4. Conclusions

coupled constitutive relation (NCCR) models are compared with DSMC data, which are generated by assuming full tangential momentum and thermal accommodation for slip and

Detailed comparisons of normalized density contours of hypersonic rarefied case Kn = 0.5 [16] are presented in Figure 4. The results of the case Kn = 0.5 show that the density contours and the stand-off shock structure predicted by the NCCR model and the DSMC are in excellent agreement, even in this high transitional regime. On the other hand, the thickness of stand-off shock structure predicted by the first-order NSF model is much smaller than that of the secondorder NCCR model and DSMC. In addition, the degree of gaseous expansion near the rear part of the cylinder predicted by the NSF model is considerably higher than that of the NCCR model and DSMC. On the whole, the results of the second-order NCCR model show better agreement

with DSMC data than the first-order NSF results in hypersonic rarefied cases studied.

As the final test case, the three-dimensional hypersonic gas flows around a suborbital re-entry vehicle, Intermediate eXperimental Vehicle (IXV) of the European Space Agency (ESA), were investigated. The computational domain is defined by unstructured meshes; tetrahedron elements of 978,445 in this three-dimensional case. The flow conditions for the hypersonic case are M = 5.0, Kn = 0.02, and an angle of attack 15 degree. Comparisons of normalized density and Mach number contours are presented in Figures 5 and 6. On the whole, there seems not much substantial difference between numerical solutions of the first-order and second-order constitutive models, since the degree of nonequilibrium is not high. However, it can be observed from the Mach number contours that some nonequilibrium effects begin to show up in the bow shock structure and in the rear part of the vehicle where rapid expansion occurs. Besides these findings, the present results demonstrate that the three-dimensional numerical simulation of the second-order constitutive model is possible for hypersonic rarefied flows like

Figure 5. Normalized density fields and contours of the three-dimensional hypersonic gas flows around a suborbital

jump boundary conditions.

16 Advances in Some Hypersonic Vehicles Technologies

re-entry vehicles with complicated geometry.

re-entry vehicle, M = 5.0 and Kn = 0.02.

A systematic derivation of the second-order constitutive equations from the kinetic Boltzmann equation is presented. The core frameworks employed in developing the thermodynamicallyconsistent constitutive models are a modified moment method, called Eu's generalized hydrodynamics, and the new closure theory, called balanced closure, recently developed by Myong. Then, multi-dimensional computational models of the second-order constitutive equations are developed. The core concepts used in developing the models are the decomposition and the method of iterations. Further, as the basic computational scheme to efficiently solve the conservation laws together with the second-order constitutive equations, a mixed explicit modal DG method is developed. In order to assess the potential of the new computational model in hypersonic flow regimes, several flow problems, including the one-dimensional shock structure and three-dimensional hypersonic gas flows around a suborbital IXV re-entry vehicle, are numerically simulated. On the whole, the new second-order model is found to enhance considerably the prediction capability of hypersonic rarefied flows in comparison with the conventional first-order model.

## Acknowledgements

This work was supported by the National Research Foundation of Korea funded by the Ministry of Science and ICT (NRF 2017-R1A2B2007634), South Korea. The author gratefully acknowledges the contributions of graduate students S. Singh, P. Raj, and A. Karchani for producing Figures 2, 5, and 6.
