Rankine-Hugoniot condition

CA 44, 1032(1950) (Penetrating or piercing jet theory of deton) 39) S.R. Brinkley Jr, "The Theory of Detonation Process , pp 83-8 in the "Summary Technical Report Division 8, NDRC , Vol 1 (1946). It includes Riemann formulation (pp 83-4 86) Rankine-Hugoniot condition ( 84 86) Chapman-Jouguet postulate (84) Becker semiempirical equation of state (85) Rayleigh, solution of the Riemann equation (86) and Hydro-thermodynamic theory, applications (87-8) 40) W. Loring H. 

As illustrated by (3.2.11), for m > 2 the first derivative of concentration at the boundary of support is discontinuous that is, a weak shock is formed at the zero concentration front. This stands in accord with the classical Rankine-Hugoniot condition that prescribes for any moving interface Xi(t)... 

Balance laws (3.67) are known to exhibit shock waves, i.e. solutions w with discontinuities of the (negative) tangent angle w at shock positions s t). See Figure 3.12 and, for some qualitative remarks, also [62]. The Rankine-Hugoniot condition specifies the shock speed s t) to satisfy... 

The equations have another bad feature which is inherited by some much more complex theories. The velocity of an interface moving at constant velocity is not uniquely determined by the obvious boundary conditions. For example, if we hold one end of a fiber fixed and pull the other at constant velocity, then we are prescribing v const. > 0 and v 0. The constant strains e and e and the velocity of the interface V are then subject only to the usual Rankine-Hugoniot conditions based on (3.1), which do not uniquely determine V. ... 

Even with isentropic flow upstream of the shock, each term in equation (3) can be written in terms of linear combinations of dM and dS where M denotes the shock Mach number. For the first terra the Rankine-Hugoniot conditions give... 

Conservation equations Expressions that equate the mass, momentum, and energy across a steady wave or shock discontinuity ((2.1)-(2.3)). Also known as the jump conditions or the Rankine-Hugoniot relations. 

The behavior of shock waves is ruled by the Rankine-Hugoniot equations, which express the conditions for conservation of mass, momentum, and energy and can be used to design suitable shock prohles. Referring to the PV diagram (see left panel of Fig. 13), the compressed state (P, V ) can be represented as... 

For a transmitted shock wave advancing into any gas at an initial pressure pe of 1 atm, the RH (Rankine-Hugoniot) equation defines a functional relationship between pressure p and particle velocity w behind the wave S3, involving initial pressure, initial specific volume v, and equations of state of the target medium. Similarly, the conditions behind the reflected wave S2 and close to the product-target interface are expressible by means either of the shock wave equations or the Rie-mann adiabatic wave equations in terms of any one such variable and the conditions... 

Theory. If p is pressure, v - specific volume, e - specific internal energy, D detonation velocity, u - particle velocity, C - sound velocity, y - adiabatic exponent and q -specific.detonation energy, the velocity of propagation and particle velocity immediately behind any plane detonation wave in an explosive, defined by initial conditions, pD, v0, eQ, and uQ, are given by the first two Rankine-Hugoniot relations ... 

As discussed in Vol 7, HI 79, the Hugoniot equations (or more correctly Rankine-Hugoniots), the simultaneous measurement of any two of the above variables is sufficient to determine all the rest provided conditions ahead of the shock (u0s P0, p o> Eo To) are known. Thus, for mathematical convenience but closely approximated in reality, the shock abruptly divides virginal (unshocked) material from shocked (compressed) material. For the reader s convenience, because we will refer to them frequently, this is illustrated in two graphs taken from the above Vol 7 article. They show the transmission of a shock from one material to another (script S s are shocks and script R s are rarefactions)... 

Rankine-Hugoniot relations are so well obeyed within about 5 nm of the front implies, via a classic argument not repeated here, that the simulated detonation is moving at the minimum velocity consistent with the conservation conditions, as predicted by ZND theory. We return to a discussion of the CJ point and related issues in Sec. 4, where several methods are presented for establishing the existence and position of the CJ point from the simulations. 

These equations relate the undisturbed explosive lying at rest with pressure Pq = 0 and specific volume Vq = to the state behind the detonation front, which is characterized by a pressure P, a specific volume V, and a particle flow velocity u. Both u and the detonation velocity, D, are measured in the reference frame of the undisturbed material. Because Pq and Vq are known, the Rankine-Hugoniot relations are a set of three equations for the four unknowns, u, D, P, and V. The first relation determines u in terms of D, P, and Vi, which leaves two equations with three unknowns. The first of the remaining equations, Eq. (4b) defines the Rayleigh line while Eq. (4c) defines the Hugoniot curve. The problem is formally determined by selecting the solution of Eqs. (4b) and (4c) that corresponds to the minimum value of D for an unsupported detonation. This additional condition is the Chapman—Jouguet hypothesis, which was put on a firmer foundation by Zel dovich. ... 

The term energy per unit area is referred to as the energy fluence (Ref 8). Recalling the Rankin-Hugoniot jump conditions, specifically the mass and momentum equations for a shock, we had derived that... 

The evolution of the test-gas is therefore represented by the system of ordinary differential equations (120), (124), (131) and (140). The initial conditions of this system are given by the RANKINE-HUGONIOT equations. 

When Ugo > af o, it is necessary to insert a discontinuity in the flow in order to obtain a solution to the inviscid conservation equations. Across the discontinuity, all interphase transfer processes are effectively frozen and hence no changes occur in droplet radius, temperature and velocity. The vapour phase conditions donwstream of the discontinuity are easily calculated by a standard Rankine-Hugoniot analysis and these provide the initial conditions for the numerical integration procedure. 

We notice of course that the consistency conditions between two equilibria and (y,W, are given by the classical Rankine-Hugoniot jump conditions ... 

We can now define the viscosity-capillarity admissibility criterion for shocks. We say a shock solution of (7) (with jj, = k, = e = 0) propagating with speed c satisfying the Rankine-Hugoniot jump conditions (13) of the form... 

If we assume that the ridge has a constant height h between the radius R of the dry area and the radius Rp of the upstream front, we can write the standard Rankine-Hugoniot shock conditions ... 

The equations for the conservation of mass, momentum and total energy have been employed by Rankine (Ref 2) and Hugoniot (Ref 3) in the formulation of three conditions relating to pressure P2, specific volume v2... 

---