All numerical cases were simulated in a finite difference framework using an in-house code developed by the authors. The three-dimensional chemical non-equilibrium NS equations in Cartesian coordinates are as follows [3]:

(1) ∂U∂t+∂E∂x+∂F∂y+∂G∂z=∂Ev∂x+∂Fv∂y+∂Gv∂z+S

where:

(2) U=(ρρuρvρwEρYs),E=(ρuρu2+Pρuvρuw(E+p)uρuYs),Ev=(0τxxτxyτxzuτxx+vτxy+wτxz+q˙xρDms∂Ys∂x),S=(00000ω˙s)

(3) τxx=−23(μl+μt)(∇⋅V⇀)+2(μl+μt)∂u∂x

(4) τxy=(μl+μt)(∂u∂y+∂v∂x)

(5) q˙x=(Kl+Kt)∂T∂x+ρ∑⁡Dmshs∂Ys∂x

In the above, u, v, w are x, y, z components of velocity, respectively. p, ρ , Y_i are the pressure, density, and mass fraction of species s, respectively. E is the total internal energy. Dms and hi are the diffusion coefficients of the species, s, and absolute enthalpy per unit mass, respectively. μl and μt are the viscosity coefficients for laminar and turbulent flows, respectively. Kl and Kt are the heat transfer coefficients for laminar and turbulent flows, respectively. A hydrogen/air reaction mechanism including 7 species (H₂, O₂, H₂O, H, O, OH, and N₂) and 8 elementary reactions [4] are used in this study.

The source term ω˙s is given with a finite rate chemical model as follows:(6) ω˙s=Ms∑j=1nr⁡(βjs−αjs)(Rj−R−j) where Ms is the molecular weight of the species, s, αjs and βjs are the stoichiometric coefficients of the related species in the jth reaction. Rj and R−j are the forward and backward net rates of the jth reaction, respectively:(7) Rj=Kj∏s=1ns⁡(ρsMs)αjs,R−j=K−j∏s=1ns⁡(ρsMs)βjs where Kj is the forward rate coefficient of the jth reaction, and K−j is the backward rate coefficient of the jth reaction.

The static enthalpy and the specific heat at constant pressure are considered with temperature and species-dependent thermodynamic properties as follows:(8) h=∑s=1ns⁡hsYs,Cp=∑s=1ns⁡Cp,sYs where hs and Cp,s are respectively the enthalpy and specific heat at a constant pressure of species, s, modeled with a polynomial function of static temperature.

Thermal conductivity and molecular viscosity are calculated using the Sutherlands formula for each species. The Wilkes formula [5] is then applied to calculate the mixture’s thermal conductivity and molecular viscosity. Moreover, mass diffusivity is calculated according to the kinetic theory as follows [6]:(9) ρDs=1−Ys1−XsμSc where Xs is the mole fraction of species s.

The total energy per unit mass of mixture E and the total enthalpy per unit mass of mixture H are defined as follows:(10) H=h+12ujuj,E=H−pρ

The high-order scheme has the advantage of less numerical dissipation in describing the flow field with high resolution and has less dependence on the grid for the same numerical accuracy. Meanwhile, both the accuracy and robustness of the scheme are required in engineering problems, making the third-order scheme a good compromise. Numerical verifications are conducted for a class of third-order schemes for engineering applications on different problem types [7]. In this study, the third-order procedures are applied to the numerical simulation of supersonic combustion to capture the flow-field structure and investigate the scheme’s rationality.

The scheme is introduced in the form of one-dimensional hyperbolic equations as follows:(11) ∂u∂t+∂f(u)∂x=0

Consider the conservative difference scheme and write it in a semi-discrete form:(12) ∂u∂t=−hj+12−hj−12Δx

The positive flux portion with the third (or higher)-order scheme in the linear condition can be written as follows:

(13) hj+12+=α0hj+12+0+α1hj+12+1+α2hj+12+2

where:

(14) hj+12+0=−12fj−1++32fj+

(15) hj+12+1=12fj++12fj+1+

(16) hj+12+2=32fj+1+−12fj+2+

(17) α0=13−α2

(18) α1=23

where α2 is an adjustable free parameter in the form of α2=16(n3rd−1n3rd) and n3rd is a control parameter. The numerical dissipation of the scheme can be adjusted by changing the value of n3rd . For example, n3rd=1 is equal to the traditional third-order WENO [8] or the WNND scheme, n3rd=1.25 is equal to the linear fifth-order WENO scheme, and n3rd→∞ is equal to the classic fourth-order center scheme.

To capture the discontinuous structure (such as the shock wave), however, the nonlinear mechanisms are needed to increase scheme dissipation in the weighted form:(19) hj+12+=∑i=02⁡ωihj+12+i where:

(20) ωi=Ωi∑k=02⁡Ωk,Ωi=αi(ε+ISi)p

(21) ISi=(fj+i+−fj−1+i+)2

A continuous variational rescale technology is proposed [7] by changing the index p. For low-speed problems: ε=10−2−10−5 , for high-speed problems: ε=10−5−10−12 .

As shown in previous investigations, the turbulence model is crucial in predicting the flow-field variables and the interaction between turbulence and combustion for supersonic combustion simulations. Menter’s shear stress transport (SST) model [9] has extensive applications in engineering and was used in this study due to its good performance with lower computational cost and more numerical robustness [9]. The SST turbulence model introduces the original model inside the boundary layer region and switches into the standard model outside the boundary layer region and in free shear flows [9]. The transport equations used for the SST turbulence model take the following form:

The governing equations of the turbulence model are as follows:

(22) ∂(ρk)∂t+∂(ρujk)∂xj=Pk−β∗ρωk+∂∂xj[(μ+σkμt)∂k∂xj]

(23) ∂(ρω)∂t+∂(ρujω)∂xj=Pω−βρω2+∂∂xj[(μ+σωμt)∂ω∂xj]+2(1−F1)ρσω2ω∂k∂xj∂ω∂xj

Details of the equations can be found in reference [9]. The decoupling method [10] is applied, implying that NS and turbulence equations are solved.

The finite difference method is used in this study, and the compressible NS equations are discretized on a single structured grid. Based on its robustness and high resolution of stationary discontinuity, the nonoscillatory nonfree-parameters dissipative difference scheme (NND) [11] is adopted for the inviscid flux vectors. The viscous flux vectors were calculated using a second-order central difference scheme. The implicit LUSGS method is employed for time marching. In order to meet the computational requirements and improve computational efficiency, a massage-passing interface is introduced for a massive parallel computation. Because of the single-block body-fitted grid, the load balancing problem can be solved simply by splitting the grid equally.

Lehr’s experiment [12] provided experimental schlieren diagrams of a supersonic combustor, flow-field temperature distributions, and experimental temperature measurement results. This is the most commonly used example for testing numerical calculation methods and combustion models in supersonic combustion simulations. Shock-induced combustion phenomena, ranging from decoupled shock-deflagration systems to overdrive oblique detonation waves, were experimentally investigated by Lehr’s experiment [12]. The flux splitting scheme, NND, is tested under the experimental conditions listed in Table 1 to estimate its accuracy and robustness on shock capture for supersonic combustion simulations.

First, the grid sensitivity analysis is studied to eliminate the influence of grid errors. Three grid spacing scales are then adopted to represent the grid’s independence: coarse, medium, and fine grids. The detailed grid information and the distribution of the medium grid are shown in Table 2 and Δ is the first grid-scale in the wall-normal direction.

Fig. 1 shows the temperature contours of the decoupled shock-deflagration overlaid on the shock location [13] obtained from the numerical results of the medium grid. Fig. 2 shows the distribution of the temperature along the X direction on the stagnation line. Combustion does not begin immediately behind the shock wave, as depicted. Instead, the combustion front appears to be separated from the shock wave by an induction zone. This behavior can also be observed clearly from the stagnation line temperature in Fig. 2. The numerical results for the three sets of grids are nearly identical in the upstream region. As the flow and combustion process develops after the detonation wave, the medium and fine grids remain constant, whereas the coarse grid begins to diverge with time.

The numerical method predicts an accurate ignition position as the temperature distribution is consistent with the experiment data. Thus, the numerical implementation can then be applied to supersonic combustion simulations.

The structure problem between fully mixed hydrogen and oxygen is discussed in the following part, and the combustion is proposed at x=1 . The freestream flow conditions are M = 2.5, T = 1500K, p = 1, bar, Y_H2 = 0.1, and Y_O2 = 0.9, and the numerical results are shown in Figs. 3–5:

Two important parameters related to the combustor temperature and pressure are given in [1], θ=T4∗T2∗ and σ=p4∗p2∗ (* represents the stagnation point), in which “2” represents the combustor’s inlet and “4” represents its outlet. The nondimensional outlet pressure and temperature are: P4=0.2608892andT4=2.124833 . Thus,

(24) θ=T4∗T2∗=T4(1+γ−12M42)T2(1+γ−12M22)=1.416623

(25) σ=P4∗P2∗=P4(1+γ−12M42)γγ−1P2(1+γ−12M22)γγ−1=0.3414326

where 1<θ<n=T4T2=2.124833 and 0<σ<0.6 . A small detonation wave occurs near the flow field’s discontinuity, followed by supersonic combustion all over the field. Therefore, supersonic combustion can be achieved using the detonation wave under appropriate flow conditions according to the theoretical analysis in [1].

It has been demonstrated that the streamwise vortex can enhance fuel mixing and improve the supersonic combustion efficiency from both flow mechanism and numerical simulations in [1,12,14,15]. Thus, a more detailed numerical simulation is conducted to investigate the effect and influence of streamwise vortex couples on supersonic combustion in this study.

Fig. 6a shows the sketch maps of the combustor and fuel injector. The length, width, and entrance height of the combustor are 170, 31, and 12.2 cm, respectively. The expansion angle on the top wall is 1°. The injection device consists of an expansion surface and two wedges on the expansion surface. Fig. 6b shows that the wedge has a thickness of 5 cm and a compression surface of 5° in the first 11 cm of the wedge, while the remaining 11-cm in length expands to the underside of the wall. The two wedges are arranged in the middle, with a 7-cm gap between them. Two injection holes are ranged on the lower surface of the wedge, from which hydrogen is ejected at an angle of 12° along the flow direction. The computational mesh is arranged 11 cm to the region upstream of the wedges to resolve the turbulent boundary layer.

A single-block body-fitted grid is used for the calculation, and compression is performed along the wall directions to meet the requirements of the viscous boundary layer for NS equations. The grid is taken as 360×181×167 ( streamwise×normal×spanwise ), with a total number of about 11 million, and the distribution near injection holes is shown in Fig. 7. The first grid-scale in the normal direction of the wall is set as 0.001 mm based on the empirical grid setting strategy [3,16]:

Parallel computing using the aforementioned third-order scheme and the SST turbulence model is implemented with 30 CPUs under the following conditions (Table 3):