Aerodynamic Response of a Serpentine Inlet to Horizontal Periodic Gusts

Abstract

Gust is a common atmospheric turbulence phenomenon encountered by aircraft and is one major cause of several undesired instability problems. Although the response of aircraft to the incoming gust has been widely investigated within the subject of external-flow aerodynamics in the past decades, little attention is paid to its effects on the internal flow within aircraft engines. In this paper, a newly implemented Field Velocity Method (FVM) in OpenFOAM is used to simulate the flow field and aerodynamic responses of a serpentine inlet exposed to non-stationary horizontal sinusoidal gusts. Validations are performed on the results obtained based on the baseline Computational Fluid Dynamics (CFD) solver and the gust modeling method. Finally, the flow field and aerodynamic characteristics of the serpentine inlet under horizontal sinusoidal gust conditions are comprehensively investigated. It is found that the gusts not only significantly change the flow structure but also play an unfavorable role in the total pressure distortion of the serpentine inlet. This finding shows the necessity to consider gust effects when designing and evaluating the performance of aircraft engines.

1. Introduction

Gust is one of the most common and important atmospheric phenomena for flight and has played an important role in many catastrophic flight accidents in the past decades [1,2,3]. Investigations of aircraft aerodynamic responses to atmospheric disturbances like gusts have demonstrated the importance of gusts in aerospace engineering because they are generally unexpected, giving the aircraft control less time to respond. Taking aircraft lifting surfaces such as wing and horizontal tail as examples, the unsteady loads caused by gusts not only decrease the fatigue lifetime of the structure but also affect the flight performance and passengers’ comfort level [4]. Severe atmospheric gusts can lead to serious effects on the flight path and propulsion system [5,6]. Therefore, gust loads have to be considered in the load cases during aircraft development and all the representative airworthiness regulations such as the FAR-25 and the CS-25 have made detailed specifications for the determination of critical gust loads [4]. Several artificial gust generators have been developed to experimentally investigate these phenomena, which include rotating slotted cylinders [7], oscillating plates [8], wind vanes [9], and active grids [10].

Using CFD (Computational Fluid Dynamics) methods, it is possible to prescribe the gust velocity using the Field Velocity Method (FVM) [11,12] and the Split Velocity Method (SVM) [13,14,15,16] for simulating the airfoil lift response to a gust of arbitrary shapes. The main differences between FVM and SVM are twofold. First, SVM requires special grid refinement for gust grid transport while FVM does not. Second, SVM can account for the mutual interaction of the gust and the research object [6] since the gust is resolved in the flow field, whereas the gust shape is unaffected from the start to the end in FVM [17]. A wide range of reduced-order models [18,19,20,21] was also developed for the efficient estimation of airfoil aerodynamic force responses to gusts [22,23,24]. These simple models may be combined with a CFD method to enhance the accuracy of the result and to provide a direct coupling with the aeroelastic equations of motions for predicting displacement responses [25].

It is well known that the engine inlet is usually located at the very front of the propulsion system and plays an important role in deciding the performance of the engine itself. Therefore, any disturbances in the incoming air may cause significant flow field deterioration, such as distortion and separation in the inlet. Although first noticed as early as in the 1960s [26], hardly any emphasis has been placed on investigating the effects of gusts on engine inlets, and only a handful of numerical studies on gust-inlet interactions were carried out so far [27,28,29]. Kozakiewicz and Frant [30,31] numerically studied the impact of changes in speed, gust angle, and sideslip angle on the development of the intake vortex for both a single and a double fuselage-shielded inlet. A common conclusion can be drawn that low-speed gust is the most dangerous because low-speed gusts contribute to the formation of the inlet vortex regardless of the gust angles for a wide range of sideslip angles [31]. However, the gusts in the aforementioned study were indeed steady crosswinds, rather than unsteady oscillating gusts as prescribed in the existing airworthiness regulations. Halwas and Aggarwal [28] numerically investigated the effect of side gust on the performance of a supersonic inlet. They found that the shock structure, flow characteristics, and inlet performance all are strongly affected by the presence of gust. The shock structures and the external and internal flows become asymmetric. The shock/boundary-layer interaction becomes stronger, resulting in strong boundary-layer separation, secondary flow, and a separated flow region in the diffuser section. In addition, the inlet performance generally deteriorates, resulting in significant reductions in the total pressure recovery and the mass flow ratio and increases in the flow distortion. However, the adverse effects of the shock waves and their interaction with the boundary layer can be effectively removed by using a bleed system [29]. Übelacker, Hain and Kähler [32] experimentally investigated the interaction between the gust generated by a pitching airfoil and the leading-edge separation bubble of a through-flow nacelle. It was found that the gust significantly influences the inlet separation region by changing the angle of attack.

From the above discussions, it is easily found that only limited gust studies have been performed concerning the inlet internal flow consideration compared to the extensive studies on the wing external flow aspect. Moreover, it is seen from the small amount of open literature that the more common unsteady gusts in nature, such as oscillating and transporting gusts, were rarely involved. The present work is motivated by these considerations. In this paper, we present new insights into the flow characteristics and mechanisms of an ultra-compact serpentine inlet subjected to periodic horizontal gusts via computational fluid dynamics (CFD) on the platform of the open-source package OpenFOAM^®. The aims of this study are primarily threefold. First, this study is thus motivated to provide a primary insight into the effects of gusts on internal flows and attract more attention to this relevant field. Second, all existing studies about gust effects on engine inlets [28,29,30,31] investigated the effect of side gusts with changing sideslips but with a constant magnitude. The present study investigates unsteady sinusoidal gusts, which may reveal different mechanisms regarding gust-inlet interactions in a real environment. Finally, most of the existing gust modeling methods are based on in-house codes. This study implements the FVM within the open-source OpenFOAM toolbox, providing better access for researchers in the field.

Considering the streamwise direction of the main flow inside the diffuser, horizontal gusts are deemed to more significantly affect the inlet flow and thus are adopted by this study. The unsteady solver for the inlet aerodynamics is validated by comparing the calculation results with the previous wind-tunnel measurement data [33]. Both the flow field characteristics and aerodynamic performance responses of the inlet are inspected to reveal the underlying mechanism for the gust-inlet interaction. The following content of this paper is structured as follows. Section 2 presents the relevant theories and details of the numerical setup. Section 3 presents and analyzes the results of the effects of periodic gusts on the serpentine inlet. Finally, a conclusion is made in Section 4 to summarize the results and findings obtained from the present study.

2. Theory and Methodology

2.1. Model Description

The inlet model in the present work is an ultra-compact serpentine inlet integrated with the forebody of a flying-wing stealth aircraft, which has been studied previously by the Inlet Research Group of Nanjing University of Aeronautics and Astronautics (NUAA) [33,34], as shown in Figure 1a. The main geometric parameters of the model are tabulated in

Table 1. Main geometric parameters of the ultra-compact serpentine inlet.

Parameter	Value
Diameter of the AIP	D (65 mm)
Distance between the forebody tip and the inlet	2.0 D
Total length of the inlet	2.5 D
Length of the diffuser	2.3 D
Vertical offset of the diffuser	0.66 D

. The entrance of the top-mounted serpentine inlet is smoothly merged with the forebody of the aircraft. The inlet is ultra-compact for the sake of the rigorous restrictions of the aircraft size and weight, with a length of 2.3 D for the compact diffuser, where D = 65 mm is the diameter of the aerodynamic interface plane (AIP). Section 1-1 shows half of the trapezoidal shape of the entrance section and Section 2-2 the unique concave shape of the middle section. The typical cross-sectional shape of the middle section of a traditional design is also shown in Figure 1a for contrast. Furthermore, the vertical offset between the entrance center and the exit center of the diffuser is as high as 0.66 D, which is highly prone to boundary layer separations at the aft part of the top surface and the fore part of the bottom surface. Therefore, a large internal bump is introduced into the top surface to suppress massive flow separations and decrease the distortion index at the AIP (Figure 1b). More descriptions of the configuration of the inlet model are accessible in our previous investigation [33]. Experimental measurements were previously done in the NH-1 high-speed wind tunnel of NUAA and will be reused here for the current numerical model validation.

2.2. Performance Parameter Definition

Two common parameters, i.e., total pressure recovery σ and circumferential total pressure distortion index, DC₆₀ are employed in this paper to characterize the inlet performance at the aerodynamic interface plane (AIP) between the inlet and the engine. σ is defined as

$$\sigma =p_{avg360}^{*}/p_{\infty }^{*}$$

(1)

where $p_{avg360}^{*}$ is the mass flow weighted average total pressure at the AIP, and $p_{\infty }^{*}$ is the total pressure of the freestream air. And DC₆₀ is calculated as

$${\mathrm{DC}}_{60}=\left(p_{min60}^{*}/p_{avg360}^{*}\right)/q_{avg360}$$

(2)

where $p_{min60}^{*}$ is the minimum value of the mass flow weighted average total pressure over any 60° sector around the center of the AIP, $q_{avg360}$ is the mass flow weighted average dynamic pressure at the AIP.

In addition, the static pressure coefficient $\mathrm{Cp}$ is used to discuss the static pressure distribution on the wall of the inlet, which is defined as

$$\mathrm{Cp}=\left(p-p_{\infty }\right)/q_{\infty }$$

(3)

where $p$ and $p_{\infty }$ are the static pressures on the inlet wall surface and of the freestream air, respectively. $q_{\infty }$ is the dynamic pressure of the freestream air.

2.3. Governing Equations

The unsteady three-dimensional (3D) compressible Reynolds averaged Navier-Stokes (URANS) equations are solved for both investigations: no-gust and gust cases. Meanwhile, the turbulence is modeled by the Menter $k-\omega$ Shear Stress Transport (SST) model [35] with a fully turbulent boundary layer assumed. The governing equations including the mass, momentum, energy, and turbulence equations can be written in a common vector form as:

$$\frac{\partial }{\partial t}{{\displaystyle \int }}_{\mathsf{\Omega }}\stackrel{\rightharpoonup }{W} d\mathsf{\Omega }+{{\displaystyle \oint }}_{\partial \mathsf{\Omega }}\left({\stackrel{\rightharpoonup }{F}}_c-{\stackrel{\rightharpoonup }{F}}_v\right)dS={{\displaystyle \int }}_{\mathsf{\Omega }}\stackrel{\rightharpoonup }{Q} d\mathsf{\Omega }$$

(4)

where $\stackrel{\rightharpoonup }{W}$ is the vector of conservative variables, ${\stackrel{\rightharpoonup }{F}}_c$ is the vector of convective fluxes, ${\stackrel{\rightharpoonup }{F}}_v$ is the vector of viscous fluxes, and $\stackrel{\rightharpoonup }{Q}$ is the source term comprising all volume sources due to body forces, volumetric heating, and turbulence. $\mathsf{\Omega }$ is the control volume bounded by the closed surface $\partial \mathsf{\Omega }$, and $dS$ denotes the surface element. The concrete expressions of all the variables can be found in Blazek’s book [36].

The velocity correction is applied throughout the flow field. A horizontal sinusoidal gust can be perceived as a superposition of a sinusoidal time-variant velocity field parallel to the freestream. As opposed to SVM, which is also named as Resolved Gust Approach (RGA) in [25,37,38], the superposed velocity field is modeled by modifying the grid time metrics without actually moving the grid. Therefore, there is no need for a high grid resolution in the whole domain to transport a gust from the inflow boundary to the aircraft to minimize the numerical losses [17]. Mathematically, FVM can be explained by considering the general flow velocity $\stackrel{\rightharpoonup }{V}$ in the computational domain, which can be rewritten as

$$\stackrel{\rightharpoonup }{V}=\left(u-x_{\tau }\right)i+\left(v-y_{\tau }\right)j+\left(w-z_{\tau }\right)k$$

(5)

where $u$, $v$, and $w$ are the components of the velocity along the coordinate directions, and $x_{\tau }$, $y_{\tau }$, and $z_{\tau }$ are the grid time metrics components. For the flow over a stationary body, these components are zero.

The velocity field in the presence of a horizontal gust can be expressed as

$$\stackrel{\rightharpoonup }{V}=\left(u-x_{\tau }+u_g\right)i+\left(v-y_{\tau }\right)j+\left(w-z_{\tau }\right)k$$

(6)

Thus, the modified time metrics are

$${\tilde{x}}_{\tau }i+{\tilde{y}}_{\tau }j+{\tilde{z}}_{\tau }k=\left(x_{\tau }-u_g\right)i+y_{\tau }j+z_{\tau }k$$

(7)

The type of gust of interest in the present study is horizontal and sinusoidal, i.e., only the velocity component in the horizontal (streamwise) direction changes with the time variable in a sinusoidal fashion. The gusty inflow velocity profile simulated in this study is illustrated in Figure 2. It consists of only one full period of sinusoidal oscillation followed by the mean velocity for t > T. The oscillation amplitude, frequency, and mean velocity of the horizontal sinusoidal gusts of interest in this study are ${\tilde{u}}_g$, $f$, and $u_0$, respectively.

The horizontal component of the gusty inflow velocity can be expressed as

$$u=u_0+{\tilde{u}}_g\cdot \sin (2\pi f\cdot t)$$

(8)

where $t=\left[\mathit{0}:\frac{\mathrm{T}}{\mathrm{N}}:\frac{\left(\mathrm{N}-1\right)\mathrm{T}}{\mathrm{N}}\right]$ is the vector of time, discretizing one period of gust $\mathrm{T}$ into $\mathrm{N}$ equal intervals. $\mathrm{N}$ decides the resolution of the discretized gust shape. A larger $\mathrm{N}$ implies that the discretized gust shape is more approximate to the physically continuous one.

Discretization of the gust velocity was done by using Matlab. As for a complete running of a gust case, an unsteady calculation of the no-gust case was first carried out by the compressible solver ‘rhoPimpleFoam’, of which the result was used to initialize the flow field for the subsequent gust computation. Then, the gust velocity, which is read from an external velocity document, is superposed with the mean velocity of the flow field for calculation of the inlet response, as illustrated in Figure 3.

2.4. Solution Algorithm

All the velocity, turbulence, and energy terms were discretized by the second-order linear-upwind scheme, as well as the pressure terms by the second-order linear differencing scheme. The dual time stepping approach was used for the temporal discretization together with a second-order backward Euler differencing scheme. Within each time step, the linearized algebraic equations were solved using a generalized Geometric-algebraic Multi-grid (GAMG) convergence acceleration method with the Gauss-Seidel smoother. In addition, the PIMPLE algorithm [39], which is a combination of the pressure-implicit split operator (PISO) and the semi-implicit method for pressure-linked equations (SIMPLE) algorithms, was used to solve the pressure-velocity equation [40]. The time step size is adjustable so that the Courant number remains below one, to satisfy the CFL condition. Solutions were computed on a 2-CPU workstation with a total of 128 “AMD-EPYC-7742” processors. All the calculations were performed using a Linux-compiled version of the OpenFOAM code (Version 1912) for multiple parallel computations. The convergence criterion was based upon a decrease in solution residual error of at least four orders of magnitude from the starting conditions.

2.5. Mesh and Boundary Condition

Due to the symmetry of the geometric model and the flow field characteristics of all simulations in this study, half of the model is adopted for numerical construction. The computational domain is set as 14 D, 16 D, and 10 D in the x, y, and z directions, which can sufficiently eliminate the boundary effect on the computed results, as shown in Figure 4. The origin of the coordinate system concerning the geometry is at the leading of the fore-body. The internal duct was also extended linearly by 1.0 D from the AIP to the actual exit of the diffuser. For mesh generation, a fully structured multi-block mesh strategy is adopted using the ANSYS ICEM CFD software [41]. The fine mesh, which will be adopted for the whole study, in total has 51 blocks and 4.20 million cells. The internal diffuser mesh employs an O-grid topology and has 201, 120, and 31 nodes in the streamwise, circumferential, and radial directions, respectively. The height of the first layer adjacent to all the walls is set to be 1 × 10⁻⁵ m, which satisfies the requirement of the normalized wall distance $y^{+}\sim 1$.

The boundary conditions are also shown in Figure 4. At the symmetry plane, colored in blue, no fluid can enter or exit the domain, and the normal gradients of the scalar variables and velocity components tangential to the symmetry plane are zero, which can be expressed as

$$\stackrel{\rightharpoonup }{v}\cdot \stackrel{\rightharpoonup }{n}=0, \stackrel{\rightharpoonup }{n}\cdot \nabla \varphi =0, \stackrel{\rightharpoonup }{ n}\cdot \nabla \left(\stackrel{\rightharpoonup }{v}\cdot \stackrel{\rightharpoonup }{t}\right)=0$$

(9)

where $\varphi$ stands for the scalar variables including $p$, $T$, $K$, and $\omega$. $\stackrel{\rightharpoonup }{t}$ denotes the unit tangential vector. The outlet of the domain (in green) and the outlet of the diffuser are applied with the Neumann boundary condition given by

$$\nabla \stackrel{\rightharpoonup }{v}=0$$

(10)

and all the walls are treated as no-slip and impermeable walls, i.e.,

$$\stackrel{\rightharpoonup }{v}\cdot \stackrel{\rightharpoonup }{t}=0, \stackrel{\rightharpoonup }{v}\cdot \stackrel{\rightharpoonup }{n}=0$$

(11)

A list of the boundary condition settings for the computational domain concerning the main variables in OpenFOAM is shown in

Table 2. Boundary condition settings for the computational domain.

Boundary	$\mathit{k}$	$\mathit{\omega }$	p	T	$\stackrel{\rightharpoonup }{\mathit{V}}$
Farfield	inletOutlet	inletOutlet	freestreamPressure	inletOutlet	freestreamVelocity
Outlet	inletOutlet	inletOutlet	fixedValue	inletOutlet	zeroGradient
Symmetry plane	symmetry	symmetry	symmetry	symmetry	symmetry
Forebody_inlet	kqRWallFunction	omegaWallFunction	zeroGradient	zeroGradient	noSlip
Diffuser_outlet	inletOutlet	inletOutlet	fixedValue	inletOutlet	zeroGradient

.

3. Results and Discussions

3.1. Mesh Independence Examination

For the CFD simulation, the first important consideration is to guarantee the results are independent of the mesh used, i.e., to conduct a mesh independence examination. To this end, a set of grids sequentially named as coarse-, fine-, and dense-mesh is checked at the freestream Mach number of M = 0.7 and the non-dimensional static pressure at the AIP p_AIP/p₀ = 1.18 where p₀ is the static pressure of the freestream air.

Table 3. Inlet performance calculated with different resolved meshes at M = 0.7 and p_AIP/p₀ = 1.18.

Data Source	Number of Grid Cells	σAIP	Error of σAIP
EXP. [33]	—	0.961	—
Coarse mesh	2.48 million	0.968566	1.039 × 10−3
Fine mesh	4.20 million	0.967687	1.597 × 10−4
Dense mesh	5.95 million	0.967527	—

tabulates the number of cells, the calculated total pressure recovery σ_AIP, the absolute error relative to the result by the dense mesh, as well as the previous wind-tunnel experimental measurement [33]. All three grids obtain satisfactory results with good accuracy relative to the experimental measurement. Enhancing the mesh resolution from the coarse mesh to the fine one has reduced the prediction error by an order of magnitude.

The differences between the total pressure contours and the wall static pressure coefficient [33] predicted by the three meshes are also very small, as shown in Figure 5 and Figure 6, respectively. The five low-total-pressure zones as depicted in our previous study by Ansys Fluent [33] are all captured by the three meshes, as well as the bat-shaped distribution, as shown in Figure 5. The predicted static pressure distributions along the top and bottom surfaces of the serpentine inlet all agree well with the experimental results for all the meshes. In summary, there is no significant difference between the results obtained by the three generated meshes both qualitatively and quantitatively. Considering the computational efficiency and accuracy for the later simulation of gusts, the fine mesh is selected for all the subsequent gust response computations.

3.2. Unsteady Aerodynamic Solver Validation

To validate the accuracy of the current solver for prediction of the fundamental aerodynamic performance with no gust, an experimental case at the test condition of M = 0.7 and p_AIP/p₀ = 1.18 was first calculated with the current solver. The time-averaged surface static pressures on both the top and bottom surfaces of the inlet along with the variation distribution of the transient values are shown in Figure 7, where our previous wind-tunnel experimental measurements and CFD results calculated by Fluent [33] are also shown for comparison. The variation bars represent the mean, peak, and valley pressures counted from an efficient period. Generally, the current solver predicts the wall static pressure well with good agreement with the experimental measurements. The adverse pressure gradients are slightly under-predicted by both solvers, which is due to the common deficiency of RANS methods in predicting three-dimensional rotating and separated flows [33,42,43]. The maximum relative errors of the results predicted by OpenFOAM and Fluent relative to the experimental measurements are about 24.5% and 20.1%, respectively. Despite these differences, the trends are consistent with that of the experimental results, and the size of the flow separation region on the lee side of the top surface is correctly predicted.

Figure 8 presents a comparison between the time-averaged total pressure contours at the AIP predicted by the current unsteady solver and the experimental observation. The typical characteristics of the flow structure are successfully simulated by the current solver. The five low-total-pressure zones as depicted in our previous study by Ansys Fluent [33] are also captured by the current solver, as well as the bat-shaped distribution. By changing the pressure at the outlet of the diffuser, the inlet performance at different operating conditions can be obtained. Figure 9 shows the total pressure recovery and circumferential total pressure distortion index [33] calculated by the two solvers. It can be seen that the results are consistent with the experimental data in trend and have a small relative error in magnitude. In conclusion, the current solver is qualified in predicting the flow field characteristics and aerodynamic performance of the serpentine inlet.

3.3. Inlet Low-Speed Performance without Gusts

The following study primarily concerns the aerodynamic performance of the serpentine inlet at low velocities, because gust encounters are more common and adverse at aircraft landing and take-off phases. Thus, for all the following gust response calculations, the freestream velocity is fixed at u₀ = 80 m/s (i.e., M = 0.235), and the backpressure p_AIP/p₀ = 0.85, leading to an AIP Mach number of 0.45.

The flow field characteristics at an inlet Mach number M = 0.235 are presented in Figure 10. The vortex tubes are visualized using the Q-criterion. Figure 10a shows the iso-surface of Q′ = 1 × 10⁻⁵, where (Q′ = Q/(Q_max − Q_min)). Similar to the high-speed behavior at M = 0.7 as observed by Sun and Tan [33], the three dominant vortex tubes are also present but more chaotic at M = 0.235 with some more minor vortex tubes accompanying the three major ones (Figure 10a). These vortex tubes are oriented both in the streamwise and lateral directions, indicating a strong three-dimensional effect of the highly curved serpentine inlet geometry on the flow field. There is only a slight flow separation at the lee side of the top surface thanks to the application of the bump (Figure 10b). The bat structure of the total pressure contours becomes less sharp at the low-speed condition (Figure 10c). Besides, regarding the five low-total-pressure regions, the ones referred to as No. 1, 2, and 5 have shrunk significantly, while the two at both sides of the bottom part (No. 3 and 4) are slightly enlarged. The changes agree well with the pattern of the secondary flow at the AIP (Figure 10d).

As there is unsteady flow separation occurring on the top surface, it is necessary to clarify the unsteadiness of flow field characteristics before investigating gust responses. Figure 11 shows the unsteady fluctuation and the spectrum of the wall static pressure probed at the location ‘p5’ in the separation zone in Figure 10b. It is seen from Figure 11a that both low- and high-frequency fluctuations of the wall static pressure are present due to the shedding of the separation bubbles. Figure 11b indicates that there are two predominant peak zones in the spectrum, one within the range of gust frequency studied in the present study (25–300 Hz) and the other one located at a much higher frequency zone between 2000 and 3000 Hz. Figure 12 and Figure 13 show the corresponding results for total pressure recovery and distortion at the AIP, respectively. Since these are surface-averaged quantities, these curves do not show fluctuations at very high frequencies. Note that the mean values of the total pressure recovery and distortion are about 0.945 and −0.36, respectively. Although the AIP has fully been out of the separation zone and about 1 D away from the vortex core at the separation zone, the unsteady shedding vortices from the top surface still cause both performance parameters to slightly fluctuate at a low peak frequency of around 100 Hz. This is an important finding for analyzing the inlet aerodynamic responses to the gusts in the last subsection.

3.4. Gust Model Validation

Before the study of the response of the serpentine inlet to sinusoidal gusts, the accuracy of the current gust modeling method should also be validated, which includes two aspects, one concerns producing a gusty flow field and the other is gust response. For the former consideration, a structured grid with a uniform cell size of 0.01 m is generated for a rectangular computational domain where a sinusoidal gust with the frequency f = 10 Hz, amplitude ${\tilde{u}}_g=20$ m/s plus a mean velocity u₀ = 80 m/s is introduced at the left boundary, as shown in Figure 14. Both the top and bottom boundaries of the domain are set as the “slip” wall. To investigate the gust velocity distribution at different locations, five probe positions are selected to monitor the local airflow velocities. Figure 15 presents a comparison of the gust velocity profile of a full period between the CFD value monitored at the origin of the coordinate, O, and the theoretical value. It is expected that the current numerical result agrees quite well with the theoretical one with no significant numerical dissipation.

To further validate the accuracy of the current FVM for gust response predictions, a NACA 0006 airfoil was selected for two-dimensional simulations. The reasons why an external flow case is chosen are twofold. The first is the fact that there is little data about internal flow gust responses like the inlet gust responses of interest in this study. The second is data availability of airfoil gust response for reference and comparison. The NACA 0006 airfoil selected here is a thin symmetric airfoil that is very suitable for reduced order modeling (ROM) such as the Küssner function and convolution integral model [25]. The computational domain and grid are shown in Figure 16. The total numbers of nodes and cells are 637,548 and 377,464, respectively. The height of the first layer adjacent to the walls is 1 × 10⁻⁵ c, where c is the chord length of the airfoil and c = 0.02 m. Two kinds of vertical gust shapes are simulated, i.e., 1-cosine and sinusoidal, whose shape definitions are also accessible in [25]. As shown in Figure 17, the gust velocity simulated by the FVM method is highly consistent with the theoretical value for both gust shapes, and the airfoil lift coefficient response also agrees generally well with the theoretical value calculated by the CIM. There is some deviation at the ending part of both lift responses because the CIM results converge to the mean no-gust value after one period, while the CFD results continue with another gust cycle. To be concluded, the currently implemented FVM method is capable of simulating various gust shapes and responses.

3.5. Gust Discretization Examination

As the internal flow characteristics of the serpentine inlet are highly sensitive to the freestream conditions, the resolution of the gust velocity profile should be guaranteed to minimize the unfavorable effect of gust shape discretization on the accuracy of the calculated results. Therefore, the influence of the number of gust discretization intervals, N, on the aerodynamic responses of the serpentine inlet during a full period of the gust is first investigated. The settings of the material and gust parameters for the following gust tests are tabulated in

Table 4. Values of the material parameters for the inlet gust response calculations.

Variable	Value
Ambient pressure, p0, Pa	101,325
Inlet exit pressure, pe, Pa	86,126.25
Air density, ρ0, kg/m3	1.17
Air dynamic viscosity, μ0, Pa·s	1.82 × 10−5
Ambient temperature, T0, K	300
Inlet exit temperature, Te, K	300
Freestream Mach number, M	0.235
Freestream velocity, u0, m/s	80
Gust frequency, f, Hz	25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300
Gust amplitude, ${\tilde{u}}_g$, m/s	4, 8, 12, 16, 20

.

The changes of the horizontal flow velocity probed at three locations in the symmetry plane and the various inlet performances at the AIP with the variation of N are shown in Figure 18. It is seen from Figure 18a to Figure 18c that the influence of N on the flow velocity becomes more and more apparent when approaching the AIP. The gusty flow velocity at locations far from the serpentine inlet coincides with the specified gust profile. However, both the mean velocity and fluctuation amplitude are considerably increased near and inside the serpentine inlet due to the suction effect of the favorable backpressure set at the exit of the serpentine inlet. It can also be seen that the flow velocity profile gradually loses the original sinusoidal shape in the proximity of the serpentine inlet and that an interaction with the intrinsically unsteady flow field within the serpentine inlet takes place, resulting in strong high-frequency oscillations at the AIP, as shown in Figure 18c. In general, the current gust discretization resolutions of interest have obtained negligible differences in the simulation results. To minimize its influence on the prediction accuracy of the gust response of the inlet, the maximum number of gust discretization intervals, i.e., N = 2000, is selected for all the later calculations.

3.6. Gust-Inlet Coupled Flow Field Characteristics

Gust responses of inlet performance are fundamentally contributed by the interaction between the gust and the inlet. Therefore, it is necessary to first have an overall understanding of the characteristics of the gust-inlet interacted flow field. Figure 19 shows the velocity vectors combined with the streamline distributions near the flow separation zone at the top surface for the gust case at four-time instants corresponding to the four extreme points of the total pressure recovery shown later. They are t1/T = 1.375 (acceleration phase), t1/T = 1.825 (deceleration phase), t1/T = 2.325 (acceleration phase) and t1/T = 2.80 (deceleration phase). Compared with the results in the absence of gust as shown in Figure 10c, a remarkable change is the enlarged flow separation zone at all the instants under the interaction of gust, suggesting that both the acceleration and deceleration phases of the gust promote the extent of flow separation in the serpentine inlet. Besides, a couple of vortexes is formed at the two instants of the deceleration phase (Figure 19b,d), which will be discussed later in more detail.

As found from the results above, both the acceleration and deceleration phases of the periodic gust have caused strong flow separations at the lee side of the top surface. However, the dominating mechanisms behind the two cases are not the same. Figure 20 presents a comparison of the static pressure distribution on the top surface in the symmetry plane between the no-gust case and the four instants of the gust case of f = 100 Hz and ${\tilde{u}}_g=8$ m/s. It is seen that the adverse pressure gradient is much larger than that in the absence of gust at the two instances of the acceleration phase (i.e., t1 and t3). This is due to the velocity increase during the acceleration phase and a resulting decreased static pressure at the lee side of the top surface upstream of the separation region (x = 0.19 m), which leads to a stronger adverse pressure gradient further downstream. The main flow characteristic during the deceleration phase (i.e., t2 and t4) is characterized by the development of two separation vortices. The vorticity of these coupling vortices is significantly higher compared to the single vortex in the absence of gust and also to the single vortex developing in the acceleration phase, as is demonstrated in Figure 21. As is shown in Figure 20b, the adverse pressure gradient in the region between x = 0.19 m and x = 0.22 m during the deceleration phase (i.e., t2 and t4) is essentially equal to that of the no-gust case. However, the kinetic energy of the flow is significantly lower, which leads to earlier and much stronger flow separation on the lee side. In conclusion, this analysis demonstrates that flow separation is significantly enhanced during both the acceleration and the deceleration phase, but the flow is more strongly affected during the deceleration phase.

Figure 22 presents a comparison of the total pressure contours at the AIP between the four instances of the gust. It is observed that both the structure and magnitude of the total pressure distribution are changed due to the interaction between the gust and the inlet. All the five low-total-pressure zones marked in Figure 10c have been enlarged by the gust, indicating greater total pressure losses within these zones. The secondary vortices are even larger at times t2 and t4, and this aspect can be detrimental to the engine further downstream. However, the average total pressure losses remain within 4% of the total pressure recovery in the absence of gust, as will be discussed in more detail later in this paper. Moreover, the difference in the total pressure distribution seems not to have changed the structure of the secondary vortices at the AIP other than the scope of the vortex pairs, as shown in Figure 23.

3.7. Inlet Aerodynamic Performance Responses to Gusts

This subsection deals with the influences of sinusoidal gusts on the inlet aerodynamic performance for both the gust frequency and amplitude. For the gust frequency influence study, the gust amplitude is fixed at ${\tilde{u}}_g=10\%u_0=8$ m/s while the frequency varies from 25 to 300 Hz. To study the influence of gust amplitude, the gust frequency is fixed at f = 50 Hz while the amplitude varies from 4 to 20 m/s, considering the higher convergence rate of the performance response at this frequency.

Figure 24 compares the inlet performance between various gust frequencies. Both performance parameters start to respond to the sinusoidal gusts immediately from the initiation of gusts and finally converge to the mean value (not shown for frequencies larger than 50 Hz). Due to the intrinsic unsteadiness of the inlet flow field without gusts, as has been discussed in Section 3.3, several sinusoidal peaks of the total pressure at the AIP are observed when the gusts are present in the flow field. However, the existence of gusts has significantly intensified the fluctuation of the total pressure, compared with that solely due to the three-dimensional effect and boundary-layer separation as well as flow separation on the top surface of the inlet in the absence of gusts. Here, we define a parameter called response time to account for the period during which gusts impart significantly large fluctuations in the inlet performance. It can be seen that the response time of both performance parameters is nearly identical to the gust period at the relatively low frequencies of f = 25 Hz and 50 Hz. However, both the amplitude and time of the two performance responses increase rapidly as the gust frequency is increased to 75 Hz. This is considered to be due to the intrinsic flow field characteristics in the absence of gusts, which have a fluctuating frequency of around 100 Hz as revealed in Section 3.3. Therefore, the interaction between the gust and the inlet may have caused a resonance in the flow field characteristics beyond 75 Hz, which magnifies the unsteady fluctuations in the flow field, as will be verified quantitatively later in

Table 5. Comparison of the maximum amplitudes of total pressure recovery and distortion fluctuations between the cases without/with gusts.

Variable	Without Gust	With Gust
σm/σ0	0.48%	3.83%
DC60m/DC600	16.00%	56.39%

.

Taking the same practice of extracting the amplitude of gust response in external flows [44], Figure 25 presents the minimum, maximum, and average values of the inlet performance responses over a wide range of frequencies, accompanied by the results of the no-gust case. Note that the minimum and maximum values represent the valley and crest of the unsteady responses shown in Figure 24. It is clear to observe that the minimum- and maximum-value curves are approximately symmetrical to the mean values which are close to the results with no gust. Despite this result, the impacts of the gust-inlet interaction are different from the two performance parameters. To the total pressure recovery, its reduction by the deceleration phase of gust yet can be remedied by the following counterpart phase. To the total pressure distortion, however, its reduction means an increasing non-uniformity of the flow field, which will directly be transferred to the engine face further downstream and may cause a sudden and instantaneous occurrence of stall and surge of the compressor in the engine. To be concrete, for σ, the most severe effect of the gusts occurs at the gust frequency f = 100 Hz, at which σ reaches the lowest, as shown in Figure 12a. This should be due to the resonance caused by the interaction between the gust and the inlet flow field possessing a natural fluctuation frequency of 100 Hz, as shown in Figure 12b. For DC₆₀, it is interesting to observe a slight enhancement of the average value between f = 75 Hz and 150 Hz. However, as analyzed above, the minimum value of the distortion has the most important significance to the inlet, which remains at a low level within the same range of frequency and generally decreases with the increasing gust frequency. More quantitative evaluations of the gust influence on the variations of the total pressure recovery and distortion will be presented later.

Figure 26 compares the time histories of inlet total pressure recovery and distortion index at the AIP for various gust amplitudes. Generally, the total pressure recovery responses show a similar sinusoidal pattern as that of the gust shape with progressive growth in amplitude with the increasing gust amplitude. In contrast, the majority of the distortion response happens in the first half period, during which the gust plays a positive role in increasing the streamwise flow velocity as well as decreasing the distortion.

Figure 27 presents the dependence of the minimum, maximum, and average values of the inlet performance responses on gust amplitude, as well as the unsteady calculation results of the no-gust case. Compared with the case of gust frequency, the dependence on gust amplitude is relatively simpler. In general, the minimum and maximum responses of total pressure recovery assume approximately symmetrical to the mean values, causing the average response to be lying in the vicinity of the mean values. Moreover, the average response of the distortion index lies above the mean value beyond ${\tilde{u}}_g=8$ m/s, indicating a positive role of the gusts in decreasing the flow distortion at the AIP of the serpentine inlet. However, it should be clarified again that the gust-induced deterioration of the flow field homogeneity is a more important response index.

The above results have shown that sinusoidal gusts induce significant oscillations in inlet aerodynamic performance. Therefore, the oscillation amplitude is also an important index to weigh the influence extent of gusts. Like the practice adopted in wing-gust response studies [9,44], two non-dimensional index parameters are defined here, i.e., total pressure recovery amplitude

$${\sigma }_m/{\sigma }_0=\frac{{\sigma }_{max}-{\sigma }_{min}}{2{\sigma }_0}\times 100\%$$

(12)

and total pressure distortion amplitude

$$\mathrm{DC}{60}_m/\mathrm{DC}{60}_0=\frac{\mathrm{DC}{60}_{max}-\mathrm{DC}{60}_{min}}{2\mathrm{DC}{60}_0}\times 100\%$$

(13)

The results of both parameters are presented in Figure 28. Both the total pressure recovery and distortion amplitudes experience an ascending, descending, and re-ascending process as gust frequency increases, with the maximum at fD/u₀ = 0.102 and 0.081(i.e, f = 125 Hz and 100 Hz, respectively). On the other hand, the total pressure recovery amplitude increases linearly with the increasing gust amplitude over the full range, while the distortion amplitude increases monotonously till ${\tilde{u}}_g/u_0=0.2$(i.e, ${\tilde{\mathrm{u}}}_{\mathrm{g}}=16$ m/s) and drops beyond that. These results indicate that the dependence of the inlet performance response on the gust frequency is much more complicated than that on the gust amplitude.

To compare the effect of gusts on increasing the fluctuations of the inlet performance with the intrinsic unsteadiness of the flow field in the absence of gusts, Table 5 lists the maximum amplitudes of total pressure recovery and distortion fluctuations for the gust cases investigated in this study as well as for the no-gust case. This demonstrates that the gusts have amplified the total pressure recovery and distortion fluctuations by several times the fluctuations caused by the intrinsic unsteadiness of the flow field. This may validate the above hypothesis of the production of resonance in the flow field by the interaction between the gusts and the inlet.

4. Conclusions and Outlook

In conclusion, the effects of horizontal sinusoidal gusts on the flow field and aerodynamic performance of a serpentine inlet were investigated based on a field velocity method implemented in OpenFOAM. Before starting the gust response simulations, preliminary work was done to guarantee the numerical solution accuracy. First, the accuracy of the baseline CFD solver was validated via a thorough comparison between the numerical results and wind-tunnel experimental data for the flow field and aerodynamic characteristics of the inlet at a high flight Mach number M = 0.7 in the absence of gusts. Second, the accuracy of the gust modeling method and the dependence of the modeling results on the model parameters were carefully examined to guarantee accurate inputs of the gust shapes of interest. Third, because gusts are more often to occur and more severe at aircraft take-off and landing phases, the flow field of the inlet at a relatively lower flight Mach number M = 0.235 was calculated to serve as a baseline for all the following gust response simulations. Finally, some gust response computations were run to investigate both the flow field and aerodynamic characteristics of the serpentine inlet under horizontal sinusoidal gust conditions.

It was found that both the flow field and aerodynamic performance of the inlet are significantly affected by the horizontal sinusoidal gusts. As for the flow field responses to the gusts, an increased area of the low-momentum fluid zone is formed by the gust, which includes not only the amplification of the flow separation zone in the absence of the gusts but also new additions in the other locations in the flow field. The deceleration phase of the sinusoidal gusts is more detrimental to the flow field characteristics of the serpentine inlet than the acceleration phase. More total pressure losses have occurred at typical locations of the AIP under gust conditions compared with the no-gust case. However, the secondary flow pattern at the AIP did not experience significant influence from the gusts, but the scopes of the vortex pairs were altered by the gust at different moments. The aerodynamic performance of the serpentine inlet has also significantly been affected by the horizontal sinusoidal gusts. Increased oscillations in both the total pressure recovery and distortion are observed, with both the amplitude and time increased rapidly beyond the gust frequency of 75 Hz. The minimum and maximum responses of both performance responses assume approximately symmetrical to the mean values, causing the average response to be lying in the vicinity of the mean value. Both the total pressure recovery and distortion amplitudes show a complex changing trend with the increasing gust frequency but an approximately linear relationship with the gust amplitude in most cases. It is worthy of special attention that the deterioration of the total pressure distortion caused by sinusoidal gusts cannot be counteracted and directly affects the quality of the air ingested by the engine, although the average responses are close and sometimes even above the mean values in the no-gust case.

From the results of this paper, it is clear that gusts can influence significantly the internal flow characteristics of the engine inlets. Therefore, it is suggested that more attention should be paid to the adverse effects of gusts on inlet performance which directly affects the quality of the air entering the gas generator of gas turbine engines. To study the aerodynamic performance of inlets under gusty conditions, considering the velocity and scale of atmospheric gusts, engine inlets possessing complex geometries and operating at subsonic flight speeds deserve special attention. It should be pointed out that the present study only provides a preliminary insight into the influence of gusts on aircraft engines. Further studies may be conducted by combing the compressor with the inlet to have a more comprehensive understanding of the inlet-compressor coupled responses to horizontal sinusoidal gusts. Last but not the least, experimental tests are necessary to characterize the gusty inflow condition and visualize the gust-affected internal flow characteristics in wind tunnels. A major disadvantage of the current study is that all the gust response results in the current study are based on pure CFD simulations, although the gust modeling method has carefully been examined before the massive calculation. The good news is that we have designed and manufactured a novel double-vane gust generator apparatus recently, which will be mounted in our low-speed wind tunnel for tests soon.