The interdisciplinary science that deals with the study of mechanical waves in gases, liquids, and solids is defined as Acoustics. These waves are oscillatory perturbations which move from a source and propagate through the medium of either gas, liquid or solid. The frequency of these oscillations ranges from 20 to 20,000 hertz, the audible range of the normal human ear. One of the many fields related to physical acoustics and deals with the aerodynamic sound generation is Aeroacoustics. The aerodynamic noise is generated by unsteady and irregular flows (turbulent flow) or aerodynamic forces interacting with surfaces as distinct from classical acoustics in which sound is generated by the vibration of solid bodies. Aeroacoustics is a part of many aspects of our modern day-to-day life. Vacuum cleaners, hair dryers, exhaust pipes, fans and ventilation systems are some examples of widely used machines which produce significant noise. Another aspect of aeroacoustics can be explored in its engineering applications, which is the basic topic of this thesis. The interaction between aerodynamically generated sound and flow field leads to flow oscillations. This self-sustained flow instability (oscillations) in the landing gear wells and bomb bay of an airplane leads to structural and aerodynamic problems. The noise generated by windows or a sunroof of an automobile are just a few aeroacoustics engineering problems.
During the 1940's and 1950's research in aeroacoustics became focused on aerospace applications with an increased need to control adverse unsteady flow effects in and around bomb bays and to a lesser extent to reduce vibration and noise radiation from landing gear wells. Results from this period were generally obtained from wind tunnel experiments and flight tests. The computational fluid dynamics approach was introduced in the research efforts of the 1970's. Due to the complexity of the physics of unsteady cavity flow much is still not understood. A thorough knowledge of the nature of unsteady flow is desirable to enable design of usable flow control techniques for reducing the adverse cavity flow effects.
The study of flow around cavities in the present thesis was primarily motivated by problems encountered in aerospace applications. In many aircrafts, weapons and landing gear require internal storage and mid-flight deployment. Once exposed to the freestream, unsteady flow around the internal storage cavity can generate high intensity acoustic tones within the cavity and produce vibration in the surrounding structures. The generation of these high intensity acoustic tones and the vibration of surrounding structures results in noise radiation, excessive heat transfer, structural fatigue, and interference with onboard avionic navigation and guidance systems . Structural damage is a particularly pertinent issue for the bomb bays of high-speed military aircraft. High subsonic flow over cavities typical of bomb bays can excite internal acoustic pressure levels of up to 170dB . These are sufficient to cause significant damage to missile fins and components of other systems prior to deployment . Open window or sunroof on a running car is a kind of cavity that generates noise and discomfort .
Cavities such as aircraft bomb bay, landing gear wells and the sunroof of an automobile can be modeled as simple rectangular cavities in both experiment and numerical simulations. Most computations reported in the literature were carried out for the flow around simple rectangular cavities. However, the typical geometrical configuration of cavities of aircraft and automobiles are far more complicated than that of the simple rectangular cavity. They are similar to the cavity with cover plates at the edges, or Helmholtz resonator shapes.
1.2 Cavity Noise Mechanism
The mechanism of noise generation by fluid flow in a cavity has been studied by numerous investigations in the past . This mechanism involves noise radiations that are induced by flow in the cavity. The cavity noise spectrum contains both broadband components, introduced by turbulence in the shear layer, and tonal components (narrowband components) due to feedback coupling between the flow field and the acoustic field.
1.2.1 Sources of Cavity Noise
Experimental investigations carried out by Rossiter , one of the first researchers to carry out research on a number of different rectangular cavities. He described the feedback mechanism based on shadow graphic observations. According to him, the periodic flow pattern in the cavity can be described by the following four-step procedure:
- The shedding of Vortices from the leading edge of the cavity is trailed downstream along the shear layer until they reach the trailing edge of the cavity.
- Near the trailing edge, interaction of the vortices with the downstream wall of the cavity leads to generation of acoustic waves. Part of these acoustic waves is radiated above the cavity into the acoustic far-field.
- The remaining part of the acoustic waves (or pressure waves) is radiated within the cavity in the upstream direction until they reach the leading edge of the cavity.
- Pressure waves that reach the upstream wall cause the shedding of a new vortex at the leading edge. Pressure waves determine the frequency of this feedback phenomenon by influencing the spacing between the different vortices.
In this way, vortices and acoustic disturbances form a feedback loop, which generates tonal noise. The total phenomenon is schematically shown in Figure 1.1.
1.3 Classification of Cavity
Cavities can be classified into different categories depending on the geometrical ratios of the length to depth (L/D) or length to width (L/W). Depending on the way the aeroacoustic noise is generated and radiated in the cavity, the different categories of cavities differ from each other.
1.3.1 Classification based on L/D
Generally a cavity with a length-to-depth ratio less than one (L/D < 1) is considered to be a deep cavity, while a cavity with L/D > 1, is classified as a shallow cavity. The reverse flow, or recirculation flow, inside a deep cavity is not as strong as in a shallow cavity. In contrast, Rossiter  made a distinction between shallow cavities (L/D > 0.4) and deep cavities (L/D < 0.4):
- In shallow cavities, there can be more than one recirculation zone and for longer cavities (L/D > 0.7); there can be a reattachment of the flow to the bottom of the cavity. The acoustic spectrum for this flow pattern is dominated by broadband noise and periodic components of relatively small amplitude.
- In deep cavities, there can be one or maximum two recirculation zones which force the above described feedback loop to dominate the broadband noise. The second- and third- cavity feedback resonances are the most dominant tones in the noise spectra.
1.3.2 Classification Based on L/W
Block  was the first researcher to classify cavity based on length-to-width ratio. The acoustic field is two-dimensional if L/W < 1 and three-dimensional if L/W > 1based on her experimental research. Ahuja and Mendoza  also confirmed these findings in their research. Changing the width of the cavity does not affect the resonance frequencies but the overall sound pressure level decreasing by as much as 15 dB for a three-dimensional cavity was also confirmed by them. Based on this research, it can be concluded that it is possible to compare 2-D computational aero acoustic (CAA) results with experimental results.
1.3.3 Classification Based on Cavity Flow Phenomenon
Much of the flow physics governing cavity behavior remains unclear. Cavity flows can be characterized in different ways. In terms of the freestream Mach number, cavity flow can be defined as either subsonic or supersonic. Based on the observed flow field patterns using flow visualization techniques, studying time-averaged static pressure distributions along cavity centerlines and depending on the length-to-depth ratio have lead to the identification of three main flow types  closed, open, and transitional. These flow types are essentially two dimensional, with three-dimensional effects superimposed upon their basic flows. The cavity L/D ratio, freestream Mach number and width-to-depth (W/D)  are a few of the parameters that strongly influences the type of cavity flow. All cases discussed in this study are transonic, three-dimensional, open cavity flows.
If the cavity ratio of L/D = 13 , then the flow phenomenon occurring in the cavity is known as Closed flow. Closed flow occurs in shallow cavities, such as missile bays on military fighter aircraft , and it is characterized by flow impingement on the cavity floor. This characteristic flow pattern is illustrated in Figure 1.2. The shear-layer separates from the leading edge of the cavity, impinges upon and attaches to the cavity floor and then separates from the cavity floor downstream to flow over the cavity trailing edge. Two enclosed regions of recirculating flow within the cavity aft of the leading wall and forward of the rear wall are created as a result of the shear-layer attachment and separation from the cavity floor. The reattachment of the shear layer near the rear wall forms a stagnation point near the trailing corner. The shear layer forms a streamline separating the recirculating flow from the freestream.
At supersonic speeds the flow regime is further characterized by expansion waves at the front and rear walls of the cavity and floor impingement and exits shock waves emanating from within the cavity as shown in Figure 1.3.
The flow phenomenon within a cavity with a length-to-depth ratio less than ten (L/D < 10)  is known as Open flow. Open flow occurs in deep cavities and is characterized by the shear layer bridging the cavity length separating the internal cavity flow from the freestream flow. The characteristic flow pattern is illustrated in Figure 1.4. The shear layer separates from the leading edge of the cavity, bridges the length of the cavity and reattaches aft of the cavity trailing edge. The shear layer forms a dividing streamline separating internal and external cavity flow and terminating in a stagnation point on or aft of the cavity trailing edge. High pressure ahead of the cavity trailing wall and low pressure aft of the cavity leading wall causes backflow within the cavity and subsequently an internal circulating cavity flow, the exact configuration of which is dependent on the cavity depth as a function of cavity length and width.
At supersonic conditions backflow within the cavity causes a weak shock wave to generate at the leading edge in order to adapt the freestream flow to the high pressure over the mouth of the cavity. A shock wave is also generated at the rear edge of the cavity to readapt the shear layer to the freestream flow as shown in Figure 1.5.
If a cavity has a length-to-depth ratio within the range of 10 < L/D < 13 , then the flow phenomenon occurring in the cavity is known as Transitional flow. These flow features demonstrate a transition between open and closed characteristics. The exact boundaries between open, transitional, and closed cavity flow are uncertain and depend on several parameters. Transitional flow features differ for subsonic and supersonic flow. For subsonic flow, as the length-to-depth ratio is incrementally increased or decreased, the transition is gradual and smooth. For supersonic flow however, the transition is divided into two distinct stages and the flow changes between these two stages are abrupt. Variations in the configuration of cavity floor impingement and separation shocks have lead to the identification by Stallings and Wilcox  of two distinct stages of supersonic transitional flow, namely transitional-open and transitional-closed flow.
- Closed cavity flow
- Transitional-closed cavity flow
- Transitional-open cavity flow
- Open cavity flow
1.3.4 Classification Based on Cavity Oscillation
The instability of the shear layer on impingement on the rear edge of the cavity is the main source of oscillation in the cavity and results in a complex flow phenomenon. The freestream flow conditions, shear-layer fluid properties and internal cavity flow field are the main parameters that affect this phenomenon. Rockwell and Naudascher  studied the flow fields for self sustaining oscillatory cavity and divided them into three classes based on observable flow features. The three classes of oscillating cavity flow are described below and are displayed in Figure 1.7:
- Fluid-dynamic oscillations are defined as those which arise from instabilities within the shear flow.
- Fluid-resonant oscillations are those which arise from shear flow oscillations coupled with standing waves within the cavity. Standing waves are created at acoustic resonant modes based on any of the cavity dimensions; length, width or depth.
- Fluid-elastic oscillations are those which arise from shear-layer oscillations coupled with elastic cavity wall motion.
Fluid-dynamic and fluid-resonant oscillations are particularly pertinent to aerospace applications and occur mainly in low and high speed cavity flows respectively. The present study is concerned with fluid-resonant oscillatory cavity flows which are particularly relevant to high speed open cavity flow applications.
The necessary condition for self sustaining oscillatory flow is the presence of both a generating mechanism and feedback mechanism. Therefore, a thorough knowledge of the mechanisms involved in oscillation generation and feedback control of the unsteady flow features may be attained by modification of relevant flow parameters.
Fluid-resonant oscillations occur when standing waves are allowed to form within a cavity [Figure 1.8]. Ideally a cavity dimension should be of the same order as, or greater than, the acoustic wavelengths (?a). Standing waves naturally form the longest dimension. Therefore, longitudinal standing waves are generated for L/D >> 1, thus causing the cavity to resonate in a longitudinal mode. High values of W/D ratio cause lateral standing waves to form, generating cavity resonance in a lateral mode. Low values of L/D cause vertical standing waves to form, generating cavity resonance in a depth mode.
1.4 Cavity Flow Properties
The cavity flow physics and its resonance attribute from numerous experimental observations largely depend on several flow parameters:
- ReT (Reynolds number based on boundary-layer momentum thickness)
- L/d (d is the turbulent inflow boundary-layer thickness)
These parameters are believed to govern the cavity flow instability and its coupling to resonant frequencies and amplitudes.
1.4.1 Mach Number Effect
Plumblee et al  proved the existence of a threshold Mach number under which acoustic tones will not be activated. Increasing the Mach number will favor the transition from shear-layer mode to wake mode. Under shear-layer mode, as the Mach number increases, the dominant frequency of the pressure fluctuations is observed to jump from one value to another. In the wake mode, Colonius et al  analyzed their DNS data and revealed that the fundamental frequency (or Strouhal number) is almost independent of the Mach number variation from M=0.4 to M=0.8. Gates et al  also noted in their experiment that the modal amplitude does not vary with the Mach number.
1.4.2 Boundary Layer Thickness (L/d) Effect of Incoming Flow
Boundary layer thickness of the incoming flow has a significant influence on cavity flow characteristics. Boundary layer thickness in cavity flow is often studied in the form of L/d or D/d. Early observations by Sarohia and Massier , Rockwell and Naudascher  indicated a critical L/d value exists for transition from the steady shear layer (no oscillation) to unsteady shear layer (oscillation). Beyond this critical value, the dominant frequency also experiences a jumping growth as L/d increases, which is similar to the case of Mach number variation in the shear-layer mode.
1.5 Pressure Spectra of Open and Closed Cavities
The discipline of aeroacoustics, which is a study of pressure waves in fluids, is intimately related to fluid dynamics. Many sounds that are technologically important in industrial applications are generated by and propagated in fluid flows. The phenomena associated with sounds can therefore be understood and analyzed in the general framework of fluid dynamics. The governing equations for acoustics are indeed the same as those that govern fluid flows.
The main challenge in numerically predicting sound waves stems from the well-recognized fact that sounds have much lower energy than fluid flows, typically by several orders of magnitude. This poses a great challenge to the computation of sounds in terms of difficulty of numerically resolving sound waves, especially when one is interested in predicting sound propagation to the far field. Another challenge comes from the difficulty of predicting the very flow phenomena (e.g., turbulence) in the near field that are responsible for generating sounds.
The unsteady cavity flow field consists of a combination of random and periodic pressure fluctuations. Magnitudes of each component vary with flow type. Closed cavity flows exhibits mostly random pressure fluctuations and does not normally exhibit the unsteady oscillatory flow types. Open cavity flows are dominated by strongly periodic pressure fluctuations with a less significant random component. Figure 1.9, obtained from research carried out by Tracy and Plentovich , illustrates the typical flow sound pressure level (SPL) spectra for closed and open cavity flows. The dominant pressure fluctuation features can be clearly seen.
Transitional and open cavity flows commonly experience periodic pressure fluctuations which cause acoustic cavity tones, as illustrated in Figure 1.9(b). Cavity flow experiences resonant phenomenon once tonal peaks attain a magnitude twice that of background pressure levels. It is this self sustaining oscillatory cavity flow that produces the undesirable high-intensity acoustic tones.
1.6 Rossiter Modes
The typical acoustic spectrum of an open cavity flow is shown in Figure 1.10. The presence of frequency peaks in the graph provides evidence of the existence of acoustic modes within the cavity. The magnitude of each frequency peak is recorded as a sound pressure level; this is the normal measurement for acoustic analysis.
Rossiter  derived the semi-empirical equation, equation (1), which is used to predict the acoustic mode frequencies within a low-speed open cavity flow. The speed of sound in the cavity is assumed to be equal to the speed of sound in the freestream flow.
Heller et al.  extended Rossiter's formula to account for the difference between the freestream Mach number and the cavity Mach number, extending the effectiveness of the formula into the high subsonic, transonic, and supersonic regimes. The modified Rossiter formula is given in equation (2). It was found that the speed of sound in the cavity is equal to the stagnation speed of sound in the freestream flow.
1.7 Methods of Simulation for Cavity Flow
Aeroacoustics field is still new compared to other aeronautical fields; innovative techniques of acoustic prediction are constantly being developed. The three main techniques currently used are computational aeroacoustics (CAA), computational fluid dynamics with an acoustic analogy (CFD-FW-H), and experimental methods. The CAA approach relies on the computational simulation of an entire flow field including the far field; hence it requires very large domains and has a high computational expense. The experimental approach may be the most accurate of the three, but is also the most expensive and hence not viable to use for all aero acoustic testing. The expense here comes from not only having to run a wind tunnel and fabricating the model, but also from using highly sensitive audio recording equipment.
In contrast to the other two approaches, the CFD with acoustic analogy or "hybrid" approach is relatively cheap, both computationally and financially. The main difficulty with hybrid approach is obtaining the required accuracy in the simulation results. Since sound propagation is directly resolved in CAA or direct method, one normally needs to solve the compressible form of the governing equations (e.g., compressible Reynolds-Averaged Navier-Stokes [RANS] equations, compressible form of filtered equations for Large-eddy simulation [LES] or Detached-eddy simulation [DES]. The direct method becomes feasible when receivers are in the near field (e.g., cavity noise). In the present simulations, the CAA or the direct method is used to simulate the 3-D open cavity with and without cover plates. The compressible form of filtered equations for DES based on the Spalart-Allmaras [S-A] turbulence model is used to simulate the entire computational domain and the data at the receivers in the near field is extracted.
1.8 Fast Fourier Transform
To study the frequency content of the pressure spectra recorded on the cavity floor, it is necessary to convert the data from the time domain to the frequency domain. Following the standard approach to study the cavity frequencies, this is accomplished using a fast Fourier transform (FFT). The number of data samples used in performing the FFT must be a power of 2, which determines the frequency resolution of the result. In order to reduce the broadband noise level of the results, several blocks of points were used for each FFT, and the results were averaged together to produce the final spectrum. Due to the non periodicity of the pressure history data the blocks uses a sliding window approach, where each set of points for the FFT overlapped the previous points by a certain amount.
1.9 Aim and Objectives
The purpose of the present analysis was to study the flow phenomenon of the 3-D (L/W > 1) shallow (L/D > 1) open cavity (Rossiter mode) for the test case of QinetiQ  with and without cover plates. This was achieved by simulating the flow phenomena using a numerical method.
The aim here was to attain an accurate aeroacoustic analysis and to understand the flow phenomenon of an open cavity with cover plates.
The objectives were as follows:
- To understand the current state of the art of aeroacoustic analysis
- To determine the efficiency of DES variants i.e., S-A ,k-omega shear stress transport [SST], realizable k-epsilon turbulence models for use in aeroacoustics applications
- To determine the most efficient mesh size for aeroacoustic simulations
- To compare the aeroacoustic data and the flow phenomenon of the open cavity with and without cover plates
- To compare the aeroacoustic data of all the DES variants for the cavity with cover plates.
The following literary survey gathers information on the problem of aircraft noise pollution, the causes of it and the current methods of predicting it both experimentally and computationally. The aero acoustic spectrum generated by flows in cavities can be divided into low frequency noise and broadband noise; this literature review includes both the low frequency part and the broadband noise of the spectrum.
The experimental method used to detect acoustic signals is a very expensive technique of using anechoic chambers with wind tunnel outlets built in. The normal set up involves the test piece being mounted on a balance inside the anechoic chamber and an open jet wind tunnel used to produce the flow. Various techniques have been used to determine the acoustic signals in the experimental configuration. The type of approach used depends on the aim of the experiment. The expense of the experimental technique for determination of the acoustic source further justifies the need for increasing the sophistication of the computational approaches.
The cavity flow interaction has been one of the benchmark experiments for acoustic analogies due to the simple geometry and commonality of the test. Different computational approaches have attempted to emulate the results of the benchmark experiment carried out by QinetiQ  on M219 cavity configuration.
The most distinguishing visual features of cavity flow are the waves outside the cavity that are generated by the large-scale structures in the cavity shear layer. These structures grow very rapidly due to the resonance in the cavity as a result of the flow acoustic coupling. Krishnamurthy  was perhaps the first to provide clear visual evidence of such waves where he observed a series of waves being swept downstream by the freestream flow.
Rossiter  based on his experiments subsequently proposed a model of acoustic environment in shallow cavity. His idea was that there is one primary source of acoustic waves which is near the trailing edge of cavity. Rossiter made an assumption that vortex shedding process in cavity has the same frequency as acoustic waves. Further Rossiter obtained the semi-empirical formula to predict the measured resonant as shown by Equation 1.
Later, the modified Rossiter equation derived by Heller et al.  as given in Equation 2 allows for the difference in the speed of sound within the cavity and in the free stream flow. Heller et al.  interpreted it as a phase-locked criterion induced by the aero acoustic feedback loop between vortices shed at upstream corner (leading edge) and the acoustic disturbances generated by vortex-trailing edge interactions. The most important in this formula is that predicted frequencies do not correspond to acoustic proper modes of the cavity and generally are not multiplies of a fundamental frequency. In this form, Rossiter equation is used to predict noise's frequencies for high subsonic Mach numbers, usually greater than 0.4. There are some other equations which have been reviewed by Oshkai et al.  for frequency predictions, however, Rossiter equation is more flexible and provides results with higher accuracy. More accurate formulation was later intoduced by Bilanin and Covert .
Heller & Delfs  observed waves outside a 3D cavity using Schlieren and classified the waves into four categories. Zhang et al.  observed shock waves generated by large-scale structures in the shear layer of a 2D cavity and determined that the angles of these shock waves correspond to a Mach angle. Bauer & Dix  conducted detailed surface pressure measurements of supersonic cavity flows over a range of Mach number from 0.6 to 5.04.
Cattafesta et al.  and Kegerise et al.  obtained fluctuating surface pressure measurements for low to high-subsonic cavity flows with the goal of implementing active closed-loop control. Similarly, Ukeiley et al.  have examined the unsteady wall-pressures to better understand the cavity dynamics and subsequently to control its behavior. These studies confirm the presence of high dynamic pressure loads inside the cavity where the fluctuating pressure spectra is dominated by discrete frequency, or cavity tones. Furthermore, the results show that the frequencies of these modes are fairly accurately predicted by the well-known Rossiter's correlation.
Unalmis et al.  measured the dynamic pressures in a high Mach number cavity flow, and visualized the cavity shear structures using a double-pulsed planar laser scattering (PLS) technique. In addition, they obtained limited PIV measurements for only a small portion of the flow field. For both the PLS and PIV measurements, images were only obtained outside the cavity. Velocity field inside the cavity were not measured. One of the main findings of their work was the lack of correlation between the cavity acoustics and the dominant structures in the shear layer.
Murray & Elliott  also examined the behavior of the cavity shear layers for supersonic flows using Schlieren and double-pulsed PLS. They found that similar to compressible free shear layers, the structures become increasingly three-dimensional and smaller in extent with increasing Mach number. In a carefully conducted experimental study, Forestier et al.  examined a Mach 0.8, deep cavity (L/D = 0.42) using unsteady pressures, high-speed Schlieren photography.
The fact that main source of disturbances is at the trailing edge of cavity was shown by a number of researchers. Chung in his work  showed that pressure fluctuations have the largest amplitude just downstream of rear wall for subsonic Mach numbers. Heller et al.  mentioned that rear wall behaves like a "pseudopiston" and generate pressure waves.
A comprehensive study of shallow cavity flow was reported by Ahuja and Mendoza . They confirmed that Rossiter equation can predict oscillation frequencies with at most 20% accuracy over all modes for Mach numbers from 0.26 to 1. Grace et al.  and Rockwell and Naudascher  made a survey of different studies of flow over shallow cavity which includes classifications of cavity flows, observed phenomena, prediction method and suppression techniques. Alternatively, it could be concluded that these parameter ranges for resonance only hold at the specific Mach numbers, whereas Ahuja and Mendoza  used Mach numbers larger than 0.25.
Intuitively, one expects that the first mode to be dominant in the flow. However, for a flow over shallow cavities it is not so. Ahuja and Mendoza  and Kegerise et al.  observed very carefully their respective frequency spectrums. The second and third mode cavity feedback resonance are typically the most dominant tones in the spectra and not necessarily first were observed. But their studies were not focused on explanation of this fact.
Oshkai et al.  proposed an idea that whether higher or lower frequency is dominant depends on which large or small scale mode is dominant in shear layer. If the cavity is sufficiently long, full development of large-scale instabilities of the shear layer flow along the cavity is allowed. And, it was shown by Rockwell et al. , that such kind of oscillations is irrespective of the thickness of the incoming boundary layer. For short cavities, the absolute frequency is relatively high, and the momentum thickness of the separating boundary layer is the appropriate length scale for characterizing the initial instability, presumably associated with small-scale vortex formation.
Alvi et al.  revealed that the presence of large highly convoluted, turbulent structures within the shear layer is responsible for significant mixing enhancement in the flow. Moreover, they suggested that global instability rather than convective instability is dominant in the structures. The occurrence of multiple discrete tones arises a question: whether these tones coexist at all time or they occur at different moments. If the later is true, a mode switching phenomenon will occur. Mode switching phenomenon is important from the flow control point of view. If a specific cavity flow has no mode switching, the flow is considered statistical stationary and the control of such a flow is simple. If there is mode switching in the flow, the control is expected to be more challenging, especially for the real-time feedback control, since it is possible that a control mechanism will suppress one mode at the expense of enhancing other modes.
Karamcheti  appears to be the first to notice the mode switching phenomenon. He wrote that at one moment when component A was present, a note could be heard which momentarily disappeared at another moment when component B was present. However, the quantitative and experimental evidence didn't appear in the literature until much later.
Cattafesta, and Kegerise et al. [26, 27] from the results of short-time Fourier transform and wavelet transform found the same phenomenon called "mode-switching" reported by Alvi et al . Mode- switching refers to a process whereby the dominant energy shifts temporally from one Rossiter mode to another. It is well known that a number of vortical structures spanning the length of cavity corresponds to the Rossiter mode number. From schlieren flow visualization, authors concluded that the shift in the dominant Rossiter mode as indicated in the wavelet and STFT analysis is reflected in cavity shear layer structure as a shift in the number of vortical structure spanning the length of cavity.
Delprat  approached the resulting aero acoustic spectrum of Rossiter from a different angle, using signal processing theory to provide additional insight into the cavity flow physics. Delprat  reformulated the Rossiter formula as a two-step amplitude modulation process in terms of what she termed the "fundamental aero acoustic loop frequency" and a very low frequency modulating signal. Rowley and Williams  explain the "peak splitting" phenomenon, where two spectral peaks form near the original peak. It is important to note that these split peaks are not necessarily at any of the cavity modes or at the forcing frequency, and they are not at the frequency of the unforced dominant tone, but nearby. Thus, the physical explanation behind these split peaks must involve an additional physical mechanism.
The main source of aero acoustic phenomenon in an open cavity is due to the shear layer instabilities in the cavity. Thus, proper modeling of the shear layer and its instabilities using the appropriate turbulence model becomes important in order to achieve reliable flow predictions. Several CFD analyses for turbulent flow over an open cavity [38-40]were performed using RANS equations because of its computational affordability. However, these computations failed to accurately capture the unsteady flow in large separated flow regions as encountered in the cavity.
Some investigations which used LES [40, 41] have shown to be computationally expensive for resolving boundary layer turbulence at high Reynolds numbers. In more recent studies with the DES [38-41] formulated by Shur et al  massive separation at realistic Reynolds number are simulated. The hybrid model captures the turbulence phenomenon by adopting LES model in the separated flow regions and RANS model in attached boundary layer regions.
When the modeling of turbulent cavity flows is coupled with aero acoustic analysis, traditional RANS methods were used with some success in modeling the narrowband Rossiter mode components of the acoustic spectra but unable to predict the broadband contribution. However, both LES and the hybrid RANS-LES approach of DES are sufficient to predict cavity acoustics of both narrowband and broadband types . However, DES is preferred over LES because of its computational affordability.
Turbulent flows over open cavities have been extensively studied by means of theoretical and numerical analysis. CFD-based numerical studies of flow over open cavities have been carried out for both laminar and turbulent flows under different flow conditions and various geometric configurations. There have been a number of investigations [38-41] on the configuration.
Larcheveque  numerically simulated M219 cavity using LES model with 2nd order spatial scheme, a 2nd order implicit temporal scheme and a mixed scale turbulence model on a grid of 3.2x106 cell. In the acoustic spectra, he predicted 3rd mode to be dominant while the 2nd mode was dominant in the experimental data. Only the 4th mode frequency was predicted within the 5% error limit.
Mendonca et al. [43, 44] also simulated the M219 cavity using a 2nd order mixed upwind and central spatial scheme and k-e turbulent DES model on a 1.1x106 hexahedral cell grid. The typical near wall y+ value was about 300. Therefore, the near wall physics was not captured. No quantitative comparisons of mode frequency and amplitude were given except the power spectrum densities. Mendonca  also provided a comparison between a coarse grid (1.1x106 cells) simulation and a fine grid (2.8x106 cells) simulation. In , a band limited RMS pressure calculation was introduced through a FFT technique.
Ashworth  used a Fluent DES model to simulate the M219 cavity with a 2nd order spatial scheme, a 2nd order implicit temporal scheme and a Spalart-Allmaras (S-A) turbulent model on a 1.68x106 cell grid. Comparisons between the URANS and the DES predictions showed that the 2nd mode was missing from the URANS prediction but was captured by the DES model.
Peng  simulated an open cavity flow using DES and unsteady RANS modeling approach based on the Spalart-Allmaras (S-A) model. Peng highlighted the importance of grid resolution particularly in DES modeling. Nayyar, Barakos and Badcock  numerically simulated the flow inside the cavity using LES and DES at M=0.85.They found that both DES and LES fare much better than URANS in resolving the higher frequencies and velocity distributions inside the cavity.
Li and Hamed  studied the effect of sidewall boundary conditions on the computed flow induced pressure fluctuations in a transonic open cavity using DES. The slip wall boundary conditions increased the resolved eddies below the fluctuating shear layer spanning the cavity, and resulted in greater span-wise variation in the unsteady flow field.
Shieh and Morris  have developed a numerical method for computational acoustics and have recently applied it to flow past a cavity. Their method embeds into a parallel RANS simulation the one-equation Spalart-Allamaras turbulence model  and the detached eddy simulation (DES) method of Spallart et al.. They use the DRP (Dispersion-relation preserving) scheme of Tam and Web  for spatial discretization and explicit fourth order Runge Kutta integration in time. They solved for two-dimensional flow past two different geometries. One shows the characteristics of oscillations due to a shear-layer mode and the other operates in the wake mode.
Baysal  has been active in the computation of transonic flows past cavities. He usually solves the RANS equations using a finite volume approach with fourth order damping. The key to his computational successes seems to be his turbulence model which is a modified Baldwin-Lomax model. The modifications account for vortex-boundary layer interaction and separation, multiple walls, and turbulent effects.
Tam et al.  use a finite volume solution of the double thin-layer Navier Stokes equations to simulate supersonic flow past a cavity. They compare results obtained using Baldwin-Lomax models previously implemented by other researchers. Fuglsang and Cain  offer yet another approach to the modeling of turbulence for the cavity geometry. They used the Baldwin-Lomax turbulence model upstream but then apply DNS in the cavity region. They focused their simulation on assessing the effect of forcing the shear layer.
2.3 Current Research
Most previous computations were carried out for shapes of simple rectangular cavities. However, the geometrical configuration of aircraft landing gear, weapon bays, automobile sunroof, windows etc. are more complicated than that of the simple rectangular cavity. They are similar to a cavity with cover plates at the edges, or Helmholtz resonator shapes. Some research on the effect of cover plates on the 2-D rectangular cavity was carried out by Heo and Lee . They numerically investigated the effects of length variation, thickness variation and on opening position variation of cover plates on the open cavity at very low subsonic Mach numbers.
Nishimura et al  examined the noise generation mechanism in an open cavity with cover plates and studied the methods to reduce the noise by experiments and numerical simulations at low subsonic Mach numbers. Massenzio et al  discussed the phenomena of flow-acoustic coupling between the unstable shear layer across a slot covering a cavity i.e., Helmholtz resonator. According to their study, the main mechanisms associated with the onset of resonances were shear tone resonance and the coupled tone "shear layer-cavity" resonance, the latter being responsible for the higher sound pressure levels. The theoretical model was developed and experimental measurements at low subsonic Mach numbers were carried out.
Bartel and Mc Avoy  conducted experiments on different combinations of missile bay configurations and Cruise Missile Carrier Aircraft (CMCA). The configurations were similar to the cavity with cover plates and the Mach number in this experiment ranged from subsonic to transonic regimes. However, sufficient amount of numerical investigation on the 3-D cavity with cover plates which has more complicated flow phenomenon have not been conducted. Therefore, the current study attempts to address the additional 3-D behavior of the flow.
2.4 Literature Conclusion
The present investigation is aimed at exploring the flow phenomena of the cavity with cover plates using the DES based one equation S-A model and comparing it with the cavity without cover plates. The three DES variants; S-A, K-Omega SST, realizable K-Epsilon are also simulated for the case of cavity with cover plates and the aero acoustic data are compared. The CFD calculations are performed on the M219 cavity at Mach 0.85. The cavity has a W/D ratio of 1 and L/D ratio of 5 exhibiting flow in shear layer mode. Two cavity configurations are assessed. The first is cavity without cover plates and the second is cavity with cover plates. The results are evaluated through comparisons of computed SPL spectra with existing experimental results and with the semi-emperical formula of the Rossiter for both the configurations.
Model and Grid Generation
3.1 Description of the models
The computational domain is displayed in Figure 3.1. The cavity has a size of 5D×D× D (length× width× depth, D=0.1016m). The inflow sections extend 7D upstream in consistent with the leading edge setting in experiment. And the outflow boundary is 5D behind the cavity to reduce disturbance from the downstream boundary. The pressure farfield boundary condition is applied at the inflow, outflow and upper boundaries in this case. Therefore, a farfield boundary is applied and set 10D above the cavity mouth to avoid any acoustic wave reflection back into the cavity flow field. The inflow/outflow floor and the cavity floor are set as viscous adiabatic walls. The sidewall boundary conditions are set to symmetric boundary. For inflow boundary, the simulation was performed with free stream conditions of M8 = 0.85, P8 = 62940 Pa, T8 =270.25 K and the Re (Reynolds number) based on the cavity length is about 8 million.
The current research activity involves the investigation of two different geometries of the open cavity. The first is same as the test case used by QinetiQ  in their experiment as shown in Figure 3.2 and the second is similar to QinetiQ  but with cover plates over the cavity. The test case is an open cavity contained in a flat plate of length 1.7272 m and width 0.3048 m. The rectangular cavity has dimensions of L=0.508 m in length, D=0.1016 m in depth and W=0.1016 m in width, giving a ratio of L : D : W = 5 : 1 : 1. The leading edge of the cavity was located 0.7112 m downstream from the plate leading edge and the cavity center-line was off-set 0.0254 m from plate center line. The length of the opening on the cavity in the second test case is 0.1524 m.
The two test case models used in this numerical simulation are shown in Figure 3.3. The first test case (Figure 3.3 (a)) is considered in order to examine the overall flow structure, pressure variation, rms pressures and SPL spectra using second order finite volume schemes available in the computer code Fluent™  and comparing it with the experimental data of QinetiQ  for M219 cavity. The second test case (Figure 3.3 (b)) is considered in order to examine the flow field and the effect of the cover plates present on the cavity on the pressure histories and on the SPL spectra and comparing the data with the first test case.
Kulite pressure measurements were made at point's k20, k24, k25, k29 respectively on the cavity ceiling as shown in Figure 3.2. Pressure histories from these points are used to compare with the experimental data of QinetiQ.
3.2 Grid generation
The geometry and structured mesh are generated using the preprocessor, Gambit , supplied in the Fluent software suite. A 3-D computational mesh of 3.54 million hexahedral elements for the first test case and 3.19 million hexahedral elements for the second test case are generated. Certain requirements were required from the mesh created for the cavity flow simulation. The main requirement was to allow easy convergence of the CFD code to supply a relatively quick solution. The second requirement of the grid was to try and ensure the stability of the CFD solution. From experience, it was known that a structured grid is the most efficient type for any solution with relatively simple geometry. It was also known that the quality of each cell was important for solution stability. With quality, the main factor to consider was equiskew of each cell. In the present analysis, the equi-skew was less than 0.25 anywhere in the grid. There are 0.84x106 cells (200x70x60) within the cavity and 2.7x106 cells (270x100x100) above the cavity in the first test case and 0.84x106 cells (200x70x60) within the cavity and 2.35x106 cells (235x100x100) above the cavity in the second test case. The grid was highly clustered towards the plate surface in order to resolve the viscous boundary layer. At the wall, the common wall-function grid has typically resulted in y+ < 2, which is sufficient to resolve the viscosity-affected near-wall region. A schematic of 3-D computational domain and some 2-D grid slices are illustrated in Figures 3.4 and 3.5.
3.3 Grid-Independency Tests
Grid-independence studies are important in all areas of computational fluid dynamics and especially so in direct simulations of sound. The resolution has to be sufficiently fine to resolve all important physical parameters, and the boundary conditions must not affect the solution significantly. The latter is critical in CAA because sound waves propagate over large distances. Although a boundary might be far away from the region of interest, any reflected sound waves will contaminate the solution in the entire domain. The real test is the sound at the observer locations because the boundary conditions can reflect or generate sound that would contaminate the sound at those (or any other) positions. In the present case, the grid independency test was performed on a 2-D mesh grid and implemented the same amount of the grid on the 3-D case in the x and y direction.
3.3.1 Test case 1
To perform the grid-independency study for test case 1, two different grids, grid 1 and grid 2 were generated. Table 3-1 lists detailed information for those grids. Grid 1 is not same as Grid 2 with grid clustering in the vicinity of wall of the cavity. Figure 3.6 illustrates an overall view of Grid 1 and grid 2 in the vicinity of the wall.
4.1 Compressible Navier-Stokes equation
Fluent  is a commercial CFD software that solves the governing flow equations (Equation 4) using a cell-centered control-volume space discretization method. In the present work, the compressible flow equations are solved using an implicit segregated solver and a second-order upwind scheme. The system of equations that govern the transonic turbulent cavity flows are the compressible Navier-Stokes equations. The conservation of mass, momentum and energy equations can be written for a control volume V enclosed by surface area A in the following format:
4.2 Turbulence Model
The fluctuating velocity fields are characteristics of turbulent flows. This fluctuation mixes with the transported quantities such as momentum, energy, and species concentration, causing them to fluctuate as well. Since these fluctuations contain small scale energy and high frequency spectrum, they are computationally expensive to simulate directly in practical applications. In order to overcome this problem, the exact governing equations are time-averaged or manipulated in order to remove the small scales, these results in a modified set of equations that are computationally less expensive to solve. These modified equations contain additional unknown variables, and thus the necessity of the turbulence models are required to determine these variables in terms of the known quantities.
FLUENT provides several options for turbulence models. In the present simulation, the three variants of detached eddy simulation (DES) model in FLUENT based on a modified version of the Spalart-Allmaras (S-A), K-? SST, Realizable K-e models are considered as a more practical alternative to LES for predicting the flow around high-Reynolds-number. The DES approach combines an unsteady RANS version of the above three variants with a filtered version of the same model to create two separate regions inside the flow domain: one that is LES-based and another that is close to the wall where the modeling is dominated by the RANS-based approach.
All variants of DES model behave in such a way that the source terms dominate the transport equations for the turbulence variables in regions where the flow becomes detached and the turbulence is in equilibrium.
1. DES model based on Spalart-Allmaras (S-A):
The S-A model uses the Boussinesq approach to relate Reynolds stresses to the mean velocity gradient of the flow. In this turbulence model, only one additional transport equation is solved. The model proposed by Spalart and Allmaras  solves a transport equation for a quantity that is a modified form of the turbulent kinematic viscosity. In the S-A model the working variable is used to form the eddy viscosity and its transport equation takes the form,
2. DES model based on K-? SST:
The SST model  differs from the standard model as follows:
- There is gradual change from the standard k-? model in the inner region of the boundary layer to a high-Reynolds-number version of the k-e model in the outer part of the boundary layer
- The modified turbulent viscosity formulation is used to account for the transport effects of the principal turbulent shear stress
FLUENT in addition to the standard k-? model, also provides a variation called the shear-stress transport (SST) k- ? model, so named because the definition of the turbulent viscosity is modified to account for the transport of the principal turbulent shear stress. It is this feature that gives the SST k- ? model an advantage in terms of performance over both the standard k- ? model and the standard k-e model. Other modifications include the addition of a cross-diffusion term in the ? equation and a blending function to ensure that the model equations behave appropriately in both the near-wall and far-field zones.
The shear-stress transport (SST) k-? model was developed by Menter  to effectively blend the robust and accurate formulation of the k-? model in the near-wall region with the free-stream independence of the k-e model in the far field. To achieve this, the k-e model is converted into a k-? formulation. According to Travin , the modifications necessary for the dissipation term of the turbulent kinetic energy as described in Menter's work such that
3. DES model based on K-e:
FLUENT provides all the three variants of K-e, i.e., the standard, RNG, and realizable k-e models. All three models have similar forms, with transport equations for k and e. The major differences in the models are as follows:
- Method of calculating turbulent viscosity
- Turbulent Prandtl numbers governing the turbulent diffusion of k and e
- Generation and destruction terms in the e equation
FLUENT provides the so-called realizable k-e model . The term "realizable" is used to signify the fact that the model satisfies certain mathematical constraints on the normal stresses and is consistent with the physics of turbulent flows. The realizable k-e model proposed by Shih et al.  was intended to address the drawbacks of traditional k-e models by adopting the following:
- A new eddy-viscosity formula involving a variable Cµ originally proposed by Reynolds .
- A new model equation for dissipation (e) based on the dynamic equation of the mean-square vorticity fluctuation.
In the DES model, the Realizable K-e RANS dissipation term is modified such that:
The realizable k-e model more accurately predicts the spreading rate of both planar and round jets. The flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation, the realizable k-e model provides superior performance. The disadvantage of the realizable k-e model is that it produces non-physical turbulent viscosities in situations when the computational domain contains both rotating and stationary fluid zones.
DES also yields a relatively low computational cost associated with the calculation of the turbulent viscosity. Also beneficial for this application, FLUENT allows the Spalart-Allmaras (S-A), K-? SST, realizable K-e models to be integrated with wall bounded functions when the resolution is not adequately fine, or is not needed to be fine. When the mesh is sufficiently fine near the wall to resolve a laminar sublayer, the shear stress is obtained from an equation relating stress and strain. When the mesh is too coarse, the law of the wall is employed.
4.3 Boundary conditions
Boundary conditions are paramount factors in influencing the accuracy of the flow computation and convergence properties of the solution. Fluent solver uses a cell vertex discretization, which results in solution points located on the boundaries of the zones. By default, boundary conditions are imposed explicitly after the interior solution has been computed for each zone after each iteration. In the present simulations, though with greater computational effort and complexity, wall boundary conditions are imposed implicitly so as to improve the stability of the numerical scheme.
- Pressure Far-Field Boundary: Pressure far-field conditions are used to model a free-stream condition at infinity, with free-stream Mach number and static conditions being specified. The pressure far-field boundary condition is often called a characteristic boundary condition, since it uses characteristic information (Riemann invariants) to determine the flow variables at the boundaries. For our simulations, static conditions were set constant for Mach number, temperature, pressure and flow angles. This boundary condition is applicable only when the density is calculated using the ideal-gas law. Using it for other flows is not permitted. To effectively approximate true infinite-extent conditions, you must place the far-field boundary far enough from the object of interest.
- Symmetry boundary: Symmetry boundary conditions are used when the physical geometry of interest and the expected pattern of the flow solution has mirror symmetry. They can also be used to model zero-shear slip walls in viscous flows. You do not define any boundary conditions at symmetry boundaries, but you must take care to correctly define your symmetry boundary locations. Symmetry boundaries are used to reduce the extent of your computational model to a symmetric subsection of the overall physical system. Fluent assumes a zero flux of all quantities across a symmetry boundary. There is no convective flux across a symmetry plane: the normal velocity component at the symmetry plane is thus zero. There is no diffusion flux across a symmetry plane; the normal gradients of all flow variables are thus zero at the symmetry plane.
- Viscous wall: Wall boundary conditions are used to bound fluid and solid regions. In viscous flows, the no-slip boundary condition is enforced at walls by default, but you can specify a tangential velocity component in terms of the translational or rotational motion of the wall boundary, or model a "slip" wall by specifying shear. The viscous wall boundary condition enforces a no-slip condition of the flow, a zero pressure gradient, and adiabatic thermal condition at the viscous solid surface for current simulations. For the case of stationary wall surface, the Cartesian velocity component become u = v = w = 0 at the surface.
4.4 Computational set-up
The simulations were conducted using Navier-Stokes equations solver FLUENT . This code is quite diverse and allows for multi-block/zone grid option that is useful for an efficient execution in a parallel mode on a cluster. It provides a variety of higher order spatial, temporal and explicit and implicit schemes. The time-accurate numerical simulation is carried out using the segregated implicit unsteady double precision solver. The prediction of acoustic spectra is very sensitive to the pressure and thus, the numerical diffusion usually found in the numerical schemes is to be reduced. Higher order schemes have lower numerical diffusions. For this reason, higher order schemes are preferable for aero-acoustic computations.
The segregated solver used in this work employs a SIMPLE-type solution algorithm. Face values required for computing the convection terms are interpolated from the cell centers using the second-order upwind scheme. A second order accurate central differencing scheme is used for the momentum equations in order to avoid the numerical diffusion of upwind schemes. The remaining flow variables are solved using the second order upwind scheme to increase the order of accuracy. The second order upwind scheme is selected to interpolate pressure values at the faces. The Pressure-Implicit with Splitting of Operators (PISO) pressure-velocity coupling scheme is applied to derive an equation for pressure from the discrete continuity equation.
Second-order time-accurate integration is used. A time step of 1x10-5 seconds is selected with non-iterative time advancement scheme, which significantly speeds up transient simulations. Air is modeled as an ideal gas. Air viscosity is defined as a function of temperature by Sutherland's viscosity law. The DES model based on S-A turbulence model is employed for the test case 1 and all the DES variants are employed for the test case 2 as a numerical approach to simulate turbulence effects. The DES models are well-suited for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation. The solution is carried out using the Fluent's enhanced wall treatment for compressible flows. The boundary layer flow is assumed to be fully turbulent within the entire computational domain.
The cavity simulation using all DES model variants based on S-A , K-? SST , realizable K-e  viscous solver was applied using a second order implicit scheme. As it is possible when using the pressure far field boundary conditions to specify flow velocity, temperature and pressure were set up, to be identical to the experimental conditions and Mach number was worked out to be 0.85. This generated a flow velocity of 270.25 m/s which is in accordance with the experimental values. In addition, the flow velocity is also appropriate to the final approach speeds of commercial aircraft, where noise levels are of paramount importance. The computation was performed for a total of 50,000 time steps or 0.5 seconds with a time step size of 1x10-5 seconds. The initial data of 0.1 seconds was neglected. The pressure measurements were selected from the remaining 0.4 seconds.
The DES transient calculations started from laminar steady-state simulations, for 5000 iterations. For the first test case, calculations of 0.5 seconds elapsed time on the 3.54 million mesh cells took approximately 28 days on six 3.6 MHz Xeon (EM64T) processors under Linux OS. The same elapsed time on the second test case with 3.19 million cells took approximately 24 days. The flow in the domain passes the cavity 275 times over a time period of 0.5 seconds in the present computation for Mach 0.85.
The simulations consist of two parts. The first part includes the time histories of the surface pressures, P (t), at locations K20, K24, K25, and K29 respectively for the test case 1 as shown in Figure 5.1 and for the test case 2 as shown in Figures 5.2, 5.3 and 5.4 which were recorded at each time step. The second part of the solution includes the instantaneous ?ow ?eld for the 3-D open cavity test case1 (without) and test case2 (with) cover plates obtained over a sufficiently long time period. The computation has been carried out for about 50,000 time steps, of which the ?rst 10,000 time steps have been discarded in the analysis of the pressure time series. The pressure oscillation is closely related to the sound resonance from the cavity. The recorded P (t) has been used to compute the power spectral density (PSD) and the sound pressure level (SPL) as functions of frequency, f. The discrete Fourier transform has been used for P (t) to compute the PSD. The sound pressure level (SPL) is subsequently obtained from the calculated PSD, which is given by Equation 3.
The Detached eddy simulation (DES) with Spalart-Allamaras (S-A) model is used to simulate the first test case and three DES variants; DES based S-A, DES based K-? SST, and DES based Realizable K-e models are simulated for the test case 2. The computational results are compared with the experimental results of QinetiQ for the test case1 (cavity without cover plates). No experimental data is available for direct comparison with the computational results for the test case 2 (cavity with cover plates). Therefore, the comparison was made with the modified semi-emperical Rossiter formula and the three DES variants are compared with each other.
5.1 Pressure variations along the cavity ceiling
The pressure oscillation for the cavity flow is attribute to the mixing layer developing over the cavity, which impacts on the aft edge of the cavity with stretching of the vortices and has consequently caused pressure feedback .The feedback, which is essentially the upstream propagation of disturbances, is enhanced by the impact of the downstream cavity edge. The pressure oscillation produces vorticity fluctuations and, consequently, amplifies the disturbance in the shear layer, forming a feedback loop. The pressure field within the cavity is thus closely coupled with the shear layer over the cavity. The interaction between the mixing layer and the pressure field is described as shear mode by Rowley et al. . The intensity of the pressure fluctuations in the cavity depends on the feature of the mixing layer and the impact of the cavity aft edge in the evolution of inherent momentum and vorticity, as analyzed by Larcheveque et al. .
In Figure 5.1 the pressure fluctuations of DES S-A model, p(t), which have been taken from the cavity floor surface of the test case1 cavity at four locations, k20 on the front part of the cavity floor, k24 and k25 on the middle part of the cavity and k29 on the rear of the cavity. Note that the data have been probed from the fully developed flow field, ruling out any effects of initial flow conditions. As seen in Figure 5.1, the pressure fluctuations are relatively low on the front part of the cavity floor and more intensive fluctuations on the rear part. Figures 5.2, 5.3 and 5.4 show the pressure fluctuations of DES S-A, DES K-? SST and DES Realizable K-e models respectively for the test case2 model. The same trend is seen as compared to the above cavity. However, the magnitude of the pressure fluctuations is much larger in the case of the test case2 cavity than compared with the test case1 cavity. This implies that there are larger disturbances in the shear layer in the cavity case2 as compared with cavity case1. Figure 5.5 compares the rms pressure variations of DES S-A model within the cavity for both the cases. The general trend for Prms has been well produced. The over-estimation in Prms on the floor near the rear wall is an indication that the part of the shear layer may have been deflected to the cavity floor, prior to the impingement to the cavity aft wall. Figure 5.6 compares the rms pressure variations of all the DES variants; DES S-A, DES K-? SST and DES realizable K-e turbulence models for the test case2 model. As seen in Figure 5.6, the result demonstrates that the flow is dominated by large-eddy structures and that the modeling parameters were properly selected.
5.2 Pressure Analysis
The pressure fluctuations are the main cause of sound resonance in the open cavity flow. To identify these acoustic tones, the time series of fluctuating pressures is analyzed by means of the discrete fourier transform, which provides PSD or SPL as a function of frequency f. In the present analysis, the PSDs were computed with MATLAB using Welch's method of averaging over periodograms, with 50% overlap between blocks and using hamming window for each block. Figure 5.7 shows the SPL of DES S-A model at 4 locations K20, K24, K25 and K29 for the test case1 model. Figures 5.8, 5.9 shows the SPL of DES S-A model and SPL of all the three DES variants at 4 locations K20, K24, K25 and K29 respectively for the test case2 model. The SPL spectrum for each test case is discussed below.
5.2.1 Test case1 (without cover plates)
Figure 5.7 illustrates the SPL, in decibels, predicted with the DES S-A model and compared with the experimental data obtained from QinetiQ. The plot refers to the pressure points K20, K24,K25 and K29, located respectively near the inflow edge, near the center of the cavity and near the exit edge. The Vertical bars correspond to the Rossiter frequencies, as computed with the modified Rossiter formula as given by Equation 2 with K=0.57 and a=0.25. The magnitude of SPL up to 1000 Hz is sufficiently predicted and the first three resonance frequencies clearly results from the computed spectrum. The first mode is least well predicted with an over-prediction ranging from 8-12 dB. The second mode is rather well predicted with an under-prediction ranging from less than 2-6 dB. The third mode is over predicted in the range of 2-4 dB and the fourth mode is over predicted by 2-8 dB. The resonant frequencies are very well predicted for the first mode within 10 Hz, second and third mode within 25 Hz and the weaker fourth mode within 60 Hz.
5.2.2 Test case2 (with cover plates)
The SPL spectrum of DES S-A model as shown in Figure 5.8 of the test case2 model have multiple distinct peaks of comparable strength which are characterize by compressible flow induced cavity oscillations. The Vertical bars correspond to the Rossiter frequencies, as computed with the modified Rossiter formula as given by Equation 1 with K=0.55 and a=0.21 as used by Delprat  in his analysis. Many authors have analyzed the SPL spectrum with strong nonlinear coupling with the Rossiter modes and have explained them in various ways. Cattafesta  and Kegerise et al.  from the results of short-time Fourier transform and wavelet transform found a phenomenon which they called "mode-switching". Mode- switching refers to a process whereby the dominant energy shifts temporally from one Rossiter mode to another. Rowley and Williams  explain the "peak splitting" phenomenon, where two spectral peaks form near the original peak. Delprat  approached the resulting aero acoustic spectrum of Rossiter from a different angle, using signal processing theory to provide additional insight in the cavity flow physics. Delprat  reformulated the Rossiter formula as two-step amplitude modulation process in terms of what she termed the "fundamental aero acoustic loop frequency" (fa) and a very low frequency modulating signal (fb).
As seen from the Figure 5.8, the spectral modes are fI = 274.673, fI I= 522.735, fIII = 802.6, fIV = 1076.1, fV = 1604.03, fVI = 2411.82, where fI, fIII and fIV are dominant peaks. The Rossiter frequencies are f1 = 519.4138, f2 = 1176.9, f3 = 1834.386 and f4 = 2491.871. Out of the four Rossiter frequencies only three are visible in the spectrum, but they are not the dominant peaks in the spectrum. There is also the possibility of peak splitting phenomenon in the third Rossiter mode of the spectrum. The above spectrum seems to have more complex nonlinear coupling phenomena and thus, require more understanding of the flow phenomenon. A low frequency component fI = 274.673 is observed in all the four locations. The origin of this low frequency component, with roughly half the frequency of the fundamental, remains unexplained.
Figure 5.9 compares the SPL spectrum of all the three DES variants for the test case2 model. As seen in Figure 5.9, the SPL spectrum of DES K-? SST and DES realizable K-e model behaves in the same way as the SPL spectrum of DES S-A model. The SPL spectrum is dominated by multiple peaks and seems to have more complex nonlinear coupling phenomenon. The dominant peak in SPL spectrum of both DES K-? SST and DES realizable K-e model is the low frequency component with roughly half the frequency of the fundamental, remains unexplained as is the case with the SPL spectrum of the DES S-A model. The other peaks occurring in the SPL spectrum of all the DES variants are comparable.
5.3 Cavity Flow Field Analysis
5.3.1 Test case1 (without cover plates)
The present cavity ?ow to a large extent is characterized by the shear-layer mode , for which the mixing layer bridges the cavity opening. The pressure oscillation is of ?uid-resonant type according to Rockwell and Naudasher , which is generated due to the coupling between the shear layer and the pressure ?eld. The self-sustaining process is closely related to the wave re?ection within the cavity due to the impingement on the aft wall, and to the addition and removal of mass at the cavity trailing edge in the case of a shallow cavity with su?ciently large length-to-depth ratio L/D = 5. Nonetheless, in the above analysis of pressure ?uctuations and of the consequent sound resonance, it has been implied that the strength of the mixing layer may have been under-estimated with a part of the ?uid de?ected from this layer and bended towards the cavity ?oor, prior to the impingement on the aft cavity wall. In order to observe the property of the 3D unsteady ?ow ?eld, Figure 5.10 illustrates the computed velocity ?ow?elds contours at a number of subsequent time instances, covering a time period of 0.4 sec simulated by using DES based S-A turbulence model. In each illustration, the left window presents the flow field in the mid sections of the cavity in the streamwise and the right one provides the flow field in the spanwise direction respectively. The velocity field has been colored by the magnitude of velocity.
Unlike the 2D simulations, the present 3D simulation does not render any periodical ?ow feature. Figure 5.10 shows that the ?ow is characterized by the shear layer over the cavity opening and by the recirculating motion within the cavity. The shear layer, emanating from the upstream boundary layer coming off from the cavity leading edge, spans the mouth of the cavity and stagnates at the rear wall. In spite of a relatively high freestream Mach number = 0.85, the ?ow is of essentially subsonic type with no shock system formed. The shear layer waves up and down with break up occurring at the aft end of the cavity as shown in Figure 5.10. It impinges on the rear wall of the cavity. The interaction between the shear layer and the aft wall gives rise to two consequences. First, a recirculation ?ow is formed in the cavity due to the entrainment of the shear layer and the ?ow reverse after the impingement below the rear wall. Second, the interaction provides an e?ective feedback that ampli?es the shear layer instability. It is shown that the impingement of the shear layer on the cavity aft wall alters the intensity and the extension of the recirculation bubble, which interacts with the mixing layer and consequently provides the ampli?cation condition for this instability. The shear-layer instability is coupled with the pressure waves generated in the cavity and producing acoustic tones at discrete frequencies as analyzed in the above section.
Moreover, the ?ow exhibits signi?cant time-dependent three-dimensionality, as can be seen from the ?ow motions on the spanwise section as shown in Figure 5.11. The recirculation bubble evolves not only in the longitudinal streamwise direction, but also presents vortical motions in the transverse spanwise direction. In Figure 5.11, the three-dimensional vortical motion has been further illustrated by plotting the unsteady velocity vector ?elds. The recirculating motion is contained within the cavity in the front part, where the vortical motion is close to the mixing layer and is relatively weak with a low level of turbulence intensity. Similar to the mass exchange process occurring over the aft edge of the cavity, these vortical motions are closely coupled with the pressure oscillations inside the cavity in terms of the pressure modes and their resonant frequencies. Figure 5.12 shows the mean velocity streamline pattern for the test case 1. Figure 5.13 shows the variation of the turbulent viscosity both in the streamwise and in the transverse spanwise direction over a period of 0.4 sec. The pattern of velocity flow field and the turbulent flow contour are in agreement with the previously performed simulations by various authors.
5.3.2 Test case2 (with cover plates)
The open cavity with cover plates also exhibits the shear layer or Rossiter mode. The flow phenomenon is same as that of the cavity without cover plates. Figures 5.14- 5.16 show the velocity flow field for 0.4 sec both in the streamwise and transverse spanwise direction for DES S-A, DES K-? SST and DES realizable K-e turbulence models respectively. The length of the shear layer is reduced because of the presence of the cover plate. In Figures 5.14- 5.16, the shear mixing layer waves up and down moving over the cavity in which a dominant recirculation motion is contained. This unsteady shear layer reaches the aft edge of the cavity and has induced vertical flow motions along the aft wall due to impingement. For the instant shown in Figures 5.17-5.19 in addition, a dominant backflow is formed due to the recirculation near the cavity floor. With the DES prediction, the instantaneous mixing layer tends to break down causing extensive local vortical motions inside the cavity. Near the cavity floor, the flow is instantaneously driven toward the cavity aft wall and a part of the flow is "detached" from the shear layer and approaching to the floor. The mean velocity streamlines of test case2 for DES S-A model is shown in Figure 5.20. The number of recirculation zones is seen inside the cavity. The DES S-A model indicates that large values of eddy viscosity arise after a short distance from the leading edge of the cavity as shown in Figure 5.21.
Figures 5.22 and 5.23 illustrate the turbulent intensity generated within the cavity by DES k-? SST and DES realizable K-e turbulence models respectively. As seen in Figure 5.23, large amount of turbulent intensity is generated by realizable k-e model as compared to other DES models, which is due to the fact that realizable k-e model produces non-physical turbulent viscosities in situations when the computational domain contains both rotating and stationary fluid zones, which is one of the disadvantage of this turbulence model. However; the instability in the shear layer is sufficiently large to cause the cavity under resonance and thus, results in the high pressure fluctuations in the cavity.
The cavity flow is validated in the present study with the experimental data of QinetiQ  and the semi-emperical Rossiter formula. The rms pressure along the cavity of the test case1 (without cover plates) is plotted in Figure 5.5 and compared with the available experimental rms pressure along the cavity floor. The numerical results follow the trend and compare well with the experimental values. Figure 5.7 shows the SPL plot of the test case1 and the results are well predicted compared to the experimental values. The rms pressure and SPL plot of DES S-A model for the test case2 is also shown in Figures 5.5 and 5.8 respectively. The results are well predicted with the semi-emperical Rossiter formula. Figures 5.6 and 5.9 compares the rms pressure and SPL plot of all the three DES variants for the test case2 model and the results follow the trend and very much comparable.
The DES computations have been conducted for a turbulent open-cavity flow using the Spalart-Allmaras one-equation model for the test case1 and S-A, K-? SST and realizable K-e models for the test case2. The open cavity has an aspect ratio of 5: 1: 1 and is immersed in a freestream at a Mach number of M8 = 0.85. Two sets of results have been analyzed, including the surface pressure fluctuations on the cavity floor and the predicted flow fields. The prediction for the pressure fluctuations has been compared with available experimental measurements for the Test case1 and the pressure fluctuations for the Test case2 are compared with the semi-emperical Rossiter formula.
In the analysis of tonal resonance derived from pressure oscillations, it is shown that DES computations are able to distinguish the second and third tonal pressure modes, as well as the fourth pressure mode at some locations on the floor surface. The first mode is not well resolved for both the test cases. The SPL spectrum of the Test case1 model predicts tonal magnitudes reasonably well compared to the experiment for the second and third modes and for the fourth mode near the aft wall of the cavity. The maximum error in predicting tonal magnitude for the test case1 model is 8% and 6% in measuring frequencies. The SPL spectrum of the test case2 model involves multiple distinct peaks of comparable strength which are characterize by compressible flow induced cavity oscillations. The presence of low frequency component and the component occurring between the first and second Rossiter mode in the spectrum is not known. To understand this complex spectrum, the role of nonlinearities in the cavity shear layer needs to be further investigated.
The rms pressure variation of DES S-A model along the cavity floor for the test case1 is well predicted compared to the experimental data. For the test case2, the rms pressure variation of all the DES variants models along the cavity floor follows the trend and the results are comparable with each other. The SPL spectrums of the test case2 model for all the DES variants models are also very much comparable with each other.
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