Multiple temperature kinetic model and gas-kinetic method for hypersonic non-equilibrium flow computations

It is well known that for increasingly rarefied flowfields, the predictions from continuum formulation, such as the Navier–Stokes equations lose accuracy. For the high speed diatomic molecular flow in the transitional regime, the inaccuracies are partially attributed to the single temperature approximations in the Navier–Stokes equations. Here, we propose a continuum multiple temperature model based on the Bhatnagar–Gross–Krook (BGK) equation for the non-equilibrium flow computation. In the current model, the Landau–Teller–Jeans relaxation model for the rotational energy is used to evaluate the energy exchange between the translational and rotational modes. Due to the multiple temperature approximation, the second viscosity coefficient in the Navier–Stokes equations is replaced by the temperature relaxation term. In order to solve the multiple temperature kinetic model, a multiscale gas-kinetic finite volume scheme is proposed, where the gas-kinetic equation is numerically solved for the fluxes to update the macroscopic flow variables inside each control volume. Since the gas-kinetic scheme uses a continuous gas distribution function at a cell interface for the fluxes evaluation, the moments of a gas distribution function can be explicitly obtained for the multiple temperature model. Therefore, the kinetic scheme is much more efficient than the DSMC method, especially in the near continuum flow regime. For the non-equilibrium flow computations, i.e., the nozzle flow and hypersonic rarefied flow over flat plate, the computational results are validated in comparison with experimental measurements and DSMC solutions.