Gradient relations at the front of shock and detonation waves

Abstract

A natural assumption on the form of the calorific equations of state (internal energy) for one-dimensional motion was used to obtain the so-called gradient relations that give a one-to-one correspondence between the first partial spatial derivatives of the pressure, density, mass velocity (gradients of parameters) at shock and detonation fronts and the time derivative (acceleration) of the front. The assumption is based on the fact that, taking into account the thermal equation of state, the total internal energy, including both the thermodynamic part and potential chemical energy, can be represented as a function of pressure and density. This holds for both inert media and reaction products in the state of chemical equilibrium.Key words: shock wave, detonation wave, gradients of parameters, acceleration of the front.The strong-discontinuity relations, which are often called the laws of conservation of mass, momentum, and energy at the shock front, are well-known [1,2]. With the thermal effect of chemical reactions taken into account, these laws of conservation are also applicable to a detonation front (a strong discontinuity with heat release) [3,4]. If the motion of the medium behind the front is described by a smooth one-dimensional solution and the parameters ahead of the front are constant, it is possible to give a one-to-one correspondence between the partial spatial derivative (gradient) of any parameter and the time derivative of the velocity (acceleration) of the front. For one-dimensional adiabatic flow of a perfect gas, such gradient relations at the shockwave front are given in [5,6]. These relations can be used to develop numerical and analytical approximate methods for solving gas-dynamic problems and establishing asymptotic laws of attenuation of shock waves [7,8]. Here it is also pertinent to note a paper on a related topic [9], in which spatial derivatives were obtained for gas-dynamic functions behind a curved stationary shock wave subjected to an oncoming uniform supersonic flow. These results were further developed in a study [10] of the behavior of the vortex velocity 1 Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090; prokh@hydro.nsc.ru.vector behind strong-discontinuity surfaces. However, in the cited papers [9, 10], the effect of acceleration of the front on parameter gradients was not considered. The range of application of the gradient relations at the shock front presented in [5, 6] is appreciably limited by the conditions of the perfect gas model [11]; for polyatomic gas and mixtures of various chemically inert gases, this model is approximately valid only in a narrow range of temperatures. Therefore, the previously obtained gradient relations cannot be used for practically important cases such as: 1) gas motion behind strong shock waves, where excitation of additional degrees of freedom and dissociation of molecules are possible; 2) equilibrium flow of reacting gases behind a detonation front propagating in a chemically active m...