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We have set ourselves the problem of calculating the laminar flow on a body of revolution in an axial flow which simultaneously rotates about its axis. The problem mentioned above, the flow about a rotating disk in a flow, which we solved some time ago, represents the first step in the calculation of the flow on the rotating body of revolution in a flow insofar as, in the case of a round nose, a small region about the front stagnation point of the body of revolution may be replaced by its tangential plane. In our problem regarding the rotating body of revolution in a flow, for laminar flow, one of the limiting cases is known: that of the body which is in an axial approach flow but does not rotate. The other limiting case, namely the flow in the neighborhood of a body which rotates but is not subjected to a flow is known only for the rotating circular cylinder, aside from the rotating disk. In the case of the cylinder one deals with a distribution of the circumferential velocity according to the law v = omega R(exp 2)/r where R signifies the cylinder radius, r the distance from the center, and omega the angular velocity of the rotation. The velocity distribution as it is produced here by the friction effect is therefore the same as in the neighborhood of a potential vortex. When we treat, in what follows, the general case of the rotating body of revolution in a flow according to the calculation methods of Prandtl's boundary-layer theory, we must keep in mind that this solution cannot contain the limiting case of the body of revolution which only rotates but is not subjected to a flow. However, this is no essential limitation since this case is not of particular importance for practical purposes.
A method based on linearized and slender-body theories, which is easily adapted to electronic-machine computing equipment, is developed for calculating the zero-lift wave drag of single- and multiple-component configurations from a knowledge of the second derivative of the area distribution of a series of equivalent bodies of revolution. The accuracy and computational time required of the method to calculate zero-lift wave drag is evaluated relative to another numerical method which employs the Tchebichef form of harmonic analysis of the area distribution of a series of equivalent bodies of revolution. The results of the evaluation indicate that the total zero-lift wave drag of a multiple-component configuration can generally be calculated most accurately as the sum of the zero-lift wave drag of each component alone plus the zero-lift interference wave drag between all pairs of components. The accuracy and computational time required of both methods to calculate total zero-lift wave drag at supersonic Mach numbers is comparable for airplane-type configurations. For systems of bodies of revolution both methods yield similar results with comparable accuracy; however, the present method only requires up to 60 percent of the computing time required of the harmonic-analysis method for two bodies of revolution and less time for a larger number of bodies.
An investigation has been conducted in the Langley 20-foot free spinning tunnel to study the relative behavior in descent of a number of homogeneous balsa bodies of revolution simulating anti-personnel bombs with a small cylindrical exploding device suspended approximately 10 feet below the bomb. The bodies of revolution included hemispherical, near-hemispherical, and near-paraboloid shapes. The ordinates of one near-paraboloid shape were specified by the Office of the Chief of Ordnance, U. S. Army. The behavior of the various bodies without the cylinder was also investigated. The results of the investigation indicated that several of the bodies descended vertically with their longitudinal axis, suspension line, and small cylinder in a vertical attitude,. However, the body, the ordinates of which had been specified by the Office of the Chief of Ordnance, U. S. Army, oscillated considerably from a vertical attitude while descending and therefore appeared unsuitable for its intended use. The behavior of this body became satisfactory when its center of gravity was moved well forward from its original position. In general, the results indicated that the descent characteristics of the bodies of revolution become more favorable as their shapes approached that of a hemisphere.
A new method is presented for obtaining the velocity potential of the flow about a body of revolution moving uniformly in the direction of its axis of symmetry in a fluid otherwise at rest. This method is based essentially on the fact that the form of the differential equation for the velocity potential is invariant with regard to conformal transformation of the meridian plane. By means of the conformal transformation of the meridian profile into a circle a system of orthogonal curvilinear coordinates is obtained, the main feature of which is that one of the coordinate lines is the meridian profile itself. The use of this type of coordinate system yields a simple expression of the boundary condition at the surface of the solid and leads to a rational process of iteration for the solution of the differential equation for the velocity potential. It is shown that the velocity potential for an arbitrary body of revolution may be expressed in terms of universal functions which, although not normal, are obtainable by means of simple quadratures. The general results are applied to a body of revolution obtained by revolving a symmetrical Joukowski profile about its axis of symmetry. A numerical example further serves to illustrate the theory.
The wake behind a two-bladed model rotor in light climb was measured using particle image velocimetry, with particular emphasis on the development of the trailing vortex during the first revolution of the rotor. The distribution of vorticity was distinguished from the slightly elliptical swirl pattern. Peculiar dynamics within the void region may explain why the peak vorticity appeared to shift away from the center as the vortex aged, suggesting the onset of instability. The swirl and axial velocities (which reached 44 and 12 percent of the rotor-tip speed, respectively) were found to be asymmetric relative to the vortex center. In particular, the axial flow was composed of two concentrated zones moving in opposite directions. The radial distribution of the circulation rapidly increased in magnitude until reaching a point just beyond the core radius, after which the rate of growth decreased significantly. The core-radius circulation increased slightly with wake age, but the large-radius circulation appeared to remain relatively constant. The radial distributions of swirl velocity and vorticity exhibit self-similar behaviors, especially within the core. The diameter of the vortex core was initially about 10 percent of the rotor-blade chord, but more than doubled its size after one revolution of the rotor. According to vortex models that approximate the measured data, the core-radius circulation was about 79 percent of the large-radius circulation, and the large-radius circulation was about 67 percent of the maximum bound circulation on the rotor blade. On average, about 53 percent of the maximum bound circulation resides within the vortex core during the first revolution of the rotor.
Low-thrust trajectories about planetary bodies characteristically span a high count of orbital revolutions. Directing the thrust vector over many revolutions presents a challenging optimization problem for any conventional strategy. This paper demonstrates the tractability of low-thrust trajectory optimization about planetary bodies by applying a Sundman transformation to change the independent variable of the spacecraft equations of motion to the eccentric anomaly and performing the optimization with differential dynamic programming. Fuel-optimal geocentric transfers are shown in excess of 1000 revolutions while subject to Earths J2 perturbation and lunar gravity.
Low-thrust trajectories about planetary bodies characteristically span a high count of orbital revolutions. Directing the thrust vector over many revolutions presents a challenging optimization problem for any conventional strategy. This paper demonstrates the tractability of low-thrust trajectory optimization about planetary bodies by applying a Sundman transformation to change the independent variable of the spacecraft equations of motion to the eccentric anomaly and performing the optimization with differential dynamic programming. Fuel-optimal geocentric transfers are shown in excess of 1000 revolutions while subject to Earths J2 perturbation and lunar gravity.
Stresses and deformations in thin shells of revolution - dynamic response of shells of revolution and bending analysis of elastic plastic circular plates
A large deflection elastic-plastic analysis is presented, applicable to orthotropic axisymmetric plates and shells of revolution subjected to monotonic and cyclic loading conditions. The analysis is based on the finite-element method. It employs a new higher order, fully compatible, doubly curved orthotropic shell-of-revolution element using cubic Hermitian expansions for both meridional and normal displacements. Both perfectly plastic and strain hardening behavior are considered. Strain hardening is incorporated through use of the Prager-Ziegler kinematic hardening theory, which predicts an ideal Bauschinger effect. Numerous sample problems involving monotonic and cyclic loading conditions are analyzed. The monotonic results are compared with other theoretical solutions.
Efforts to develop a near sonic transport have placed renewed emphasis on obtaining accurate aerodynamic force and pressure data in the near sonic speed range. Comparison of wind-tunnel and flight data obtained for a blunt-nose body of revolution showed significant discrepancies in drag levels near Mach 1 - apparently due to wind-tunnel wall interference. Subsequent tests of geometrically similar bodies of revolution showed that increasing the model-to-test-section blockage ratio from 0.00017 to 0.0043 resulted in altered drag curve shapes, delayed drag divergence, and 'transonic creep' from subsonic drag levels due to increased wall interference.
A low-speed wind-tunnel investigation was conducted in order to determine the cause of asymmetric yawing moments produced by long pointed fuselage nose shapes at high angles of attack. Force tests were conducted with a cone, tangent-ogive model, and a paraboloid of revolution over a range of Reynolds numbers from 0.15 million to 0.35 million for an angle-of-attack range from 0 deg to 75 deg and an angle-of-sideslip range of plus or minus 30 deg. Tuft and smoke flow-visualization tests were also conducted to aid in the analysis. Large asymmetric yawing moments were obtained for the cone and tangent-ogive body at high angles of attack (of the order of 40 to 60 deg). These large moments were caused by asymmetric shedding of vortex sheets off the long pointed nose. The asymmetric moments could be eliminated by use of symmetrically arranged strikes on the nose. The paraboloid of revolution did not produce a strong asymmetric flow field at high angles of attack and did not exhibit asymmetric moments.
A procedure for the structural analysis of stiffened shells of revolution is presented. A digital computer program based on the Love-Reissner first order shell theory was developed. The computer program can analyze orthotropic thin shells of revolution, subjected to unsymmetric distributed loading or concentrated line loads, as well as thermal strains. The geometrical shapes of the shells which may be analyzed are described. The shell wall cross section can be a sheet, sandwich, or reinforced sheet or sandwich. General stiffness input options are also available.
A large deflection elastic-plastic analysis is presented applicable to orthotropic axisymmetric plates and shells of revolution subjected to monotonic and cyclic loading conditions. The analysis is based on the finite-element method. It employs a new higher order, fully compatible, doubly curved orthotropic shell-of-revolution element using cubic Hermitian expansions for both meridional and normal displacements. Both perfectly plastic and strain hardening behavior are considered. Strain hardening is incorporated through use of the Prager-Ziegler kinematic hardening theory, which predicts an ideal Bauschinger effect. Numerous sample problems involving monotonic and cyclic loading conditions are analyzed.
Asymptotic solutions of Novozhilov's equations of shells of revolution are derived for axisymmetric and first harmonic loadings. The solutions obtained are valid throughout the shallow and nonshallow regions. Stresses in dome shaped shells of revolution with a discontinuity in the form of a circular hole, a circular rigid insert, or a nozzle at the apex have been investigated. Numerical results are obtained for spheres, ellipsoids, and paraboloids, containing a discontinuity under a internal pressure and moment. Curves depicting stress distributions are given. The influence of three types of discontinuity on the stresses of the shells is also given.
This paper describes the latest addition to the STARS system of computer programs, STARS-2P, for the plastic, large deflection analysis of axisymmetrically loaded shells of revolution. The STARS system uses a numerical integration scheme to solve the governing differential equations. Several unique features for shell of revolution programs that are included in the STARS-2P program are described. These include orthotropic nonlinear kinematic hardening theory, a variety of shell wall cross sections and discrete ring stiffeners, cyclic and nonproportional mechanical and thermal loading capability, the coupled axisymmetric large deflection elasto-plastic torsion problem, an extensive restart option, arbitrary branching capability, and the provision for the inelastic treatment of smeared stiffeners, isogrid, and waffle wall constructions. To affirm the validity of the results, comparisons with available theoretical and experimental data are presented.
Programs, using Love-Reissner first-order shell theory, can analyze orthotropic thin shells of revolution subjected to unsymmetric distributed loading or concentrated line loads and thermal strains. They can perform stability or vibration analysis of thin shells of revolution subjected to axisymmetric distributed loading or concentrated line loads and thermal strains.
An experimental investigation was conducted to determine the effect of forebody geometry, a grit ring around the nose, Reynolds number, Mach number, and angle of attack on the aerodynamic characteristics of a body of revolution. Aerodynamic force and moment characteristics were measured for a cylindrical body with tangent ogive noses of fineness ratio 2.5, 3.0, 3.5, and 5.0. The cylindrical body was tested with an ogive nose having a rounded tip and an ogive nose with two different nose strake arrangements. Aerodynamic configurations were tested at various Mach numbers, angles of attack, and Reynolds numbers. The data demonstrate that the aerodynamic characteristics for a body of revolution can be significantly affected by changes in nose fineness ratio, nose bluntness, Reynolds number, Mach number, and, of course, angle of attack. Nose strakes increased the normal forces but had little effect on the side forces that developed at subsonic Mach numbers for alpha greater than about 25. A grit ring around the nose had little or no effect on the aerodynamic characteristics.