

Figures

TABLE I

FIG. 8. Vorticity contours at t = 0.6 on 512 x 512 grid for varying viscosity.

FIG. 4. Kinetic energy versus time for smooth shear layer.

FIG. 5. Vorticity contours at t = 0.3 on 256 x 256 grid for varying viscosity.

FIG. I. Vorticity contours for smooth shear layer on 128 x 128 grid.

FIG. 6. Vorticity contours at 1=0.3 on 512 x 512 grid for varying viscosity.

FIG. 2. Vorticity contours for smooth shear layer on 256 x 256 grid.

FIG. 7. Vorticity contours at t = 0.6 on 256 x 256 grid for varying viscosity.

FIG. 3. Vorticity contours for smooth shear layer on 512 x 512 grid.