Unsteady, Transonic Flow Around Delta Wings Undergoing Coupled and Natural Modes Response: A Multidisciplinary Problem

An experimental study of the breakdown of leading-edge vortices on diamond, cropped, delta, and double delta wings during dynamic pitching

A general theory is presented for the calculation of the total forces acting on purely supersonic wings. The method applies to wings having an arbitrary downwash distribution (stationary or non-stationary) and is valid whenever all of the wing edges are supersonic. The general three-dimensional non-stationary problem is reduced to an equivalent two-dimensional problem. In the case of harmonic oscillations the aerodynamic coefficients are expressed in terms of known or tabulated functions. The specific example of an oscillating delta wing is considered and values of the aerodynamic coefficients for plunging, pitching, and rolling oscillations are calculated for two Mach numbers.

The present paper supplies some general theorems with which periodic supersonic motions of a thin wing of fairly general planform may be analyzed to yield valuable three-dimensional results. It is shown that the method developed by Evvard (Refs. 1,2) for treating the steady supersonic motion of a thin wing with subsonic leading or side edges is valid for an oscillating wing of similar planform. Illustrations of the application of these general theorems are furnished by a careful study of several types of periodic oscillations of a rectangular wing. The present report includes a complete analysis for the case of plunging oscillations. Important steps have also been taken towards solution of the cases of pitching and rolling oscillations. The essential results are presented in a number of vector diagrams giving the magnitudes and phase angles of the lift and moment. Computations are made for several aspect ratios at two Mach numbers (M=10/7,2) when the reduced frequency (k) ranges from 0 to 2.0. It is found that the lift and moment vectors acting on a rectangular wing with supersonic plunging oscillations have positive phase angles within certain ranges of Mach numbers and aspect ratios, while the corresponding vectors acting on a wing of infinite span with the same kind of motion have negative phase angles for every Mach number. This new discovery indicates strongly the necessity of revising present day wing flutter calculations.

Injection of jets from nozzles embedded in a delta wing were used to add momentum into the core of the leading edge vortices and thus delay their breakdown. The tests were performed on a delta wing with a 60° sweep angle at a Reynolds number of 260,000 and angles of attack varying from 12 to 20°. At a typical angle of attack of 15°, the vortex breakdown location was moved from its natural location at 55% of the wing chord to 90% near the trailing edge, using a blowing coefficient of Cu = 0.007. Though optimized for a = 15°, the jet blowing was effective in the entire range of angles of attack studied. Flow visualizations, which were performed by illuminating smoke with a laser sheet, showed the effect of the jet blowing on the vortex core and were used to determine the vortex trajectory and breakdown location. The mean axial velocity and turbulence intensity were mapped across the vortex at several axial locations using Laser Doppler Velocimetry. Vortex breakdown resulted in strong deceleration of the mean velocity in the core region, the appearance of negative flow and increased turbulence intensity. Fluctuations in the vortex breakdown location at a Strouhal number of 0.13 were recorded. The control jet injection accelerated the core velocity and stabilized the vortex as evidenced by a reduction in the turbulence intensity and a substantial shift of the vortex breakdown location towards the trailing edge. Nomenclature

Using linearized theory a close approximation to the optimum camber and twist distribution of a Delta wing of zero thickness with sonic leading edges has been obtained. The optimum wing has a camber line that is negative over the front quarter of the total planform area and is positive over the remaining area. A certain amount of washout is exhibited. Compared to the flat plate a saving in induced drag of slightly over 7 percent is shown. The leading edges are relieved of a considerable portion of the total lift. It is shown that this lift has been re-distributed over the central part of the wing.

A delta wing configuration with a sweep angle of 65 ° has been investigated numerically by detached eddy simulation and compared to experimental results. In the simulations different refinement settings of the grid as well a s time different time step sizes have been applied in order to reproduce the formation of the primary vortex correctly. For the comparison experimental data is available from steady and unsteady surface pressure measurements, hot wire anemometry and several other measurement techniques. I. Introduction avourable configuration for supersonic aircraft com monly are delta wings. There has been intense work on investigation of the complex flow field around delt a wings during the past decades. Two major projects concerning these flows were the international vorte x flow experiments (VFE) one and two. Both featured experimental as well as computational investigation s of delta wing wind tunnel models providing a larg e database of reference material for validation purposes. In VFE- 2 a delta wing with a sweep angle of 65° was invest igated featuring a sharp and several rounded leading edges 1 , of which only the sharp leading edge has been use d in the present study. In the simulations of the round lead ing edges the challenge is reproducing the correct separation behaviour behind the leading edge in order to achie ve the correct vortical flow field further downstre am. At higher angles of attack another phenomenon becomes present, the two primary vortices break down and thus the character of the flow field downstream of the b reakdown location changes drastically. The present work is focusing stronger on this vortex breakdown than on the flow separation. Thus it is limited to the shar p leading edge, which dictates the position for the separation, sin ce the flow cannot stay attached after passing the edge. Though its position thus is fixed the actual process of separa tion still needs to be handled carefully in the sim ulation, since it prescribes the formation of the vortices building t he fundament for the vortex breakdown. Common eddy viscosity RANS models without extensions for vortical flow usually are not able to reprodu ce these flows with vortex breakdown. Thus for the pre sent study an approach of detached eddy simulation (DES) has been selected. The underlying turbulence model is t he SST model 2 and the grid shield model is standard DES 3 . Several numerical meshes have been used to investigate the solution sensitivity to different refinemen t regions. Also the influence of time step size has been addressed. The simulations are compared with wind tunnel measurement results obtained in the low speed wind tunne l at the Technical University Munich.

An experimental study of leading edge vortex breakdown on a family of delta wings is currently being conducted at the University of Notre Dame. Flow visualization and imaging processing techniques were used to measure the location of vortex breakdown as a function of angle of attack or roll angle. Both static and dynamic experiments were conducted. The models were driven through dynamic pitch sweeps for a zero roll angle, and for a fixed angle of attack the models were driven through dynamic roll sweeps. This paper reviews the results obtained for a 65 swept wing. Two models were used in the study, both models had the same leading edge sweep but different leading edge geometries. A centerbody was added to the delta wing to examine its influence on vortex breakdown location. Results are presented that show the influence of the leading edge geometry and the centerbody on the static and dynamic location of vortex breakdown. For the dynamic models, vortex breakdown was found to lag the corresponding location from static experiments. Data is presented that shows the influence of the motion amplitude and reduced frequency on the vortex hysteresis.

The unsteady, three-dimensional, Reynolds averaged Navier Stokes equations (NS) are used to simulate and study the flow response around a delta wing undergoing a pitch-up motion up to 90 degrees amplitude. The governing equations are solved time-accurately using the upwind. Roe flux-difference splitting, finite volume scheme. The primary model under consideration consists of a 76 degree swept, sharp-edged delta wing of zero thickness. The Reynolds number and Mach number are 0.45 X 10(exp 6) and .03 respectively. The wing is forced to pitch-up through a ramp function, using a wide range of reduced frequency to provide a good understanding of reduced frequency effect on high amplitude pitch maneuvers. Three reduced frequency values of .0834, 0.134 and 0.04, are used in this study. The computed results are compared with each other as well as the existing experimental data.

It is well known that aircraft undergoing high angle of attack excursions (e.g., the Cobra maneuver) may experience asymmetric forebody vortex shedding under certain conditions. The detachment of these vortices from the forebody is similar to that observed from von Karman vortex shedding from cylinders. A series of experiments, where a von Karman vortex street wake was made to impinge upon a 70-degree delta wing, was conducted at Wichita State University. The aim is to better understand the mixing mechanism that occurs at these high angle of attack flight regimes in a less Reynolds number sensitive environment. A von Karman wake having a frequency similar to the forebody shedding process is used. As the von Karman vortex filaments are entrained by the shear layer, they appear to wrap themselves around the core. There is a temporal correlation to the burst location of the delta wing’s leading-edge vortices, indicating modulation of the burst location. Additionally, the core becomes distorted. It has been found that the vortex burst location is moved forward towards the apex when subjected to the von Karman wake. Nomenclature c Wing root chord d Cylinder diameter FOV Field of View LE Leading Edge Rec Chord Reynolds number, U∞c/ν ReD Diameter Reynolds number, U∞d/ν s Coordinate along root chord U∞ Freestream Velocity x Coordinate along horizontal direction α Angle of attack (deg) Introduction Delta wings have evolved over the years and are now used primarily in the form of leading edge extensions on many fighter aircraft. As these aircraft become more and more maneuverable, the understanding of the physics of time-dependent unsteady flows is becoming more important. In particular, if accurate computational models of these maneuvers are to be developed, it is necessary to understand the mechanisms involved in features such as vortex bursting and mixing on the lee side of the delta wing. This is true of maneuvers such as the Cobra and other “hyper-agile” maneuvers. To understand the situation, it is important to briefly discuss some of the experimental findings in the areas of vortex bursting behavior, the Cobra maneuver physics, and forebody shedding. On the Impingement of a von Karman Vortex Street on a Delta Wing Ismael Heron and Roy Y. Myose Department of Aerospace Engineering Wichita State University, Wichita KS 67260-0044 It is well documented that delta wings at a fixed angle of attack generate lift by separating a shear layer of air (or fluid, such as water) at the leading edge, and this shear layer forms two strong counter-rotating vortices on either side of the wing. These leadingedge (LE) vortices undergo small fluctuations in space, but remain relatively fixed over the suction side of the delta wing, and are critical to the generation of lift, as they produce a large suction peak on the surface. Two much smaller vortices, the secondary vortices, are also formed, as seen in Figure 1. In other words, LE vortices are the result of a balance between vorticity being generated at the leading edge, and the ability of the flow field to convect said vorticity along the vortex core. The LE vortices are not stable, and at some point their coherent structure will undergo a dramatic change, expanding around the core, slowing down axially, and either forming a bubble or a spiral, with the spiral form being more predominant at Reynolds numbers of interest to delta wing designers. This change, called vortex burst or breakdown, is dependent on the aspect ratio of the wing, angle of attack, pressure gradients, yaw angle, and swirl angle of the vortex, among others. The exact reason for this bursting is not known, but research has focused on two general areas. (a) The flows upstream and downstream of the vortex burst are two separate and very different flows, and the vortex burst is a necessary feature, similar to a hydraulic jump. (b) The core of the LE vortex serves as a mechanical waveguide for longitudinal waves; these waves either coalesce, or they become critical, thereby triggering the burst. Regardless of the burst triggering mechanism, the effect of the vortex burst is to reduce the lift generated by the delta wing. If the delta wing is pitched to a given angle of attack (α) and then maintained at that angle until the transient flow features die down, it is said to be tested under “static” conditions. As this 22nd Applied Aerodynamics Conference and Exhibit 16 19 August 2004, Providence, Rhode Island AIAA 2004-4731 Copyright © 20 * Graduate Research Assistant, Student Member AIAA. † Associate Professor, Associate Fellow AIAA. Copyright ©2004 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

Three-dimensional steady and unsteady vortex-dominated flows around sharp-edged delta wings are considered in this paper. The problem is formulated by using the unsteady conservative Euler equations for the flow relative motion with respect to a moving frame of reference. An implicit approximately-factored finite volume scheme is used to solve the resulting equations on a three-dimensional computational grid which is generated by using a modified Joukowski transformation in cross-flow planes at the grid chord stations. The scheme is applied to a delta wing undergoing pitching oscillation around a large angle of attack. The initial conditions correspond to a steady flow around a delta wing of aspect ratio of one, freestream Mach number of 0.3 and mean angle of attack of 20.5. The steady flow results are compared with those of an explicit computational scheme and the experimental data, and they are in good agreement.

A flux-difference splitting scheme is employed to compute low-speed flows over a delta wing for angles of attack from 0 to 40 deg as steady-state solutions to the three-dimensional, Reynolds-averaged Navier-Stokes equations in their thin-layer approximation. The finite-difference scheme is made spatially second-order accurate by applying a total variation diminishing-like discretization to the inviscid fluxes and central differencing to the viscous shear fluxes. Using first-order accurate Euler backward-time differencing, an efficient implicit algorithm is constructed, which combines approximate factorization in cross planes with a symmetric planar Gauss-Seidel relaxation in the remaining third spatial direction. The geometry of the thin (maximum thickness is 0.021), slender (aspect ratio is unity), sharp-edged delta wing is taken from Hummel's (1967, 1978) wind tunnel model. Over the entire angle-of-attack range, the computed values of lift and pitching moment are in good agreement with the experimental data. Also details of the flow-fieldlike spanwise surface pressure distributions compare well with the experiment. Computed flow-field results with a bubble-type vortex burst are analyzed in detail.

Computational results are presented for transient vortex breakdown above a delta wing subject to a pitch-and-hold maneuver to high angle of attack. The flows are simulated by solving the full three-dimensional Navier-Stokes equations on a moving grid using the implicit Beam-Warming algorithm. An assessment of the effects of numerical resolution, and favorable comparison with experimental data suggest the computational approach captures the basic dynamics of this transient breakdown. The pressure gradient along the vortex axis is found to play a dominant role in the initiation of breakdown. A description of the 3-D instantaneous structure of the flow field is provided for the first time using critical-point theory. The reversed-flow region in the vortex core is associated with pairs of opposite spiral/saddle critical points. At its onset, the vortex breakdown is fairly axisymmetric, however, as it proceeds upstream and a stronger transition takes place along the axis, asymmetric effects become important and result in the formation of a bubble-type breakdown. This bubble structure is open and contains within itself a pair of stagnation points which are diametrically opposed and which rotate in the same sense as the base flow. These critical points suggest the existence of azimuthal disturbances, and their rotation might be linked with the coherent oscillations observed in vortex breakdown. N o m e n c l a t u r e

This paper presents an experimental study on the effects of roll on the loading and rolling moment of a 60 deg swept delta wing. The parameters studied were the incidence angle (including those in which vortex breakdown was present), roll angle, and ground proximity. Results indicate that when vortex breakdown is not present, the effects of roll on the wing loading and rolling moment are in reasonable agreement with predictions. When vortex breakdown is present, the magnitude of the rolling moment coefficient is reduced considerably and its sign can even be altered. The changes in the rolling moment characteristics due to vortex breakdown can be attributed to the roll-induced spanwise displacement of the leading-edge vortices. Effects of ground proximity on roll characteristics were shown to be small.

The limiting factor in simulating flows past realistic configurations of interest has been the discretization of the physical domain on which the governing equations of fluid flow may be solved. In an attempt to circumvent this problem, many Computational Fluid Dynamic (CFD) methodologies that are based on different grid generation and domain decomposition techniques have been developed. However, due to the costs involved and expertise required, very few comparative studies between these methods have been performed. In the present work, the two CFD methodologies which show the most promise for treating complex three-dimensional configurations as well as unsteady moving boundary problems are evaluated. These are namely the structured-overlapped and the unstructured grid schemes. Both methods use a cell centered, finite volume, upwind approach. The structured-overlapped algorithm uses an approximately factored, alternating direction implicit scheme to perform the time integration, whereas, the unstructured algorithm uses an explicit Runge-Kutta method. To examine the accuracy, efficiency, and limitations of each scheme, they are applied to the same steady complex multicomponent configurations and unsteady moving boundary problems. The steady complex cases consist of computing the subsonic flow about a two-dimensional high-lift multielement airfoil and the transonic flow about a three-dimensional wing/pylon/finned store assembly. The unsteady moving boundary problems are a forced pitching oscillation of an airfoil in a transonic freestream and a two-dimensional, subsonic airfoil/store separation sequence. Accuracy was accessed through the comparison of computed and experimentally measured pressure coefficient data on several of the wing/pylon/finned store assembly's components and at numerous angles-of-attack for the pitching airfoil. From this study, it was found that both the structured-overlapped and the unstructured grid schemes yielded flow solutions of comparable accuracy for these simulations. This study also indicated that, overall, the structured-overlapped scheme was slightly more CPU efficient than the unstructured approach.

: The goal of this research is the control of leading-edge vortex breakdown location utilizing along-the-core blowing near the apex on the leeward surface of sharp-edged, slender, delta wings at high angles of attack. In the S2Ch subsonic wind tunnel at ONERA Chalais Meudon, two delta wing models with 70-deg sweep angles and root chords of 950 mm have been configured to collect qualitative and quantitative surface and flowfield data. First, an examination of the streamwise, time-dependent oscillation of the leading-edge vortex breakdown locations without active flow control is presented. These results further the understanding of the vortex breakdown phenomena and provide a more precise basis for evaluating the effectiveness of various flow control methods. Second, open-loop blowing along one of the vortex cores on the leeward surface of the delta wing demonstrates the ability to displace downstream the controlled, time-averaged, vortex breakdown location by 20% of the root chord.