Joseph L. Steger

Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods☆

The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included.

Automatic mesh-point clustering near a boundary in grid generation with elliptic partial differential equations

Elliptic partial differential equations are used to generate a smooth grid that permits a one-to-one mapping in such a way that mesh lines of the same family do not cross. Problems that arise due to lack of clustering at crucial points or intersections of mesh lines at highly acute angles, are examined and various forcing or source terms are used (to correct the problems) that are either compatible with the maximum principle or are so locally controlled that mesh lines do not intersect. Attention is given to various schematics of unclustered grids and grid detail about (highly cambered) airfoils.

Numerical Simulation of Steady Supersonic Viscous Flow

A noniterative, implicit, space-marching, finite-difference algorithm is developed for the steady thin-layer Navier-Stokes equations in conservation-law-form. The numerical algorithm is applicable to steady supersonic viscous flow over bodies of arbitrary shape. In addition, the same code can be used to compute supersonic inviscid flow or three-dimensional boundary layers. Computed results from two-dimensional and three-dimensional versions of the numerical algorithm are in good agreement with those obtained from more costly time-marching techniques.

On the use of composite grid schemes in computational aerodynamics

Abstract In finite difference flow field simulations the use of a single well-ordered body-conforming curvilinear mesh can lead to efficient solution procedures. However, it is generally impractical to build a single grid of this type for complex three-dimensional aircraft configurations. As a result, a trend in computational aerodynamics has been toward the use of composite grids. Composite grids use more than one grid to mesh an overall configuration with each individual subgrid of the system patched or overset together. Because each individual subgrid in the system is well ordered, the overall grid is suitable for efficient finite difference solution using vectorized or multitasking computers. Some of the advantages and difficulties of using various composite grid schemes are reviewed in this paper.

Navier-Stokes computations of projectile base flow with and without mass injection

A computational capability has been developed for predicting the flowfield about projectiles, including the recirculatory base flow at transonic speeds. In addition, the developed code allows mass injection at the projectile base and hence is used to show the effects of base bleed on base drag. Computations have been made for a secant-ogive-cylinder projectile for a series of Mach numbers in the transonic flow regime. Computed results show the qualitative and quantitative nature of base flow with and without base bleed. Base drag is computed and compared with the experimental data and semiempirical predictions. The reduction in base drag with base bleed is clearly predicted for various mass injection rates. Results are also presented that show the variation of total aerodynamic drag both with and without mass injection for Mach numbers of 0.9 < M< 1.2. The results obtained indicate that, with further development, this computational technique may provide useful design guidance for projectiles. MAJOR area of concern in shell design is the accurate prediction of the total aerodynamic drag. Both the range and terminal velocity of a projectile (two critical factors in shell design) are directly related to the total aerodynamic drag. The total drag for projectiles can be divided into three components: 1) pressure drag (excluding the base region), 2) viscous (skin friction) drag, and 3) base drag. At transonic speeds, base drag constitutes a major portion of the total drag. For a typical shell at M = 0.90, the relative magnitudes of the aerodynamic drag components are: 20% pressure drag, 30% viscous drag, and 50% base drag. The critical aerodynamic behavior of projectiles, indicated by rapid changes in the aerodynamic coefficients, occurs in the transonic speed regime and can be attributed in part to the complex shock structure existing on projectiles at transonic speeds. Therefore, in order to predict the total drag for projectiles, computation of the full flowfield (including the base flow) must be made. There are few reliable semiempirical procedures that can be used to predict shell drag; however, these procedures cannot predict the effects of mass injection. The objective of this research effort was to develop a numerical capability, using the Navier-Stokes computational technique, to compute the flowfield in the base region of projectiles at transonic speeds and thus to be able to compute the total aerodynamic drag with and without mass injection. The pressure and viscous components of drag generally cannot be reduced significantly without adversely affecting the stability of the shell. Therefore, recent attempts to reduce the total drag have been directed toward reducing the base drag. A number of studies have been made to examine the total drag reduction due to the addition of a boattail.1 Although this is very effective in reducing the total drag, it has a negative impact on the aerodynamic stability, especially at transonic

A numerical study of three-dimensional separated flow past a hemisphere cylinder

Separated and vortical flow about a hemisphere-cylinder body has been investigated. An algorithm featuring two implicit factors, and partial flux splitting has been used to solve the thin-layer Navier-Stokes equations. In analyzing the complex flow patterns, experimental data and topological concepts are used to complement the numerical results in interpreting the surface-flow patterns as well as the flowfield structures. Basic issues concerning the three-dimensional separation characteristics and the leeward vortical structures are examined.

Numerical Solution of the Azimuthal-Invariant Thin-Layer Navier-Stokes Equations

NUMERICAL solutions have been obtained for a twodimensional azimuthal- (or circumferentially) invariant form of the thin-layer Navier-Stokes equations. The governing equations which have been developed are generalized over the usual two-dimensional and axisymmetric formulation by allowing nonzero velocity components in the invariant direction. The equation formulation along with the solution method is described, and results for spinning and nonspinning bodies are presented. Contents The three-dimensional flow field equations are frequently simplified for flowfields which are invariant in one coordinate direction. In the usual axisymmetric approximation, the azimuthal velocity is assumed to be zero, and the momentum equation in that direction can be eliminated. Thus, only four equations are required to be solved for four unknowns. However, for a variety of interesting flowfields, the velocity component in the invariant direction (here taken as TJ) is not zero although the governing equations are still twodimensional. Examples include viscous flow about an infinitely swept wing, the viscous flow about a spinning axisymmetric body at 0-deg angle of attack, and axisymmetric swirl flows. Each of these flows can be solved as a twodimensional problem although all three momentum equations have to be retained, and source terms replace the derivative of the flux terms in the rj-direction. Azimuthal-invariant equations are obtained from the threedimensional equations1 by making use of two restrictions: 1) all body geometries are of axisymmetric types and 2) the state variables and the contravariant velocities do not vary in the azimuthal direction. Here, TJ is used for the azimuthal coordinate, and the terms azimuthal and rj-invariant will be used interchangeably. A sketch of a typical axisymmetric body is shown in Fig. la. In order to determine the circumferential variation of typical flow and geometric parameters, we first establish correspondence between the

Comparison Between Navier-Stokes and Thin-Layer Computations for Separated Supersonic Flow

In the numerical simulation of high Reynolds-number flow, one can frequently supply only enough grid points to resolve the viscous terms in a thin layer. As a consequence, a body-or stream-aligned coordinate system is frequently used and viscous terms in this direction are discarded. It is argued that these terms cannot be resolved and computational efficiency is gained by their neglect. Dropping the streamwise viscous terms in this manner has been termed the thin-layer approximation. The thin-layer concept is an old one, and similar viscous terms are dropped, for example, in parabolized Navier-Stokes schemes. However, such schemes also make additional assumptions so that the equations can be marched in space, and such a restriction is not usually imposed on a thin-layer model. The thin-layer approximation can be justified in much the same way as the boundary-layer approximation; it requires, therefore, a body-or stream-aligned coordinate and a high Reynolds number. Unlike the boundary-layer approximation, the same equations are used throughout, so there is no matching problem. Furthermore, the normal momentum equation is not simplified and the convection terms are not one-sided differenced for marching. Consequently, the thin-layer equations are numerically well behaved at separation and require no special treatment there. Nevertheless, the thin-layer approximation receives criticism. It has been suggested that the approximation is invalid at separation and, more recently, that it is inadequate for unsteady transonic flow. Although previous comparisons between the thin-layer and Navier-Stokes equations have been made, these comparisons have not been adequately documented.

Coefficient matrices for implicit finite difference solution of the inviscid fluid conservation law equations

Abstract Implicit finite difference methods for the inviscid fluid conservation law equarions are described. Conservative for coefficient matrices are found which are similar to Jacobian linearization and which reduce the computational work of various implicit schemes for the Euler equations of inviscid fluid flow. Extension of the basic ideas is suggested in order to achieve a quite different computational advantage.

Numerical investigation of a jet in ground effect with a crossflow

One of the flows inherent in V/STOL operations, the jet in ground effect with a crossflow, is studied using the Fortified Navier-Stokes (FNS) scheme. Through comparison of the simulation results and the experimental data, and through the variation of the flow parameters (in the simulation) a number of interesting characteristics of the flow have been observed. For example, it appears that the forward penetration of the ground vortex is a strong inverse function of the level of mixing in the ground vortex. An effort has also been made to isolate issues which require additional work in order to improve the numerical simulation of the jet in ground effect flow. The FNS approach simplifies the simulation of a single jet in ground effect, but will be even more effective in applications to more complex topologies.