Mathematics for solving supersonic flow problems

Mach cone and path (stream) line, sketched in Figure 23 are singular surfaces called ‘character­istic surfaces”; their generating lines are Mach lines or ‘’characteristics’*, hi viscid supersonic (or transient) flow is completely described by a set of partial differential equations (PDEs). valid only along characteristics, called "compatibility equations"; they do not contain any derivatives across the characteristic surfaces, but allow for undefined jumps in these derivatives 1581. (59J, {601, (61). {62). This means:

Any other set of PDEs describing supersonic (or transient ) flow contains derivative com­ponents normal to characteristic surfaces which are not defined by the PDEs’. Solutions of those equations may use invalid information or produce solutions containing random

pans. This may prohibit accurate or even useful solutions.

Inviscid supersonic (or transient) flow equations arc hyperbolic. They describe radiation prob­lems. The FDEs itself allow for discontinuities in the derivatives of die variables (like velocity, pressure, entropy). If, for a given problem, the initial conditions do not contain discontinuities of the derivatives, discontinuities may evolve in the flow field. Furthermore, any solution to these equations (except for the tnvial identity solution, i. e. all derivatives are given everywhere as zero and remain zero) is composed only by discontinuous elementary solution parts, maybe for higher derivatives.

Л straightforward formulation for characteristic directions and compatibility equations was developed in the early 50ics by C. Heinz (63] at 1SL: Focusing on the essential normal direction, the number of equations used was reduced to the necessary minimum. This formula­tion is available in (61].

The above mentioned set of variables (velocity, pressure, entropy; total energy beeing dependent of pressure, velocity and entropy) is selected for decoupling of the variables in the compatibility equations. For other sets analogous formulations and discontinuities hold.

Viscous and heat conducting flow equations (Navier-Stokes equations) contain addi­tional derivatives in all space directions without any preference. Those additional derivatives are of elliptical type; the resulting Navier-Stokes equations arc of mixed or parabolic type. Viscous and heat flux influence is limited to thin layers (boundary layer, shear layers, shocks) and sepa­ration regions.

Shocks can develop in the flowtield by steeping up of solutions and at boundaries, where sudden changes of boundary conditions occur. Shocks arc described by the Rankine – Hugoniot equations (58]. (59), (61] which arc derived by surface integration over a flat volume along the shock surface. Even the "inviscid" Rankine-Hugoniot equations contain viscous and heat fluxes across the shock surface. So the whole flowfield can be described by the inviscid equations, except boundary layer, shear layers and separation zones. In the free flow field, the discontinuous solution properties of the "inviscid" equations must be respected. Even when solving the Navier-Stokes equations for shocks, the thickness of a shock (less than 10 molecule free path length) is below numerical resolution; within this small layer the number of molecules is not sufficient to establish state variables as required for the continuum formulation of the Navier-Stokes equations. Therefore validity of solutions must be carefully checked.

In frequent case studies the capturing of shocks in numerical volutions is improved by selection of so called conservative variables which should be conserved when passing a shock. Caution is needed, though: In the Rankine-Hugoniot equations, basically not the variables are conserved, but their fluxes normal to the shock. For example, normal to a stationary shock not density p is conserved, but pvns. with ns the shock normal vector; only by chance, the conserv­ative velocity pv (i. e. momentum) is the flux of p. On the other hand. v,. the velocity component parallel to the shock surface, is conserved across the shock, but not pv,.