Unsteady Kutta Condition Research Articles

A lightweight vortex model for unsteady motion of airfoils

A low-order vortex model has been developed for analysing the unsteady aerodynamics of airfoils. The model employs an infinitely thin vortex sheet in place of the attached boundary layer and a sheet of point vortices for the shed shear layer. The strength and direction of the vortex sheet shed at the airfoil trailing edge are determined by an unsteady Kutta condition. The roll-up of the ambient shear layer is represented by a unique point vortex, which is consistently fed circulation by the last point vortex of the free vortex sheet. The model's dimensionality is reduced by using three tuning parameters to balance representational accuracy and computational efficiency. The performance of the model is evaluated through experiments involving impulsively started and heaving and pitching airfoils. The model accurately captures the dynamics of the development and evolution of the shed vortical structure while requiring minimal computational resources. The validity of the model is confirmed through comparison with experimental force measurements and a baseline unsteady panel method that does not transfer circulation in the free vortex sheet.

The unsteady Kutta condition on an airfoil in a surging flow

This work presents an experimental validation study of Isaacs’ incompressible unsteady-airfoil theory at Reynolds numbers above $10^{6}$ , and explores the validity of the classical Kutta condition applied to surging flows. Harmonic variation of the free-stream velocity was produced by rotating choke vanes in an unsteady transonic wind tunnel, with time-resolved lift coefficients determined from surface pressure measurements on a NACA 0018 airfoil. Unsteady lift results demonstrate the same trends with reduced frequency and velocity amplitude ratio that are predicted by Isaacs’ theory. However, significant deviations of the lift magnitude and phase angle are observed. In order to understand the cause of these deviations, the background-oriented schlieren technique was used to visualize density gradients in the immediate vicinity of the airfoil trailing edge. The time-resolved background-oriented schlieren displacement field indicates oscillatory behaviour of the trailing-edge stagnation streakline, which violates the classical Kutta condition for this unsteady surging flow.

An inviscid model of unsteady separated vortical flow for a moving plate

An inviscid vortex shedding model for separated vortices from a solid body is studied. The model describes the separated vortices by vortex sheets and the attached flow via conformal mapping. We develop a computational model to simulate the vortex shedding of a moving body, with varying angle. An unsteady Kutta condition is imposed on the edges of the plate to determine the edge circulations and velocities. The force on the plate is obtained by integrating the unsteady Blasius equation. We apply the model to two representative cases of an accelerated plate, with impulsive start and uniform acceleration, and investigate the dynamics for large angles of attack. For both cases, the vortex force is dominant in the lift over times. The lift coefficients are initially high and decrease in four chord lengths of displacement, in general. For large angles of attack, the appearance of a peak of lift at an early time depends on the power-law velocity, which differs from the behavior for small angles of attack. The lift and drag from the model are in agreement with the Navier–Stokes simulation and experiment for moderate Reynolds numbers. We also demonstrate the vortex shedding of hovering and flapping plates. In the hovering motion, the large increase in lift at the early backward translation is due to the combined effect of the vortex force and added mass force. In the flapping plate, our model provides an improvement in the prediction for the induced force than other shedding models.

Unsteady aerodynamic model of flexible flapping wing

Bio-inspired flapping wing has potential application to micro air vehicles (MAV). Due to the nature of lightweight and flexibility of micro flapping wing structures, elastic deformation as a result of aeroelastic coupling is inevitable in flapping motion. This effect can be significant and beneficial to the aerodynamic performance as revealed in the present investigation for a flexible flapping wing of variable camber versus a rigid one. Firstly a two dimensional (2D) unsteady aerodynamic model (UAM) based on potential flow theory has been extended from previous study. Both leading and trailing edge discrete vortices are included in the model with unsteady Kutta condition satisfied to fully characterize the unsteady flow around a flapping wing. A wall function is created to modify the induced velocity of the vortices in the UAM to solve the vortices penetration problem. The modified UAM is then validated by comparing with CFD results of a typical insect-like flapping motion from previous research. Secondly the UAM is further extended for a flexible flapping wing of camber variation. Comparing with a rigid wing in a prescribed plunging and pitching motion, the results show lift increase with positive camber in upstroke by mitigating negative lift. The results also agree well with CFD simulation. Thirdly the 2D UAM is extended to calculate the aerodynamic forces of a 3D wing with camber variation, and validated by CFD results. Finally the model is applied to aerodynamic analysis of a 3D flexible flapping wing with aeroelastic coupling effect. Significant increase of lift coefficient can be achieved for a flexible flapping wing of positive camber and twist in upstroke produced by the structure elastic deformation.

Unsteady aerodynamics and vortex-sheet formation of a two-dimensional airfoil

Unsteady inviscid flow models of wings and airfoils have been developed to study the aerodynamics of natural and man-made flyers. Vortex methods have been extensively applied to reduce the dimensionality of these aerodynamic models, based on the proper estimation of the strength and distribution of the vortices in the wake. In such modelling approaches, one of the most fundamental questions is how the vortex sheets are generated and released from sharp edges. To determine the formation of the trailing-edge vortex sheet, the classical steady Kutta condition can be extended to unsteady situations by realizing that a flow cannot turn abruptly around a sharp edge. This condition can be readily applied to a flat plate or an airfoil with cusped trailing edge since the direction of the forming vortex sheet is known to be tangential to the trailing edge. However, for a finite-angle trailing edge, or in the case of flow separation away from a sharp corner, the direction of the forming vortex sheet is ambiguous. To remove anyad hocimplementation, the unsteady Kutta condition, the conservation of circulation as well as the conservation laws of mass and momentum are coupled to analytically solve for the angle, strength and relative velocity of the trailing-edge vortex sheet. The two-dimensional aerodynamic model together with the proposed vortex-sheet formation condition is verified by comparing flow structures and force calculations with experimental results for several airfoil motions in steady and unsteady background flows.

Computation of the Starting Vortex Flow Past a Flat Plate

This paper compares two numerical methods applied to compute the starting vortex flow past a flat plate. The plate is inclined relative to a constant background flow at angle α, with α = 90°, 60°, 30°. The numerical methods considered are (1) direct numerical simulation of the viscous flow (DNS), and (2) an inviscid vortex sheet model. The viscous DNS solves the Navier- Stokes equations by an operator splitting finite-difference method, for Reynolds numbers Re = 250, 500, 1000, 2000. The inviscid flow is computed by a regularized vortex sheet method, with the unsteady Kutta condition imposed at the edges of the plate, for regularization parameters δ = 0.2, 0.1, 0.05. We present viscous vorticity contours, and compare streaklines and shed circulation obtained with both methods. Good agreement is found in the large-scale features of the separated spiral streaklines and the shed circulation as Re increases and δ decreases. For small inclination angle α, secondary separation on the downwind side of the plate introduces small-scale features in the viscous flow that are absent in the inviscid model. The vortex sheet model is much less costly than the viscous DNS, but it is limited by the omission of the boundary layers present in the viscous flow.

The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction

The Kutta condition is applied to aerofoils with sharp trailing edges to allow for viscous effects to be considered within a simplified system of equations that are inviscid. This paper discusses in detail the inclusion of an unsteady Kutta condition at a sharp trailing edge during gust–aerofoil interaction and illustrates how the analytic solution for the far-field noise generated by this interaction changes if the unsteady Kutta condition is neglected, or more precisely, if the unsteady pressure is permitted to be singular at the trailing edge. The analytic solution, both with and without the unsteady Kutta condition, is compared with numerical results that have no imposed unsteady Kutta condition. Importantly the results agree well only when the unsteady Kutta condition is neglected in the analytic solution. This paper highlights where the far-field acoustics are most affected by neglecting the unsteady Kutta condition for a variety of singularities that can occur in the unsteady pressure at the trailing edge and shows that results permitting different behaviour in the unsteady surface pressure at the trailing edge could give significantly different far-field noise predictions, even though the surface pressure elsewhere along the aerofoil surface agrees with benchmark solutions.

Note on the Physical Basis of the Kutta Condition in Unsteady Two-Dimensional Panel Methods

Force generation in avian and aquatic species is of considerable interest for possible engineering applications. The aim of this work is to highlight the theoretical and physical foundations of a new formulation of the unsteady Kutta condition, which postulates a finite pressure difference at the trailing edge of the foil. The condition, necessary to obtain a unique solution and derived from the unsteady Bernoulli equation, implies that the energy supplied for the wing motion generates trailing-edge vortices and their overall effect, which depends on the motion initial parameters, is a jet of fluid that propels the wing. The postulated pressure difference (the value of which should be experimentally obtained) models the trailing-edge velocity difference that generates the thrust-producing jet. Although the average thrust values computed by the proposed method are comparable to those calculated by assuming null pressure difference at the trailing edge, the latter (commonly used) approach is less physically meaningful than the present one, as there is a singularity at the foil trailing edge. Additionally, in biological applications, that is, for autonomous flapping, the differences ought to be more significant, as the corresponding energy requirements should be substantially altered, compared to the studied oscillatory motions.

Numerical Simulations of Fast-start Motions of Fish

Fast-start swimming is a very special and important type of locomotion of fish. Basically, fast-star motion can be classified into two types: C-type and S-type. It is generally thought that the C-type fast-start is adopted by fish for escape while the S-type fast-start is used in prey capture. Previous experimental data have shown that the maximum acceleration and maximum velocity of fish performing C-start are significantly greater than that for S-type motion. The present research investigates numerically the fluid flows generated by empirically modeled fast-start movements of fish. For simplicity, a 2-D potential flow is assumed, and the unsteady Kutta condition is applied to handle the vortex-shedding phenomenon at fish tail. The panel method is adopted to solve the flow field numerically, and the force and moment acting on fish body are calculated by using the unsteady Bernoulli equation. Once the force and moment acting on fish are obtained at each instant, motion of fish can then be tracked by applying Newton's second law. Effects of force and moment exerting on fish, the moving speed and distance traveled by fish under different types of S-start and C-start movements are compared and discussed. Also presented are the fluid-dynamics interpretation of the fast-start motions and explanation of key mechanisms that lead to different performances of S-start and C-start modes. Basically in S-start mode, fish receives significant thrust from the momentum of accelerating fluid enhanced by the starting vortex near its tail. While in C-start mode, fish speeds up via a turning motion induced by the fluid force and moment acting on it. Since in C-start mode, fish utilizes the lateral force and moment to assist its turning and speeding rather than resists them, fish moving with C-start mode has faster moving speed and better efficiency of performance than with S-start mode. This can explain why fish normally adopt C-type fast-start mode in their attempt to escape from danger.

Flow effects on the acoustic end correction of a sudden in-duct area expansion

For scattering of plane waves at a sudden area expansion in a duct, the presence of flow may significantly alter the reactive properties. This paper studies the influence of a mean flow field and unstable separated flow on the reactive properties of the expansion, formulated as an end correction. Theoretical and experimental results show that the expansion end correction is significantly affected by the flow and hydrodynamic waves excited at the edge of the expansion. The effects are different in three regions where the Strouhal number is small, of order 1, and large. The influence is most significant at Strouhal numbers of the order 1, with specific limiting values for large and small Strouhal numbers, respectively. In the analytic model, an important feature is the shear layer at the edge modeled as a vortex sheet with the unsteady Kutta condition applied at the edge. The influence of Mach number, Helmholtz number, and area expansion ratio is studied, and a quasistationary, small Strouhal number, approximation yields an expression for the end correction. Further, the influence of edge condition is explored, emphasizing the importance of interaction between sound and unsteady vorticity shedding at the edge of the area expansion.

Unsteady panel method for flapping foil

A computer program based on a potential-flow panel method was developed to evaluate the hydrodynamic forces acting on a harmonically heaving and pitching two-dimensional rigid foil. A new formulation of the unsteady Kutta condition, postulating a finite pressure difference at the trailing edge of the oscillating foil, is proposed and implemented in the numerical procedure. A comparison with published experimental data [Anderson JM, Streitlien K, Barrett DS, Triantafyllou MS. Oscillating foils of high propulsive efficiency. J Fluid Mech 1998;360(1):41–72; Read DA, Hover FS, Triantafyllou MS. Forces on oscillating foils for propulsion and maneuvering. J Fluids Struct 2003;17(1):163–83; Schouveiler L, Hover FS, Triantafyllou MS. Performance of flapping foil propulsion. J Fluids Struct 2005;20(7):949–59] showed good agreement with the computational results. The analysis of the experimental results indicates clearly that robust computational procedures are necessary for reliable data processing in terms of calibration, noise removal, correction factors, etc. The proposed two-dimensional approach to the problem is currently being developed as a three-dimensional procedure that will address the unsteady flow effects such as inertia-related ones that are currently omitted.

Direct computation of the noise induced by a turbulent flow through a diaphragm in a duct at low Mach number

A direct computation of the noise radiated by a turbulent flow in a duct obstructed by a diaphragm is performed by compressible large-eddy simulation. For the low Mach number configuration considered, a two-step hybrid method could be less expensive, but would fail at characterizing sound-flow interactions. The application of direct calculation is thus challenging but gives both the acoustic and aerodynamic fields in the same computation in order to shed light on the noise generation mechanisms. Acoustic energy is produced and dissipated in the asymmetric jet-type flow issuing from the diaphragm. A low-frequency radiation is correlated with the breakdown of the coherent jet-column structures as the plane jet reattaches the upper wall. The Kelvin–Helmholtz vortices in the shear layers of the jet constitute the primary instabilities but do not radiate sound. On the contrary, this vorticity shedding seems to dissipate waves propagating upstream by enforcing the unsteady Kutta condition.

The separated flow of an inviscid fluid around a moving flat plate

In this study we consider the separated flow of an inviscid fluid around a moving flat plate. The motion of the plate, which is initially started from rest, is prescribed and unconstrained and we set ourselves the task of fully characterizing the resulting motion in the surrounding fluid. To do this we use a boundary integral representation for the complex-conjugate velocity field $\Phi(z,t)$ and require that the force and torque on the plate be determined as part of the solution. The flow solution is assumed to consist of a bound vortex sheet coincident with the plate and two free vortex sheets that emanate from each of the plate's two sharp edges. The time evolution of these vortex sheets is then considered in general. For physical reasons, the flow solution is required to satisfy the unsteady Kutta condition, which states that $\Phi(z,t)$ must be bounded everywhere, and the rigorous imposition of this condition then yields two types of additional constraint. The first governs the rate at which circulation is shed from the plate's edges and the second ensures that the free vortex sheets are shed tangentially. In fact, all the familiar flow characteristics associated with the imposition of the steady Kutta condition are rigorously shown to have exact parallels in the unsteady case. In addition, explicit expressions for the normal force and torque on the plate are derived. An asymptotic solution to the full system of evolution equations is developed for small times $t > 0$ and a simplified version of this solution is used as an initial condition for a desingularized numerical treatment of the full problem. A fast numerical algorithm is proposed and implemented and the results of several example calculations are presented. The featured examples are limited to high effective angles of attack due to the occurrence of a specific type of event that prevents further time-integration of the evolution equations using the current numerical method. The event corresponds physically to a situation in which a Lagrangian point placed at one of the plate's edges moves onto instead of away from the edge.

Viscous–inviscid method for the simulation of turbulent unsteady wind turbine airfoil flow

A Viscous–inviscid interaction method is presented that allows for the simulation of unsteady airfoil flow in the context of wind turbine applications. The method couples a 2-D external unsteady potential flow to a 2-D unsteady turbulent boundary layer. The separation point on the airfoil leeward side is determined in a self-consistent way from the boundary-layer equations, and the separated flow region is modelled independently. Wake shape and motion are also determined in a self-consistent way, while an unsteady Kutta condition is implemented. The method is able to deal with attached flow and light stall situations characterised by unsteady turbulent boundary-layer separation size up to 50% of the airfoil chord length. The results of the validation campaign show that the method could be used for industrial design purposes because of its numerical robustness, reasonable accuracy, and limited computational time demands.

Hydrofoil sound generation at high Reynolds number

The unsteady separated turbulent flow near the trailing edge of a loaded hydrofoil is often a source of hydroacoustic noise. Intense turbulence in this region may produce noise directly, but, at low Mach number, hydrodynamically forced structural motions may also radiate sound. In fact, interactions between the foil’s unsteady vortical wake flow and the hydrofoil structure may produce undesired self-excited hydrodynamic-vibratory resonances sometimes called ‘‘singing.’’ This presentation reports experimental results from a series of recent experiments focused on understanding and documenting these phenomena at chord-based Reynolds numbers up to 60 million. The measurements include foil surface static and dynamic pressures, foil vibration, LDV-determined average flow velocities and turbulence quantities, and PIV flow fields in the immediate vicinity of the foil’s trailing edge. The experiments are conducted at the US Navy’s Large Cavitation Channel with a two-dimensional test-section-spanning hydrofoil (2.1 m chord, 3.0 m span) at flow speeds from 0.5–18.3 m/s. Special interest is focused on the trailing edge of the foil where the unsteady Kutta condition is investigated. [Significant assistance provided by personnel from NWSC-CD. Sponsored by Code 333 of ONR.]

David Crighton and the unsteady Kutta condition

Why do airplanes fly? Because of the Kutta condition. This is a facetious response to the popular explanation (Bernoulli effect) of the lift on an airplane wing. The Kutta condition is that the (inviscid flow) near any distinguished sharp point (such as the trailing edge) should be finite; it invariably requires that the flow further downstream should have a discontinuity in fluid velocity. True potential flows have no discontinuities, but they can have singularities at edges, and result in no lift (d’Alembert’s paradox). Acousticians live comfortably without invoking the Kutta condition when they use Sommerfeld’s theory of the diffraction of sound by a knife edge, but it is invoked without hesitation by most aerodynamicists. Crighton made the distinction of steady versus unsteady flows, acoustic waves being an example of unsteady flows, and wrote an article in 1985 in Annual Reviews of Fluid Mechanics on the Kutta condition in unsteady flow. This summarized and appraised works by himself and others up to that time on when one should preferably use the Kutta condition; the article ends with the optimistic statement that a firm basis existed for understanding the applicability of the condition. The present paper attempts to explain Crighton’s reasoning, but suggests that the subject is still murkily understood.

Receptivity to harmonic forcing of a finite-width mixing layer just downstream of the trailing edge of a flat plate

Analytical solutions for the receptivity of a piecewise linear mixing layer to harmonic disturbances at the trailing edge of a flat plate are presented. The solution is for an ideal fluid, with viscous effects at the trailing edge being accounted for by imposing an unsteady Kutta condition. The wake from the flat plate is included in the model. Initial mixing layer response for the discrete and continuous modes of instability is determined as a function of forcing strength, Strouhal number, and cross stream location. Forcing frequencies throughout the unstable regime are considered. The various components of the perturbation field and their contributions to the perturbation velocity and vorticity are investigated in the near field of the trailing edge. In the limit that the forcing frequency approaches zero, the amplification of the streamwise velocity and vorticity associated with the growing instability mode is shown to be inversely proportional to the forcing frequency, in agreement with previous results obtained using the vortex sheet approximation. In contrast, the initial cross stream velocity of the disturbance is quite small for low frequencies, and reaches a peak at the most amplified frequency of the mixing layer profile. The analysis also shows that at higher frequencies the continuous modes of instability become increasingly important channels whereby vorticity is carried away from the trailing edge. The differences in wave number between the various competing components at these higher frequencies leads to the appearance of nodes in the time-averaged perturbation quantities downstream. The entire perturbation is examined at the most amplified frequency.

Calculation of Three-Dimensional Unteady Sheet Cavitation by a Simple Surface Panel Method

This paper presents a calculation method for the 3-D unsteady sheet cavitating hydrofoil problem. The method is based on a simplified surface panel method “SQCM”. Hess and Smith type source panels are distributed on the hydrofoil and cavity surfaces. Discrete vortices are distributed on the camber surface according to Lan's QCM (Quasi-Continuous vortex lattice Method). The boundary conditions to determine these singularities are the constant pressure condition on the cavity surface and the zero normal velocity condition on the hydrofoil and camber surfaces (except the trailing edge). We consider the unsteady Kutta condition that the pressures must coincide with each other on the upper and lower surfaces at the trailing edge. Then we introduce the normal velocity at the trailing edge. This normal velocity is obtained by iteration for the equal pressure condition. The cavity shape in each spanwise section is determined so that the zero normal velocity condition is satisfied on the cavity surface.In the present method, a cavity length for each spanwise section is given first. Then the singularities and the cavity shapes are determined. The cavity length is corrected in order that the opening at the cavity end will get closer to the target value. By using the corrected cavity length, the calculation is repeated from the beginning. These steps are repeated until the opening at the cavity end agrees with the target value in each section and then we move to next time step.This method is applied to the Wagner problem, heaving hydrofoil and a hydrofoil in a sinusoidal gust. We show some calculated results for partially cavitating and super cavitating hydrofoil and compare them with another calculated results.

Variable-domain variational finite element solution of IA inverse problem of 2-D unsteady compressible flow around oscillating airfoils

In this paper, on the basis of the variational principles developed the finite element method (FEM) is employed for numerical solution of the inverse problem of 2-D unsteady compressible flow around oscillating airfoils by incorporating the non-reflecting far-field boundary conditions and a new unsteady Kutta condition. All unknown boundary (airfoil contour) and discontinuities(shocks and free trailing vortex sheets) are determined via the functional variation with variable domain and artificial density concept. For the numerical realization of the variable-domain variation, a special finite element with self-adjusting nodes is also suggested herein. The numerical results show that the present method is effective for the design of unsteady airfoil.

WALL-BOUNDED BLADE-TIP VORTEX INTERACTION NOISE

An analytical treatment is given of the canonical problem of the generation of sound when a locally rectilinear vortex is drawn through the clearance gap between a wall and the square tip of a rigid blade at low Mach number. The mean flow in the neighborhood of the blade tip is assumed to be locally two-dimensional, and is modelled by free streamline potential flow theory, according to which the flow into the gap on the pressure side emerges as a wall jet on the suction side. The additional vorticity shed from the tip (in accordance with the unsteady Kutta condition) during the passage of the vortex is assumed to convect at constant speed along the free streamline boundary of the jet. Sound is generated by a dipole source, the magnitude and orientation of which are determined by the unsteady lift experienced by the blade. We calculate the separate contributions to the radiation (i.e., to the dipole strength) from the vortex and the shed vorticity. The sound is generated while the vortex is within a distance from the blade tip comparable to the clearance, and has wavelength which is order 1/Mlarger, whereMis the characteristic Mach number of the flow. The conclusions of this analysis are supported by an alternative, simplified treatment in which the vortex motion through the gap is assumed to be steady and the shed vorticity convects at a fixed distance from the duct wall rather than along the free streamline.