Harvard Lomax

High-Accuracy Finite-Difference Schemes for Linear Wave Propagation

Two high-accuracy finite-difference schemes for simulating long-range linear wave propagation are presented: a maximum-order scheme and an optimized scheme. The schemes combine a seven-point spatial operator and an explicit six-stage low-storage time-march method of Runge--Kutta type. The maximum-order scheme can accurately simulate the propagation of waves over distances greater than five hundred wavelengths with a grid resolution of less than twenty points per wavelength. The optimized scheme is found by minimizing the maximum phase and amplitude errors for waves which are resolved with at least ten points per wavelength, based on Fourier error analysis. It is intended for simulations in which waves travel under three hundred wavelengths. For such cases, good accuracy is obtained with roughly ten points per wavelength.

Computation of Space Shuttle Flowfields Using Noncentered Finite-Difference Schemes

Second- and third-order, noncentered finite-difference schemes are described for the numerical solution of the hyperbolic equations of fluid dynamics. The advantages of noncentered methods over the more conventional centered schemes are: simpler programming logic, nonhomogeneous terms are easily included, and generalization to multidimensional problems is direct. Second- and third-order methods are compared with regard to dissipative and dispersive errors and shock-capturing ability. These schemes are then used in a shock-capturing technique to determine the inviscid, supersonic flow field surrounding space shuttle vehicles (SSV). Resulting flow fields about typical pointed and blunted, delta-winged SSVs at angle of attack are presented and compared with experiment.

Second- and Third-Order Noncentered Difference Schemes for Nonlinear Hyperbolic Equations

Second- and third-order, explicit finite-difference schemes are described for the numerical solution of the hyperbolic equations of fluid dynamics. The schemes are uncentered in the sense that spatial derivatives are generally approximated by forward or backward difference quotients. The advantages of noncentered methods over the more conventional centered schemes are: programing logic is simpler, nonhomogeneous terms are easily included, and generalization to multidimensional problems is direct. The von Neumann stability analysis for the proposed methods is reviewed and second- and third-order methods are compared with regard to dissipative and dispersive errors and shock-capturing ability.

Computation of shock wave reflection by circular cylinders

The nonstationary shock wave diffraction patterns generated by a blast wave impinging on a circular cylinder are numerically simulated using a second-order hybrid upwind method for solving the two-dimensional inviscid compressible Euler equations of gasdynamics. The diffraction was followed through about 6 radii of travel of the incident shock past the cylinder. A broad range of incident shock Mach numbers are covered. The complete diffraction patterns, including the transition from regular to Mach reflection, trajectory of the Mach triple point, and the complex shock-on-shock interaction at the wake region resulting from the Mach shocks collision behind the cylinder are reported in detail. Pressure-time history and various contour plots are also included. Comparison between the work of Bryson and Gross, which included both experimental schlieren pictures and theoretical calculations using Whithams ray/shock theory, and results of the present finite-difference computation indicate good agreement in every aspect except for some nonideal gas and viscous effects that are not accounted for by the Euler equations.

A new approximate LU factorization scheme for the Reynolds-averaged Navier-Stokes equations

A new approximate LU factorization scheme is developed to solve the steady-state Reynolds-averaged NavierStokes (NS) equations. Central differencing is used for both implicit and explicit operators, and special care is taken to obtain well-conditioned factors on the implicit side. The scheme is then analyzed and optimized according to a simple linear analysis. It is unconditionally stable for the model hyperbolic equation in both two and three dimensions. However, the requirement for well-conditioned factors has essentially limited the effective time step that the scheme can achieve. Supersonic and transonic 3-D flows past a hemisphere cylinder are computed to demonstrate the convergence characterstics of the scheme. A good convergence rate is achieved in the inviscid case. Finally, an explicit eigenvector annihilation procedure is employed successfully to remove the stiffness caused by the fine grid spacing for viscous flows.

Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

Abstract A fast direct (noniterative) “Cauchy-Riemann Solver” is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver (Bunemans method, but with the cyclic reduction of simple tridiagonal matrices replaced by the Thomas algorithm). The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.

Some Prospects for the Future of Computational Fluid Dynamics

For the oncoming decade it is anticipated that new generations of high-capacity and high-speed computers will appear and that some of them will be dedicated to fluid dynamics applications. An attempt is made here to examine the potential for a new generation of numerical techniques that will accompany the computer advances. Possibilities for improvement in solution techniques, grid adaptations, turbulence approximations, language constructions, and general code robustness are considered.

Generalized relaxation methods applied to problems in transonic flow

Generalized relaxation methods application to transonic flow problems, combining with numerical integration theory for ordinary differential equations

A Fully Implicit, Factored Code for Computing Three-Dimensional Flows on the ILLIAC IV

Publisher Summary Many fluid-flow phenomena are one- or two-dimensional in their behavior and can be analyzed by equations formulated in these dimensions, flows in the real world are three-dimensional and are unsteady. Many two-dimensional fluid flows of interest to aerodynamicists can be resolved to a high degree of accuracy and with reasonable run times by the standard scientific computers of the 1970s. This is not true for unsteady three-dimensional flows. This chapter discusses the problems that arise in constructing efficient programs to be used for the large three-dimensional flow simulations. It presents a computational procedure designed for making time-accurate simulations of general compressible fluid flows. It is designed specifically for the Illinois automatic computer (ILLIAC), however, the concepts are useful for other computers with similar architectures. The solution of three-dimensional, time-accurate problems based on the Reynolds-averaged, Navier–Stokes equations for compressible fluid flows can easily require several million words of storage.

The computation of supersonic flow fields about wing-body combinations by “shock-capturing” finite difference techniques

Supersonic flow field computation for wing-body combinations by shock-capturing finite difference techniques, discussing improvement based on Runge-Kutta method