Gas-kinetic unified algorithm for hypersonic aerothermodynamics covering various flow regimes solving Boltzmann model equation

Abstract

How to solve hypersonic aerothermodynamics and complex flow mechanism covering various flow regimes from high rarefied free-molecular flow of outer-layer space to continuum flow of near-ground is one of the frontier basic problems in the field of fluid physics. In this work, the unified Boltzmann model equation based on the molecular velocity distribution function is presented for describing complex hypersonic flow transport phenomena covering all flow regimes by physics analysis and model processing of the collision integral to the Boltzmann equation. The discrete velocity ordinate method is developed to simulate complex flows from low Mach numbers to hypersonic flight, and the gas-kinetic coupling-iteration numerical scheme is constructed directly to solve the evolution and updating of the molecular velocity distribution function by employing the unsteady time-splitting method and the NND finite-difference technique. Then, the gas-kinetic unified algorithm (GKUA) is presented to~simulate the three-dimensional hypersonic aerothermodynamics and flow problems around space vehicles covering various flow regimes from free-molecule to continuum. To verify the accuracy and reliability of the present GKUA and simulate gas thermodynamic transport phenomena covering various flow regimes, firstly, the two-dimensional supersonic flows around a circular cylinder are simulated in the continuum regime of Kn= 0.0001 and in the high rarefied regime of Kn= 0.3 through the comparison between the Navier-Stokes (N-S) solution and the direct simulation Monte Carlo (DSMC) result, respectively. It is indicated that the GKUA can exactly converge to the N-S solution in the continuum flow regime, and the computed results of the GKUA are consistent with the DSMC simulation with a small deviation of 0.45% in the high rarefied flow regime. Then, the three-dimensional complex hypersonic flows around reusable satellite shape are studied as one of the engineering applications of the GKUA with a wide range 0.002 Kn 1.618 of the free-stream Knudsen numbers and different Mach numbers during re-entering Earth atmosphere with the flying altitudes of 110-70~km. The computed results are found to be in high resolution of the flow fields and in good agreement in a deviation range of 0.27%-8.56% by comparison among the relevant reference data, DSMC and theoretical predictions. The complex flow mechanism, flow phenomena and changing laws of hypersonic aerothermodynamics are revealed for spacecraft re-entry into the atmosphere, and the effects of rarefied gas and wall temperature on the aerothermodynamics characteristics of re-entry satellite shape are compared and analysed with different Knudsen numbers and wall temperature ratios of Tw/T = 1.6, 10 and 15.6. It is validated that the non-dimensional heat flux coefficient in the rarefied transitional flow regime is higher than that of the continuum and near-continuum flow regimes, the high wall temperature results in the enlarging amplitude of temperature variation on the stagnation line and the serious effect on the heat flux of the stagnation point, and wall temperature becomes lower, the heat flux coefficient of wall surface becomes bigger, and the friction force and pressure coefficients decrease. The non-equilibrium level of flow velocity slip and temperature jump on the surface of space vehicle becomes severer, and the stronger heat transfer effect between the space vehicle and the gas flow is produced as the Mach number or Knudsen number of the free-stream flow increases. It can be realized from this study that the gas-kinetic unified algorithm directly solving the Boltzmann model velocity distribution function equation may provide an important and feasible way that complex hypersonic aerothermodynamic problems and flow mechanisms from high rarefied free-molecule to continuum flow regimes can be solved effectively and reliably.