THE NUMERICAL SIMULATION TECHNIQUES RESEARCH AND PRELIMINARY EXPERIMENTAL VALIDATION OF THE START CHARACTERISTICS FOR A TWO-DIMENSIONAL HYPERSONIC INLET

The paper presents the research of the numerical simulation techniques and the preliminary experimental validation on the start characteristics for a typical two-dimensional (2D) hypersonic inlet. In order to obtain the start and unstart hysteresis loop, numerical simulations methods using the incoming §ow ¦eld, the zero-velocity §ow ¦eld, and the latest convergent §ow ¦eld as the initialized §ow ¦eld separately are adopted to calculate the inlet start characteristics until the §ow ¦elds converge. The calculation software is AHL3D, a parallel computational §uid software self-developed by CARDC. The numerical methods also include the method from unstart §ow ¦eld to start one with Mach-numbers gradually increasing till the inlet is start and the method from start §ow ¦eld to unstart one with Mach-numbers decreasing till the inlet is unstart. According to the comparison with di¨erent initialized §ow ¦eld and with di¨erent step-size of Mach-number, the inlet start characteristics are obtained and the numerical techniques are analyzed. Based on the comparison and the analysis, it is suggested that the numerical techniques of inlet start characteristics should use the zero-initialized §ow ¦eld with one step to the ¦nal Mach-numbers to predict the minimal self-start Mach-number of the inlet and use the incoming §ow-initialized §ow ¦eld to predict the minimum


INTRODUCTION
To predict the start characteristics for a ¦xed-geometric hypersonic inlet with a wide working range of Ma 0 , especially to predict the self-start characteristics, is an important step for an air-breathing hypersonic inlet in practical applications. The main purpose of thus prediction is to obtain the start hysteresis loop and the start boundary as well. Researches [1,2] show that the hysteresis loops at present are obtained not only by wind tunnel tests [1,35], but also by numerical simulation [612] as follows.
Neaves et al. [6] present a time-accurate high-speed §ow calculation using a three-dimensional (3D) adaptive mesh algorithm with a time accurate implicit NavierStokes solver. The calculations they accomplished are the steady-and unsteady-state ones for a 3D inlet unstart due to a combustor perturbation simulated by closing a di¨user ramp which raises exit pressure. The computational results demonstrate the ability to calculate 3D high-speed inlet unstarts with a solution adaptive mesh algorithm. However, the document does not focus on the start characteristics of the inlet without combustor working.
Timofeev et al. [7] provided a summary of their researches on the application of locally adaptive unstructured unsteady 2D and 3D Euler numerical codes to the starting of hypersonic inlets. They carried out REST inlet computations for a range of Mach numbers under a sudden insertion into a hypersonic §ow. The method of di¨erent modes of inlet starting under di¨erent initial conditions can be a good reference for current paper.
Veillard et al. [11] demonstrated a method for obtaining the limiting contraction for supersonic intake-starting via overboard spillage for a simple ramp-type intake family. The document predicted starting characteristics compared favorably with 2D inviscid numerical simulations, or the Kantrowitz starting theory. Analytical self-starting predictions were well con¦rmed by numerical simulation. But the inviscous results in the paper may overvaluate the theoretical guidance to the designs of practical intake.
Barber et al. [9] used two di¨erent computational §uid dynamics (CFD) codes, Choi£s UTNS and NASA£s CFL3D, to model complex hypersonic inlet starting phenomena. The document simulated various free-stream Machnumbers to determine whether the codes would predict consistent starting results. In order to obtain convergent, high-speed, viscous solutions, the paper points out that inviscid or Euler calculations should not be used to assess starting performance and that the choice of turbulence model can have a signi¦cant impact on the prediction of shock-waveboundary-layer interaction. Yet, the research needs to be investigated further before the codes can be reliably used for time accurate inlet start access.
Yuan and Liang [8] adopted quazi-steady method to calculate the unstart performance for a hypersonic side-wall inlet model. The paper uses NAPA CFD software to calculate the inlet start characteristics with setting a series of Mach-numbers as incoming §ow Mach-numbers (i. e., Ma 0 , see the Nomenclature above). The result of the document shows that the Mach-number of the inlet restart of the research may probably not be the self-started Mach-number. Yuan and Liang [10] also analyzed the characteristics of restart performance for a hypersonic inlet, adopting a method of gradually increasing Ma 0 . This method is believed to be a safe one to obtain restart Mach-number, or the self-start Mach-number. In fact, this method is adopted in the present paper too.
He et al. [12] launched a lot of work and pointed out that for self-start simulation of a hypersonic inlet with variable incoming §ow conditions, steady calculations give the same result as the unsteady ones do. The CFD software used in the research is AHL3D, which will be used in the present work too.
Based on the documents mentioned above, it can be inferred that the method of using steady numerical simulation to predict start characteristics for hypersonic inlets is nearly the same reasonable as that of using unsteady numerical simulation, that the initial §ow of the inlet plays a prominent role in the numerical prediction of inlet start characteristics, and that the gradually changing Ma 0 has a signi¦cant e¨ect on the ¦nal result. For example, the Mach-number of incoming §ow(i. e., Ma 0 ) can be set to a low supersonic value and after the §ow ¦eld converges, an unstart §ow ¦eld can be achieved. Then, with the gradually increasing Ma 0 and with the marching of unsteady calculation, the unstart inlet §ow ¦eld will change little by little till the inlet starts. Thus, the minimal self-start Ma is obtained. Subsequently, the Ma 0 can be decreased gradually and with steady or unsteady calculating till the inlet turn back to unstart, thus, the minimal Ma for the inlet to be able to start is obtained too. Considering that the unsteady calculation has weakness including the massive workload and the long-term calculation, which are usually very intolerant, the steady calculation will be carried out instead in the present paper.
On the summarization of the work in [13,14], the present work ¦nds that the steady calculation can also be used to predict the inlet start characteristics, particularly, for any 2D hypersonic inlets, through a large amount research practice. It is shown that not only Ma 0 determines the calculation result of the inlet state but also the initialized §ow ¦eld does the same e¨ect. How to a¨ect the inlet state will be discussed in the current paper. Based on the analysis and research, the present work brings about many de¦nitions of Mach-numbers relevant to inlet start and to calculating process, as well as the numerical techniques to quickly predict inlet start characteristics. At last, some experiments are designed and launched to validate the numerical techniques of predicting inlet start characteristics.

THE START HYSTERESIS LOOP AND THE RELEVANT MACH-NUMBERS
The start hysteresis loop is a typical and common aerodynamic phenomenon for any hypersonic inlets so long as internal contraction exists, no matter how little the internal contraction is. In general, the start hysteresis loop of the inlet can be shown in the following forms: Ma hysteresis loop, angle of attack hysteresis loop, variable-geometric hysteresis loop, etc. Among them, the Ma hysteresis loop is the typical and basic one, which is shown in Fig. 1.

Figure 1 Start Ma hysteresis of a hypersonic inlet
In the ¦gure, the horizontal axis is the Ma axis, and the loop with dashed line of which half is in grey and half is in black is the start Ma hysteresis loop mentioned above. Along the loop in the clockwise direction, with the increase of Ma, an inlet will quit unstart and enter a status of start. The Mach-mumber at the division point, shown as Ma R in the ¦gure, corresponds to the minimal self-start Ma. Meanwhile, along the same loop in the same direction, with the decrease of Ma, a started inlet will switch to unstart. At this time, Ma at the division point, shown as Ma L in the ¦gure, is the minimal start Ma for an inlet to keep starting.
There are several signi¦cances about a start Ma hysteresis loop of an inlet. Firstly, it gives two boundaries. One is the minimal Ma, say, Ma L in Fig. 1, at which the inlet can keep starting. The other is the minimal Ma, say, Ma R in Fig. 1, at which the inlet can self-start. Obviously, Ma L is less than Ma R ; so, an inlet which can start at a lower Ma not less than Ma L , will not start until the Ma 0 is increased to Ma R . This is the very reason why the inlet start hysteresis happens.
Secondly, Ma L and Ma R divide the whole Ma axis into three parts. In the left part with Ma < Ma L , the inlet will always unstart, while in the right part with Ma > Ma R , the inlet will always start. In the middle part with Ma L < Ma < Ma R , the inlet may either start or unstart.
Thirdly, the hysteresis loop is separated into two parts by Ma L and Ma R , corresponding to unstarted state and started state, respectively. When the Ma is increased or decreased, so long as it does not cross over Ma L or Ma R , whether the inlet is started or unstarted is decided by the nearest state just before, in fact, the state of the inlet is the same as the previous one.
Fourthly, only when Ma 0 increases and crosses over Ma R , an unstarted inlet can switch to the started state. If Ma 0 decreases and crosses over Ma L , a starting inlet will switch to unstart at once.
To predict the start characteristics of an inlet is just to predict the two Mach-numbers: the minimal Ma for an inlet to keep starting and the minimal Ma for an unstarting inlet to turn to start. In order to be convenient to refer in the present work, the terms Ma L and Ma R are used here to represent them, respectively.

THE GEOMETRIC MODEL AND THE CALCULATION METHOD
The model of the geometry is a hypersonic inlet self-designed by CARDC, with a designing software based on parallel CFD calculation environment and multiple objectives optimization [15]. The model is a 2D inlet. Figure 2a gives the main dimension at the symmetry. The calculation software used in the paper is AHL3D, a parallel CFD software self-developed by CARDC independently. AHL3D solves the time-dependent, Reynolds-averaged Navier-Stokes equations for turbulent, compressible §ows using a ¦nite-volume, time-marching approach on multizone, structured grids. Spatial accuracy is formally second order using the steger-warmming §ux-di¨erence splitting upwind formulation for inviscid §ux-mode. Steady §ows are simulated through an iterative process using local time stepping. Unsteady §ows are simulated through a second-order marching in time. Turbulence models can be chosen from SpalartAllmaras (S-A) one-equation model, or ke/kω two-equation models. AHL3D is capable of solving the §ows of speeds ranging from low sub- Figure 2 The main dimensions at the symmetry (a) and the 2D mesh of the model (b) sonic to hypersonic, especially, for supsonic combustion §ow. At high temperature, AHL3D has models to simulate high-temperature e¨ects in air with speci¦c heat as a function of temperature. For the current calculation, the solver used is based on NavierStokes coupled solution algorithm, adopting time-dependent method to march to a steady state with a turbulent kω shear stress transport model and a material of ideal gas.
In order to reduce the calculation workload, the 2D multiblock structured mesh at the symmetry plane of the model is generated. The total number of the rectangular cells is about 54.3 thousand. Figure 2b shows the structured mesh generated for the prediction of Ma L and Ma R in the hysteresis (a 3D mesh will be generated and used later to guide the verifying experiment). The calculation domain to be discretized is also shown in Fig. 2b. The boundary conditions include free incoming §ow, nonslip, and adiabatic wall condition.
In the calculation, three processes are carried out in the present paper. The ¦rst process is to initialize the whole domain using the incoming §ow and then march the calculation till the §ow converges. The second process is to use a certain convergent §ow as its initialized one and set the incoming §ow condition, then march the calculation till the §ow converges. The third process is to initialize the whole §ow with a velocity of zero, and set the incoming §ow condition, then march the calculation till the §ow ¦eld converges. Whether the inlet starts or not can be judged by the convergent §ow ¦eld; thus, the relationship between Ma 0 at present and the state of the inlet can be built.

The Solution of Direct-Start Ma and Direct-Unstart Ma
The present paper chooses a speci¦c Ma as the incoming §ow Ma, or Ma 0 , and initializes the inlet domain with it. The calculation does not stop until the §ow converges. Judged from the convergent §ow, if the inlet is starting, Ma 0 which is used to initialize the domain is de¦ned to be direct-start Ma. Otherwise, if the inlet is unstarting, then the Ma 0 is de¦ned to be direct-unstart Ma. It is obvious that the direct-start Ma corresponds to a Ma at which the inlet can keep starting and that the direct-start Ma has a minimal value while the direct-unstart Ma has a maximal value. Although the maximal direct-unstart Ma is less than the minimal direct-start Ma, they are almost the same as each other. It can be inferred that the minimal direct-start Ma may possibly be the same as the value of Ma L in Fig. 1.
The direct-start Ma has a minimal value. To predict this value, many calculations with di¨erent Ma 0 are needed. Here, in the current paper, Ma 0 represents the incoming §ow Mach-number(see the Nomenclature above). Figure 3 gives an example for the solution of minimal direct-start Ma.
From the calculation, it can be seen that the minimal direct-start Ma is between 3.5 (Fig. 3a) and 4.0 (Fig. 3b).

The Solution of Ma L
Ma L means a minimal Mach-number at which the inlet still keeps starting. In the Ma start hysteresis loop, Ma L corresponds to the boundary Mach-number for a hypersonic inlet to switch from starting to unstarting.
From Fig. 1, Ma L can be solved as follows. Firstly, try the solution of directstart Ma using a Mach-number, as Ma 0 , which should be greater than Ma R , usually through trial-and-error solution. Thus, a direct-start Ma and a convergentstarted §ow ¦eld can be obtained. Then, decrease Ma 0 but adopt the last convergent-start §ow ¦eld as the initial §ow ¦eld to calculate till it converges. Here, the initialized §ow ¦eld is de¦ned as Converged-Init. Continue decreasing Ma 0 and calculating till the §ow ¦eld converges. Thus, decelerating Ma 0 will result to an unstart state of the inlet, i. e., decelerate-unstart happens. Hence, the relationship of a series of Mach-numbers and starting or unstarting inlet states is achieved. From analyzing the relationship Ma L can be acquired. In practice, Ma L always lies within a range. Figure 4 gives an example process for the solution of Ma L .
From the calculation process and the result shown in Fig. 4, it can be seen that Ma L is between 3.5 and 4.0. It can also be inferred that the range of Ma L is the same as that of direct-start Ma.

The Solution of Ma R
As is pointed out above, Ma R is a boundary Mach-number over which the increasing Ma 0 crosses, the inlet will switch from unstarting to starting state. It is also di©cult to obtain the exact value of Ma R . Instead, the range of Ma R is obtained in practice.
From the analysis of the start Ma hysteresis loop shown in Fig. 1, the solution of Ma R can be explained as following process. First, a direct-unstart Ma and the corresponding convergent-unstart §ow ¦eld can be obtain. Second, use the convergent-unstart §ow ¦eld obtained just now as the initial §ow ¦eld but gradually increase the Ma 0 . Calculate the §ow ¦eld till the solution converges. The research shows that there are two ways to set the initial §ow ¦eld. One is single-step method with variable-step sizes and the other is multiple-steps method with a ¦xed-step size. The way of single-step method with variable-step sizes means that taking a convergent-unstarting §ow ¦eld as the same initial one and parallel changing the step size to set a series of Ma 0 and calculating them till the range of Ma R is attained. The way of multiple-step method with a ¦xed-step size means that for every §ow calculation, the initial §ow ¦eld changes to the latest convergent-unstarting §ow ¦eld with Ma 0 increasing gradually and that all the calculations are the serial ones till the last §ow ¦eld converges to a starting state. Whichever the way is, the process for the solution of Ma R is a process using trial-error-solution. Figure 5 gives an example of single-step method with variabl-step sizes for the solution of Ma R . From the result of Fig. 5, it can be seen that Ma R is in the range between 5.0 and 5.1.   Figure 6 gives an example of muliple-step with a ¦xed-step size for the solution of Ma R .
From the result shown in Fig. 6, it can be seen that Ma R using muliple-step with a ¦xed-step size way is also in a Mach-number range between 5.0 and 5.1.
Comparing the results of these two ways, it is shown that Ma R from the solution of multiple-step method with a ¦xed-step size way is almost the same as that from the solution of single-step method with variable-step sizes way.

The Solution of Zero-Start and Zero-Unstart Mach-Numbers
The zero-start Ma or zero-unstart Ma solution derives from the practice.
The term zero-start Ma is de¦ned as follows. First, initialize the §ow ¦eld with a velocity of zero and then, set the Ma 0 as a speci¦c one, followed by using the steady solver to calculate the inlet §ow ¦eld.
On the one hand, if the §ow becomes starting, a lower Mach-number is set to be Ma 0 and the calculation continues. The process goes on till the §ow converges to an unstarting state.
On the other hand, if the §ow becomes unstarting, a higher Mach-number is set to be Ma 0 and the calculation continues too. With the process going on, a starting state §ow ¦eld will be obtained sooner or later. When the §ow ¦eld of the inlet is starting in these processes, Ma 0 is called zero-start Ma. Otherwise, when the §ow ¦eld of the inlet is unstarting, Ma 0 is called zero-unstart Ma. The following calculation shows that the zero-start Ma solution can replace that of single-step or multiple-step method.
Both the zero-start Ma and the zero-unstart Ma exist in ranges. The minimal zero-start Ma is almost the same as the maximal zero-unstart Ma (in fact, former is a little greater than the latter). Figure 7 gives an example process for the solution of zero-start Ma.
The calculation shows that the zero-start Ma is between 5.0 and 5.1 and that the range of the zero-start Ma is the same as that of step method. This means that the minimal zero-start Ma is identical to the minimal value of self-start Mach-number, or Ma R in the present paper.

THE FURTHER DISCUSSION OF MACH-NUMBERS DEFINED ABOVE
Through the results researched above, the minimal starting Ma of these solutions and processes can be listed here with each range in Table 1.
From Table 1, it can be shown that Ma L is identical to the minimal value of direct-start Ma and that Ma R is identical to the minimal value of zerostart Ma. Since the process of direct-start Ma solution is simpler than that of Ma L solution, and the process of zero-start Ma solution is simpler than that of Ma R solution and both the process for the solution of direct-start Ma and for the solution of zero-start Ma can be parallel calculated at the same time, the present research suggests that the Ma L and Ma R in the inlet start Ma hysteresis loop should be obtained by the solution of direct-start Ma and zerostart Ma.
The following content will further analyze the feasibility and the reliability of replacing Ma L and Ma R with the direct-start Ma and zero-start Ma, respectively, through the comparison of the pressure distribution on the body wall of the converged §ow ¦elds with di¨erent solutions. Figure 8 gives the pressure distributions along the body wall of the convergent §ow ¦elds of the di¨erent solutions. From the ¦gure, it can be seen that all the pressure distributions compared are totally identical with each other. Hence, (i) when Ma 0 is less than the minimal direct-start Ma or Ma L , the in-  let §ow ¦eld will not start and the pressure distribution is the same as each other at the same Ma 0 < Ma L , no matter whether the process used is the solution of direct-start Ma or the solution of Ma L ; (ii) when Ma 0 > Ma L , the inlet §ow ¦eld will always start both by the solution of direct-start Ma and by the solution of Ma L , and the two pressure distributions are the same as each other at the same Ma 0 > Ma L ; (iii) when Ma L < Ma 0 < Ma R , the pressure distribution using step-method by solution of Ma R is the same as that by the solution of zero-start Ma, which means the solution of zero-start Ma can converge to a correct unstarting result; and (iν) when Ma 0 > Ma R , the inlet §ow ¦eld will always start for all the solutions, and the distributions for the solution of zero-start Ma are identical to those for the solution of direct-start Ma, which reveals that the solution of zero-start Ma can converge to a correct starting result.
The double-solution phenomena with Ma L < Ma 0 < Ma R have been veri¦ed by the current research too. Figure 9 gives two §ow ¦elds at Ma 0 of 4.0. Figure 9a

VALIDATION OF THE INLET START CHARACTERISTICS
In order to validate the calculation techniques mentioned in this paper, some experiments are conducted to obtain the start characteristics.

Test Model and Numerical Simulation
The model used in the wind tunnel has the same dimension at symmetry plane as that in Fig. 2. The model has a width in the internal §ow path as 8 times height of the throat and a width in the external §ow path as 13.3 times height as the throat. Figure 10a gives a photograph of the model in the test segment of the wind tunnel. In order to provide the data for validation, the start characteristics are calculated using the numerical simulation techniques by solution of direct-start Ma and of zero-start Ma. The mesh generated here is 3D because the test model has 3D e¨ects in the wind tunnel. The total amount of the multiblock structured mesh cells is about 5.4 · 10 6 shown in Fig. 10b.

The Wind Tunnel Experimental Results
The experiments are conducted in 0.6×0.6 m blowdown sub-tran-sup-sonic wind tunnel in CARDC (China Aerodynamics Research and Development Center). The nominal Mach-number is 4.0, the nominal total pressure is 6.25 atm, and the nominal total temperature is 288 K. Because both the start and unstart states are needed, a rotatable cowl [16] is adopted which can rotate at real-time in the experimental model.
The start and unstart states have been achieved successfully in the experiments. The typical pressure data along the §ow direction are shown in Fig. 11. The results of CFD are shown in the ¦gure too. Figure 11a gives the CFD pressure distribution by the solution of direct-start Ma, and Fig. 11b gives the Figure 11 The comparison of pressure distribution between the calculation and experiment: (a) the inlet is starting (1 ¡ wall: centerbody; 2 ¡ 1st = 7 • ; and 3 ¡ 2nd = 7 • ); and (b) the inlet is unstarting (1 ¡ test = 7 • ; 2 ¡ CFD; and 3 ¡ geometry) Figure 12 The wave structure comparison with calculation and experiments: (a) the inlet is starting; and (b) the inlet is unstarting CFD pressure distribution by the solution of zero-unstart Ma. Comparing the pressure distribution from the experiments with that from calculation, it can be seen that the distributions are almost consistent with each other, which means that the numerical simulation techniques of direct-start Ma and zero-start Ma for the evaluation of inlet start characteristics are of feasibility and credibility. Figure 12 gives the schlieren photographs of the shock structure. The ¦gures also give the numerical schieren. Comparing these two shock wave structures in Figs. 12a and 12b, it can be seen that the numerical simulation techniques by solution of direct-start Ma and zero-start Ma are surely precise and exact enough to predict the inlet start characteristics. This also means that the techniques of direct-start Ma and zero-start Ma presented in the paper can be used to predict the inlet start characteristics and can be developed for the future research and application.
These experiments and their data are preliminary, but the result is exciting. Further research will be carried out at the critical point between start and unstart states and on the 3D e¨ect.

CONCLUDING REMARKS
Based on the analysis of the start Mach-number hysteresis loop and researches of numerical simulation processes, the present paper brings forward several concepts and solutions of some starting Mach-numbers. The conclusion drawn from the current research is as follows: (1) the minimal Mach-number for a hypersonic inlet to keep starting and the minimal Mach-number for the inlet to self-start, corresponding to Ma L and Ma R in the start Ma hysteresis loop, respectively, can be calculated by using AHL3D, a CFD software self-developed by CARDC independently; (2) Ma L and Ma R in the start Ma hysteresis loop can be obtained by the solution of direct-start Ma and by the solution of zero-start Ma, respectively; and (3) the experimental results preliminarily show that the solution of Ma L and Ma R in the current paper can provide a quick and convenient method to predict starting characteristics for hypersonic inlets, especially, for the 2D ones.