Publications

We present an approach to the design of feedback control laws that stabilize relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. We present two different approaches of control design. The first approach assumes that the closed loop system is affine in the control and that the actuation is equivariant. We derive feedback laws for the reduced system that minimize a quadratic cost function. The second approach is more general; here the actuation need not be equivariant, but the actuators can be translated in the symmetry direction. The controller resulting from this approach leaves the dynamics associated with the symmetry variable unchanged. Both approaches are simple to implement, as they use standard tools available from linear control theory. We illustrate the approaches on three examples, a rotationally invariant planar ODE, an inverted pendulum on a cart, and the Kuramoto-Sivashinsky equation with periodic boundary conditions.

We use Direct Lyapunov Exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that, despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.

Flow over an impulsively started low-aspect-ratio flat plate at angle-of-attack is investigated for a Reynolds number of 300. Numerical simulations, validated by a companion experiment, are performed to study the influence of aspect ratio, angle of attack, and planform geometry on the interaction of the leading-edge and tip vortices and resulting lift and drag coefficients. Aspect ratio is found to significantly influence the wake pattern and the force experienced by the plate. For large aspect ratio plates, leading-edge vortices evolved into hairpin vortices that eventually detached from the plate, interacting with the tip vortices in a complex manner. Separation of the leading-edge vortex is delayed to some extent by having convective transport of the spanwise vorticity as observed in flow over elliptic, semicircular, and delta-shaped planforms. The time at which lift achieves its maximum is observed to be fairly constant over different aspect ratios, angles of attack, and planform geometries during the initial transient. Preliminary results are also presented for flow over plates with steady actuation near the leading edge.

When an airfoil is pitched up rapidly, a dynamic stall vortex forms at the leading edge and produces high transient lift before shedding and stall occur. The aim of this work is to develop low-dimensional models of the dynamics of these leading-edge vortices, which may be used to develop feedback laws to stabilize these vortices using closed-loop control, and maintain high lift. We first perform a numerical study of the two-dimensional incompressible flow past an airfoil at varying angles of attack, finding steady states using a timestepper-based Newton/GMRES scheme, and dominant eigenvectors using ARPACK. These steady states may be either stable or unstable; we develop models linearized about the stable steady states using a method called Balanced Proper Orthogonal Decomposition, an approximation of balanced truncation that is tractable for large systems. The balanced POD models dramatically outperform models using the standard POD/Galerkin procedure, and are used to develop observers that reconstruct the flow state from a single surface pressure measurement.

Understanding fish-like locomotion as a result of internal shape changes may result in improved underwater propulsion mechanisms. We use a geometric framework to consider the simplified problem of an articulated two-dimensional body in a potential flow. This paper builds upon the current geometric theory by showing that although the group of Euclidean transformations is non-Abelian, certain tools available for Abelian groups may still be exploited, making use of the semidirect-product structure of this group. In particular, the holonomy in the rotation component may be explicitly computed as a function of the area enclosed by a path in shape space. We use this tool to develop open-loop gaits for an articulated body with two shape variables, using plots of the curvature of the mechanical connection, which relates motion in the shape space to motion of the overall body. Results from numerical computations of the mechanical connection are compared to theoretical results assuming the joints are hydrodynamically decoupled. Finally, we consider a simple method for trajectory tracking in the plane, using a one-parameter family of gaits.

The response of a supersonic cavity to open-loop forcing with a pulsed-blowing actuator is explored experimentally. It is shown that excitation at frequencies near the Rossiter modes are amplified, while frequencies between the first two Rossiter modes are attenuated. The results clearly demonstrate that the Rossiter modes in the supersonic cavity are weakly damped (not self-excited) modes. The supersonic modes are not saturated, and do not show the kind of nonlinear interactions with the forcing modes observed in subsonic flow. These differences between supersonic and subsonic flows are consistent with previously developed models of cavity oscillations, and the results suggest that linear techniques for the design of closed-loop controllers may be particulary effective for supersonic flows. For the flow regime studied, the oscillatory component of open-loop forcing does not play a significant role in the suppression mechanism in supersonic cavity flows.

We present plans and preliminary results for a recently initiated multidisciplinary research effort aimed at closed-loop control of three-dimensional leading edge and tip vortices on low aspect ratio wings relevant to micro and small unmanned air vehicles. The goal of control is to extend the parameter space for which steady lift can be maintained at high angles of attack and to regulate leading edge and tip vortex formation to improve maneuverability and gust response. Initial efforts toward model-based control design and sensor-actuator configuration are presented. Dynamically dominant modes will be identified by integrating proper orthogonal decomposition (POD) and balanced truncation, and tunable empirical models will be based on energy exchange between the mean flow and POD modes and the addition of shift modes to track structures along natural and controlled transients. Plans for experimental and computational cross-validations of closed-loop control strategies are discussed, and preliminary results assessing actuator authority are presented. The differing stability characteristics of a leading-edge vortex in two-dimensional computations and finite-aspect ratio three-dimensional experiments are also highlighted.

We present an approach to the design of feedback control laws that stabilize the relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. Assuming equivariant actuation, we derive feedback laws for the reduced system that are optimal in the sense that they minimize a quadratic cost function. We illustrate the method by stabilizing unstable traveling waves of a dissipative PDE possessing translational invariance: the Kuramoto-Sivashinsky equation with periodic boundary conditions.

Fluid systems often exhibit inherently low-dimensional behavior, even though the governing equations are complex and high-dimensional. At this time, full 3D discretizations of the Navier-Stokes equations are too computationally intensive to be used for control synthesis, so low-order models of the flow physics are desirable. This paper compares two different model-reduction procedures for the linearized flow in a plane channel: the method of Proper Orthogonal Decomposition and Galerkin projection, popular in the fluid mechanics community; and balanced truncation, a common method for model reduction of linear systems. Standard methods of computing balancing transformations are computationally intractable for systems of this size, so we use a numerical approximation of balanced truncation using empirical Gramians computed from simulations of the linearized and adjoint systems. For the channel flow considered here, the subspaces spanned by POD modes and balancing modes are very close, but reduced-order models from approximate balanced truncation perform much better than the standard POD models, for the same order of truncation.

Reduced-order models of the flow in a plane channel flow are constructed in two regimes: a minimal flow unit in which turbulence is sustained, and a transitional flow linearized about the laminar profile. Proper orthogonal decomposition (POD) of data from a direct numerical simulation of a channel flow in a minimal flow unit is performed in order to examine the coherent structures and their dynamical interactions. Empirical basis functions are obtained both using Fourier modes in the streamwise and spanwise directions, with POD modes in the wall-normal direction, and using 3D (non-Fourier) POD modes that can translate in the streamwise direction (traveling modes). Traveling modes capture more energy than standard Fourier-POD for the same number of modes. In addition, models of a channel flow linearized about a laminar profile are constructed using both POD and balanced POD, an approximation to balanced truncation that is tractable for very large systems. Balanced POD models significantly outperform standard POD, especially at including the effects of actuation.

We develop low-dimensional models for the evolution of a free shear layer in a periodic domain. The goal is to obtain models simple enough to be analyzed using standard tools from dynamical systems theory, yet including enough of the physics to model nonlinear saturation and energy transfer between modes (e.g., pairing). Recently, experiments have suggested that high-frequency forcing of shear layers over open cavities may provide a mechanism for suppression of tones in cavities, and a long-term goal of this work is to study the dynamics of forced shear layers, to better understand these effects. In the present paper, 2D direct numerical simulations of a spatially periodic, temporally developing shear layer are performed. Low-dimensional models for these dynamics are obtained using a modified version of proper orthogonal decomposition/Galerkin projection, in which the basis functions can scale in space as the shear layer spreads. Equations are obtained for the rate of change of the shear layer thickness. When scaling is included in the shear flow dominated by k=1 only, the first POD mode of wave number k=1 captures 93% of the energy, which is impossible to obtain by regular POD analysis without scaling. For the flow dominated by both k=1 and k=2, when scaling is included, the first POD mode of wave number k=1 and k=2 together capture 95% of the total energy. Projection of incompressible Navier-Stokes equations to the first two POD modes of k=1 gives a simple 2-mode model. If the projection is onto the first two POD modes of both k=1 and k=2, a more complex 4-mode model can be built to describe more complex flows. The 2-mode model can describe certain single-frequency features of the system, such as vortex roll-up, nonlinear saturation, and viscous damping. The 4-mode model can describe interactions between two frequencies (vortex merging) as well. The relation between the phase difference of the first (symmetric) and second (asymmetric) POD modes of the same wave number and the shear layer spreading rate can be clearly observed in both direct numerical simultions and model computations.

Active flow control of cavity tones has been successful in controlling Rossiter tones in both subsonic and supersonic flows. While closed-loop control promises the lowest actuator power requirements and the ability to adapt to changing flow conditions, so far it has only been demonstrated on subsonic cavities. Suppression of Rossiter tones in supersonic cavities has been demonstrated using open-loop techniques. Progress and issues related to closed-loop control and controlling supersonic cavities are reviewed.

Models for understanding and controlling oscillations in the flow past a rectangular cavity are developed. These models may be used to guide control designs, to understand performance limits of feedback, and to better understand experimental results. Traditionally, cavity oscillations are assumed to be self-sustained: no external disturbances are necessary to maintain the oscillations, and amplitudes are limited by nonlinearities. We present experimental data which suggests that in some regimes, the oscillations may not be self-sustained, but rather lightly damped: oscillations are sustained by external forcing, such as boundary layer turbulence. In these regimes, linear models suffice to describe the behavior, and the final amplitude of oscillations depends on the characteristics of the external disturbances. These linear models are particularly appropriate for describing cavities in which feedback has been used for noise suppression, as the oscillations are small and nonlinearities are less likely to be important. It is shown that increasing the gain too much in such feedback control experiments can lead to a peak-splitting phenomenon, which is explained by the linear models. Fundamental performance limits indicate that peak splitting is likely to occur for narrow-bandwidth actuators and controllers.

We review recent advances in understanding, modeling, and controlling oscillations in the flow past a cavity. The fundamental mechanisms underlying cavity flow oscillations have been known for at least 40 years, but suppressing these oscillations in a reliable and robust way is still a challenge today. Interest in controlling the flow past a cavity is motivated by aerospace applications, but in addition, cavity flows provide an attractive canonical problem for exploring general flow control techniques. The focus is on recent advances in modeling these flows, and in controlling them, using both open-loop and closed-loop techniques. A relatively new perspective is that cavity oscillations may not always be self-sustained, but under some flow conditions may be lightly-damped resonances, sustained by external disturbances such as boundary layer turbulence. Areas in which our understanding is incomplete, and which deserve further study are discussed, in particular the effects of high-frequency open-loop forcing, fundamental limitations of feedback control for a given configuration of sensors and actuators, and the development of a feedback design methodology that respects the limited range of validity of the available dynamical models.

Techniques for feedback control of oscillations in the flow past a cavity are presented. Low-order models are obtained using two methods (empirical Galerkin models, and a simple nonlinear oscillator model), and validated against 2D direct numerical simulations of the flow, which is actuated by a body force at the leading edge of the cavity. The models are used to construct dynamic observers, which reconstruct the flow state from a single pressure sensor, and perform much better than static estimators commonly used for flow estimation. Several control approaches are compared, including simple proportional control with a phase lag, LQG control using Galerkin models, and a dynamic phasor approach based on the work of Noack et al (2003). All three controllers are implemented in the full simulation, and able to reduce the amplitude of oscillations. The LQG regulator requires careful tuning, and the closed-loop behavior often does not match that predicted by the model, but the dynamic phasor approach eliminates the oscillations completely in the full simulation, with a transient response that matches that predicted by the low-order model.

This paper is concerned with modeling the dynamics of $N$ articulated solid bodies submerged in an ideal fluid. The model is used to analyze the locomotion of aquatic animals due to the coupling between their shape changes and the fluid dynamics in their environment. The equations of motion are obtained by making use of a two-stage reduction process which leads to significant mathematical and computational simplificiations. The first reduction exploits particle relabelling symmetry: that is, the symmetry associated with the conservation of circulation for ideal, incompressible fluids. As a result, the equations of motion for the submerged solid bodies can be formulated without explicitly incorporating the fluid variables. This reduction by the fluid variables is a key difference with earlier methods and it is appropriate since one is mainly interested in the location of the bodies, not the fluid particles. The second reduction is associated with the invariance of the dynamics under superimposed rigid motions. This invariance corresponds to the conservation of total momentum of the solid-fluid system. Due to this symmetry, the net locomotion of the solid system is realized as the sum of geometric and dynamic phases over the shape space consisting of allowable relative motions, or deformations, of the solids. In particular, reconstruction equations that govern the net locomotion at zero momentum, that is, the geometric phases, are obtained. As an illustrative example, a planar three-link mechanism is shown to propel and steer itself at zero momentum by periodically changing its shape. Two solutions are presented: one corresponds to a hydrodynamically decoupled mechanism and one is based on accurately computing the added inertias using a boundary element method. The hydrodynamically decoupled model produces smaller net motion than the more accurate model, indicating that it is important to consider the hydrodynamic interaction of the links.

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output projection method, which allows tractable computation even when the number of outputs is large. The output projection method requires minimal additional computation, and has a priori error bounds that can guide the choice of rank of the projection. Connections between POD and balanced truncation are also illuminated: in particular, balanced truncation may be viewed as POD of a particular dataset, using the observability Gramian as an inner product. The three methods are illustrated on a numerical example, the linearized flow in a plane channel.

Feedback control is used to suppress oscillations in the subsonic flow past a rectangular cavity. A heuristic feedback law is introduced into 2D direct numerical simulations, using zero-net-mass forcing at the leading edge of the cavity and a pressure sensor at the wall, and the oscillations are reduced by -13 dB. Reduced-order models are obtained from full-order direct numerical simulations, and used to design observers to reconstruct the full flow information from a single pressure measurement at the wall, and these observers are shown to be more effective than Linear Stochastic Estimation (LSE), which is commonly used for this purpose. Initial results of a new experiment are also given, and a model for designing zero-net-mass actuators with a desired bandwidth is presented, along with experimental data which supports the model predictions.

We present a framework for applying the method of Proper Orthogonal Decomposition (POD) and Galerkin projection to compressible fluids. For incompressible flows, only the kinematic variables are important, and the techniques are well known. In a compressible flow, both the kinematic and thermodynamic variables are dynamically important, and must be included in the configuration space. We introduce an energy-based inner product which may be used to obtain POD modes for this configuration space. We then obtain an approximate version of the Navier-Stokes equations, valid for cold flows at moderate Mach number, and project these equations onto a POD basis. The resulting equations of motion are quadratic, and are much simpler than projections of the full compressible Navier-Stokes equations.

We present a general method for analyzing and numerically solving partial differential equations with self-similar solutions. The method employs ideas from symmetry reduction in geometric mechanics, and involves separating the dynamics on the shape space (which determines the overall shape of the solution) from those on the group space (which determines the size and scale of the solution). The method is computationally tractable as well, allowing one to compute self-similar solutions by evolving a dynamical system to a steady state, in a scaled reference frame where the self similarity has been factored out. More generally, bifurcation techniques can be used to find self-similar solutions, and determine their behavior as parameters in the equations are varied. The method is given for an arbitrary Lie group, providing equations for the dynamics on the reduced space, for reconstructing the full dynamics, and for determining the resulting scaling laws for self-similar solutions. We illustrate the technique with a numerical example, computing self-similar solutions of the Burgers equation.

A review of active control of flow-induced cavity oscillations is presented. This paper is motivated by two factors. First, the search for solutions to the practical problem of suppressing oscillations caused by flow over open cavities has generated significant interest in this area. Second, cavity oscillation control serves as a model problem in the growing multidisciplinary field of flow control. As such, we attempt to summarize recent activities in this area, with emphasis on experimental implementation of open- and closed-loop control approaches. In addition to describing successes, failures, and outstanding issues relevant to cavity oscillations, we highlight the characteristics of the various actuators, flow sensing and measurement, and control methodologies employed to date in order to emphasize the choices, challenges, and potential of flow control in this and other applications.

This paper presents a series of experiments aimed at model-based feedback control of oscillations in the compressible flow past a rectangular cavity. Zero-net-mass air injection is used with feedback from pressure transducers located inside the cavity to suppress the oscillations, and measurements of the shear layer above the cavity are made for both the natural and suppressed state. A system identification experiment is performed, and used to determine a linear model that describes the oscillations, and this model is used to derive feedback laws for acoustic suppression. The model accurately predicts the behavior of different feedback controllers, including frequencies where suppression occurs, as well as new resonant frequencies that may be introduced by the controller.

We develop discrete mechanics and variational integrators for a class of degenerate Lagrangian systems, and apply these integrators to a system of point vortices. Excellent numerical behavior is observed. A longer term goal is to use these integration methods in the context of control of mechanical systems, such as coordinated groups of underwater vehicles. In fact, numerical evidence given in related problems, such as those in Kane et al. (Int.\ J.\ Num.\ Math.\ Eng.\ 49, 2000) shows that in the presence of external forces, these methods give superior predictions of energy behavior.

We use POD/Galerkin projection to investigate and derive reduced-order models of the dynamics of compressible flows. We examine DNS data for two flows, a turbulent M=0.9 jet and self-sustained oscillations in the flow over an open cavity, and show how different choices of norm lead to different definitions of the energetic structures, and, for the cavity, to different reduced-order models of the dynamics. For the jet, we show that the near-field dynamics are fairly well represented by relatively few modes, but that key quantities of interest, such as acoustic radiation, are not well captured by norms that are defined based on volume integrals of pressure and velocity. For the cavity flow, we demonstrate that vector-valued POD modes lead to reduced-order models that are much more effective (accurate and stable) than scalar-valued modes defined independently for different flow variables.

Numerical simulations are used to investigate the resonant instabilities in two dimensional flow past an open cavity. The compressible Navier-Stokes equations are solved directly (no turbulence model) for cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field, which is shown to be in good visual agreement with Schlieren photographs from experiments at several different Mach numbers. The results show a transition from a shear-layer mode, primarily for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The shear-layer mode is well characterized by the acoustic feedback process described by Rossiter (1964), and disturbances in the shear layer compare favorably with predictions based on linear stability analysis of the Kelvin-Helmholtz mode. The wake mode is characterized instead by a large-scale vortex shedding with Strouhal number independent of Mach number. The wake mode oscillation is similar in many ways to that reported by Gharib and Roshko (1987) for incompressible flow with a laminar upstream boundary layer. Transition to wake mode occurs as the length and/or depth of the cavity becomes large compared to the upstream boundary-layer thickness, or as the Mach and/or Reynolds numbers are raised. Under these conditions, it is shown that the Kelvin-Helmholtz instability grows to sufficient strength that a strong recirculating flow is induced in the cavity. The resulting mean flow is similar to wake profiles that are absolutely unstable, and absolute instability may provide an explanation of the hydrodynamic feedback mechanism that leads to wake mode. Predictive criteria for the onset of shear-layer oscillations (from steady flow) and for the transition to wake mode are developed based on linear theory for amplification rates in the shear layer, and a simple model for the acoustic efficiency of edge scattering.

Experiments using active control to reduce oscillations in the flow past a rectangular cavity have uncovered surprising phenomena: in the controlled system, often new frequencies of oscillation appear, and often the main frequency of oscillation is split into two sideband frequencies. The goal of this paper is to explain these effects using physics-based models, and to use these ideas to guide control design. We present a linear model for the cavity flow, based on the physical mechanisms of the familiar Rossiter model. Experimental data indicates that under many operating conditions, the oscillations are not self-sustained, but in fact are caused by amplification of external disturbances. We present some experimental results demonstrating the peak-splitting phenomena mentioned above, use the physics-based model to study the phenomena, and discuss fundamental performance limitations which limit the achievable performance of any control scheme.

An experimental investigation of acoustic mode noise suppression was conducted in a cavity using a digital controller with a linear control algorithm. The control algorithm was based on flow field physics similar to the Rossiter model for acoustic resonance. Details of the controller and results from its implementation are presented in the companion paper by Rowley, et al (2002). Here the experiments and some details of the flow field development are described, which were done primarily at Mach number 0.34 corresponding to single mode resonance in the cavity. A novel method using feedback control to suppress the resonant mode and open-loop forcing to inject a non-resonant mode was developed for system identification. The results were used to obtain empirical transfer functions of the components of resonance, and measurements of the shear layer growth for use in the design of the control algorithm.

We investigate nonlinear dynamical models for self-sustained oscillations in the flow past a rectangular cavity. The models are based on the method of Proper Orthogonal Decomposition (POD) and Galerkin projection, and we introduce an inner product and formulation of the equations of motion which enables one to use vector-valued POD modes for compressible flows. We obtain models between 3 and 20 states, which accurately describe both the short-time and long-time dynamics. This is a substantial improvement over previous models based on scalar-valued POD modes, which capture the dynamics for short time, but deviate for long time.

We present a method for applying the Karhunen-Loève decomposition to systems with continuous symmetry. The techniques in this paper contribute to the general procedure of removing variables associated with the symmetry of a problem, and related ideas have been used in previous works both to identify coherent structures in solutions of PDEs, and to derive low-order models via Galerkin projection. The main result of this paper is to derive a simple and easily implementable set of reconstruction equations which close the system of ODEs produced by Galerkin projection. The geometric interpretation of the method closely parallels techniques used in geometric phases and reconstruction techniques in geometric mechanics. We apply the method to the Kuramoto-Sivashinsky equation and are able to derive accurate models of considerably lower dimension than are possible with the traditional Karhunen-Loève expansion.

The goal of this work is to provide accurate dynamical models of oscillations in the flow past a rectangular cavity, for the purpose of bifurcation analysis and control. We have performed an extensive set of direct numerical simulations which provide the data used to derive and evaluate the models. Based on the method of Proper Orthogonal Decomposition (POD) and Galerkin projection, we obtain low-order models (from 6 to 60 states) which capture the dynamics very accurately over a few periods of oscillation, but deviate for long time.

Many compressible flow and aeroacoustic computations rely on accurate nonreflecting or radiation boundary conditions. When the equations and boundary conditions are discretized using a finite-difference scheme, the dispersive nature of the discretized equations can lead to spurious numerical reflections not seen in the continuous boundary value problem. Here we construct discretely nonreflecting boundary conditions, which account for the particular finite-difference scheme used, and are designed to minimize these spurious numerical reflections. Stable boundary conditions that are local and nonreflecting to arbitrarily high order of accuracy are obtained, and test cases are presented for the linearized Euler equations. For the cases presented, reflections for a pressure pulse leaving the boundary are reduced by up to two orders of magnitude over typical ad hoc closures, and for a vorticity pulse, reflections are reduced by up to four orders of magnitude.

The modes of oscillation and radiated acoustic fields of compressible flows over open cavities are investigated computationally. The compressible Navier-Stokes equations are solved directly (no turbulence model) for two dimensional open cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field. The results show a bifurcation from a shear layer mode, for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The shear layer mode is well characterized by Rossiter modes and these oscillations lead to intense upstream acoustic radiation dominated by a single frequency. The wake mode is characterized instead by a large-scale vortex shedding with Strouhal number nearly independent of the Mach number. The vortex shedding causes the boundary layer to periodically separate upstream of the cavity. Acoustic radiation is more intense, with multiple frequencies present. The wake mode oscillation is similar to that reported by Gharib and Roshko (1987) for incompressible cavity flows with laminar upstream boundary layers.

Numerical simulations are used to investigate the resonant instabilities in the flow past an open cavity. The compressible Navier-Stokes equations are solved directly (no turbulence model) for two-dimensional cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field. The results show a transition from a shear layer mode, for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The shear layer mode is well characterized by Rossiter modes. The wake mode is characterized instead by a large-scale vortex shedding with Strouhal number independent of the Mach number. The vortex shedding causes the boundary layer to periodically separate upstream of the cavity. The wake mode oscillation is similar to that reported by Gharib and Roshko (J. Fluid Mech., 177, 1987) for incompressible flow with a laminar upstream boundary layer. The results suggest that laminar separation upstream of the cavity edge is the cause of the transition to wake mode.

Many compressible flow and aeroacoustic computations rely on accurate nonreflecting or radiation boundary conditions. When the equations and boundary conditions are discretized using a finite-difference scheme, the dispersive nature of the discretized equations can lead to spurious numerical reflections not seen in the continuous boundary value problem. These reflections can lead to poor convergence to a stationary state, and can lead to self-forcing of flows. We have constructed numerically nonreflecting boundary conditions which account for the particular finite-difference scheme used, and are designed to minimize these spurious numerical reflections. These extend our earlier work on one-dimensional boundary conditions to the multidimensional case. Stable boundary conditions which are nonreflecting to arbitrarily high-order-of-accuracy are obtained. Various test cases are presented which show excellent results.