**1. Introduction**

Numerical simulation has become one of important approaches in academia and industry to research-complicated reactive flows in recent years, though it is still a fast developing subject. For combustion in low-speed flows, usually the low-Mach-number assumption is used, which assumes that the change of temperature is unrelated to the hydrodynamic pressure [1,2]. While for transonic or supersonic reactive flows, such as in the scramjet combustor, the compressi‐ ble effects could not be ignored and the set of fully compressible Navier-Stokes equations should be solved.

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There are three numerical simulation methods for compressible reactive flows via solving Navier-Stokes equations, including Reynolds-averaged Navier-Stokes (RANS) equations, large eddy simulation (LES), and direct numerical simulation (DNS). RANS simulates all scales of fluctuations using turbulent models. However, there is still lack of a general model suitable for all flow types, since RANS ignores the details and impacts of turbulent fluctuations. LES divides the flow scales into large and small scales via a low-pass filter. The large-scale flows are solved using the governing equations, and thus LES is able to give the temporal and spatial evolution of large-scale eddies. The small-scale flows are simulated using a sub-grid-scale (SGS) model, but there is also no general SGS model. DNS solves the Navier-Stokes equations for all scales of flows without introducing any models and assumptions, and thus it can accurately predict the full-scale flow characteristics. However, its main limitation in practice is the high cost of calculation.

In the early days, RANS or LES was used to simulate the compressible reactive flows consid‐ ering the computing cost. Dauptain et al. [3] simulated the supersonic hydrogen flame using the large eddy simulation, and their numerical results agreed well with the experiment measurements. LES was also used to simulate the supersonic flow and combustion in a model scramjet combustor [4], and the velocity and temperature at different cross-sections were compared with the experimental data. Edwards et al. used the hybrid method of LES/RANS to simulate the supersonic reacting wall-jet flow [5]. There are also other examples using RANS or LES [6–9]. Recently, some researchers have used DNS to simulate the compressible reactive flows. For example, O'Brien et al. [10] simulated the time-developing, H2/air-reacting mixing layers and investigated the dynamics of backscatter of kinetic energy using DNS; Diegelmann et al. [11] did the direct simulations of reactive shock-bubble interaction with detailed chem‐ istry under varying initial pressure.

One of the main difficulties in the simulation of compressible flows is how to distinguish the discontinuities which may widely exist in supersonic flows. Jiang and Shu [12] proposed the weighted essentially non-oscillatory (WENO) scheme to capture shock waves with almost no oscillations. But the WENO scheme is too dissipative in regions with large shear rates. To overcome this drawback, the hybrid scheme was hence developed, which switches the WENO scheme and the linear (high-order) scheme according to a discontinuity detector. Pirozzoli [13] proposed a hybrid compact-WENO scheme to obtain high resolutions for simulation of shockturbulence interaction. Furthermore, Ren et al. [14] proposed a Roe type, characteristic-wise compact-WENO scheme. Considering overcoming the complexity and program paralleliza‐ tion of compact schemes, Hu et al. [15] proposed a simple WENO scheme, which switches between the WENO scheme and its optimal linear scheme. A detailed review of numerical methods used for high-speed flows can be seen in [16].

The spray combustion can occur in the scenario of compressible flows [17–24]. Therefore, the methods to treat the dispersed droplets in the flows will be concerned by different approaches, such as Eulerian and Lagrangian methods. The Eulerian one needs complicated modelling of two-phase correlation terms, which is yet to be done. Even though the high computation cost will be taken by the Lagrangian method to track each particle released to the flow fields, it is the physically clear and mathematically simple method in simulations of two-phase reactive flows. This chapter will also consider the spray combustion in mixing layer flows as a com‐ putation example.

This chapter tries to contribute the numerical simulation in the direction of compressible reactive flows. It is organized as follows. In section 2, the governing equations are presented; the methods to calculate the multispecies physical parameters and to deal with the chemical stiffness are also presented. In section 3, the discretization schemes and time integration methods are discussed, including the discretization of inviscid/viscous terms. In section 4, three examples of numerical simulations of compressible flows are presented. In section 5, the conclusions are drawn finally.
