Numerical simulation of the leading edge bump for the cavity flow

Based on the IDDES method, the M219 cavity is simulated in flux-mixed scheme, with the main purpose of studying the mechanism of the noise generation in the cavity flow and proposing corresponding suppression measures. The study mainly includes the grid resolution study and the numerical simulation of the leading edge bump. Through the comparative analysis with the experimental results provided by Henshaw et al. and the LES results from Haase et al. it is known that the different grids in this paper can all obtain the flow characteristics of the cavity flow, and the different grids show the convergence. The leading edge bump can significantly raise the shear layer, reduce the disturbance flowing into the cavity, and play a role in suppressing both the cavity tones and the broadband of the cavity flow.


Introduction
In the aerospace field, there are a large number of cavity structures, such as: aircraft landing gear, engine combustion chamber, weapons bay, etc..The cavity structure can bring obvious resonance noise to the flow, not only damage the fragile parts, but also affect the electronic elements or the experience and operation of the pilots, thus bringing uncontrollable and serious consequences.In order to suppress the impact of cavity noise, researchers have conducted a lot of study on it.
As early as the 1980s, researchers have used CFD methods to simulate cavity flow problems [1].Rossiter [2] conducted numerous experiments on passive control measures of spoilers for weapon bay and found that leading-edge spoiler were effective in reducing the amplitude of pressure pulsations in the cavity under subsonic conditions.Shaw [3] conducted flight tests to evaluate the effectiveness of passive control devices such as leading edge spoilers, slant trailing edge, and trailing edge airfoils in suppressing oscillations in cavity flow with L/D=2.The test shows that: the slant trailing edge can only suppress a single tone; the combination of slant trailing edge and leading edge spoiler can suppress two tones, and was the best one.Zhou et al [4] conducted tests under various leading edge plate to define the selection method and optimal parameters of the plate.It was shown that the leading edge plate reduced the flux rate and intensity in the cavity by raising the shear layer and thus suppressed the noise.Wu et al [5][6][7] used experimental methods to study comprehensively the effects of flow control approaches such as the leading edge vertical saw-tooth fence, the floor pressure balance tubes, the leading edge blowing, leading edge rod, and slant trailing edge on the flow characteristics in the bay and store separation characteristics of the weapon bay.And Yang et al [8][9][10][11] studied the effect of perpendicularity sawtooth, boundary layer thickness, slant trailing wall, pressure pipe on the floor, and jet at the fore edge on the cavity noise in a more comprehensive way by using experimental and numerical methods.Zhang et al [12] studied the effect of the cavity noise by using the numerical method of porous rear wall combined with the dissipative cavity.The dissipative cavity lenth is considered to have a great role on the suppression, and if the dissipative cavity is too short the noise level in open cavity will increase, while a larger size can effectively reduce noise.
In this paper, we study the suppression of the M219 cavity.On the one hand, the mechanism is studied and the numerical method is validated, and on the other hand, a new flow control approach is proposed.The paper consists of the following parts: the second section gives a brief description of the simulation method; the third section presents the validation of the method and the grid resolution; the forth section discusses the noise reduction effect of the bump.

CFD methodology
In this study, the finite volume method and structured grid are used to solve the NS equations in curvilinear coordinate system.The control equations are as follows: The above equation is solved using a dual-time stepping method [13].And the flux-mixed scheme is used for the inviscid term, while the viscous term is calculated using the central difference scheme.The IDDES method is also used for obtaining the turbulent structures around the cavity.

The flux-mixed scheme
The IDDES [14] method is a modification of the DES [15] method, which implements the switching between the RANS method and the LES method with grid scale filtering near the wall by a length scale, and can capture the small scales in the flow without a significant increase in computation.
For the DES methods, the LES method is used for the separation region and the RANS method is used for the rest of the solution.Based on the mixed characteristics of the control equations, the hybrid central/upwind approximation of the inviscid fluxes is considered, which is as follows:   Where, cent E denote the central approximation, upwind E denote the upwind approximation, and  is the blending function designed by Travin et al [16] with the ratio of the length scale from the LES mode to the RANS mode.In the LES mode, the value is close to 0, resulting in an "almost centered" scheme with low dissipation, while in the RANS mode, the value is close to 1, resulting in an "almost upwind" scheme with robustness.The specific form is as follows: where,

Geometric and mesh
The M219 cavity is an open cavity with a length-to-depth ratio L/D = 5.Henshaw [17] conducted experiments in 2000 at the ARA wind tunnel at the Bedford, UK, and tested at Ma = 0.6, 0.85, and 1.35, respectively.The cavity is located in a plate with an x-direction 1.8288 m and a y-direction 0.4318 m.The leading edge of the cavity is 0.7874 m from the leading edge of the plate, and the cavity centerline is offset by 0.0254 m.The length of the cavity is L=0.508 m, with the depth D=0.1016 m, and the width W=0.1016 m.The geometry of the cavity is shown in Figure 1.This case was also chosen by the DESider as an standard case to test the numerical art on simulating separation flows [18].
In order to reduce the consume, the computational domain does not include the all model but only the model surface and the cavity surface, with a length of 20 m in the x-direction, a width of 2.45 m in the y-direction, and a height of about 1 m.In order to capture the vortex structure in the flow, the mesh is finer at the edge of the cavity and in the shear layer.Three sets of grids, coarse, medium and fine, were generated for the grid resolution study.The mesh topology and surface mesh are shown in Fig. 2 and Fig. 3.In the wind test, 10 Kulite transducers were installed along the centerline of the rig, (which did not coincide with the centerline of the cavity itself) to obtain the unsteady pressure, which were labeled as K20~K29.The position of the transducers is shown in Figure 4, and the coordinate values of each transducer are shown in Table 1.

Grid resolution study
In this section, grid sesolution was carried out using the three sets of meshes described previously.The flow conditions is from the test: Ma=0.85,T=266.53K, Re=13.47×10 6 .The flow direction is parallel to the surface of the cavity.The turbulence model was chosen as SST turbulence model with Roe scheme   The flow velocity profiles at three stations in the mid-plane of the cavity from different grids are shown in Figure 6.From the figure, it can be observed that the flow velocity profiles obtained from different grids are in good agreement with the reference results [19] and possess good consistency.It shows that the method used in this paper has high accuracy in simulating unsteady flow.
The instantaneous Q criterion iso-surface at t=0.15s from three grids, is shown in Figure 7. From the figure, it can be observed that the coarse grid can also resolve the small-scale structures in the cavity, but the grid is finer the turbulent structures are richer.It is obvious observed from the fine grid's result that amount of vortex clusters from the leading edge impinge on the aft wall.The vortex breaks at the impinge point.Most of the vortex flows into the cavity, and the other flow out of the cavity.For the points on the cavity wall, we can get the root mean square value of the pressure.And we can caculate the OverAll Sound Pressure Level (OASPL) on the wall.The OASPL is a significant parameter of the noise.8.It can be observed that the OASPL contour on the wall of the cavity is similar for different grids.The OASPL contour at the rear part of the floor is significantly higher than that at the front part.There are some differences in the distribution size for different grids.
In the test, pressure pulsations were captured by the Kulite transducers on the floor.The distribution curves of the OASPL at different locations on the floor are shown in Figure 9.The OASPL on the last transducer K29 is the highest, with a value of 166 dB.The lowest one is on the second transducer K21, with a value of 153 dB.The difference between the total sound pressure level at each measurement point and the experimental results is smaller as the grid density increases: the total sound pressure level at the measurement points K22-K26 shows good grid convergence, and the total sound pressure level at the measurement points K20-K26 shows good grid convergence.The total sound pressure levels at K22-K26 showed good grid convergence, while no significant grid convergence was found at K20, K21, K27-K29.In addition, the OASPL at each measurement point of the coarse grid is consistent with the trend of the corresponding measurement points of the experimental results, and each measurement point is about 2 dB higher than the experimental results.Figure 10 gives the spectral curves of the sound pressure level at the relevant measurement points at the bottom of the cavity.It can be seen that the peak frequencies at the measurement points at the bottom of the cavity with different grid densities do not differ much from the experimental curves, and the overall trend is in good agreement; in general, the peak frequencies decrease in amplitude with the encryption of the grid, reflecting a certain degree of grid convergence.The numerical simulation results and the experimental results capture four obvious frequency peaks at K20 and K21 at the front edge of the cavity and K28 and K29 at the rear edge of the cavity.According to Rossiter's analysis, there are four resonant acoustic pressure level modes in the cavity flow, which are from 1st to 4th order modes, and the amplitude of the 1st order mode in the low frequency band and the 4th order mode in the high frequency band are smaller than that of the 2nd and 3rd order modes in the middle frequency band.This is reflected in both numerical simulation results and experimental results.
In addition, since the total simulation time length of the numerical simulation is 0.2s and the time length for the spectral analysis is 0.16s (the result of the non-constant calculation, the flow field with constant flow in the initial field needs to pass through a certain time length of flow before the whole flow field enters the statistically credible non-constant flow state.Therefore the non-constant results need to be processed by excluding the flow field results for this period of time), and the resolution for low frequency pulsation information is not sufficient.Therefore the error between the numerical simulation results and the experiment will be larger for the 1st order mode peak in the low frequency band.However, it is clear from the results that the 2nd and 3rd order modes in the flow are the dominant modes in the cavity flow, so the difference in the 1st order mode does not make much difference to the overall accuracy of the results.A density gradient fluctuations in the mid-plane (z = -20 mm) at sequential times is given in Figure 11.The fluctuations show the shear layer shedding and its impingement on the aft wall: first,at t=0s, at t=0.00036s,A becomes a larger vortex and is about to shed from the shear layer, while the shear layer oscillates downward at this moment to produce a downward shedding vortex B. At the moment of 0.00072s, the shedding vortex A finishes shedding from the shear layer and moves downstream and away from the cavity, the shedding vortex B receives energy transport from the shear layer and the cavity, and the vortex energy increases and is about to finish shedding from the shear layer; at the moment of 0.00108s, the shedding vortex A continues to flow downstream and away from the cavity and is about to finish separating from the shedding vortex B, and the shedding vortex B continues to receive the energy transferred from the flow in the cavity 0.00144s moment, shedding vortex A complete separation from shedding vortex B, in the process of flowing downstream continuously release energy, began to weaken, shedding vortex B and the cavity structure interaction continuous energy exchange; 0.0018s moment, shedding vortex A in contact with the vortex transported out of the cavity is continuously weakened (energy transfer to the contact vortex), shedding The vortex structure of vortex B is about to disappear by the influence of the complex vortex structure inside the cavity; at 0.00216s, the shedding vortex A reaches the top of the trailing edge of the cavity and the structure of shedding vortex B completely disappears; at 0.00252s, the shedding vortex A flows through the cavity.

The leading edge bump measure
It is found in the literature that raising the shear layer can effectively reduce the cavity noise.In this paper, we choose to install a smooth bump at the leading edge to evaluate the effect.The geometry of the bump at the leading edge is shown in Figure 12.The previous study shows that the coarse grid results already have good convergence at the current grid density.Therefore, coarse meshes are used for calculations.The flow of the cavity are simulated at two hights: 5 mm and 10 mm, which will be denoted with bump5 and bump10.
Figure 13 shows the instantaneous Q criterion iso-surface at t=0.15s from bump5 and bump10, with contour map being colored by the pressure coefficient.It can be seen observed that after the shear layer is lifted by the bump, and the shear layer produces significantly shedding vortices.However, numerous broken vortex structures are still visible inside the cavity.After increasing the height of the bulge, the height of the shear layer increases, and the flow structure inside the cavity still contains numerous smallscale structures.
Figure 14 gives the OASPL contour on the surface of the cavity with diffefent bumps.It can be observed that the region that its OASPL is greater than 170 dB exists only in a small area on the rear wall.The area that decrease by increasing the height of the bump.Figure 16 shows the sound pressure spectral curves of sample points on the floor in the mid-plane simulated with different bumps.It can be observed that, with bums being mounted in front of the leading edge, except for the third mode frequency at the two locations (K20, K21) on the floor, there are no significant mode frequencies at the rest locations.That is meaning, the bump eliminates the resonant noise in most parts of the cavity flow.From the modal amplitude, it can also be found that the modal amplitude is lower than the cavity flow with bumps being mounted, which means that the bump is valid on reducing the modal amplitude.
To sum up, the bump is effective to suppress on the cavity flow noise.And it's both effective on the modal frequency and modal amplitude in the cavity flow, resulting in a OASPL reduction of about 4 dB at the loations.A density gradient fluctuations in the mid-plane (z = -20 mm) at sequential times is given in Figure 17.The shedding vortex generated from the shear layer is shown throughout the flow through the cavity.0.00036s, A becomes a larger vortex and is about to be completely dislodged from the shear layer, while the shear layer oscillates downward and generates a downward shedding vortex B. At the moment of 0.00072s, the shedding vortex A finishes shedding from the shear layer and moves downstream and away from the cavity, the shedding vortex B receives energy transport from the shear layer and the cavity, and the vortex energy increases and is about to finish shedding from the shear layer; at the moment of 0.00108s, the shedding vortex A continues to flow downstream and away from the cavity and is about to finish separating from the shedding vortex B. The subsequent shedding vortex C comes into contact with the shedding vortex The subsequent shedding vortex C is in contact with the shedding vortex B and starts to absorb the energy of the shedding vortex B. At 0.00144s, the shedding vortex A completes the separation from the shedding vortex B and is somewhat strengthened by the energy supplement from the cavity in the process of flowing downstream, and the shedding vortex B continuously conveys the energy to the structure and shedding vortex C in the cavity, and weakens itself.At the moment of 0.00216s, shedding vortex A continues to exchange energy with the vortex structure inside the cavity and moves downstream, the vortex structure of shedding vortex B has disappeared, and shedding vortex C is strengthened and moves downstream; at the moment of 0.00216s, shedding vortex A continues to exchange energy with the vortex structure inside the cavity, and shedding vortex C gains enough energy to be detached from the contact with the flowing structure inside the cavity; at the moment of 0.00252s, shedding vortex A reaches above the back edge of the cavity, and The shedding vortex C carries enough energy to break away from the contact of the flowing structure inside the cavity.From the vortex flow process, we can know that the vortex from the shear layer to the outflow process, experienced a complex energy exchange process, due to a large amount of energy from the shear layer to the shedding vortex C, and finally the shedding vortex C is carried out of the cavity by the mainstream, so the cavity with only a small amount of energy for oscillation.

Conclusion
In this paper, the M219 cavity configuration is numerically simulated by solving the RANS equations in the curvilinear coordinate system using the IDDES method.Through theoretical analysis and comparison with experimental results, the conclusions are as follows.
(1) The method used in this paper can simulate the noise environment of the M219 cavity accurately.
(2) The different grid densities used in this paper can obtain relatively accurate numerical simulations, and the results of different grids reflect convergence.
(3) The smooth bump measures of the leading edge proposed in this paper can effectively lift the shear layer, reduce the inflow of disturbances in the cavity, and play a role in suppressing the cavity noise.

Figure 1 .
Figure 1.The geometric of the M219 cavity.

Figure 2 .
Figure 2. The mesh topology of the M219 cavity.

Figure 3 .
Figure 3.The surface mesh of different grids for the M219 cavity.

Figure 4 .
Figure 4.The position of the transducers in the M219 cavity.

FMIA- 2023
Journal of Physics: Conference Series 2599 (2023) 012033 for initial calculation and IDDES method was chosen for unsteady calculation, with flux-mixed scheme.The time step is 2 × 10 -5 s with subiteration being 100.The pressure contour and streamlines of the mean flow in the mid-plane for different grids are shown in Figure 5.It is observed that along x-direction the flow impinge on the aft wall of the cavity.A part of the flow from the shear layer at the leading edge flows into the cavity.Two large circular streamlines were formed in the cavity.One located in the middle of the cavity beside the shear layer, and the other located at the floor near the aft wall.It flows like that when the shear layer flow impinge on the aft wall, part of the flow flows downward along the aft wall and the flow will roll up after flowing to the floor.A part of the flow forms a circular vortex here, and the other continues to flow forward and forms another circular vortex in the middle of the cavity.

Figure 5 .
Figure 5.The pressure contour and streamlines of the mean flow in the mid-plane for different grids.

Figure 6 .
Figure 6.Comparisons of different flow profiles at three stations in the mid-plane of the cavity from different grids.

Figure 7 .
Figure 7.The instantaneous Q criterion iso-surface at t=0.15s from different grids (colored by the pressure coefficient).
the reference pressure.The OASPL contour of different grids are plotted together for comparison, as shown in Figure

Figure 9 .
Figure 9.The distribution curve of OASPL at each locations on the floor with different grids.

Figure 10 .
Figure 10.Comparisons of sound pressure spectrum of sample points on the floor in the mid-plane with different grids.

Figure 11 .
Figure 11.Density gradient fluctuations in the mid-plane at sequential times.

Figure 12 .
Figure 12.The geometry of M219 cavity with the bump at the leading edge.

Figure 13 .
Figure 13.The instantaneous Q criterion isosurface at t=0.15s for different bumps.

Figure 14 .
Figure 14.The OASPL contour on the surface of the cavity with different bumps.

Figure 15 .
Figure 15.The distribution curve of OASPL on the floor with different bumps.

Figure 15
Figure15shows the distribution curves of OASPL on the floor with different bumps.It can be observed that the OASPL on the floor decreases significantly with bumps being mounted in front of the leading edge.Bump5 decreases about 1 dB at the locations in the front of the cavity, and decreases about 4 dB at the rest locations.Bump10 decreases about 4 dB at all locations.Figure16shows the sound pressure spectral curves of sample points on the floor in the mid-plane simulated with different bumps.It can be observed that, with bums being mounted in front of the leading edge, except for the third mode frequency at the two locations (K20, K21) on the floor, there are no significant mode frequencies at the rest locations.That is meaning, the bump eliminates the resonant noise in most parts of the cavity flow.From the modal amplitude, it can also be found that the modal amplitude is lower than the cavity flow with bumps being mounted, which means that the bump is valid on reducing the modal amplitude.To sum up, the bump is effective to suppress on the cavity flow noise.And it's both effective on the modal frequency and modal amplitude in the cavity flow, resulting in a OASPL reduction of about 4 dB at the loations.

Figure 16 .
Figure 16.Comparisons of sound pressure spectrum of sample points on the floor in the mid-plane with different bumps.

Figure 17 .
Figure 17.Density gradient fluctuations in the mid-plane of bump5 at sequential times.

Table 1 .
The coordinate values of each Kulite transducer.