Discrete Vortex Model Research Articles

Iterative Maneuver Optimization in a Transverse Gust Encounter

This paper presents a framework based on either iterative simulation or iterative experimentation for constructing an optimal, open-loop maneuver to regulate the aerodynamic force on a wing in the presence of a known flow disturbance. The authors refer to the method as iterative maneuver optimization and apply it in this paper to regulate lift on a pitching wing during a transverse gust encounter. A candidate maneuver is created by performing an optimal control calculation on a surrogate model of the wing–gust interaction. Execution of the proposed maneuver in a high-fidelity simulation or experiment provides an error signal based on the difference between the force predicted by the surrogate model and the measured force. The error signal provides an update to the reference signal used by the surrogate model for tracking. A new candidate maneuver is calculated such that the surrogate model tracks the reference force signal, and the process repeats until the maneuver adequately regulates the force. The framework for iterative maneuver optimization is tested on a discrete vortex model as well as in experiments in a water towing tank. Experimental results show that the proposed framework generates a maneuver that reduces the magnitude of lift overshoot by 92% for a trapezoidal gust with peak velocity equal to approximately 0.7 times the freestream flow speed.

Energy Harvesting Performance of Thick Oscillating Airfoils Using a Discrete Vortex Model

Abstract The energy harvesting performance of thick oscillating airfoils is predicted using an inviscid discrete vortex model (DVM). National Advisory Committee for Aeronautics (NACA) airfoils with different leading-edge geometries are modeled that undergo sinusoidal heaving and pitching with reduced frequencies, k=fc/U∞, in the range 0.06−0.14, where f is the heaving frequency of the foil, c is the chord length, and U∞ is the freestream velocity. The airfoil pitches about the midchord with heaving and pitching amplitudes of h0=0.5c and θ0=70 deg, respectively, known to be in the range of peak energy harvesting efficiencies. A vortex shedding initiation criteria is proposed based on the transient local wall stress distribution determined from computational fluid dynamics (CFD) simulations and incorporates both timing and location of leading-edge separation. The scaled shedding times are shown to be predicted over the range of reduced frequencies using a timescale based on the leading-edge shear velocity and radius of curvature. The convection velocity of the shed vortices is also modeled based on the reduced frequency to better capture the dynamics of the leading-edge vortex. An empirical trailing-edge separation correction is applied to the transient force results using the effective angle of attack modified to include the pitching component. Impulse theory is applied to the DVM to calculate the transient lift force and compares well with the CFD simulations. Results show that the power output increases with increasing airfoil thickness and is most notable at higher reduced frequencies where the power output efficiency is highest.

A leading-edge vortex initiation criteria for large amplitude foil oscillations using a discrete vortex model

A leading-edge vortex initiation criterion of an oscillating airfoil is shown to provide a means to predict the onset of leading-edge separation. This result is found to collapse the occurrence of separation during the oscillation cycle when scaled using the leading-edge shear velocity determined from the foil oscillating kinematic motion. This is of importance in developing low order models for predicting energy harvesting performance, as is shown in this study using an inviscid discrete vortex model (DVM). Results are obtained for a thin flat airfoil undergoing sinusoidal heaving and pitching motions with reduced frequencies of k=fc/U∞ in the range of 0.06–0.16, where f is the heaving frequency of the foil, c is the chord length, and U∞ is the freestream velocity. The airfoil pitches about the mid-chord, and the heaving and pitching amplitudes of the airfoil are ho=0.5c and θ0=70°, respectively, illustrating results for conditions near peak efficiency for energy harvesting. The DVM uses a panel method, applicable to a wide range of foil geometries. An empirical trailing-edge separation correction is also applied to the transient force results. The vortex shedding criterion is based on the transient local wall stress distribution determined using a computational fluid dynamics (CFD) simulation, indicating the time and location of zero stress at the foil surface. In addition, the local pressure gradient minimum is also used as a local indicator. The effects of a wide range of Reynolds numbers on separation are shown for the given range of reduced frequencies. The use of the effective angle of attack, when modified to include the pitching component, is also shown to correlate the leading-edge vortex initiation time. The advantage of the proposed separation criteria is that it can be fully determined from the motion kinematics and then applied to a wide range of low order models. Model results are given for the transient lift force and compare well with the CFD simulations. It is noted that at higher reduced frequencies the DVM overpredicts portions of the transient loads possibly caused by the reversed viscous flow from the trailing edge.

Method of Unsteady Hydrodynamic Characteristics Determination in Turbulent Boundary Layer

This paper presents the method of the turbulent flow simulation. The method may be used to address the computational aeroacoustics (CAA) problems, where the vortex noise’s sources have to be determined. This method is an alternative to both large-eddy simulation (LES) methods and stochastics turbulence simulation techniques. The proposed method is more computationally efficient compared to LES and, unlike stochastic approaches, it does not require empirical constants. The simulation according to this method is achieved in two main stages. During the first step the averaged flow’s properties are obtained using the RANS simulation. These properties are used for the formulation of the discrete vortex model on the second step. Vortices’ intensities are oscillating with amplitudes and frequencies obtained from the RANS simulation with random phase shifts. Turbulent velocity field is then determined as the sum of averaged flow velocities, velocities induced by the pulsing vortices and velocities induced by the trailing vortices (Kelvin circulation theorem). The method is verified by considering the test problem. The developed turbulent boundary layer near the horizontal wall is simulated by means of both the presented method and the LES method. A good agreement between these two methods indicates on the viability of the approach presented in this paper. However, a thorough investigation of the method is still yet to be accomplished.

Unsteady aerodynamics of lift regulation during a transverse gust encounter

A discrete vortex model is utilized to simulate the unsteady aerodynamics of a wing encountering a large-amplitude transverse gust. A closed-loop control law based on pitching input is selected to regulate lift during the gust encounter. The unsteady flow physics of the gust encounter under closed-loop pitch actuation is studied and the total lift force is decomposed into individual contributions to study the various mechanisms with which closed-loop control achieves lift regulation.

Shock-driven hydrodynamic instability of a sinusoidally perturbed, high-Atwood number, oblique interface

A shock incident on an interface will cause any initial perturbations on that interface to grow. When the shock front is parallel to the interface, the perturbations grow due to the Richtmyer-Meshkov (RM) process. When there is some tilt between the shock front and the interface, shear flow will result across the postshock interface. Recent experiments on the OMEGA EP laser have studied the hydrodynamic instability growth which results from a supported shock interacting with a sinusoidally perturbed, oblique interface. The observed instability growth was dominated by Richtmyer-Meshkov at early times but became Kelvin-Helmholtz (KH)-like at late times. Previously, this instability growth was described using an analytic model for the deposition of baroclinic vorticity on the interface by a shock combined with a discrete vortex model. Here, we utilize the same baroclinic vorticity deposition model in conjunction with a desingularized, periodic Birkhoff-Rott equation to model instability evolution. The Birkhoff-Rott equation takes into account the vorticity distribution along the interface, whereas the discrete vortex model assumed that all vorticity over each wavelength of the perturbation is confined to a point. We compare the new model to xRAGE simulations and experiments. The model is found to overpredict both the instability growth and shear across the interface by about a factor of two, but correctly predicts that the growth is RM-like at early times and KH-like at late times.

Shock-driven discrete vortex evolution on a high-Atwood number oblique interface

We derive a model describing vorticity deposition on a high-Atwood number interface with a sinusoidal perturbation by an oblique shock propagating from a heavy into a light material. Limiting cases of the model result in vorticity distributions that lead to Richtmyer-Meshkov and Kelvin-Helmholtz instability growth. For certain combinations of perturbation amplitude, wavelength, and tilt of the shock, a regime is found in which discrete, co-aligned, vortices are deposited on the interface. The subsequent interface evolution is described by a discrete vortex model, which is found to agree well with both RAGE simulations and experiments at early times.

The ‘hanging flag’ problem: on the heaving motion of a thin filament in the limit of small flexural stiffness

We investigate the fluid–structure interaction of a vertically hanging filament immersed in uniform incompressible high Reynolds number flow. The filament is subject to small-amplitude harmonic heaving at its upstream edge, and to a gravity-induced (‘hanging chain’) tension force. We focus on the limit of small bending rigidity to examine the differences between a highly elastic beam (where bending rigidity is small but finite) and a membrane (where bending rigidity vanishes). The problem is analysed by means of thin airfoil theory, in conjunction with a discrete vortex model for the downstream wake. Denoting the filament non-dimensional rigidity (normalized by the tension force) by $\bar{\unicode[STIX]{x1D700}}$, it is first verified that the beam deflection and associated flow field converge to the membrane solution at $\bar{\unicode[STIX]{x1D700}}\rightarrow 0$. At low actuation frequencies, the differences between the membrane and beam motions are small, and both follow a nearly rigid-body motion parallel to the upstream-edge actuation. With increasing frequency, the differences between the beam and membrane become visible at increasingly lower values of $\bar{\unicode[STIX]{x1D700}}$, and the stabilizing effect of beam flexural rigidity, resulting in reduced flapping amplitudes, is apparent. Examining the beam motion near its edge points at non-small frequencies, semi-analytic approximations for the associated time-periodic displacements are obtained. Close to the actuated end, a layer of width $\bar{\unicode[STIX]{x1D700}}^{1/2}$ is found, where the flexural rigidity term in the equation of motion is balanced by the tension term. Here, the differences between the beam and membrane deflections are attributed to the additional zero-slope condition satisfied by the former. In the vicinity of the free end, a local Taylor expansion is carried out. A balance between the bending and inertia terms results in a layer of width $\propto \bar{\unicode[STIX]{x1D700}}^{1/4}/\bar{\unicode[STIX]{x1D714}}_{h}^{1/2}$, where $\bar{\unicode[STIX]{x1D714}}_{h}$ denotes the scaled heaving frequency. The layer is therefore thicker than the upstream layer for $\bar{\unicode[STIX]{x1D714}}_{h}\approx 1$, and becomes thinner with increasing $\bar{\unicode[STIX]{x1D714}}_{h}$. Within the layer, the beam deflects linearly with the distance from the edge, in marked difference from a membrane and in accordance with the free-end conditions satisfied by the former.

Prediction of vortex-shedding noise from the blunt trailing edge of a flat plate

A time-domain hybrid approach for aerodynamic noise prediction is developed based on a discrete vortex model (DVM) for the unsteady incompressible flow simulation and the acoustic perturbation equations (APE) for the acoustical field computation. The aim is to assess the applicability of the present DVM-APE method to the problems where sound is generated by the large-scale coherent flow structures. The hybrid DVM-APE approach is employed to predict the vortex-shedding noise from the blunt trailing edge of a flat plate. Simulations are implemented on flat plates with different thicknesses in a certain range of low Mach numbers, in order to identify the scaling dependence of the vortex-shedding noise on the freestream speed as well as the plate thickness. Acoustical directivity patterns at different Helmholtz numbers are presented, and agreements are achieved when compared with previous studies. A comparison of the sound pressure level spectrum between the present DVM-APE simulation and the published experimental results is also presented, showing good agreements for both the peak frequencies and the sound pressure levels.

On the attenuating effect of permeability on the low frequency sound of an airfoil

The effect of structure permeability on the far-field radiation of a thin airfoil is studied. Assuming low-Mach and high-Reynolds number flow, the near- and far-field descriptions are investigated at flapping-flight and unsteady flow conditions. Analysis is carried out using thin-airfoil theory and compact-body-based calculations for the hydrodynamic and acoustic fields, respectively. Airfoil porosity is modeled via Darcy׳s law, governed by prescribed distribution of surface intrinsic permeability. Discrete vortex model is applied to describe airfoil wake evolution. To assess the impact of penetrability, results are compared to counterpart predictions for the sound of an impermeable airfoil. Considering the finite-chord airfoil as “acoustically transparent”, the leading-order contribution of surface porosity is obtained in terms of an acoustic dipole. It is shown that, at all flow conditions considered, porosity causes attenuation in outcome sound level. This is accompanied by a time-delay in the pressure signal, reflecting the mediating effect of permeability on the interaction of fluid flow with airfoil edge points. To the extent that thin-airfoil theory holds (requiring small normal-to-airfoil flow velocities), the results indicate on a decrease of ~10percent and more in the total energy radiated by a permeable versus an impermeable airfoil. This amounts to a reduction in system sound pressure level of 3dB and above at pitching flight conditions, where the sound-reducing effect of the seepage dipole pressure becomes dominant. The applicability of Darcy׳s law to model the effect of material porosity is discussed in light of existing literature.

Flow-excited acoustic resonance of a Helmholtz resonator: Discrete vortex model compared to experiments

The acoustic resonance in a Helmholtz resonator excited by a low Mach number grazing flow is studied theoretically. The nonlinear numerical model is established by coupling the vortical motion at the cavity opening with the cavity acoustic mode through an explicit force balancing relation between the two sides of the opening. The vortical motion is modeled in the potential flow framework, in which the oscillating motion of the thin shear layer is described by an array of convected point vortices, and the unsteady vortex shedding is determined by the Kutta condition. The cavity acoustic mode is obtained from the one-dimensional acoustic propagation model, the time-domain equivalent of which is given by means of a broadband time-domain impedance model. The acoustic resistances due to radiation and viscous loss at the opening are also taken into account. The physical processes of the self-excited oscillations, at both resonance and off-resonance states, are simulated directly in the time domain. Results show that the shear layer exhibits a weak flapping motion at the off-resonance state, whereas it rolls up into large-scale vortex cores when resonances occur. Single and dual-vortex patterns are observed corresponding to the first and second hydrodynamic modes. The simulation also reveals different trajectories of the two vortices across the opening when the first and second hydrodynamic modes co-exist. The strong modulation of the shed vorticity by the acoustic feedback at the resonance state is demonstrated. The model overestimates the pressure pulsation amplitude by a factor 2, which is expected to be due to the turbulence of the flow which is not taken into account. The model neglects vortex shedding at the downstream and side edges of the cavity. This will also result in an overestimation of the pulsation amplitude.

Suppression of Helmholtz resonance using inside acoustic liner

When a Helmholtz resonator is exposed to grazing flow, an unstable shear layer at the opening can cause the occurrence of acoustic resonance under appropriate conditions. In this paper, in order to suppress the flow-induced resonance, the effects of inside acoustic liners placed on the side wall or the bottom of a Helmholtz resonator are investigated. Based on the one-dimensional sound propagation theory, the time domain impedance model of a Helmholtz resonator with inside acoustic liner is derived, and then combined with a discrete vortex model the resonant behavior of the resonator under grazing flow is simulated. Besides, an experiment is conducted to validate the present model, showing significant reduction of the peak sound pressure level achieved by the use of the side-wall liners. And the simulation results match reasonably well with the experimental data. The present results reveal that the inside acoustic liner can not only absorb the resonant sound pressure, but also suppress the fluctuation motion of the shear layer over the opening of the resonator. In all, the impact of the acoustic liners is to dampen the instability of the flow-acoustic coupled system. This demonstrates that it is a convenient and effective method for suppressing Helmholtz resonance by using inside acoustic liner.

Two-Dimensional Potential Method Simulations of a Body–Strake Configuration

The lee-side flow over a cruciform wing–body configuration at supersonic speeds in the “” orientation up to angles of 25 deg have been simulated using the single concentrated vortex and discrete vortex model methods. The wings or strakes are of very low aspect ratio of order 0.025 with taper ratio with a length of mounted on a tangent ogive body. The simulations revealed that the single concentrated vortex model is only suitable at low angles, while the discrete vortex model can predict both the flowfield and normal force reasonably when the lee-side boundary-layer separation characteristics are modeled. The resulting secondary vortex due to the wing–body junction has been identified to significantly influence the lee-side flow topology, and its exclusion results in higher normal forces and incorrect prediction of vortex positions.

On the acoustic radiation of a pitching airfoil

We examine the acoustic far field of a thin elastic airfoil, immersed in low-Mach non-uniform stream flow, and actuated by small-amplitude sinusoidal pitching motion. The near-field fluid-structure interaction problem is analyzed using potential thin-airfoil theory, combined with a discrete vortex model to describe the evolution of airfoil trailing edge wake. The leading order dipole-sound signature of the system is investigated using Powell-Howe acoustic analogy. Compared with a pitching rigid airfoil, the results demonstrate a two-fold effect of structure elasticity on airfoil acoustic field: at actuation frequencies close to the system least stable eigenfrequency, elasticity amplifies airfoil motion amplitude and associated sound levels; however, at frequencies distant from this eigenfrequency, structure elasticity acts to absorb system kinetic energy and reduce acoustic radiation. In the latter case, and with increasing pitching frequency ωp, a rigid-airfoil setup becomes significantly noisier than an elastic airfoil, owing to an \documentclass[12pt]{minimal}\begin{document}$\omega _p^{5/2}$\end{document}ωp5/2 increase of its direct motion noise component. Unlike rigid airfoil signature, it is shown that wake sound contribution to elastic airfoil radiation is significant for all ωp. Remarkably, this contribution contains, in addition to the fundamental pitching frequency, its odd multiple harmonics, which result from nonlinear interactions between the airfoil and the wake. The results suggest that structure elasticity may serve as a viable means for design of flapping flight noise control methodologies.

Discrete vortex model of a Helmholtz resonator subjected to high-intensity sound and grazing flow

In this paper, a theoretical model is developed to study the acoustical response of a Helmholtz resonator as a duct-branched acoustic absorber subjected to both high-intensity sound and grazing flow. The present model is comprised of a discrete vortex model in combination with a one-dimensional duct sound propagation model. The present work is to study the overall effect of incident sound interacting with grazing flow but putting emphasis on the nonlinear or intermediate regime where the sound intensity has a marked or non-negligible influence on the acoustic behavior of the Helmholtz resonator. The numerical results reveal that the flow field around the orifice is dominated by the evolution of the vortex sheet and the flow pattern is influenced by the ratio of the orifice flow velocity to the grazing flow velocity. When the incident sound pressure is high or the resonance occurs, the resonator shows nonlinearity, i.e., the acoustic impedance and absorption coefficient vary not only with duct flow Mach number buy also with incident frequency and incident sound pressure level.

A discrete-vortex model for the arbitrary motion of a thin airfoil with fluidic control

A low-order model for the arbitrary motion of a thin airfoil with trailing edge fluidic control is derived from basic fluid mechanics principles. The model consist of solving a single ordinary differential equation with a special treatment of a vortex shedding criteria. The model is compared with experimental and high-order numerical simulations and the results give a reasonable means of predicting the lift and moment on a thin airfoil. Furthermore, the model is extended to account for the actuation and control due to the synthetic jet actuation near the trailing edge. The model response is compared to experimental results.

Discretization of the vorticity field of a planar jet

In data assimilation, information from sensors is used to correct the state variables of a numerical model. This has been used to great advantage by the weather prediction community in the context of direct numerical simulation (DNS) models, but has seen comparatively little use in point-vortex models. This is due in large part to data-processing issues. In order to keep up with the speeds necessary for effective data assimilation, one must extract and discretize the vortex structures from velocity field data in a computationally efficient fashion—i.e., using as few discrete vortices as possible to model the measured flow. This paper describes a new strategy for accomplishing this and evaluates the results using data from a laboratory-scale vortex-dominated planar jet. Large-scale vortex structures are found using a family of variants on traditional vortex extraction methods. By augmenting these methods with simple computational topology techniques, one obtains a new method that finds the boundaries of the coherent structures in a manner that naturally follows the geometry of the flow. This strategy was evaluated in the context of two standard vortex extraction methods, vorticity thresholding and Okubo–Weiss, and tested upon velocity field data from the experimental fluid flow. The large-scale structures found in this manner were then modeled with collections of discrete vortices, and the effects of the grain size of the discretization and the parameters of the discrete vortex model were studied. The results were evaluated by comparing the instantaneous velocity field induced by the discrete vortices to that measured in the jet. These comparisons showed that the two extraction techniques were comparable in terms of sensitivity and error, suggesting that the computationally simpler vorticity thresholding method is more appropriate for applications where speed is an issue, like data assimilation. Comparisons of different discretization strategies showed that modeling each large-scale vortex structure with a single discrete vortex provided the best compromise between mean-squared error and computational effort. These results are of potential interest in any situation where one must balance accuracy and expense while extracting vortices from a snapshot of a flow field; data assimilation is only one example.

Elementary Vortex Filament Model of the Discrete Nonlinear Schrödinger Equation

The discretization of a vortex filament is studied. A discrete vortex model and a discrete version of the local induction approximation technique (LIA), which guarantees that the vortex filament obeys the discrete nonlinear Schrodinger equation, is proposed.

The optical distortion mechanism in a nearly incompressible free shear layer

The aero-optical distortions caused by compressible flows have been used by researchers for flow diagnostics and accepted by designers of airborne optical systems as a performance penalty. In order to estimate these distortions, an understanding of the optical distortion mechanism is required. This article examines the mechanisms which produce a variable-density field (and accompanying index-of-refraction field) in a nearly incompressible shear-layer flow. The two-dimensional-shear-layer velocity field was approximated using a discrete vortex model. From this ‘known’ velocity field, the pressure and density fields were determined by iteratively solving the unsteady Euler equations. The resulting index-of-refraction field produced simulated schlieren images which closely resemble experimental schlierens. Optical wavefronts computed from the simulation reasonably match the behaviour of large-scale aberrations measured in a transonic wind tunnel. Small-scale distortions in the experimental data may have been caused by boundary layers on the splitter plate and tunnel walls or by three-dimensional effects that were not simulated.

Simplified models of flue instruments: influence of mouth geometry on the sound source.

Flue instruments such as the recorder flute and the transverse flute have different mouth geometries and acoustical response. The effect of the mouth geometry is studied by considering the aeroacoustical response of a simple whistle. The labium of a transverse flute has a large edge angle (60 degrees) compared to that of a recorder flute (15 degrees). Furthermore, the ratio W/h of the mouth width W to the jet thickness h can be varied in the transverse flute (lips of the musician) while it is fixed to a value W/h approximately 4 in a recorder flute. A systematic experimental study of the steady oscillation behavior has been carried out. Results of acoustical pressure measurements and flow visualization are presented. The sharp edge of the recorder provides a sound source which is rich in harmonics at the cost of stability. The larger angle of the labium of the flute seems to be motivated by a better stability of the oscillations for thick jets but could also be motivated by a reduction of broadband turbulence noise. We propose two simplified sound source models which could be used for sound synthesis: a jet-drive model for W/h>2 and a discrete-vortex model for W/h<2.