1 Introduction

The weapon bays on fourth-generation fighters such as Jian-20 as shown in Fig. 1 and unmanned aircraft as shown in Fig. 2 are facing with a serious aeroacoustic environment, which is a risk causing structural failure (Dobrzynski 2010). The sound pressure level inside this kind of weapon bay can reach up to 170 dB possibly making the electronic equipment malfunction. This type of weapon bay as a cavity structure has a length–depth ratio of 3 to 10. The cavity is characterized by highly unsteady flow, and the complex vortex structure in the cavity leads to acoustic resonance, which generates the problem of excessive aerodynamic noise. The weapon bay doors obstruct the flow from getting out of the weapon bay, so the combat unmanned aerial vehicles, X-47b in Fig. 2 for example, have much more serious aeroacoustic problems. The complex configuration brings challenges on aerodynamic noise.

Fig. 1
figure 1

Weapon bay on Jian-20

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Fig. 2
figure 2

Weapon bay on X-47b

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The mechanism research on cavity aeroacoustic is the foundation of all research, and the simulation method provides the way to recognize the phenomenon. It is an effective way to discuss the mechanism by investigating the influence of different cavity configuration, flow velocity, thickness of shear layer etc. Rossiter (1962) put forward the definition of open flow and close flow for shallow cavity, and pointed out that the critical length to depth ratio that determine the flow type of the cavity was influenced by the boundary layer thickness, the altitude and the presence weapon bay doors. Eleonra et al. (2018) investigated on different cavities using computational fluid dynamic method and verified the results by comparing the wind tunnel tests. Hajczak (2019) studied the characteristics of compressible rectangular landing gear cavity flows. Most simulation methods on pressure fluctuation were based on hybrid LES/RANS methods considering the accuracy and efficiency. Nichols (2006) examined the use of the Euler equations, N–S equations with no turbulence model in the bay, unsteady RANS with a transport turbulence model, and N–S equations with hybrid RANS/LES turbulence models for bay flow simulations on three bay configurations. Barakos (2010) present DES for the M219 experimental cavity geometry and the uninhabited combat air vehicle cavity geometry. It is suggested that advanced multi-block topologies had to be used to properly represent the planform of the uninhabited combat air vehicle and all the details of the cavity, including doors and hinges, while results with an empty cavity were encouraging for such complex configurations. Shaw (1982) carried out wind tunnel test on cavity of F-111 and investigated three kinds of passive control method. Zhao et al. (2018) investigated an active control method called air curtains. Liu et al. (2019) simulated weapon bay configuration and studied passive and active control methods such as spoilers and leading edge blowing.

Because the structure of weapon bay on unmanned aerial vehicle is different from regular cavity in theoretical research, the depth is variable and it has a hatch cover by the side of the weapon bay. Previous research (Liu and Tong 2015) simulated the internal aerodynamic noise level on standard cavity and cavity with opened door, and it is shown that the SPLs inside the cavity as well as the magnitude of tones are amplified by the side doors. The main purpose of this study is to explore the aerodynamic noise level of complex cavities with slant ceiling and irregular length–depth ratio. The aerodynamic noise environment is studied based on a standard open cavity with slant ceiling using DDES firstly. Then, a weapon bay which has variable depth and opened doors by the side on a certain type of UAV is analyzed, and the sound pressure level inside the weapon bay is obtained providing reference for the design of this type of UAV.

2 DDES

The aerodynamic noise of the cavity is theoretically caused by the interaction of the internal multi-scale vortex structure, (Liu et al. 2019). The Reynolds-Averaged Navier–Stokes method (RANS) cannot predict the sound pressure level inside the cavity accurately, as it is not able to resolve all scales of the vortex structures especially the vortex near the shear layer, (Li et al. 2022). Large Eddy Simulation (LES) is a feasible way to solve accuracy issue in the boundary layer or shear layer of cavity, but according to the research of Tomimatsu (2012), the cost is unacceptable under the condition of high Reynolds number. Detached Eddy Simulation (DES) as a hybrid of RANS and LES, (Squires 2004) can simulate the large-scale separation region with high precision and high simulation. In previous study, (Liu et al. 2019; Liu and Tong 2015), the DES method was applied on different kinds of cavity and the accuracy comparing with the experiment was acceptable.

The governing equations for LES and RANS simulation take the same form:

$$\frac{\partial \rho }{ \partial t}+\frac{ \partial \rho \overline{{u }_{i}}}{ \partial {x}_{i}}=0$$
(1)
$$\frac{\partial (\uprho \overline{{u }_{i}})}{ \partial t}+\nabla \cdot \left(\rho \overline{{u }_{i}}\widetilde{\upnu }\right)=\rho {f}_{i}+\frac{\partial {\uptau }_{ij}}{\partial {x}_{j}}-\frac{\partial \overline{p}}{\partial {x }_{i}}$$
(2)
$$\frac{\partial (\rho T)}{ \partial t}+\nabla \cdot \left(\rho \overline{{u }_{i}}T\right)=\nabla \cdot \left(\frac{k}{{C}_{p}}\mathrm{grad}T\right)\rho {f}_{i}+{S}_{\mathrm{T}}$$
(3)

where and overline denotes a quantity averaged over one grid cell and time step, and \({\uptau }_{ij}=\overline{{u }_{i}{u}_{j}}-\overline{{u }_{i}}\overline{{u }_{j}}\) are either the subgrid-scale (SGS) stresses or the Reynolds stresses that must be modeled.

The one-equation Spalart–Allmaras (S–A) model can be written as

$$\frac{\partial \widetilde{\upnu }}{ \partial t}+{u}_{i}\frac{\partial \widetilde{\upnu }}{\partial {x}_{i}}=\frac{1}{\sigma }\left[\nabla \cdot \left(\left(\upnu +\widetilde{\upnu }\right)\nabla \widetilde{\upnu }\right)+{\mathrm{C}}_{b2}{\left(\nabla \widetilde{\upnu }\right)}^{2}\right]+P\left(\widetilde{\upnu }\right)-D\left(\widetilde{\upnu }\right)$$
(4)

where the dissipation is defined as

$$D\left(\widetilde{\upnu }\right)={C}_{\mathrm{w}1}{f}_{\mathrm{w}}{\left(\frac{\widetilde{\upnu }}{d}\right)}^{2}$$
(5)

and the production term

$$ P\left( {\tilde{\nu }} \right) = C_{b1} \tilde{v}\tilde{\Omega } $$
(6)

\(\widetilde{\nu }\) is turbulence variable, and ν is kinematic viscosity.\(\sigma \), \({C}_{b1}\), \({C}_{b2}\) and κ are constants. The vorticity variable is given by \(\tilde{\Omega } = \left| \Omega \right| + \frac{{\tilde{\nu }}}{{\kappa^{2} d^{2} }}f_{v2}\). Functions \({f}_{\mathrm{w}}\) and \({f}_{v2}\) are defined to induce turbulence viscosity near the wall. \(d\) refers to the distance from wall.

When we turn this standard S–A model to DES, \(d\) in Eq. (5) will be replaced by

$$\widetilde{d}=\mathrm{min}\left(d,{C}_{\mathrm{DES}}{L}_{\mathrm{g}}\right)$$
(7)

The empirical constant \({C}_{\mathrm{DES}}\) has a value of 0.65. \({L}_{\mathrm{g}}\) is a grid length scale and defined as

$${L}_{\mathrm{g}}=\mathrm{max}\left(\Delta x,\Delta y,\Delta z\right)$$
(8)

\(\Delta x\), \(\Delta y\) and \(\Delta z\) are the local grid lengths. As a consequence, RANS method based on one-equation S–A model will be adopted in the region near the wall including the whole boundary layer as \(\widetilde{d}=d\). When the region is far from the wall, \(\widetilde{d}={C}_{\mathrm{DES}}{L}_{\mathrm{g}}\), the dissipation will be determined by local grid length. Once the dissipation term and production term reach to a balance, it is found from Eq. (4) that \(\widetilde{\nu }\) is in direct proportion to \(\tilde{\Omega }d^{2}\), that is,

$$ \tilde{\nu } \propto \tilde{\Omega }L_{{\text{g}}}^{2} $$
(9)

Equation (9) has the same characteristic with the model defined by Smagorinsky (1963) as a subgrid scale model used in LES method. For a typical RANS grid with a high aspect ratio in the boundary layer, the wall-parallel grid spacing usually exceeds the boundary layer thickness, so Eq. (8) will ensure that the DES model is in the RANS mode for the entire boundary layer. However, in case of a dense grid in all directions, the DES limiter can activate the LES mode inside the boundary layer, where the grid is not fine enough to sustain LES requirement. Therefore, a new formulation called Delayed Detached Eddy Simulation (DDES) is presented to preserve the RANS mode throughout the boundary layer.

The length scale \(\widetilde{\mathrm{d}}\) in DDES is re-defined as

$$\widetilde{d}=d-{f}_{\mathrm{d}}\mathrm{max}(0, \mathrm{d}-{C}_{\mathrm{DES}}{L}_{\mathrm{g}})$$
(10)

where \({f}_{\mathrm{d}}\) is called tke-dissipation-multiplier and given by

$${f}_{\mathrm{d}}=1-\mathrm{tan}h\left({\left[8{r}_{\mathrm{d}}\right]}^{3}\right)$$
(11)

\({r}_{\mathrm{d}}\) can be considered as a ratio of the turbulence length scale and the wall distance \({f}_{\mathrm{d}}\) is designed to be 1 in the LES region where \({r}_{\mathrm{d}}\ll 1\), and \({f}_{\mathrm{d}}=0\) elsewhere. Therefore, when flows transport from a region with a large value of eddy viscosity into a region of relatively small strain, this could cause the DDES model to switch the mode early than DES which means a transition from LES to RANS mode away from the body.

The simulation used a Green-Gauss cell based finite volume scheme and chosen second-order implicit time integration and third-order MUSCL spatial discretization to improve accuracy. The DDES transient calculations with S-A model were started using a time step of 10–5 s with 30 iterations after a steady RANS computation finished.

Sound pressure levels (SPLs) were obtained from the simulation results by the Eq. (12),

$$\mathrm{SPL}=20\mathrm{lg}(\frac{{P}_{\mathrm{rms}}}{{P}_{\mathrm{ref}}})$$
(12)

The minimum audible pressure variation \({P}_{\mathrm{ref}}\) usually takes the value of \(2\times {10}^{-5}\mathrm{Pa}\). Overall Sound Pressure Levels (OASPLs) were obtained by the Eq. (13),

$$\mathrm{OASPL}=10{\mathrm{log}}_{10}({10}^{\frac{{\mathrm{SPL}}_{f1}}{10}}+{10}^{\frac{{\mathrm{SPL}}_{f2}}{10}}+\dots +{10}^{\frac{{\mathrm{SPL}}_{fn}}{10}})$$
(13)

3 Standard test cases

3.1 Model and mesh

Foster et al. (2005) conducted an aerodynamic noise wind tunnel test on a cavity called M219 (Henshaw 2000), and the test results can be used to verify the results of this simulation analysis. The size of the cavity is shown in Fig. 3a, which is a typical open cavity. There are 10 pressure sensors at the bottom of the cavity for recording the pulsating pressure change, and the sample frequency is 6 kHz.

Fig. 3
figure 3

a Geometries for clean cavity and the distribution of pressure sensors (mm). b Distribution of pressure sensors and geometry of cavity with slant ceiling

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The cavity with slant ceiling is obtained by rotating the ceiling of the M219 cavity along the central line by 23.5°. As a result, the cavity depth changes from 80 to 123.2 mm, and the length–depth ratio of the cavity ranges from 4.12 to 6.35. The configuration of clean cavity and cavity with slant ceiling are shown in Fig. 3.

The calculation area of the two simulation cases is consistent with the size of the M219 cavity, and the upper boundary of the simulation domain is 10D away from the cavity as pressure far field. All the pressure far field boundaries including inflow and outflow were set as Ma = 0.85, P = 62940 Pa, T = 270.25 K, \(\mu_{t} /\mu_{0} = 10\), Re = 6.785 × 105. The physical meaning of the pressure far field boundary condition is to simulate the flow field at far away by making the disturbance wave propagating to the boundary without reflection. The wall was set as adiabatic and no-slip conditions. The grid near the wall is dense with y+ < 2, which is capable to resolve the small vortex structure generated by the viscous effect. The whole computational domain for clean cavity and cavity with slant ceiling are shown in Fig. 4

Fig. 4
figure 4

Computational mesh for all cases

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The simulation was carried out for a total of 0.5 s, and only the last 0.2 s data was selected. Both cases have about 4.5 million meshes costing 110 days total using a cluster with 144 processors.

3.2 Results discussion

Figure 5 shows the overall SPLs in experiment and DDES from the front to the rear of the clean cavity and slant ceiling cavity (Slant-S represents K00-K09 from shallow side, Slant-D represents K10-K19 from deep side).

Fig. 5
figure 5

OASPL of passive control cases

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The overall sound pressure level at the downstream of the cavity is very high, reaching to 167 dB, while the overall sound pressure level upstream of the cavity reaches 153 dB. The DDES method can meet the requirements of pressure fluctuation simulation on this kind of cavity in accuracy, as the simulation results on OASPL are in good agreement with the experimental results. It can be seen from Fig. 5 that both the deep side and shallow side have the same distribution of OASPL as the standard cavity. It is found that the OASPL on the deep side is generally higher than that on the shallow side by 1 dB.

Figure 6 shows the sound pressure level at x/L = 0.05 and x/L = 0.95.

Fig. 6
figure 6

Sound pressure level at x/L = 0.05 and 0.95

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With reference to the previous research, (Zhao et al. 2018; Liu et al. 2019), the sound pressure level characteristics inside the open cavity are generally considered to have four modes which are four distinct peaks, and the second-order and third-order modes play a leading role. The sound pressure levels at deep side and shallow side of the slant ceiling cavity are almost consistent. The frequency of tone noise is completely the same, but the amplitude at deep side is slightly higher than the shallow side. The frequency and amplitude of the tone noise at position of x/L = 0.95 are summarized in Table 1 for all cases.

Table 1 Tone noise for all cases at x/L = 0.95
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All cases show peak splitting phenomenon on the fourth mode of the cavity. There are multiple peaks near the frequency of the fourth mode, and the amplitudes of each peak are very close. As a result, it makes the determination of the fourth mode somewhat difficult. Comparing the modes at the shallow side of the slant ceiling cavity to the deep side, the frequency of all modes for both sides are completely consistent, but the amplitude at the deep side increases slightly, and the main reason is that transverse flow from shallow side to deep side was generated by the slant ceiling, causing the pressure fluctuations at deep side more intensive. Comparing the slant ceiling cavity to clean cavity, the amplitude of the first order mode is significantly higher than the other three modes, becoming the dominant resonant mode. In addition, the amplitudes of the other three modes maintain the same level with the difference less than 2 dB, and frequencies of all modes are almost the same.

4 UAV

4.1 Model and mesh

The structure of weapon bay calculated in this paper is similar to UAV X-47b in Fig. 2. The weapon bay is not a standard rectangular cavity, and it has curvature leading edge and hatch door, so the size of weapon bay changes in the longitudinal direction and especially in the transverse direction. The layout of the UAV’s weapon bay is shown in Fig. 7, and two bays are arranged in parallel with a frame in the middle and two hatch doors at both sides which is similar to the weapon bay of Jian-20 fighter. The length, width and depth of each bay are L = 2.6 m, W = 0.4 m and D = 0.3–0.5 m, so the length-to-depth ratio varies from 5.2 to 8.7. The mission altitude of the weapon bay is 6 km, with the Mach number of incoming flow from 0.6 to 0.8.

Fig. 7
figure 7

Geometry of UAV and grid distribution of simulation model

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The simulation model is based on the weapon bay shown in Fig. 7 and takes half of the body to shorten computation time. The computational domain is extended 0.8 L to the inlet boundary, 1.5 L to the outlet boundary, 1.2 W to the right side boundary, and 12 W to the upper boundary to minimize the reflection effect of sound waves. The inlet, outlet and right side boundary were set as pressure far field boundary conditions with Ma = 0.6 and 0.8, P = 47417 Pa, T = 249.187 K and eddy viscosity ratio \(\mu_{t} /\mu_{0} = 10\). Symmetry boundary condition was applied on the left side boundary, and adiabatic, no-slip wall conditions were applied on the surface of the weapon bay and hatch door. The Reynolds number is 2.7 × 107.

The mesh around weapon bay and hatch door was generated meticulously with about 4.9 million cells. At the wall, the grid results in y+ < 3, which is sufficient to resolve the viscosity-affected near-wall. The whole computational domain and mesh is shown in Fig. 7.

The simulation was performed for a total of 0.8 s with the first 0.4 s of data discarded to eliminate any transients leaving 0.4 s of computational pressure history. There are 30 equally spaced data collection points named by D0 (M0, S0) to D9 (M9, S9) on the ceiling shown in Fig. 8 to measure the time histories of pressure. The simulation was performed on a cluster with 144 processors, taking 65 days for each case.

Fig. 8
figure 8

Distribution of pressure collection points

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4.2 Results discussion

The distribution of \({\mathrm{f}}_{\mathrm{d}}\) in the cavity is shown in Fig. 9. The value of \({f}_{\mathrm{d}}\) is close to 0 near the wall, so the RANS mode is adopted according to the analysis in Sect. 2. In the middle of the cavity and the area away from the wall, \({f}_{\mathrm{d}}\) is close to 1, which means the LES mode is activated. Generally, the DES method is implemented.

Fig. 9
figure 9

Distribution of tke-dissipation-multiplier

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It is seen from Fig. 10 that after passing the front edge, the incoming flow forms a shear layer above the weapon bay. Part of the flow enters the interior of the weapon bay and then impacts on the rear wall. However, when Ma = 0.6, the shear layer is thicker and the impact on the rear wall is smooth. When Ma = 0.8, it can be observed that the shear layer has an obvious oscillation effect, and the impact effect on the back wall is more intensive.

Fig. 10
figure 10

Instantaneous velocity profile of weapon bay at t = 0.4 s

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Figure 11 shows the pressure profiles for the two cases at t = 0.4 s. Because the bottom of the UAV is a curved surface, the front wall of the weapon bay has a high-pressure area. As the incoming flow go through the weapon bay, the high-pressure area at the rear of the cavity is formed due to the impact of the shear layer. With the Ma number increasing to 0.8, the range of the high-pressure area becomes significantly larger, indicating that the unsteady effect of the flow in the weapon bay is enhanced.

Fig. 11
figure 11

Instantaneous pressure profile of weapon bay at t = 0.4 s

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Figure 12 shows the overall SPLs from all collection points including shallow side, middle and deep side in the weapon bay on Ma = 0.6 and 0.8 cases.

Fig. 12
figure 12

OASPL distribution along the weapon bay of UAV

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As the inflow velocity Mach number increasing from 0.6 to 0.8, the overall sound pressure level in the weapon bay increases significantly by 16–23 dB. The main reason is that as the Mach number increased, the shear layer over the weapon bay oscillates apparently, with more flow striking the rear wall, which causes the flow inside the weapon bay significantly unsteady and generates more pressure waves. As a result, the aerodynamic noise generated from the weapon bay is greatly enhanced. By comparing three transverse positions S, M and D in the weapon bay, it is found that the OASPLs at every position are very close and have the same trend from the front to the back. According to the research in last chapter, the airflow on the shallow side will flow to the deeper side, resulting in a slightly higher sound pressure level of about 1 dB on the deeper side. However, according to previous investigation, (Liu et al. 2019), the sound pressure levels on the shallow side intend to increase due to the obstruction by hatch door. As a result, the sound pressure levels in different lateral positions inside the weapon bay have little difference under the influence by these two factors above.

Figure 13 shows the sound pressure level at D0 and D9. It can be seen by comparing two positions that the sound pressure level at position D9 is significantly higher than that at position D0 about 20 dB. After the Mach number increased from 0.6 to 0.8, the curves of sound pressure level at both positions move upward. The main reason as mentioned before is that the oscillation effect of the free shear layer become intensive, and the impact on the rear wall is stronger, leading to higher sound pressure levels inside weapon bay.

Fig. 13
figure 13

Sound pressure level at D0 and D9

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The sound pressure levels at two positions show that the resonance tone noise occurs in the low-frequency region and can be distinguished by three peaks especially on Ma = 0.8. The second-order mode could be the dominant mode, and the frequency is shifted forward slightly by about 10 Hz from Ma = 0.6 to 0.8. The amplitude of the second-order mode increases significantly by more than 15 dB.

5 Conclusions

The DDES computations have been conducted for an open cavity flow using S–A one-equation model. The open cavity immersed in a free stream at a Mach number of 0.85 has two configurations, standard cavity with an aspect ratio of 5:1:1 and cavity with slant ceiling. The overall sound pressure level in the deeper side is generally higher than that in the shallow side about 1 dB. The reason could be the transverse flow from shallow side to deep side generated by the slant ceiling causing the pressure fluctuations at deep side more intensive.

A pressure fluctuation investigation using DDES was carried on a UAV weapon bay which has curvature leading edge and hatch door. The results show that the sound pressure level has three peaks as three tone noise, and the dominant tone noise in the weapon bay is located in the low-frequency region. The overall sound pressure level in the weapon bay is up to 157 dB, which is a big challenge on separation safety for the missile in weapon bay. When the Mach number turns from 0.6 to 0.8, the amplitude of all modes increases about 16–23dB. The pressure fluctuation characteristics obtained by the DDES method inside the weapon bay of UAV could be a reference for the loadings analysis and safety design for the typical unmanned aerial vehicle.