Shock structure and multiple sub-shocks in Grad 10-moment binary mixtures of monoatomic gases (original) (raw)
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Computation of Hypersonic Shock Waves in Inert Gas Mixtures Using the Generalized Boltzmann Equation
AIP Conference Proceedings, 2011
For numerical solution of the Generalized Boltzmann Equation (GBE) for simulating rarefied hypersonic flows in a gas mixture of multiple species, the GBE is formulated in the impulse space. The gas mixtures may consist of both monatomic gases and diatomic gases with arbitrary constituents, concentrations, and mass ratios. The conservative discrete ordinates method of Tcheremissine is applied to validate the solutions against the existing simulations for shock waves in an inert binary mixture of monatomic gases. The method is then exercised for various concentration ratios, mass ratios, and density ratios to evaluate its ability to simulate a wide range of binary gas mixtures. It is also applied to simulate two of the three primary constituents of air (N 2 , O 2 , Ar) in a binary mixture at 1:1 concentrations and at the relative concentrations found in air. These solutions can serve as validation test cases for other methods as well as an important building block in developing complex 3D simulations for shock waves in a mixture of multiple gases.
Ab Initio Simulation of Shock Waves Propagating Through Gaseous Mixtures
31st International Symposium on Shock Waves 1, 2019
A structure of planar shock wave propagating through a helium-argon mixture is calculated applying the direct simulation Monte Carlo method based on ab initio interatomic potential for several values of the Mach number in the range between 1.5 and 10. To characterize the density and temperature variations along the shock wave, dimensionless slopes of these quantities, defined through their maximum derivative with respect to the spatial coordinate, are calculated. A comparison of the slopes and density distributions for different values of molar fraction shows that the chemical composition strongly affects the shock wave characteristics.
47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition
This paper describes the computational methodology for computing hypersonic nonequilibrium shock wave (SW) flows in a mixture of non reacting diatomic gases such as Nitrogen and Oxygen using the Generalized Boltzmann Equation (GBE) at Knudsen numbers in transitional and rarefied flow regimes. In the GBE, the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively. The computational framework available for the standard Boltzmann equation (for a monoatomic gas with translational degrees of freedom) is extended by including both the rotational and vibrational degrees of freedom in the GBE. The solution of GBE requires modeling of transition probabilities, elastic and inelastic cross-sections etc. of a diatomic gas molecule, needed for the solution of the collision integral. The whole problem that includes both the vibrational-translational (VT) and rotational-translational (RT) energy transfers is solved by applying a three-stage splitting procedure to the GBE. The three stages consist of free molecular transport, VT relaxation, and RT relaxation. For computation of shock structure in a mixture of gases, the GBE needs to be formulated in impulse space instead of the standard velocity space. Furthermore, till now, the computations of SW in neutral gas mixtures have been performed only for 1D problem and they assume cylindrical symmetry of the solution in the velocity space. In this paper, we describe the development of a general numerical method for multicomponent neutral mixtures applicable to 2D and 3D flows. A 3D code has been developed and applied to compute the SW in neutral binary mixture.
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010
Hypersonic flows about space vehicles produce flow fields in thermodynamic nonequilibrium with local Knudsen numbers L K n / (where is the mean free path of gas molecules and L is a characteristic length) which may lie in all the three regimescontinuum, transition and rarefied. Flows in continuum regime can be modeled accurately by the Navier-Stokes (NS) equations; however the flows in transition and rarefied regimes require a kinetic approach such as the Direct Simulation Monte Carlo (DSMC) method or the solution of the Boltzmann equation. This paper describes the development of a computational methodology and a code for computing hypersonic non-equilibrium shock wave flows of diatomic gases using the Generalized Boltzmann Equation (GBE) at Knudsen numbers in transitional and rarefied flow regimes. The GBE solver has been validated by computing the 1D shock structure in nitrogen for Rotational-Translational (R-T) relaxations and comparing the numerical results with the experimental data for Mach numbers up to 15. The solver has been exercised successfully for computing the 2D blunt body flows in nitrogen and 3D flow from a rectangular jet of nitrogen in vacuum for R-T relaxations. The issues of stability of the algorithm and the possibility of reducing the number of rotational levels in the computations without compromising the accuracy of the solutions have been rigorously addressed. A new two-level kinetic model has been developed for computing the RT relaxations in a diatomic gas and has been validated by comparing the results with the solutions of complete GBE. The model is about twenty times more efficient than the GBE in computing the shock structure. It should be noted that the model is different than the BGK model; it accounts for both elastic and inelastic collisions. The computational methodology has been extended to compute the hypersonic shock structure in diatomic gases including both the RT and Vibrational-Translational (V-T) relaxations. 1-D shock structure in nitrogen has been computed including both R-T and V-T relaxations and has been validated by comparing the results with the experimental data. A computational methodology has also been developed to compute the hypersonic shock structure in a non-reactive mixture of two diatomic gases. 1-D shock structure has been computed in an inert mixture of nitrogen and oxygen for R-T relaxations. To accomplish this, the GBE is formulated and solved in "impulse space" instead of velocity space.
Dalton’s and Amagat’s laws fail in gas mixtures with shock propagation
Science Advances, 2019
A shock propagating through a gas mixture leads to pressure, temperature, and density increases across the shock front. Rankine-Hugoniot relations correlating pre-and post-shock quantities describe a calorically perfect gas but deliver a good approximation for real gases, provided the pre-shock conditions are well characterized with a thermodynamic mixing model. Two classic thermodynamic models of gas mixtures are Dalton's law of partial pressures and Amagat's law of partial volumes. We measure post-shock temperature and pressure in experiments with nonreacting binary mixtures of sulfur hexafluoride and helium (two dramatically disparate gases) and show that neither model can accurately predict the observed values, on time scales much longer than that of the shock front passage, due to the models' implicit assumptions about mixture behavior on the molecular level. However, kinetic molecular theory can help account for the discrepancy. Our results provide starting points for future theoretical work, experiments, and code validation.
Effects of molecular diffusivity on shock-wave structures in monatomic gases
Physical Review E, 2021
We present a full investigation into shock wave profile description using hydrodynamics models. We identified constitutive equations that provide better agreement for all parameters involved in testing hydrodynamic equations for the prediction of shock structure in a monatomic gas in the Mach number range 1.0 − 11.0. The constitutive equations are extracted from a previously derived thermomechanically consistent Burnett regime continuum flow model. The numerical computations of the resulting hydrodynamic equations along with classical ones are performed using a finite difference global solution (FDGS) scheme. Compared to previous studies that focussed mainly on the density profile across the shock, here we also include temperature profiles as well as nonnegativity of entropy production throughout the shock. The results obtained show an improvement upon those obtained previously in the bi-velocity (or volume/mass diffusion) hydrodynamics and are more accurate than in the hydrodynamic models from expansions method solutions to the Boltzmann equation.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2015
Previous experiments have revealed that shock waves driven through dissipative gases may become unstable, for example, in granular gases and in molecular gases undergoing strong relaxation effects. The mechanisms controlling these instabilities are not well understood. We successfully isolated and investigated this instability in the canonical problem of piston-driven shock waves propagating into a medium characterized by inelastic collision processes. We treat the standard model of granular gases, where particle collisions are taken as inelastic, with a constant coefficient of restitution. The inelasticity is activated for sufficiently strong collisions. Molecular dynamic simulations were performed for 30 000 particles. We find that all shock waves investigated become unstable, with density nonuniformities forming in the relaxation region. The wavelength of these fingers is found to be comparable to the characteristic relaxation thickness. Shock Hugoniot curves for both elastic and...
Aspects of planar, oblique and interacting shock waves in an ideal dissociating gas
Physics of Fluids, 2003
We develop a compact dimensionless framework for the analysis of canonical thermo-chemical nonequilibrium flow fields involving normal, oblique and interacting shock waves. Discontinuous solutions of the conservation equations are coupled with thermodynamic and kinetic models for an ideal dissociating gas. Convenient forms are provided for the variation of the relevant dimensionless parameters across shock waves in dissociating gases. The treatment is carried through in a consistent manner for the pressure-flow deflection angle plane representation of shock wave interaction problems. The contribution of the current paper is a careful nondimensionalization of the problem that yields a tractable formulation and allows results with considerable generality to be obtained.
Weak shock waves for the general discrete velocity model of the Boltzmann equation
Communications in Mathematical Sciences, 2007
We study the shock wave problem for the general discrete velocity model (DVM), with an arbitrary finite number of velocities. In this case the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this paper we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998. In this paper, we give a constructive, more straightforward, proof that suits the discrete case. Our approach is based on earlier results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points of general dynamical systems of the same type as in the shock wave problem for DVMs. The same approach can also be applied for DVMs for mixtures.