Numerical calculation of supersonic wake flow.

The particle-in-cell method has been used to obtain numerically the full, time-dependent development of the plane or cylindrical wake behind the flat base of a projectile moving with supersonic speed through air. Viscous and real-gas effects were neglected, but it is indicated how they could be incorporated into the calculation. Particular attention is paid to the case of Mach 1.89, arising from the passage of a very strong shock over a projectile at rest. Many features are shown of the transient build-up to statistically steady state, as well as of the late-time flow behavior. Comparisons between plane and cylindrical projectiles reveal a number of differences. Effects at higher Mach numbers are discussed briefly. I. Introduction H IGH-SPEED photographs of the flow of air about a supersonic projectile usually show the prominent structure indicated in Fig. 1. From these and other visible features, a number of details about the flow can be calculated or inferred, especially in the region outside of the wake. Within the wake itself, however, the behavior is less easily amenable to precise experimental analysis. Such properties as the timeaveraged fields of velocity, temperature, and density are difficult to obtain. Recent experimental papers1 ~3 show some of the techniques that have been successfully tried and include references to much of the earlier work. Other authors have introduced idealized models of the wake flow whose properties could be examined numerically,4 or analytically.5"7 The advantage of such an approach is that the achievement of agreement with some of the measured flow properties tends to confirm the physical basis of the model and to strengthen faith in the prediction of unmeasurables. The disadvantage of most models is that the restrictions required for analytical tractability are too stringent to reveal some of the most interesting flow features. Models for which numerical solution is accomplished by high-speed computer may not be so restricted, and it is the purpose of the present paper to show that such a model can be quite detailed in its presentation of the time-dependent flow throughout the entire near-wake region. We have used the particle-in-cell (PIC) method for numerical, multidimensional, time-dependent fluid dynamic studies to calculate several typical wake-flow problems. Transient effects were investigated for those examples in which flow was initiated by the diffraction of a shock about the projectile base. In those and higher Mach-number flows, the late-time statistically steady state was analyzed in detail. The calculations were performed without the inclusion of viscous or real-gas effects, but it is shown that one of the great advantages of the numerical model is that these effects are almost trivially simple to include. Use has been made of the symmetry of both the plane and cylindrical flows to reduce the number of spatial variables to two. The PIC method is well suited to hydrodynamic calculations in which multidimensional compressible flow involves large fluid distortions. We made use of a PIC code called "VALLE" which was written for the IBM 7030 (Stretch) Data Processing System. The computing method has been described in detail in previous literature8"11; thus, only a brief description will be given here. , The differential equations on which the computing method is based are those of mass, momentum, and energy conservation (dp/5*) + (u-V) P = -pV-u p(5u/50 + p(u-V) u = -Vp P(57/50 -f P(U-V)/ = -pV-u in which all symbols have their usual meanings. In the version of the method reported here, the effects of true viscosity and heat conduction have been neglected. The calculational system is based upon a rectangular mesh of fixed Eulerian cells through which the fluid moves. The fluid is represented by Lagrangian mass points called "particles," each of which carries a fixed mass of fluid. The mass of each cell is represented by the sum of the particle masses within that cell. The calculation proceeds through a sequence of finite time steps in which the field variables are changed as follows: first, cell pressures are calculated from cellwise values of mass, volume, and specific internal energy. (For this problem we chose the simple gas equation of state with a fixed specific heat ratio.) From the gradients of these pressures, tentative new values of the two components of velocity are calculated, each of which is compared with a maximum allowable value. If the new value is too large, the calculation would become unstable, and so the time cycle interval is cut and the velocity calculation is restarted. Tentative new values of specific internal energy are also computed. In both cases, the values are tentative because o'f omission of transport