Supersonic Flow Past Bodies of Revolution with Thin Wings of Small Aspect Ratio

1951 ◽  
Vol 3 (1) ◽  
pp. 61-79 ◽  
Author(s):  
P. M. Stocker
Keyword(s):  
Supersonic Flow ◽  
Aspect Ratio ◽  
The Body ◽  
Supersonic Stream ◽  
List Type ◽  
Body Of Revolution ◽  
Small Aspect Ratio ◽  
Flow Past ◽  
Special Cases ◽  
Ratio Set

SummaryThe method developed by G. N. Ward for the treatment of slender pointed bodies in a uniform supersonic stream is applied to three special cases. (i)Supersonic flow past a body of revolution with thin wings of symmetrical section and of small aspect ratio at zero incidence.(ii)Supersonic flow past a body of revolution with plane wings of small aspect ratio set at incidence to the body, the whole being at incidence to the stream.(iii)Supersonic flow past a body of revolution with a plane fin of small aspect ratio set at incidence, the whole being at incidence to the stream.The pressure distribution on the wing has been calculated for a special case of (i) and is given in the Appendix.

Supersonic Flow Past Wing-Body Combinations

1953 ◽  
Vol 4 (3) ◽  
pp. 287-314 ◽  
Author(s):  
W. Chester
Keyword(s):  
Supersonic Flow ◽  
Aspect Ratio ◽  
Leading Edge ◽  
The Body ◽  
Infinite Cylinder ◽  
Main Stream ◽  
Exact Equation ◽  
Body Of Revolution ◽  
Flow Past ◽  
General Conditions

SummaryThe supersonic flow past a combination of a thin wing and a slender body of revolution is discussed by means of the linearised equation of motion. The exact equation is first established so that the linearised solution can be fed back and the order of the error terms calculated. The theory holds under quite general conditions which should be realised in practice.The wing-body combination considered consists of a wing symmetrically situated on a pointed body of revolution and satisfying the following fairly general conditions. The wing leading edge is supersonic at the root, and the body is approximately cylindrical downstream of the leading edge. The body radius is of an order larger than the wing thickness, but is small compared with the chord or span of the wing.It is found that if the wing and body are at the same incidence, and the aspect ratio of the wing is greater than 2 (M2-1)-½, where M is the main stream Mach number, the lift is equivalent to that of the complete wing when isolated. If the wing only is at incidence then the lift is equivalent to that of the part of the wing lying outside the body.The presence of the body has a more significant effect on the drag. If, for example, the body is an infinite cylinder of radius a, and the wing is rectangular with aspect ratio greater than 2(M2-1)-½, then the drag of the wing is decreased by a factor (1-2a/b), where 2b is the span of the wing.When these conditions do not hold the results are not quite so simple but are by no means complicated.

The Supersonic Flow Past a Slender Ducted Body of Revolution with an Annular Intake

1950 ◽  
Vol 1 (4) ◽  
pp. 305-318
Author(s):  
G. N. Ward
Keyword(s):  
Supersonic Flow ◽  
Velocity Potential ◽  
Operational Calculus ◽  
The Body ◽  
Body Of Revolution ◽  
Internal Flows ◽  
Flow Past ◽  
External Forces ◽  
Internal Pressures ◽  
Slender Body Of Revolution

SummaryThe approximate supersonic flow past a slender ducted body of revolution having an annular intake is determined by using the Heaviside operational calculus applied to the linearised equation for the velocity potential. It is assumed that the external and internal flows are independent. The pressures on the body are integrated to find the drag, lift and moment coefficients of the external forces. The lift and moment coefficients have the same values as for a slender body of revolution without an intake, but the formula for the drag has extra terms given in equations (32) and (56). Under extra assumptions, the lift force due to the internal pressures is estimated. The results are applicable to propulsive ducts working under the specified condition of no “ spill-over “ at the intake.

Stokes flow past a slender body of revolution

1976 ◽  
Vol 78 (3) ◽  
pp. 577-600 ◽  
Author(s):  
James Geer
Keyword(s):  
Stokes Flow ◽  
Slenderness Ratio ◽  
Slip Boundary Condition ◽  
The Body ◽  
Slender Body ◽  
Total Force ◽  
Body Of Revolution ◽  
Flow Past ◽  
Special Cases ◽  
Slender Body Of Revolution

The complete uniform asymptotic expansion of the velocity and pressure fields for Stokes flow past a slender body of revolution is obtained with respect to the slenderness ratio ε of the body. A completely general incident Stokes flow is assumed and hence these results extend the special cases treated by Tillett (1970) and Cox (1970). The part of the flow due to the presence of the body is represented as a superposition of the flows produced by three types of singularity distributed with unknown densities along a portion of the axis of the body and lying entirely inside the body. The no-slip boundary condition on the body then leads to a system of three coupled, linear, integral equations for the densities of the singularities. The complete expansion for these densities is then found as a series in powers of ε and ε log ε. It is found that the extent of these distributions of singularities inside the body is the same for all the singular flows and depends only upon the geometry of the body. The total force, drag and torque experienced by the body are computed.

The behaviour of supersonic flow past a body of revolution, far from the axis

1950 ◽  
Vol 201 (1064) ◽  
pp. 89-109 ◽  
Keyword(s):  
Supersonic Flow ◽  
General Theory ◽  
Body Of Revolution ◽  
Flow Past ◽  
Physical Importance ◽  
Special Case ◽  
Pointed Body ◽  
Asymptotic Forms

A theory is developed of the supersonic flow past a body of revolution at large distances from the axis, where a linearized approximation is valueless owing to the divergence of the characteristics at infinity. It is used to find the asymptotic forms of the equations of the shocks which are formed from the neighbourhoods of the nose and tail. In the special case of a slender pointed body, the general theory at large distances is used to modify the linearized approximation to give a theory which is uniformly valid at all distances from the axis. The results which are of physical importance are summarized in the conclusion (§ 9) and compared with the results of experimental observations.

Force and moment characteristics of supersonic flow past a cylindrical body of revolution with a fluid wing

1988 ◽  
Vol 22 (5) ◽  
pp. 743-746 ◽  
Author(s):  
V. F. Zakharchenko ◽  
Yu. Kh. Kardanov ◽  
P. V. Sidorov
Keyword(s):  
Supersonic Flow ◽  
Cylindrical Body ◽  
Body Of Revolution ◽  
Flow Past

On the propagation of weak shock waves

1956 ◽  
Vol 1 (3) ◽  
pp. 290-318 ◽  
Author(s):  
G. B. Whitham
Keyword(s):  
Supersonic Flow ◽  
Delta Wing ◽  
Thickness Distribution ◽  
Leading Edge ◽  
Weak Shock ◽  
Wave Drag ◽  
Main Step ◽  
Body Of Revolution ◽  
Flow Past ◽  
Cone Field

A method is presented for treating problems of the propagation and ultimate decay of the shocks produced by explosions and by bodies in supersonic flight. The theory is restricted to weak shocks, but is of quite general application within that limitation. In the author's earlier work on this subject (Whitham 1952), only problems having directional symmetry were considered; thus, steady supersonic flow past an axisymmetrical body was a typical example. The present paper extends the method to problems lacking such symmetry. The main step required in the extension is described in the introduction and the general theory is completed in §2; the remainder of the paper is devoted to applications of the theory in specific cases.First, in §3, the problem of the outward propagation of spherical shocks is reconsidered since it provides the simplest illustration of the ideas developed in §2. Then, in §4, the theory is applied to a model of an unsymmetrical explosion. In §5, a brief outline is given of the theory developed by Rao (1956) for the application to a supersonic projectile moving with varying speed and direction. Examples of steady supersonic flow past unsymmetrical bodies are discussed in §6 and 7. The first is the flow past a flat plate delta wing at small incidence to the stream, with leading edges swept inside the Mach cone; the results agree with those previously found by Lighthill (1949) in his work on shocks in cone field problems, and this provides a valuable check on the theory. The second application in steady supersonic flow is to the problem of a thin wing having a finite curved leading edge. It is found that in any given direction the shock from the leading edge ultimately decays exactly as for the bow shock on a body of revolution; the equivalent body of revolution for any direction is determined in terms of the thickness distribution of the wing and varies with the direction chosen. Finally in §8, the wave drag on the wing is calculated from the rate of dissipation of energy by the shocks. The drag is found to be the mean of the drags on the equivalent bodies of revolution for the different directions.

Airscrews at Supersonic Forward Speeds

1951 ◽  
Vol 3 (1) ◽  
pp. 23-50
Author(s):  
J. C. Burns
Keyword(s):  
Boundary Conditions ◽  
Working Conditions ◽  
Point Of View ◽  
Supersonic Stream ◽  
Main Stream ◽  
Small Perturbations ◽  
Potential Equation ◽  
Flow Past ◽  
Special Cases ◽  
Approximate Expressions

SummaryThe flow past an airscrew rotating with uniform angular velocity in a uniform supersonic stream is considered from the point of view of axes fixed in the airscrew, the motion relative to these axes being steady. A linearised equation and boundary conditions are found for the potential of the disturbance flow caused by airscrews which produce only small perturbations in the main stream. This equation is solved by expanding the potential in powers of the ratio, χ, of the tip speed of the airscrew to the speed of sound in the undisturbed stream. Equations and boundary conditions are found for the coefficients of each term of the series and it is seen that the term independent of χ satisfies the ordinary linearised potential equation for fixed axes. By subtracting suitable special integrals, the more complicated equations for the coefficients of the various powers of χ can be reduced to this ordinary potential equation. Hadamard's methods have been applied by Puckett, Evvard and, finally, Ward to the problem of finding the flow past a thin wing fixed in a supersonic stream and their methods are applied to find the terms of the series in our case. The first two terms are found explicitly for a particular type of windmill. Approximate expressions are then developed for the torque and the drag on such a windmill and the results applied in special cases. The best working conditions for windmills of this type are found by considering the effect on efficiency and power of varying the shape and the speed of rotation, within the limits imposed by the strength of the material of the airscrew.

Axisymmetric slow viscous flow past an arbitrary convex body of revolution

1972 ◽  
Vol 55 (4) ◽  
pp. 677-709 ◽  
Author(s):  
Michael J. Gluckman ◽  
Sheldon Weinbaum ◽  
Robert Pfeffer
Keyword(s):  
Integral Equation ◽  
Convex Body ◽  
Aspect Ratio ◽  
Boundary Value Problem ◽  
Stokes Flow ◽  
Flow Equation ◽  
The Body ◽  
Slow Flow ◽  
Body Of Revolution ◽  
Arbitrary Convex

Considerable advances have been made in the past few years in treating a variety of problems in slender-body Stokes flow (Taylor 1969; Batchelor 1970; Cox 1970, 1971; Tillett 1970). However, the problem of treating the creeping motion past bluff objects, whose boundaries do not conform to a constant co-ordinate surface of one of the special orthogonal co-ordinate systems for which the Stokes slow-flow equation is simply separable, is still largely unsolved. In the slender-body Stokes flow studies mentioned above, the viscous-flow boundary-value problem is formulated approximately as an integral equation for an unknown distribution of Stokeslets over a line enclosed by the body. The theory is valid for only very extended shapes, since the error in drag decays inversely as the logarithm of the aspect ratio of the object. By contrast, the present authors show that the boundary-value problem for the axisymmetric flow past an arbitrary convex body of revolution can be formulated exactly as an integral equation for an unknown distribution of ring-like singularities over the surface of the body. The kernel in this integral equation is closely related to the fundamental separable solutions of the Stokes slow-flow equation when written in an oblate spheroidal co-ordinate system of vanishing aspect ratio. The two lowest-order appropriate spheroidal singularities are found to provide a complete description for all surface elements, except those perpendicular to the axis. Higher-order singularities of all orders are required to describe axially perpendicular surfaces, such as the ends of a cylinder or the blunt base of an object. The newly derived integral equation is solved numerically to provide the first theoretical solutions for low aspect ratio cylinders and cones. The theoretically predicted drag results are in excellent agreement with experimentally measured values.

Supersonic flow over an inclined wing of zero aspect ratio

1950 ◽  
Vol 46 (2) ◽  
pp. 307-315 ◽  
Author(s):  
K. Stewartson
Keyword(s):  
Supersonic Flow ◽  
Aspect Ratio ◽  
Potential Theory ◽  
Asymptotic Expression ◽  
Supersonic Stream

ABSTRACTAn asymptotic expression is found for the lift distribution on a long, narrow, laminar wing, at incidence in a supersonic stream. The approximations of the linearized potential theory are used.

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