The quasi-stationary structure of radiating shock waves. I. The one-temperature fluid

We calculate the quasi-stationary structure of a radiating shock wave propagating through a spherically symmetric shell of cold gas by solving the time-dependent equations of radiation hydrodynamics on an implicit adaptive grid. We show that this code successfully resolves the shock wave in both the subcritical and supercritical cases and, for the first time, we have reproduced all the expected features – including the optically thin temperature spike at a supercritical shock front – without invoking analytic jump conditions at the discontinuity. We solve the full moment equations for the radiation flux and energy density, but the shock wave structure can also be reproduced if the radiation flux is assumed to be proportional to the gradient of the energy density (the diffusion approximation), as long as the radiation energy density is determined by the appropriate radiative transfer moment equation. We find that Zel'dovich and Raizer's (1967) analytic solution for the shock wave structure accurately describes a subcritical shock but it underestimates the gas temperature, pressure, and the radiation flux in the gas ahead of a supercritical shock. We argue that this discrepancy is a consequence of neglecting terms which are second order in the minimum inverse shock compression ratio [, where is the adiabatic index] and the inaccurate treatment of radiative transfer near the discontinuity. In addition, we verify that the maximum temperature of the gas immediately behind the shock is given by , where is the gas temperature far behind the shock. Received 21 September 1998/ Accepted 2 February 1999     

Effect of surroundings on the transient energy transfer in a layer of radiating gas

Summary An analysis is presented for the transient cooling or heating of a stagnant layer of hot radiating gas surrounded by a cold gas capable of absorbing and emitting radiation. Scattering of radiation is neglected, and energy transfer by conduction and convection is considered to be negligible compared with radiation. The gas is assumed to be perfect and in local thermodynamic equilibrium. The heating of a cold gas by a diffuse and a collimated radiation flux incident on the boundary of the gas from some external source is considered, and the dependence of physical and radiative properties on temperature is taken into account. The problem is formulated exactly using radiative transfer theory. A scheme is developed for the numerical solution of the nonlinear integrodifferential equations of energy conservation. Starting with arbitrary, but given, initial temperature profiles, temperature distributions and local radiative fluxes are predicted as a function of time for a wide range of physically interesting conditions.     

Simulating radiative shocks with the CRASH laser package

We present the latest improvements in the Center for Radiative Shock Hydrodynamics (CRASH) code, a parallel block-adaptive-mesh Eulerian code for simulating high-energy-density plasmas. The implementation can solve for radiation models with either a gray or a multigroup method in the flux-limited-diffusion approximation. The electrons and ions are allowed to be out of temperature equilibrium and flux-limited electron thermal heat conduction is included. We have recently implemented a CRASH laser package with 3-D ray tracing, resulting in improved energy deposition evaluation. New, more accurate opacity models are available which significantly improve radiation transport in materials like xenon. In addition, the HYPRE preconditioner has been added to improve the radiation implicit solver. With this updated version of the CRASH code we study radiative shock tube problems. In our set-up, a 1 ns, 3.8 kJ laser pulse irradiates a 20 micron beryllium disk, driving a shock into a xenon-filled plastic tube. The electrons emit radiation in the shocked xenon. This radiation preheats the unshocked xenon. Photons traveling ahead of the shock will also interact with the plastic tube, heat it, and in turn this can drive another shock off the wall into the xenon.     

Grey-body surface radiation coupled with conduction and convection for general geometries

This paper presents a numerical technique for the simulation of the effects of grey-diffuse surface radiation on the temperature field of fluid flows using FIDAP, a general purpose incompressible, viscous fluid code. The radiating surface relationships assume a non-participating medium, constant surface temperature and heat fluxes at the discretized elemental level. The technique involves the decoupling of energy and radiation exchange equations. A concept of macrosurfaces, each containing a number of radiating boundary surfaces, is introduced. These boundary macroelements then carry the information from the radiating boundary into the fluid regime. A number of simulations illustrating the algorithm are presented.     

The flow of a nonequilibrium ionized radiating gas around a body with consideration of temperature difference between electrons and ions

The flow around a blunt body at hypersonic speed by a current of nonequilibrium ionized monatomic nonviscous radiating gas is studied, with consideration of temperature difference between the electron gas and the ion-atom gas. Atomic excitation due to collisions with electrons and subsequent ionization, as well as photoionization, are taken into consideration. Since the value of the shock wave separation is small in comparison with the characteristic dimension of the body, the radiation transfer equation is written in the local onedimensional planar layer approximation. The influence of incident flow parameters upon the flow field across the shock wave and the distribution of radiation thermal flux are studied.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 9–14, January-February, 1972.     

Thermally developing laminar flows with radiative interaction using the total band absorptance model

The problem of energy transfer in a thermally developing, radiating and conducting medium is studied. In particular, laminar flow of carbon monoxide is considered although the results may be interpreted more generally as referring to any infra red radiating diatomic gas. The effects of radiation and conduction on the temperature profile and the Nusselt number are presented for slug flow and for the parabolic velocity distribution. On leave from the University of California, Berkeley. On leave from the University of Arizona, Tucson.     

Heat transfer in a radiating fluid with slug flow in a parallel-plate channel

Summary As a step towards a better understanding of combined conduction, convection, and radiation, fully developed heat transfer in slug flow in a flat duct between two parallel plates is considered. The flat duct consists of two diffuse, nonblack, isothermal surfaces. The gray radiating fluid between them is capable of emitting and absorbing thermal radiation. The problem is formulated in terms of a nonlinear integrodifferential equation, and the solution is obtained by an approximate method. The differences between heating and cooling the fluid are examined. The effects of the optical thickness of the fluid, the ratio which determines the relative role of energy transport by conduction to that by radiation, and the emissivity of the duct surfaces on the temperature distribution and the heat transfer characteristics are investigated. An approximate method for calculating the radiant heat flux at the wall is presented, and the accuracy of the approximation is tested.Work done under the auspices of the U.S. Atomic Energy Commission.     

Flow over blunt cones during entry into an atmosphere of carbon dioxide and nitrogen

An investigation has been made of hypersonic flow over spherically blunted cones in an atmosphere consisting of carbon dioxide and nitrogen. Local thermodynamic equilibrium is assumed in the shock layer. Account is taken of viscosity, diffusion, heat conduction, and radiative energy transport. The problem is solved using equations for dynamics of a viscous radiating gas without isolating inviscid flow and boundary-layer regions in the shock layer. The selectivity of the radiation is allowed for by using a two-stage approximation for the spectral dependence of the absorption coefficient, obtained by processing detailed data on absorption cross sections. The solution is found by a flow establishment method. Results are presented for flow over blunt cones with different semiangles.     

Distribution of radiant thermal flux over the surface of a sphere for hypersonic flow of a nonviscous radiating gas past the sphere

In the present study using the Newtonian approximation [1] we obtain an analytical solution to the problem of flow of a steady, uniform, hypersonic, nonviscous, radiating gas past a sphere. The three-dimensional radiative-loss approximation is used. A distribution is found for the gasdynamic parameters in the shock layer, the withdrawal of the shock wave and the radiant thermal flux to the surface of the sphere. The Newtonian approximation was used earlier in [2, 3] to analyze a gas flow with radiation near the critical line. In [2] the radiation field was considered in the differential approximation, with the optical absorption coefficient being assumed constant. In [3] the integrodifferential energy equation with account of radiation was solved numerically for a gray gas. In [4–7] the problem of the flow of a nonviscous, nonheat-conducting gas behind a shock wave with account of radiation was solved numerically. To calculate the radiation field in [4, 7] the three-dimensional radiative-loss approximation was used; in [5, 6] the self-absorption of the gas was taken into account. A comparison of the equations obtained in the present study for radiant flow from radiating air to a sphere with the numerical calculations [4–7] shows them to have satisfactory accuracy.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 44–49, November–December, 1972.In conclusion the author thanks G. A. Tirskii and É. A. Gershbein for discussion and valuable remarks.     

Shock wave structure for a fully ionized plasma

We study planar shock wave structure in a two-temperature model of a fully ionized plasma that includes electron heat conduction and energy exchange between electrons and ions. For steady flow in a reference frame moving with the shock, the model reduces to an autonomous system of ordinary differential equations which can be numerically integrated. A phase space analysis of the differential equations provides an additional insight into the structure of the solutions. For example, below a threshold Mach number, the model produces continuous solutions, while above another threshold Mach number, the solutions contain embedded hydrodynamic shocks. Between the threshold values, the appearance of embedded shocks depends on the electron diffusivity and the electron–ion coupling term. We also find that the ion temperature may achieve a maximum value between the upstream and downstream states and away from the embedded shock. We summarize the methodology for solving for two-temperature shocks and show results for several values of shock strength and plasma parameters in order to quantify the shock structure and explore the range of possible solutions. Such solutions may be used to verify hydrodynamic codes that use similar plasma physics models.     

Efficient Thermo-Chemistry Tabulation for Non-Premixed Combustion at High-Pressure Conditions

Propellant injection and turbulent combustion in high-pressure engines is often dominated by real-gas effects. However, previous studies suggested that the departure of the fluid properties from an ideal gas behavior has only a limited effect on the laminar flame structure. This is due to the fact that chemical reactions take place in the flame zone where the temperature is sufficiently high and molecular interactions are negligible, i.e., the ideal gas assumption is valid. On the other hand, various experimental and numerical studies of injection processes at high-pressure conditions demonstrated that real-gas effects can have a strong impact on the turbulent flow. Mixing is influenced by the rapid change of fluid properties. In this work, we exploit the gap in the fidelity of the thermodynamics model needed to describe the laminar flame structure and that needed to describe the turbulent flow field. We then propose a new real-gas flamelet model with increased numerical performance. The computational cost of the new formulation is not significantly higher than that of an ideal gas simulation. The performance of the method is analyzed and the error that is introduced by our assumptions is assessed by comparison to more complete modeling. Finally, the method is used to simulate a turbulent jet flame emanating from a coaxial injector at supercritical pressure and cryogenic oxidizer temperature. The results are compared with experimental OH^? images giving evidence of the suitability of the present method.     

Calculation of inviscid radiating gas flow past a blunt body

During hypersonic gas flow past a blunt body with a velocity on the order of the escape velocity or more, the gas radiation in the disturbed region behind the shock wave becomes the primary mechanism for aerodynamic heating and has a significant effect on the distribution of the gasdynamic parameters in the shock layer. This problem has been considered from different points of view by many authors. A rather complete review of these studies is presented in [1–4].In earlier studies [5, 6] the approximation of bulk emission was used. In this approximation, in order to account for the effect of radiative heat transfer a term is added in the energy equation which is equivalent to the body efflux, whose magnitude depends on the local thermodynamic state of the gas. However, the use of this assumption to solve the problem of inviscid flow past a blunt body leads to a singularity at the body [7, 8]. To eliminate the singularity, account is taken of the radiation absorption in a narrow wall layer [7], or the concept of a viscous and heat-conductive shock layer is used [8]. A further refinement was obtained by Rumynskii, who considered radiation selectivity and studied the flow of a radiating and absorbing gas in the vicinity of the forward stagnation point of a blunt body.In the present paper we study the distribution of the gasdynamic parameters in the shock layer over the entire frontal surface of a blunt body in a hypersonic flow of a radiating and absorbing gas with account for radiation selectivity.     

The radiative Knudsen layer

The analogy between the kinetic theory of gases and photon gas dynamics (see [1], for instance) means that many qualitative results obtained in kinetic theory can be extended to photon gas dynamics. This includes the behavior of a radiating optically thick gas near solid surfaces. It is known that in an optically thick gas the radiative heat conduction approximation is inapplicable at distances of the order of the mean free path of the radiation from the surface of a body [1, 2]. The behavior of a photon gas near a surface can be predicted on the basis of the complete radiative transfer equation. In the present paper the problem of such wall layers in an optically thick gas (which have much in common with the well-known Knudsen layers) is solved in the approximation of a gray gas in local thermodynamic equilibrium. For simplicity only the steady-state case is considered. Expressions for the temperature jumps at the surface in a wide range of gas temperatures are obtained.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 4, pp. 113–121, July–August, 1973.In conclusion the author thanks M. N. Kogan for the topic and guidance during the conduction of this work.     

Thermohydrodynamic characteristics of combined double-diffusive radiation convection heat transfer in a cavity

This work deals with the numerical analysis of a radiating gas flow caused by both temperature and buoyancy concentration gradients in a square cavity; in this regard, the set of governing equations, including conservation of mass, momentum, species, and energy are solved by a numerical technique. In terms of radiation, since the fluid is considered as a semitransparent medium, the radiative term in the energy equation appears and is calculated by numerical solving of the radiative transfer equation (RTE). Furthermore, all of the surrounding cavity walls are considered to be opaque, gray, and diffuse with constant emissivity. All of the flow equations are solved by the finite difference method (FDM) and the RTE by the discrete ordinate one (DOM). In the present study, an attempt is made to verify the optical thickness effects on flow, thermal behavior, and mass transform in a cavity flow, such that reciprocating trends were seen in this manner. Our numerical results show that the thermal field in double-diffusive convection flow reaches very fast its steady-state situation in comparison to the concentration distribution. Besides, it is found that the thermohydrodynamic characteristics of a double-diffusive convection flow of a radiating gas are much affected by optical thickness.     

Structure of shock waves with emission in a gas flow impinging on a barrier

The parameters of emitting shock waves generated upon high-velocity rarefied-gas flow impingement on a plane barrier are investigated with account for the reflecting capacity of the barrier. In the case of aluminum plasma impingement the unsteady radiation-gasdynamic flows are numerically calculated using the Lax-Wendroff scheme at different values of the radiation reflection coefficient. The radiation transport is accounted for in the gray gas approximation. In the quasi-stationary stage the structure of compressed shock layer on the wall is considered in the radiant heat conduction approximation.     

Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas

Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture \(K_p\) , and the ratio of the density of solid particles to the initial density of the gas \(G_a\) are investigated. A comparison between the solutions for the isothermal and adiabatic cases is also made.     

Radiative and convective heat transfer in a plane layer of absorbent curtain

An examination is made of the thermal state of a plane layer of gray gas injected into a turbulent stream of high temperature gas flowing over a permeable flat plate.Similarity-type formulations of problems are encountered both in examination of flow near a stagnation point, and also in analysis of the lifting of the boundary layer by intense injection through a porous plate [1]. The examination described relates to the following idealized formulation of the problem (Fig. la).In a plane layer of gray absorbing medium, formed by plane-parallel diffusely radiating surfaces (1-porous plate; 2-boundary of high temperature turbulent gas stream), heat transfer is accomplished by radiation and convection of the layer normal to the surfaces and by molecular heat conduction. All the physical and optical properties of the medium and of the boundary surfaces are assumed to be constant, independent of temperature.The temperature of the wetted surface of the specimen and also that of the fictitious surface determining the upper bound of the lift-off region, are given.Also assumed given is the velocity of the injected medium, which is constant throughout the entire lift-off layer. This idealization appreciably facilitates our examination without in principle changing its features.A very simplified examination of this problem was given in [2]. The special case of a medium with low optical thickness was examined in [3,4].The problem was examined in [5] under the assumption that molecular heat conduction in the medium is negligibly small.In the formulation considered the generalized energy equation has the form     

Prandtl number effects in MRT lattice Boltzmann models for shocked and unshocked compressible fluids

This paper constructs a new multiple relaxation time lattice Boltzmann model which is not only for the shocked compressible fluids, but also for the unshocked compressible fluids. To make the model work for unshocked compressible fluids, a key step is to modify the collision operators of energy flux so that the viscous coefficient in momentum equation is consistent with that in energy equation even in the unshocked system. The unnecessity of the modification for systems under strong shock is analyzed. The model is validated by some well-known benchmark tests, including thermal Couette flow, Riemann problem. The first system is unshocked and the latter is shocked. In both systems, the Prandtl number effects are checked. Satisfying agreements are obtained between new model results and analytical ones.     

Radiation of a shock layer during hypersonic flow of air about a spherical segment

The problem on the flow of radiating air about a spherical segment is solved. A comparison of results obtained with investigations of the flow of a radiating gas about a sphere and the flow of gas about a spherical segment, with radiation left out of account, is made. The influence of radiation in the neighborhood of the rim of the segment on the flow in the shock layer is considered, and it is shown that it does not exert a significant influence on the fields of the gas-dynamical parameters because the latter are determined by processes occurring near the axis of symmetry, due to the phenomenon of radiation freezing.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 101–106, January–February, 1972.