Self-Excited Supersonic Cavity Flow Instabilities as Aerodynamic Noise Sources

A turbulent cavity flow at Mach 1.5 and 2.5 is modelled to study the flow instability and the associated aerodynamic noise generation. The short-time averaged Navier-Stokes equations, coupled with a k – ω turbulence model, are solved to predict the large-scale time-dependent flow. Values of the cavity wall pressure, drag, streamwise velocity and density are in good in agreement with past measurements and the results of other computations. The noise generation physics of the unsteady flow is addressed by estimating the noise source strength in a Lighthill acoustic analogy. The time-dependent flow predictions highlighted the upstream and downstream cavity edges as areas of large flow unsteadiness. The same areas are identified by the acoustic analogy as the dominant noise source regions in this flow.


Introduction
A numerical investigation is performed on the time-mean and time-dependent characteristics of the compressible turbulent flow over a rectangular cavity or enclosure. At certain flow regimes [1], the flow-geometry interaction occurring in a cavity onsets large-scale self-excited instabilities that dominate the aerodynamic flow field. In this field, the time-dependent pressure is oscillatory along the solid boundaries and generates time-dependent drag. Strong tonal noise is also generated by the aerodynamic instabilities. The physics of noise production is based on Lighthill's acoustic analogy. This study addresses by numerical investigation the noise source characteristics based on the timeaveraged Lighthill stress tensor distribution, derived from the time-dependent aerodynamic results. This approach attempts to capture through the acoustic analogy the aerodynamic noise sources responsible for the sound perceived in the acoustic far-field. This investigation complements the available literature on the aerodynamics and near-field acoustics of self-sustained open cavity flows [2][3][4][5], giving some further information on the source strength of the noise perceived at more than 20 cavity lengths away from the enclosure.
Zhang [6,7] discussed the flow unsteadiness developing over a cavity or enclosure. A turbulent boundary layer approaches the enclosure towards the upstream edge (Fig. 1). At this location, the geometry induced flow separation forms a shear layer. The latter is unstable, developing large-scale self-sustained instabilities that characterize the unsteady aerodynamic field. Aerodynamic pressure fluctuations inside the enclosure accompany the observed shear layer unsteadiness and provide the feed-back for the onset of a stable self-sustained oscillatory flow regime. The flow instability is responsible for wall pressure fluctuations, pressure drag and aerodynamic noise radiation.
In this study, a rectangular enclosure of length to depth ratio 3 is tested at free stream Mach numbers (M ∞ ) of 1.5 and 2.5. Measurements and past numerical predictions by Zhang [6,7] are available for comparison. Charwat et al. [8]  Even the simple geometry of a rectangular cavity develops a complex flow pattern that is the object of on-going fundamental research (e.g. Jeng & Payne [9] and Zhang et al. [10]). Practical applications of related cavity flows include slotted wall wind tunnels, slotted flumes, bellows-type pipe geometries, gate slots, spacecraft, and aircraft components. The subject is reviewed by Rockwell & Naudascher [2], Blake & Powell [3], Colonius [4], Grace [11] and by Rowley & Williams [12].
In the time-averaged flow, the presence of the shear layer and its rate of growth, the pressure distribution around the walls, the characteristics of the in-coming boundary layer and of the re-attached flow downstream of the cavity all influence the cavity mean drag. A quantitative prediction is attempted of the salient time-averaged features to estimate the cavity mean drag.
In the time-dependent flow, a qualitative analysis is presented on the convective amplification of the large-scale instability in the shear layer and of the accompanying unsteady convected vorticity. The flow-geometry interaction at the downstream cavity edge dominates the pressure fluctuations throughout the cavity. Past studies have used the knowledge of such interaction to reduce the unsteady aerodynamic loads of a model enclosure [13,14]. This study aims to relate such flow physics to the sources of the aerodynamic noise perceived in the acoustic far-field.
Aerodynamic noise is generated as a by-product of the flow instability. Cavity noise measurements by Block [15] and numerical studies by Hardin & Pope [16] highlighted the tonal characteristics of the radiated sound field at certain flow regimes. For the selected test cases the noise characteristic wavelength (λ) is greater than the cavity characteristic dimension (D). Flow-acoustic interaction inside the cavity is limited by the size of the enclosure (λ < D) and hence the unsteady flow can be regarded as a compact noise source. In the acoustic far-field, at a distance of 20 or more cavity lengths away from the enclosure, the pressure fluctuation perceived by an observer as aerodynamic noise is the integral effect of the Lighthill stress tensor field. This can be extracted from the time-dependent aerodynamic predictions.
The present study focuses on the large-scale turbulent structure in the cavity responsible for aerodynamic noise generation. Lighthill's acoustic analogy is followed to identify the acoustically active regions in the flow field. The physics of noise generation is addressed as a by-product of the time-dependent flow.
These results contribute to the understanding of cavity noise generation, which is the basis for developing successful noise control strategies.

Flow Conditions
The geometry and flow conditions describe a longitudinal rectangular cavity driven by a turbulent shear layer, studied in Zhang [6,7]. The depth of the cavity D is 15mm and the length of the cavity L is 45mm. The geometry is tested at free stream Mach numbers (M ∞ ) of 1.5 and 2.5. At M ∞ = 1.5 the free stream temperature (T ∞ ), pressure (p ∞ ), density (ρ ∞ ), stagnation temperature (T s ) and stagnation pressure (p s ) are 200K, 53.801kN/m 2 , 0.9373kg/m 3 , 288.5K, and 197.51kN/m 2 respectively. At M ∞ = 2.5 the corresponding values are 128K, 17.390kN/m 2 , 0.4701kg/m 3 , 288.5K, and 297.12kN/m 2 . A turbulent boundary layer approaches the enclosure upstream edge. The boundary layer thickness (δ) and momentum thickness (δ 2 ) are respectively 5mm, 0.417mm at M ∞ = 1.5, and 5mm and 1.290mm at M ∞ = 2.5. The flow Reynolds number (Re) based on the cavity depth is 4.5 × 10 5 in both cases.

Aerodynamic model
A combined deterministic and stochastic approach is followed to model the cavity aerodynamic flow. The flow visualization by Zhang [6] in Fig. 9(a) indicates that the aerodynamic field can be regarded as large-scale time-evolving structures in a background of random turbulence. These large-scale flow instabilities are obtained from the selective amplification of flow disturbances in the cavity. The excited modes are the eigenmodes of the cavity flow, as shown by the analysis of Bilanin & Covert [17]. The enclosure geometry and the inlet boundary layer profile are the boundary conditions that determine the eigenmodes and, more generally, the frequency and wavenumber response characteristics of the cavity. The observed large-scale structures develop in the shear layer spanning the cavity open surface and are Kelvin-Helmoltz type instabilities. They are therefore problem-specific and their initial growth is essentially a deterministic inviscid process. The large-scale instability growth can be modelled by inviscid instability analysis methods as in Tam [20].
Measurements by Zhang [6] show that the large-scale instabilities are gener- The large-scale instabilities reach the downstream edge having covered just one fundamental mode wavelength, which is unlikely to be sufficient to allow the dominant instability modes to decay in smaller eddies and establish a fully developed turbulent flow. The large scale structure can still be readily identified on the trailing edge plate in Zhang [6], at a streamise distance greater than 6D from the upstream edge. The accompanying wall pressure measurements in Zhang [6] confirm a power spectral density dominated by low frequency isolated tones over a lower amplitude broad-band turbulent contribution, decaying with frequency. Further downstream, away from the enclosure, the convected instabilities are expected to decay, creating a tur-bulent kinetic energy cascade to the higher wavenumbers and the turbulent kinetic energy is eventually dissipated by viscous stresses in the dissipation sub-range, at the Kolmogorov length scale. As the flow is supersonic, this downstream flow regime does not significantly affect the upstream cavity.
A separation of kinetic energy length scales approach is adopted to obtain a  [23]) written in vector form. The turbulent stress tensor is estimated using the k−ω two-equation model of Wilcox [21]: whereŨ = ρ,ρũ,ρ ẽ s +k ,ρk,ρω T , The relationship for a perfect gasp =ρT /γM 2 ∞ completes the governing equa-tions. The eddy viscosity (μ t ), dynamic viscosity (μ l ), viscous stress tensor (τ ), turbulent stress tensor t , heat flux vectors (q,q t ), specific stagnation energy (ẽ s ), and specific internal energy (ẽ) are defined in the following auxiliary relations: The The Roe [24] flux difference split approximate Riemann method estimates the inviscid fluxes F i at the unit volume boundaries. The second-order space accurate extension is implemented. The MinMod inviscid flux limiter function is adopted to preserve monotonicity. The inviscid flux components of the k − ω equations are included as in Rona [25], following the same procedure of Roe. Second-order central difference is used to estimate the turbulent fluxes F t that are integrated with the inviscid ones in finite-volume form.
This compact integration strategy upgrades the operator split approach by Zhang [7], enhancing the efficiency of the numerical quadrature in the model. At the inflow boundary (b1) the mass flux remains constant throughout the computation and is similar to the upstream boundary condition used by Bastin [27] for the case of a supersonic uniform stream. In the unsteady cavity flow, upstream propagating disturbances are present in the boundary layer.
The upstream effect is limited below the sonic line to a distance less than one boundary layer thickness. This is difficult to quantify, but since the distur-bances are damped while propagating from the upstream cavity edge (y 1 = 0) to the inflow boundary (y 1 = −3D), the selected boundary condition appears to be adequate (Rona [25]). Along the solid walls (b2 − b6) a no-slip condition is imposed. At the outflow (b7) all conservative variables are extrapolated assuming constant gradients in space (first order extrapolation). Non-reflecting boundary conditions (Zhang [7]) apply on (b8). Along this latter computational boundary, the flow is assumed uniform and parallel and the conservative variables are set equal to their interior values and are constant along the outgoing Mach wave.
The flow state at the beginning of the computation is described in Fig. 2.
The inflow condition is imposed in the flow field above the cavity. Inside the cavity, stagnation temperature conditions apply and u = 0. As in Zhang [7],

Noise source prediction
Lighthill's acoustic analogy is used to determine the noise generation from the predicted unsteady flow. The main contributions to the far-field acoustic radiation are expected from the momentum flux fluctuations in the largescale structure. The sources of noise can be found by short-time averaging the Lighthill [28] governing equation for aerodynamic noise. This equation is where the tensorT =ρũũ − τ +t + (p − c 2 ∞ρ ) I is the short-time averaged Lighthill stress tensor,t = −ρ u u being the short-time averaged Reynolds stress tensor. This is derived following the procedure of Lighthill [28] from the short-time averaged Navier-Stokes equations and is exact. where is the geometric scaling factor for acoustic propagation. The surface S bounds the flow domain V and n is its outward surface normal unit vector.
In equation (12), R * accounts for the convection of the acoustic waves by a constant Mach number M ∞ that approximately models the flow above the enclosure boundary layer for the purpose of acoustic convection. Other flowacoustic interactions within the unsteady flow have not been modelled.
The volume integral in equation (12) operates on the second order Lighthill stress tensor differential that represents the aerodynamic sound sources from shearing flow. Specifically, noise is produced by momentum flux accelerations in the direction of the far-field observer at x. These acoustic quadrupoles were shown by Lighthill [28] to radiate with an acoustic intensity ∝ U 8 ∞ /c 5 ∞ . The surface integral term, evaluated along the cavity solid boundary, models the contributions to noise by the aerodynamic flow interacting with the enclosure walls. This first order differential term is a dipole noise source model that was shown by Curle [29] to give an acoustic intensity ∝

Pressure fluctuation
The self-sustained characteristic of the flow instability is evident in the nor-

Aerodynamic noise sources
The physics of aerodynamic noise production from compressible turbulent nonuniform flows is complex. In this study, some characteristics of this process are disclosed, which relate to the large-scale flow instability.
The unsteady pressure predictions highlight the flow region at the downstream cavity edge as potentially the most unsteady. The accelerating fluid in the direction of the observer produces aerodynamic sound, as described by the application of the Lighthill solution by Proudman [32]. Estimating the rootmean-square of the streamwise and normal flow velocities therefore gives some description of the noise production physics at the downstream cavity edge.
The prediction at M ∞ = 1.5 in Fig. 14(a) shows that (ũ 1 ) rms , normalized as in Section 2, is maximum at the downstream cavity edge and is 0.272. The region of maximum (ũ 1 ) rms extends mainly in the streamwise direction. This suggests that a noise source of streamwise longitudinal and lateral quadrupole type is present at this location in the aerodynamic field. A second localized (ũ 1 ) rms maximum is located at the upstream edge.
The root-mean-square normal velocity near-field is mainly related to the shear layer normal displacement. Two regions of maximum (ũ 2 ) rms are shown in The lateral quadrupole type source contribution to the far-field noise is gen-T 12 rms is 0.130. This is located close to the upstream cavity edge where a localized large (ũ 2 ) rms was identified in Fig. 14 An enhanced computational fluid dynamic method was adopted to further the analysis by Zhang [7] and address cavity noise generation.

Acknowledgements
The author wishes to thank Prof. X. Zhang