This paper addresses the oblique detonation engine intake as the object of study (Fig. 1); the study selected air incoming Mach numbers of 5, 7, and 9, respectively. The calculation area as shown in Fig. 2, the red dotted line within the region, the coordinates of the calculation domain to match the schematic, the direction of air incoming flow for the positive direction of the x-axis, and the y-axis is perpendicular to the direction of air incoming flow up.

This paper addresses the fuel mixing characteristics of a slant-detonation engine intake in a restricted space and the structural complexity in engineering applications. The model is simplified, retaining the main structure of the intake tract, and constructed in two dimensions.

In order to ensure the grid quality and orthogonality, the blank area in the lower part of the wall in the red dashed box in Fig. 1 is used as the computational domain, as shown in Fig. 2. The physical model is 620 mm in total length, 450 mm in x-direction projection of the first wedge, and 150 mm in x-direction projection of the second wedge, and a 20 mm long computational domain is set at the front of the inlet to reflect the more realistic hypersonic incoming flow, considering the escape of the incoming air into the inlet. The angle between the first wedge surface and the *x*-positive direction *θ**1* is 15°, and the angle between the second wedge surface and the x-positive direction *θ**2* is 30°. In the study of the mixing of hydrogen and air with different parameters, the control variable method is used to change one of the parameters of hydrogen-to-oxygen equivalent ratio, incoming Mach number, hydrogen nozzle position, and hydrogen injection angle, and keep the other parameters constant.

When studying the degree of fuel blending, it is necessary to consider the basic parameters of the operating environment (30 km altitude): the atmospheric temperature is T0 = 226.51 K, and the atmospheric pressure p0 = 1197.0 Pa. The sound velocity of the air in this operating environment can be calculated according to the following Eq. 1:

$${\alpha }_{air}=\sqrt{r\cdot R{g}_{air}\cdot {T}_{0}}$$

1

Where *r* is the specific heat ratio of diatomic molecules, i.e., 1.4; RgAir =R/M is the gas constant of air, R = 8.314 J/(mol-K) is the ideal gas constant, and M = 0.02896 kg/mol is the molar mass of air. The specific parameters are shown in Table 1.

Table 1

Atmospheric parameters at 30 km

H[km] | T0 [K] | P0 [Pa] | ρ0 [kg/m3] | α[m/s] |

30 | 226.51 | 1197.0 | 1.84E-2 | 301.68 |

Depending on the incoming Mach number and the angle of hydrogen injection, the static pressure at the hydrogen nozzle and the static temperature will also vary, and the static pressure at the hydrogen nozzle will be different when using single, double, and triple hydrogen nozzles, which is calculated using the controlled flow method, i.e., the air flow mass \({\dot{m}}_{air}\) in the inlet tract is controlled by the Mach number of the incoming airflow Mair for a given inlet tract cross-section Lair, the required flow rate at the hydrogen nozzle is given by the number of moles of hydrogen required for the hydrogen-oxygen reaction and combined with the equivalence ratio ER\({\dot{m}}_{H2}\), and thus the total fuel pressure required in the sonic throat is calculated as

$${\dot{m}}_{air}=\rho \cdot {\alpha }_{air}\cdot {M}_{air}\cdot {L}_{air}$$

2

$${\dot{m}}_{max}=\sqrt{\frac{r}{R}}{\left(\frac{2}{r+1}\right)}^{\frac{r+1}{2(r-1)}}\frac{{p}_{0}}{\sqrt{{T}_{0}}}{A}^{*}=0.01062\frac{{p}_{0}}{\sqrt{{T}_{0}}}{A}^{*}$$

3

Where \({\dot{m}}_{max}\) is the maximum hydrogen fuel flow rate of the sonic throat, **r** and **R** are the adiabatic index and gas constant of the fuel, respectively, p0 and T0 are the total pressure and total temperature of the fuel, respectively, and\({A}^{*}\) is the normal area of the acoustic throat, i.e., the hydrogen nozzle, and since the model is a two-dimensional model, the\({A}^{*}\) is replaced by LH2. By changing the definition of the incoming Mach number and the hydrogen injection angle, the static pressure of the hydrogen nozzle can be obtained for different incoming Mach numbers and injection angles. The conversion equations of static pressure to total pressure and static temperature to total temperature are shown in Eqs. 4 and 5.

$$\frac{P}{{P}_{0}}={\left(1+\frac{r-1}{2}{M}_{a}^{2}\right)}^{-\frac{r}{r-1}}$$

4

$$\frac{T}{{T}_{0}}={\left(1+\frac{r-1}{2}{M}_{a}^{2}\right)}^{-1}$$

5

The hydrostatic pressure at the hydrogen nozzle for an equivalent ratio of 1.2 is calculated as shown in Table 2, and the angle of injection is the angle with the positive direction of the x-axis.

Table 2

Static pressure at the hydrogen orifice at an equivalent ratio of 1.2

Spraying Structure | incoming flow Mach number [Ma] | Spraying Angle | Static pressure [Pa] |

Single spray | 5 | 90° | 89367.9 |

| 7 | 45°,165° | 241699 |

| | 90° | 125115 |

| 9 | 90° | 160862 |

Double spray | 7 | 35°,175° | 176670 |

| | 55°,155° | 94004.5 |

| | 90° | 62557.5 |

Triple spray | 7 | 30°,180° | 155642 |

| | 45°,165° | 80566.3 |

| | 60°,150° | 56969.3 |

| | 90° | 41705 |

The calculated hydrostatic pressure at the hydrogen orifice for an equivalent ratio of 4 is shown in Table 3.

Table 3

Static pressure at the hydrogen orifice at an equivalent ratio of 4

Spraying Structure | incoming flow Mach number [Ma] | Spraying Angle | Static pressure [Pa] |

Single spray | 5 | 90° | 297659 |

| 7 | 45°,165° | 569254 |

| | 90° | 416723 |

| 9 | 90° | 535787 |

Double spray | 7 | 35°,175° | 350889.5 |

| | 55°,155° | 245695.5 |

| | 90° | 208461.5 |

Triple spray | 7 | 30°,180° | 268349 |

| | 45°,165° | 189751.33 |

| | 60°,150° | 154931.33 |

| | 90° | 138907.67 |

In this paper, we study the mixing of hydrogen in the supersonic incoming flow in the two-dimensional viscosity-free case, and we need to verify whether the mixing process occurs with combustion, so the controlling equation of the flow is the Euler equation with coupled chemical reaction source terms. The equations are in the form of Eqs. 2–7 as follows.

$$\frac{\partial U}{\partial t}+\frac{\partial F}{\partial x}+\frac{\partial G}{\partial y}=S$$

6

Among them.

$$U=\left[\begin{array}{c}{\rho }_{1}\\ \begin{array}{c}⋮\\ \begin{array}{c}{\rho }_{n}\\ \rho u\end{array}\\ \rho v\end{array}\\ E\end{array}\right]\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{F}=\left[\begin{array}{c}{\rho }_{1}u\\ \begin{array}{c}⋮\\ \begin{array}{c}{\rho }_{n}u\\ \rho {u}^{2}+p\end{array}\\ \rho uv\end{array}\\ \left(E+p\right)u\end{array}\right]\hspace{0.33em},$$

$${G}=\left[\begin{array}{c}{\rho }_{1}v\\ \begin{array}{c}⋮\\ \begin{array}{c}{\rho }_{n}v\\ \rho vu\end{array}\\ \rho {v}^{2}+p\end{array}\\ \left(E+p\right)v\end{array}\right]\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{S}=\left[\begin{array}{c}{\omega }_{1}\\ \begin{array}{c}⋮\\ \begin{array}{c}{\omega }_{n}\\ 0\end{array}\\ 0\end{array}\\ 0\end{array}\right].$$

7

The subscript i represents the components in the primitive reaction, and n *is the* number of components; ρ is the mixture density, which can be expressed as the sum of the component densities ρi. u and v *are the* airflow velocities in the *x* and *y* directions, respectively, and p is the pressure. E is the total energy per unit volume, which can be expressed as

$$E={\rho }\text{h}-\text{p}+\frac{1}{2}{\rho }\left({\text{u}}^{2}+{\text{v}}^{2}\right)$$

8

where h *is the* specific enthalpy. Where Ru is the universal gas constant, and W**i** is the molar mass of component i. ω**i** is the mass production rate per unit volume of the *i-th* component.

For the engine to work properly, in addition to considering whether there will be premature detonation in the combustion chamber, it is equally necessary to investigate whether in the intake tract, where the incoming air and hydrogen jets have been compressed by two wedge surfaces, the rise in the post-wave temperature of the oblique excitation wave formed by the mutual coupling of the two and the bow-shaped excitation wave formed by the two will cause premature fuel combustion to occur. The 9-component 19-reaction of the Jachimowsk reaction model[26] was used in testing the presence of premature combustion in the intake tract. The product of combustion of hydrogen with oxygen in water, so it is possible to observe, based on the molar fraction distribution of each component in the calculated results, whether there is premature combustion. The results are shown in Fig. 3. In the whole flow field, components OH and H2 O are not present, which can indicate that this model does not occur in the intake tract of the possibility of premature combustion.

In this paper, the above equations are solved by a second-order TVD-type finite volume method, where the HLLC (Harten-Lax-van Leer Contact) approximate Riemann solver for the interface fluxes is used for calculation. The time advance is performed by the fourth-order Runge-Kutta method, and the CFL number controls the time step.

In order to verify the grid independence, three grids with different coarse and fine mesh numbers of 570000, 2280000, and 4560000 are used under the same initial conditions. From their pressure and temperature clouds, the results obtained from numerical simulations using grids with two grid resolutions of different coarseness and fineness are the same, so the small grid model is sufficient for calculation.