SUMMARY Numerical calculations of the 2-D steady incompressible driven cavity flow are presented. The NavierStokes equations in streamfunction and vorticity formulation are solved numerically using a fine uniform grid mesh of 601 × 601. The steady driven cavity solutions are computed for Re ≤ 21,000 with a maximum absolute residuals of the governing equations that were less than 10 −10 . A new quaternary vortex at the bottom left corner and a new tertiary vortex at the top left corner of the cavity are observed in the flow field as the Reynolds number increases. Detailed results are presented and comparisons are made with benchmark solutions found in the literature.

Numerical solutions of 2‐D steady incompressible driven cavity flow at high Reynolds numbers

SUMMARY We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity fonn, can be approximated to fourth--order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low~to-mediwn Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re $1500 point-SOR iteration is used and the convergence is fast.

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A compact fourth‐order finite difference scheme for the steady incompressible Navier‐Stokes equations

A study of finite difference approximations to steady-state, convection-dominated flow problems

In this paper the development and behaviour of a new finite element algorithm for viscous incompressible flow is presented. The stability and background theory are discussed and the numerical performance is considered for some benchmark problems. The Taylor–Galerkin approach naturally leads to a time-stepping algorithm which is shown to perform well for a wide range of Reynolds numbers (1 ⩽ Re ⩽ 400). Various modifications to the algorithm are investigated, particularly with respect to their effects on stability and accuracy.

A Taylor–Galerkin‐based algorithm for viscous incompressible flow

High accuracy solutions of incompressible Navier-Stokes equations

Boundary approximations and accuracy in viscous flow computations

(32) and where R is the radius of curvature of the nozzle contour at the throat. The velocity distributions (30) and (31) are both the solutions in the second approximation of Hall's perturbation method for the nozzle contour f(x)=l+-x2 (33) The numerical results of the nozzle contour and the sonic line for the axial velocity distributions, Eqs. (30) and (31), are shown in Figs. 1 and 2 by the solid lines that are compared with the results obtained by the quasi-one-dimensional approximation and by the inverse application of Hall's method. For the small value of e( = 0.2), for which Hall's method is believed to be well applicable and accurate, the present result of the nozzle shape is in good agreement with that of his result, Eq. (33). In the subsonic portion of the throat (x 0). A small difference in the location of the sonic line is found between the present and Hall's results. For the larger value of e( =0.5), for which the validity of Hall's method is in doubt, the nozzle shape of the present method deviates appreciably from his result, but is relatively in good agreement with the quasi-one-dimensional result. Finally, it must be noted that in order to adequately check the validity of our scheme, it would be necessary to carry out the calculations for larger values of n ( > 10) and to investigate the convergency of the results with n. These calculations are now being systematically carried out on the electronic digital computer FACOM 230 at the computing center of Kyoto University. The results will be reported in the near future.

A critique of a second-order upwind scheme for viscous flow problems

SUMMARY Two-dimensional stagnation point flow of a viscous fluid is modified by the introduction of a thin plate placed symmetrically at the stagnation point. Separation of the boundary layer results in back flow near the plate even at small Reynolds numbers. The Navier-Stokes equations governing the flow are replaced by finite differences using unequal net spacing which is finer in the boundary layer. The choice of finite differences guarantees convergence of the iterative procedures used. Only those relaxation parameters, which are necessary to obtain convergence and also to make the method computationally efficient, have been used. Numerical results for Reynolds numbers 9, 100 and 400 are obtained.

Numerical solution of a separtated viscous flow problem by a non‐uniform finite‐difference net

SUMMARY A new finite differenoe scheme for the convection-diffusion equation with variable coefficients is proposed. The difference scheme is defined on a single square cell of size 2h over a 9-point stencil and has a truncation error of order h4. The resulting system of equations can be solved by iterative methods. Numerical results of some test problems are given.

Murli M. Gupta

Papers

Boundary approximations and accuracy in viscous flow computations

A critique of a second-order upwind scheme for viscous flow problems

Numerical solution of a separtated viscous flow problem by a non‐uniform finite‐difference net

Convection-diffusion equation with