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. Even with the inclusion of higher-order terms, such as Burnett or high-order moment equations, a single temperature assumption is still assumed, which limits their success in the flow applications in the transition regime. 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 derived macroscopic equations are different from the standard Navier-Stokes-type continuum formulation, where the second viscosity coefficient is replaced by the temperature relaxation term. Also, due to the introduction of the rotational temperature, one more governing equation for the rotational energy evolution is introduced. In the continuum flow regime, where the particle collision time is much smaller than the characteristic time scale, the generalized macroscopic governing equations go back to the standard Navier-Stokes forms. 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. The advantage of developing a multiscale method is that the flow physics in the microscopic description is much simple and can be easily implemented in a numerical scheme. For example, the slip in velocity and temperature can be automatically obtained in the multi-scale gas-kinetic method through the modeling of gas-solid surface interaction, and the non-equilibrium effect can be captured through the generalization of particle collision time. 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. In other words, the multiscale kinetic scheme is much more efficient than the DSMC method, especially in the near continuum flow regime. This paper concentrates on the non-equilibrium flow computations, such as the nozzle flow and hypersonic rarefied flow over flat plate for diatomic gases. The computational results are validated in comparison with experimental measurements and DSMC solutions. Since the gas-kinetic scheme presented in this paper for the multiple temperature model has the similar efficiency as the the standard Navier-Stokes method, it provides an indispensable new tool for the study of non-equilibrium flow, especially in the continuum transition flow regime.
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