Classification of Gas-dynamic Discontinuities and their Interference Problems

The aim of the study is to give a common classification of gasdynamic discontinuities, shock-wave structures and shock-wave processes. We have considered the classification of gas-dynamic discontinuities, shockwave processes, shock-wave structures, discontinuity interaction problems. We have considered different classification criteria: thermodynamic, cinematic, transiency, discontinuity direction, arriving and outgoing discontinuities. A comprehensive list of discontinuity interference problems is given, as well as classification of possible transformations and wave front reorganization up to the case of the problem two-dimensional transient definition.


INTRODUCTION
Objects of study are gas-dynamic discontinuities, shock-wave structure and shock-wave processes, as well as their classification.
The Shock-Wave Process (SWP) should be understood as the process of transformation of gasdynamic variables in waves and discontinuities: f→f 0 (1) Variables f are a great number of cinematic (uspeed, w-acceleration), thermodynamic ( p-pressure, pdensity, t-temperature), f o -parameters of deceleration, entropy change ΔS = Cvlnϑ/ϑ, where ϑ = p/p γ -Laplace-Poisson invariant and h and ho-enthalpy, as well as thermal and physical parameters (thermal capacity c p and c v , γ = c p /c v -adiabatic index, viscosity index etc.), which can change during the SWP.
We have set the problem to classify all possible types of interaction of all types of waves and gasdynamic discontinuities. Let us remind briefing information about the Gas-Dynamic Discontinuities (GDD). As we know, supersonic streamline can include areas where parameters change leap, abruptly. In such case, within the perfect gas model they say about existence of gas-dynamic discontinuities.
Gas-dynamic discontinuities in supersonic streamline can be zero-order Ф 0 : the depression/compression wave center, shock wave and sliding surface where the streamline gas-dynamic parameters endure discontinuity (pressure P, total pressure Р 0 , speed u, velocity vector angle ϑ) and of the first order called as weak discontinuities (discontinuity characteristics, weak tangential discontinuities) Ф 1 , where first-order derivatives of gas-dynamic variables endure discontinuity. It is possible to specify features (discontinuities) Ф i of the space of any order gasdynamic variables.
Dynamic compatibility conditions (DCC)at GDD Ф 0 (Uskov and Mostovykh, 2010a), connecting the streamline parameters before discontinuity and after it, are deduced from the conservation laws for streamline of matter, power streamline, momentum flux component written before discontinuity and after it. This ratio`s parameter is discontinuity intensity J (more often it is specified as the ratio of pressure after discontinuity and pressure before it).
Differential conditions of dynamic compatibility (DCDC) Ф0 connect the streamline irregularity before discontinuity and after it (Uskov and Mostovykh, 2012): Coefficients А ij , с i are published in the works by Uskov and Mostovykh (2012) and Uskov et al. (1995). For generality, equation included N 4 = δ/y (δ = 0 in two-dimensional streamline) and N 5 = K σ (compression shock curvature). DCDC for known streamline field before discontinuity, discontinuity intensity and curvature allow to calculate derivatives from gasdynamic variables after the discontinuity (Uskov and Mostovykh, 2010b). If one of irregularities is known, you can find the discontinuity curvature in the given point. It allows, in some cases, to calculate the streamline field with clear selection of gas-dynamic discontinuities and to calculate their geometry by means of DCDC. For example, at the supersonic jet hese derivative ontinuity char edges of isen n intense disco not only firsther-order deriv inuities are t fication: ding to the c c principle, w ary and transi ed in superson yer (  ) wav shock waves, F and surface of mlines going thr rogressive wav space and time ties ratio of halpy H 0 = h 02 / ). In steady str cribes the total waves (comp ndtl-Meyer wav amlines, the tot mann waves an dH 0 are related        The first line covers the designation of the world shock wave features. The wave can arise at some moment (endpoint), propagate in space (regular points) and decay with formation of three waves (triple points). At typical moments of time, an instantaneous shock wave cane be reorganized according to black arrows in Fig. 5. Particularly, any triple point of the world shock wave gives a couple of instantaneous shock wave points (shown in the second line) and the world shock wave end point generates a new point of instantaneous shock wave. The set wore organizations totally include all reorganizations of the general position (Bogaevsky, 2002). Figure 6 illustrates a flat case (d = 2). World shock wave is a surface with features; all these features are shown in the second line. Instantaneous shock wave is a curve which can have triple points and end points (the same features which world shock waves have in case of dimensionality d = 1. Instantaneous shock wave can be reorganized according to black arrows in Fig. 6.
All reorganizations of wave fronts and shock-wave given in Fig. 5 and 6 totally cover possible types of interference of one-dimensional transient waves and two-dimensional stationary and transient waves and discontinuities. In the same wave, the classification of triple stationary and transient waves may be introduced. In this case, the world wave will be a hyper surface in four-dimensional space-time. And its isochront crosssections will be instantaneous triple SWCs. This case is not covered by the work.

CONCLUSION
The given classification is the most common. It consists of formulae following the aggregate SWC and shows the direction of wave possible interaction and reorganization. We have considered different classification signs: thermodynamic, cinematic, transiency, discontinuity direction, incoming and outgoing е discontinuities, aggregate SWC which configuration is specified by parameters of incoming discontinuities and DCC at the outgoing in the interference point tangential discontinuity. On the other hand, DCC execution at the tangential discontinuity after SWC is necessary, but insufficient condition of shock-wave structure existence. For SWC existence the following structural stability conditions given in i. 2.2 must be executed. In relation to dimensionality, time is just a coordinate; cinematically transient problems are equal to stationary problems with one more dimensionality. It is necessary to take into consideration the important fact that even dimensionality and odd dimensionality is described by different geometrical types (simplistic and contact), accordingly and the SWC problem classification is separated into two sets: even and odd space-time.
The problems cinematic equivalence does not mean the dynamic equivalence existence. At transient discontinuities, the total enthalpy changes, but at stationary compression shocks (standing wave) it remains constant.