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# Some problems concerning the rotational motion of a perfect fluid

## Citation

Marble, Frank Earl (1948) Some problems concerning the rotational motion of a perfect fluid. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2JGF-0Z67. https://resolver.caltech.edu/CaltechETD:etd-11122003-104144

## Abstract

In an effort to obtain some understanding of the processes involved in the rotational motion of a perfect fluid several particular linearized examples of rotational flow are solved in detail. The first part discusses some types of boundary value problem which arise. The solution of the non-linear partial differential equation by a particular iteration process is considered and the process is shown to converge for an extended version of the problem when the vorticity distribution is sufficiently smooth. The first step of the iteration process may constitute a good approximation in these cases and is taken as the basis of linearized solutions studied in the remainder of the work. The process of straightening a non-uniform velocity profile by means of an idealized screen is considered in Part II as a problem in rotational motion of an ideal fluid with the screen replaced by an appropriate non-conservative force field. The detailed solution is given for both the linearized problem and the second approximation, The complete second order correction is less than 6 percent of the local velocity given by the linear solution for a rather severe case, The corrections arising from the various physical processes involved are analyzed and found to exceed 6 percent in same cases but are inherently compensating. The two-dimensional rotational flow about a closed body is obtained in Part III by utilizing the Green's function method of solving the inhomogeneous differential equation involved. The conformal transformation which maps the given contour into a circle is used to find the appropriate Green's function for the contour. Solutions are then written down for any body, the Riemann mapping function of which is known, The Blasius force and moment formula are extended to include the case of general rotational motion, the relations of Kuo appearing as special forms where the vorticity distribution is uniform. In the final part the theory of the three-dimensional. flow through an axial turbomachine, associated with variation of circulation along the blade length, is described as an extension of the classical theory of finite wings and is simplified to a problem in axially symmetric rotational fluid motion by considering an infinite number of blades in each row. The linearized problem is solved for the radial, tangential, and axial velocity components induced by a single row of stationary or rotating blades with finite chord and prescribed loading. The particular case for which the blade chord approaches zero, and the tangential velocity changes discontinuously, is associated with the theory of the Prandtl lifting line for finite wings, The complete solution is given for a single stationary or rotating blade row of given loading with a hub/tip ratio of 0.6 and blade aspect ratio of 2. The corresponding discontinuous approximation is compared with the more nearly exact solution and is shown to constitute a useful approximation to the solution for a finite blade chord when the discontinuity is located appropriately. An exponential approximation for the velocity components, deduced from the analysis, allows rapid estimation of the rate at which the equilibrium velocity profiles develop ahead of and behind a blade row and, using the superposition principle, provides a simple means or approximating the velocity distribution in a multistage turbomachine and of discussing mutual interference of blade rows.

Item Type: Thesis (Dissertation (Ph.D.)) Aeronautics California Institute of Technology Engineering and Applied Science Aeronautics Public (worldwide access) Liepmann, Hans Wolfgang GALCIT Unknown, Unknown 1 January 1948 CaltechETD:etd-11122003-104144 https://resolver.caltech.edu/CaltechETD:etd-11122003-104144 10.7907/2JGF-0Z67 No commercial reproduction, distribution, display or performance rights in this work are provided. 4518 CaltechTHESIS Imported from ETD-db 12 Nov 2003 21 Dec 2019 02:49

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