Slender bodies of revolution with minimum wave drag in nonequilibrium supersonic flow

We construct analytical solutions for the problem of nonlinear supersonic flow past slender bodies of revolution due to small amplitude oscillations. The method employed is based on the splitting of the time dependent small perturbation equation to a nonlinear time independent partial differential equation (P.D.E.) concerning the steady flow, and a linear time dependent one, concerning the unsteady flow. Solutions in the form of three parameters family of surfaces for the first equation are constructed, while solutions including one arbitrary function for the second equation are extracted. As an application the evaluation of the small perturbation velocity resultants for a flow past a right circular cone is obtained making use of convenient boundary and initial conditions in accordance with the physical problem.

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On non-steady motion of slender bodies

Aeronautical Quarterly ◽

10.1017/s000192590000041x ◽

1950 ◽

Vol 2 (3) ◽

pp. 183-194 ◽

Cited By ~ 19

Author(s):

John W. Miles

Keyword(s):

Mach Number ◽

Supersonic Flow ◽

Original Work ◽

Slenderness Ratio ◽

Steady Motion ◽

Order Of Approximation ◽

Bodies Of Revolution ◽

Kutta Condition ◽

Unsteady Supersonic Flow ◽

Slender Bodies

SummaryFollowing the original work of Munk, Jones, and Ward for steady flow, a solution is given for unsteady, supersonic flow over very slender bodies of revolution and wings. The results are subject to the restriction (M2- l)δ2loge[(M2- 1)½δ]«1 where δ is the slenderness ratio,Mis the Mach number and, in addition, to the usual restrictions imposed by linearisation. As examples, the lifts and pitching moments on flat wings and bodies of revolution executing pitching motion and the damping moment on a rolling wing are calculated.It is shown that the order of approximation is consistent with the limitations already imposed by linearisation, at least in supersonic flow, where no Kutta condition is required.

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Nonlinear unsteady supersonic flow analysis for slender bodies of revolution: Series solutions, convergence and results

Mathematical Problems in Engineering ◽

10.1155/s1024123x97000641 ◽

1998 ◽

Vol 3 (6) ◽

pp. 481-501

Author(s):

M. Markakis ◽

D. E. Panayotounakos

Keyword(s):

Supersonic Flow ◽

Initial Conditions ◽

Flow Analysis ◽

Circular Cone ◽

Series Solutions ◽

Bodies Of Revolution ◽

Unsteady Supersonic Flow ◽

Slender Bodies ◽

Specific Geometry ◽

Limiting Values

In Ref. [6] the authors constructed analytical solutions including one arbitrary function for the problem of nonlinear, unsteady, supersonic flow analysis concerning slender bodies of revolution due to small amplitude oscillations. An application describing a flow past a right circular cone was presented and the constructed solutions were given in the form of infinite series through a set of convenient boundary and initial conditions in accordance with the physical problem. In the present paper we develop an appropriate convergence analysis concerning the before mentioned series solutions for the specific geometry of a rigid right circular cone. We succeed in estimating the limiting values of the series producing velocity and acceleration resultants of the problem under consideration. Several graphics for the velocity and acceleration flow fields are presented. We must underline here that the proposed convergence technique is unique and can be applied to any other geometry of the considered body of revolution.

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