Linearized Stability Analysis And Numerical Computation Of Burnett Equations In Hypersonic Flow

For computation of rarefied flows in continuum-transition regime with Knudsen number Kn of O(1), Burnett equations, which were originally derived in1936, have been proposed about a century ago as a set of extended hydrodynamics equations (EHE) that represent the second-order departure from thermodynamic equilibrium in the Chapman-Enskog expansion of Boltzmann equation. Although a number of variations of Burnett equations have been proposed to overcome the problems of instability under small perturbations and boundary conditions in the literatures known as the Conventional Burnett (CB) equations, the Augmented Burnett (AB) equations and the BGK-Burnett equations over the years, the Burnett equations are still hard to be applied successfully to solve a number of rarefied flow problems in continuum-transition regime.In this paper, another simpler set of conventional Burnett equations, which are designated as simplified conventional Burnett (SCB) equations, is proposed by order of magnitude analysis in the limit of high Mach numbers for hypersonic flow applications. We perform the linearized stability (known as the Bobylev Stability) analysis of one-dimensional Burnett equations including rotational temperature and three-dimensional Burnett equations for all variants in the literatures on this subject. By introducing small perturbations in the steady state flow field, the trajectory curve and the variation in attenuation coefficient with wave frequency of the characteristic equation are obtained for all variants of Burnett equations to determine their stability. The results show that SCB equations are unconditionally stable under small wavelength perturbations and are much simpler to solve numerically compared to AB equations without compromising accuracy.The1D shock structure and quasi1D Couette flow are simulated using variant Burnett equations to validate the high-order constitutive relationships and boundary conditions. Through the comparison with the flow variables of DSMC method and experiments, SCB calculations are validated in1D flow and the1st order Maxwellian slip boundary conditions are employed considering the computational efficiency and stability.An implicit numerical solver is developed for the solution of3D SCB equations for calorically perfect gas. The SCB equations are applied to compute the hypersonic flow past2D and3D blunt bodies in continuum and continuum-transition regime. To simulate the rotational non-equilibrium effect in a diatomic gas, both the Navier-Stokes (NS) and the SCB equations are modified by including a rotational non-equilibrium relaxation model. The results of SCB solutions are compared with the NS and DSMC solutions. There is no significant difference between SCB and NS calculations in continuum regime, which demonstrates that the applicable range of SCB equations covers that of continuum method. It is also shown that the SCB calculations are in close agreement with the DSMC results in hypersonic rarefied flow.Considering the high temperature effect encountered in hypersonic flow, the energy exchange between transitional and vibrational energy is considered in3D SCB equations to simulate accurately. The flow variables (density, translational and vibrational temperature, Mach number and species) for flow past2D and3D blunt bodies in literatures are simulated using SCB and NS equations simultaneously and compared in continuum-transition regime.