Gust response analysis needs to be carried out at detailed design stage for any aircraft to meet airworthiness certification standards.

This calculation is based on a definition of an atmospheric gust model from the airworthiness regulations like the Federal Aviation Regulation, CCAR-25 (China Civil Aviation Regulation) regulations, the vertical continuous gust model is usually taken as the discrete gust load analysis model. In them, the “1-cos” discrete gust model, which has been adopted as the most common standard model.

To conduct gust response calculation in time domain, the mathematical representation of the gust model should be derived in the time domain at first. The representation determines exactly what kind of form will the gust velocity change has and is also the input foundation driving all dynamics response analyses. Eq.(8) shows the corresponding “1-cos” gust’s time domain representation.

Equation (9) is the time-domain expression for the sinusoidal (“sin”) gust.

where: U g is the vertical gust velocity, U d s is is the gust gradient distance (characterizing the gust wavelength), V is the aircraft velocity, t is time.

The gust ratio GR is defined as follows:

where V denotes the gust amplitude and U ∞ represents the free-stream velocity.

The internal flow field of an S-shaped inlet is complex. While appropriate numerical simulation methods can enhance simulation accuracy, it is challenging for any single numerical approach to guarantee absolute precision in the results. Different computational methods may lead to variations in the outcomes. Therefore, before conducting in-depth research and analysis, it is necessary to validate the chosen computational method to ensure its reliability.

The physical model of the S-shaped inlet used for validation is derived from the reference “Experimental Study on Flow Control Technology for S-Shaped Inlets” [14]. The centerline shape of this model is as follows:

where: X c l , Z c l are the x , z coordinates of the centerline; Δ Z c l is the offset distance from the centerline; L is the length of the inlet.

The radius distribution of the inlet (measured perpendicular to the centerline) is as follows:

where: R i is the radius at the throat; R e f is the radius of the inlet exit cross-section.

The geometric parameters of this inlet model are as follows: throat radius R i = 0.0744   m , exit radius R e f = 0.088   m , total length L = 0.5282   m , and offset distance Δ Z c l = 0.1584   m . A bell-shaped lip with a length equal to one times the throat radius is designed at the inlet entrance.

The inlet boundary condition of the S-shaped duct is set as a pressure inlet. For comparison with experimental results, the inflow conditions are configured to match the experimental free-stream velocity V = 60   m / s , with a total temperature of 298 K and a total pressure of 101,000 Pa at the inlet. The angle of attack is α = 8 ˚ and the sideslip angle is β = 0 ˚ . The outlet of the S-shaped inlet is set as a pressure outlet with a specified back pressure, while the walls of the S-shaped duct are treated as no-slip walls. An unstructured mesh is employed for the computational domain, with boundary layer refinement near the walls.

Before conducting numerical simulations, it is essential to determine the appropriate mesh size. In this study, three grids with varying densities were designed to conduct a grid independence study. The specific grid cell counts are listed in Table 1.

Table 1. Mesh size.

The CFD method described earlier was employed to simulate the S-shaped inlet, and the computational results obtained from the three grids were compared with experimental data. The comparative metrics selected are the total pressure recovery coefficient and the distortion index, which are defined as follows:

Taking the center point of the inlet exit plane as the reference point, p t 2 is the average total pressure at the inlet exit, p t ∞ is the free-stream total pressure upstream of the inlet entrance, ( p min ) 60 is the minimum average total pressure within a 60˚ angular range, q a v is the average dynamic pressure at the exit, and p a v is the average total pressure at the exit.

Figure 1. Variation of distortion index with mass flow coefficient.

Figure 2. Variation of total pressure recovery coefficient with mass flow coefficient.

As shown in Figure 1 and Figure 2, based on the comparison, the relative error between the experimental total pressure recovery coefficient and the simulation result is 0.5%, while the relative error for the distortion index is 1.0%. In summary, the methodology employed in this study can effectively simulate the flow field of the S-shaped inlet.

This study employs the commercial CFD software Star-CCM+ as the solver. For steady flows, an implicit steady solver is adopted for steady-state computations; for unsteady gust responses, a first-order implicit unsteady formulation is utilized. The SST k-ω turbulence model is selected to ensure resolution accuracy in regions with adverse pressure gradients. Spatial discretization is performed using a second-order upwind scheme, while transient terms are handled with a second-order implicit formulation.

The research object here is the intake gust response, the geometrical size of the physical model is that the fuselage length is 13.5 m and the wing span is 25.983 m. The set-up flowfield is rectangular flowfield whose dimension is 260 m (length) × 200 m (width) × 100 m (height), and the nose of the model is located at 60m away from the flowfield inlet, at a distance of 50 m from the upper and lower boundary, as well as 100 m away from the left and right boundaries. The model is presented in Figure 3.

Figure 3. Computational model.

The inlet is set as a velocity inlet, the outlet as a pressure outlet, the aircraft surfaces as no-slip walls, and the remaining faces are also set as pressure outlets. The distribution of these boundary conditions is shown in Figure 4.

Figure 4. Computational domain.

The mesh quality and the mesh quantity have a great influence on the calculation result and efficiency of the calculation program. In order to verify that the meshing method adopted in this paper can meet the calculation requirement, a grid independency test has been carried out. In the present investigation for that purpose we have used three series of different mesh refinements (coarse, medium, and fine—were created. The simulations for the same boundary conditions have been conducted on all of them. In particular, at the working point Ma = 0.71; H = 5 km are used three meshes sets composed by about 35.25 millions, 17.76 million, and 10.54 million elements was verified. The verification results are as Figure 5 shows.

(a) (b)

(c) (d)

Figure 5. Grid independence study validation. (a) Variation of lift coefficient with varying mesh counts; (b) Variation of drag coefficient with varying mesh counts; (c) Total Pressure recovery coefficient; (d) Distortion index.

The lift coefficient and drag coefficient change almost not at all with increasing element quantity among the three mesh sets. And the deviation of both the lift coefficient and drag coefficient among the three sets is less than 5%. It can be seen that the mesh resolution has enough ability to reflect the details of the flow field and satisfies the accuracy requirement of numerical simulation.

Therefore, we chose the mesh scheme which has about 17.76 million elements as the final study. The reason is that it can get a better compromise in terms of calculation speed and precision. It not only improves the computing stability but also ensures the accuracy and reliability of the simulation results. Then, for this mesh, the boundary layer is solved by using 30 layers. The thickness of the first layer near the wall is 9.396 × 10⁻⁶ m, the total boundary layer thickness is 8.129 × 10⁻² m. The number of elements in the final mesh is 17.76 M.

The operating conditions for the steady ingesting inflow case have been fixed as mentioned in the following Table 2 so that we can investigate how does the aerodynamic performance of the UAV’s inlet changes with change in angle of attack, and to systematically study the aerodynamics behavior of the over-wing mounted intake in case of a blended-wing-body UAV with a uniform inflow, so as to give some benchmark results which can be compared to future investigations in case of a gust disturbance. The main studied parameters are the exit Mach number, total pressure recovery coefficient and total pressure distortion index in this paper.

Table 2. Calculation conditions table.

The Total Pressure Recovery Coefficient can be defined as a ratio between the mean total pressure on the exit of the intake and the freestream total pressure up-stream from the inlet of the intake, denoted as σ ：

The non-uniformity of the total pressure on the exit plane of the intake, known as distortion, can be measured by a quantity called Distortion Index, denoted by D C ( θ c r ) , and it can be defined as：

where: ( p min ) θ c r is the minimum sector average total pressure within the θ c r sector, θ c r is the critical distortion angle (in this paper, is θ c r = 60 ˚ selected),； q a v is the average dynamic pressure,； p a v is the average total pressure across the entire exit plane.

The overall pressure recovery coefficient ( σ ) and inlet distortion index ( D C ( θ c r ) ) for the developed intake at varying angles of attack, are shown in the following picture.

As shown in Figure 6, with the increase of the angle of attack, the total pressure recovery coefficient decreases and the distortion index is increased accordingly. It can be seen that there is an overall decline on the aerodynamic performance for the intake at larger angles of attack, with increased flow loss, and reduced uniformity of the exiting flow.

(a) (b)

Figure 6. Total pressure recovery coefficient and distortion index under steady-state conditions. (a) Variation of total pressure recovery coefficient with angle of attack; (b)Variation of distortion index with angle of attack.

The change in the distribution of pressure contours can also indicate the mechanism of field flow behind such a performance change, as indicated by the following Figure 7, with a growing AoA, the high pressure zone in the middle of the intake exit plane is greatly compressed, and it’s pressure maximum is slowly decreasing at the same time as the low-pressure area above the plane keeps expanding. This immediately translates into a lower overall total pressure level at the exit (i.e. lower total pressure recovery coefficient), as well as higher pressure gradient over the plane (i.e. a deterioration (increase) of the distortion index), and thus explains the inherent degradation of intake performance with angle of attack, from the viewpoint of flow field structure. We can also notice that with an increasing angle of attack, there is a corresponding increase in the inlet velocity of the incoming flow. It will further enhance the compression effect and the tendency of boundary layer separation, which plays an important role on deteriorating total pressure recovery performance, as well as increasing flow distortion.

Figure 7. Intake exit pressure contour.

As shown in Figure 8, the streamline plot indicates that main stream inside the intake is remained attached for all of the calculated range of angle of attack, with no large-scale flow separation occurring. As the angle of attack increases however, more low-energy boundary layer flow from the forebody is ingested into the intake, and it plays a much stronger role. This low-total-pressure flow entrainment directly dilutes the total pressure level at the exit plane and decreases the total pressure recovery coefficient. Meanwhile, the increased mixing with, and interaction of this low energy flow with the main core stream increases the pressure gradient in the exit plane, which manifests itself by increasing of a distortion index. This behavior also makes physical sense given what we see from the pressure contours—the narrowing of the high pressure region and widening of the low pressure region—together explain how the increased AoA causes a degradation to intake performance through its effects on internal flows.

Figure 8. Streamline plots at different angles of attack.

The above-mentioned steady state situations have been analyzed to reveal how systematically the aircraft attitude (angle of attack) affects the intake performance, but, in real flights, atmospheric turbulence/gust is inevitable source of unsteady disturbance. The disturbance subjects the intake inlet to extremely fluctuated, non-homogeneous boundary condition in time. This transitory inlet condition will not only cause transitory flow separation and reattachment within the inlet, but it will also more seriously, propagates pressure perturbations downstream that can endanger the operating margin of the engine.

Thus, a steady-state-only assessment of its performance cannot be sufficient for representing the intake’s applicability in real, complex environments. To explore its dynamic disturbance rejection ability, as well as the transient performance of its flow field quality, unsteady numerical simulation for the whole intake in case of normal gust excitation would be performed. The main research objective is to investigate the multi-dimensional dynamic characteristics of intake performance parameter (total pressure recovery coefficient and distortion index) caused by gust disturbance, examining in depth how each one of the following factors influences:

Different Mean Angles of Attack: Studies the variation in the intake’s sensitivity and stability to dynamic disturbances under different baseline flight attitudes.

Gust Frequency Variation: Studies the dependence on the perturbation timescale (from slow low frequency fluctuations, up through fast high frequency transients), with respect to the resulting interior flow response in the inlet.

Varying Gust Strengths (Gust Ratio): The effect of gust magnitude on the extent of variation in control response variables, as well as on the severity at which a stall condition occurs.

Different Gust Types (Sinusoidal Gust, 1-cos Gust): Compares the differing mechanisms by which disturbances with distinct spatial-temporal distribution characteristics affect the transient loads and performance degradation of the intake.

With an integrated study on the aforementioned high-dimensional parameters space, this paper is intended for revealing systemically the law about the development of the intake performance under the unsteady inflow, provides detailed data support and theoretical basis to the dynamic stability evaluation and robustness design of.

The purpose of present work is to improve the evaluation of the operating envelope of the intake in terms of dynamics so as to provide necessary foundation on predicting the performance and designing the stability of the intake at unstably flying states.

The calculation conditions for the intake under gust environments are as follows Table 3.

Table 3. Calculation conditions under gust environment.

The gust response results of the intake are as follows.

The curves in Figure 9 and Figure 10 show the transient response characteristic of main aerodynamics parameter of intake with different angle-of-attack, and on total pressure recovery coefficient: in a region of small angle of attack values where the influence of the incoming gust disturbance onto the inner flow pattern is rather low. The mean-value of total-pressure-recovery-coefficient is still relatively large and has a small amplitude of fluctuation, which presents a relative-fluctuation level as 2.05%, 0.81%, and 2.39%, respectively. The variations are clearly well temporally correlated to the gust velocity wave shape suggesting that at these conditions the overall evolution of the pressure recovery is mainly driven by the local variations of the inflow speed.

Figure 9. Total pressure recovery at different angles of attack.

Figure 10. Distortion index at different angles of attack.

With increasing angle of attack, intensified flow separation and boundary layer distortion make the effect of gust disturbance much more pronounced. In a large angle-of-attack regime, the standard deviation value of overall pressure recovery factor is increased up to 4.37%. It is also worth mentioning here that there is higher fluctuation level for 8o AoA as compared to other two low AoAs which implies that with increasing AoA, the flow becomes less stable.

For the distortion index itself, it can be seen from the dynamic response also has the feature of following with respect to the incoming gust, but there is an obvious difference in terms of angle of attack: the amplitude of the distortion index’s vibration at large angles of attack is much larger than that of small angles of attack. This result again proves that a larger angle of attack will deteriorate the aerodynamic performance of the intake, and it also intensifies the unsteady features of the flow non-uniformity. So it has larger effect to the steady running of lower stream engine.

Noteworthy also is that after passing the gust, it can be clearly observed that the total pressure recovery coefficient still exhibits fluctuations after 1 second. This phenomenon reveals that the reconstruction and stabilization of the internal flow field within the inlet require a certain amount of time, and the flow recovery lags behind the external free-stream disturbances. The fact indicates that the reorganization/stabilization process of the internal flow field inside the intake takes some time to complete, i.e., the return to its aerodynamic condition is delayed with respect to the end of the external perturbation.

As shown in Figure 11 and Figure 12, at various gust ratios in excitation conditions, the intake’s dynamic responses features show obvious rules: The transient trend for the overall pressure recovery coefficient as well as the distortion index is identical, but that their dynamic response amplitudes change dramatically as a function of the gust magnitude (gust ratio). In particular, higher gust ratio represents stronger input of dynamic pressure disturbance that increases greatly the unsteady characteristics of the internal flow inside the intake as well as the aerodynamic/structural coupling effect; thus, causing the sharp increase in the oscillation amplitude of both the total pressure recovery coefficient and the distortion index. On the other hand, at low gust ratio condition, the energy of the incoming flow perturbation is smaller, leading to a small modulation for the flow state; and so do the amplitude of the dynamical fluctuations of the above quantities.

Figure 11. Total pressure recovery at different gust ratios.

Figure 12. Distortion index at different gust ratios.

This also means that the gust ratio determines the magnitude of the dynamic response gain of this system, namely it controls how much the response of the intake will amplify an incident flow disturbance and when this becomes unstable.

As shown in Figure 13 and Figure 14, in the response to the gust excitation with various frequency, although the evolution trend of total pressure recovery coefficient and distortion index in the time domain are quite similar, their response amplitudes are strongly dependent on the frequency. With an increasing excitation frequency, a decreasing gust wavelength is associated with it. In particular, since the wavelength of a 1 Hz and 10 Hz gust is about 189.6 m and 18.96 m, respectively. This discrepancy in wavelength is basically reflecting on the spatial scale of the flow perturbation: low frequency perturbations possess a large wavelength, leading to rather mild spatial velocity gradient; while the high frequency perturbations possess small wavenumbers, resulting in a much larger velocity gradient that reduces the adaptive adjustment and equilibration timescale of the flow structure against the perturbation. Therefore, at a high frequency excitation, the amplitudes of dynamic fluctuation of the two quantities (total pressure recovery coefficient and distortion index) have been greatly weakened.

Figure 13. Total pressure recovery at different frequencies.

Figure 14. Distortion index at different frequencies.

As shown in Figure 15 and Figure 16, the unsteady aerodynamic response under the excitation from different kinds of gusts shows obvious difference in modes for the intake. Under the 1-cos gust, without having a continuous vertical component of motion in its excitation cycle, has lesser degree of dynamic loading on the intake inlet flow field. Hence, it excites aerodynamical response of low frequencies, while both overall pressure recovery factor and distortion level have just one clear minimum in this course. On the contrary, the sinusoidal gust contains one full cycle of vertical velocity change. It induces more intense response because of its larger flow field excitation. During the same time period, the overall pressure recovery coefficient presents evident valleys and peaks that lead to a more compact dynamic behavior.

Figure 15. Total pressure recovery for different gust types.

Figure 16. Distortion index for different gust types.

A further observation is that in the case of the 1-cos gust the instantaneous value of the total pressure recovery coefficient remains lower than its corresponding value obtained under the gust-free baseline state for most part of the time; indicating therefore that such an excitation form has a persistent attenuating effect upon the time-averaged performance of the intake. The dynamic evolution of the distortion index displays high synchronicity to that of the total pressure recovery coefficient, their patterns of change to be approximately similar. This together proves that there exists a good correlation between the inside non-steady flow field characteristics and aerodynamics performance of the intake under the disturbance of various gusts.

During the investigation on the influence from sinusoidal gust excitation, it is found that the total pressure recovery coefficient can, at some times to be higher than its steady state (gust free) level. This happens in the time when the gust has a vertical downwards component which means that instantaneous inflow disturbances with a certain orientation can, to some extent and for a limited time period to alleviate the flow condition in the intake and improve the total pressure recovery performance. This result indicates that the impact of gusting on the inlet can be non-negative in some cases, with some relationship between the directional character of the gust and transient aerodynamic gain.

Figure 17 shows the pressure contour distributions on the exit plane during one full cycle of the gust (Sinusoidal gust, 5 Hz, Gust Ratio 0.1). It can be seen from the pressure contours that the overall pressure field pattern changes as the gust progresses through its cycle: During the early period (first ¼ cycle), the higher the gust strength is, the smaller area of pressure peak appears to be. Second 1/4-cycle: With the weakening of the gust, during the second half of this 1/4-cycle the HP region grows again with a simultaneous large growth of its pressure maximum.

Figure 17. Exit pressure contours at different time instants (Condition: Sinusoidal gust, 0˚ angle of attack, 5 Hz, Gust Ratio 0.1).