chapter  3
34 Pages

Shock Waves

In the previous chapter the process of the propagation of disturbances was considered in the linear approximation. However, as was shown in Section 2.6, successive compression waves overtake each other with the result that the initial disturbance is enhanced and a shock wave forms. The process of the generation of a shockwave froman initially continuous compressionwave is an important element of the theory andwill be described in Chapter 4. However, in this chapter we will consider a shock wave as an already formed and isolated flow element. Examples of shock waves near bodies in supersonic flows are shown in Figure 3.1. With

respect to their outward appearance they could be subdivided into bow shocks, which represent a forward front of all disturbances, and internal ones. With respect to their origin, shock waves can be subdivided into attached (to the body’s leading edge or a bend in the body surface as in Figure 3.1c and d), detached (from the body as in Figure 3.1a, b, and d), submerged (formed by compression waves inside the flow region as in Figure 3.1d), and other which will be considered in what follows. When a steady discontinuity is formed, the mass, momentum, and energy conservation

laws 1.7.12 derived in Section 1.7 must be satisfied on either side of the discontinuity. We will first consider normal shock waves whose fronts are orthogonal to the velocity

vector of the gas flow across the shock. In this case, the velocity vector direction does not change across the shock, so that the gas flow is one-dimensional and relations 1.7.12 can be simplified somewhat. Moreover, we will consider (up to Chapter 11) only adiabatic shock waves with qm = 0 in equations 1.7.12. We will further assume that the dissipative terms in these equations can be neglected in end sections 1 and 2 (Figure 1.16d); this gives relations between flow parameters on either side of an inviscid shock

ρ1u1 = ρ2u2 (u = vn), p1 + ρ1u21 = p2 + ρ2u22 H1 = h1 + u

2 = h2 + u

2 = H2 (3.1.1)

Bringing sections 1 and 2 together we obtain in a limit a discontinuity front of zero thickness with the quantities on the two sides of the front being related by formulas 3.1.1. The internal structure of the front is of no importance if only there are no sources of mass, momentum, and energy inside. Thus, if the flow parameters are preassigned, say, at left, an equation of state should be invoked to close the system of three equations in four unknown parameters u2, ρ2, p2, and h2. We will assume for a while that this equation has the form ρ = ρ(p, h, λ) where λ is

a parameter. Such equations are encountered in studying nonequilibrium flows, where λ of a in of state 1.3.4. In this case condition for the λ is required. However, for an

Real Gas Flows with

equilibrium gas state behind the shock, the equation of state ρ = ρ(p, h) is a two-parameter equation (Section 1.3); thus, the formulation of the problem of an inviscid shock wave is closed. However, the question arises (which is beyond the framework of the inviscid theory),

whether the concept itself of a zero-thickness shock could be an element of an inviscid flow, since in substitutingadiscontinuous solution3.1.1 in theNavier-Stokes equationsweobtain unbounded dissipative terms, which were omitted in passing to the Euler equations. This question of a fundamental nature will be answered in the next section in consider-

ing examples of viscous shock structures. It will be shown that the physical thickness δ of a shock front considered as a zone of dissipative effect concentration is extremely small, being of the order of molecular free path; this makes it possible to consider the shock wave front as a mathematical surface, at least, for flows inviscid as a whole. This postulate will be used starting from Section 3.3 up to Chapter 12. The smallness of the shock thickness makes it possible to consider shocks as quasisteady

in coordinate systems fitted to the shocks themselves (i.e., to neglect time-dependent terms in the integral conservation laws of Section 1.7). In fact, for a control surface enclosing the viscous shock transition zone of thickness δ, the ratio of the time-dependent term, say, in the mass conservation law 1.7.1 to the rate of the gas flowing at a velocity D through this zone is of the order of (ρδ/t0)/(ρD) ∼ δ/Dt0 1 due to the assumption on a relative smallness of the shock thickness. Here t0 is a time scale of the gas dynamic problem. Thus, in a shock-fitted coordinate system, the flow parameters on either side of the

shock are related by inviscid steady conservation laws 1.7.12 or by relations 3.1.1 for one-dimensional shocks. The second critical point of shock wave theory is related to two physically opposite

situations, which are admitted by relations (Equation 3.1.1)

u2 < u1, p2 > p1, ρ2 > ρ1, h2 > h1, e2 > e1 u2 > u1, p2 < p1, ρ2 < ρ1, h2 < h1, e2 < e1 (3.1.2)

h− p/ρ is the The for e written previously follow 3.3.5 or 3.4.22, In both cases this is flow 1 that

flows in the discontinuity front. In otherwords, if the conservation laws alone are involved, then both compression and rarefaction shocks are admissible. However, for the reasons outlined in Section 2.6 these situations are not physically

equivalent. In Sections 3.3 and 3.4 this question will be considered in more detail; until the rigorous substantiation would be done, we will assume that only compression shocks can exist in gases. Media anomalous in this regard will be considered in Section 4.12. In connection with gas properties we note (returning to Section 1.3) that equations of

state for high-temperature gases may have rather complicated forms. Therefore, all further considerations will be performed for gases with the most general properties restricted only by conditions necessary for compression waves to exist. Finally, in this chapter wewill consider equilibrium adiabatic shock waves for which the gas

states on either side of the shock are equilibrium and are associatedwith the same equation of state in the sense that there exists a reversible equilibriumprocess transforming one state into the other. Thus, air heated in a shock up to high temperatures triggering gas dissociation returns to its initial state if the pressure and temperature are restored to their initial values. The simplest example of this kind is provided by a perfect gas. An opposite example is furnished by a shock wave propagating in a nonequilibrium

gas. The well-known detonation and deflagration waves, ahead of which a medium is in a metastable rather than equilibrium state, are also phenomena of this kind. For example, as a result of oxyhydrogen gas explosionwith subsequent cooling, water vapors are formed but not the initial mixture. The properties of these nonequilibrium shock waves, as well as those ofnonadiabatic shocks,will be considered inChapter 11,which is devoted to nonequilibrium flows.