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Analysis of coupling aerodynamics and acoustics are performed to investigate the self-sustained oscillation and aero dynamic noise in two-dimensional flow past a cavity with length to depth ratio of 2 at subsonic speeds. The large eddy simulation (LES) equations and integral formulation of Ffowcs-Williams and Hawings (FW-H) are solved for the cavity with same conditions as experiments. The obtained density-field agrees well with Krishnamurty’s experimental schlieren photograph, which simulates flow-field distributions and the direction of sound wave radiation. The simulated self-sustained oscillation modes inside the cavity agree with Rossiter’s and Heller’s predicated results, which indicate frequency characteristics are obtained. Moreover, the results indicate that the feedback mechanism that new shedding- vortexes induced by propagation of sound wave created by the impingement of the shedding-vortexes in the shear-layer and rear cavity face leads to self-sustained oscillation and high noise inside the cavity. The peak acoustic pressure oc curs in the first oscillation mode and the most of sound energy focuses on the low-frequency region.

High-speed compressible flows past open cavities induce complex unsteady aerodynamic characteristics, such as flow separation in the cavity front-face, shear-layer instabilities, vortex shedding, shock waves/boundary-layer interactions and self-sustained flow oscillations within the cavity. Under certain conditions, the strong self-sustained flow oscillations associated with large-amplitude noises around 150 dB for high-speed flows can occur in a flow past an open cavity [

Oscillation in the flow past an open cavity has been investigated for decades, still there remain many open questions about even the basic physical mechanisms underlying the self-sustained oscillations. Cavity oscillations in compressible flows are typically described as a flow-acoustic resonance phenomenon, and its first detailed description is credited to Rossiter (1964) who proposed a semiempirical formula for predicting the frequency peaks in high subsonic compressible flows over shallow cavities, named open cavities, with a length-todepth ratio [

Nowadays, numerical simulation investigations have been performed for cavity oscillations and aeroacoustic characteristics based on the development in CFD technique. Previous numerical studies of compressible cavity flows have used the two dimensional unsteady RANS (Reynolds averaged Navier-Stokes) equations with a k-w turbulence model (Lamp & Chokani 1997; Zhang, Rona & Edwards 1998; Fuglsange & Cain 1992). The effectiveness of compressible turbulence models for separated oscillating flows, and especially their radiated acoustic field (which, as noted above, is an integral part of the resonant instability modes) remains an open question. Direct numerical simulation (DNS) is an accurate numerical method to simulate turbulence, but it requires high cost and computational capacities. Fortunately, Large Eddy Simulations (LES) provide a means to study the details of the modes of oscillation and the basic physical mechanisms of sound generation [8,9]. Computational Aero-Acoustics (CAA) also presents some advantages in studies on aeroacoustic characteristics of cavity flow.

The purpose of the study is to indicate the self-sustained oscillations and the physical mechanisms of sound generation inside an open cavity. The numerical simulation method is to analyze the unsteady flow characteristics in the near flow-flied by utilizing LES, and to predict sound radiation and acoustic-flied by solving Ffowcs Williams-Hawking (FW-H) equations [

A LES model was initially employed to run the two dimensional cavity flow simulations. The numerical model is a compressible flow LES model. In Cartesian coordinates, the box-filtered equations are utilized to obtain the equations [11,12] , shown as follows,

where symbols overbar, tilde and superscripts denote Reynolds, box-filtered quantities and sub-grid scale quantities respectively. are density, velocity components, pressure, temperature, total energy, viscous stress tensor and sub-grid stress tensor, respectively. There are auxiliary relations,

where constants, The molecular viscosity and eddy viscosity is modeled by sub-grid (SGS) models. In a mixedscale SGS model [

where

Advantages of mixed-scale model are no artificial averaging and wall-damping function required. A 2nd order central finite-difference scheme is employed for the spatial discretization [14,15]. The time integration scheme is a 3rd order Runge-Kutta scheme [16,17]. By placing a buffer-zone combined together with Giles characteristic condition as a non-reflecting boundary condition (NRC) in inflow and outflow regions this code performs better than without buffer zone conditions [

Finally, complete content and organizational editing before formatting. Please take note of the following items when proofreading spelling and grammar: Ffowcs Williams and Hawkings derived the most general form of wave equation using the technique of generalized function theory which allowed them to utilize the free-space Green function in constructing the integral solution. The aeroacoustic prediction method is presented in reference [

According to reference [

where,

Details of those parameters can be found in reference [

It is often the case that the flow is computed with a two-dimensional cavity model, such as reference [

Some acoustic parameters analyzed are sound pressure level (SPL), sound pressure frequency spectrum (SPFS) and Strouhal number (St). Herein SPL denotes pressure fluctuation magnitude, which was calculated by Equation (12). SPFS denotes sound pressure spectral energy on the different discrete frequencies, which was calculated by Equation (13). St calculated by Equation (14) is a nondimensional number which denotes oscillation frequency inside cavities. is the sound pressure spectral density function, which was calculated by FFT and defined by was Equation (15). Therein P_{rms} is the rootmean-square of the pressure fluctuation, which was obtained by integrating the power spectral density in the frequency band of 0 - 10 kHz and extracting the square root. P_{ref} is the benchmark sound pressure, 20 mPa. f denotes oscillation frequency, and L is the cavity length, and is free-stream velocity, and T is a special time to collect experimental data, and denotes frequency range to analyze SPFS characteristics.

A quintessential code validation example is the aerodynamic noise generated by turbulent flow past a circular cylinder. Predicting the noise radiated from this seemingly simple configuration is not an easy task, mainly due to the difficulty of obtaining an accurate prediction of the flow when it is turbulent. We took here the case experimentally investigated by Revell et al. [_{D}) and Mach number (Ma) are 90,000 and 0.2, respectively. Brentner et al. computed the flow and noise for the same case using the FW-H approach and the near-field flow obtained from URANS predictions. In the present work, the flow was computed by LES with the Smagorinsky’s subgrid-scale viscosity model using a two-dimensional CFD model on a 100,000-cell quadrilateral mesh (See

According to reference [

The contours of instantaneous vorticity at a different time instant (T is computational period) are depicted in _{D}), respectively. Its time-averaged value C_{D}) was found to be approximately 1.418 which is larger than the experimentally measured value of 1.320 [_{D} by 25% - 55%. The predicted Strouhal number also agrees quite better with the measured one than others. The predicted time-averaged drag coefficient and Strouhal number are summarized in

and the corresponding power spectral density, respectively. The aeroacoustic characteristics, SPFS and PSD, predicted at the two receivers are presented in

The geometrical parameters of the cavity model in the paper are same as that researched by Rowley [^{6} based on cavity length, and the sound signals are received on receiver-1 at the cavity front wall and receiver-2 at the cavity rear wall, respectively. It should be noted that the quoted values of the boundary-layer momentum thickness at the upstream cavity edge, are taken from the initial condition for each run. The oscillations that develop in the cavity can alter this value. The boundary-layer thickness to cavity depth ratio is 0.038. The computational domain is 300 thousand structure mesh, shown in

accurate sound source signal, the sound source correlation lengths is selected to 5L for two-dimensional cavity model according to some references.

separation appears and the vortex sheds and induces a new separated-vortex at the front wall of the cavity at 0.5T (see

the cavity are induced by flow past the open cavity, shown in

A set of non-dimensional modal peak oscillation frequencies within a open cavity at which acoustic tone amplitude occurs can be predicted by a Rossiter’s semiempirical Eqation (16) determined by Rossiter in 1964. Heller modified the equation in 1975 who presented radiate velocity of turbulence sound wave upward should be local sound velocity, and the modified equation is given as Equation (17). is Strouhal number which indicates flow oscillation modes and peak frequencies of cavity acoustic tones. is n mode peak oscillation frequency, and U is free-stream velocity, and constant and. We performed a Fast Fourier Transform Algorithm (FFT) of 150 samples (every 300 time units) of the computational data over a period of time, corresponding to 15 periods of the lower frequency, and 25 periods of the higher frequency. The resulting data record is approximately periodic in time, and any drift in the data is removed prior to taking the FFT.

_{1} and St_{2}, within the cavity are about 0.29 and 0.73, respectively, which is similar with the results predicted by Rossiter and Heller (see

generation, development and shedding and impingement of the vortex and the cavity wall at subsonic speeds. The receiver-2 locates in a region near the cavity rear wall, aerodynamic noise generation appear in the region. Therefore, the SPFS at the self-sustained oscillation mode 1 of the receiver-2 is more than that of the receiver-1. For flow self-sustained oscillation characteristics inside the cavity, the mode 1 is a key self-sustained oscillation and on which peak frequency of cavity tone and sound pressure amplitude occurs, and the oscillation energy mostly focuses on mode 1.

A study on physical mechanism of sound generation inside an open cavity with length to depth ratio of 2 was conducted by numerical simulation method. There are the near field cavity CFD simulations by LES and the far field aero-acoustic predictions by FW-H integral equation. The results about the aerodynamic noise generated by turbulent flow past a two-dimensional circular cylinder indicate that a quintessential code validation is feasible. Acoustic energy focuses on self-sustained flow oscillation mode 1, and the oscillation mode predicted by the method is good agreement with Rossiter and Heller’s predicted results. The self-sustained flow oscillation and intense aerodynamic noise within the open cavity are induced by free-stream shear-layer separation, vortexgeneration-development-shedding, interaction of the shearlayer and flow within the cavity and impingement of flow and the cavity wall. Moreover, a sound feedback mechanism and self-sustained flow oscillation of vortex generation, shedding-vortex, aerodynamic noise generation, vortex regeneration and re-shedding and aerodynamic noise regeneration within the open cavity.

We should please to thank Pro Weimin JIANG in CARDC and Pro Cunbiao LI in Peking University for many helpful comments, and for suggesting the windtunnel test methodology and theoretical analysis. We should also thank High-speed Aerodynamics Institute of China Aerodynamics Research and Development Center for affording experimental apparatuses. The researched work was supported by Government fund “973” project (Number: 2009CB723802) and CARDC research fund.