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On Similarity Rules for Transonic Flows

Description: "A method used by Tsien to derive similarity rules for hypersonic flows is utilized to derive Von Karman's similarity rules for transonic flows. At the lower limit of the transonic region of flow the theory yields a formula for the critical stream Mach numbers of a given family of symmetrical profiles. It is further shown that this formula can also be obtained by means of the Prandtl-Glauert small-perturbation method" (p. 83).
Date: December 9, 1947
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On Transonic Flow Past a Wave-Shaped Wall

Description: "This report is an extension of a previous investigation (described in NACA rep. 1069) concerned with the solution of the nonlinear differential equation for transonic flow past a wavy wall. In the present work several new notions are introduced which permit the solution of the recursion formulas arising from the method of integration in series. In addition, a novel numerical tests of convergence, applied to the power series (in transonic similarity parameter) representing the local Mach number distribution at the boundary, indicates that smooth symmetrical potential flow past the wavy wall is no longer possible once the critical value of the stream Mach number has been exceeded" (p. 1).
Date: 1953
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On the Particular Integrals of the Prandtl-Busemann Iteration Equations for the Flow of a Compressible Fluid

Description: "The particular integrals of the second-order and third-order Prandtl-Busemann iteration equations for the flow of a compressible fluid are obtained by means of the method in which the complex conjugate variables are utilized as the independent variables of the analysis. The assumption is made that the Prandtl-Glauert solution of the linearized or first-order iteration equation for the two-dimensional flow of a compressible fluid is known. The forms of the particular integrals, derived for subsonic flow, are readily adapted to supersonic flows with only a change in sign of one of the parameters of the problem" (p. 909).
Date: May 29, 1950
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On a New Method for Calculating the Potential Flow Past a Body of Revolution

Description: "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" (p. 7).
Date: May 18, 1942
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On the Use of Residue Theory for Treating the Subsonic Flow of a Compressible Fluid

Description: "A new mathematical technique, due to Milne-Thomson, is used to obtain an improved form of the method of Poggi for calculating the effect of compressibility on the subsonic flow past an obstacle. By means of this new method, the difficult surface integrals of the original Poggi method can be replaced by line integrals. These line integrals are then solved by the use of residue theory. In this way an equation is obtained giving the second-order effect of compressibility on the velocity of the fluid" (p. 39).
Date: September 8, 1941
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

Effect of Compressibility at High Subsonic Velocities on the Lifting Force Acting on an Elliptic Cylinder

Description: An extended form of the Ackeret iteration method, applicable to arbitrary profiles, is utilized to calculate the compressible flow at high subsonic velocities past an elliptic cylinder. The angle of attack to the direction of the undisturbed stream is small and the circulation is fixed by the Kutta condition at the trailing end of the major axis. The expression for the lifting force on the elliptic cylinder is derived and shows a first-step improvement of the Prandtl-Glauert rule. It is further shown that the expression for the lifting force, although derived specifically for an elliptic cylinder, may be extended to arbitrary symmetrical profiles.
Date: May 16, 1946
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On a Solution of the Nonlinear Differential Equation for Transonic Flow Past a Wave-Shaped Wall

Description: "The Prandtl-Busemann small-perturbation method is utilized to obtain the flow of a compressible fluid past an infinitely long wave-shaped wall. When the essential assumption for transonic flow (that all Mach numbers in the region of flow are nearly unity) is introduced, the expression for the velocity potential takes the form of a power series in the transonic similarity parameter. On the basis of this form of the solution, an attempt is made to solve the nonlinear differential equation for transonic flow past the wavy wall" (p. 1).
Date: 1952
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

The Flow of a Compressible Fluid Past a Curved Surface

Description: "An iteration method is employed to obtain the flow of a compressible fluid past a curved surface. The first approximation which leads to the Prandtl-Glauert rule, is based on the assumption that the flow differs but little from a pure translation. The iteration process then consists in improving this first approximation in order that it will apply to a flow differing from pure translatory motion to a greater degree. The method fails when the Mach number of the undisturbed stream reaches unity but permits a transition from subsonic to supersonic conditions without the appearance of a compression shock" (p. 305).
Date: September 13, 1943
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

The Flow of a Compressible Fluid Past a Circular Arc Profile

Description: "The Ackeret iteration process is utilized to obtain higher approximations than that of Prandtl and Glauert for the flow of a compressible fluid past a circular arc profile. The procedure is to expand the velocity potential in a power series of the camber coefficient. The first two terms of the development correspond to the Prandtl-Glauert approximation and yield the well-known correction to the circulation about the profile" (p. 385).
Date: July 15, 1944
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On Subsonic Flow Past a Paraboloid of Revolution

Description: Note presenting the use of the Janzen-Rayleigh method of expansion in powers of the stream Mach number to calculate the velocity potential for steady subsonic flow past a paraboloid of revolution. Only the first two terms of the expansion are calculated.
Date: February 1957
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

The Small-Disturbance Method for Flow of a Compressible Fluid With Velocity Potential and Stream Function as Independent Variables

Description: Note presenting the equations of two-dimensional compressible flow as treated according to the Prandtl-Busemann small-disturbance method. The independent variables are the compressible velocity potential and stream function and the dependent variables are the rectangular Cartesian coordinates in the plane of flow. The six first-order differential equations corresponding to the first three iteration steps are put into complex-vector form.
Date: August 1954
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

Incompressible Flow Past a Sinusoidal Wall of Finite Amplitude

Description: Note presenting an investigation of some of the possible mathematical troubles that may occur in the iterative procedure so much in use in present-day aerodynamics. The analysis indicates the general undesirability of determining the velocity potential or the stream function in the form of a series whose coefficients are analytically developed in a power series of some parameter.
Date: February 1954
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

Potential Flow About Elongated Bodies of Revolution

Description: "This report presents a method of solving the problem of axial and transverse potential flows around arbitrary elongated bodies of revolution. The solutions of Laplace's equation for the velocity potentials of the axial and transverse flows, the system of coordinates being an elliptic one in a meridian plane, are given. The theory is applied to a body of revolution obtained from a symmetrical Joukowsky profile, a shape resembling an airship hull" (p. 189).
Date: November 12, 1934
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

Compressible Flow About Symmetrical Joukowski Profiles

Description: "The method of Poggi is employed for the determination of the effects of compressibility upon the flow past an obstacle. A general expression for the velocity increment due to compressibility is obtained. The general result holds whatever the shape of the obstacle; but, in order to obtain the complete solution, it is necessary to know a certain Fourier expansion of the square of the velocity of flow past the obstacle. An application is made to the case flow of a symmetrical Joukowski profile with a sharp trailing edge, fixed in a stream of an arbitrary angle of attack and with the circulation determined by the Kutta condition" (p. 197).
Date: November 19, 1937
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

Two-Dimensional Subsonic Compressible Flow Past Elliptic Cylinders

Description: "The method of Poggi is used to calculate, for perfect fluids, the effect of compressibility upon the flow on the surface of an elliptic cylinder at zero angle of attack and with no circulation. The result is expressed in a closed form and represents a rigorous determination of the velocity of the fluid at the surface of the obstacle insofar as the second approximation is concerned. Comparison is made with Hooker's treatment of the same problem according to the method of Janzen and Rayleight and it is found that, for thick elliptic cylinders, the two methods agree very well" (p. 245).
Date: February 11, 1938
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

A Theoretical Study of the Moment on a Body in a Compressible Fluid

Description: "The extension to a compressible fluid of Lagally's theorem on the moment on a body in an incompressible fluid and Poggi's method of treating the flow of compressible fluids are employed for the determination of the effect of compressibility on the moment of an arbitrary body. Only the case of the two-dimensional subsonic flow of an ideal compressible fluid is considered. As examples of the application of the general theory, two well-known systems of profiles are treated; namely the elliptic profile and the symmetrical Joukowski profiles with sharp trailing edges" (p. 581).
Date: March 24, 1939
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On the Particular Integrals of the Prandtl-Busemann Iteration Equations for the Flow of a Compressible Fluid

Description: Note presenting the particular integrals of the second-order and third-order Prandtl-Busemann iteration equations for the flow of a compressible fluid that have been obtained by means of the method in which the complex conjugate variables are utilized as the independent variables of the analysis. The forms of the particular integrals, derived for subsonic flow, are readily adapted to supersonic flows with only a change in sign of one of the parameters of the problem.
Date: August 1950
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

The Flow of a Compressible Fluid Past a Sphere

Description: "The flow of a compressible fluid past a sphere fixed in a uniform stream is calculated to the third order of approximation by means of the Janzen-Rayleigh method. The velocity and the pressure distribution over the surface of the sphere are computed and the terms involving the fourth power of the Mach number, neglected in Rayleigh's calculation, are shown to be of considerable importance as the local velocity of sound is approached on the sphere. The critical Mach number, that is, the value of the Mach number at which the maximum velocity of the fluid past the sphere is just equal to the local velocity of sound, is calculated for both the second and the third approximation and is found to be, respectively, Mcr=0.587 and Mcr=0.573" (p. 1).
Date: May 1940
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

Circular Motion of Bodies of Revolution

Description: The circular motion for airship-like bodies has thus far been calculated only for a prolate ellipsoid of revolution (reference 1, p.133 and reference 2). In this paper, however, the circular motion of elongated bodies of revolution more nearly resembling airships will be investigated. The results will give the effect of rotation on the pressure distribution and thus yield some information as to the stresses set up in an airship in circular flight.
Date: February 1936
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

Effect of Compressibility at High Subsonic Velocities on the Moment Acting on an Elliptic Cylinder

Description: Note presenting the utilization of an extended form of the Ackeret iteration process to calculate the compressible flow at high subsonic velocities past an elliptical cylinder. The expression for the moment acting on the elliptical cylinder is derived and shows a first-step improvement of the Prandtl-Glauert approximation. A second-step improvement is obtained in the Prandtl-Glauert approximation for the lifting force acting on the elliptic cylinder.
Date: March 1947
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On the Small-Disturbance Iteration Method for the Flow of a Compressible Fluid With Application to a Parabolic Cylinder

Description: Note presenting the assumptions of the Prandtl-Busemann small-disturbance method, together with the requirements of continuity and irrotationality, which lead to a recursive system of first-order partial-differential equations. The results of the analysis are applied to the case of subsonic flow past a parabolic cylinder.
Date: January 1955
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department

On a New Method for Calculating the Potential Flow Past a Body of Revolution

Description: Report presenting a new method for obtaining the velocity potential of the flow about a solid of revolution moving uniformly in the direction of its axis of symmetry in a fluid otherwise at rest. The method is based essentially on the fact that the form of the differential equation for the velocity potential is invariant with regard to conformal transformations in a meridian plane. The general results are applied to a body of revolution obtained by revolving a symmetrical Joukowski profile about its axis of symmetry.
Date: July 1942
Creator: Kaplan, Carl
Partner: UNT Libraries Government Documents Department
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