Flow Coefficient and Starting Performance Prediction of Variable Geometry Curved Axisymmetric Inlet

Abstract

With the development of combined cycle engines, it is urgent to estimate more quickly and accurately the flow capture capacity and starting performance of variable geometry inlets over a wide Mach number range. Based on the flow field and parameter fitting, two prediction methods for the curved axisymmetric inlet with lip translation scheme have been proposed. The method based on the flow field of the reference inlet is more efficient than the parameters-based prediction method, as it can accurately predict the lip translation distance and the corresponding flow coefficient over the entire working range of the inlet without additional numerical calculations. Moreover, the starting Mach number is accurately predicted by the fitting method based on the throat Mach number of the reference inlet, with a relative error of only 0.95% compared to the numerical simulation. The flow coefficient-based method is simple and accurate for predicting lip translation distances with a known starting Mach number, with a relative error of only 1.65% compared to numerical simulations. The prediction approaches can overcome the drawbacks of the standard iterative algorithms and significantly enhance computational accuracy and efficiency.

1. Introduction

With the continuous expansion of hypersonic flight speed and airspace, the combined-cycle engine has become a research hotspot, with examples including the Rocket Based Combined Cycle engine (RBCC) [1], the Air Turbo Rocket engine (ATR) [2,3], the Turbine Based Combined Cycle engine (TBCC) [4,5,6], the Pre-cooled Cycle engine [7,8,9,10], and so on. The inlet is one of the core components of the combined cycle engine, and its operating range needs to be further extended. In this case, a fixed geometry inlet can neither take into account the performance requirements of high Mach numbers and the starting requirements of low Mach numbers at the same time nor satisfy the engine flow requirements over a wide range of Mach numbers. Therefore, the adoption of variable geometry in the design of inlets is an inevitable trend [11,12,13,14].

The axisymmetric inlet has good engineering design advantages due to its simple structure, convenient manufacturing, and high utilization of windward areas. Since the 1950s, many studies have been carried out and the results have been successfully applied to hypersonic vehicles, such as the US Hypersonic Research Engine (HRE) [15], the Hyfly project [16], and the CIAM-NASA Scramjet Flight Program [17]. Xie et al. [18] systematically introduced the design method of the axisymmetric inlet and conducted wind tunnel experiments. Wang et al. [19] carried out a multi-objective optimization design of the axisymmetric inlet profile. Due to the narrow operating range of the engine at the early stage, the studies focused on the fixed geometry inlets, while for the later-developed engine with a wide operating range, especially combined cycle engines, the variable geometry inlet was required.

Most variable geometry inlets adopt the scheme of moving the center cone, such as the classic TBCC engine inlet of J58 [20], the RBCC engine inlet of the GTX plan [21], and the SABRE (Synergistic Air-Breathing Rocket Engine) single-cone multistage adjustable axisymmetric inlet [22]. Maru et al. [23] proposed a multi-row disk (MRD) inlet which can automatically change the distance between multi-row conical disks to ensure that the shock wave intersects at the lip under different inflow Mach numbers, but there are extra losses between the disks. Kojima et al. [24] used a movable cone to control the conical shock before the inlet in the air-turbo ramjet expander-cycle engine (ATREX) to keep the throat Mach number at about 1.3, thereby facilitating high compression performance. Cheng et al. [25] proposed a variable geometry inlet concept combined with local sub-flow circulation which can improve the flow capture capability of the inlet at off-design points and keep the total pressure recovery coefficient unchanged. Yuan et al. [26] proposed a three-dimensional aerodynamic design scheme of the inlet operating at Ma_i = 1.5–4.0 and control laws of the bleed slots are gained. Cai et al. [27] proposed an axisymmetric adjustable inlet that can meet both flow distribution and compression requirements of two channels in a wide working range for the precooled combined-cycle engine. The inlet is controlled by the coordinated translation of the central cone and the splitter plate. Nan et al. [28] performed geometric adjustment by moving the central cone to match the aerodynamic compression surface so as to achieve full flow capture.

Most of the above axisymmetric inlets are designed using the traditional straight-cone method, so the performance of the off-design points is significantly decreased. Recently, Zhang and co-workers [29] proposed a new design method of a curved compression inlet and carried out a series of experiments. The results show that the inlet performs well in the whole operating range. Under the condition of given Mach number distribution, Li et al. [30] used the method of characteristics to inversely design the axisymmetric inlet with good performance. In addition, the variable geometry inlet usually has a complex adjustment structure, which will induce sealing and cooling problems and add additional structural mass. A new variable geometry scheme with simple control and convenient adjustment, called the lip adjustment scheme, was developed to overcome the shortcomings. It can not only ensure the self-starting performance under a low Mach number but also can easily adjust the flow coefficient [30,31,32].

In this paper, the methods for fast and accurate prediction of the flow capture capability and starting performance of the inlet are studied for the curved compression axisymmetric inlet with lip translation over a wide range of Mach numbers. The structure of the flow field in the external pressure section of the traditional binary multi-wedge inlet is simple, and the inviscid calculation can be carried out through the shock relations, which yield lip-adjusted positions corresponding to the desired flow coefficients. The flow fields behind conical and curved shocks are too complex to be calculated by those methods, so the iterative trial calculation method is used [33]. In addition, there are barely published literature reports about the prediction of the starting performance of the curved compression axisymmetric inlet [34]. Therefore, the prediction methods based on the flow field and parameters fitting are proposed; these are verified by numerical simulation. It is shown that the prediction accuracy and computational efficiency are very high and can meet prediction requirements of the flow and starting performance of the variable geometry inlet over a wide range of Mach numbers.

2. Aerodynamic Design of the Reference Inlet

The reference inlet (named Inlet1), which is the reference configuration of the variable geometry axisymmetric inlet in the range of Ma_i from 3.0 to 7.0, was designed. The design Mach number is Ma_i = 6.0 and the external pressure section was inverse-designed with the arctangent Mach number distribution law. This distribution law can weaken the intensity of the leading edge shock and increase the isentropic compression ratio, thereby improving the overall performance of the inlet. Based on the parameter studies of the external pressure section and the internal contraction section, a set of optimized parameters was selected to complete the design of the reference inlet, as shown in

Table 1. Design parameters of the reference inlet.

Parameters	Value	Unit
Intake capture radius; Rc	200.0	mm
Semi-cone angle of the leading edge; δc	7.6	°
Total compression angle; δt	21.0	°
Inner angle of the lip; θi	12.0	°
Total length of the inlet; Lt	978.0	mm
Length of the external pressure section; Lc	730.0	mm
Height of the throat; Hth	16.0	mm
Length of the isolator; Liso	112.0	mm
Total contraction ratio; Rct	6.01	-
Internal contraction ratio; Rci	1.71	-

. The structure and wave system of the reference inlet is shown in Figure 1, where a simple lip translation scheme is used and the green part can be moved. Compared with the central cone movement scheme, this scheme is simpler and needs a smaller driving force [30].

3. Numerical Calculation Method

3.1. Numerical Details

Numerical simulation of the steady and unsteady flow fields in an axisymmetric inlet with an angle of attack of 0 degree was performed with the Fluent solver. The effectiveness of this solver has been widely verified by calculating hypersonic flow, which can meet the steady and unsteady calculation accuracy of the hypersonic inlet [35,36,37]. The Reynolds averaged Navier-Stokes equations in two dimensions were solved using a finite volume spatial discretization method. During the computation, the inviscid flux scheme was Roe’s method, and the splitting format of the convective flux was the AUSM (Advection Upstream Splitting Method) format. The discrete scheme is the second-order upwind scheme, and the turbulence model is the SST (Shear Stress Transport) k - ω with some extensions for modeling high-speed, wall-bounded flows and separation regions [38,39]. The calculation process regards the gas as ideal and considers the variation of the specific heat capacity and thermal conductivity with temperature and solves the viscosity using Sutherland’s formula. In addition, the unsteady flow field was solved by a two-time-step implicit time-discrete method, where the physical time-step Δt is given according to the following equation [35]:

$$\mathsf{\Delta }t<\frac{\delta }{\lambda }$$

(1)

where δ and λ are the minimum mesh size and characteristic wave velocity of the unsteady computational domain, respectively. In the present study, the physical time step is set as 10⁻⁶ s with 50 inner iterations in each time step. In the calculation, the residual and throat pressure were monitored. The convergence criterion is that the residuals of each equation and the monitored parameters are reduced to 10⁻⁴ or they no longer change; additionally, the import and export flow remains stable. The inflow parameters are listed in

Table 2. Different inflow parameters.

Mai	Hi (km)	pi (Pa)	Ti (K)
6.0	25.0	2549.22	221.55
7.0	27.0	1879.98	223.54
5.0/4.5	23.0	3466.87	219.57
4.0/3.5	20.0	8849.73	216.65

, where Ma_i and H_i are the inflow Mach number and height; p_i and T_i are the inflow static pressure and static temperature.

3.2. Turbulence Model Validation

The simulation accuracy of the SST k - ω model for the hypersonic inlet flow field was verified by the experimental data in the open-circuit wind tunnel at the Aachen Jet Propulsion Laboratory [40]. The experimental model is a 2D mixed compression inlet, as shown in Figure 2. In this original model geometry, two consecutive ramps deflect the incoming flow. However, due to the restrictions of the wind tunnel, the first ramp of the inlet was omitted in the experimental tests. The given angle of attack of the freestream, α = 10°, arises from the missing first ramp. The outline of the inlet and the definition of the main dimensions are given in Figure 2 and

Table 3. Main dimension parameters of the inlet [40].

Parameters	Value	Unit
Inner width	52.0	mm
Overall length	400.0	mm
Isolator height	15.0	mm
Second ramp angle δ2	21.5	°
Cowl angle δ3	9.5	°
Divergence angle δ4	5.0	°

. The presented two-dimensional numerical simulations belong to a configuration with an isolator length of l = 79.3 mm at Ma_i = 2.41 without throttling. The total pressure and total temperature are 540 kPa and 305 K, respectively. The grids near the wall are refined to ensure y+ below 1.0, and the total grid is 141,000 structured cells.

Compared to the experimental schlieren image, the computational results give a good prediction of the flow structure of the hypersonic inlet, as shown in Figure 3. The shock pattern, the appearance and extension of the separation packages, and the development of the boundary layer are in good agreement with the experiment. In addition, the calculated static pressure distributions of ramp and cowl surface (see Figure 4 and Figure 5) agree well with the experimental results. Overall, the applied numerical simulation yields satisfying results, so the SST k - ω turbulence model can be used in the current study.

3.3. Unsteady Numerical Method Validation

In order to evaluate the ability of Fluent code to solve unsteady compressible flow, an experimental study of the shock-focusing reflection process was conducted by Izumi et al. [41]. The experimental device is shown in Figure 6. The section size of the shock tunnel is 44 mm × 44 mm. The expression of the shock reflector surface is X = CY². The height of the parabolic reflector is 28 mm. In the unsteady numerical verification calculation, the conditions of C = 0.5 and the Mach number of the moving shock Ma_s = 1.5 are selected. The computational domain mesh and boundary conditions are shown in Figure 7 with about 60,000 cells.

Assuming the specific heat ratio γ is 1.4, the vacuum pressure and temperature of the shock tunnel are set to 101,000 Pa and 300.00 K, respectively. The velocity of the moving shock-front V₁ is zero and the parameters after the shock are calculated by the shock relation (Equations (2)–(4)), as shown in

Table 4. Boundary conditions of the numerical calculation.

Boundary Type	Mai	pi (Pa)	Ti (K)
Pressure far field	0.6044	248,290	396.06
Pressure outlet	-	101,000	300.00

.

$$\frac{p_2}{p_1}=1+\frac{2\gamma }{\gamma +1}(M{a_{\mathrm{s}}}^2-1)$$

(2)

$$\frac{T_2}{T_1}=\frac{(\frac{2\gamma }{\gamma +1}M{a_{\mathrm{s}}}^2-\frac{\gamma -1}{\gamma +1})(1+\frac{\gamma -1}{2}M{a_{\mathrm{s}}}^2)}{\frac{\gamma +1}{2}M{a_{\mathrm{s}}}^2}$$

(3)

$$\frac{V_2-V_1}{a_1}=\frac{2}{\gamma +1}(M{a}_{\mathrm{s}}-\frac{1}{M{a}_{\mathrm{s}}})$$

(4)

Numerical calculations were conducted by solving the two-dimensional unsteady Euler equations. The time step for unsteady numerical calculation is 10⁻⁸ s, with 50 iterations per time step. Figure 8 shows the comparison between the experiment and the computation at different times during the shock-focusing reflection process. The incident shock propagates from left to right and the reflected shock propagates from right to left. The time t is nondimensionalized with respect to γ^1/2D/a₁, where γ is a specific heat ratio, a₁ is the speed of sound in air ahead of the shock, and the half height of the shock reflector D is 14 mm. The upper part comprises the experimental schlieren images at different times, and the lower part comprises the numerical schlieren images at corresponding times. At the dimensionless time t’ = 1.68, the incident shock disappears, and only the reflected shock remains. The numerical calculation results capture well the whole shock reflection process and the flow field structure is highly consistent, which verifies that the Fluent software is reliable for solving unsteady compressible flows.

3.4. Grid Analysis

Considering the existence of shock/boundary layer interaction in the hypersonic inlet, the mesh is refined near the wall and in regions that require focusing, such as near the lip and in the internal pressure section. The mesh near the wall is refined in the boundary layer to ensure that the wall y⁺ is less than 1.0. The mesh and boundary conditions for the steady and unsteady computations are shown in Figure 9. The inflow conditions of the inlet are set as the pressure far field. The solid wall boundary is set to be a no-slip adiabatic wall. The exit is set as a pressure outlet. The performance of a grid-independent analysis confirms that the grid resolution used in this paper is sufficient to capture the physically relevant features. In Figure 10, the static pressure distribution along the central cone surface is shown for three different grid-refinement levels: coarse (30,320 structured cells), medium (127,000 structured cells), and fine (434,400 structured cells), indicating the grid at medium and beyond is sufficient to obtain the accurate solution. Therefore, a medium grid was selected for further numerical analysis.

4. Results and Analysis of the Reference Inlet

4.1. Flow Features

The two-dimensional steady calculation was carried out for the axisymmetric reference inlet in the range of Ma_i = 3.0–7.0, and Figure 11a–e illustrates the flow fields of the reference inlet. At the design point of Ma_i = 6.0, as shown in Figure 11a, the curved leading edge shock just hits the lip position, and the flow coefficient is 1.00. After the leading edge shock, the Mach number decreases continuously in the radial direction. The lip-curved shock is weak, with almost no separation at the shoulder of the inlet. As the compression increases, the Mach number decreases with the streamline. The isentropic compression wave is located between the leading edge shock and the lip shock. As shown in Figure 11b, the leading edge shock is more curved and penetrates into the lip at Ma_i = 7.0. Some expansion waves are generated near the lip, and the flow is accelerated to Mach number 9.0. The curved lip shock becomes stronger, hitting the front of the throat and producing a small separation package. At Ma_i = 5.0 and 4.0, as shown in Figure 11c,d, the leading edge shock waves become straighter and the overflow windows become larger, the lip shock waves become weaker, and there is no separation near the shoulder. At Ma_i = 3.0, Figure 11e shows that the inlet is not starting, but the flow field is generally stable, and there is no periodic oscillation phenomenon. At this time, the separation package and the induced oblique shock appear near the lip. The upper part of the internal pressure section is mainly a supersonic zone and the actual flow area gradually decreases. There is a large, closed separation package near the cone which occupies more than half of the channel, and the exit Mach number is close to 1.0.

In order to perform a careful analysis of the unstarted flow field for an axisymmetric reference inlet with a low Mach number, the unsteady calculation was performed at Ma_i = 3.0. The flow field shown in Figure 12a was selected as the initial flow field at t = 0 ms. The calculation results are shown in Figure 12b–g. Since Ma_i = 3.0 is lower than the starting Mach number, the captured flow entering the inlet cannot all pass through the throat, which is choked at this time. Figure 12a shows that the supersonic flow cannot propagate disturbance forward and the captured flow accumulates continuously at the throat, which will increase the pressure continuously. The high-pressure perturbation propagates upstream through the subsonic region of the boundary layer, forcing the boundary layer to thicken and bend the streamlines. Since the Mach number here is low, the interaction between the shock boundary layer is not serious. There is no significant separation package, only a small subsonic region at the upper and lower walls.

As the pressure further increases due to accumulation, the pressure perturbation continues to propagate upstream with the boundary layer. The separation packages appear on both the cowl wall and the cone (Figure 12b), and they as well as the induced shock wave continue moving forward (Figure 12c). Due to the increasing Mach number of the wavefront, the induced shock becomes stronger and the interaction between the shock wave and the boundary layer is enhanced. Separation packages are getting larger and longer. The actual flow capacity was further weakened. In addition, the interaction between the induced shock waves further aggravates the complexity of the flow field, with the highest separation package at the incident point of the induced shock. On the one hand, the interaction between the shock waves and the separation packages makes the induced shock waves move forward and strengthen with the forward movement of the separation packages (Figure 12d) until they reach the lip (Figure 12e). On the other hand, the induced shock enhances the separation package, which generates a dynamical coupling property. It can also be seen from Figure 12e that the induced shock waves near the cowl wall move to the front of the lip and gradually become a normal shock. The induced shock on the cone intersects with the normal shock wave, forming a typical “λ” shock, whose slip layer makes the flow field more complex, and the flow near the throat approaches sound velocity. The induced shock and separation package significantly increase the total pressure loss, which further reduces the actual flow capacity of the throat. Therefore, the induced shock at the cone can only stop far from the lip. Figure 12f shows the farthest position of the separation package from the lip at t = 3.30 ms. Stronger separation-induced shock appears in front of the lip, further enhancing the overflow capability of the inlet. The disturbance inside the lip can overflow from the outside of the lip after the induced shock leaves the lip, so the pressure disturbance can only spread forward from the separation zone on the cone, resulting in a significant increase in the separation package, which occupies most of the intake of the internal pressure section. Then there is a flow-balancing process. The separation package on the cone starts to move backward and decrease, and the induced shock starts to become weaker. The unstarted flow field is generally stable and no longer changes after t = 5.05 ms (Figure 12g). There is a separation-induced shock in front of the lip of the stable unstarted flow field. The separation package at the incidence point of the lip-induced shock is the largest, and a local subsonic flow appears after the shock. The flow in the supersonic region behind the lip-induced shock accelerates to Mach 1.8, the channel in the supersonic region further back becomes smaller, and the flow continues to decelerate towards 1.0.

In summary, the unstarted process with the low Mach number is a complex, unsteady process. Over time, the separation package adapts to the increasing or decreasing back pressure by the simultaneous adjustment of shape, position, and induced separation shock, and then reaches a steady state, which is known as the dynamic self-sustaining stability of the separation package.

4.2. Performance Parameters

The performance parameters of the reference inlet are calculated using the mass-weighted method, as stated in

Table 5. Overall performance parameters of the reference inlet.

Mai	φ	σth	pth/pi	Math	1/q(Math)	σe	pe/pi	Mae
7.0	1.000	0.431	25.4	3.42	6.300	0.359	27.5	3.21
6.0	1.000	0.641	22.2	3.12	4.747	0.533	23.1	2.93
5.0	0.881	0.763	17.1	2.68	3.123	0.650	18.7	2.50
4.5	0.761	0.803	14.4	2.43	2.471	0.687	16.2	2.25
4.0	0.699	0.821	13.2	2.10	1.837	0.715	13.8	1.97
3.5	0.610	0.834	11.4	1.74	1.376	0.750	12.1	1.64
3.0	0.308	0.464	11.0	0.82	1.030	0.438	7.63	1.08

. These parameters could be used to quantify the inlet performance, including the flow coefficient φ, total pressure recovery coefficient σ, pressure ratio p/p_i, and so on. Subscripts: th is the throat, and e is the exit.

The definition of flow coefficient φ is defined as:

$$\phi =\frac{{\dot{m}}_{\mathrm{e}}}{{\dot{m}}_{\mathrm{c}}}$$

(5)

wherein ${\dot{m}}_{\mathrm{e}}$ indicates the mass flow at exit; ${\dot{m}}_{\mathrm{c}}$ indicates the mass flow captured by the inlet. The coefficient of σ is defined as:

$$\sigma =\frac{p_{\mathrm{t}}}{p_{\mathrm{ti}}}$$

(6)

herein, $p_{\mathrm{t}}$ signifies the mass-weighted total pressure at the section; $p_{\mathrm{ti}}$ denotes the inflow total pressure. The specific flow density function q (Ma) is given below:

$$q\left(Ma\right)=Ma\left[\frac{2}{\gamma +1}\right(1+\frac{\gamma -1}{2}M{a}^2){]}^{-\frac{\gamma +1}{2(\gamma -1)}}$$

(7)

wherein γ is the specific heat ratio. The inlet shows good performance for Ma_i = 3.5–7.0, especially the high flow coefficients at low Mach numbers, with flow coefficients greater than 0.60 for Ma_i = 3.5. At Ma_i = 3.0, the inlet is unstarted and the flow coefficient and the total pressure recovery coefficient decrease significantly, with the flow coefficient only half of its value at Ma_i = 3.5.

Ref. [23] presents an optimized high-performance tri-cone inlet (named Inlet2), and the geometry and performance parameters of the curved compression reference inlet (Inlet1) proposed are compared with it. First, the external pressure section of the reference inlet is 5.20% shorter than that of Inlet2. As the lengths of the isolator are not equal, the throat parameters are chosen for comparison, as shown in Figure 13. At the design point with Ma_i = 6.0, the performance of the reference inlet is basically equivalent to that of the tri-cone inlet, but at the off-design point with Ma_i less than 5.2, the performance of the reference inlet, especially the flow coefficient, is higher than that of Inlet2. The flow coefficient and total pressure recovery coefficient of the reference inlet are 10.4% and 1.36% higher than those of Inlet2 at Ma_i = 4.0, respectively. Thus, the comprehensive performance of the reference inlet is better over a wide range of Mach numbers.

5. Prediction Method of Flow Coefficient

The inlet must satisfy the engine flow requirement over a wide range of Mach numbers, so two prediction methods have been proposed: one based on the flow field and the other on parameter fitting.

5.1. Prediction Method Based on Flow Field

There is a one-to-one mapping relationship between the lip translation distance and the flow coefficient in the reference inlet flow field at a specified Mach number. According to this characteristic, by knowing the flow fields in a wide Mach number range the lip translation distance (defined L_m) under the target flow coefficient (defined φ_t) can be calculated. Figure 11 shows the flow fields at the reference inlet, and an example is given to illustrate the specific prediction method. Compared with the flow coefficient of the reference inlet, the flow coefficient at the mode transition Mach number (Ma_i = 4.0) requires an increase of 10%, and the flow coefficient at other Mach numbers such as Ma_i = 5.0 requires an increase of 6%.

Step 1 Based on the flow coefficient requirement, the target flow coefficient corresponding to one of the inflow Mach numbers is firstly selected for the preliminary prediction of the lip translation distance. In this paper, the reference inlet flow field with modal conversion Mach number Ma_i = 4.0 (see Figure 14) was selected. AB is the curved leading edge shock, the flow before AB is the free flow, CD is the lip translation line, R_c is the capture radius, R_t is the radius of the streamline before AB, and EC is the lip translation distance. Equation (8) is derived from the definition of flow coefficient (Equation (5)), which shows the relationship between R_t and R_c. In this example, R_c = 200 mm, φ = φ_t = 0.769, R_t = 175.374 mm can be obtained according to Equation (8), and the corresponding streamline is made according to R_t. The intersection point of the streamline and CD is E, then EC is the lip translation distance corresponding to the target flow coefficient. In this example, L_m = EC = 11.28 mm.

$$R_{\mathrm{t}}=R_{\mathrm{c}}\sqrt{\phi }$$

(8)

Step 2 Besides the mode transition Mach number (Ma_i = 4.0), continue to calculate the flow coefficient of the inlet at other Mach numbers after translating the lip distance by the value of EC, then evaluate whether it meets the requirements. In this example, Ma_i = 5.0 is selected for illustration, as shown in Figure 15. In the flow field of the reference inlet at Ma_i = 5.0, the position of point E is determined from the lip translation line at EC = 11.28 mm. Then, a streamline is made through point E, resulting in a radius R_t = 195.35 mm. The corresponding flow coefficient φ = 0.953 can be calculated by Equation (8), which is higher than the target flow coefficient φ_t = 0.934 at Ma_i = 5.0, indicating that the lip translation distance EC = 11.28 mm meets the design requirements.

Step 3 follows Step 2 by traversing all the other Mach numbers, computing the actual flow coefficients corresponding to the lip translation distance EC, and evaluating whether they meet the requirements. If all the requirements are met, the obtained lip translation distance EC is the final value. If not, choose one of the Mach numbers and the corresponding target flow coefficient that does not satisfy the requirement, and repeat the above procedure until all flow coefficients meet the requirement.

This method can quickly obtain the lip translation distance that meets the entire flow coefficient requirement in a given Mach number range, avoiding massive numerical simulations and thus greatly saving compactional time and resources.

5.2. Prediction Method Based on Parameter Fitting

Step 1 The first two-dimensional viscous numerical calculations of the above variable geometry inlet were performed at the mode transition Mach number Ma_i = 4.0, and the specific performance parameters are given in

Table 6. Performance parameters of the inlet with lip moving forward at Ma_i = 4.0.

Lm (mm)	φ	σth	pth/pi	Math	σe	pe/pi	Mae
−20	0.557	0.793	9.68	2.27	0.678	9.86	2.13
−15	0.596	0.804	10.46	2.23	0.693	10.77	2.09
−10	0.632	0.804	11.45	2.18	0.698	11.76	2.04
−5	0.666	0.816	12.32	2.15	0.710	12.74	2.01
0	0.699	0.821	13.17	2.10	0.715	13.79	1.97
5	0.730	0.817	14.32	2.05	0.713	14.99	1.92
10	0.759	0.819	15.29	2.01	0.717	16.03	1.88
15	0.786	0.820	16.18	1.98	0.720	16.96	1.85
20	0.812	0.820	17.00	1.94	0.724	17.86	1.82

, with a positive sign for L_m indicating forward motion and a negative sign for backward motion. It can be seen that as the lip moves forward, the total pressure recovery coefficient and the pressure ratio of the exit increase. This is because the increase of the internal contraction ratio improves the compression efficiency, which is exactly the opposite when the lip moves backward.

Step 2 Polynomial curve fitting is performed on the flow coefficient φ and the translation distance L_m in Table 6, and the fitting formula is obtained as Equation (9), the correlation coefficient of which is R² = 1.000, as is shown in Figure 16. Given the target flow coefficient φ_t = φ = 0.769, the translation distance L_m = 11.79 mm can be obtained according to Equation (9). The moving distance of the lip increases from −20 mm to 20 mm, and the flow coefficient increases from 0.557 to 0.812, indicating that the actual capture flow of the inlet can be effectively adjusted by changing the lip position.

$$\phi =-3.61\times 10^{-5}{L_{\mathrm{m}}}^2+6.36\times 10^{-3}{L}_{\mathrm{m}}+0.699$$

(9)

Step 3 Except for Ma_i = 4.0, continue with numerical simulations to calculate the flow coefficients corresponding to other inflow Mach numbers with lip forward L_m = 11.79 mm to evaluate whether they can meet the requirements. If all the requirements are satisfied, then the L_m = 11.79 mm is the final value. If not, one of the inflow Mach numbers and the corresponding target flow coefficient that does not satisfy the requirement is selected and the above steps are repeated until the flow coefficients satisfy the requirements for all Mach conditions.

5.3. Numerical Verification

The accuracy of the above prediction methods is verified by numerical simulations. Two-dimensional numerical calculations were performed for an axisymmetric inlet with a forward lip of 11.28 mm. The performance parameters at Ma_i = 4.0 and 5.0 are shown in

Table 7. Overall performance parameters of the inlet with lip moving forward.

Mai	φ	σth	pth/pi	Math	σe	pe/pi	Mae
4.0	0.769	0.820	15.5	2.00	0.718	16.2	1.87
5.0	0.953	0.767	19.2	2.58	0.652	20.9	2.43

. The predicted values of the flow coefficients are exactly equal to the numerical values, which proves that the first prediction method is very accurate. The second method is computationally intensive, less accurate, and does not predict the flow coefficients at other Mach numbers. Using a desktop computer (4-core X64 processor, Intel i7-8700 CPU, 8 GB memory), the numerical simulation of a case in Table 6 takes 1890 s. As a result, the additional eight cases in the second method take a total of 15,120 s, while the time can be reduced to 180 s by using the first method.

Figure 16 shows the flow field of the inlet with L_m = 11.28 mm at Ma_i = 4.0 and 5.0. Compared with the reference inlet (Figure 11), the shape of the leading edge shock is unchanged. The forward translation of the lip reduces the overflow and further improves compression efficiency.

Figure 16. The flow fields of the inlet with the lip moving forward at Ma_i = 4.0 and 5.0; (a) Ma_i = 4.0; (b) Ma_i = 5.0.

Figure 16. The flow fields of the inlet with the lip moving forward at Ma_i = 4.0 and 5.0; (a) Ma_i = 4.0; (b) Ma_i = 5.0.

6. Prediction Method of Starting Mach Number

In this paper, the reference inlet is unstarted at Ma_i = 3.0. There are two important issues that need to be addressed: one is the prediction of the starting Mach number, and the other is the prediction of the critical translation distance of the lip if the inlet can be started at Ma_i = 3.0.

6.1. Prediction of Starting Mach Number of Reference Inlet

Step 1 The root reason for the reference inlet at low Mach numbers not starting is throat choking [34,42]; the flow state of the throat affects the starting performance of the inlet directly. Therefore, the throat Mach number is chosen to predict the starting Mach number of the reference inlet in this paper. The throat Mach numbers corresponding to the low Mach number of Ma_i = 3.5–4.5 are fitted by an exponential function (Equation (10)), and the correlation coefficient is R² = 0.9999:

$$\frac{1}{q\left(M{a}_{\mathrm{th}}\right)}=0.1774e^{0.585M{a}_{\mathrm{i}}}$$

(10)

where q(Ma_th) is the specific flow density function of the throat.

Step 2 Assuming that the reference inlet is in a critical starting state, without considering the viscosity and the non-uniformity of the throat, the ideal throat Mach number Ma_th = 1.0 is introduced into Equation (10) and the starting Mach number Ma_s = 2.95 is obtained. However, the numerical result shows that the inlet is unstarted at Ma_i = 3.0 (Figure 12), indicating that the predicted value is smaller, and 1/q(Ma_th) = 1.0 needs to be corrected. The starting Mach number Ma_s = 3.15 is obtained by trial calculation of numerical simulation; the throat Mach number Ma_th = 1.41 and the corresponding 1/q(Ma_th) = 1.12, so 1.10 is chosen as the modified value of 1/q(Ma_th). Substitute this modified value into Equation (10) and the starting Mach number Ma_s = 3.12. The relative error of predicted Ma_s with the numerical result is −0.95%, which meets the requirements of prediction accuracy.

6.2. Prediction of Lip Translation Distance at the Specified Starting Mach Number

Due to the low Mach number at which the reference inlet is unstarted, the starting of the inlet can be achieved by reducing the actual capture flow through a backward translation of the lip. At the specified starting Mach number Ma_s = 3.0, the key to accurately predicting the lip backward distance is to quickly determine the critical flow coefficient. Two prediction methods are proposed in this paper.

6.2.1. Prediction Method Based on Fitting of Flow Coefficient

The flow coefficients of low inflow Mach numbers (Ma_i = 4.5, 4.0, and 3.5) in Table 5 are selected for polynomial fitting (Equation (11)), and the correlation coefficient is R² = 1.000:

$$\phi =-0.054M{a_{\mathrm{i}}}^2+0.583M{a}_{\mathrm{i}}-0.769$$

(11)

The flow coefficient φ is obtained by fitting the inflow Mach numbers greater than the starting Mach number, so Equation (11) can be used to predict the critical flow coefficient corresponding to the starting Mach number. By taking Ma_i = 3.0 into Equation (11), φ = 0.494 can be obtained. According to the flow coefficient prediction method based on the flow field in Section 5.1, the lip translation distance L_m = −6.15 mm when φ = 0.494.

6.2.2. Prediction Method Based on Fitting of Total Pressure Recovery Coefficient

The total pressure recovery coefficients of the throat of Ma_i = 4.5, 4.0, and 3.5 in Table 3 are selected for polynomial fitting (Equation (12)), and the correlation coefficient is R² = 1.000:

$${\sigma }_{\mathrm{th}}=-0.01M{a_{\mathrm{i}}}^2+0.049M{a}_{\mathrm{i}}+0.785$$

(12)

Therefore, the fitting value of σ_th is 0.842 at Ma_i = 3.0. According to the definition of the flow coefficient under zero-dimensional conditions, φ_c also can be calculated by:

$${\phi }_{\mathrm{c}}={\sigma }_{\mathrm{th}}\ast \frac{q\left(M{a}_{\mathrm{th}}\right)}{q\left(M{a}_{\mathrm{i}}\right)}/R_{\mathrm{ct}}$$

(13)

where the values of parameters at Ma_i = 3.0 are given in

Table 8. The parameter values of the Equation (13) at Ma_i = 3.0.

Parameters	Value
Mai	3.0
q(Mai)	0.236
q(Math)	1.10
σth	0.842
Rct	6.01

. It can be seen from Section 6.1 that 1/q(Ma_th) is not equal to 1.0, and the flow coefficient φ_c = 0.539 can be obtained by Equation (13).

According to the numerical result of the external pressure section of the reference inlet at Ma_i = 3.0, as shown in Figure 17, the flow coefficient φ is 0.526 calculated by Equation (5), thus the flow coefficient φ_c calculated by Equation (13) is unreasonable. Comparing the flow coefficient calculated by Equation (13) φ_c with φ in Table 5, it was found that φ/φ_c is less than 1.0. The distribution of throat parameters is not uniform, so the correction coefficient η is introduced to correct the flow coefficient φ_c by Equation (14).

$$\phi =\eta {\phi }_{\mathrm{c}}$$

(14)

wherein η corresponds to the known inflow Mach numbers shown in

Table 9. The correction coefficient η corresponds to the inflow Mach numbers.

Mai	η
3.5	0.891
4.0	0.877
4.5	0.850
5.0	0.866
6.0	0.837

. According to the linear interpolation at Ma_i = 4.0 and 3.5, η = 0.911 is obtained at Ma_i = 3.0, so φ = 0.491 is calculated by Equation (14). According to the flow coefficient prediction method based on the flow field in Section 5.1, the lip translation distance L_m = −6.49 mm when φ = 0.491.

6.2.3. Numerical Simulation Verification

In Section 6.2.1 and Section 6.2.2, the critical flow coefficients φ at the starting Mach number Ma_s = 3.0 are obtained by two different prediction methods, which are 0.494 and 0.491, and the corresponding lip translation distances L_m are −6.15 mm and −6.49 mm, respectively. The numerical results are shown in Figure 18, where the inlet with lip translation distance L_m = −6.15 mm is started at Ma_i = 3.0, and its critical translation distance corresponding to Ma_s = 3.0 is −6.05 mm. The relative errors of the prediction methods based on the fitting of flow coefficients and the total pressure recovery coefficients are 1.65% and 7.27%, respectively, indicating the higher accuracy of the first method.

7. Conclusions

A curved compressible axisymmetric reference inlet with controllable Mach number distribution was designed, and then prediction methods were proposed for the flow coefficient and starting performance of the inlet lip translation scheme. The results were analyzed and verified by numerical simulation. The conclusions are as follows:

(1)

The overall performance of the reference inlet is good and the flow fields have almost no separation within the range of Ma_i = 3.5–7.0. The flow coefficient remains above 0.60 at Ma_i = 3.5. Compared to the optimized high-performance traditional tri-cone inlet, the reference inlet has better comprehensive performance over a wide range of Mach numbers, with 10.4% and 1.36% higher flow coefficient and total pressure recovery coefficient at Ma_i = 4.0, respectively. At Ma_i = 3.0, the inlet is unstarted, the flow field is basically stable after t ≥ 5.05 ms, and its separation package shows obvious self-sustaining stability.

(2)

The prediction method based on the flow field can accurately predict the flow coefficients of the inlet with lip translation in the whole working range only by using the flow field of the reference inlet, while the prediction method based on parameter fitting needs to calculate the performance of multiple lip positions, which is computationally expensive and has low accuracy.

(3)

The starting Mach number of the reference inlet can be accurately predicted based on the throat Mach number fitting method, and the relative error with the numerical simulation result is less than 1%. In the lip translation distance prediction for the specified starting Mach number, the prediction method based on the fitting of the flow coefficient is simple and accurate and the relative error with the numerical simulation result is 1.65%, while the prediction method based on the fitting of the throat total pressure recovery coefficient fitting is relatively complex with more undetermined parameters, and the relative error is 7.27%.

The present study shows that the prediction methods for flow coefficient and starting performance can overcome the shortcomings of the traditional iterative algorithms. These methods can be considerably more computationally efficient and can quickly satisfy the design requirements of variable geometry inlets.