Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations

Some Preliminary Thoughts * Part I: Inviscid Hypersonic Flow * Hypersonic Shock and Expansion-Wave Relations * Local Surface Inclination Methods * Hypersonic Inviscid Flowfields: Approximate Methods * Hypersonic Inviscid Flowfields: Exact Methods * Part II: Viscous Hypersonic Flow * Viscous Flow: Basic Aspects, Boundary Layer Results, and Aerodynamic Heating * Hypersonic Viscous Interactions * Computational Fluid Dynamic Solutions of Hypersonic Viscous Flows * Part III: High-Temperature Gas Dynamics * High-Temperature Gas Dynamics: Some Introductory Considerations * Some Aspects of the Thermodynamics of Chemically Reacting Gases (Classical Physical Chemistry) * Elements of Statistical Thermodynamics * Elements of Kinetic Theory * Chemical Vibrational Nonequilibrium * Inviscid High-Temperature Equilibrium Flows * Inviscid High-Temperature Nonequilibrium Flows * Kinetic Theory Revisited: Transport Properties in High-Temperature Gases * Viscous High-Temperature Flows * Introduction to Radiative Gas Dynamics.

Hypersonic and high temperature gas dynamics

A diagonal form of an implicit approximate-factorization algorithm

Boundary-conforming coordinate transformations are used widely to map a flow region onto a computational space in which a finite-difference solution to the differential flow conservation laws is carried out. This method entails difficulties with maintenance of global conservation and with computation of the local volume element under time-dependent mappings that result from boundary motion. To improve the method, a differential ''geometric conservation law" (GCL) is formulated that governs the spatial volume element under an arbitrary mapping. The GCL is solved numerically along with the flow conservation laws using conservative difference operators. Numerical results are presented for implicit solutions of the unsteady Navier-Stokes equations and for explicit solutions of the steady supersonic flow equations.

Geometric Conservation Law and Its Application to Flow Computations on Moving Grids

Abstract An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.

An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. [application to Eulerian gasdynamic equations

Implicit finite-difference procedures for the primitive form of the incompressible Navier-Stokes and the compressible Euler equations are used to compute vortex wake flows. The partial differential equations in strong conservation-law form are transformed to cluster grid points in regions with large changes in vorticity. In addition to clustering, fourth-order accurate, spatial difference operators are used to help resolve the flowfield gradients. The use of implicit time-differencing permits large time steps to be taken since temporal variations are typically small. Computational efficiency is achieved by approximate factorization. Both two-dimensional and preliminary three-dimensional calculations are described. I. Introduction T HE concentrated vorticity in the near wake of a large aircraft can pose a destructive threat to smaller aircraft within the same airspace. Consequently, experimental and theoretical efforts have been under way to understand, predict (for use in avoidance systems), and possibly reduce the vortex wake hazard. Most theoretical models developed to treat this problem rely on tracing discrete vortices,1"3 but an alternate and potentially more powerful approach is to use finitedifference procedures.4'5 Computer programs based on such methods can ultimately account for flowfield nonlinear effects with few ad hoc assumptions. Implicit finite-difference procedures are developed here to solve the incompressible Navier-Stokes equations and compressible Euler equations for simplified two- and threedimensional, unsteady vortex wake flows. This paper is divided into five interdependent sections. The flowfield and its numerical implications are discussed in Sec. II. The incompressible equations are developed for simulation as a system of first-order partial-differential equations in Sec. Ill, and the finite-difference algorithms are described in Sec. IV. Simulation based on the compressible flow equations is discussed in Sec. V. and, finally, simple wake-vortex flow calculations using both kinds of modeling are presented in Sec. VI.

Implicit Finite-Difference Procedures for the Computation of Vortex Wakes

This program determines the supersonic flowfield surrounding three-dimensional wing-body configurations of a delta wing. It was designed to provide the numerical computation of three dimensional inviscid, flowfields of either perfect or real gases about supersonic or hypersonic airplanes. The governing equations in conservation law form are solved by a finite difference method using a second order noncentered algorithm between the body and the outermost shock wave, which is treated as a sharp discontinuity. Secondary shocks which form between these boundaries are captured automatically. The flowfield between the body and outermost shock is treated in a shock capturing fashion and therefore allows for the correct formation of secondary internal shocks . The program operates in batch mode, is in CDC update format, has been implemented on the CDC 7600, and requires more than 140K (octal) word locations.

Multishocked, Three-Dimensional Supersonic Flowfields with Real Gas Effects

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.

Computation of Space Shuttle Flowfields Using Noncentered Finite-Difference Schemes

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.

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

O heighten his understanding of the fluid dynamics and aerodynamics pertinent to the early stages of the atmospheric flight-vehicle design process, the aerodynamicist has at his disposal three standard tools: analytical methods, computational procedures, and experimentation. Some of the advantages and disadvantages of each design tool are summarized in Fig. 1. Analytical methods provide quick, closed-form solutions, but they require unduly restrictive assumptions, can treat only simple configurations, and capture only the idealized aerodynamics. Through experimentation, representative or actual configurations can be tested and representative and complete aerodynamic data can be produced. Experimentation is costly, however, both in terms of the model and actual test time. In addition, the limited conditions that can be attained in wind tunnels restrict the scope of ex

Paul Kutler

Papers

Implicit Finite-Difference Procedures for the Computation of Vortex Wakes

Multishocked, Three-Dimensional Supersonic Flowfields with Real Gas Effects

Computation of Space Shuttle Flowfields Using Noncentered Finite-Difference Schemes

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

A perspective of theoretical and applied computational fluid dynamics