Shock Waves. Rankine–Hugoniot Jump Conditions

S.12.5. Shock Waves. Rankine–Hugoniot Jump Conditions

In general, the basic hyperbolic equations (1) are obtained from continuity equations, i.e., balances of mass, momentum, and energy for continuous flows. Continuous solutions of these equations were studied in Subsections S.12.3 and S.12.4. We now derive balance conditions on shocks.

Let us consider a discontinuity along a trajectory x f ( t) and obtain the mass balance condition along a discontinuity (shock wave). The region x>x f ( t) is conventionally assumed to lie ahead of the shock, and the region x<x f ( t) is assumed to lie behind the shock. The shock speed D is determined by the relation

A, behind the shock, the minus superscript will be used, A − , since this value corresponds to negative x in the initial value formulation. Likewise, the value of A ahead of the shock will be denoted A + . In particular, the density and the velocity ahead of the shock are denoted ρ + and v + , while those behind the shock are ρ − and v − . For an abitrary system of the form (1), the balance equations for a shock can be represented as

To refer the value of a quantity,

[ G i (u)]

D = [F i (u)],

i = 1, . . . , n,

A at the shock. The equations of (25) are called the Rankine–Hugoniot jump conditions.

where [ A] = A + − A − stands for the jump of a quantity

Example 10. The Rankine–Hugoniot condition for the single equation (2) reads as follows:

[ u]D = [F ].

The shock speed is equal to the slope of the line connecting the points ( u − ,

+ )) in the ( F (u u, F ) plane (see Fig. 9).

− F (u )) and ( u + ,

Example 11. The Rankine–Hugoniot conditions for an isentropic gas flow (3)–(4) follow from (25). We have

[ ρ]D = [ρv],

[ ρv]D = [ρv 2

+ p(ρ)].

Eliminating the shock speed

D from (27), we obtain the equation

Each of the signs before the square root in (28) corresponds to a branch of the locus of points that can be connected with a given point ( v − , ρ − ) by a shock (see Fig. 10a). For an ideal polytropic gas ( p = Aρ γ ), relations (28) can be transformed to

s A(ρ + − ρ − )( ρ + ) γ −( ρ − ) γ

Let us determine the set of states ( v + , ρ + ) reachable by a shock from a given point ( v − , ρ − ). Express the point ( v + , ρ + ) ahead of the shock via the solution of the transcendental system (27) to obtain

v − ( v , ρ , D), ρ = ρ ( v , ρ , D).

() b

Figure 10. Loci of points that can be connected by a shock wave: (a) with a given state ( − − ( v + , ρ +

v , ρ ) and (b) with a given state ). The solid lines correspond, respectively, to the minus and plus sign in formula (29) before the radical for cases (a) and (b).

The graphs of the solution determined by (29), or (28), are shown in Figs. 10a and 10b. The solid lines correspond to the minus sign before the radical and represent stable (evolutionary) shocks, while the dashed lines correspond to the plus sign and represent unstable (nonevolutionary) shocks; see the next subsection.

Consider the locus of points u + and a rarefaction wave trajectory near a point u − in the space u=( u 1 , ...,u n ) T . These two curves have the same tangent vector at the point u − . In order to prove

this fact, let us consider small-amplitude shocks. Setting G i (u) = u i in (25) and retaining only the leading term in the expansion in powers of |u + −u − | ≪ 1, we obtain

(e F− DI)[u] = 0,

where the same notation as in (18) is used. Hence, the vector [u] is a right eigenvector of the matrix

D tends to an eigenvalue at the point u + (or u − ).

e F=e F(u). Therefore, it coincides with the rarefaction wave vector. The shock speed

D) is a solution of the transcendental system of n equations (25). In general, the transcendental system has n roots. It follows that there should exist n shock curves u + =u + (u − , D). We call a curve the ith shock if it is tangent to the ith rarefaction wave at u − .

The set of points, or the locus of states, u + =u + (u − ,

References for Subsection S.12.5: O. A. Oleinik (1957), I. M. Gelfand (1959), P. Lax (1973), A. G. Kulikovskii (1979), C. M. Dafermos (1983), B. L. Rozhdestvenskii and N. N. Yanenko (1983), J. Smoller (1983), R. Courant and R. Friedrichs (1985), L. D. Landau and E. M. Lifshitz (1987), D. J. Logan (1994), E. Godlewski and P.-A. Raviart (1996), A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov (2001), A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux (2002).