**5. Conclusions**

This chapter introduced the state of art of the numerical simulation on compressible flows. The high-order hybrid WENO scheme, as one of the important issues for simulating the supersonic flows, is required to compute the numerical fluxes, in which the WENO scheme is used near the discontinuities and linear scheme is used in the smooth regions of the flows. In the solver developed by the present authors, the point-implicit method is utilized to treat the chemical stiffness problem.

A TGV problem, defined to investigate the transports of passive scalar, is simulated by the present numerical procedures. The decaying of the kinetic energy and variance of passive scalar due to initial gradients are analyzed based on the numerical results. The spatially developing mixing layer flows are simulated, taking the hydrogen and liquid *n*-decane droplets as fuel and air as oxidizer in the scenario of supersonic inflows. The stable combustion of hydrogen/air and the flame structure are analyzed by the numerical results. The ignition considering the eddy turnover effects is carefully analyzed, since the large-scale structures and droplet dispersion can be captured by the present numerical methods.

This chapter tries to contribute the field of numerically analyzing compressible reactive flows. The new high-order schemes and numerical strategy will be developed in the future, so that the stabilities and robustness will be enhanced by the solver based on the Navier-Stokes equations. Thus, deeply understanding the flow physics of compressible reactive flows will be benefited from results obtained by numerical simulations.
