Jump condition

To reiterate, the development of these relations, (2.1)-(2.3), expresses conservation of mass, momentum, and energy across a planar shock discontinuity between an initial and a final uniform state. They are frequently called the jump conditions" because the initial values jump to the final values as the idealized shock wave passes by. It should be pointed out that the assumption of a discontinuity was not required to derive them. They are equally valid for any steady compression wave, connecting two uniform states, whose profile does not change with time. It is important to note that the initial and final states achieved through the shock transition must be states of mechanical equilibrium for these relations to be valid. The time required to reach such equilibrium is arbitrary, providing the transition wave is steady. For a more rigorous discussion of steady compression waves, see Courant and Friedrichs (1948). 

For shock waves in solids, the shock pressure P is typically much greater than the initial pressure Pq, which is normally ambient atmospheric conditions, so that Pq is usually neglected. Eq can also be taken to be zero, sinee internal energy is a thermodynamie state funetion and ean be refereneed to any initial state. Removing Eq and Pq from the jump conditions results in their eommon form... 

The various forms of the jump conditions are summarized in Table 2.1. 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

It is instructive to collect the important relations here for comparison to the jump conditions derived in Section 2.4. When the bead parameters are replaced with the properties of particle and shock velocities, force and internal energy, the relations can be written as... 

The jump conditions (2.1)-(2.2) can be combined and solved for the shock and particle velocities... 

The form of the jump conditions also depends on the coordinate system. Substituting (2.40) into the general Eulerian form of the momentum jump condition (Table 2.1) yields the Lagrangian jump condition... 

The introduction of Lagrangian coordinates in the previous section allows a more natural treatment of a continuous flow in one dimension. The derivation of the jump conditions in Section 2.2 made use of a mathematical discontinuity as a simplifying assumption. While this simplification is very useful for many applications, shock waves in reality are not idealized mathematical... 

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. 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

Equation of state An equation that deseribes the properties of a given material, and distinguishes one material from another. It defines a surfaee in thermodynamie variable spaee on whieh all equilibrium states lie. In shoek-wave applieations, the initial and final states are frequently assumed to lie on the equation of state surface, and this equation ean be eombined with the jump conditions to define the Huqoniot curve whieh is material speeific. 

Steady wave A propagating transition region that connects two uniform states of a material. The wave velocities of all parts of the disturbance are the same, so the profile does not change with time, and the assumptions that go into the jump conditions are valid. 

Unsteady wave A loading or unloading wave whose profile changes with time. The jump conditions cannot be rigorously applied to such a wave. 

Solution. The jump conditions at the elastic shock front require... 

The jump conditions across a discontinuity moving into undisturbed material at rest are... 

The time rates of change of the shock amplitude can also be obtained for other variables by differentiating the jump conditions, (A.6) and (A.7), along a shock path to obtain... 

Once the piston-driven flow field is known, the flame-driven flow field is found by fitting in a steady flame front, with the condition that the medium behind it is quiescent. This may be accomplished by employing the jump conditions which relate the gas-dynamic states on either side of a flame front. The condition that the reaction products behind the flame are at rest enables the derivation of expressions for the density ratio, pressure ratio, and heat addition... 

The code reproduced shock-jump conditions well, but many details in the solution were lost because of the smearing effect of artificial viscosity. 

Whitaker, S, The Species Mass Jump Condition at a Singular Surface, Chemical Engineering Science 47, 1677, 1992. 

As both phases occupy the full flow field concurrently, two sets of conservation equations correspond to these two phases and must be complemented by the set of interfacial jump conditions (discontinuities). A further topological law, relating the void fraction, a, to the phase variables, was needed to compensate for the loss of information due to model simplification (Boure, 1976). One assumption that is often used is the equality of the mean pressures of the two phases, ... 

In a steady state, the shock velocity 100 km s-1 and the pre-shock number density — 1 cm-3. The jump conditions are crudely illustrated in Fig. 3.28 and results of a specific model calculation in Fig. 3.29. 

The [O ii] or [S n] doublet ratio gives the ambient ISM density p or n, normally after division by 4 in accordance with the jump condition for a strong shock. 

The only way to determine all five constants w+, w., v+,Di, D2 in the above problem is to supply an additional jump condition. This jump condition cannot be universal since, if applied at both discontinuities, it leads to an overdetermined system. We must therefore differentiate between the waves moving with the speeds and A. Notice that only the first... 

Cook et al (Ref 6) considered that the jump condition of NDZ theory is not a satisfactory solution, but that pressure in the reaction zone is limited to values no greater than the pressure P at the C-J plane, except possibly for an extremely short distance of perhaps several mean free paths at the extreme front where thermal equilibrium may not exist (Ref 7, p 174)... 

Eulerian equations for the dispersed phase may be derived by several means. A popular and simple way consists in volume filtering of the separate, local, instantaneous phase equations accounting for the inter-facial jump conditions [274]. Such an averaging approach may be restrictive, because particle sizes and particle distances have to be smaller than the smallest length scale of the turbulence. Besides, it does not account for the Random Uncorrelated Motion (RUM), which measures the deviation of particle velocities compared to the local mean velocity of the dispersed phase [280] (see section 10.1). In the present study, a statistical approach analogous to kinetic theory [265] is used to construct a probability density function (pdf) fp cp,Cp, which gives the local instantaneous probable num-... 

The discussion given here has been qualitative and does not constitute a correct stability analysis. Even for the simplified model adopted, consideration should be given to jump conditions for interactions at the shock and... 

Equation (77) and (dT/dx) = Q+ = 0 provide first approximations to the results obtained by a more formal derivation of jump conditions across the thin reaction zone. Equation (56) applies on each side of the reaction zone in this three-zone problem. To the lowest order in the ZePdovich number, T remains constant for x > 0. The problem for x < 0 becomes identical to that described by equations (56)-(58), with the replacements E, = EJ 2R ... 

Appropriate integrations of the differential equations across the reaction sheet serve to eliminate the rate term w from the system, thereby providing a set of reaction-free equations with reaction-sheet jump conditions for analyzing the dynamics of the flame. The reaction-zone equations... 

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