Unsteady separated flow from deployed weapons bay doors can interact with the highly unsteady flow in the open bay cavity, which is known to exhibit strong acoustic content and could lead to fluid-resonance and high-intensity acoustic noise. The culmination of these unique flow physics can potentially excite structural modes of the doors, aircraft surfaces, or externally carried munitions and fuel tanks and can ultimately lead to aeroelastic instabilities, such as buffet, flutter, limit-cycle oscillations, or fatigue-induced failures. A hybrid Reynolds-averaged Navier–Stokes large eddy simulation (RANS/LES) method with low-dissipation schemes is developed to improve flow and acoustics predictive capabilities for supersonic weapons bays. Computational simulations are conducted for a weapons cavity with different deployed bay doors configurations, including the effect of dynamically moving doors, to assess the tonal content and unsteady aerodynamic loads on the doors. Wind tunnel testing is also carried out to provide unsteady experimental data for use in validating the high-fidelity simulation capability. The simulation results in terms of unsteady pressure, velocity fluctuations, and pressure resonant frequencies are computed and presented. The results suggest that the deployed doors energize the shear layer and cause it to go deeper into the cavity and produce higher unsteady fluctuations on the weapons cavity floor and aft wall. The deployed doors also cause a shift in the dominant resonant modes.

## Introduction

Modern high-speed aircraft offers numerous circumstances in which doors often must be opened and closed during flight. Applications range from low-speed deployment of landing gear, application of speed brakes, deployment of weapons bay doors at high speeds, and many others. Complex unsteady flow is typically observed in these applications, with unsteady aerodynamic loading and flow oscillations often present on the bay doors. The open cavity flow by itself is inherently unsteady and there is no practical mechanism to produce steady flow. In the open cavity, the leading-edge boundary layer separates and a shear layer is formed between the freestream flow and the flow inside the cavity. This shear layer spans the length of the cavity where the Kelvin–Helmholtz instability is driven by the mean shear in the velocity profile. This mean shear results in shear layer growth and the organization of the underlying vortical structures. These small-scale structures convect downstream and evolve into large-scale coherent structures. Consequently, the shear layer rolls up and impinges on the aft wall of the cavity.

Heller and Bliss [1] provided detailed analysis of the shear layer in an attempt to explain the unsteadiness found in open cavities, and it was argued that a considerable curvature of the shear layer over the cavity mouth occurs before the shear layer impinges on the aft wall at an oblique angle causing high-intensity acoustic tones to develop. This impingement generates resonant conditions which influence the shear layer through complex interactions with the upstream advancing acoustic waves. Discrete tones are generated by both laminar and turbulent boundary layers, however, laminar boundary layers tend to be thicker than turbulent boundary layers resulting in a thicker initial shear layer and lower frequency tones [2]. The acoustic tones can also induce vibrations on the store as well as on the aircraft structures in the vicinity of the cavity [3].

The loading and aeroacoustics problem is magnified with deployed doors of the weapons bay where unsteady aerodynamic loading and flow oscillations are often present on the bay doors. The separated flow from the doors can interact with the unsteady flow in the bay, which is known to exhibit strong acoustic content, and can lead to fluid-resonance and high-intensity acoustic noise through complex interactions with the upstream advancing acoustic waves and the shear layer. The culmination of these unique flow physics can potentially excite structural modes of the doors, aircraft surfaces, or externally carried munitions and fuel tanks, and can ultimately lead to aeroelastic instabilities, such as buffet, flutter, limit-cycle oscillations, or fatigue-induced failures.

Understanding the underlying physics and essential triggering mechanisms of the complex aeroacoustic content in high-speed cavities is of great importance to airspace supremacy since aircraft weapons are essential military technologies that provide strategic and tactical supremacy. Captive-carry weapons, inside weapons bay cavities in particular, provide an appealing technology for aircraft agility because external stores can account for up to 30% of the total aircraft drag [4]. However, safety procedures for store deployment are of critical concern due to the complex cavity flow environment at the cavity opening. The high-intensity acoustic noise and pressure fluctuations can damage the stores and the aircraft structures near the weapons bay cavity.

While computational fluid dynamics (CFD) methods can be readily used to simulate unsteady flow physics, traditional numerical schemes employed for aerodynamic simulation are generally too dissipative for use in predicting the acoustic content. This is due to the fact that acoustic frequencies are typically orders-of-magnitude higher than the frequencies present in the mean flow, and the numerical dissipation included to maintain stability in CFD simulations has the effect of rapidly dissipating the smaller-scale high-frequency acoustic features. It follows that traditional flux schemes such as the Roe solver may be overly dissipative to accurately resolve the weapons bay acoustic content. To accurately predict and propagate acoustic information, improved low-dissipation schemes are essential.

The focus of this study is to better understand the effect of deployed doors and different door configurations on the unsteady pressure loading and acoustic tonal content. A hybrid RANS/LES solution method is enhanced with the development and integration of a low-dissipation scheme to improve prediction capability for weapons bay acoustic tonal content and unsteady loads on deployed weapons bay doors. In Sec. 2, the hybrid solution method and low-dissipation scheme are discussed. Section 3 presents the details of the weapons cavity model with deployed doors and wind tunnel setup, and a discussion of the simulation results with a focus on the unsteady pressure and velocity fluctuations and pressure resonant frequencies. Section 4 explains the geometrical and computational model in the study. Section 5 presents the results and discussion of the study and Sec. 6 concludes the paper.

## Solution Methodology

A hybrid RANS/LES solution method coupled with a low-dissipation kinetic energy consistent (KEC) scheme is used to improve the prediction capability for weapons bay unsteady loads and acoustic tonal content. The solution methodology used in this effort is implemented in the loci/chem flow solver. loci [79] is a software framework that simplifies the development of complex multiphysics high-resolution numerical models. The framework was originally developed in the late 1990s with funding from the National Science Foundation with the goal of simplifying the development of complex numerical models that can take advantage of massively parallel high-end computing systems. The framework incorporates concepts from artificial intelligence and database systems to develop a seamless approach for developing complex simulation software. The framework provides a rule-based programing model whereby an application is described in terms of a collection of simple computational kernels. The loci framework can assemble these kernels and optimize their scheduling on parallel high-performance architectures. As a result, the loci framework makes an excellent platform for the development and integration of a wide range of computational models. The framework supports the development of run-time loadable modules that allow loci applications to be extended to support new physics and models with ease. Additionally, the verification capabilities of the loci framework can provide assurances that the composition of these models satisfies rules of internal logical consistency. The framework also provides a logical deduction computation phase that can automatically find bugs in numerical models caused by logical inconsistencies in their specification.

The loci/chem code [810] was originally developed as a technology demonstrator for the loci framework and has become a mature software for complex multiphysics simulations. The chem solver is a density-based Navier–Stokes solver employing high-resolution approximate Riemann solvers implemented for predicting multicomponent mixing and chemically reacting flows using unstructured grids. These approaches make the chem solver very well suited for compressible flow simulation in complex geometries. In addition, the core algorithms have been extended to accurately model flows at low speeds through the use of preconditioning techniques. The loci/chem code has a variety of turbulence models with high-speed compressibility corrections [11], and the numerical models have been demonstrated to be at least second-order accurate in space and time through rigorous verification using the method of manufactured solutions (MMSs) [12].

### Hybrid Reynolds-averaged Navier–Stokes Large-Eddy Simulation Solution Method.

The hybrid RANS/LES model [13] is an implementation of a multiscale turbulence model in which the eddy viscosity is a function of two turbulent length scales as opposed to just one for the traditional turbulence models. The basic idea of the current hybrid model is that the largest turbulent scales are resolved on the computational mesh while the smallest, unresolved scales continue to be modeled. This requires the definition of appropriate length scales, a filtering mechanism to determine which scales are modeled and which are resolved, and a blending function to smoothly match the eddy viscosity between the two regimes.

The turbulent length scale is defined as
$LT=max(6.0υtRANSΩ,lT)$
(1)

where Ω is the local mean flow vorticity that helps define an algebraic length scale, and lT is a turbulent length scale associated with two equation RANS models. The subscript RANS indicates that the values are from the unfiltered RANS model, while the factor 6.0 is recommended by Nichols and Nelson [13] to make the two length scales approximately equal.

When the grid is substantially refined outside of the boundary layer, additional turbulent scales will be resolved, and the eddy viscosity will be decreased compared to its usual RANS value. The subgrid value for the turbulence kinetic energy is thus taken as a fraction of the RANS value
$kLES=kRANSfd$
(2)
where the damping function is defined as
$fd=12{1+tan h[2π(Λ−0.5)]}$
(3)
The function Λ determines which scales are locally resolved by comparing the turbulent length scale LT to the local grid size
$Λ=11+(LT2LG)4/3$
(4)
where
(5)
The blended eddy viscosity is computed from
$υt=υtRANSfd+(1−fd)υtLES$
(6)
where the LES-based subgrid eddy viscosity is given by
$υtLES=min(0.0854LGkLES,υtRANS)$
(7)

When the turbulent scales (LT) are much smaller than local grid scale (LG), implying that they cannot be adequately resolved, the RANS mode is used in the usual single scale turbulence model approach. On the other hand, when the turbulent scales are much larger than the local grid scale and can be resolved, the model switches smoothly to the LES mode. This results in a smaller eddy viscosity which is necessary to resolve the unsteady turbulent fluid motion that was originally damped out by the larger RANS eddy viscosity. The above hybrid RANS/LES model must be associated with a two equation turbulence model. In this work, the Menter's shear stress transport turbulence model is used.

### Low-Dissipation Model.

To accurately resolve the tonal content in the weapons bay, the numerical schemes employed must provide sufficiently low dissipation to resolve the frequency ranges commonly observed in weapons bays from subsonic through hypersonic regimes. While the existing numerical schemes in loci/chem may be sufficient for resolving frequency content for a variety of high-speed applications, low-dissipation schemes may be necessary to efficiently resolve very high-frequency tonal content for some applications.

In the low-dissipation approach employed here, the second-order scheme is upgraded to provide higher resolution of high-wave number components of the flowfield by reducing the numerical dissipation. Specifically, these upgrades modify the discretization of the time integration term as well as the convective terms in the governing equations, based on the kinetic-energy-consistent finite-volume scheme recently presented by Subbareddy and Candler [14]. Key aspects of that approach include a time integration procedure analogous to a density-weighted Crank–Nicolson method, and a convective flux formulation that consistently represents kinetic energy in all the equations. The method has been shown to be stable without addition of any dissipation operators in smooth flow regions. In regions with discontinuities (shocks), it is necessary to add a dissipation component to the convective flux similar to the Roe scheme.

The time integration scheme described in this section is adopted from the work of Subbareddy and Candler [14] in which a kinetic energy consistent, low-dissipation method for compressible flow was presented. This scheme uses a midpoint scheme, which is given by the equation
$V(Qn+1−Qn)=ΔtR(qn+1/2)$
(8)

where n is the current iteration number, V is the cell volume, Δt is the time step, and Q represents the vector of conservative variables. To satisfy the energy consistency for the time derivative, there is a special treatment in the right-hand side: A special averaging is used to compute the primitive variables at the half time-step. This half time-step value, denoted as q= qn, is computed using an averaging of the primitive variables as described by Sheta et al. [15], which is discussed next.

#### Pressure and Density Averaging.

According to the pressure and density primitive variable formulation, the average state is computed using the relation
$q*=[(ρin+1+ρin)/2(ρn+1un+1+ρnun)/(ρn+1+ρn)(ρn+1vn+1+ρnvn)/(ρn+1+ρn)(ρn+1wn+1+ρnwn)/(ρn+1+ρn)((1+ϵ)Pgn+1+(1−ϵ)Pgn)/2]$
(9)
In the above equation, Pg is the gage pressure, $ρ=∑ ρi$, and $ϵ$ is a pressure damping term used to stabilize the equations when larger time steps are employed. Using this averaging, the right-hand side of Eq. (8) can be rewritten as a function of q* by expressing $Qn+1$ as a function of q* and $Qn$ by employing the relations in Eq. (9) and thus deriving the nonlinear time-stepping operator given by
$L(q*)=VΔt(Qn+1(q*,Qn)−Qn)−R(q*)=0$
(10)
Equation (10) is solved using a Newton iteration method, as follows:
$L′(q*,p)(q*,p+1−q*,p)=−ℓ(q*,p)$
(11)
for p ≥ 0, where the Newton iteration is initialized using the previous time-step value $(q*,p=0=qn)$, and the numerical flux Jacobian is given by
$L′(q*)=VΔt∂Q∂q∂qn+1∂q*−[∂∂q*R(q*)]$
(12)
where ∂Q/∂q and ∂R/∂q exist in the standard solution method in which three-point backward Euler time integration scheme is used. To obtain ∂qn+1/∂q, we rearrange Eq. (9) to express qn+1 as a function of q as follows:
(13)

Equation (13) is also used to derive the term $Qn+1(q*,Qn)$ in Eq. (10) that relates the next time-step conservative vector to the half-step value q used in the Newton iteration.

## Wind Tunnel Model and Setup

The wind tunnel testing was conducted in the blow down supersonic tunnel at the Fluid Mechanics Laboratory of the University of Florida, Gainesville, FL. The tunnel, shown in Fig. 1, offers a modular design and includes optical access through the test section to allow for the use of particle image velocimetry (PIV) and Schlieren imaging techniques. The air for the facility is fed from a compressor system capable of generating compressed air at 210 psi through multiple rotary screw compressors. In the tunnel, subsonic air is brought into the plenum and filtered through a honeycomb substrate used to breakdown any remaining organized large-scale turbulent motions in the flow conditioning section. The air then passes through the first contraction designed to smoothly transition the cross section from circular to rectangular, which matches the inlet face of the nozzle. The nozzle is a two-dimensional converging–diverging assembly with an exit cross section of dimensions 76.2 mm wide by 101.6 mm tall and was designed using the method-of-characteristics. The nozzle produces a Mach 1.44 flow. The air next passes through a constant area test section with a uniform rectangular cross section before exiting the tunnel through the diffuser. The test section has a height of 101.6 mm and a width of 76.2 mm, and it should be noted that the aspect ratio of this test section is somewhat unconventional in supersonic wind tunnel design with the height being greater than the width. The additional vertical height allows for a maximum length cavity to be inserted by giving the anticipated oblique shock wave at the leading edge of the cavity a longer streamwise distance to traverse before being reflected off the top. The wind tunnel facilities are described in detail by Dudley et al. [16].

Fig. 1
Fig. 1
Close modal

The test section, as illustrated in Fig. 2, was symmetrically designed so either a top or bottom mounted cavity could be installed. The design consists of a one-piece 6061-aluminum frame with four rectangular cutouts for optical access. The primary optical inserts were fabricated with 19 mm thick N-BK-7 Schott optical grade glass machined and polished to 0.5 through 1 wave flatness per 25.4 mm 60/40 scratch/dig and a maximum allowable chip of 0.25 mm. Plexiglas and aluminum inserts are available when visual access to the tunnel is not required. The rectangular cavity model and the doors in the fully open position are also illustrated in Fig. 2. The cavity was mounted on the floor of the test section and instrumented with Kulite XCQ-062 pressure transducers along the centerline of the floor with one transducer in the center of the aft wall. The sensors in the floor were differential transducers with a 5 psi range, while the one at the aft wall was an absolute pressure sensor with a 30 psi range. All the transducers were sampled at a rate of 90 kHz with a PXI4472 24 bit data acquisition card, and 2,097,152 points were acquired at each transducer. The experiments were conducted with a freestream Mach number of 1.4 at a stagnation pressure of 25 psi. The freestream unit Reynolds number in the test section was approximately 1.7 × 104 per millimeter. The tunnel controller has been empirically determined to hold the Mach number constant to within 1% of nominal value over the entire run.

Fig. 2
Fig. 2
Close modal

## Geometrical and Computational Model

Details of the weapons bay cavity geometry with deployed bay doors are shown in Fig. 3. The figure shows the doors in a fully open configuration, which is one of the three different configurations considered in this study. The weapons cavity is a rectangular open cavity model with L/D = 6 and L/W = 3, and the physical dimensions of the cavity and wind tunnel model are tabulated in Table 1. The boundary conditions used in the computational simulations are also illustrated shown in Fig. 3. All the solid surfaces of the cavity, doors, and the wind tunnel are treated with the classical no-slip boundary conditions, except the test section side walls where a slip inviscid boundary condition is used. The top boundary of the wind tunnel is treated with a characteristic nonreflective boundary condition, and extrapolated boundary conditions are applied at the supersonic exit boundary.

Fig. 3
Fig. 3
Close modal
Table 1

List of cavity and wind tunnel dimensions (in millimeter)

LDWlwLehδ
76.212.725.4196.8576.244.4550.83.8
LDWlwLehδ
76.212.725.4196.8576.244.4550.83.8

Three different door configurations are considered in this study, corresponding to two fully open doors at 90 deg, two partially open doors at 45 deg, and one door closed and one fully open at 90 deg. The surface models of the doors over the cavity as well as the dimensions of the leading-edge profile are shown in Fig. 4. The doors were designed with a double chamfer on the leading edge to weaken the shocks at the door leading edges. The thicknesses of the door leading and trailing edges are about three thousandths of an inch. The inner walls of the bay doors are in-plane with the inner side walls of the cavity.

Fig. 4
Fig. 4
Close modal

To reduce the complexity of the computational model and the grid cell count, the wind tunnel nozzle is not modeled. In this study, the one-dimensional nozzle exit flow profile extracted from a steady-state boundary layer profile obtained from a separate RANS solution of the nozzle by Dudley and Ukeiley [17] is interpolated onto the desired grid inlet plane for prescribed inlet boundary conditions.

A computational unstructured grid system is employed to model the flow inside the wind tunnel including the cavity and open doors. The overall computational grid is comprised of about 21 × 106 cells, and most of the cells are inside the cavity and between and around the bay doors to capture all of the unsteady flow features of this problem. Three different slices through the computational grid system are shown in Fig. 5. The grid is wrapped around the corners of the cavity to maintain consistent clustering over all the cavity edges, and a boundary layer region is uniformly maintained over all the viscous surfaces of the present model, as shown in the figure, with a minimum wall spacing of y/δ = 4 × 10−4 leading to a y+< 2, where δ is the boundary layer thickness at the leading edge of the cavity.

Fig. 5
Fig. 5
Close modal

The computational simulations are conducted at similar conditions to the wind tunnel experiments at a freestream Mach number of 1.44, dynamic pressure of 74.4 kPa, and stagnation pressure of 172 kPa. The freestream velocity is 415 m/s, and the Reynolds number is approximately 1.29 × 106 based on the cavity length. The boundary layer at the leading edge of the cavity was measured to be approximately 3.8 mm thick. First, an initial quasi-steady-state solution is established to wash out any initial disturbances and pressure fluctuations due to the startup. After that, the time-accurate unsteady flow simulation is carried out as a restart from the quasi-steady-state solution. The data in the first 25 ms of the time-accurate simulation are basically ignored and used only to derive the unsteady physical flow nature of the problem and to absorb the startup disturbances in the simulation. The computations are advanced in time using a constant time step of Δt = 5 × 10−6 s, which is equivalent to nondimensional time step of 0.02723 based on the cavity length. The solution is allowed to develop for about 120 flow periods based on the lowest predicted Rossiter tone to allow the flow to become statistically stationary.

The simulations are conducted with the full Navier–Stokes CFD code loci/chem using the hybrid RANS/LES and low-dissipation models described above, and the k–ω turbulence model near the walls. The second-order accurate Roe scheme is used with 10 N subiterations and five Gauss–Seidel linear iterations. The pressures are monitored at the floor of the cavity at nine different locations along the centerline of the cavity and at the center point of the cavity aft wall, as shown in Fig. 6.

Fig. 6
Fig. 6
Close modal

The power spectral density of the unsteady pressure data was computed with a record length of 214 using 75% window overlap which resulted in a frequency resolution of about 12 Hz with a sampling frequency of 200 kHz corresponding to Strouhal number of 2.2 × 10−3 and 36.7, respectively. Ensemble averaging was used to compute the mean and root-mean-square (RMS) of the flow and turbulent quantities. Complete solution files were saved every 0.0005 s over the considered 120 flow periods for use in computing the averages. The sensors of the experiments were sampled simultaneously at 90 kHz with high pass filtering of 100 Hz and low pass filtering of 30 kHz. The spectra for the experimental data use a record length of 8192 points resulting in a frequency resolution of 11 Hz.

## Results and Discussion

The focus of this study is to better understand the effect of deployed doors and different door configurations on the unsteady pressure loading and acoustic tonal content. The results of this study are discussed next.

### Open Weapons Cavity.

First, grid independence and validation studies are conducted on the baseline open weapons cavity without doors to assess the numerical and grid parameters suitable for the problem. Three different grid levels are considered; coarse, medium, and fine grid resolution for the total of 5, 9.5, and 14.5 × 106 grid cells, respectively, for the open cavity problem without doors. It is particularly noteworthy that the three grid levels are developed using the same minimum grid spacing near the walls. Increasing the minimum spacing near the walls would violate the requirements of the LES and turbulence models and would consequently affect the results negatively in a way that is different than the objective of the grid resolution study. All the cases were run with the same time step and the same simulation time discussed in Sec. 4. Figure 7 shows the effect of grid resolution on the fluctuating RMS surface pressure on the cavity floor. The figure also shows the experimental data for this case. The RMS values are representative of the fluctuating pressure levels integrated over all the frequencies.

Fig. 7
Fig. 7
Close modal

The figure shows the high sensitivity of the results at the coarse grid level. The medium grid level is closer to the fine grid level except at the second half of the cavity where the flow separates from the cavity floor. It is believed that these results emphasize the importance of finer grid resolution near the trailing edge of the cavity where the fluctuating flow and turbulent quantities increase. The grid parameters of the fine grid level are used on the cavity and on the doors for all the cases considered in this study.

A limited time-step study was also conducted using a time step of Δt = 2.0 × 10−6 s and 1.0 × 10−6 s. The study showed insignificant changes to flow field parameters and therefore is not presented in this article. It is believed that the adopted time step of 5.0 × 10−6 s that was selected for numerical stability was low enough to provide sufficient time scales to capture the flow physics resolved by the fine grid used in this study.

The pressure spectrum at the center point of the aft wall is shown in Fig. 8. The spectrum is nondimensionalized using the dynamic pressure. The tones from the modified Rossiter equation are also shown in the figure. The original Rossiter equation [3] was modified by Heller and Bliss [1] by assuming that the cavity temperature is similar to the freestream stagnation temperature and is given by

Fig. 8
Fig. 8
Close modal
$St=fLU∞=m−αM∞1+γ−12M∞2+1k$
(14)

where m is the integer mode number, α is the phase delay constant, γ is the ratio of specific heats, and M is the freestream Mach number. The constant k represents the ratio of the convection speed to the freestream velocity. The default values of the constants k and α are 0.57 and 0.25, respectively, which are typical values for flow under M = 2 with a thin approaching boundary layer.

The spectra exhibit multiple peaks that are reasonably well estimated using the modified Rossiter equation. These tones are a function of the feedback mechanism that dominates the cavity flows as discussed in Ref. [18]. The cavity is second mode dominant as observed by Dudley et al. [16].

The nondimensional mean streamwise velocity contours are qualitatively compared to the experimental data of Dudley and Ukeiley [17] in Fig. 9. The flow penetrates deep into the cavity due to flow spillage over the cavity sidewalls. It is evident that the mean streamwise velocity in the shear layer and the flow penetration depth inside the cavity are in good agreement with the experimental data.

Fig. 9
Fig. 9
Close modal

Mean streamwise velocity profiles and mean streamwise turbulent velocities, normalized by U, at different streamwise sections along the cavity are shown in Fig. 10. The figure shows the progression of the shear layer along the cavity. The results are qualitatively comparable to the results published by Dudley and Ukeiley [17].

Fig. 10
Fig. 10
Close modal

### Fluctuating and Average Pressure Field on Deployed Doors.

The unsteady fluctuating surface pressures (in terms of RMS pressure nondimensionalized by freestream dynamic pressure Q) along the cavity floor centerline for multiple configurations are presented in Fig. 11. The results of the baseline cavity with no doors agree well with the experimental data. The figure also shows that the doors, and consequently the formed shocks, have caused larger fluctuations inside the cavity as expected. The 90 deg open-door case shows fluctuations that are consistently larger than the no doors case at all the points, and the difference in pressure fluctuation is largest around the footprint of the leading-edge shocks.

Fig. 11
Fig. 11
Close modal

The fluctuations on the cavity floor for the 45 deg open-door case, although still higher than the baseline cavity, are lower than those observed for the 90 deg open-door case. For the one-closed door (port side) case, the pressure fluctuations are higher at the upstream end of the cavity rather than at the downstream end of the cavity near the aft wall. This is consistent with the observation of Murray et al. [19].

The fluctuating surface pressures along the inboard-surface centerline of the open door (starboard side) for the one-door open case and for the 45 deg open and 90 deg open-door cases are presented in Fig. 12. The figure shows that the pressure fluctuations for the one-door open case are higher at the upstream end of the door than in both the 45 deg and 90 deg open-door cases. At the downstream end, the pressure fluctuations are higher than the 45 deg open-door case but lower than the 90 deg open-door case. The fluctuating surface pressures along the centerline of the port and starboard side doors for the 45 deg and 90 deg open-door cases are presented in Fig. 13. The figure shows a nearly symmetric distribution between the port and starboard doors for both opening positions, especially for the 45 deg open doors. The slight asymmetry shown for the 90 deg open doors at x/L of 0.05 and 0.75 is significantly reduced for the 45 deg open doors. The authors believe that the asymmetry at x/L of 0.05 for the 90 deg open doors is created due to a slight asymmetric shock wave created by the leading edges of the doors as discussed in Sec. 5.3.

Fig. 12
Fig. 12
Close modal
Fig. 13
Fig. 13
Close modal

The fluctuating and mean surface pressures over the cavity aft wall along the centerline are shown in Fig. 14. The fluctuating surface pressure for the baseline cavity compares well with the published data of Dudley and Ukeiley [17]. The figure shows that the leading-edge shock wave causes the pressure to be higher over the aft wall of the cavity at all the locations, and the fluctuating pressure is also higher for the open-door cases across most of the aft wall, except near the upper region of the aft wall.

Fig. 14
Fig. 14
Close modal

Additional wind tunnel experiments were later conducted on multiple door configurations to increase confidence on the prediction method for the deployed doors configurations. The unsteady fluctuating surface pressures on the cavity floor for the case of 45 deg open doors and for the case of one-open and one-closed doors, along with the no-door case, are shown in Fig. 15. The pressures for the new experimental cases were measured at x/L of 0.27, 0.73, and 1.0. The case of 45 deg open doors shows matching results at the two aft points of the cavity, but the results at the forward point were overpredicted. Reasonable agreement is shown in the figure for the case of single open door.

Fig. 15
Fig. 15
Close modal

Mean surface pressure contours, normalized by the dynamic pressure, on the cavity floor and port side wall are shown in Fig. 16. The symbols W, L, and D in the figure represent the width, length, and depth of the cavity, respectively. The overall pressure levels over all the cavity surfaces are increased because of the open doors as shown in the figure. The pressures on the cavity floor and side walls for the open-door case are higher for most of the cavity. For the no-door case, the mean surface pressure is only higher near the trailing-edge of the cavity.

Fig. 16
Fig. 16
Close modal

The mean surface pressures, normalized by the dynamic pressure, on the 90 deg open doors and over the middle section across the doors (y/D = 0.5) above the cavity surface are shown in Fig. 17 for the one-open and one-closed door case, the 45 deg open-door case, and 90 deg open-door case. The symbols w and l in the figure represent the width and length of the test section, respectively. The figure shows the formation of leading-edge shocks that trail downstream and reflect multiple times, including reflections from the wind tunnel side walls onto the door outer surfaces. The figure clearly shows stronger shocks and large expansions near the trailing edge of the doors for the 90 deg open-door case compared to the other cases, which facilitate the larger unsteadiness of the pressure signal for that particular case. A slight shock asymmetry is also observed near the leading edges of the doors for the 90 deg open-door case indicated by slightly smaller blue area near the starboard door.

Fig. 17
Fig. 17
Close modal

### Transient Velocity Field.

The nondimensional mean streamwise velocity contours for the case of cavity with no doors and the case of deployed bay doors are shown in Fig. 18. The results demonstrate that the open doors cause the flow to penetrate considerably deeper into the cavity after crossing the shock wave near the door leading edges. Furthermore, the results suggest that the compressed flow between the two doors, caused by the shock wave, forces the air to go into the cavity. The mean values of the turbulent eddy viscosity at the mid-depth inside the cavity (y/D = −0.5) are shown in Fig. 19, where trails of the shock wave can be clearly seen inside the cavity.

Fig. 18
Fig. 18
Close modal
Fig. 19
Fig. 19
Close modal

The nondimensional fluctuating streamwise velocity at the mid-depth inside the cavity (y/D = −0.5) is shown in Fig. 20, and the same plots across the middle sections of the doors are shown in Fig. 21. The figures show the increase in the unsteady fluctuations inside and outside the cavity for the deployed doors cases. The fluctuating components are significantly higher just downstream of the shocks, and the velocity fluctuations trail downstream affecting most of the area inside and above the cavity especially near the cavity aft wall. As the doors become fully open, the fluctuations concentrate near the doors with fewer disturbances near the center of the cavity. These fluctuations are obviously a concern for weapons separation from the cavity.

Fig. 20
Fig. 20
Close modal
Fig. 21
Fig. 21
Close modal

### Instantaneous Snapshots of Transient Flowfield.

Instantaneous snapshots every 2.0 ms of the transient flowfield in the cavity are shown in Fig. 22 for the 90 deg open-door case. The figure shows the nondimensional surface pressure, vorticity magnitude, and vorticity isosurfaces. The nondimensional vorticity isosurfaces are shown for a value of |ω| = 75, and they are colored and shaded by the nondimensional streamwise velocity. The snapshots show the highly unsteady pressure field across the cavity and door surfaces, and it is evident that the flow inside the cavity is dominated by the presence of complex unsteady vortical structures. The figures also show the large expansion region downstream of the cavity aft wall as well as the frequent impingement of the shear layer on the aft wall indicated by frequent red spots.

Fig. 22
Fig. 22
Close modal

### Power Spectra of the Pressure Signal.

The power spectra at the aft wall for the wind tunnel experimental data of some doors configurations are presented in Fig. 23, and the power spectra for the configurations at two points in the cavity floor are presented in Fig. 24. For reference, the Rossiter modes are included on these figures. The Rossiter modes are calculated with the same standard coefficients presented earlier. From these figures, one can observe how the frequency content contributed to the overall changes presented previously. The figures show that the baseline cavity has a dominant tone in the second resonant mode. However, for the cases with doors, we observe a shift in the dominant tone to the third resonant tone mode. Comparing the baseline case to the case with the doors fully open, there is also an overall shifting of the resonant frequencies.

Fig. 23
Fig. 23
Close modal
Fig. 24
Fig. 24
Close modal

One of the dominant features observed here is that the half-open door cases did not exhibit strong resonant features but the broadband levels were increased. However, there is evidence that the weak resonant behavior appears to be at consistent frequencies with the baseline cavity. The behavior with both of the one door closed cases exhibited strong peaks in the power spectra. The two different cases exhibited different peak frequencies implying that it is not only completely dictated by the geometric features of the closed cavity volume but also an interaction with the open side.

### Effect of Dynamically Moving Doors.

The substantial changes observed previously in the pressure signals and the resonant tones due to the presence of deployed doors showed large influence of the doors on the weapons bay cavity flow. The current research team felt it would be very interesting to characterize the point at which these changes occur during the dynamic opening and closing of the bay doors.

To enable the study of the effect of dynamically moving doors on the unsteady pressure signals inside the weapons bay, an overset computational model is developed consisting of three computational grids. Two computational grids are developed for the port and starboard doors separately, and the two door grids are overlapped with a background grid for the cavity and the wind tunnel internal region. The overset domain connectivity computations and data exchanges between the different grids in the model are performed automatically using the overset/chimera capabilities of loci/chem. The grid quality and minimum wall spacing of this simulation are consistent with the values considered before.

These dynamic simulations are conducted in two steps: (1) solving for a physical repeated solution inside the cavity with the doors fixed at a 5 deg open position to generate the initial condition for the dynamic case and (2) solving for the transient flowfield with dynamically moving doors. The doors are assumed to be moving with a speed of 1.5 Hz which is equivalent to a Strouhal number of 2.75 × 10−4, a representative figure for fighter aircraft. For example, weapons bay doors of Northrop Grumman's YF-23 fighter open in about 2.1 s. For a weapons bay door of 1 m at the same freestream speed, the Strouhal number would be about 2.87 × 10−4. The computational simulations of the cavity with deployed bay doors are conducted at the same flow conditions considered with the weapons cavity without doors. The flow conditions correspond to a freestream Mach number of 1.44, dynamic pressure of 74.4 kPa, and stagnation pressure of 172 kPa. The computations are run at a constant time step of Δt = 5 × 10−6 s.

Instantaneous snapshots of the nondimensional surface pressure on the doors and cavity for the dynamically moving doors are shown in Fig. 25. The figure includes snapshots at increments of 5 deg as the doors open from 5 deg to 35 deg position. Instantaneous snapshots of the nondimensional surface pressure inside the cavity at the same opening positions of the doors are shown in Fig. 26. The figures show that initially, when the doors are slightly open, the variations of the pressure on the doors and cavity aft wall are very small. Pressure variations start to develop as the doors open up and allow for a shear layer to develop and impinge on the aft wall. This observation is also supported by the fluctuating pressure distribution on the cavity floor and doors. The reader should notice that the pressure variations on the outside of the doors are actually small and that they are only visible due to the narrow range of pressure scale used in this figure.

Fig. 25
Fig. 25
Close modal
Fig. 26
Fig. 26
Close modal

The fluctuating (RMS) pressure on the cavity floor and doors is shown in Figs. 27 and 28, respectively. In these figures, the RMS is computed for all the recorded signals within 5 deg increments, which shows the transient variation of the pressure at different states of the doors. The figures clearly show slight pressure fluctuations on the cavity floor and doors as the doors open from 5 deg to 10 deg position. The pressure fluctuation slightly increases in the next 5 deg increment until the doors are about 20–25 deg open. The figures show strong pressure fluctuations on the cavity and doors occur when the doors move between the 25 and 30 deg segment. This indicates larger shear layer build up, strong impingement on the aft wall, and strong interaction with the upstream advancing acoustic waves.

Fig. 27
Fig. 27
Close modal
Fig. 28
Fig. 28
Close modal

## Conclusion

A hybrid RANS/LES method with low-dissipation schemes is developed and presented to improve flow and acoustics predictive capabilities for weapons bays and to characterize the effects of deployed doors and doors configuration on the unsteady flow nature inside and outside the weapons cavity. Computational simulations were conducted for a rectangular weapons cavity of length-to-depth ratio of 6 at Mach number of 1.5. Three different door configurations were considered; two open doors, two 45 deg open doors, and one fully open and one closed doors, and an additional case for dynamic opening of weapons bay doors was also investigated.

The results showed that the deployed doors increased the unsteady pressure loading on the cavity and that the unsteady shear layer travels deeper into the cavity causing higher pressure fluctuations at the floor and near the aft wall of the cavity. The results also showed high unsteady pressure fluctuations on the door surfaces and that the pressure loadings were nearly symmetric between the port and starboard doors. A slight asymmetry is observed near the leading edges of the doors for the 90 deg open-door configuration. The pressure fluctuations were also generally higher near the trailing edge of the cavity. The pressure fluctuations for the 90 deg open doors were higher than the 45 deg open-door configuration, and the pressure fluctuations for the one-door open case were higher near the upstream end of the door than both the 45 deg and 90 deg open-door cases. At the downstream end, the pressure fluctuations were higher than the 45 deg open-door case but lower than the 90 deg open-door case.

A shifting of the dominant tone of the cavity from a second resonant mode for the baseline case to a third resonant mode for the case with doors was also observed. In addition, an overall shifting of the resonant frequencies was observed due to the presence of doors. The half-open door cases did not exhibit strong resonant features, but the broadband levels were increased. However, there is evidence that the weak resonant behavior appears to be at consistent frequencies with the baseline cavity. The cases with one door closed exhibited strong peaks in the power spectra, but the peak frequencies were different implying that the tone is not only completely dictated by the geometric features of the closed cavity volume but also is affected by the interaction with the open side.

A study for the effect of dynamically moving doors was also conducted. The speed of the doors is selected to produce a time scale that is roughly equivalent to fighter aircraft. The results showed that strong pressure variations start to develop as the doors open up which allow for a shear layer to develop and impinge on the aft wall. A strong jump in pressure fluctuations was observed when the doors moved between the 25 and 30 deg open position, roughly indicating larger shear layer build up, strong impingement on the aft wall, and strong interaction with the upstream advancing acoustic waves. These initial results for the dynamically moving doors case show some interesting phenomena, but the results are not conclusive and further study is required to confirm the findings.

## Acknowledgment

The authors would like to thank James Grove and Rudy Johnson of the U.S. Air Force Research Laboratory (AFRL) for their support and useful discussions during this study. The authors would also like to thank Jonathan Dudley of Eglin Air Force Base for several helpful discussions during the course of this effort. The work presented in this paper was supported by the U.S. AFRL. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. Air Force (USAF).

## Nomenclature

• f =

frequency

•
• $fd$ =

damping function

•
• $k$ =

turbulent kinetic energy

•
• l, h, w =

length, height, and width of wind tunnel test section, mm

•
• L, D, W =

length, depth, and width of weapons cavity, mm

•
• LG =

local grid scale

•
• LT =

turbulent length scale

•
• q =

vector of primitive variables

•
• Q =

vector of conservative variables

•
• St =

Strouhal number (f L/U)

•
• u, v, w =

Cartesian velocity components

•
• U =

freestream velocity

•
• x, y, z =

Cartesian coordinates components

•
• δ =

length of boundary layer at the leading edge of weapons cavity

•
• Δt =

time step, s

•
• =

function to determine the resolved turbulent scales

•
• Ω =

mean flow vorticity

•
• $L′$ =

Jacobian of $ℓ$ with respect to q*

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