The Minimum Drag of Thin Wings in Frictionless Flow

This is a calculation of the chordwise lift distribution, total lift, and moment on a two-dimensional wing striking a sharp-edged gust a t supersonic speed. A direct solution is established by considering a distribution of sources in a fluid at rest. An alternate method using Busemann's conical flow is also shown to be applicable. The time history of the total lift and mid-chord moment is discussed. I t is shown tha t the total lift increases with time and reaches a maximum that corresponds to the steadystate phase of the flow. The mid-chord moment goes through a maximum independent of the Mach Number if the latter value is larger than 4/7r, while this maximum can become infinite for a range of Mach Numbers between 4/TT and 1. (1) INTRODUCTION CONSIDERABLE ATTENTION has been given lately to nonstationary flow problems of wings flying at supersonic speeds. Most of the work, however, has been concerned with the aerodynamic forces on an oscillating airfoil from the standpoint of flutter analysis. The problem of the wing hitting a sharp-edged gust is of a different nature and turns out to be actually much simpler than the oscillating airfoil problems. It is shown in section 2 that it may be treated by a distribution of sources of a simple type along the chord and that the pressure distribution may be derived by elementary methods. The procedure does not introduce a moving fluid but considers a fluid at rest in which nonstationary sources are distributed in a layer of variable extent. This point of view, which is closer to acoustics than to aerodynamics, is somewhat novel and seems to present advantages of simplicity and closeness to physical reality in certain categories of problems. The pressure distribution derived by this method is applied to the calculation of the time history of lift and moment on the wing in section 3. Particular attention is given to the value of the mid-chord moment;.-which starts from zero, rises to a maximum,,and goes back to zero. The value of this maximum and related 4ata is evaluated in section 4. These results are of particular interest to the designer. The derivation of the pressure as given in section 1 is only one of the methods that may be used in this problem. As an independent check and as an illustration of the application of Busemann's method of conical flow to a nonstationary problem, an alternate derivation is given for the pressure distribution in section 4. Received August 15, 1948. * Member of Cornell Aeronautical Laboratory consulting staff. Also Professor of Applied Physical Sciences, Brown University. In a paper by Schwarz procedures used in oscillating airfoil theory are extended to the problem of a wing striking a sharp-edged gust at supersonic speed. Results for an oscillating down-wash lead to the gust problem by a Fourier integral representation. This method constitutes a considerable detour and introduces intermediate results of a transcendental nature which are actually not needed and are more complicated than the result. I t may be verified that the expression derived in the present paper for the pressure distribution is equivalent to that derived by Schwarz. He does not, however, discuss the physical aspects of the problem or derive expressions for lift and moment. (2) DERIVATION OF THE PRESSURE DISTRIBUTION The wing of chord / enters a uniform gust of upward velocity i/0 at the supersonic velocity V (Fig. 1). The velocity component normal to the wing must remain zero, and this condition is equivalent to the generation of a velocity normal to the wing which cancels the gust velocity (Fig. 2). This may also be considered as a reflection of the gust on the wing. Because the velocity is supersonic, the pressure distribution on. one side does not influence the pressure on the other, and therefore we need only consider the bottom side. The pressure distribution on top will be the same except for a reversal of sign. For the same reason the pressure distribution is not influenced by the trailing edge, and

Simulation of unsteady motion of a propeller in a fluid including free wake modeling

SummaryIn this paper lower bounds are obtained for the intensity of the sonic bang resulting from a slender aircraft flying straight and level at supersonic speeds. The lower bounds are obtained for an aircraft subject to various constraints on its total lift, volume, area distribution and centre-of-pressure position. The work described proves the results quoted in an earlier paper, corrects the results given in that paper for the general configuration, and extends the results to cover more constraints, mainly the centre-of-pressure position.

Low aspect ratio control surfaces may be used on supersonic airplanes and guided missiles. Their deflections will generally create gaps which are approximately in the flight direction (e.g., all-movable tails and variable-incidence wings). The aerodynamic effects of such gaps have nearly always been neglected in theoretical investigations. These gap effects are properly neglected for high aspect ratio control surfaces but, since these effects increase with decreasing aspect ratio, they may be important for low aspect ratio control surfaces. The purpose of this paper is to obtain some understanding of the effects of streamwise gaps in low aspect ratio lifting surfaces at supersonic speeds by investigating a simplified configuration whose gap effects are expected to be typical. The configuration consists of a flat lifting half-delta wing which is adjacent and normal to a vertical wall such that the wing side edge is parallel to the wall and forms a streamwise gap. The usual assumptions of linearized supersonic wing theory are made and the following wing characteristics are studied as functions of gap size, Mach Number, and leading edge sweep: spanwise lift distribution, lift, center of pressure, drag due to lift, and vorticity at the trailing edge. In addition, in connection with wing-tail interference, the strengths and initial positions of the rolled-up trailing vortices are investigated. An illustrative example of the gap effects on wingtail interference is also presented. Viscous effects in the gap, which are important for very small gaps, are discussed qualitatively. E{k', v) — incomplete elliptic integral of second kind =

AREFLEX spanwise cambered delta wing with a conical camber designed for M= 1.4, using the method of Ref. 1, was tested at the design Mach number as well as off-design Mach number M=0.15 and 2.3, respectively. The test results are compared with those of a plane wing and also with the available theoretical results at the design condition. At subsonic speed, the cambered wing has less lift at a given incidence and higher lift-to-drag ratio at a given lift than the plane wing, while at supersonic speeds, both of these quantities were less on the cambered wing. At supersonic speed, at the design incidence and Mach number, there is good agreement between results from theory and experiment. The center of pressure on the cambered wing is ahead of that on the plane wing at subsonic speed, while the reverse is true at supersonic speeds. Finally, it is found that over a useful range of lift the cambered wing is aerodynamically more efficient at subsonic speeds, and less so at supersonic speeds, than the plane wing.

Exact solution of an unsteady flow of elastico-viscous fluid through a porous media in a tube of spherical cross section under constant pressure gradient has been obtained in this paper. Initially, the flow is generated by a constant pressure gradient. After attaining the steady state, the pressure gradient is suddenly withdrawn and the resulting fluid motion in a tube of spherical cross section by taking into account of the porosity factor of the bounding surface is investigated. The problem is solved in two-stages: the first stage is a steady motion in tube under the influence of a constant pressure gradient, the second stage concern with an unsteady motion. The problem is solved employing separation of variables technique. The results are expressed in terms of a non-dimensional porosity parameter, elastico-viscosity parameter, which depends on the Non-Newtonian coefficient. The flow parameters are found to be identical with that of Newtonian case as elastic-viscosity parameter and porosity tends to infinity. Numerical results were simulated in MATLAB software to analyze the effect of Elastico-viscous parameter, porosity parameter on the velocity profile. Boundary conditions were satisfied. It is seen that the effect of elastico-viscosity parameter, porosity parameter of the bounding surface has significant effect on the velocity parameter.

: A number of characteristics of an unsteady turbulent boundary layer have been measured on the after part of the upper surface of an 0012 airfoil (X/ c = 0.69 and 0.94). The data is taken at chord Reynolds number of 700,000 and over a range of reduced frequencies (based upon the semichord) of 0.5 to 6.4. Mean and unsteady velocity profiles, as well as Reynolds stress profiles, are presented, as is the nonlinear coupling from the unsteady motion into the steady motion. Data is presented and discussed which shows that for this experiment the periodic unsteady turbulent velocity profile tends toward a universal shape.

A lift distribution based section design method has been proposed. Through Newton-Raphson iterations, the section geometry is efficiently designed to meet the requirements for total lift and lift distribution. The effect of fluid viscosity on total lift is taken into account by coupling Reynolds-averaged Navier-Stokes (RANS) simulation with the potential-flow based design procedure. The present method avoids the difficulty of assigning velocity or pressure distributions on section surfaces. As the loading and thickness distributions are expressed in parametric forms, it is easy to ensure that the designed geometry is continuous and smooth. The effects of lift and thickness distributions on cavitation bucket are numerically investigated. A shift of lift loading towards the aft part of section tends to decrease the margin of back cavitation, while the width of cavitation bucket can be kept almost unchanged. To have a wider cavitation bucket, one can increase the leading edge radius, move properly the location of maximum thickness towards the leading edge, or decrease the curvature at the location of maximum thickness.

Odd viscous liquids are endowed with an intrinsic mechanism that tends to restore a displaced particle back to its original position. Since the odd viscous stress does not dissipate energy, inertial oscillations and inertial-like waves can become prominent in such a liquid. In this paper, we show that an odd viscous liquid in three dimensions may give rise to such axially symmetric waves and also to plane-polarized waves. We assume tacitly that an anisotropy axis giving rise to odd viscous effects has already been established, and proceed to investigate the effects of odd viscosity on fluid flow behaviour. Numerical simulations of the full Navier–Stokes equations show the existence of inertial-like waves downstream of a body that moves slowly along the axis of an odd viscous liquid filled cylinder. The wavelength of the numerically determined oscillations agrees well with the developed theoretical framework. When odd viscosity is the dominant effect in steady motions, a modified Taylor–Proudman theorem leads to the existence of Taylor columns inside such a liquid. Formation of the Taylor column can be understood as a consequence of helicity segregation and energy transfer along the cylinder axis at group velocity, by the accompanying inertial-like waves, whenever the reflection symmetry of the system is lost. A number of Taylor column characteristics known from rigidly rotating liquids are recovered here for a non-rotating odd viscous liquid. These include counter-rotating swirling liquid flow above and below a body moving slowly along the anisotropy axis. Thus in steady motions, odd viscosity acts to suppress variations of liquid velocity in a direction parallel to the anisotropy axis, inhibiting vortex stretching and vortex twisting. In unsteady and nonlinear motions, odd viscosity enhances the vorticity along the same axis, thus affecting both vortex stretching and vortex twisting.

Slender wing vortex flows at subsonic, transonic, and supersonic speeds were investigated in a 6 x 6 ft wind tunnel. Test data obtained include off-body and surface flow visualizations, wing upper surface static pressure distributions, and six-component forces and moments. The results reveal the transition from the low-speed classical vortex regime to the transonic regime, beginning at a freestream Mach number of 0.60, where vortices coexist with shock waves. It is shown that the onset of core breakdown and the progression of core breakdown with the angle of attack were sensitive to the Mach number, and that the shock effects at transonic speeds were reduced by the interaction of the wing and the lead-edge extension (LEX) vortices. The vortex strengths and direct interaction of the wing and LEX cores (cores wrapping around each other) were found to diminish at transonic and supersonic speeds.

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.

The passive flight of a thin wing or plate is an archetypal problem in flow–structure interactions at intermediate Reynolds numbers. This seemingly simple aerodynamic system displays an impressive variety of steady and unsteady motions that are familiar from fluttering leaves, tumbling seeds and gliding paper planes. Here, we explore the space of flight behaviours using a nonlinear dynamical model rooted in a quasisteady description of the fluid forces. Efficient characterisation is achieved by identification of the key dimensionless parameters, assessment of the steady equilibrium states and linear analysis of their stability. The structure and organisation of the stable and unstable flight equilibria proves to be complex, and seemingly related factors such as mass and buoyancy-corrected weight play distinct roles in determining the eventual flight patterns. The nonlinear model successfully reproduces previously documented unsteady states such as fluttering and tumbling while also predicting new types of motions, and the linear analysis accurately accounts for the stability of steady states such as gliding and diving. While the conditions for dynamic stability seem to lack tidy formulae that apply universally, we identify relations that hold in certain regimes and which offer mechanistic interpretations. The generality of the model and the richness of its solution space suggest implications for small-scale aerodynamics and related applications in biological and robotic flight.

The reaction zone structure of spontaneous reaction wave at steady subsonic and supersonic speeds is discussed using numerical simulation and theoretical approaches. The main objective of this study is to develop the mathematical model of the spontaneous reaction wave propagating at a steady speed and to clarify the relationship between the reaction zone structure and non-dimensional parameters. A one-dimensional direct numerical simulation (DNS) of the spontaneous reaction wave propagation is performed, and the modeling assumptions are discussed based on the numerical results. The non-dimensional parameters, the Damköhler number and a parameter related to compressibility defined by the Mach number of the spontaneous reaction wave velocity, are introduced into the modeling. The mathematical model is developed using asymptotic techniques, and the solution of the reaction zone structure is analytically derived using Newton’s method. The relationship between reaction zone structure of spontaneous reaction wave and non-dimensional parameters is discussed based on the analytical solutions.

In this paper, we presented the results of experimental studies of the diffusion combustion of a plain hydrogen microjet flowing from a slit micronozzle at subsonic and supersonic speeds. For the first time, four scenarios of diffusion combustion of a plain hydrogen microjet including supersonic combustion in the presence of supersonic cells in both air and hydrogen are presented. The stabilization of the subsonic combustion of a hydrogen microjet was established to be due to the presence of a «bottleneck flame region» while the stabilization of the supersonic combustion of a microjet was found to be associated with the presence of supersonic cells. The observed hyster­esis of diffusion combustion of a plain hydrogen microjet depends on both the method of igniting the microjet (near or far from the nozzle exit) and the direction of change in the rate of its outflow (growth or reduction).

A new method of detecting boundary layer transition at supersonic speeds is described. rity of methods such as schlieren and those of maximum surface temperature and peak surface pitot pressure,locate positions near the end of transition, which, as will be shown in the paper, have a strong Mach number and unit Reynolds number dependence. A more complete picture of transition dependence on these parameters has been obtained by measuring the change in the root mean square of the voltage fluctuation across surface thin film gauges operated at constant temperature. effects of Mach number and unit Reynolas number on the beginning and length of transition to be established more precisely than with previously used methods. The use of thin film gauges to infer local laminar and turbulent skin friction a t supersonic speeds is also described. The majoThis technique enables the