Maurice Holt

The design of plane and axisymmetric nozzles by the method of integral relations

The problem of nozzle design in transonic flow is difficult because the governing equations are of mixed type, changing from elliptic in the low speed region near the nozzle entry to hyperbolic in the supersonic region near the exit.

Interaction of an oblique shock wave with turbulent hypersonic blunt body flows

A shock-on-shock interaction near the cowl lip of an engine inlet is studied numerically by solving the compressible full Navier-Stokes equations. The present study focuses on Edneys type III shock interference pattern in which the turbulent free shear layer impinges on the cowl lip (modeled as a cylinder). Van Leers flux-vector splitting upwind MUSCL scheme is used in the solver with algebraic turbulence models. Turbulent solutions of surface pressures and heat fluxes along the cowl lip are in reasonably good agreement with experiment except in the local shear layer impingement region. The present study showed some indications that higher order turbulent models may be required to improve the solution near the free shear layer impingement region.

The Godunov Schemes

In one of his earliest papers concerned with numerical schemes for solving equations of Gas Dynamics Godunov (1959) seeks an alternative to the Method of Characteristics. He proposes three main requirements for such schemes. Firstly, they should retain the simplicity of Characteristics Methods while overcoming the inconveniences introduced by shearing and distortion of characteristics networks. Secondly, they should be able to include consideration of surfaces of discontinuity such as shock waves and fluid interfaces. Thirdly, when applied to linearized equations they should predict a solution for physical variables which is in qualitative agreement with analytical solutions.

A Review of Numerical Techniques for Calculating Supercritical Airfoil Flows

Three numerical methods for calculating transonic supercritical flow past airfoils are reviewed. The first is the complex characteristics approach of Garabedian, the second is the method of integral relations developed mostly by Tai, and the third is a development of Telenin’s method initiated by Chattot in application to the double wedge problem.

Interaction of an oblique shock wave with supersonic turbulent blunt body flows

A numerical study of shock-on-shock interactions near a cylindrical body, representative of the engine inlet cowl of the National Aerospace Plane (NASP), is presented. Among the six principal shock interference patterns depending upon the intersection point, noted by Edney, the most critical cases, of types III and IV, are considered in the present study. In these cases, anomalous amplifications of peak pressure and heat flux occur at the shear layer and supersonic jet impingement points, respectively. The primary goal of this study is to calculate the entire flow field numerically, capturing all the interacting shocks and complicated shocklayer flows. The finite volume formulation of Van Leers flux-vector splitting MUSCL scheme, in generalized coordinates, is used to solve the full Navier-Stokes equations in strong conservative form.

Supersonic flow past circular cones at high angles of yaw, downstream of separation

The calculation of viscous supersonic flow over circular cones at high angles of yaw was partially carried out by Fletcher and Holt (1976) and by Holt and Chan (1979). The flow field was calculated as the interaction between the outer inviscid flow and an inner conical boundary layer flow. The latter was treated by the orthonormal version of the Method of Integral Relations (M.I.R.) (Fletcher and Holt, 1975) and continued up to the cross flow separation line.

Telenin’s Method and the Method of Lines

The Method of Integral Relations, described in Chapt. 5, is one technique for reducing the amount of finite difference computation in the numerical solution of partial differential equations. The reduction is achieved by integrating the governing equations in one or more coordinate directions and representing unknowns in integrands by polynomials or trigonometrical expansions in the respective coordinates. We then solve ordinary or partial differential equations (of lower order) for the unknown coefficients in these expansions.

The Method of Integral Relations

The Method of Integral Relations was originally formulated by Dorodnitsyn in 1956 and subsequently applied to a wide variety of current problems in fluid dynamics. It is designed for the solution of problems governed by partial differential equations of elliptic, mixed elliptic-hyperbolic, or parabolic type. It has been used mostly for problems in two independent variables but in recent years has been extended to deal with three dimensional problems.

The BVLR Method

An alternative finite difference method for calculating steady high speed now past bodies of general shape was proposed by Babenko and Voskresenskii (1961) and applied to circular conical flows by Babenko, Voskresenskii, Lyubimov and Rusanov (1963). Gonidou (1967), in generalizing this application to elliptic cones, uses the shorter description, BVLR method, and this has now been generally adopted. The method was first developed for purely supersonic flows but was later extended by Rusanov (1968) and by Lyubimov and Rusanov (1970) to mixed subsonic-supersonic flows and applied to calculate a series of complex blunt body flow fields. This extension is discussed in Sect. 3.2.

The Method of Characteristics for Three-Dimensional Problems in Gas Dynamics

Although both the Godunov and BVLR methods for solving compressible flow problems in three dimensions use finite difference formulas in rectangular networks their foundations rest on the properties of characteristics of the governing equations. Both before and during the development of these methods, over the past twenty five years, a series of methods of characteristics, as such, have been proposed for three dimensional problems. Many of these compete with finite difference techniques, especially when applied to such problems as unsteady transonic flow, and it is appropriate to include a survey of such methods here. They may be classified under four headings: 1) Methods based entirely on bicharacteristics (Thornhill (1952), Fowell (1961), Butler (1960), Ransom et al. (1970)). 2) Methods based on two bicharacteristic families and one noncharacteristic family (Coburn and Dolph (1949), Holt (1956)). 3) Optimal methods of type 2) (Bruhn and Haack (1958), Schaetz (1961)). 4) Methods based on characteristic lines not in the bicharacteristic direction (Albrecht and Urich (1961). Sauer (1962)).