Shock-Fitting Principles

When air is compressed by a pressure wave its temperature raises, thus increasing the speed of sound of the air left behind. Subsequent waves traveling through this compressed region travel faster than the front wave and begin to catch up with it. As waves overtake each other, the front of the wave becomes steeper, eventually developing very high gradients. Within this narrow region of high gradients the steepening effect of convection reaches a balance with the smearing effect of diffusion and a shock wave is formed. The thickness of the shock wave is typically just a few mean free paths. This is illustrated in Figure 2.1 where the shock wave thickness in multiples of the mean free path, l, is plotted as a function of free stream Mach number following a definition of shock wave thickness by Prandtl [170]. When this flow phenomenon is modeled with the Euler equations, the thin shock layer is replaced by a discontinuity. These shock discontinuities are one of the most important features of the Euler equations and present a great challenge to their numerical representation. Over the last 40 years, much work has been devoted to the numerical treatment of these discontinuities with the purpose of developing schemes that represent them accurately. The prevalent approach today is shock capturing, which has its roots in the work of von Neumann and Richtmyer [235] as we discussed in Section 1.3. In shock capturing, by means of a local modification to the governing equations, the discontinuity is replaced by a thin viscous-like layer which is computed using the same discretization used elsewhere in the flow. The other approach called shock-fitting is the subject of this book. In shock-fitting the discontinuities are treated as true discontinuities governed by their own set of partial differential equations. Shock-fitting was used by many investigators from the late 1940s through the 1960s, and unlike shock capturing there is no one original

manifesto that we can single out for the origin of shock-fitting. In the United States, Gino Moretti is the undisputed virtuoso of shock-fitting and, although occasionally we might present our own variations, it is Moretti’s theme that we try to follow.