Structure Of Normal Shock Waves Research Articles

気体論による垂直衝撃波の構造と相互作用の数値解析

気体論による垂直衝撃波の構造と相互作用の数値解析

Shock wave structure in gas mixtures with large mass disparity

The structure of normal shock waves is considered when the ratio of molecular masses mp/m of a binary mixture of inert monatomic gases is large and the density ratio ρp/ρ is of order unity or below. Generalized hydrodynamic equations, valid for arbitrary intermolecular potentials, are obtained from a hypersonic closure of the kinetic equation for the heavy gas and a near-equilibrium closure for the light component. Because the Prandtl number of the light gas and the Schmidt number of the mixture are nearly constant, the only independent transport coefficient arising in the model is the viscosity μ of the light gas, which is absorbed into a new independent position variable s. Knowledge of μ as a function of temperature thus determines the shock structure independently from the details of the intermolecular potential, allowing comparison with experiments in the complete absence of free parameters. In terms of the ratio M (frozen Mach number) between the speed of propagation and the sound speed of the light gas in the unperturbed medium, one finds that: (i) When M > 1, the behaviour is similar to that of a ‘dusty gas’, with a broad relaxation layer (outer solution) following a sharp boundary layer through which the speed of the heavy gas is almost constant (a shock within a shock). (ii) When (1 + ρp/ρ)s−½ < M < 1, the boundary layer disappears, yielding a so-called ‘fully dispersed wave’. (iii) Because the internal energy of the heavy gas is negligible, the present problem differs from previous shock studies in that, for the first time, the structure of the relaxation region is obtained algebraically in phase space, thus permitting an exhaustive study of the behaviour. From it, the overshooting solution found by Sherman (1960) is related to the unphysical degenerate branch of the outer solution arising when M > 1, showing a failure of the Chapman–Enskog theory, even for weak shocks, when the heavy gas is dilute. Also, an algebraic explanation arises for the ‘double hump structure’ observed in He–Xe shocks. (iv) When M is nearly unity, the initial boundary layer spreads out, and the structure must be obtained by integration of a numerically unstable system of three differential equations. However, the reduction of order brought about by the weak variation of the light-gas entropy at the head of the shock, results in a stable system of equations that we integrate numerically. Excellent phase-space agreement with recent shock-tube experiments of Tarczynski, Herczynski & Walenta (1986) is found for both weak and strong shocks.

Structure of normal shock waves in a gas-particle mixture

The structure of normal shock wave propagating in a gas particle mixture has been obtained by including the particle-volume effect.

Sonic Flow Through an Abrupt Change in Cross-Section

The steady, transonic flow in a rectangular duct following an abrupt change in section has been studied by measuring the density with a Mach-Zehnder interferometer and velocity with a laser-Doppler velocimeter. The flow structure was controlled either by a single, normal shock wave or by a series of reflected oblique shocks. In the case of the normal shock wave structure the one-dimensional compressible flow theory was found to apply adequately to the overall duct. Within the duct the flow was not one-dimensional, but had a minimum velocity in the centre and four shear layers.

The structure of normal shock waves in polytropic gases

The formation and structure of normal shock waves in compressible, polytropic gases is investigated using the continuum gas dynamical equations. A phenomenological model for temperature-dependent viscosity and thermal conductivity is preassumed. The fundamental resulting nonlinear differential equation is solved exactly, and the complete structure of the shock wave is determined. It is found that the flow quantities and characteristics depend upon one parameter, the upstream Prandtl number Pr1, which satisfies the inequality 3γ/(γ+1)≤2Pr1<3, where γ is the ratio of specific heats. For smaller values of Pr1, the strength and compression ratio of the shock increase. Special cases of interest cannot be obtained from this solution and are to be treated independently. Numerical results are given and discussed.

An Asymptotic Analysis of Strong Radiation-Resisted Normal Shock Waves

An asymptotic analysis is presented for the structure of a strong radiation-resisted normal shock wave in a gray gas. Previous solutions for the weak and strong radiation cases are reviewed and modified; then, new analytical and numerical solutions for the moderate radiation case are developed. Analytical solutions for the flow quantities at the imbedded discontinuity as functions of the radiation parameter, valid over the range of the parameter, are derived.

The use of the loaded-sphere molecular model for computer simulation of diatomic gases

An approximate binary collision procedure is devised for the loaded-sphere molecular model and is used with Bird's direct simulation technique for computer experiments on diatomic gases. The collision procedure is tested by simulating the approach to equilibrium of a constant-speed gas of loaded spheres and good agreement with Jeans's (1904) theoretical prediction is obtained. The procedure is then used to study the structure of normal shock waves in the diatomic gas, giving mean velocity, density and temperature profiles. It is found that the present procedure consumes considerably more computer time than comparable simulations of monatomic gases.

Structure of Normal Shock Waves in Gas Mixtures

A theoretical analysis of steady, normal shock structure in binary gas mixtures of inert, monatomic gases is given. The analysis is based on the moment method of kinetic theory. The governing equations of this two-temperature-two-velocity model are in no way restricted by initial concentration of the species, mass ratio of the species, or Mach number. The system of equations which describes the structure includes the conservation of mass of each species, the conservation of momentum and energy of the mixture, the transfer of momentum and peculiar energy equations, the evolution of the stress moment of each species, and the heat flux moment of each species. The nondimensional parameters which characterize the phenomenon are identified as the number density ratio in the free stream η = (nj/ni)−∞, the mass ratio θ = mi/mj, and the mixture Mach number in the free stream M−∞. By choosing various combinations of these parameters, five different types of structures are proposed. By further restricting the light species and heavy species upstream Mach number, a given type of structure may be shown to require more than one scale for a complete transition. The anomalies resulting from some continuum theories and numerical calculations are investigated. The proper application of the Mott-Smith ansatz to normal shock waves in binary gas mixtures is clarified. Comparison with existing experimental and theoretical results is made.

Radiation-resisted shock waves in gas-particle flows

The effect of thermal radiation on the structure of normal shock waves in gas-particle flows is analyzed. The study is restricted to conditions at which the gas flow can be decoupled from the particle flow and where the gas contribution to radiation is negligible. An approximate closed-form solution, derived for a gray, absorbing particle cloud, demonstrates that, within the limits of the analysis and for a given shock Mach number, radiative influences increase with increasing stagnation temperature and particle diameter and decreasing pressure and particle loading. The radiation-affected regions were found to extend large distances upstream of the gas phase discontinuity. Results from numerical calculations are in substantial agreement with those of the approximate theory. Decreasing the shock Mach number, at constant pressure, stagnation temperature, particle loading, and particle diameter, increased the fraction of the particle temperature rise across the shock occurring upstream of the gas phase discontinuity. Nomenclature

Structure of a strong normal shock wave due to nonequilibrium radiation and collisional ionization

The general theory about gas-dynamics with nonequilibrium radiative and collisional ionization is applied to the strong normal shock wave. It is shown that the problem is reduced to the solution of a system of two non-linear integral equations. It is proved that, if certain conditions are satisfied, an iteration method, to resolve these equations, can be applied; it turns out then that the fundamental parameter controlling the convergence or non convergence of the procedure is the value of the degree of ionization a just on the front of the shock. Numerical results are given and the physical complete structure of the shock is analized.

The structure of normal shock waves in a binary gas mixture

A previous study of normal shock waves through numerical experiments with a simulated gas on a digital computer is extended to binary gas mixtures. The mixture is simulated by two sets of rigid elastic sphere molecules with the appropriate mass and diameter ratios. Velocity profile results for medium strength waves in a mixture of equal parts argon and helium are in qualitative agreement with the continuum calculations of Sherman (1960), but there is no initial acceleration of the argon in mixtures containing a very small initial mole fraction of this gas. The temperature profiles are similar to those for the velocity in that the argon profile lags behind the helium profile. However, when there is a small proportion of heavy gas, the profiles cross-over and the temperature of the heavy gas overshoots the Rankine-Hugoniot downstream value. For very strong shock waves, the overall shock thickness expressed in upstream mean free paths becomes larger, but the profiles are generally similar to those for the medium strength waves.

Newtonian hypersonic theory for normal shock wave structure.

Newtonian hypersonic theory for normal shock wave structure, discussing asymptotic solutions and three layer models

The velocity distribution function within a shock wave

The structure of normal shock waves in a gas composed of rigid sphere molecules is investigated by numerical experiments with a simulated gas on a digital computer. The non-equilibrium between the temperatures based on the longitudinal and lateral velocity components is studied and the results compared with the theory of Yen (1966). Details of the velocity distribution function are presented for a shock of Mach number 10. The distribution functions for both the longitudinal and lateral velocity components are plotted for a number of locations in the shock profile and are compared with the equilibrium distribution.

Influence of Viscosity on Shock Waves Structured by Radiation

The structure of normal shock waves with radiation, viscosity, and thermal conductivity is treated in the physically relevant limit of large radiation absorption length (or photon mean free path) compared to molecular transport length (or molecular mean free path). If radiation is not too intense or the shock too weak, then a viscous, conducting shock is found embedded in a radiation structured inviscid shock. Under very general conditions, this viscous ``shock within a shock'' has precisely the same structure as a classical (radiationless) viscous, conducting shock when reinterpreted as to initial and final states, to lowest order in an expansion based on the ratio of the two transport lengths. This lowest order solution is no longer trivial when the optically thin region considered is embedded in a much thicker region. The next order corresponds to the usual optically thin case but the well known formula (accounting for emission but not absorption by the gas) does not apply; the divergence of the radiation flux is actually given by the emission minus absorption of the radiation from the outer region. Since the relative importance of these two effects changes markedly through the inner viscous shock, the usual formula is meaningless here.