Special and extreme triple shock-wave configurations

Abstract

SPECIAL AND EXTREME TRIPLE SHOCK-WAVE CONFIGURATIONSUDC 533.6.011.72 V. N. Uskov and M. V. ChernyshovTriple shock-wave configurations in steady supersonic flows of an inviscid perfect gas are considered. Triple configurations with special intensities of shock waves and extreme ratios of various flow parameters behind these configurations are determined.1. Object of the Study. Formulation of the Problem. A triple configuration (TC) of shock waves is the structure of gas-dynamic discontinuities in a supersonic flow, which consists of three shock waves (j i , where i = 1, 2, 3) and a tangential discontinuity τ possessing a common (triple) point T (Fig. 1). The initial-flow streamline that arrives at the triple point divides the flow into two parts, one of them passing through a system (sequence) of shock waves j 1 and j 2 , and the other passing only through one (basic or resultant) shock wave j 3 . The tangential discontinuity τ separates flows moving along it behind the shock waves j 2 and j 3 ; therefore, this tangential discontinuity is sometimes called the slip line.The classification proposed by Landau [1] for discontinuities with respect to the point T allows us to consider the shock wave j 1 to arrive at the triple point. The flow behind the shock wave j 1 is supersonic if there is the next shock wave j 2 , which can either arrive at the point T (Fig. 1a) or emanate from this point (Fig. 1c-e). In the case of a subsonic flow behind the shock waves j 2 (Mach number M 2 < 1) or j 3 (M 3 < 1), the directions of these shock waves in the vicinity of the point T do not allow us to state whether they are the reason for TC formation or the consequence of the latter.TC calculation in a flow with a certain Mach number normally requires setting the parameters of the incoming shock wave j 1 (e.g., the angle of its inclination to the free stream σ 1 or intensity J 1 , which is the ratio of static pressures ahead of the shock wave and behind it). TC formation is sometimes called the "branching" of the incoming shock wave. The triple configuration is determined by the following conditions of dynamic compatibility at the discontinuity: 1) equal static pressures (p 2 = p 3 ); 2) identical directions of flows separated by the discontinuity.Thus, a continuous supersonic flow in the TC transforms to a discontinuous flow consisting of two flows with different gas-dynamic parameters f 2 and f 3 behind the TC, which are separated by a discontinuity. The ratios I f = f 2 /f 3 and the differences ∆f τ = (f 2 − f 3 )/f (f stands for flow parameters ahead of the triple point) characterize flow nonuniformity caused by different degrees of transformation of initial-flow parameters in a system of two shock waves and in a single basic shock wave. If the TC is formed in a nonuniform initial flow or the shock waves j i are curved, the conventional conditions on the discontinuity τ are supplemented by differential conditions [1]: 1) equal curvature of the streamlines adjacent to the discontinuity τ ; 2) equal derivatives of static pressure o...

Triple–Shock–Wave Configurations: Comparison of Different Thermodynamic Models for Diatomic Gases

“…The solution with the maximal pressure behind the reflected shock p 2 value is called the main one; other solutions are called the alternative ones and are numbered in the value p 2 descending order [3,8,9]. In this work the following TC parameters were found: the flows deflection angles β i on the three shocks, the shock inclination angles σ i , as well as the gas pressures p i , temperatures T i , enthalpies h(T i ), Mach numbers M i behind the three shocks.…”

The interaction of a Prandtl–Meyer wave and a quasi-one-dimensional flow region

“…2b) exists at the beginning of the flow considered (x ¼ 0). This slipstream can occur, in particular, at the Mach reflection of another interaction of the shocks that form the triple configuration [6][7][8]. The slipstream τ divides the flow regions 1 and 2.…”

Triple shock configurations with negative angle of reflection

“…If the quantities of М 1 , γ and ω 1 =φ 1 are given, then it is possible to determine all flow parameters behind the shock waves and angles of the triple configurations (see, for example, [11] for the full system of equations).…”

“…In this case incident shock wave occurs as attached shocks at the wedges. The triple-wave configurations can be calculated independently of boundary problems of their origin using the triple point theory [8][9][10][11][12][13]. Let us consider a plane supersonic flow with gas specific heats ratio γ and March number М 1 (Fig.…”