Derivation and validation of a similarity law for free-flight wind tunnel tests of parallel stage separation

Zhiming CHEN,Fei XUE,Hihun YU,Yuho WANG,Zenghui JIANG,Wei LU, Lei DONG

a China Academy of Aerospace Aerodynamics, Beijing 100074, China

b Beijing Institute of Astronautical Systems Engineering, Beijing 100076, China

c China Academy of Launch Vehicle Technology, Beijing 100076, China

KEYWORDS Drop testing in wind tunnel;Free-flight wind tunnel test;Multi-body separation;Parallel stage separation;Similarity law

Abstract Aiming at the safety problem of the stage separation of parallel reusable high-speed air vehicles,this paper studies the unsteady test method and focuses on deriving a similarity law of parallel stage separation free-flight wind tunnel tests.The new similarity law considers the influences of aerodynamic force and gravity on the motions of the two stages,as well as the influence of aerodynamic interference between the two stages on each other’s motion.From the perspective of multiangle physical equations,the conditions to ensure that the two-stage separation trajectory of a wind tunnel test is similar to that of a real air vehicle are derived innovatively,so as to ensure the authenticity and credibility of wind tunnel test results.The similarity law is verified by an HIFiRE-5 air vehicle, and the separation trajectories of wind tunnel tests and the real air vehicle are obtained by numerical simulation.The research shows that the similarity law derived in this paper can ensure that wind tunnel free-flight tests have the ability to predict the two-stage separation characteristics of real parallel vehicles.By analyzing the separation trajectory curve of the typical state,it is found that the new method can ensure that the trajectory error of a wind tunnel test does not exceed 1%,which indicates that this method is credible.The establishment of the new method lays the foundation for subsequent wind tunnel tests and provides support for research on the safety of the stage separation of parallel reusable air vehicles.

1.Introduction

Reusable air vehicles can reduce the cost of shuttle missions and shorten the mission cycle, which are highly economical.1Therefore, many countries have focused on researching reusable air vehicles, including the United States, Russia,Japan, and many European countries, who have produced many works since the 1980 s.2,3Using two methods,i.e.,Single Stage to Orbit (SSTO) and Two Stages to Orbit (TSTO), a variety of conceptual schemes have been proposed.4,5In the two-stage orbit entry scheme, the first-stage air vehicle that has completed the mission is separated from the second-stage air vehicle at the right time and returns to the ground.Thus,propellant consumption is significantly reduced, as are launch costs.Many countries have put forward their own concept of TSTO,6which does have many advantages, but whether the two stages can be safely separated directly determines the success of a flight mission.For such an air vehicle,the separation characteristics of the two stages are extremely important.In a typical two-stage air vehicle arrangement scheme, the parallel arrangement has higher practicability than that of the tandem arrangement, and, at the same time, the parallel arrangement also brings more practical problems.For example, a TSTO vehicle is in a high-dynamic pressure environment with a high Mach number when performing interstage separation, and there is complex aerodynamic interference between the two stages during separation,which makes the safety of separation uncertain7.

In order to ensure the safety of separation,it is necessary to study the separation characteristics of high-speed air vehicles.Cheng et al.8studied the thermal environment of real parallel vehicles and found that shock waves could cause a variety of heat flux.The interaction between shock waves can be roughly divided into two categories.A non-reflected part of the incident shock wave will cause a V-shaped heat flux distribution.Jia et al.9studied the aerodynamic interaction of a hypersonic TSTO air vehicle, where the interaction between the shock wave and the boundary layer was caused by a bow shock wave.In addition, decreasing the stage spacing was shown to increase the pressure and heat flux.Zhang et al.10developed a thermodynamic analysis tool to study RBCC(Rocket Based Combined Cycle) engines and characterize the influences of initial launch conditions on flight trajectory.Maddock et al.11introduced the conceptual design and performance analysis of a small-payload two-stage space shuttle carrier rocket.The wing area and aerodynamic characteristics were evaluated, as well as the mass and engine selection.Yang et al.12,13studied the unsteady aerodynamics of an air launch-into-orbit system and proposed a new reduced-order modeling method; the longitudinal dynamic response of the air launch to the orbital system during stage separation was simulated.Li et al.14studied the interference flow between rockets under various incoming flow conditions; the interference flow field was analyzed, and the action mechanism of a lateral jet was clarified.

The aerodynamic force of static interference between the two stages has always been a focus of TSTO research.Decker and Wilhite15,16conducted a wind tunnel test on the parallel interstage separation characteristics of Mach 3 and Mach 6,which showed that both had obvious aerodynamic interference on the other side and significantly affected its movement.Bordelon et al.17improved a previous test method and carried out relevant experimental research.The research showed that the aerodynamic force of the interference between the two stages was mainly generated by the shock wave of the air vehicle.It was found that the aerodynamic force at the key position between the two stages was unsteady,and the static force measurement error was large.Therefore, dynamic research methods have more advantages than static ones.However,compared with the static aerodynamic interference research of parallel vehicles, experimental research on the dynamic aerodynamic force between the two stages has been very rare,and dynamic research has only appeared sporadically in numerical calculation articles,18–20which has obtained some interesting results.In the existing public literature, articles using dynamic research methods to study the characteristics of parallel stage separation have not been found, and even the data of dynamic research methods has been very few.However, the theory and methods of wind tunnel testing are very important and directly related to the reliability of test results,and they urgently require further research.

According to the characteristics of parallel stage separation,compared with the previous static force measurement and Captive Trajectory Simulation (CTS) test technology, the dynamic wind tunnel test method has more advantages: the mass of the two-stage separator of parallel vehicles is closer,as is the two-stage volume, the aerodynamic interference is serious, and the stage separation greatly influences the twostage attitude.The former two test methods of static force measurement and CTS test technology are not competent.To solve these problems, the multi-body separation freeflight wind tunnel test method is advantageous, which is very suitable for parallel stage separation research.Free-flight wind tunnel testing is an unsteady test method that is different from steady wind tunnel testing,21–23and the test method is becoming increasingly mature.24–25Furthermore, there is also a certain research foundation for the multi-body separation problem,which includes a free drop test,16–21cluster munition separation,22and tandem vehicle separation.Therefore,multibody separation free-flight wind tunnel testing is a hot spot in the field of wind tunnel testing.However, because the twostage position of the parallel stage is significantly different from previous free-flight tests, the constraint equation of two-stage separation is also different.Therefore, the similarity law of parallel stage separation free-flight wind tunnel testing has not been established, and special theoretical derivation is needed.

The differences between and difficulties of parallel stage separation and previous free-drop test/cluster munition separation/tandem vehicle separation are as follows: firstly, wind tunnel tests and real air vehicles must simulate many parameters, which leads to many similar equations of the parallel stage separation free-flight wind tunnel test.Secondly, both stages are lifting body shapes, and the aerodynamic force received by each stage is no longer small compared with its gravity.These two parameters need to be considered when deriving similar conditions.The main work of this paper is as follows.Combined with the characteristics of parallel stage separation, through the simultaneous establishment and derivation of multiple physical equations,a similarity law suitable for parallel stage separation free-flight wind tunnel tests is finally obtained.The international HIFiRE-5 shape is used to design the two-stage shape by numerical simulation.The separation of high-altitude flight conditions and reduced-model wind tunnel test conditions is simulated, and the separation trajectories of a real flight and a reduced-model wind tunnel test are obtained.The accuracy of this wind tunnel test method is verified by comparing the separation trajectories of these two.A similarity law of parallel stage separation free-flight wind tunnel testing is obtained, which provides conditions for future research of parallel stage separation and lays a theoretical foundation for researching the safety of separation characteristics between the two stages of orbit entry.

2.Conditions for similarity of trajectory between two stages

The xy plane passes through the centroid of each stage.Subscript ‘‘1”represents first-stage parameters, subscript ‘‘2”represents second-stage parameters, subscript ‘‘m”represents wind tunnel test model parameters, and subscript ‘‘s”represents real air vehicle parameters.Therefore, x1mrepresents the x-coordinate value of the centroid of the first-stage wind tunnel test model, y1mrepresents the y-coordinate value of the centroid of the first-stage wind tunnel test model, x2mrepresents the x-coordinate value of the centroid of the secondstage wind tunnel test model, and y2mrepresents the ycoordinate value of the centroid of the second-stage wind tunnel test model.

2.1.Wind tunnel test

a1xmrepresents the horizontal acceleration of the first-stage wind tunnel test model, and a1ymrepresents the vertical acceleration of the first-stage wind tunnel test model.a2xmrepresents the horizontal acceleration of the second-stage wind tunnel test model,and a2ymrepresents the vertical acceleration of the second-stage wind tunnel test model.g is the acceleration of gravity.We let the first- and second-stage models have an overall forward launch speed of v0m.The models are placed in a wind tunnel test,where the two stages begin to separate at tm= 0.At a certain tm, the primary centroid coordinates are(x1m, y1m) and the secondary centroid coordinates are (x2m,y2m) as follows:

S1mrepresents the reference area of the first-stage wind tunnel test model, C1ymrepresents the lift coefficient of the firststage wind tunnel test model, C1xmrepresents the drag coefficient of the first-stage wind tunnel test model, and m1mrepresents the mass of the first-stage wind tunnel test model.S2mrepresents the reference area of the second-stage wind tunnel test model, C2ymrepresents the lift coefficient of the secondstage wind tunnel test model, C2xmrepresents the drag coefficient of the second-stage wind tunnel test model,and m2mrepresents the mass of the second-stage wind tunnel test model.

Linear acceleration:

We define the relationship between the first and second stages as follows: N = m1m/m2mand R = S1m/S2m.When the real air vehicle parameters are determined, R and N are constants: N is the mass ratio of the first stage to the second stage, and R is the area ratio of the first stage to the second stage.When the angle of attack of the first and second stages does not change much, C1xm, C2xm, C1ym, C2ymcan be considered to be constant and independent of time.Then,

2.2.Real flight

The xy plane passes through the centroid of each stage.Subscript ‘‘1”represents first-stage parameters, subscript ‘‘2”represents second-stage parameters, subscript ‘‘m”represents wind tunnel test model parameters, and subscript ‘‘s”represents real air vehicle parameters.Therefore, x1srepresents the x-coordinate value of the centroid of the first-stage real air vehicle,y1srepresents the y-coordinate value of the centroid of the first-stage real air vehicle, x2srepresents the x-coordinate value of the centroid of the second-stage real air vehicle, and y2srepresents the y-coordinate value of the centroid of the second-stage real air vehicle.

a1xsrepresents the horizontal acceleration of the first-stage real air vehicle, and a1ysrepresents the vertical acceleration of the first-stage real air vehicle.a2xsrepresents the horizontal acceleration of the second-stage real air vehicle,and a2ysrepresents the vertical acceleration of the second-stage real air vehicle.We let the first and second stages have an overall forward speed of v0s,and ts=0 when the two stages begin to separate.At a certain ts, the primary centroid coordinates are (x1s, y1s)and the secondary centroid coordinates are (x2s, y2s) as follows:

S1srepresents the reference area of the first-stage real air vehicle,C1ysrepresents the lift coefficient of the first-stage real air vehicle,C1xsrepresents the drag coefficient of the first-stage real air vehicle, and m1srepresents the mass of the first-stage real air vehicle.S2srepresents the reference area of the second-stage real air vehicle, C2ysrepresents the lift coefficient of the second-stage real air vehicle, C2xsrepresents the drag coefficient of the second-stage real air vehicle, and m2srepresents the mass of the second-stage real air vehicle.

Linear acceleration:

2.3.Conditions for similarity of trajectory

To ensure that the separation trajectory of the wind tunnel test is similar to that of the real flight,Eqs.(10)and(19)need to be equal.Like other wind tunnel tests, the influence of Reynolds number on the transition, that is, the influence of Reynolds number on the resistance, is neglected in the force analysis of the air vehicle.Under the conditions of geometric similarity,basically the same flight conditions, consistent flight attitude angle, and within a certain range, there exist

When the two stages are not far away,we can conclude that Eqs.(20)and(23)are valid.Therefore,the following identity is true:

It can be seen that for the simulation of the parallel stage separation of a real air vehicle with two stages without initial separation speed and initial separation force, when the freeflight wind tunnel test method is used,it is necessary to ensure the geometric similarity of the two stages of the wind tunnel test, and that the mass ratio is the same as the parameters of the real air vehicle (the same R and N values as those of the real flight are used in the wind tunnel test).Thus, the twostage separation trajectory obtained by the free-flight wind tunnel test is similar to that of a real air vehicle.Of course,in order to ensure the similarity of angular displacement in the wind tunnel test, it is necessary to adopt the light model method in the following equation, which was introduced in detail in previous articles:26–32

where kmis the ratio of the wind tunnel test model mass to the real air vehicle mass, kρis the ratio of the wind tunnel airflow density to the real air vehicle airflow density,and klis the ratio of the wind tunnel test model reference length to the real air vehicle reference length.

3.Verification

In order to verify the feasibility of the theoretical derivation in this paper, the numerical simulation method is used to simulate the two-stage separation trajectory of a real parallel stage separation.At the same time,the CFD method is used to simulate the wind tunnel test environment in this paper, and the separation trajectory of the wind tunnel test is obtained.The separation trajectory curves of the real flight and the reduced wind tunnel test are obtained and compared to verify the accuracy of this method.

3.1.HIFiRE-5

The air vehicle used in this verification is the HIFiRE-5 model in Ref.33Because this study is aimed at parallel stage separation, two HIFiRE-5 models are used.The relative positions and centroid parameters of the two stages are shown in Figs.1 and 2.

The length of the second stage is 0.4 times that of the first stage, and the line between the two centroids is parallel to the vertical direction.

Fig.1 Centroid parameters of two stages.

Fig.2 Outline dimensions of two stages.

Fig.3 provides a description of the coordinate system.The coordinate origin is located at the vertex of the head of the first stage at the time of two-stage separation, which points backward along the body axis of the first stage in the x-positive direction and vertically upward in the y-positive direction,and the right-hand rule determines the z-positive direction.

In order to realize the relative motion of multiple bodies,the dynamic overlapping unstructured mesh method is used.In the solver, the unstructured-mesh finite-volume method is used for spatial discretization, and the implicit LU-SGSbased dual-time stepping scheme is used for time discretization to solve the unsteady compressible Reynolds-averaged Navier-Stokes equation.For the separation problem,the trajectory of the object can be obtained by coupling the six-degree-offreedom motion equation.

Near the second stage, there are 800000 nodes of tetrahedral unstructured mesh,forming 370000 mesh elements.There are 2000000 grid nodes near the first stage, and 1400000 grid cells have been created.The time step of the wind tunnel test is 1.0 × 10-4s, the internal step size is set to approximately 200, and the actual flight time step size is 2.0 × 10-3s, the internal step size is set to approximately 220, and the convergence is determined by reducing the residual of three orders of magnitude.The volume of the grid is 601000 times that of the first stage.

3.2.States of numerical simulation

Fig.3 Coordinate system description.

The parameters of the real air vehicle and the wind tunnel test model are shown in Table 1,where m represents mass of stage,Ix,Iy,Izrepresent the inertia values of stage around x,y,and z axes respectively.The two stages in this paper are separated at a high altitude and a high speed.At such a high speed and such a high altitude,all are provided by the first stage.At this time,the first stage will soon end its flight mission, so it has consumed most of the fuel and its weight has been greatly reduced.Also at this time, the second-stage flight mission has not yet started, and the fuel is still unused.Compared with the first stage, the density of the second stage is much larger.In this paper, the mass ratio of the second stage to the first stage is 0.47.The parameters of the wind tunnel model are chosen according to the method used in this paper.Table 1 also shows the states of this simulation.Among them, 30 km and 40 km are selected as the separation altitudes.In addition, the mass ratios of the first stage to the second stage in these two states are equal to the mass ratio of the real air vehicle to ensure that N in each equation is consistent (Eqs.(10) and (19)).The shapes of the models in these two states are geometrically similar, and the scales of the two levels are equal, so as to ensure that R in each equation is equal (Eqs.(10) and (19)); their respective masses follow Eq.(25) to ensure that the angular displacement in the test is similar.In order to ensure safe separation, the included angle between the body axes of the first and second stages is 4°, that is, the second stage rotates its head upward 4° around its own center of mass on the basis of Fig.1.At the time of separation, the angle of attack of the first stage is - 4°, and the angle of attack of the second stage is 0° relative to the air flow.Referring to Fig.1, at the time of separation, the air flow blows from the top left of the air vehicle vertex.The scale of the wind tunnel is 1:21.5,and the real air vehicle does not scale.

In designing the wind tunnel test model, the flow field parameters of the FD 07 wind tunnel are used to simulate the effect of a wind tunnel test in the FD 07 wind tunnel,where q∞is the incoming flow pressure,P∞is the incoming flow static pressure,T0is the total incoming flow temperature,and ρ∞is the incoming air flow density, as seen in Table 2.

4.Result analysis

4.1.Numerical simulation results

Fig.4 shows the separation characteristics at 30 km for a real air vehicle simulated by CFD.It can be seen from the figure that the two stages can be separated safely and smoothly.The separation of the two stages mainly depends on thedownward movement of the first stage,and the first stage has a larger angular movement.

Table 2 FD 07 parameters.

Table 1 Real vehicle and model parameters.

Fig.4 Separation characteristics of a real air vehicle at 30 km.

Fig.5 shows the separation characteristics of a wind tunnel test using CFD simulation.The two-stage separation is consistent with the real air vehicle separation in Fig.4, and the two stages can also be separated smoothly and safely.The separation also mainly depends on the downward movement of the first stage, and the first stage has a larger angular movement.On the whole, the wind tunnel test results obtained from the wind tunnel test designed by the design method in this paper are consistent with the separation law of the real air vehicle at 30 km and have high accuracy.

Fig.6 shows the separation characteristics of a real air vehicle at 40 km simulated by CFD.Similar to the 30 km separation characteristics in Fig.4, the two stages can be separated smoothly and safely.The separation mainly depends on the downward movement of the first stage, and the first stage has a larger angular movement.Compared with the 30 km separation in Fig.4,because the aerodynamic pressure at 40 km is lower, the overall separation time is longer, which also conforms to the physical law.

Fig.7 shows the separation characteristics of a wind tunnel test at 40 km simulated by CFD.The two-stage separation is consistent with the separation of the real air vehicle in Fig.6, and the two stages can also be separated smoothly and safely.The separation mainly depends on the downward movement of the first stage, and the first stage has a larger angular movement.On the whole, the wind tunnel test results obtained from the wind tunnel test designed by the design method in this paper are consistent with the separation law of the real air vehicle at 40 km and have high accuracy.Compared with the 30 km wind tunnel test separation in Fig.5,because the aerodynamic pressure at 40 km is lower, the overall separation time is longer,which also conforms to the physical law.

Fig.5 Simulated wind tunnel test separation characteristics at 30 km.

Fig.6 Separation characteristics of a real air vehicle at 40 km.

Fig.7 Simulated wind tunnel test separation characteristics at 40 km.

Because the dynamic pressure of the air flow is different under different conditions, the pressures in Figs.4-7 cannot be compared.In order to facilitate pressure comparison,different pressure gauges are designed for Figs.4-7 so that the background of each state is blue, highlighting the red highpressure area.Figs.5 and 7 are both in the wind tunnel test state, and both are Mach 7.Therefore, the flow field parameters in Figs.5 and 7 are not significantly different,so their pressure gauges are the same, and the maximum value is 5 × 103Pa.

From the separation location and flow field diagrams,it can be seen that the parallel stage separation free-flight wind tunnel test similarity law derived in this paper can well predict the separation characteristics of a real parallel vehicle’s twostage separator through the wind tunnel test method and has significant advantages.

4.2.Separation trajectory analysis

In order to quantitatively analyze the characteristics and advantages of this method, the following is a detailed analysis of the above simulated data.We obtain the separation error between the wind tunnel test and the real air vehicle.In addition, it should be noted that the aerodynamic interference between the first and second stages takes their distance into account.In other words,if the two stages exceed a certain distance, aerodynamic interference can be ignored.This distance is generally equal to the length of the reference length.In this paper, the reference length is the length of the first-stage air vehicle.In other words,whether the trajectory obtained by this method is similar to that of the real air vehicle is mainly determined in the range that the distance between the two stages is less than the length of the first stage.

It is specified here that Δy is the vertical displacement difference between the second and first stages,Δymis the vertical displacement difference between the second and first stages during the wind tunnel test, and Δysis the vertical displacement difference between the second and first stages during the real flight.Δx is the horizontal displacement difference between the second and first stages, Δxmis the horizontal displacement difference between the second and first stages during the wind tunnel test,and Δxsis the horizontal displacement difference between the second and first stages during the real flight.l0is the reference length, and l0mis the reference length in the wind tunnel test.Here,the length of the first-stage wind tunnel test model is taken as l0m.l0sis reference length of the real flight.Here, the length of the real air vehicle of the first stage is taken as l0s.We denote the trajectory error as η.Therefore, according to the above description,vehicle, especially the main two-stage separation relative position relationship, which ensures that the wind tunnel test results are similar to the trajectory of the real air vehicle and achieves the expected design goals.

Fig.8 Line displacement separation trajectory curves of wind tunnel test and real air vehicle at 30 km.

Fig.9 Line displacement separation trajectory curves of wind tunnel test and real air vehicle at 40 km.

Table 3 Trajectory error at different positions and altitudes.

According to Fig.9, it can be found that the wind tunnel free-flight test method designed in this paper can well simulate the parallel stage separation characteristics, ensure that the wind tunnel test results are similar to the trajectory of the real air vehicle, and achieve the expected design goals.In order to better reflect the advantages of this method, the following is a quantitative analysis of the curve data of the simulation results in Figs.8 and 9.Table 3 shows the trajectory error (η) at different locations and altitudes using the method in this paper,which is the difference between the abscissa of the two curves in the figure.

It can be seen from Table 3 that the parallel stage separation can be simulated with high accuracy using the wind tunnel free-flight test method designed in this paper within the whole interference area of the two stages.In the two typical states mentioned in this paper,the maximum error of trajectory simulation does not exceed 1%,which effectively ensures the accuracy of wind tunnel tests and has great practicality.

5.Conclusions

Due to unsupported interference, wind tunnel free-flight test technology is a very suitable test method to study the multibody interference and unsteady effect of air vehicles.In order to ensure the accuracy of subsequent wind tunnel tests, this paper deduces the similarity criteria for free-flight tests in parallel interstage separation wind tunnels, which have not been established in the past.The new similarity law not only considers the influences of aerodynamics and gravity on the motion of each first-stage air vehicle,but also focuses on the influence of aerodynamic interference between the two stages on each other’s motion.The derivation process of this paper combines the trajectory similarity equation between the two stages and obtains the necessary conditions to ensure the similarity between wind tunnel tests and the separation trajectory of real air vehicles, so as to ensure the accuracy of wind tunnel tests and improve their reliability.Using the parallel HIFiRE-5 configuration,the CFD method is used to simulate the separation trajectory of a wind tunnel test and a real air vehicle, so as to verify the similarity law in this paper.Results show the following:

(1) It can be seen from the separation position diagrams and flow field diagrams of the two stages at different times that by using the similarity law derived in this paper,the wind tunnel free-flight test can better predict the separation characteristics of the two stages of the real parallel vehicle;

(2) By observing the curve, it can be found that the wind tunnel free-flight test method designed in this paper can ensure that wind tunnel test results are similar to the trajectory of the real air vehicle and achieve the expected design goals.

(3)In the quantitative analysis of the results of the two typical states proposed in this paper, the maximum error of the test trajectory does not exceed 1%.

Therefore, this similarity law can effectively ensure the accuracy of wind tunnel testing and has great practicality.This similarity law can take the shape of the lifting body into account and has broad application prospects; in addition, it lays a theoretical foundation for subsequent research on parallel stage separation free-flight wind tunnel tests and provides strong technical support for research on the separation characteristics of parallel stages of reusable air vehicles.

Research prospect: Based on rigorous mathematical derivation, this paper simplifies the 8 aerodynamic parameters of Eqs.(20)-(23) into constants, and obtains the conditions for the establishment of Eq.(24) in this paper.However, these 8 parameters actually vary with the angle of attack.When the authors treat these 8 parameters as variable functions of time, Eq.(9) cannot be further simplified.Therefore,it is a conservative method to treat these 8 parameters as constants in this paper.Of course, there is no evidence to show that the similarity law in this paper does not hold when these 8 parameters are not constant.Moreover,the 8 parameters are not regarded as constants in the numerical examples in this paper, but the similarity law in this paper still shows good results.Whether the similarity law in this paper is still effective when the angle of attack changes greatly needs to be further studied.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Nos.U21B2054 and 11772317).