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Kaplan, Carl

Publications and source records attributed to Kaplan, Carl.

33 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1944

On the flow of a compressible fluid by the hodograph method II : fundamental set of particular flow solutions of the Chaplygin differential equation

The differential equation of Chaplygin's jet problem is utilized to give a systematic development of particular solutions of the hodograph flow equations, which extends the treatment of Chaplygin into the supersonic range and completes the set of particular solutions. The particular solutions serve to place on a reasonable basis the use of velocity correction formulas for the comparison of incompressible and compressible flows. It is shown that the geometric-mean type of velocity correction formula introduced in part I has significance as an over-all type of approximation in the subsonic range. A brief review of general conditions limiting the potential flow of an adiabatic compressible fluid is given and application is made to the particular solutions, yielding conditions for the existence of singular loci in the supersonic range. The combining of particular solutions in accordance with prescribed boundary flow conditions is not treated in the present paper.

Garrick, I E↗
NASA NTRS1944

The flow of a compressible fluid past a circular arc profile

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. The second approximation, involving the square of the camber coefficient, improves the velocity and pressure fields but yields no new results with regard to the circulation, since the circulation about the profile is an odd function of the camber coefficient. The third approximation, involving the cube of the camber coefficient, permits the use of higher values of the camber coefficient and furthermore yields an improvement to the Prandtl-Glauert rule with regard to the effect of compressibility on the circulation of the circular arc profile. Numerical examples with tables and graphs illustrate the results of the analysis.

Kaplan, Carl↗
NASA NTRS1944

On the flow of a compressible fluid by the hodograph method I : unification and extension of present-day results

Elementary basic solutions of the equations of motion of a compressible fluid in the hodograph variables are developed and used to provide a basis for comparison, in the form of velocity correction formulas, of corresponding compressible and incompressible flows. The known approximate results of Chaplygin, Von Karman and Tsien, Temple and Yarwood, and Prandtl and Glauert are unified by means of the analysis of the present paper. Two new types of approximations, obtained from the basic solutions, are introduced; they possess certain desirable features of the other approximations and appear preferable as a basis for extrapolation into the range of high stream Mach numbers and large disturbances to the main stream. Tables and figures giving velocity and pressure-coefficient correction factors are included in order to facilitate the practical application of the results.

Garrick, I E↗
NASA NTRS1944

On the Flow of a Compressible Fluid by the Hodograph Method. II - Fundamental Set of Particular Flow Solutions of the Chaplygin Differential Equation

The differential equation of Chaplygin's jet problem is utilized to give a systematic development of particular solutions of the hodograph flow equations, which extends the treatment of Chaplygin into the supersonic range and completes the set of particular solutions. The particular solutions serve to place on a reasonable basis the use of velocity correction formulas for the comparison of incompressible and compressible flows. It is shown that the geometric-mean type of velocity correction formula introduced in part I has significance as an over-all type of approximation in the subsonic range. A brief review of general conditions limiting the potential flow of an adiabatic compressible fluid is given and application is made to the particular solutions, yielding conditions for the existence of singular loci in the supersonic range. The combining of particular solutions in accordance with prescribed boundary flow conditions is not treated in the present paper.

Garrick, I. E.↗
NASA NTRS1943

On a new method for calculating the potential flow past a body of revolution

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.

Kaplan, Carl↗
NASA NTRS1943

The flow of a compressible fluid past a curved surface

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. The limiting value at which potential flow no longer exits is indicated by the apparent divergence of the power series representing the velocity of the fluid at the surface of the solid boundary.

Kaplan, Carl↗
NASA NTRS1942

On the use of residue theory for treating the subsonic flow of a compressible fluid

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. The method is practicable for obtaining the higher-order effects of compressibility on the velocity field. As an illustration of the general result, the flow past an elliptic cylinder is discussed.

Kaplan, Carl↗
NASA NTRS1940

The flow of a compressible fluid past a sphere

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.

Kaplan, Carl↗
NASA NTRS1939

A theoretical study of the moment on a body in a compressible fluid

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. The effect of compressibility on the position of the center of pressure is also discussed. Several numerical examples of both the elliptic and the Joukowski profiles are given.

Kaplan, Carl↗
NASA NTRS1938

Compressible flow about symmetrical Joukowski profiles

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. The results are obtained in a closed form and are exact insofar as the second approximation to the compressible flow is concerned, the first approximation being the result for the corresponding incompressible flow. Formulas for lift and moment analogous to the Blasius formulas in incompressible flow are developed and are applied to thin symmetrical Joukowski profiles for small angles of attack.

Kaplan, Carl↗
NASA NTRS1938

Two-dimensional subsonic compressible flow past elliptic cylinders

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. The labor of computation is considerably reduced by the present solution.

Kaplan, Carl↗
NASA NTRS1936

Circular motion of bodies of revolution

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.

Kaplan, Carl↗
NASA NTRS1936

Potential flow about elongated bodies of revolution

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. The pressure distribution and the transverse-force distribution are calculated and serve as examples of the procedure to be followed in the case of an actual airship. A section on the determination of inertia coefficients is also included in which the validity of some earlier work is questioned.

Kaplan, Carl↗