Numerical Simulation Study on Aeroacoustic Characteristics within Deformable Cavities

Cavity flow phenomena are encountered in many kinds of aviation vehicles. ,e flow-induced noise can easily cause structure resonance and fatigue damages. ,erefore, the study on the mechanism and effective control methods of cavity noise are very important to engineering applications. A new active control method was proposed based on the deformable cavity in order to mitigate the cavity noise. Large eddy simulation (LES) and computational aeroacoustics (CAA) are combined to simulate a typical open cavity noise. ,e results show the first mode sound-pressure level (SPL) of tonal noise decreases gradually while the first mode frequency sharply jumps within a small range of the slant angle of the trailing and bottomwall. In addition, with the increase in the slant angle, the decrease of the first mode SPL of tonal noise at Mach 0.6 is more significant than that at Mach 0.85, but the increase of the first mode frequency at Mach 0.85 is more dramatical than that at Mach 0.6. ,e proposed method can not only reduce the first mode SPL obviously but also increase the first mode frequency dramatically, which makes it different from the natural frequency of the cavity structure and sequentially helps the cavity avoid fatigue damages from resonance.


Introduction
Flow over cavities has been widely studied during the last several years because of its practical and academic value.In aviation industry, cavity flow phenomena widely exist in different positions of aircraft, such as landing gear wheel wells of civil aircraft, the crevices in the surface of aircraft, and the weapon bays of modern combat aircraft.Especially for the modern combat aircraft, in order to reduce low radar cross section, the weapons are enclosed in weapon bays.However, the weapon bays can induce high self-sustaining oscillations which in turn can generate flow-induced noise arising from the cavities.e strong noise can exert severe damage on both sensitive parts of weapons and equipment within the cavities.
e control methods for cavity noise can be classified into passive control methods and active control methods.Passive control methods are to control cavity flow with geometric modifications and do not use external energy/ momentum sources to control the flow.
e passive control methods may demonstrate good performance at design flow conditions without much complexity but may lose efficacy at off-design conditions.
Active control methods employ external energy/ momentum sources to control the cavity flow.e common active control methods include mass injection [17][18][19][20], slot blowing [21], microjet actuators [22], steady blowing [23,24], plasma actuators [25][26][27][28][29], miniature fluidic actuators, speakers and resonant tubes [30], and oscillating ramps and fences [31].e active control methods can offer attractive noise suppression and can adjust control parameters according to different flow conditions.All of these active control methods have demonstrated control of cavity resonance under subsonic flow conditions, but only a few of these have been successful under supersonic-free stream conditions.In addition, Shaw [32] found that the acoustic feedback phenomenon and shear layer receptivity are very sensitive to the state of the boundary layer at the point of separation.us, the selection of the most effective active suppression concept should be based on a configuration as close to the full-scale one as possible.
In this work, a new active noise control method is proposed.A mechanism installed within the cavity is to construct a deformable cavity.Figure 1 shows the structures of the deformable cavity.In the deformable cavity, 1 is the fixed leading edge, 2 is the fixed leading wall, 3, 5, and 7 are the hinges, 4 is the bottom of the cavity, 6 is the trailing wall, and 8 is the trailing edge-contained sliders; sliders which can slide horizontally are placed in the guide rail 9 and structures indicated by 10 are the panels of both sides.Figure 2 shows a deformed situation of the cavity with the slant angle α of the trailing wall 6. e mechanism controls the deformation of the cavity by horizontally sliding the trailing edge 8 to control the change of the slant angle α of the trailing wall 6.At the same time, the bottom 4 is also slanted.
According to different flow conditions, the proposed method can adjust the slant angles of the bottom and trailing wall of the deformable cavity for suppressing the cavity noise.In order to study aeroacoustic characteristics within the deformable cavity under different flow conditions, a deformable cavity based on the geometry of M219 cavity [33][34][35][36][37] has been simulated in the work.e cavity flow and flow-induced noise are studied by large eddy simulation (LES) and computational aeroacoustics (CAA).

Cavity Configuration and Flow
Parameters.In the present numerical simulation, the M219 cavity has dimensions of L � 0.508 m in length and D � 0.1016 m in depth, giving a ratio of L : D � 5 : 1.For comparison with the experimental results from the work of Chen et al. [34], the simulations were performed with free stream conditions of M � 0.85, P � 62940 Pa, and T � 270.25 K, and the Re (Reynolds number) based on the cavity length is 7,950,810.At the same time, the pressure across the cavity floor was monitored.ese monitoring points are shown in Figure 3.It is shown that the monitoring locations are related to the length of the cavity in Table 1.
e inflow, outflow, and upper boundaries of the domain are applied with the pressure far-field conditions with M � 0.85, pressure P � 62940 Pa, and temperature T � 270.25 K. Adiabatic wall conditions are applied on the surface of the cavity walls.

Numerical Method.
Because direct numerical simulation (DNS) is too expensive and unsteady Reynolds-averaged Naiver-Stokes (URANS) is unable to predict the unsteadiness within the cavity correctly, many computational fluid dynamics (CFD) researchers usually use LES to study various complex flow problems.e philosophy behind LES is to resolve the larger turbulence eddy scales, while a subgrid scale model is used to model the smaller turbulence eddy scales.Several recent efforts [38][39][40][41] that have applied LES to cavity flow have recently been published.Sinha et al. [38] found that engineering-oriented LES with modest grids provided a good representation of the interactions between narrow band, acoustic tones, and broadband, vortical turbulent structures.Levasseur et al. [39] simulated the cavity with the LES method in transonic and illustrated the ability of LES to capture the Rossiter frequencies at the cavity floor center.ornber and Drikakis [40] simulated the deep cavity with the LES method and the results obtained showed that flow fields and sound-pressure   Other LES studies have also attempted to model the important acoustics that occur in the supersonic regime over rectangular cavities [41], to develop a model to correlate the aspect ratio and Mach number.erefore, the LES with the subgrid scale model is adopted to study the deformable cavity noise.
In LES, the motion is separated into the larger and the smaller turbulence eddies, and the separation is achieved by means of a low-pass filter.
e filter function, G(x, x ′ ), implied here is then as follows: After filtering, a transient flow variable is divided into two parts: where ϕ is the mean component of the larger eddies, which can be calculated directly, and ϕ ′ is the component of the smaller eddies, which is represented with the subgrid scale model.ϕ is filtered as where D is the flow domain, x ′ is a spatial term in the real flow domain, and x is a spatial coordinate of the larger eddies after filtering.e formulas can be obtained with equation (3) and Navier-Stokes equations: where ρ is the fluid density, u i is the mean flow speed, p is the mean pressure, μ is the viscosity coefficient of fluid molecules, and τ ij is the subgrid stress tensor.
To solve equations ( 4) and ( 5), the basic SGS stress model [42,43] is defined as where μ t is the subgrid viscosity, δ ij is the Kronecker delta (when i � j, δ ij � 1, i ≠ j, and δ ij � 0), τ kk is an isotropic part of the subgrid scale, and S ij is the strain rate tensor of the subgrid scale.en, we can calculate where S is the stretch rate tensor, |S| � , Δ i is the grid scale in the i direction, and C s is the Smagorinsky constant.
In the work, LES with the dynamic Smagorinsky model (DSL) is performed.In the DSL subgrid model, the Smagorinsky constant C s is dynamically computed based on the information provided by the resolved scales of motion.e gradient discrete scheme uses the least-squares cell-based gradient evaluation.e convective flux type adopts the Roe flux-difference splitting scheme.
e inviscid convection term uses the second-order upwind scheme, and the viscous diffusion term uses the second-order central difference scheme.
e time-dependent solution selects the secondorder implicit, also known as the dual-time formulation.
e simulation includes the time histories of the pressure, p(t), at locations K20-K29, respectively, which were recorded at each time step.
e computation has been carried out for about 25,000 time steps, of which the first 5000 time steps have been discarded in the analysis of the pressure time series.e pressure oscillation is closely related to the sound resonance from the cavity.e recorded p(t) has been used to compute the power spectral density (PSD) and the SPL.e SPLs are an indication of the intensity of noise generated in the cavity.
e fast Fourier transform (FFT) has been used for p(t) to compute the PSD.
e SPL is then obtained from the calculated PSD, which is defined by where P ref � 2 × 10 −5 Pa is the value adopted as the minimum audible sound pressure variation.

Computational Mesh.
e computational domain consists of the cavity region itself and the external flow region.Structured grids are adopted, and the grid structure of the deformable cavity used in this work is shown in Figure 4. e fineness of the grid structure near the wall is evaluated by y + : where Δy is the initial grid spacing normal to the wall, ρ is the fluid density, u τ is the wall friction velocity, τ w is the wall shear stress, and μ is the fluid viscosity.In the wall normal direction, the mesh is stretched according to smooth stretching ratios.In terms of wall units, y + near the wall in the wall normal direction is about y + ≈ 1.In order to verify the independence of the grid structure, three grid structures were simulated, which were coarse, medium, and fine grid structures.e cell numbers of these are 65,000, 106,000, and 154,000, respectively.e fine grid structure is shown in Figure 4. Since the cavity itself is the main research object of the aerodynamic noise problem, the differences between various grid structures are mainly embodied in the different cell numbers of the cavity itself.e cell numbers of coarse, medium, and fine grid structures are 260 × 140, 380 × 200, and 500 × 260, respectively.After simulations, we found that the differences of the results among the three grid structures are very small, and the results of fine grid structures are the Shock and Vibration best compared with the experimental results.erefore, we adopted the fine grid structure in this work.
e Courant number is given by the following equation: where u is the free stream velocity, Δt is the time step, and Δx is the length of the grid in the direction of velocity.
During the simulation of cavity noise, it is necessary to ensure that the Courant number is less than or equal to 1 in the entire flow field in order to achieve numerical stability and time calculation accuracy, so the time step is Δt ≈ 2 × 10 −5 s based on the grid structure we adopted.

Validations.
Figure 5 illustrates the SPL, in decibels, predicted with LES with DSL subgrid model and compared with the experimental and simulation data obtained from Chen et al. [34].e data represent the pressure points at K29, located near the trailing wall.e magnitude of SPL is sufficiently predicted, and the resonance frequencies clearly result from the computed spectrum.Among the simulation results of three grid structures, the magnitude of SPL of fine grid structure is the closest to the experimental results, so the fine grid structure is adopted in the paper.

Volume Changes of Deformable Cavity.
In Figure 6, D is the depth of the cavity near the leading wall, d is the depth of the cavity near the trailing wall, and α is the slant angle of the trailing wall.By calculation, when the bottom and trailing walls are on a straight line, α takes the maximum value of 80.406 °.In Figure 7, we can see that the space of cavity near the trailing wall is gradually compressed as α gradually increases.When α increases to 60 °, d has been reduced to half of D. Considering the practical engineering applications, the range of α is limited in the range of 0 ∘ ≤ α ≤ 60 ∘ in this work.α is divided into sixteen equal parts.at is, α gradually increases from 0 °to 60 °.en, we establish the deformable cavity model and grid structures, respectively.

Flow Field Analysis.
e property of the unsteady flow field is closely related to the sound resonance from the cavity, so the instantaneous vorticity contours of the flow field was observed.Figure 8 shows the comparison of instantaneous vorticity contours of the rectangle cavity (α � 0 °) at t � 4/6T and deformable cavity (α � 60 °) at t′ � 4/6T′.Figure 9 shows the comparison of instantaneous vorticity contours of the rectangle cavity (α � 0 °) at t � 5/6T and deformable cavity (α � 60 °) at t′ � 5/6T′.In Figures 8 and 9, t and t′ are the current time of the instantaneous vorticity contours.T is the period related to the first Rossiter frequency of the rectangle   Shock and Vibration cavity.T′ is the period related to the first Rossiter frequency of the deformable cavity with α � 60 °.Compared with the rectangle cavity flow field, the flow field of deformable cavity with α � 60 °has the following differences: (i) In Figure 8, we can clearly find that the vortices separating from the leading edge in the deformable cavity (α � 60 °) are smaller than those in the rectangular cavity, which can make the impingement force between the vortices and trailing edge decrease in the deformable cavity.(ii) When α � 60 °, the corner of the trailing wall becomes more blunt than that of the rectangle cavity, which can also make the impingement force decrease and make the cavity noise radiate to exterior greatly.(iii) In Figures 8(b) and 9(b), the vortices gradually upraised from the leading edge to the trailing edge.
In Figure 8(b), the vortices are almost above the trailing edge.In Figure 9(b), most of the vortice forms are also clearly discernible after impingement.e above phenomena are quite different from that shown in Figures 8(a In view of the three points above, we can conclude that the flow field of the deformable cavity has been improved greater than that of the rectangle cavity.Taking into account the cavity noise mechanism of impingement, the strong noise environment in the original rectangle cavity may be improved inevitably.

Sound Field Analysis.
Figure 10 shows the comparison of the first mode SPL of tonal noise at K29 when Mach is 0.6 and 0.85, respectively.e first mode SPL of tonal noise at K29 decreases as α gradually increases at Mach 0.6 and Mach 0.85.When Mach is 0.6, the first mode SPL of tonal noise at K29 decreases 10.43 dB at α � 60 °, but when Mach is 0.85, the first mode SPL of tonal noise at K29 decreases 6.02 dB at α � 60 °.It is obvious that the decrease of the first mode SPL of tonal noise at Mach 0.6 is more significant than that at Mach 0.85 as α gradually increases.
In the practical engineering applications, the cavities of aircraft are usually used to transport the weapons and other equipment.erefore, we should ensure that the cavity has enough space for carriage.For this reason, when we select the slant angle of the trailing wall α, we should fully consider α and its potential interaction with the space of cavity.In addition, we found an interesting phenomenon that not only the SPL of the first mode of the tonal noise decreases significantly but also the frequency of the first mode of the tonal noise increases dramatically when the deformation of the cavity has changed.
Figure 11 shows the comparison of the first mode frequency of tonal noises at K29 when Mach is 0.6 and 0.85, respectively.When Mach is 0.6, the first mode frequency of tonal noise at K29 remains 128.20 Hz with the increase of α from 0 °to 33 °(rectangular cavity at α � 0 °), then increases dramatically to 174.85 Hz as α increases from 33 °to 34 °, and lastly remains 174.85 Hz as α increases from 34 °to 60 °.
Finally, the first mode frequency of the tonal noise increases 46.65 Hz.When Mach is 0.85, the first mode frequency first decreases slightly from 160.25 Hz and then remains 157.33 Hz with the increase of α from 0 °to 24 °; then it decreases to 139.85 Hz with the increase of α from 24 °to 28 °and remains 139.85 Hz with the increase of α from 28 °to 34 °; later, it increases dramatically to 227.26 Hz with the increase of α from 34 °to 35 °and remains 227.26Hz with the increase of α from 35 °to 44 °; then, it increases to 244.74 Hz with the increase of α from 44 °to 52 °and remains 244.74 Hz with the increase of α from 52 °to 60 °.Finally, the first mode frequency of tonal noise increases 84.49Hz.It is obvious that the increase of the first mode frequency of tonal noise at Mach 0.85 is more dramatical than that at Mach 0.6.
In fact, the dramatical increase of the first mode frequency will make the sound resonance frequency different from the natural frequency of cavity structure, which can

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Shock and Vibration mitigate the damage of resonance to the cavity structure.e phenomenon of dramatical increase in the resonance frequency is also found by Rowly et al. [44].ey concluded that the resonance frequency decreases as L (the length of cavity) gradually increases, but once a critical value of L is reached, the frequency jumps up, as the cavity switches to a high mode.According to their conclusion, the phenomenon of dramatical increase in the first mode frequency that appeared in the deformable cavity can be explained as follows: the actual effective length of cavity has changed due to the change of the cavity shape.When the actual effective length is changed to a critical value, the first mode frequency increases dramatically.
In the practical engineering applications, the SPL and frequency of first mode can be fully considered to the selection of α. α should be located in the range where the first mode frequency has increased dramatically.At the same time, we can select α to make the first mode SPL reduce greatly according to the actual needs of the cavity volume.

Conclusions
In this work, the numerical simulation of LES is used to verify the feasibility of the proposed active control method for cavity noise at Mach 0.6 and 0.85.e following conclusions are obtained through comprehensive analysis: (i) When Mach is 0.85, the first mode SPL of tonal noise at K29 decreases as α gradually increases, and the reduction is 6.02 dB.When Mach is 0.6, the first mode SPL of tonal noise at K29 also decreases as α gradually increases, but the maximum reduction is 10.43 dB. is shows that the decrease of the first mode SPL of tonal noise at Mach 0.6 is more significant than that at Mach 0.85 as α gradually increases.e decrease of the first mode SPL of tonal noise has an important positive effect on improving the noise environment and protecting the devices within cavity.(ii) e first mode frequency sharply jumps within a small range of the slant angle α of the trailing wall.When Mach is 0.85, the first mode frequency of tonal noise at K29 increases Hz dramatically with the increase of α from 34 °to 35 °.When Mach is 0.6, the first mode frequency of tonal noise at K29 increases 46.65 Hz dramatically with the increase of α from 33 °to 34 °.Obviously, the increase of the first mode frequency of tonal noise at Mach 0.85 is more dramatical than that at Mach 0.6 as α gradually increases.e dramatical increase of the first mode frequency of tonal noise can make the sound resonance frequency different from the natural frequency of cavity structure, which can mitigate the damage of resonance to the cavity structure.

Figure 3 :
Figure 3: Monitoring points at the bottom of the deformable cavity.

Figure 7 :
Figure 7: e depth d as a function of the slant angle α.

Figure 10 :Figure 11 :
Figure 10: Comparison of the first mode SPLs of tonal noise at K29.

Table 1 :
Positions of monitoring points.
levels (SPLs) correlated very well with the experiment.