Integral Relations in the Linearized Theory of Wing-Body Interference

Lifting-line theory has formed the basis of wing-design understanding for nearly a century. This theory has previously been used to develop a class of optimal lift distributions that minimize induced drag for finite wings under specific structural and aerodynamic constraints. This class of lift distributions was first introduced by Prandtl in 1933 and includes the elliptic lift distribution and the bell-shaped lift distribution, among others, as special cases. Here, it is shown that, for all lift distributions in this class except the elliptic lift distribution, an antisymmetric twist distribution can be added to the wing to control the adverse yawing moment during roll. The requisite amount of this additional twist distribution depends on the prescribed yawing-moment sign and magnitude, and it can be used to produce zero adverse yaw or even proverse yaw during roll. Of course, employing this methodology to control the roll–yaw coupling produces additional induced drag relative to the case without roll–yaw coupling control. This induced-drag penalty is studied in detail and shows that a symmetric lift distribution within this class can be selected to minimize induced drag for a desired roll–yaw coupling. Roll initiation and steady rolling rate are both considered. Results can be applied to the design of morphing aircraft to minimize the use of a vertical control surface for yaw control.

Circulation control has been applied to airfoils since the late 1960's, and has been proven to change the aerodynamic performance by altering the interaction of the streamlines without changing the physical characteristics of the airfoil itself. This has many applications in fluid dynamics; the focus of this application is for the replacement of the conventional helicopter rotor blade system with a fly-by-wire, active circulation controlled system. Conventional helicopters use a swashplate and a series of mechanical linkages, bearings, and dampers to create a fully articulated rotor hub system. This system is required to achieve the blade characteristics required for stable flight. The need for such a system stems from the asymmetric lift developed in maneuvering flight conditions, which requires the angle of attack of the blades to be changed based on the rotational position within the rotor plane, also known as the azimuth angle. By alternatively activating blowing slots along the leading and trailing edges of the airfoil, the aerodynamic parameters (i.e. lift and drag) can be changed, effectively changing the angle of attack through streamline alteration thus eliminating the need for physical blade pitch changes.;Mathematical models/codes are used to model and simulate the complex blade dynamics of a full-scale rotorcraft. Many of these codes use a blade element method that separates the rotor into small segments and evaluates the aerodynamic characteristics of these elements as two-dimensional airfoils at different local conditions. These local conditions include, but are not limited to, extreme angles of attack, reverse flow, compressibility effects, dynamic effects, and other aerodynamic phenomena.;This research investigated the reverse flow aerodynamic characteristics of a 10:1 elliptical airfoil with various leading and trailing edge blowing pressures. The testing conditions for the aerodynamic investigation were: angle of attack [154 to 196 degrees]; blowing coefficient [0 to 0.009] and [0 to 0.014] for the leading and trailing edge blowing slots, respectively; and Mach number [0.073 and 0.109].;It is concluded that the potential exists for increasing forward flight speeds for helicopters using circulation control in the reverse flow regions of the helicopter rotor environment. Specifically, it is concluded that positively altering the aerodynamic characteristics, primarily the lift coefficient, in reverse flow, is possible through circulation control. Through this investigation, the general trends were found for the aerodynamic characteristics of a 10:1 circulation controlled elliptical airfoil in reverse flow. These trends led to the selection of blowing configurations to decrease the asymmetric loading condition based

Introduction N methods for predicting generalized unsteady aerodynamic forces usually do not provide identical results. Experimental methods are rarely available to verify the accuracy of predicted generalized forces for complex configuration, and the flutter analyst must rely on calculated results. Convergence criteria generally are not available to enable the user to obtain the best results from a particular numerical method. This important issue was addressed recently by Rowe and Cunningham, and in a Comment by Stark followd by a Reply by Lottati. The usual way to tackle the accuracy of the method is by applying the code on several test cases and comparing these results against results obtained by other methods of computation (see Ref. 1). Another way is to use the existing relations for the aerodynamic forces developed on the configuration in direct and reverse flow. By comparison of the obtained results computed by direct and reverse flow, it is possible to assess the accuracy of the method. The general reverse-flow theorems for lifting-surface theory may also be applied to wings that undergo harmonic oscillation. The results of Ref. 1 emphasize the wavy character of the oscillatory pressure developed on an oscillating wing. The pressure distribution displayed in Ref. 1 aimed to alert users to the possible potential difficulties in obtaining an accurate solution due to the wavy character of the aerodynamic pressure on an oscillating wing. The pressure distribution becomes extremely wavy and more difficult to predict accurately as the reduced frequency k and the Mach number M get larger values. Reference 1 defined a wave number to assess the wayiness of the pressure distribution which might be useful in the estimation of relative difficulties in predicting converged aerodynamic forces. In this Note, the reverse-flow theorem will be applied to assess the accuracy of the computed unsteady aerodynamic forces for interference configuration. The computer program used to evaluate the unsteady aerodynamic forces on the configuration is the nonplanar piecewise continuous kernel function method (PCKFM).

This paper presents the development of a nonlinear aerodynamic model that describes the impact of forebody tangential blowing (FTB) on the generated roll moment for a fixed aircraft attitude. This model uses Nonlinear Indicial Responses (NlR) and incorporates physical insight gained from experimental data. The model accounts for the nonlinear aerodynamic behavior observed when FTB is applied at high angles of attack. Its predictions for the roll moment generated by FTB are compared to actual data and a model that uses a tirstorder lag of the static loads. The NlR model is found to provide a better prediction of the measured transient response. Experimental results used in the model’s development are discussed. They include flow visualization as well as static and dynamic measurements of the aerodynamic loads for various roll angles and blowing mass flow rates. The experimental apparatus used in this study is described and consists of an FTB-equipped wind tunnel model which is free to roll. The model is a wing-body combination with slots on the forebody for tangential blowing and is nominally set at 45 degrees angle of attack. To augment the control of a flight vehicle at high angles of attack, several methods of active flow control have been proposed and explored.4*5 They provide a means to alter the flow structure in a rational manner and increase the controllability of the vehicle under these flight conditions. The injection of thin sheets of air tangentially to the forebody of the vehicle has been found to be an extremely promising method6-8 and progress has been made in demonstrating the feasibility of using Forebody Tangential Blowing (FTB) to control the roll-yaw motion of flight vehicles through both experimental demonstrationsg*10 and simulations.” Experimental investigations into the use of FTB have been conducted at the low-speed wind tunnel of the Department of Aeronautics and Astronautics at Stanford University. The experimental apparatus consists of a wind tunnel model that is free to ro11.‘2 The model has a cone-cylinder fuselage and a sharp leading-edge delta wing. It is set at a nominal incidence angle of 45 degrees and is equipped with side slots to support the injection of a thin sheet of air tangentially to the forebody of the vehicle.

Compressible flow past thin wings and slender bodies are studied. First, subsonic flow past thin wings are analyzed by means of potential flow theory. The kernel function method is introduced for arbitrary planforms undergoing simple harmonic oscillations. The spanwise polynomial approximation and the chordwise trigonometric function approximations which automatically satisfy the Kutta condition and inherit the leading edge singularity are used. The Doublet-Lattice method as a more general numerical approach is given for the analysis of the flow past additional surfaces like tail or store surfaces, which are not necessarily in plane with the wing surface or surfaces having spanwise deflections. Airfoil response to the arbitrary unsteady motion is also given for subsonic flows. A brief review of shock waves and Mach waves in a supersonic flow is given. Afterwards, unsteady supersonic potential flow is studied for a simple harmonically oscillating point source. First, unsteady flow past an airfoil is considered. Then, supersonic flow about thin wings are analyzed using the Mach box technique. Introduction to supersonic kernel function method is briefly provided. Arbitrary unsteady motion of an airfoil in a supersonic flow is presented. Slender body theory is introduced to analyze the cross flow past bodies of considerable fineness where the cross flow is shown to be approximately incompressible under certain conditions. In connection with the slender body approximation, the Munk’s airship theory is utilized to predict the stability derivatives of missile like bodies.

Geometric and/or aerodynamic wing twist can be used to produce a lift distribution that results in a rolling moment. A decomposed Fourier-series solution to Prandtl’s lifting-line theory is used to develop analytic spanwise antisymmetric twist distributions for roll control that minimize induced drag on wings of arbitrary planform in pure rolling motion. Roll initiation, steady rolling rate, and the transition between the two are each considered. It is shown that if these antisymmetric twist distributions are used, the induced drag is proportional to the square of the rolling moment, and the induced drag during a steady rolling rate is equal to that on the wing at the same lift coefficient with no rolling rate or antisymmetric twist distribution. Results also show that if these antisymmetric twist distributions are used on straight, tapered wings without symmetric twist, any rolling maneuver for which the rolling rate and rolling moment have the same sign will always produce a yawing moment in the opposite direction. Computational results are also included, which were obtained using a gradient-based optimization algorithm in combination with a modern numerical lifting-line algorithm to find the optimum twist solutions. The resulting twist, induced drag, and yawing moment solutions compare favorably with the analytic solutions developed in the text. The solutions presented here can be used to inform the design of morphing aircraft.

Few studies have been devoted to effects of canard oscillations on the flowfield over the wing in supersonic flow and the associated time lags. In this article, an in-depth experimental study on a closely coupled canard-wing-body configuration in supersonic flow has been undertaken to mark the zone of influence of the oscillatory canard on the wing. Surface pressure measurements have been performed on the wing downstream of canard in both static and oscillatory cases at various wing angles of attack and canard deflection angles as well as the canard oscillation amplitudes and frequencies. The results show that the upwash and downwash flowfields due to an oscillatory canard periodically change the effective angle of attack seen by the wing and have different effects on various regions. In supersonic flow, the pressure rise due to the oblique shock waves causes the primary flow separation and the vortex break down onsets to occur at higher angles of attack, comparing to those in the lower speed regimes. According to the results, the mid-span region on the wing, which is directly in the shadow of the oscillating canard, receives more disturbances. During upstroke, the canard induces a downwash field to the front half of the wing, while in downstroke motion, the front regions of the wing experience upwash flow, giving rise to the effective angle of attack seen by the wing. Among various oscillation parameters, the pitching amplitude was observed to have a strong impact on the time lag in flowfield over the wing.

Transonic flows over several bodies of varying complexity, including a simple wing-body combination and the Shuttle Orbiter, have been modeled with the use of slender body theory. In the theory, the original three-dimensional problem is divided into two simpler component problems, the near field and the far field. The near field problem represents a crossflow that is almost incompressible and is defined by Laplace's equation with a flow tangency boundary condition on the body. A nonlinear transonic small disturbance approximation is the basis for the far-field problem and describes the flow over a body of revolution having the same longitudinal area distribution as the asymmetric body. Both the nearand far-field problems are two-dimensional boundary value problems. A boundary-element method is used to solve the near-field problem. The solution for the axisymmetric far-field problem is obtained by a successive line overrelaxation method. The two component solutions are combined to obtain the complete solution. Flows with various angles of attack and zero sideslip have been considered. Also, some off-body flow calculations were conducted. For attached flows, the calculations predicted the pressure results with good accuracy.

Tomographic particle image velocimetry was used to explore the evolution of three-dimensional flow structures of revolving low-aspect-ratio flat plates in combination with force measurements at a Reynolds number of 10,000. Two motion kinematics are compared that result in the same terminal condition (revolution with constant angular velocity and $$45^{\circ }$$ angle of attack) but differ in the motion during the buildup phase: pitching while revolving at a constant angular velocity; or surging with a constant acceleration at a fixed angle of attack. Comparison of force histories shows that the pitching wing generates considerably higher forces during the buildup phase which is also predicted by a quasi-steady model quite accurately. The difference in the buildup phases affects the force histories until six chords of travel after the end of buildup phase. In both cases, a vortex system that is comprised of a leading-edge vortex (LEV), a tip vortex and a trailing edge vortex is formed during the initial period of the motion. The LEV lifts off, forms an arch-shaped structure and bursts into substructures, which occur at slightly different phases of the motions, such that the revolving–surging wing flow evolution precedes that of the revolving–pitching wing. The delay is shown to be in accordance with the behavior of the spanwise flow which is affected by the interaction between the tip vortex and revolving dynamics. Further analysis shows that the enhanced force generation of the revolving–pitching wing during the pitch-up phase originates from: (1) increased magnitude and growth rate of the LEV circulation; (2) relatively favorable position and trajectory of the LEV and the starting vortex; and (3) generation of bound circulation during the pitching motion, whereas that of the revolving–surging wing is negligible in the acceleration phase.

Swirl and buoyancy effects on mixing performance of baffle-plate-type miniature confined multijet

Supersonic flows about conical bodies

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.

In this study, the dynamic stall evolutions were investigated using particle image velocimetry (PIV) in a water channel with Reynolds number Re = 4.5 × 103 based on the chord length. The airfoil pitching waveform was performed under the condition calculated from the angle of attack histogram of a vertical axis wind turbine (VAWT). Using PIV, the instantaneous vorticity contours and streamlines can be revealed. Based on the formation of the leading edge vortex, the stall angle can be explored at reduced frequency k = 0.09, 0.18, and 0.27. It was found that the stall angle was delayed from the angle of attack α = 16° to α = 30° as reduced frequency increased from k = 0.09 to 0.27. The hysteresis effect of stall angle delay was more pronounced for high reduced frequency. Moreover, the freestream turbulence effect on the pitching airfoil was investigated with turbulence intensity TI = 0.5 and 6.9 %. As found, the stall angles were postponed to higher angles of attack for the high turbulence intensity. The phase difference between TI = 0.5 and 6.9 % were ∆α = 8°, 4°, and 4° for k = 0.09, 0.18, and 0.27, respectively. For TI = 6.9 %, enhanced turbulence mixing reduces the velocity deficit (u/U < 1) and flow reversal (u/U < 0). In addition, the maximum velocity is reduced from u/U = 1.8 to 1.2 and the S-shaped velocity profile is diminished or weakened for TI = 6.9 %. Thus, the dynamic stall is further delayed to the downstroke. The circulation values increase rapidly to maximum and then drop quickly after dynamic stall for k = 0.18 and 0.27.

The accuracy of high-alpha slender-body theory (HASBT) for bodies with elliptical cross-sections is presently demonstrated by means of a comparison with exact solutions for incompressible potential flow over a wide range of ellipsoid geometries and angles of attack and sideslip. The addition of the appropriate trigonometric coefficients to the classical slender-body theory decomposition yields the formally correct HASBT, and results in accuracies previously considered unattainable.

The method of matched asymptotic expansions is applied to the problem of a wing of finite span in very close proximity to the ground. The general lifting surface problem is shown to be a direct problem, represented by a source-sink distribution on the upper surface of the wing and wake, with concentrated sources around the leading and side edges plus a separate confined channel flow region under the wing and wake. The two-dimensional flat plate airfoil is examined in detail and results for upper and lower surface pressure distribution and lift coefficient are compared with a numerical solution. A simple analytic solution is obtained for a flat wing with a straight trailing edge which has minimum induced drag. To lowest order, this optimally loaded wing has an elliptical planform and a lift distribution which is linear along the chord, resulting in a parabolic spanwise lift distribution. An expression for the lift coefficient at small clearance and angle of attack, valid for moderate aspect ratio, is derived. The analytic results show reasonable agreement when compared with numerical results from lifting surface theory.