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3D inverse method of characteristics for hypersonic bump-inlet integration
Acta Astronautica 166 (2020) 11–22
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Research paper
3D inverse method of characteristics for hypersonic bump-inlet integration Zonghan Yu, Guoping Huang , Chen Xia
T
∗
College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, People's Republic of China
ARTICLE INFO
ABSTRACT
Keywords: Waverider-inlet integration Inverse method of characteristics Hypersonic inlet Internal/external coupling flow Mass capturing ratio
Waverider-inlet integration design is an important approach to realize hypersonic flight. The inlet lip is the key factor for internal/external flow field coupling. This paper proposes a novel 3D inverse method of characteristics (MOC) to design the inlet lip with generalized 3D shock. The unit process and marching procedure of the inverse MOC in 2D and 3D focus on obtaining the unique coordinates of the solution points. The accuracy of inverse MOC is verified through its comparison with the analytical solution of a conical flow field. The approach is then applied for a hypersonic bump-inlet integration (The freestream Mach number M∞ = 6.0), where the inlet lip is inversely generated by a prescribed elliptic-conical shock wave. Inviscid results reveal good performance (The mass capturing ratio = 0.813, the Mach number of inlet exit Mexit = 3.71, the total pressure recovery coefficientσ = 0.752). The incident shock is well attached on the inlet lip, which is of high φ. Viscous results show relatively low performance ( = 0.738, Mexit = 2.94, σ = 0.471), which indicates that the new method is a promising solution for the hypersonic internal/external coupling flow. Although the viscous effects should be further considered to improve the design, the proposed method can be applied to 3D surface design with generalized shock shapes.
1. Introduction The hypersonic air-breathing flight vehicle is an important future trend in aeronautical research [1]. Effectively integrating the propulsion system into the airframe is the main challenge to realize hypersonic flight [2]. Nonweiler [3] first proposed the waverider concept as a lifting configuration in 1959. The early waveriders are based on planar or axisymmetric shock. Jones and Sobieczky [4] extended the waverider design based on generalized shock geometries. Afterward, abundant studies were conducted on waverider-based flight vehicles. Lewis et al. [5,6] developed an analytical method to optimize the thrust margin and lift-drag ratio of waveriders. A waverider-derived Mach 5 airplane design was developed by NASA [7], in which all major disciplines are considered and coupled. Takashima and Lewis [8] developed the MAXWARP code to design waveriders, studied the rounding effects of leading edges and off-design performance, and successfully designed a hypersonic vehicle that optimizes the hypersonic trajectory from Mach 6–10 [9]. A modified osculating-cone waverider design was also developed to correspond predicted analytical and actual computed flow fields [10]. Javaid et al. [11,12] developed an integration design method for turbine-based combined cycle. The high lift-drag ratio model allows the operating from take-off to Mach 10 (transition speeds at approximately Mach 5). According to the published researches, the
∗
waverider configuration design has a great development during recent years. The waverider concept increases the lift-to-drag ratio significantly, while it also create a non-uniform inflow for the inlet. It leads to low mass capturing ability. Hence, the design of inlet and the integration part (inlet lip) is also a challenge to the whole propulsion system. The importance of an inlet to the propulsion system in high-speed condition is revealed by the inlet-combustor ratio [13], the converted energy of the inlet over the generated energy of combustion. When the flight Mach number increases from 1.8 to 4.5, the inlet-combustor ratio also increases from 0.12 to 2.3. Meanwhile, the inlet start performance, the back-pressure capacity are two crucial aspects for the hypersonic propulsion system [14], which are closely related to the shock train structure in the inlet isolater. The unstart mechanism of hypersonic inlets (scramjet) is evidently different from that of supersonic inlets (ramjet). Li and Chang [15,16] constructed a prediction dynamic model to study the qualitatively analyze the shock train behavior. They also studied the influences of the incident shocks on the close-loop control and proposed a method which can effectively bypass the negative effects of the incident shocks by experiments [17]. In recent years, the inlet-waverider integration design has received increasing attention. Wang [18] proposed a new design method that employs the practical integration of multistage compression osculating-cones waverider and
Corresponding author. E-mail address: [email protected] (G. Huang).
https://doi.org/10.1016/j.actaastro.2019.09.015 Received 28 April 2019; Received in revised form 12 August 2019; Accepted 15 September 2019 Available online 24 September 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
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Nomenclature MOC 2D 3D M β x y z K λ θ α π p ρ V u v
w c n σ k φ
method of characteristics two-dimensional three-dimensional Mach number shock wave angle freestream direction height direction crosswise direction intercepting height line slope The angle of flow direction Mach cone static pressure ratio static pressure density overall velocity x-direction velocity y-direction velocity
z-direction velocity speed of sound generalized flow properties total pressure recovery coefficient specific heat ratio mass capturing ratio
Superscripts ’
stagnation properties reverse characteristics
Subscripts local + – o
freestream properties true value of pressure left-running characteristics right-running characteristics streamline
inverse method of characteristics (MOC) is proposed to realize this target. This new technique calculates the flow properties of upstream by using that of downstream in the 3D domain, making it different from the previously developed MOC. Thus, the inlet lip surface can be generated inversely by the prescribed incident shock wave. The accuracy of the new inverse MOC is verified, and the result is applied for a bumpinlet integration design. The mass capturing ability is analyzed in detail, and the overall flow properties are identified based on inviscid/ viscous simulation.
Busemann inlet as the third-stage compression surface with good overall performance. The influence of bluntness effects to the waverider flow field was studied by Li [19], who obtained good compromise between aerodynamic and aero-heating performance. Wang [20] reported the integration capability of a wing-body configuration that has an irregular-lip inlet and exhibits a lift-drag ratio of 5.2, high compression efficiency, and flow uniformity. Li and You [21] proposed a dual-waverider design method that could jointly create the waverider lower surface and inward turning inlet configuration. The internal and external flow fields of the inlet are well coupled to increase the mass capturing ability. They also presented an integration design of curvedconical forebody and 3D inward turning inlet [22]. The mass capturing ratio φ reaches 0.93 at Mach 6 and is maintained at 0.86 at Mach 5.0. Qiao [23] proposed a novel waverider/inlet integration design that obtains the φ of 0.976 at Mach 7.0. As demonstrated above, different integration patterns have been proposed in previous studies. While there are also challenges for the integration design. Most studies use pre-compression surface as forebody to keep a good aerodynamic performance, which will occupy a certain space of the airframe. As a result, the space of aircraft decreases and the inlet presetting position is restricted to the most upstream of the aircraft. Thereby, an integration design method with more flexible forebody configuration and inlet presetting position is needed. Considering the serious effects of boundary layer in hypersonic speeds [24], previous studies had placed the inlet at the most upstream of the waverider airframe. The treatment of the boundary layer and non-uniform inflow are two inevitable issues. In addition, the inflow experiences a strong compression because of the bluntness effects [25–27], thereby inducing an entropy surge around the leading edge. The low kinetic energy flow, which includes the entropy and boundary layers, accounts for approximately 50% of the inlet entrance domain [28,29]. Thus, the pressure-controllable bump [13,30] was proposed to efficiently divide the boundary layer. The convex shape creates a longitude and crosswise pressure gradient. The longitude pressure gradient decelerates and pre-compresses the inflow. The crosswise pressure gradient diverts the thick boundary layer asides. Given that the prescribed shock is well attached on the leading edge, the flow of the lower surface is isolated from that of the upper surface. Therefore, the bump provides pre-compressed and high-kinetic-energy flow for the inlet. The typical inlet design method based on uniform flow is not applicable because the flow field around bump is non-uniform. This study focuses on the integration part (i.e., inlet lip) design of the bump-inlet configuration based on the non-uniform inflow. The 3D
2. Development of the inverse MOC 2.1. 2D inverse MOC in the axial-symmetry condition The original MOC is a simplified solver for the steady 2D, planer, irrotational supersonic flow field [31]. During calculation, characteristics are first emitted from known points. The coordinates of solution points are determined by the intersection of the characteristics. The flow properties of solution points are then obtained through compatibility equations. For the inverse MOC, the marching direction is the opposite of the original MOC. The uniqueness of the solution is one of the biggest challenges to the inverse design; thus, the new method is not a simple formal transformation of the original one. For continuous flow field, the flow properties can be obtained by using the characteristic equations according to the original MOC. For a flow field installed with shock wave, the flow properties after the shock boundary are determined by the freestream and the shock strength. Fig. 1 illustrates the unit process of 2D inverse MOC on the shock boundary. The solid line in the figure stands for the physical truth of characteristic lines. The dotted line (straight line) stands for the discretized mesh, which is used in calculation. The relative positions of the given and solution points differ between two subplots. In the left subplot, point 2 is at the left-hand side of Co, 1, whereas in the right subplot, point 2 is at the right-hand side of Co, 1. Point 2 is in the influence region of point 1. The characteristics emitted from point 1 (C±, 1 ) cannot intersect with C±, 2 , that is, the original MOC is not applicable near the shock boundary. Points 1 and 2 are given, whereas points 3 and 3’ are pending. With the left subplot as an example, point 3’ is on C , 1 , and point 3 is on Co, 1. The values (location and flow properties) of points 1 and 2 are needed to obtain the values of points 3 and 3’. Compared with the original MOC that calculates the flow field along characteristics, the proposed MOC uses the downstream values to obtain upstream values. Therefore, an 12
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Fig. 1. Unit process of 2D inverse MOC on the shock boundary.
inverse procedure is required to obtain the shock dependent domain (i.e., the surface that can generate the prescribed shock). The methodology of 2D inverse MOC is presented. All equations of the inverse MOC are deduced in this study, while the equations obey the compatibility of characteristic relations, which are presented in Ref. [27]. With the left subplot as an example, C+, 2 (the blue chain-line) intersects with C , 1 at point 3’. The coordinates of point 3’ are obtained, and the flow properties of point 3’ are calculated by the compatibility relation of Mach lines. The characteristic and compatibility equations are presented below.
dy dx
=
±
= tan( ± ),
±
+
sin dx ± yM cos( ± )
p = RT.
= 0.
(2)
C+, 2 intersects with the Co, 1 at point 3. The values of point 3 are obtained by the compatibility equation of the streamline. The characteristic and compatibility relations are presented below. dy dx
u , v
(3)
VdV + dp = 0 . dp c 2d = 0
(4)
= o
o
=
(5)
Three equations are needed to calculate the flow properties of point 3. Point 4 is on line 12. The location of point 4 is obtained by iteration, which converges when C , 4 passes through point 3. The flow properties can then be calculated by the compatibility equations of Co, 1, C+, 2 , and C , 4 . Accuracy is confirmed by Euler predictor–corrector numerical method. The overall marching direction of inverse MOC is shown in the right subplot of Fig. 2. The shock shape is given and shown as a dark-blue solid line. Points A1, A2, …, An are obtained by the shock relations and presented as mesh grids on the shock shape. The values of points B1, B2, …, Bn-1 are then calculated. A characteristic mesh (dashed line in the right subplot), which includes streamlines and characteristics, is obtained. The streamline A1B1 … X1 is the shock-dependent domain that can induce shock wave A1A2 … An. The calculation does not refer to the gradient of the entropy. The characteristic mesh is used to immediately obtain the target surface. Hence, robustness and accuracy are in good level.
(1)
±
M2 1 dp± ± d V2
Furthermore, the identification of solution points is different from that in the original MOC. With the understanding of the basic unit process of the inverse MOC in 2D condition, the calculation of solution points and the marching process to obtain the shock dependent domain are presented in Fig. 2. In the left subplot, points 1 and 2 are given to calculate point 3. The coordinates of point 3 are obtained by the intersection of Co, 1 and C+, 2 . For the flow properties, θ, p, ρ and T are pending. The flow obeys the assumption of ideal gas, which has the following relation:
The inverse MOC is based on the spacing prior to the solution points along the characteristics. The governing equations are similar to those in the original MOC, whereas the marching direction is opposite.
Fig. 2. Calculation of the solution point (left) and marching process to obtain the corresponding shock dependent domain (right). 13
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2.2. 3D inverse MOC based on generalized shock
Thus, five compatibility equations of three characteristics are needed for the 3D inverse MOC calculation. The calculation procedure of 3D inverse MOC is proposed and shown in Fig. 3. Points A–F are known, and point P is pending. The detailed procedure is revealed as follows (Fig. 4). 1) The location of P is determined by the intersection of ConoidE and Co, B . The equation of the streamline Co, B is shown as
The inverse MOC in 3D condition is an extension of that in 2D condition. However, determining the location of solution points is more complex than that in the 2D case. Given that the isentropic compression is used in the inlet design, the equation of sound speed is used to replace the energy equation. The governing equations are shown as follows:
u x + vy + wz + u
x
+v
y
+w
z
xP = xB + uB t yP = yB + vB t . zP = zB + wB t
=0
uu x + vu y + wuz + px = 0 uvx + vvy + wvz + py = 0
.
uwx + vwy + wwz + pz = 0 c 2 (u
upx + vpy + wpz
x
+v
y
For ConoidE , point P obeys the difference equation as
+ w z) = 0
(6)
[u2
In supersonic condition, the equation set above is hyperbolic. Instead of the two characteristics in 2D domain, the characteristics in 3D condition are in a cone shape. The parameters on the characteristics and streamlines follow the relations below:
streamlines: un x + vn y + wn z = 0,
(7)
characteristics: un x + vn y + wnz = c.
(8)
dx dt
= u,
Mach cone: [u2 y2
[w 2
dy dt
= v,
(V 2
(V 2
dz dt
= w,
c 2)] dx 2 + [v 2
(yP (yP
c 2)] d
c 2)] dz 2,
+ + 2uvdxdy + 2uwdxdz + 2vwdydz = 0
(10)
uut + vvt + wwt + pt = 0 pt
c2
t
=0
[(n x2
1) u x + (n y2
yB )(zP
streamlines:
pt +
1) vy + (nz2
xB ) 2 + [v 2 (V 2
(V 2
c 2)](z
yB ) + 2uw (xP
zB
P
xB )(zP
c 2)] )2 zB ) + 2vw
. (14)
zB ) = 0
uB ) +
B vB (vP
wB (wP wB ) + pP (pP pB ) cB ( P
k ck nxk (uP
uk ) + k ck nyk (vP
2 + tk k ck2 (n xk
vB ) +
B
pB = 0 B) = 0 vk ) + k ck n zk (wP
2 1) uxk + n yk
2 1 v yk + nzk
(15) wk )
(pP
pk )
1 wzk
(k = x , y, z )
+(uyk + vxk ) nxk nyk + (vzk + w yk ) n zk nyk + (w xk + x zk ) nwk nxk ] = 0
(11)
cn x ut + cn y vt + cnz wt characteristics:
+
xB )(yP
characteristics:
,
[w 2
B uB (uP
where V is the resultant velocity. For a 3D flow field, the values of solution points can be solved through the intersection area between the Mach cone and the stream surface, which obeys the following equations:
streamlines:
yB
c 2)](xP )2
2) As analyzed before, five independent compatibility equations are needed to obtain the flow properties of P, and points Am and Bm are introduced to satisfy the definite condition. The flow properties of point P are initialized by those of point B. 3) ConoidP is then emitted and intersects with line AB at point Am. The flow properties of point Am are linearly interpolated by points A and B. 4) The flow properties of P are then reassigned by those of Am. Steps 3) and 4) are used in iteration to obtain the steady results. Point P is completely solved when the streamline (Co, B ) and the characteristics (C+, E and C±, P ) are determined. The corresponding equations are shown as follows:
(9)
(V 2
(V 2
+ 2uv (xP
The deducing equation is completed in Ref. [31]. The corresponding compatibility equations are shown as follows:
streamlines:
(13)
(16)
c2 2.3. Verification of the inverse method
,
1) wz
+ (u y + vx ) n x n y + (uz + wx ) n x nz + (vz + wy ) n y n z ] = 0
The efficiency and the accuracy of the new inverse method is verified in this section. The flow field around the 40° cone, which was subjected to Mach 2 freestream, is used for verification. The analytical solution to the conical flow is provided by solving Taylor–Maccoll equation for comparison. The shock shape is given to obtain the
(12) where subscript t stands for the partial derivative of the value. Rusanov [32] proved that in 3D steady isentropic supersonic condition, every three characteristics on the Mach cone are dependent on each other.
Fig. 3. Schematic of 3D inverse MOC. 14
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Fig. 4. Detailed calculation process of 3D inverse MOC.
corresponding shock-dependent curve. Fig. 5 shows the result of the inverse design under 2D condition. The Mach distribution corresponds well with the features of the flow field around the cone. The cone curve is generated accurately except for the first grid where a small inflection is observed. This small inflection is the intersection point between the shock and its dependent domain, thus resulting in singularity. The relative error can be relieved by increasing the mesh density. The conical flow with the 40° cone angle and three different Mach numbers are used to further verify the accuracy. The results in Table 1 show that the relative error between inverse design and the analytical solution is less than 3% for every condition. In addition, the accuracy increases with the mesh density and Mach number. Hence, the inverse method is accurate for 2D design. Under 3D condition, the cone angle is given as 30° to calculate the cone. The analytical solution is provided by solving the Taylor–Maccoll equation, which is also used in 2D verification. The results of 3D inverse design are shown in Fig. 6. According to the Mach contour of the longitudinal section, the flow field of the design corresponds well with conical flow field. The relative error is less than 4.5%. The accuracy of 3D inverse design decreases compared with that of 2D design but can also be relieved by increasing the mesh density. Thus, the accuracy of the 3D inverse MOC is at an acceptable level for engineering design.
Fig. 5. Mach contour of 2D axisymmetric cone solved by the inverse MOC (Mach 2.0, cone angle 40°).
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Table 1 Relative error between inverse design and Taylor–Maccoll solution (cone angle 40°). Mesh number
Mach 2
Mach 5
Mach 7
100 200 300
2.00e-3 1.01e-3 6.78e-4
1.55e-4 8.01e-5 5.37e-5
7.92e-5 4.12e-5 2.72e-5
3. Bump-inlet integration based on the inverse MOC 3.1. Inlet lip (shock-dependent domain) design The non-uniform flow treatment is important for propulsion/airframe integration design. The typical inlet design is based on uniform freestream. Two challenges are faced by the inlet design when subjected to non-uniform flow. On the one hand, a thick boundary layer develops along the airframe. Large energy loss occurs in the propulsion system, and the low kinetic energy flow is swallowed into the inlet. On the other hand, typical osculating theory is not applicable because the inflow is not uniform. Maintaining a high φ while using the typical inlet design is difficult. In a previous study, a bump configuration was introduced to efficiently remove the boundary layer. The bump generates an expanding pattern flow field, which has a large crosswise flow transference (Fig. 7). The flow field leads to the overlarge side-compression of the inlet. Thus, the inlet design must consider the features of non-uniform flow. This paper proposes an innovative 3D inverse MOC, in which the surface is designed based on the prescribed shock shape and the given non-uniform flow field. The elliptic cone is selected as the incident-shock shape in the inlet design. Compared with the normal cone, the elliptic cone can control the shock shape by adjusting the major/minor axis and thus has great flexibility in the bump-inlet integration design. The shape of incident shock is determined by the inlet lip. Thus, the design of the inlet lip influences the mass capturing and flow properties of inlets. The incident shock design subjected to non-uniform flow is shown in Fig. 8. Central cylinder is used in the shock design to ensure the uniqueness of the reflection shock, thus avoiding the problem of center-line singularity. The central cylinder is located between points O and G. The design procedure of the inlet lip based on the 3D inverse MOC is presented in this section (Fig. 9). The projection of the inlet lip on the yoz-plane is first designed. Red-solid line ABC is the projection of the front lip curve, and black-solid line ADC is the projection of the rear lip curve. Curve AGC is obtained by projecting ADC to the elliptic cone. Curve AHC is obtained by the intersection of the elliptic shock and the
Fig. 7. Crosswise-deflection-angle (β) distribution on bump surface.
bump. The incident shock is obtained by extracting the area between AGC and AHC and is shown in red color. Under the design condition, the incident shock is induced from upstream position (i.e., points A and C), and finally converges at point G. Thus, the incident shock is well attached to the inlet lip and consequently increases the φ of inlet. At low speeds, the incident shock deflects away from point G, thereby inducing flow spillage to ensure the inlet starting performance. After the shock shape is determined, the final step is the design of the shock-dependent area (i.e., inlet lip). According to the shock position, the flow upstream properties on the incident shock are obtained by inviscid simulation of the bump. The inlet lip surface is generated by the 3D inverse MOC. The inlet wall is generated by classical osculatingflow theory. The bump-inlet integration model is obtained as illustrated in Fig. 10. The light-blue line (i.e., leading edge) around the bump is the intersection part with airframes. The red surface (i.e., inlet lip) is the shock-dependent area generating the prescribed incident shock. The installation view is shown in the left-top subplot and is integrated with the airframe with the leading edge. The freestream is developed along the airframe and impinges on the bump. The boundary layer is removed, and high kinetic energy flow fills up the near-wall region. The
Fig. 6. Results of 3D inverse design (left) and comparison with the analytical solution (right). 16
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Fig. 8. Schematic of incident shock design subjected to non-uniform flow.
front view shows the inlet internal surface. On the one hand, the surface conducts a 3D compression to the inflow, which is more efficient than the typical 2D compression. On the other hand, the surface adjacent to the bump is geometrically continuous with the bump, thus ensuring the balanced compression of the captured flow in every direction.
3.2. Analysis of aerodynamic characteristics based on inviscid/viscous simulation A structured mesh is generated for inviscid and viscous numerical simulation to verify the integration performance of the design. The flat
Fig. 9. Design progress of inlet lip based on the new 3D inverse MOC. 17
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Fig. 10. Inlet lip design based on 3D inverse MOC.
Fig. 11. Structured mesh of the bump-inlet integration model.
plane is used to integrate with bump. The surface mesh elements are shown in Fig. 11. A C-shaped mesh (copper colored) is set around the bump leading edge. Another C-shaped mesh (blue colored) is set on the bump surface to accurately fit the bump leading edge. A cowl lip (red colored) is used to avoid overlarge expanding external-flow. An Oshaped mesh is set for every cross-section of inlet to fit the elliptic crosssection of the inlet. The freestream atmosphere condition is at 24 kmhigh altitude and Mach 6. For the viscous simulation, a 3.5 m longitude flat plane is set to produce the boundary layer. The minimum mesh height of near-wall grid element is set at 0.15 mm to meet the requested standard k–ε turbulence model. The grids in y-direction are stretched with the increasing ratio of 1.2 refined with geometric proportion rule. The entire wall is adopted with no slip and adiabatic wall conditions. All results are calculated by the CFX commercial software based on high resolution in numeric turbulence and advection scheme. The numerical methods in this study have been effectively validated by You and Liang [33], who used wind tunnel experiments and showed that these methods can calculate reasonable and reliable results. The numerical methods have also been validated in the previous study [30] through a comparison with an experimental study of bump-inlet integration shown in Fig. 12. The results show that the difference of static pressure between numerical and experimental results is less than 2.7, which indicates that the two results agree with each other. Moreover, the numerical methods effectively calculate the true flow field. The overall performance of the integration model is shown in Table 2, which includes the total pressure recovery coefficient (σ), Mach number of the inlet exit (Mexit), and mass capturing ratio (φ). The parameter φ is the mass flow of the inlet divided by that of the virtual free-flow tube. The free-flow tube is a projective plane that x-reversely extends the inlet lip to the xoz-plane. The energy loss of inviscid simulation is in an acceptable range for an integration design. According to φ value and the shock structure in Fig. 13, the 3D inverse MOC
Fig. 12. Longitude static pressure distribution of the bump-inlet integration.
Table 2 Inlet flow parameters of inviscid and viscous simulation. Simulation type
σ
Mexit
φ
Inviscid condition Viscous condition
0.752 0.471
3.71 2.94
0.813 0.738
efficiently solves the integration design problem. The viscous simulation considers the boundary layer effects when the Mexit decreases correspondingly. The φ decreases by 9.2% according to the shock/ boundary layer interaction. Hence, the 3D inverse MOC can realize the high mass capturing in inviscid condition, whereas the boundary layer removal ability of the bump and the correction of boundary layer still need improvement. The inlet mass capturing performance is studied based on the inviscid numerical simulation shown in Fig. 13. The incident shock shape is depicted by the isolines of Mach number. The shock is induced from the leading edge of inlet and thoroughly covers the inlet lip. In addition, the incident shock is almost kept in one plane for the design. The 3D inverse design generates the corresponding lip surface that can induce the prescribed incident shock. The flow compression is observed in the inlet entrance and the semisection views. The inflow enters the inlet and experiences a pressure increase. Afterward, a pressure drop, which results from the overlarge side compression, occurs downstream the incident shock and leads to flow acceleration again. The bump configuration creates an expanding flow pattern. The inflow changes the flow direction from expanding (bump) to compressing (inlet). Hence, the 18
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Fig. 13. Inlet mass capturing performance of the integration design (inviscid condition).
flow experiences larger side compression than the inlet with parallel inflow. As a result, a side-embedded compression wave is induced inside the inlet. The structure was analyzed in detail in Ref. [13]. The inlet performance in viscous condition is studied, and the results are shown in Fig. 14. Two major flow compression waves, namely, the bump and inlet incident shocks, are on the bump surface. The bump shock is induced when the inflow encounters the bump leading edge. The second compression is the incident shock. Flow spillage occurs at the central part of the inlet entrance due to the shock/boundary layer interaction. Thus, a small bulging distortion of incident shock emerges. The distorted shock is still within the bump shock, indicating that the distortion affects the mass capturing performance of inlet. However, the shock structure is simple, and no shock/shock interaction occurs. The incident and the reflection shock are illustrated in the side view of the inlet. In the right subplot, the 3D shock structure is observed according to the pressure isolines of the inlet wall. The shock structure inside the inlet is overall simple. The compression level of the upper inlet wall is stronger than that of the lower wall due to the length of boundary layer development. For the upper surface, the boundary layer starts to develop at the central position of the inlet lip. The low kinetic energy fluid is in a small level. For the lower surface, the boundary layer starts to develop from the flat plate. Although most boundary
layers have been diverted by bump, low kinetic energy fluid occupies the near-wall region of the lower wall. Thus, the shock/boundary interaction near the lower wall makes the compression weaker than that on the upper surface. The π and σ contours of inlet throat and exit are shown in Fig. 15. The incident shock profile of the shock profile of inlet is depicted by the π isolines in subplot a. The incident shock attaches well at two sides of the inlet lip, while there is flow spillage at the compression center, which leads to the distortion of the shock shape. Besides, the sideembbeded shock (SES) is also observed at the inlet bottom surface in viscous results. The overlarge side compression induces a pressure surge at the bottom surface, which leads to the flow capacity of inlet and the energy loss at the inlet throat (subplot b). The flow spillage occurs at the compression center (labeled in red color), which distorts the incident shock shape. Thus, the shock distortion and the SES effects are both related to the overlarge side compression. The π and σ contours of inlet exit are shown in subplot c and d. The flow compression and energy of the half bottom keep in a lower level than those of upper half part, which reveals that the energy loss by SES effects reduce the flow performance of the inlet half bottom part. Besides, the back-pressure capability of the bump-inlet configuration is preliminarily studied in this paper. The π contours of the
Fig. 14. Shock structure of the bump-inlet integration (viscous condition).
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Fig. 15. π and σ contours of inlet throat and exit.
longitude cross-section at various back-pressure (Pb) is shown in Fig. 16. The π of inlet exit in the through-flow condition is 17.0, which is shown in the subplot a of Fig. 16. The end shock is formed while increasing Pb. The position of the end shock moves upstream while Pb increasing. The low kinetic flow accumulates near the inlet wall. It prevents the flow to pass through smoothly by reducing the effective flow area. When Pb/P∞ reaches 100, the end shock locates at approximately x = 2.5 m. The detail flow properties under various back-pressures are demonstrated in Table 3. The three parameters decrease significantly with the increase of the Pb. The ηKE is the kinetic energy efficiency of inlet, which evaluates energy loss under different freestream conditions overall [13]. It is defined as follows. KE
1
=
1 2
M
2
T T
1+
1 2
M2
4. Conclusion This study developed an innovative inverse MOC that can design the generalized 3D surface based on the prescribed shock shape. The unit process and marching procedure of the inverse MOC are presented. This new method adopts reverse spacing to obtain the flow properties of the backward characteristics. The auxiliary points are used to acquire the unique location of solution points. The accuracy of the method is verified through a test of a conical flow field with analytical solution for comparison. The accuracy of 2D axisymmetric design is high, but that of 3D design is relatively low. Nonetheless, the result satisfies the requirement for engineering design. The inverse MOC is applied in a hypersonic bump-inlet integration design (M∞ = 6.0). The inlet lip is generated by a prescribed elliptic-conical shock wave. Inviscid results showed good performance of the inverse MOC ( = 0.813, Mexit = 3.71, σ = 0.752). The incident shock is well attached on the inlet lip, which is of high φ. By contrast, viscous results showed relatively low performance ( = 0.738, Mexit = 2.94, σ = 0.471) of the inverse MOC due the interaction effects between the unremoved boundary layer and the incident shock. The side-embedded shock effect is observed in the inlet bottom surface, which influence the inlet starting performance in lower Mach number condition. The max back-pressure capacity of the inlet is validated as Pb/P∞ = 100 in Mach 6.0 condition.
1
(17)
When Pb/P∞ is larger than 100, the periodic fluctuation of mass flow is observed in the numerical simulation. The mass flow of exit cannot keeps at a steady value, thus making the calculation not able to converge. It reveals the flow capacity of inlet has reached its maximum limit at approximately Pb/P∞ = 100.
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Fig. 16. Back-pressure capability of the bump-inlet configuration. Table 3 Flow properties of back-pressure capability study. Pb/P∞
σ
φ
ηKE
17 38 62 83 89 100
0.471 0.364 0.289 0.220 0.201 0.175
0.738 0.735 0.730 0.726 0.724 0.722
0.967 0.954 0.941 0.925 0.919 0.910
[4] K.D. Jones, F.C. Dougherty, A.R. Seebass, H. Sobieczky, Waverider design for generalized shock geometries, J. Spacecr. Rocket. 32 (1971) 957–963. [5] M.K.L. O'Neill, M.J. Lewis, Optimized scramjet integration on a waverider, J. Aircr. 29 (29) (1992) 1114–1121. [6] M.J. Lewis, N. Takashima, Engine/airframe Integration for Waverider Cruise Vehicles, (1993), https://doi.org/10.2514/6.1993-507. [7] R.J. Pegg, J.L. Hunt, D.H. Petley, L. Burkhardt, D.R. Stevens, P.L. Moses, S.Z. Pinckney, H.Z. Kabis, K.A. Spoth, W.M. Dziedzic, Design of a hypersonic waverider-derived airplane, Aiaa Aerospace Sciences Meeting & Exhibit, 1993 AIAA 1993-0401. [8] N. Takashima, M.J. Lewis, Navier-Stokes computation of a viscous optimized waverider, J. Spacecr. Rocket. 31 (No. 3) (1994) 383–391. [9] N. Takashima, M.J. Lewis, Optimization of Waverider-Based Hypersonic Cruise Vehicles with Off-Design Considerations, (1999), https://doi.org/10.2514/2.2430. [10] M.J. Lewis, M.-L. Chauffour, Shock-based waverider design with pressure gradient corrections and computational simulations, J. Aircr. 42 (No. 5) (2005) 1350–1352. [11] K.H. Javaid, V.C. Serghides, Thrust-matching requirements for the conceptual design of hypersonic waverider vehicles, J. Aircr. 42 (No. 4) (2011) 1055–1064. [12] K.H. Javaid, V.C. Serghides, Airframe-propulsion integration methodology for waverider-derived hypersonic cruise aircraft design concepts, J. Spacecr. Rocket. 42 (No. 4) (2015) 663–671. [13] Z. Yu, G. Huang, C. Xia, J. Sesterhenn, A pressure-controllable bump based on the pressure-ridge concept, Aero. Sci. Technol. 87 (2019) 133–140. [14] J. Chang, L. Nan, K. Xu, B. Wen, D. Yu, Recent research progress on unstart mechanism, detection and control of hypersonic inlet, Prog. Aerosp. Sci. 89 (2017) 1–22, https://doi.org/10.1016/j.paerosci.2016.12.001. [15] N. Li, J.T. Chang, K.J. Xu, D.R. Yu, W. Bao, Y.P. Song, Oscillation of the shock train in an isolator with incident shocks, Phys. Fluids (2018), https://doi.org/10.1063/1. 5053451. [16] N. Li, J.T. Chang, K.J. Xu, D.R. Yu, W. Bao, Y.P. Song, Prediction dynamic model of shock train with complex background waves, Phys. Fluids 29 (No. 11) (2017) 116103, https://doi.org/10.1063/1.5000876. [17] N. Li, J. Chang, K. Xu, D. Yu, Y. Song, Closed-loop control of shock train in inletisolator with incident shocks, Exp. Therm. Fluid Sci. 103 (2019) 355–363, https:// doi.org/10.1016/j.expthermflusci.2019.01.033. [18] X.-d. Wang, J.-f. Wang, Z.-j. Lyu, A new integration method based on the coupling of mutistage osculating cones waverider and Busemann inlet for hypersonic airbreathing vehicles, Acta Astronaut. 126 (2016) 424–438. [19] S. Li, Z. Wang, W. Huang, S. Xu, L. Yan, Aerodynamic performance investigation on waverider with variable blunt radius in hypersonic flows, Acta Astronaut. 137 (2017) 362–372. [20] J. Wang, J. Cai, C. Liu, Y. Duan, Y.U. Yaojie, Aerodynamic configuration integration
Compared with convnetional integration design method, the new method designs the inlet lip with regard of the flow structure of the airframe forebody, which is non-uniform and boundary layer derived. The new technique is a promising solution for the coupling design of hypersonic internal/external flow. Nonetheless, the viscous effects (including boundary layer) should be further considered to improve the design. Acknowledgments This study is supported by the national Natural Science Foundation of China (No.11372134) and the Funding of Jiangsu Innovation Program for Graduate Education (KYLX15_0259). References [1] D. Sziroczak, H. Smith, A review of design issues specific to hypersonic flight vehicles, Prog. Aerosp. Sci. 84 (2016) S0376042115300087. [2] F. Ding, J. Liu, C.-b. Shen, W. Huang, Z. Liu, S.-h. Chen, An overview of waverider design concept in airframe/inlet integration methodology for air-breathing hypersonic vehicles, Acta Astronaut. 152 (2018) 639–656. [3] T.R.F. Nonweiler, Aerodynamic problems of manned space vehicles, Aeronaut. J. 63 (585) (1959) 521–528.
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design of hypersonic cruise aircraft with inward-turning inlets, Chin. J. Aeronaut. 30 (No. 4) (2017) 1349–1362. Y. Li, A. Ping, C. Pan, R. Chen, Y. You, Integration Methodology for WaveriderDerived Hypersonic Inlet and Vehicle Forebody, (2014), https://doi.org/10.2514/ 6.2014-3229. L.I. Yiqing, X. Zhou, C. Zhu, Y. You, Integration design and analysis for curved conical forebody and three-dimensional inward turning inlet, J. Aerosp. Power 33 (01) (2018) 87–96 (in chinese). Q. Wenyou, Y.U. Anyuan, Y. Dawei, L.E. Jialing, Integration design of inwardturning inlets based on forebody shock wave, Acta Aeronautica Astronautica Sinica 39 (10) (2018) 65–76 (in chinese). D. Knight, M. Mortazavi, Hypersonic shock wave transitional boundary layer interactions - a review, Acta Astronaut. 151 (2018) S0094576518301012. K.Y. Lau, Hypersonic boundary-layer transition: application to high-speed vehicle design, J. Spacecr. Rocket. 45 (No. 2) (2008) 176–183. J.X. Liu, Z.X. Hou, G.H. Ding, X.Q. Chen, X.Q. Chen, Numerical and experimental study on waverider with blunt leading edge, Comput. Fluid 84 (No. 19) (2013)
203–217. [27] J.M. Luckring, Initial experiments and analysis of blunt-edge vortex flows for VFE-2 configurations at NASA Langley, USA, Aero. Sci. Technol. 24 (No. 1) (2013) 10–21. [28] P.L. Lawing, C.B. Johnson, Inlet boundary-layer shapes on four aircraft forebodies at Mach 6, J. Aircr. 15 (No. 1) (2015). [29] C.B. Johnson, P.L. Lawing, Mach 6 flowfield survey at the engine inlet of a research airplane, J. Aircr. 14 (1977), https://doi.org/10.2514/3.44604. [30] Z. Yu, G. Huang, C. Xia, Inverse design and Mach 6 experimental investigation of a pressure controllable bump, Aero. Sci. Technol. 81 (2018) 204–212. [31] M.J. Zucrow, J.D. Hoffman, 2, Gas Dynamics vol. 1, John Wiley and Sons, New York, 1976 68:69. [32] V.V. Rusanov, Characteristics of the general equations of gas dynamics, USSR Comput. Math. Math. Phys. 3 (No. 3) (1963) 674–698. [33] Y. You, D. Liang, R. Guo, High enthalpy wind tunnel tests of three-dimensional section controllable internal waverider hypersonic inlet, Aiaa Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2013, https://doi.org/10.2514/6.2009-31.
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Research paper
3D inverse method of characteristics for hypersonic bump-inlet integration Zonghan Yu, Guoping Huang , Chen Xia
T
∗
College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, People's Republic of China
ARTICLE INFO
ABSTRACT
Keywords: Waverider-inlet integration Inverse method of characteristics Hypersonic inlet Internal/external coupling flow Mass capturing ratio
Waverider-inlet integration design is an important approach to realize hypersonic flight. The inlet lip is the key factor for internal/external flow field coupling. This paper proposes a novel 3D inverse method of characteristics (MOC) to design the inlet lip with generalized 3D shock. The unit process and marching procedure of the inverse MOC in 2D and 3D focus on obtaining the unique coordinates of the solution points. The accuracy of inverse MOC is verified through its comparison with the analytical solution of a conical flow field. The approach is then applied for a hypersonic bump-inlet integration (The freestream Mach number M∞ = 6.0), where the inlet lip is inversely generated by a prescribed elliptic-conical shock wave. Inviscid results reveal good performance (The mass capturing ratio = 0.813, the Mach number of inlet exit Mexit = 3.71, the total pressure recovery coefficientσ = 0.752). The incident shock is well attached on the inlet lip, which is of high φ. Viscous results show relatively low performance ( = 0.738, Mexit = 2.94, σ = 0.471), which indicates that the new method is a promising solution for the hypersonic internal/external coupling flow. Although the viscous effects should be further considered to improve the design, the proposed method can be applied to 3D surface design with generalized shock shapes.
1. Introduction The hypersonic air-breathing flight vehicle is an important future trend in aeronautical research [1]. Effectively integrating the propulsion system into the airframe is the main challenge to realize hypersonic flight [2]. Nonweiler [3] first proposed the waverider concept as a lifting configuration in 1959. The early waveriders are based on planar or axisymmetric shock. Jones and Sobieczky [4] extended the waverider design based on generalized shock geometries. Afterward, abundant studies were conducted on waverider-based flight vehicles. Lewis et al. [5,6] developed an analytical method to optimize the thrust margin and lift-drag ratio of waveriders. A waverider-derived Mach 5 airplane design was developed by NASA [7], in which all major disciplines are considered and coupled. Takashima and Lewis [8] developed the MAXWARP code to design waveriders, studied the rounding effects of leading edges and off-design performance, and successfully designed a hypersonic vehicle that optimizes the hypersonic trajectory from Mach 6–10 [9]. A modified osculating-cone waverider design was also developed to correspond predicted analytical and actual computed flow fields [10]. Javaid et al. [11,12] developed an integration design method for turbine-based combined cycle. The high lift-drag ratio model allows the operating from take-off to Mach 10 (transition speeds at approximately Mach 5). According to the published researches, the
∗
waverider configuration design has a great development during recent years. The waverider concept increases the lift-to-drag ratio significantly, while it also create a non-uniform inflow for the inlet. It leads to low mass capturing ability. Hence, the design of inlet and the integration part (inlet lip) is also a challenge to the whole propulsion system. The importance of an inlet to the propulsion system in high-speed condition is revealed by the inlet-combustor ratio [13], the converted energy of the inlet over the generated energy of combustion. When the flight Mach number increases from 1.8 to 4.5, the inlet-combustor ratio also increases from 0.12 to 2.3. Meanwhile, the inlet start performance, the back-pressure capacity are two crucial aspects for the hypersonic propulsion system [14], which are closely related to the shock train structure in the inlet isolater. The unstart mechanism of hypersonic inlets (scramjet) is evidently different from that of supersonic inlets (ramjet). Li and Chang [15,16] constructed a prediction dynamic model to study the qualitatively analyze the shock train behavior. They also studied the influences of the incident shocks on the close-loop control and proposed a method which can effectively bypass the negative effects of the incident shocks by experiments [17]. In recent years, the inlet-waverider integration design has received increasing attention. Wang [18] proposed a new design method that employs the practical integration of multistage compression osculating-cones waverider and
Corresponding author. E-mail address: [email protected] (G. Huang).
https://doi.org/10.1016/j.actaastro.2019.09.015 Received 28 April 2019; Received in revised form 12 August 2019; Accepted 15 September 2019 Available online 24 September 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.
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Nomenclature MOC 2D 3D M β x y z K λ θ α π p ρ V u v
w c n σ k φ
method of characteristics two-dimensional three-dimensional Mach number shock wave angle freestream direction height direction crosswise direction intercepting height line slope The angle of flow direction Mach cone static pressure ratio static pressure density overall velocity x-direction velocity y-direction velocity
z-direction velocity speed of sound generalized flow properties total pressure recovery coefficient specific heat ratio mass capturing ratio
Superscripts ’
stagnation properties reverse characteristics
Subscripts local + – o
freestream properties true value of pressure left-running characteristics right-running characteristics streamline
inverse method of characteristics (MOC) is proposed to realize this target. This new technique calculates the flow properties of upstream by using that of downstream in the 3D domain, making it different from the previously developed MOC. Thus, the inlet lip surface can be generated inversely by the prescribed incident shock wave. The accuracy of the new inverse MOC is verified, and the result is applied for a bumpinlet integration design. The mass capturing ability is analyzed in detail, and the overall flow properties are identified based on inviscid/ viscous simulation.
Busemann inlet as the third-stage compression surface with good overall performance. The influence of bluntness effects to the waverider flow field was studied by Li [19], who obtained good compromise between aerodynamic and aero-heating performance. Wang [20] reported the integration capability of a wing-body configuration that has an irregular-lip inlet and exhibits a lift-drag ratio of 5.2, high compression efficiency, and flow uniformity. Li and You [21] proposed a dual-waverider design method that could jointly create the waverider lower surface and inward turning inlet configuration. The internal and external flow fields of the inlet are well coupled to increase the mass capturing ability. They also presented an integration design of curvedconical forebody and 3D inward turning inlet [22]. The mass capturing ratio φ reaches 0.93 at Mach 6 and is maintained at 0.86 at Mach 5.0. Qiao [23] proposed a novel waverider/inlet integration design that obtains the φ of 0.976 at Mach 7.0. As demonstrated above, different integration patterns have been proposed in previous studies. While there are also challenges for the integration design. Most studies use pre-compression surface as forebody to keep a good aerodynamic performance, which will occupy a certain space of the airframe. As a result, the space of aircraft decreases and the inlet presetting position is restricted to the most upstream of the aircraft. Thereby, an integration design method with more flexible forebody configuration and inlet presetting position is needed. Considering the serious effects of boundary layer in hypersonic speeds [24], previous studies had placed the inlet at the most upstream of the waverider airframe. The treatment of the boundary layer and non-uniform inflow are two inevitable issues. In addition, the inflow experiences a strong compression because of the bluntness effects [25–27], thereby inducing an entropy surge around the leading edge. The low kinetic energy flow, which includes the entropy and boundary layers, accounts for approximately 50% of the inlet entrance domain [28,29]. Thus, the pressure-controllable bump [13,30] was proposed to efficiently divide the boundary layer. The convex shape creates a longitude and crosswise pressure gradient. The longitude pressure gradient decelerates and pre-compresses the inflow. The crosswise pressure gradient diverts the thick boundary layer asides. Given that the prescribed shock is well attached on the leading edge, the flow of the lower surface is isolated from that of the upper surface. Therefore, the bump provides pre-compressed and high-kinetic-energy flow for the inlet. The typical inlet design method based on uniform flow is not applicable because the flow field around bump is non-uniform. This study focuses on the integration part (i.e., inlet lip) design of the bump-inlet configuration based on the non-uniform inflow. The 3D
2. Development of the inverse MOC 2.1. 2D inverse MOC in the axial-symmetry condition The original MOC is a simplified solver for the steady 2D, planer, irrotational supersonic flow field [31]. During calculation, characteristics are first emitted from known points. The coordinates of solution points are determined by the intersection of the characteristics. The flow properties of solution points are then obtained through compatibility equations. For the inverse MOC, the marching direction is the opposite of the original MOC. The uniqueness of the solution is one of the biggest challenges to the inverse design; thus, the new method is not a simple formal transformation of the original one. For continuous flow field, the flow properties can be obtained by using the characteristic equations according to the original MOC. For a flow field installed with shock wave, the flow properties after the shock boundary are determined by the freestream and the shock strength. Fig. 1 illustrates the unit process of 2D inverse MOC on the shock boundary. The solid line in the figure stands for the physical truth of characteristic lines. The dotted line (straight line) stands for the discretized mesh, which is used in calculation. The relative positions of the given and solution points differ between two subplots. In the left subplot, point 2 is at the left-hand side of Co, 1, whereas in the right subplot, point 2 is at the right-hand side of Co, 1. Point 2 is in the influence region of point 1. The characteristics emitted from point 1 (C±, 1 ) cannot intersect with C±, 2 , that is, the original MOC is not applicable near the shock boundary. Points 1 and 2 are given, whereas points 3 and 3’ are pending. With the left subplot as an example, point 3’ is on C , 1 , and point 3 is on Co, 1. The values (location and flow properties) of points 1 and 2 are needed to obtain the values of points 3 and 3’. Compared with the original MOC that calculates the flow field along characteristics, the proposed MOC uses the downstream values to obtain upstream values. Therefore, an 12
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Fig. 1. Unit process of 2D inverse MOC on the shock boundary.
inverse procedure is required to obtain the shock dependent domain (i.e., the surface that can generate the prescribed shock). The methodology of 2D inverse MOC is presented. All equations of the inverse MOC are deduced in this study, while the equations obey the compatibility of characteristic relations, which are presented in Ref. [27]. With the left subplot as an example, C+, 2 (the blue chain-line) intersects with C , 1 at point 3’. The coordinates of point 3’ are obtained, and the flow properties of point 3’ are calculated by the compatibility relation of Mach lines. The characteristic and compatibility equations are presented below.
dy dx
=
±
= tan( ± ),
±
+
sin dx ± yM cos( ± )
p = RT.
= 0.
(2)
C+, 2 intersects with the Co, 1 at point 3. The values of point 3 are obtained by the compatibility equation of the streamline. The characteristic and compatibility relations are presented below. dy dx
u , v
(3)
VdV + dp = 0 . dp c 2d = 0
(4)
= o
o
=
(5)
Three equations are needed to calculate the flow properties of point 3. Point 4 is on line 12. The location of point 4 is obtained by iteration, which converges when C , 4 passes through point 3. The flow properties can then be calculated by the compatibility equations of Co, 1, C+, 2 , and C , 4 . Accuracy is confirmed by Euler predictor–corrector numerical method. The overall marching direction of inverse MOC is shown in the right subplot of Fig. 2. The shock shape is given and shown as a dark-blue solid line. Points A1, A2, …, An are obtained by the shock relations and presented as mesh grids on the shock shape. The values of points B1, B2, …, Bn-1 are then calculated. A characteristic mesh (dashed line in the right subplot), which includes streamlines and characteristics, is obtained. The streamline A1B1 … X1 is the shock-dependent domain that can induce shock wave A1A2 … An. The calculation does not refer to the gradient of the entropy. The characteristic mesh is used to immediately obtain the target surface. Hence, robustness and accuracy are in good level.
(1)
±
M2 1 dp± ± d V2
Furthermore, the identification of solution points is different from that in the original MOC. With the understanding of the basic unit process of the inverse MOC in 2D condition, the calculation of solution points and the marching process to obtain the shock dependent domain are presented in Fig. 2. In the left subplot, points 1 and 2 are given to calculate point 3. The coordinates of point 3 are obtained by the intersection of Co, 1 and C+, 2 . For the flow properties, θ, p, ρ and T are pending. The flow obeys the assumption of ideal gas, which has the following relation:
The inverse MOC is based on the spacing prior to the solution points along the characteristics. The governing equations are similar to those in the original MOC, whereas the marching direction is opposite.
Fig. 2. Calculation of the solution point (left) and marching process to obtain the corresponding shock dependent domain (right). 13
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2.2. 3D inverse MOC based on generalized shock
Thus, five compatibility equations of three characteristics are needed for the 3D inverse MOC calculation. The calculation procedure of 3D inverse MOC is proposed and shown in Fig. 3. Points A–F are known, and point P is pending. The detailed procedure is revealed as follows (Fig. 4). 1) The location of P is determined by the intersection of ConoidE and Co, B . The equation of the streamline Co, B is shown as
The inverse MOC in 3D condition is an extension of that in 2D condition. However, determining the location of solution points is more complex than that in the 2D case. Given that the isentropic compression is used in the inlet design, the equation of sound speed is used to replace the energy equation. The governing equations are shown as follows:
u x + vy + wz + u
x
+v
y
+w
z
xP = xB + uB t yP = yB + vB t . zP = zB + wB t
=0
uu x + vu y + wuz + px = 0 uvx + vvy + wvz + py = 0
.
uwx + vwy + wwz + pz = 0 c 2 (u
upx + vpy + wpz
x
+v
y
For ConoidE , point P obeys the difference equation as
+ w z) = 0
(6)
[u2
In supersonic condition, the equation set above is hyperbolic. Instead of the two characteristics in 2D domain, the characteristics in 3D condition are in a cone shape. The parameters on the characteristics and streamlines follow the relations below:
streamlines: un x + vn y + wn z = 0,
(7)
characteristics: un x + vn y + wnz = c.
(8)
dx dt
= u,
Mach cone: [u2 y2
[w 2
dy dt
= v,
(V 2
(V 2
dz dt
= w,
c 2)] dx 2 + [v 2
(yP (yP
c 2)] d
c 2)] dz 2,
+ + 2uvdxdy + 2uwdxdz + 2vwdydz = 0
(10)
uut + vvt + wwt + pt = 0 pt
c2
t
=0
[(n x2
1) u x + (n y2
yB )(zP
streamlines:
pt +
1) vy + (nz2
xB ) 2 + [v 2 (V 2
(V 2
c 2)](z
yB ) + 2uw (xP
zB
P
xB )(zP
c 2)] )2 zB ) + 2vw
. (14)
zB ) = 0
uB ) +
B vB (vP
wB (wP wB ) + pP (pP pB ) cB ( P
k ck nxk (uP
uk ) + k ck nyk (vP
2 + tk k ck2 (n xk
vB ) +
B
pB = 0 B) = 0 vk ) + k ck n zk (wP
2 1) uxk + n yk
2 1 v yk + nzk
(15) wk )
(pP
pk )
1 wzk
(k = x , y, z )
+(uyk + vxk ) nxk nyk + (vzk + w yk ) n zk nyk + (w xk + x zk ) nwk nxk ] = 0
(11)
cn x ut + cn y vt + cnz wt characteristics:
+
xB )(yP
characteristics:
,
[w 2
B uB (uP
where V is the resultant velocity. For a 3D flow field, the values of solution points can be solved through the intersection area between the Mach cone and the stream surface, which obeys the following equations:
streamlines:
yB
c 2)](xP )2
2) As analyzed before, five independent compatibility equations are needed to obtain the flow properties of P, and points Am and Bm are introduced to satisfy the definite condition. The flow properties of point P are initialized by those of point B. 3) ConoidP is then emitted and intersects with line AB at point Am. The flow properties of point Am are linearly interpolated by points A and B. 4) The flow properties of P are then reassigned by those of Am. Steps 3) and 4) are used in iteration to obtain the steady results. Point P is completely solved when the streamline (Co, B ) and the characteristics (C+, E and C±, P ) are determined. The corresponding equations are shown as follows:
(9)
(V 2
(V 2
+ 2uv (xP
The deducing equation is completed in Ref. [31]. The corresponding compatibility equations are shown as follows:
streamlines:
(13)
(16)
c2 2.3. Verification of the inverse method
,
1) wz
+ (u y + vx ) n x n y + (uz + wx ) n x nz + (vz + wy ) n y n z ] = 0
The efficiency and the accuracy of the new inverse method is verified in this section. The flow field around the 40° cone, which was subjected to Mach 2 freestream, is used for verification. The analytical solution to the conical flow is provided by solving Taylor–Maccoll equation for comparison. The shock shape is given to obtain the
(12) where subscript t stands for the partial derivative of the value. Rusanov [32] proved that in 3D steady isentropic supersonic condition, every three characteristics on the Mach cone are dependent on each other.
Fig. 3. Schematic of 3D inverse MOC. 14
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Fig. 4. Detailed calculation process of 3D inverse MOC.
corresponding shock-dependent curve. Fig. 5 shows the result of the inverse design under 2D condition. The Mach distribution corresponds well with the features of the flow field around the cone. The cone curve is generated accurately except for the first grid where a small inflection is observed. This small inflection is the intersection point between the shock and its dependent domain, thus resulting in singularity. The relative error can be relieved by increasing the mesh density. The conical flow with the 40° cone angle and three different Mach numbers are used to further verify the accuracy. The results in Table 1 show that the relative error between inverse design and the analytical solution is less than 3% for every condition. In addition, the accuracy increases with the mesh density and Mach number. Hence, the inverse method is accurate for 2D design. Under 3D condition, the cone angle is given as 30° to calculate the cone. The analytical solution is provided by solving the Taylor–Maccoll equation, which is also used in 2D verification. The results of 3D inverse design are shown in Fig. 6. According to the Mach contour of the longitudinal section, the flow field of the design corresponds well with conical flow field. The relative error is less than 4.5%. The accuracy of 3D inverse design decreases compared with that of 2D design but can also be relieved by increasing the mesh density. Thus, the accuracy of the 3D inverse MOC is at an acceptable level for engineering design.
Fig. 5. Mach contour of 2D axisymmetric cone solved by the inverse MOC (Mach 2.0, cone angle 40°).
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Table 1 Relative error between inverse design and Taylor–Maccoll solution (cone angle 40°). Mesh number
Mach 2
Mach 5
Mach 7
100 200 300
2.00e-3 1.01e-3 6.78e-4
1.55e-4 8.01e-5 5.37e-5
7.92e-5 4.12e-5 2.72e-5
3. Bump-inlet integration based on the inverse MOC 3.1. Inlet lip (shock-dependent domain) design The non-uniform flow treatment is important for propulsion/airframe integration design. The typical inlet design is based on uniform freestream. Two challenges are faced by the inlet design when subjected to non-uniform flow. On the one hand, a thick boundary layer develops along the airframe. Large energy loss occurs in the propulsion system, and the low kinetic energy flow is swallowed into the inlet. On the other hand, typical osculating theory is not applicable because the inflow is not uniform. Maintaining a high φ while using the typical inlet design is difficult. In a previous study, a bump configuration was introduced to efficiently remove the boundary layer. The bump generates an expanding pattern flow field, which has a large crosswise flow transference (Fig. 7). The flow field leads to the overlarge side-compression of the inlet. Thus, the inlet design must consider the features of non-uniform flow. This paper proposes an innovative 3D inverse MOC, in which the surface is designed based on the prescribed shock shape and the given non-uniform flow field. The elliptic cone is selected as the incident-shock shape in the inlet design. Compared with the normal cone, the elliptic cone can control the shock shape by adjusting the major/minor axis and thus has great flexibility in the bump-inlet integration design. The shape of incident shock is determined by the inlet lip. Thus, the design of the inlet lip influences the mass capturing and flow properties of inlets. The incident shock design subjected to non-uniform flow is shown in Fig. 8. Central cylinder is used in the shock design to ensure the uniqueness of the reflection shock, thus avoiding the problem of center-line singularity. The central cylinder is located between points O and G. The design procedure of the inlet lip based on the 3D inverse MOC is presented in this section (Fig. 9). The projection of the inlet lip on the yoz-plane is first designed. Red-solid line ABC is the projection of the front lip curve, and black-solid line ADC is the projection of the rear lip curve. Curve AGC is obtained by projecting ADC to the elliptic cone. Curve AHC is obtained by the intersection of the elliptic shock and the
Fig. 7. Crosswise-deflection-angle (β) distribution on bump surface.
bump. The incident shock is obtained by extracting the area between AGC and AHC and is shown in red color. Under the design condition, the incident shock is induced from upstream position (i.e., points A and C), and finally converges at point G. Thus, the incident shock is well attached to the inlet lip and consequently increases the φ of inlet. At low speeds, the incident shock deflects away from point G, thereby inducing flow spillage to ensure the inlet starting performance. After the shock shape is determined, the final step is the design of the shock-dependent area (i.e., inlet lip). According to the shock position, the flow upstream properties on the incident shock are obtained by inviscid simulation of the bump. The inlet lip surface is generated by the 3D inverse MOC. The inlet wall is generated by classical osculatingflow theory. The bump-inlet integration model is obtained as illustrated in Fig. 10. The light-blue line (i.e., leading edge) around the bump is the intersection part with airframes. The red surface (i.e., inlet lip) is the shock-dependent area generating the prescribed incident shock. The installation view is shown in the left-top subplot and is integrated with the airframe with the leading edge. The freestream is developed along the airframe and impinges on the bump. The boundary layer is removed, and high kinetic energy flow fills up the near-wall region. The
Fig. 6. Results of 3D inverse design (left) and comparison with the analytical solution (right). 16
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Fig. 8. Schematic of incident shock design subjected to non-uniform flow.
front view shows the inlet internal surface. On the one hand, the surface conducts a 3D compression to the inflow, which is more efficient than the typical 2D compression. On the other hand, the surface adjacent to the bump is geometrically continuous with the bump, thus ensuring the balanced compression of the captured flow in every direction.
3.2. Analysis of aerodynamic characteristics based on inviscid/viscous simulation A structured mesh is generated for inviscid and viscous numerical simulation to verify the integration performance of the design. The flat
Fig. 9. Design progress of inlet lip based on the new 3D inverse MOC. 17
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Fig. 10. Inlet lip design based on 3D inverse MOC.
Fig. 11. Structured mesh of the bump-inlet integration model.
plane is used to integrate with bump. The surface mesh elements are shown in Fig. 11. A C-shaped mesh (copper colored) is set around the bump leading edge. Another C-shaped mesh (blue colored) is set on the bump surface to accurately fit the bump leading edge. A cowl lip (red colored) is used to avoid overlarge expanding external-flow. An Oshaped mesh is set for every cross-section of inlet to fit the elliptic crosssection of the inlet. The freestream atmosphere condition is at 24 kmhigh altitude and Mach 6. For the viscous simulation, a 3.5 m longitude flat plane is set to produce the boundary layer. The minimum mesh height of near-wall grid element is set at 0.15 mm to meet the requested standard k–ε turbulence model. The grids in y-direction are stretched with the increasing ratio of 1.2 refined with geometric proportion rule. The entire wall is adopted with no slip and adiabatic wall conditions. All results are calculated by the CFX commercial software based on high resolution in numeric turbulence and advection scheme. The numerical methods in this study have been effectively validated by You and Liang [33], who used wind tunnel experiments and showed that these methods can calculate reasonable and reliable results. The numerical methods have also been validated in the previous study [30] through a comparison with an experimental study of bump-inlet integration shown in Fig. 12. The results show that the difference of static pressure between numerical and experimental results is less than 2.7, which indicates that the two results agree with each other. Moreover, the numerical methods effectively calculate the true flow field. The overall performance of the integration model is shown in Table 2, which includes the total pressure recovery coefficient (σ), Mach number of the inlet exit (Mexit), and mass capturing ratio (φ). The parameter φ is the mass flow of the inlet divided by that of the virtual free-flow tube. The free-flow tube is a projective plane that x-reversely extends the inlet lip to the xoz-plane. The energy loss of inviscid simulation is in an acceptable range for an integration design. According to φ value and the shock structure in Fig. 13, the 3D inverse MOC
Fig. 12. Longitude static pressure distribution of the bump-inlet integration.
Table 2 Inlet flow parameters of inviscid and viscous simulation. Simulation type
σ
Mexit
φ
Inviscid condition Viscous condition
0.752 0.471
3.71 2.94
0.813 0.738
efficiently solves the integration design problem. The viscous simulation considers the boundary layer effects when the Mexit decreases correspondingly. The φ decreases by 9.2% according to the shock/ boundary layer interaction. Hence, the 3D inverse MOC can realize the high mass capturing in inviscid condition, whereas the boundary layer removal ability of the bump and the correction of boundary layer still need improvement. The inlet mass capturing performance is studied based on the inviscid numerical simulation shown in Fig. 13. The incident shock shape is depicted by the isolines of Mach number. The shock is induced from the leading edge of inlet and thoroughly covers the inlet lip. In addition, the incident shock is almost kept in one plane for the design. The 3D inverse design generates the corresponding lip surface that can induce the prescribed incident shock. The flow compression is observed in the inlet entrance and the semisection views. The inflow enters the inlet and experiences a pressure increase. Afterward, a pressure drop, which results from the overlarge side compression, occurs downstream the incident shock and leads to flow acceleration again. The bump configuration creates an expanding flow pattern. The inflow changes the flow direction from expanding (bump) to compressing (inlet). Hence, the 18
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Fig. 13. Inlet mass capturing performance of the integration design (inviscid condition).
flow experiences larger side compression than the inlet with parallel inflow. As a result, a side-embedded compression wave is induced inside the inlet. The structure was analyzed in detail in Ref. [13]. The inlet performance in viscous condition is studied, and the results are shown in Fig. 14. Two major flow compression waves, namely, the bump and inlet incident shocks, are on the bump surface. The bump shock is induced when the inflow encounters the bump leading edge. The second compression is the incident shock. Flow spillage occurs at the central part of the inlet entrance due to the shock/boundary layer interaction. Thus, a small bulging distortion of incident shock emerges. The distorted shock is still within the bump shock, indicating that the distortion affects the mass capturing performance of inlet. However, the shock structure is simple, and no shock/shock interaction occurs. The incident and the reflection shock are illustrated in the side view of the inlet. In the right subplot, the 3D shock structure is observed according to the pressure isolines of the inlet wall. The shock structure inside the inlet is overall simple. The compression level of the upper inlet wall is stronger than that of the lower wall due to the length of boundary layer development. For the upper surface, the boundary layer starts to develop at the central position of the inlet lip. The low kinetic energy fluid is in a small level. For the lower surface, the boundary layer starts to develop from the flat plate. Although most boundary
layers have been diverted by bump, low kinetic energy fluid occupies the near-wall region of the lower wall. Thus, the shock/boundary interaction near the lower wall makes the compression weaker than that on the upper surface. The π and σ contours of inlet throat and exit are shown in Fig. 15. The incident shock profile of the shock profile of inlet is depicted by the π isolines in subplot a. The incident shock attaches well at two sides of the inlet lip, while there is flow spillage at the compression center, which leads to the distortion of the shock shape. Besides, the sideembbeded shock (SES) is also observed at the inlet bottom surface in viscous results. The overlarge side compression induces a pressure surge at the bottom surface, which leads to the flow capacity of inlet and the energy loss at the inlet throat (subplot b). The flow spillage occurs at the compression center (labeled in red color), which distorts the incident shock shape. Thus, the shock distortion and the SES effects are both related to the overlarge side compression. The π and σ contours of inlet exit are shown in subplot c and d. The flow compression and energy of the half bottom keep in a lower level than those of upper half part, which reveals that the energy loss by SES effects reduce the flow performance of the inlet half bottom part. Besides, the back-pressure capability of the bump-inlet configuration is preliminarily studied in this paper. The π contours of the
Fig. 14. Shock structure of the bump-inlet integration (viscous condition).
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Fig. 15. π and σ contours of inlet throat and exit.
longitude cross-section at various back-pressure (Pb) is shown in Fig. 16. The π of inlet exit in the through-flow condition is 17.0, which is shown in the subplot a of Fig. 16. The end shock is formed while increasing Pb. The position of the end shock moves upstream while Pb increasing. The low kinetic flow accumulates near the inlet wall. It prevents the flow to pass through smoothly by reducing the effective flow area. When Pb/P∞ reaches 100, the end shock locates at approximately x = 2.5 m. The detail flow properties under various back-pressures are demonstrated in Table 3. The three parameters decrease significantly with the increase of the Pb. The ηKE is the kinetic energy efficiency of inlet, which evaluates energy loss under different freestream conditions overall [13]. It is defined as follows. KE
1
=
1 2
M
2
T T
1+
1 2
M2
4. Conclusion This study developed an innovative inverse MOC that can design the generalized 3D surface based on the prescribed shock shape. The unit process and marching procedure of the inverse MOC are presented. This new method adopts reverse spacing to obtain the flow properties of the backward characteristics. The auxiliary points are used to acquire the unique location of solution points. The accuracy of the method is verified through a test of a conical flow field with analytical solution for comparison. The accuracy of 2D axisymmetric design is high, but that of 3D design is relatively low. Nonetheless, the result satisfies the requirement for engineering design. The inverse MOC is applied in a hypersonic bump-inlet integration design (M∞ = 6.0). The inlet lip is generated by a prescribed elliptic-conical shock wave. Inviscid results showed good performance of the inverse MOC ( = 0.813, Mexit = 3.71, σ = 0.752). The incident shock is well attached on the inlet lip, which is of high φ. By contrast, viscous results showed relatively low performance ( = 0.738, Mexit = 2.94, σ = 0.471) of the inverse MOC due the interaction effects between the unremoved boundary layer and the incident shock. The side-embedded shock effect is observed in the inlet bottom surface, which influence the inlet starting performance in lower Mach number condition. The max back-pressure capacity of the inlet is validated as Pb/P∞ = 100 in Mach 6.0 condition.
1
(17)
When Pb/P∞ is larger than 100, the periodic fluctuation of mass flow is observed in the numerical simulation. The mass flow of exit cannot keeps at a steady value, thus making the calculation not able to converge. It reveals the flow capacity of inlet has reached its maximum limit at approximately Pb/P∞ = 100.
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Fig. 16. Back-pressure capability of the bump-inlet configuration. Table 3 Flow properties of back-pressure capability study. Pb/P∞
σ
φ
ηKE
17 38 62 83 89 100
0.471 0.364 0.289 0.220 0.201 0.175
0.738 0.735 0.730 0.726 0.724 0.722
0.967 0.954 0.941 0.925 0.919 0.910
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