Wenyou QIAO , Anyun YU , Wei GAO , Weixing WANG

a Research Center of Combustion Aerodynamics, Southwest University of Science and Technology, Mianyang 621010, China

b Jiangsu Province Key Laboratory of Aerospace Power System, Nanjing 210016，China

c Science and Technology on Scramjet Laboratory of Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

d College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

KEYWORDS Hypersonic;Inverse design;Inward-turning inlet;Method of characteristic;Shock wave

Abstract In the design of a hypersonic inward-turning inlet by applying the traditional basic f lowf ield, a ref lected shock-wave is formed in the isolator due to the continuous ref lection of the cowlref lected shock wave in the basic f low-f ield, which interacts with the boundary layer to produce a considerable inf luence on the performance of the inlet. Here, a basic f low-f ield design method that can control the velocity direction at the throat section is developed, and numerical simulations are conducted to demonstrate the effectiveness of this method.The method presented in this paper can achieve the absorption of the ref lected waves at the shoulder of the basic f low-f ield by adjusting the variation law of the center radius in the basic f low-f ield,and a smooth transition between the compression surface and the isolator can also be produced.The Mach number and total pressure recovery coeff icient of the inlet designed according to this method are 3.00 and 0.657, respectively, at design point (the incoming f low Mach number Ma∞=6.0). The results show that with this method, the inlet can eff iciently weaken both the ref lection of the shock wave and the interaction between the boundary layer and the ref lected shock waves,which improves the aerodynamic performance of the inlet.

1. Introduction

Under a hypersonic condition, the aerodynamic performance of the inlet has a great inf luence on the propulsion of the entire air-breathing propulsion system.1The inward-turning inlet has a higher total pressure recovery coeff icient and a smaller size and external drag2than those of the traditional twodimensional, axis-symmetric, sidewall compression inlets;therefore, this inlet conf iguration has received increasing attention in air-breathing propulsion systems. However, the f low-f ield of the inward-turning inlet, especially the shock shape, should be reasonably organized to achieve eff icient compression, which places higher requirements on the design technology. At present, the design of the inward-turning inlet often adopts the osculating f low method. In this method, the basic f low-f ield is designed through an inverse method, with the inlet surface obtained through the streamline tracing,cross-section transition and viscosity correction methods.Therefore, the basic f low-f ield plays a decisive role in aerodynamic performance of the inlet designed according to this method.

In the 1960s, Mo¨lder and Szpiro3f irst applied the Busemann f low-f ield4to the design of the inward-turning inlet. As the isentropic compression in this f low-f ield occupies an excessively large share,the inlet length and the contraction ratio are too large,which makes the inlet diff icult to start,and the excessively long basic f low-f ield increases the viscosity loss to a certain degree. Consequently, scholars adopted the truncated Busemann f low-f ield to design the inward-turning inlet.5-7Matthews and Jones8designed a basic f low-f ield with an equal slope and equal-pressure boundary and applied it to the modular inlet design. You9utilized an axis-symmetric basic f low-f ield by taking the cubic curve as the generatrix and, in combination with the osculating f low method, proposed an internal waverider-derived inlet design method. To ensure the uniformity of the f low-f ield parameters at the entrance and exit of the basic f low-f ield, Guo et al.10proposed the Internal Conical Flow ‘‘C” (ICFC) shock wave f low-f ield by joining the Internal Conical Flow ‘‘A” (ICFA) f low-f ield11with the Busemann f low-f ield.4Zhang et al.12-15from Nanjing University of Aeronautics & Astronautics (NUAA) proposed a basic f low-f ield design method with controllable section compression rules (pressure or Mach number distribution).

In a hypersonic inlet, the shock train in the isolator can strongly affect its aerodynamic performance16and even result in an inlet unstart. Especially for the inward-turning inlet designed with the abovementioned basic f low-f ields with a f ixed radius of the center body, the cowl-ref lected shock wave at the shoulder of the inlet cannot be achieved with a wave absorption design at the compression surface; therefore, the waves will continue to be ref lected, generating a series of ref lected shock in the isolator. All those shock waves and the secondary f low produced by the Shock-Wave/Boundary-Layer Interactions (SWBLI) will greatly affect the performance of the inlet.15-19

To restrain SWBLI,20-22the design method can be improved according to the f low mechanism of the inwardturning inlet,so that the design objective can be achieved without requirements for additional devices.Reducing the intensity of the cowl-ref lected shock wave is an effective improvement in the hypersonic inward-turning inlet.There are two reasons for this benef it: on one hand, the weaker cowl-ref lected shock wave can dramatically restrain SWBLI in the internal compression section, and on the other hand, it can be used to reduce the intensity of the continuous ref lected shock.Accordingly, the performance of the hypersonic inward-turning inlet can be improved by altering the radius of the center body.13,23However, this method cannot be used to eliminate the continued ref lection of the cowl-ref lected shock. Fang24and Liu25proposed a Method Of Characteristic (MOC) that designed a two-dimensional inlet according to the throat parameters.You et al.26proposed a basic f low-f ield design method based on the parameters behind the ref lected shock wave for the hypersonic inward-turning inlet.However,there are two types of problems in these methods:

(1) The existence of the exit parameters still needs further study.For an axis-symmetric condition,the f low parameters have a strong non-linear feature along the radial direction, especially after the axis-symmetric curved shock waves and isentropic compression, and those methods impose too many constraints on the exit parameters. All these limits will directly affect the existence of a solution for the basic f low-f ield and will even result in a calculation divergence.

(2) The design has poor f lexibility.Ignoring the existence of the solutions, although it is convenient to determine all parameters in the basic f low-f ield according to the exit f low parameters, it is diff icult to f lexibly determine the section compression rule of the inlet by applying this method. Furthermore, it is diff icult to suff iciently account for the contraction ratio distribution of the inlet. In addition, the inlet section compression rule directly affects the development of the boundary layer.Therefore,it is very diff icult to apply this method to control the development of the boundary layer. Further study is needed to adjust the f low compression rule according to the exit parameter distribution. Therefore,the method of designing the basic f low-f ield based only on the exit f low parameters cannot be applied to the design of the inward-turning inlet.

Based on a comprehensive analysis of the above design methods, this paper proposes a basic f low-f ield design method with a controllable velocity direction angle at the throat section. The main improvements of this new methods are listed below. First, this method can improve the robustness of the algorithm by loosening the constraints on the given design parameters.Second,this method can achieve the wave absorption design of the ref lected shock waves by adjusting the radius change rule of the center body. Finally, the f low-f ield downstream of the ref lected shock wave is rectif ied so that the air f low is f lattened at the exit section to avoid the strongly ref lected shock wave generated in the isolator.

The paper is organized as follows. The basic f low-f ield and inward-turning inlet design method is proposed, with a controllable velocity direction angle at throat section, at f irst,and based on this method, the detailed design process of the inward-turning inlet is then presented. Following this, the accuracy of the basic f low-f ield design method is verif ied next through the inviscid numerical calculation. At last, the f low mechanism of the inward-turning inlet is analyzed according to the viscous numerical simulation results,with the discussion of the method proposed in this paper.

2. Basic f low-f ield design with velocity direction controllable at throat section

Fig. 1 Pressure contours and structure of basic f low-f ield.27

The basic f low-f ield structure of the early hypersonic inwardturning inlet design is shown in Fig. 1,27in which x/R0, y/R0and p/p∞are axial coordinate,radical coordinate and pressure ratio,respectively.As shown in Fig.1,the cowl-ref lected shock continuously ref lected in an isolator forms a series of ref lected shock. The continuous ref lection of the cowl-ref lected shock wave in the inward-turning inlet of the compression surface,and the interaction between the boundary layer and these shock waves will signif icantly inf luence the inlet performance.Furthermore, the transition of the inlet surface and isolate is not smooth if the inlet is designed according to the streamline tracing method because the velocity directions at the throat section are not completely horizontal. Considering this, this paper develops a new basic f low-f ield design method. In this method,the wave absorption design is realized at the compression surface of the ref lected shock waves by adjusting the radius change rule of the center body, and at the same time,the downstream of the cowl-ref lected shock wave is rectif ied to avoid the continuous ref lection in the isolator and the formation of a ref lected shock.In this paper,the f low mechanism and the improvement of aerodynamic performance of the hypersonic inlet designed according to this method is described.

2.1. Design method

According to the structural features of the basic f low-f ield,the basic f low-f ield is divided into separate regions for the solution in this paper, as shown in Fig. 2, in which the area in front of the ref lected shock wave is divided into the after-incidentshock dependent domain ①and the isotropic compression domain in front of the ref lected shock ②;the f low downstream of the ref lected shock wave is divided into the after-ref lectedshock dependent domain ③and the rectif ication domain ④.First,domain ①and domain ②are determined under the condition of a given Mach number distribution along AD. Then,by adopting the secant method, the shape of the boundary CE is adjusted to make the after-wave f low parameters consistent with the given target parameters at Point D of the ref lected shock wave CD,and at the same time,the f low parameter distribution in domain ③is determined. Finally, taking the boundary DE as the inlet boundary and setting the shape and the velocity direction angle (where the angle is 0°) of the boundary DF, the f low parameter distribution in domain ④is determined. In this paper, the basic f low-f ield is designed through the MOC which is introduced in Refs.28,29,and based on which,the improvement of the design method is explained.

Fig. 2 Flow-f ield structure of basic f low-f ield.

2.2. Mach number distribution of domains ①and ②

As stated above, domain ①and domain ②are designed according to a given Mach number distribution. To design the Mach number distribution, a cubic spline function is adopted, in which the design parameters are positions of the starting point,middle control point and ending point together with the slopes of the starting and ending point. The detailed design parameters are shown in Fig. 3.

xsand xeare respectively the x coordinates of the starting position and ending position of the basic f low-f ield. Masis the Mach number at the starting position obtained by calculating the f low deviation angle δ at the starting position. Maeis the Mach number at point xe,used to control the compression strength of the basic f low-f ield and set according to the design requirements of the basic f low-f ield. θsis the inclination angle of the Mach number distribution curve at the starting position,used to control the intensity of the incident shock wave at the starting position. θeis the inclination angle of the Mach number distribution curve at point xe, used to adjust the internal contraction ratio of the basic f low-f ield. xmand Mamare respectively the coordinate and Mach number at the central control point of the basic f low-f ield,of which the value is determined from Eq. (1), and are mainly used to control the external and internal contraction ratio distributions of the basic f low-f ield.

where ? and ψ are the adjusting coeff icients of the position and the Mach number of the middle control point in the Mach number distribution curve ranging from 0 to 1, respectively.For the convenience of application, coeff icients Csand Ceare used to adjust the slope at the starting point and the terminal point of the Mach number distribution curve, as shown in Eq. (2).

where Csand Ceonly need to be given according to the demand of the incident shock intensity and the internal contraction ratio.

Fig. 3 Design parameters of Mach number distribution along compression boundary.

As determined according to the above parameters, in designing the basic f low-f ield, xs, Mas, xeand Maeare set according to the design requirements. Based on the def inition of these parameters we can f ind that θsand θe,directly related to the parameter uniformity in the basic f low-f ield and the start-up performance, should be given a parameter as close to ‘0' as possible; xmand Mam, used to adjust the weight of the external and internal contraction ratios of the basic f lowf ield to ensure the start-up performance of the inlet, should be ensured that the xmis close to the xsbut the Mamapproaches to the Maeseparately. However, the values of these parameters must be ensured that the basic f low-f ield physically exists. To sum up, the adoption of the cubic spline curve can control the Mach number distribution rule in a more accurate and more direct way than the method shown in Refs.12-15and reduce the iteration time during the design of the basic f low-f ield.

2.3. Solution of after-ref lected-shock dependent domain ③

As shown in Fig.2,the end point D of the ref lected shock wave in domain ③is the upper boundary of the exit section; therefore, to solve the ref lected shock wave shape, the after-shock f low-f ield parameters at point D must be consistent with the given f low-f ield parameters at the boundary DF. As the f lowf ield parameters at point D cannot be shown directly in the prof ile CE, which will produce the cowl-ref lected shock wave CD,and the f low-f ield parameters in domain ③can be solved directly by applying the MOC, this paper combines the MOC and the solution method of non-linear equations (here, the secant method is adopted)to determine the f low-f ield parameter distribution in domain ③.The detailed solution principle is shown in Fig.4,in which the C+is the left running characteristic line. The shape of the cowl-ref lected shock wave is corrected by adjusting the boundary CE to make the f low-f ield parameters at the end point of the cowl-ref lected shock wave(point D) meet the design requirement and to obtain the f low-f ield parameter distribution in domain ③. Under a given incoming f low condition, the MOC that determines the shape of the shock wave according to the prof ile is given in Refs.9,13,28. This paper adopts the inverse MOC to obtain the f low parameters and the boundary CE of domain③. The detailed solution process is shown below:

Step 1. By using a cubic polynomial, a control equation of the aerodynamic prof ile CE that can generate the ref lected shock wave will be controlled by the coordinates and the slope of the starting point C of the ref lected shock wave and the preset control point M, and the specif ic form for this relation is

Fig. 4 Principle of ref lected shock wave and f low-f ield of its dependent zone.

Step 2. Preset L, θds, Rcand Rd, and adjust θdeto correct the shape of the curve CE, and take the f low-f ield parameter distribution in domain ②as the wave-front f low-f ield parameters of the ref lected shock wave. Determine the shape of the ref lected shock wave generated by the curve CE and the f low-f ield parameter distribution in domain ③with the MOC.

Step 3. Apply the secant method, compare the after-shock parameters at the vertex D (in this paper, the velocity angle at the point D, θd, is set to 0°) with the target parameters at the boundary DF in Step 2,and adjust θdeaccording to the difference of parameters, and then return to Step 2 to repeat the calculation until both parameters are consistent.

As stated above,the shape of the ref lected shock wave that meets the requirements can be obtained by applying the solution method in Fig.4,and the f low-f ield parameter distribution of the after-ref lected-shock dependent domain can be determined.Then,the exit boundary DE in domain ③(the boundary of the leftward characteristic line) can be taken as the entrance boundary of the rectif ied domain ④, after which the solution can continue.

2.4. Solution principle of rectif ied domain ④

As shown in Fig.2,the key to solving domain ④is to obtain a boundary EF that can adjust the f low-f ield parameter distribution on the boundary DF in domain ④to make the distribution consistent with the given condition. The inverse characteristic method developed in Ref.29is referred to for the solution process in the domain. The solution schematic of domain ④is shown in Fig. 5, and the marching process of the method mainly contains the marching of the boundary grid cells(e.g.,the grid unit DA1D11)and the internal grid cells(the cells without the boundary grid cells on the boundary DF).The C0, C+and C-in the Fig. 5 are streamline, left-running and right running characteristic line, respectively.

Fig. 5 Schematic of MOC in rectif ied domain.

The principle that determines the streamlined shape according to the f low-f ield parameter on the downstream boundary is shown in Fig.6,in which point A1is the grid point on the exit boundary DE in domain ③and D11is the solution grid point at the boundary DF.The aim of the algorithm is to determine the shape of the streamline A1D11when the starting boundary DA1, the shape of the exit section DD11and the f low-f ield parameter distribution are given.The detailed solution process is given as follows:

First, the streamline starting from point A1intersects with the boundary DF at the solution point D11.Then,on the initial boundary DA1, the grid pointthat can send a rightrunning characteristic line to the solution point D11is determined, and then the f low-f ield parameters at point D11are determined by solving the compatibility equations along the streamline A1D11, the right-running characteristic lineas well as the f low-f ield parameter distribution rule on the boundary DF. Finally, the calculation precision of the algorithm is determined through corrector calculations. Known from the solution principle,in this method,the starting boundaries DA1and DD11should be simultaneously specif ied,which is a Goursat problem of MOC, and moreover, as the compatibility equation is absent, the f low-f ield parameter distribution rule should be adopted in the solution process. However, the characteristic line grids are similar to those of the inverse characteristic method in Ref.29Therefore, the internal grid points in domain ④can be solved with the inverse characteristic method in Ref.29

As stated above, the solution process of the rectif ied domain ④needs both the MOC shown in Fig. 6 and the inverse characteristic method proposed in Ref.29The detailed solution process is as follows:

Step 1. Starting from the vertex A1of the ref lected shock wave, determine the f low-f ield parameter at point D11on the exit section by applying the MOC in Fig. 6.

Fig. 6 Schematic of MOC at exit boundary cell.

Step 2. Taking D11A2as the initial boundary, point D12is solved with the inverse characteristic method in Ref.29,and in the same way,the points D13,D14,...,D1nare solved,and the aerodynamic boundary ED1nis also determined, where the solution point D12meets the design requirements.

Step 3. Taking the left running characteristic line D11D12...D1nas the initial boundary and D11F as the exit boundary,repeat the calculation in Step 1 and Step 2 to obtain an aerodynamic prof ile EF that makes the f low-f ield parameter distribution rule at the exit section DF meet the design requirements.

It should be stressed that the parameter described in the parameter distribution at the cowl-ref lected shock wave vertex(point D)and the throat section(boundary DF)is a parameter such as pressure, Mach number, velocity magnitude and velocity direction. This paper only studies the aerodynamic performance of the inlet when the direction of the velocity is 0°, and the inf luences of other design parameters will be studied in future works.

3. Design of basic f low-f ield and inward-turning inlet

3.1. Basic f low-f ield

In the design program of this paper,the incoming Mach number, pressure, and temperature under the design condition are 6.0, 1.0 Pa and 300 K, respectively. In order to test the application effect of this method, two basic f low-f ields, Case A and Case B, are designed in this paper. Their design parameters are shown in Table 1, all these parameters are dimensionless except the angle parameters. The basic f low-f ield is determined with the aforementioned design parameters in Table 1, as shown in Fig. 7, in which the x, y and π are the axial and radial dimensionless coordinate and pressure ratio,respectively. The total contraction ratio and internal contraction ratio of the Case A are 7.60 and 2.59, and for the Case B are 6.97 and 2.17. However, the pressure at the throat of Case A is more uniform, which is advantageous for suppressing the generation of the ref lected-shock in the isolator.

3.2. Inward-turning inlet

To verify the application effect of the design method proposed in this paper, the inlet prof ile will be designed with a directstreamline tracing method instead of controllable entrance and exit shapes. Due to the higher total pressure coeff icient and lower internal contraction ratio of the basic f low-f ield, Case B is adopted to design the inlet. After the basic f low-f ield is obtained with the method shown in Fig.7,the inviscid aerodynamic surface of the inward-turning inlet is obtained with the streamline tracing method. First, the Inlet Capture Curve(ICC) and osculating planes are given in Fig. 8, where the ICC can be viewed as the front view of the inlet entrance lip,and the ICC given in this paper is a circle with a diameter of 0.5 m. It is necessary to enlarge the basic f low-f ield by 625 times so that the shape of the ICC can satisfy the geometry of Fig. 8. Second, the ICC is discretized as a series of points in the osculating planes. Finally, a series of streamlines are traced from these discrete points within the osculating planes,and the inlet inviscid surface is composited by these streamlines.

Table 1 Basic f low-f ield design parameters.

Fig. 7 Basic f low-f ield designed with parameters in Table 1.

After the inviscid inlet surface is obtained as shown in Fig.9,the viscosity aerodynamic surface is obtained by applying the boundary-layer correction. The key of the boundarylayer correction is to obtain the distribution of the boundary-layer displacement thickness on the actual inlet

Fig. 8 Schematic of inlet entrance.

Fig.9 Outline of basic f low-f ield at mid-section and inviscid inlet conf iguration.

Fig. 10 Computational domain, surface meshes and boundary conditions of inward-turning inlet in this paper.

Table 2 Incoming f low conditions.

surface by simulating the full viscous f low.30Ref.30proposed a boundary-layer displacement thickness distribution function as shown in Eq. (4):

where δhx( ) is the correction thickness of the aerodynamic boundary,aδand bδare the coeff icients.In this paper,the coeff icients aδand bδof the upper boundary are 4.780×10-3and 6.973×10-3respectively, and the coeff icients of the center body are -4.085×10-3and 1.340×10-3, respectively. Then,the boundary-layer displacement thickness of the inlet boundaries in each osculating plane is calculated by using Eq. (4),and the inlet boundaries are corrected by superimposing the boundary-layer displacement thickness onto the boundaries.The viscosity aerodynamic surface of the inlet is composited by these corrected boundaries. The isolator is obtained by stretching the boundary of the throat section of the inlet viscosity surface in the direction of the free f low, and the length of the isolator is 7 times the equivalent diameter of the throat section. The total contraction ratio is 5.26 in the basic f lowf ield, and the internal contraction ratio is 1.97. Blunt leading edge is not added for simplicity, and the shape of the cowl in Fig. 10 does not affect the internal f low-f ield of the inlet.The incoming f low Mach number Ma∞, f light height H,incoming static pressure p∞and static temperature T∞in the calculation are listed in Table 2.

4. Introduction and validation of numerical method

4.1. Introduction of numerical method

Both the inviscid and viscous numerical simulations are separately used to validate the concept of the basic f low-f ield and the inward-turning inlet, which are designed with a controllable velocity direction at the throat section.All the numerical simulations are conducted with the commercial software ANSYS FLUENT. For the inviscid simulation, a solver that uses the cell-centered f inite-volume method to solve the twodimensional compressible Euler equation is adopted. The material uses ideal gas,and the specif ic heat at a constant pressure is calculated using the 8th-order piecewise polynomials of local temperature, in which the coeff icients are set by the default setting of FLUENT software.The convective f lux splitting uses the 2nd-order upwind Roe-Flux-Difference Splitting(Roe-FDS) scheme,31and the least squares cell-based method is used to evaluate the gradient of f low parameters.

The software ANSYS ICEM is used to generate the computational domain meshes, and boundary conditions are shown in Fig. 10. For the inviscid simulation in this paper, the horizontal and longitudinal grids are evenly distributed. The free-f low condition of the inviscid simulation is the atmospheric parameters at a f light altitude of 25 km of Ma∞=6.0. For the starting Mach number tests, the velocity of the initial f low-f ield is set to 0.

For the viscous simulation, the ReNormalized Group(RNG) k-ε model is used to simulate the turbulence f low,32and the near-wall treatment adopts scalable wall functions.The viscosity is calculated by using the Sutherland formulas.33The rest of the calculation options are consistent with the inviscid simulation.

The computational domain mesh is generated by using the ANSYS ICEM software.The grids in the near-wall region are ref ined so that the non-dimensional y+＜10, which satisf ies the requirements of the RNG k-ε model with scalable wall functions, as shown in Fig. 10.

4.2. Numerical validation and grid sensitivity analysis

For the axisymmetric, steady and inviscid basic f low-f ield of the hypersonic inward-turning inlet,there are only simple f low structures such as shock waves, weakly compressible waves and a small amount of the expansion waves. In the development of CFD,these f low-f ield structures are often used as standards for the validation of the convective f lux.34,35Currently,high-order computation schemes with the Total Variation Diminishing (TVD) limiter36can accurately simulate these f low-f ield structures, and inviscid numerical simulation methods are often used as the verif ication standard of MOC.Therefore, this paper only gives the computation method of the inviscid simulation without further validation.

The experimental results of the inward-turning inlet with a bumpy surface are adopted to validate the viscous simulation method. The test model is shown in Fig. 11, and the experiments are conducted at Ma∞=6.0 in the wind tunnel of Nanjing University of Aeronautics and Astronautics.18The attack angle of the test model is 0°. The computational domain and boundary conditions are set in Fig. 12. A sequence of four grids is generated to demonstrate the sensitivity of the simulation grids, in which the four grids contain 1.06×106,2.02×106, 4.02×106and 7.78×106cells, respectively. The spacing of these grids in the near-wall region meets the accuracy requirement of the turbulence model. Fig. 13 presents the numerical simulation and experimental results of the pressure ratio distribution along the top and bottom boundaries of the inlet in the symmetric plane. It can be seen from Fig. 13 that all the calculated results are basically the same and agree well with the experimental results (EXP). To obtain the detailed calculation result, we choose a number larger than 7.5 million for the computational grids for the subsequent CFD numerical simulation.

Fig. 11 Experimental model18 for numerical verif ication.

Fig.12 Computational domain and boundary conditions of test model.

Fig. 13 Experimental18 and numerical pressure distribution on top and bottom boundaries of symmetrical plane.

5. Numerical simulation

5.1. Inviscid simulation results of basic f low-f ield

Fig. 14 Comparison of CFD and MOC results of Case A.

As shown in Fig. 14, in the f low-f ield of the compression boundary and the throat, the results of MOC is basically the same as the CFD results. In Fig.14(a), the distribution of the Mach number (Ma)of the compression boundary in the CFD results is basically consistent with the MOC results. Furthermore, as shown in Fig. 14 (b), the relative maximum errors of the pressure ratio π and Mach number Ma at the throat are less than 2.0% and 0.4%, respectively, and the error of the velocity direction (θ) is less than 0.03°. According to the principle of aerodynamics, the pressure is more sensitive to the shape of the aerodynamic surface. Therefore, although the error of the pressure distribution is relative large,it has little inf luence on the application of this design method.

Fig. 15 shows the CFD and MOC results of Case A and Case B, respectively. For the comparison of the CFD and MOC results, Mach number contours of the upstream throat are basically the same as the MOC results,which indicates that the accuracy of the design method in this paper can meet the requirements of the inlet design. Moreover, it can be found from the CFD results of Case A, the ref lected-shocks have been completely eliminated in the isolator. According to the comparison of performance parameters of Case A in Table 3,in which σ is total pressure recovery coeff icient, the f low-f ield parameters at the throat and the exit of the CFD results and of the MOC results are almost the same. This shows that the design method of this paper completely realizes the design goal of the basic f low-f ield.

Fig. 15 Comparison between CFD and MOC results of basic f low-f ield.

Table 3 Performance parameters of basic f low-f ield at design point.

However, the ref lected-shocks of the Case B in the isolator are not absorbed completely, and the pressure and the Mach number in the CFD results between the throat and the exit have signif icant differences. These differences arise due to the continuous development of the airf low in the isolator. However, according to the change in the total pressure recovery coeff icient at the throat and the outlet in the CFD results of the Case B, the total pressure recovery coeff icient drops by only 0.005, which indicates that the total pressure loss due to the oblique shock wave in the isolator is very small.The design method of the basic f low-f ield in this paper has a high precision.

5.2. Viscous simulation results of inward-turning inlet

Fig. 16 Mach number contours of symmetrical plane and X-sections at design point (Case B).

In the design condition, known from Fig. 16, the incident shock wave basically f its the inlet lip prof ile, and the ref lected shock wave in the symmetrical section falls near the inlet throat section, which indicates that the viscous correction method adopted in this paper has a good effect on the correction of the incident shock wave. Furthermore, there is no strong ref lected shock wave in the isolator, which shows that the wave structure in the basic f low-f ield is almost consistent with the design condition. From Fig. 17, the streamlines near the inlet wall start to def lect towards the top of the isolator near the ref lected shock wave because of the interaction between the ref lected shock wave and boundary layer, while the streamlines in the vicinity of the symmetric plane def lect to the bottom of the isolator, forming a stream-wise vortex structure near the wall of both sides of the inlet.It can be seen from Figs. 16 and 18 that there is an approximately -2° airf low def lection angle near the throat section, which indicates that the viscous correction method adopted in this paper has a limited correction effect on the internal contraction section.

However, the design method in this paper can weaken the ref lected shock wave and will not produce a strongly ref lected shock in the isolator,as shown in Fig.1.This capability is very useful for reducing the secondary f low loss in the inlet.According to the total pressure recovery coeff icient contours of the equal X-section in Fig. 17, the f low in the large region outside the boundary layer and the secondary f low area has a higher total pressure recovery coeff icient, which shows that the f low in the inlet has a good quality.

Fig.17 Total pressure recovery coeff icient of X-sections of inlet,near-wall streamlines and surface streamlines at throat section(Ma∞=6.0, Case B).

Fig. 18 Vertical def lected angle in symmetric section (Case B).

The Mach number contours at the symmetric plane,throat and exit section of the inlet under different incoming f low conditions are shown in Fig. 19. The inlet f low features vary with the incoming Mach number,and they can be analyzed with the osculating method. As shown in Fig. 8, the entrance of the inward-turning inlet can be divided into two parts:the incident shock lip and waverider lip. In the design condition, the incident shock wave generated by the incident shock lip just passes through the waverider lip. However, when the Mach number of the incoming f low changes from 6.0 to 4.0, the shock wave at the entrance gradually moves far away from the lip prof ile,and the position at which the ref lected shock wave in the symmetric plane intercepts with the compression surface also gradually moves far away from the throat section. As shown in Fig. 19 (a)-(c), the wave structures in the symmetric plane of the inlet are basically the same as the design condition. This similarity occurs mainly because when the Mach number of the incoming f low is lower than the design Mach number,the intensity of the ref lected shock decreases and the ref lected shock wave falls at the upstream of the throat section, which weakens the intensity of interaction between the ref lected shock wave and the boundary layer; thus, the effect on the f low-f ield parameter distribution at the throat section is relatively low.

When the Mach number of the incoming f low changes from 6.0 to 7.0,the incident shock wave moves into the inside of the inlet lip and interacts with both the curved surface at the downstream of the waverider lip and the expansion wave generated from this lip; in this case, the f low structure of the inlet changes greatly, as shown in Figs. 19 (d) and 20. The reasons for these f low structures are further analyzed as follows:

Combining with the three-dimensional shock relationship,the incident shock produces a ref lection shock at the surface downstream of the lip GQ and the ref lection shock detaches from the lip GQ because of its very large aft-swept angle.A lateral pressure gradient in the direction forming the symmetry plane to both sides will be produced in the main f low under the inf luence of the strongly detached ref lection shock, and a lateral pressure gradient with the inverse direction is produced on the surface near the downstream of the lip GQ because the lip GQ emits a three-dimensional expansion wave.The streamwise vortices are generated by the lateral pressure gradient of these two directions,and the stream-wise vortices on both sides of the lip are converged in the isolator to form a vortex pair in the opposite direction to the bottom of the isolator (as shown in Fig.17).Since the aft-swept angle of the lip QS is small,the ref lection shock gradually moves and converges to the downstream of the lip QS.At this time,the expansion wave emitted from the lip QS is located at the upstream of the ref lection shock,which increases the loss of the ref lection shock to a certain extent. In addition, due to the absence of the compensation of the lateral pressure gradient in the symmetry plane, a slip surface is generated by the interaction between the ref lection shock and the incident shock.As shown in Fig.19(d),the slip surface together with the vortices pair as described above constitutes a large low-kinetic f low region,which will affect the aerodynamic performance of the inlet.

Furthermore, the cowl-ref lected shock wave produces a strong continuously ref lected shock in the isolator, and the interaction between that shock and the low-kinetic f low region further increases the f low loss of the inlet.However,the kinetic eff iciency under Ma∞=7.0 inf low conditions is equivalent to that of Ma∞=5.0 inf low conditions in Table 4, which indicates that the inlet still has a high aerodynamic performance under Ma∞=7.0 inf low conditions.

Fig. 19 Mach number contours of inlet center, throat, and exit section under different incoming f low conditions (Case B).

Fig. 20 Pressure contours of symmetrical plane and X-sections near entrance (Case B).

Table 4 Performance parameters of inlet throat and exit section under different incoming f low conditions.

The performance parameters of the inlet under different incoming f low conditions are listed in Table 4. In Table 4, φ,π, σ, Ma and ηKEare mass f low capture coeff icient, pressure ratio, total pressure recovery coeff icient, Mach number and kinetic energy eff iciency respectively, and the subscript ‘th'and‘exit'represent the throat and exit of isolator respectively.Further analysis of the performance parameters of the inlet throat section and the exit section reveals that the pressure ratio and exit Mach number vary little from the throat to the exit section, but the total pressure recovery coeff icient loss is very large. In combination with the Mach number contours from the inlet throat section to the exit section, this loss arises because the cross-section areas in the inlet isolator are unchanged, there is no strong ref lected shock in the isolator,and the pressure ratio and Mach number along the isolator are basically unchanged when there is no strong compression process. However, the secondary f low area produced by the interaction between the ref lected shock wave and the boundary layer in the inlet rapidly enlarges along the f low direction,which leads to a large total pressure loss in the isolator.

According to the change regulation of the inlet kinetic eff iciency,the eff iciency is the highest under the design conditions.When the Mach number of the incoming f low changes from 4.0 to 6.0, the kinetic eff iciency enlarges with the increase of the Mach number. This change indicates that the ability of the inlet to transform the kinetic energy of the incoming f low is simultaneously enhanced. However, under the incoming f low conditions of Ma∞=7.0 and Ma∞=4.0, the kinetic energy eff iciency from the throat section to the exit section drops the most. This response needs to be analyzed in two aspects: (A) under the incoming f low condition of Ma∞=7.0,the complicated waves and strong secondary f low in the inlet will cause a large loss to the aerodynamic performance of the inlet; (B) under the incoming f low condition of Ma∞=4.0, the loss is caused by a large def lection of the incoming f low Mach number from the design Mach number.This behavior shows that when there is a small change in the inlet wave structure, the secondary f low inside the isolator is the main factor that affects the inlet performance. According to the mass f low capture coeff icient of the inlet, though the kinetic eff iciency in Ma∞=7.0 inf low conditions is equivalent to that of Ma∞=5.0 inf low conditions, the inlet still has a high aerodynamic performance under Ma∞=7.0 inf low conditions.

To compare with the conventional design method, a standard inlet designed with the conventional method should be selected as the comparison object in this paper. In Ref.13, a standard example of the inward-turning inlet, optimized by the software ISIGHT,with a circle project shape was designed by the conventional design method. It can be found from the comparing of the Mach number contours at the throat and exit, the distortion of the inlet in this paper is lower than that of the standard inlet. Furthermore, as shown in Table 4, the total pressure coeff icient of the inlet in this paper at the design point is 0.657, which is much higher than 0.581 of the traditional one.

To further test the performance of the inlet in this paper,the maximum pressure and the starting Mach number are also obtained from the numerical simulation.The numerical results show that the maximum pressure of the inlet in this paper is 155 times the incoming static pressure, and the self-starting Mach number is 5.2. This shows that the inlet designed by the method in this paper can basically meet the actual engineering needs.

However,it should be noted that the inlet designed according to this method also increases the external drag along with improving the total pressure recovery coeff icient. However, it is diff icult to obtain the external drag only based on the inlet prof ile. Because the calculation of the external drag is a category of the internal and external f low coupling, and the shape of the aircraft is required,at least the prof ile of the cowl should be given.In a conclusion,it can still be found through the comparison of aerodynamic performance,the inlet designed by this method is better than the conventional one.

6. Conclusions

The following four main conclusions can be drawn from this study:

(1) The basic f low-f ield design method with a controllable velocity direction at the throat section is proposed in this paper. The section compression rule and the throat parameters are decoupled. This conf iguration reduces the restraints on the existence of a solution and balances the section compression rule and the parameter distribution at the throat section. The f lexibility of the design method is enhanced indirectly.

(2) The basic f low-f ield design method is verif ied by the numerical simulation. The numerical results show that the inlet designed according to this method can weaken the intensity of the cowl-ref lected shock wave and restrain the generation of the ref lected shock in the isolator,which is benef icial to the improvement of the total pressure at the inlet. Under the design condition of Ma∞=6, when the outlet Mach number is reduced to 3.00, the exit pressure ratio is 24.20 and the total pressure recovery coeff icient reaches 0.657.

(3) According to the change in the aerodynamic performance parameter at the inlet throat section and the exit section, the secondary f low generated from the interaction between the ref lected shock wave and the boundary layer has a large inf luence on the aerodynamic performance, which provides a feasible direction for the improvement of the inlet design.

(4) To thoroughly absorb the ref lected shock at inlet shoulder, further iteration on the base of the method in this paper is needed to maximize the uniformity of the pressure distribution at the throat section. Because the uneven pressure distribution at the throat section may induce the ref lected shock wave in the isolator, though the stream direction at the throat is tangent to the surface of the isolator. Tt is necessary to further study the effect of the uniformity of the pressure distribution on the intensity of the ref lected shock in the isolator in the future work.

The design method proposed in this paper can improve the aerodynamic performance of inward-turning inlet.However,it should be noted that there are too many design parameters in this method, and setting their values is still complicated. It is necessary to further study the inf luence of these parameters on the aerodynamic performance of the inlet. Furthermore,the inlets designed according to other target parameter distributions at exit sections require further study.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 11702229, 11602207 and 91641103). The authors wish to thank Feng WEI from China Aerodynamics Research and Development Center and the anonymous referees for help in improving the quality of the manuscript.