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STATIONARY SOLUTIONS OF THE EULER-EQUATIONS IN TWO DIMENSIONS, SINGLY-AND DOUBLY-CONNECTED V-STATES

IP.com Disclosure Number: IPCOM000128599D
Original Publication Date: 1980-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 12 page(s) / 34K

Publishing Venue

Software Patent Institute

Related People

M. Landau: AUTHOR [+4]

Abstract

In this paper we present stationary numerical solutions of the Euler equations that are composed of singly and doubly-connected regions of constant vorticity density, w 0 In the 19th century, Kirchhoff showed that an elliptic region of constant vorticity was a stationary state in a rotating frame of reference 2 , with angular velocity of rotation [Equation ommitted] 3 Its stability properties were investigated by Love in 1898 Recently, Deem and Zabusky 4,5 studied the stationary and dynamical properties of piecewise constant finite-area-vortex-regions (or "character-istic" functions) in the planes. We henceforth call them FAVR's. They found a class of isolated m-fold symmetric uniformly rotating FAVR's and one of a set of uniformly translating FAVR's, which they called "V-states". Preliminary results were presented tersely at the SIAM meeting in Alexandria Virginia, June 1980 and at the Second International Congress on Numerical Methods for Eygineering, G.A.M.N.I. 2, December 1980, Chatenay-Malabry, Paris, France

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

STATIONARY SOLUTIONS OF THE EULER-EQUATIONS IN TWO DIMENSIONS, SINGLY-AND DOUBLY-CONNECTED V-STATES

by M. Landau and N. J. Zabusky

Institute for Computational Mathematics and Applications Department of Mathematics and Statistics University of Pittsburgh Pittsburgh, PA 15261 USA 111.

+Preliminary results were presented tersely at the SIAM meeting in Alexandria Virginia, June 1980 and at the Second International Congress on Numerical Methods for Engineering, G.A.M.N.I. 2, December 1980, Chatenay-Malabry, Paris, Francel.

Permanent Address: Gulf Science and Technology, P.O. Drawer 2038, Pittsburgh, PA, U.S.A. nstitute for Computational Mathematics and Applications Department of Mathematics and Statistics University of Pittsburgh Pittsburgh, PA U.S.A.

1. INTRODUCTION

In this paper we present stationary numerical solutions of the Euler equations that are composed of singly and doubly-connected regions of constant vorticity density, w 0 In the 19th century, Kirchhoff showed that an elliptic region of constant vorticity was a stationary state in a rotating frame of reference 2 , with angular velocity of rotation

(Equation Omitted)

3 Its stability properties were investigated by Love in 1898

Recently, Deem and Zabusky 4,5 studied the stationary and dynamical properties of piecewise constant finite-area-vortex-regions (or "character-istic" functions) in the planes. We henceforth call them FAVR's. They found a class of isolated m-fold symmetric uniformly rotating FAVR's and one of a set of uniformly translating FAVR's, which they called "V-states".

Preliminary results were presented tersely at the SIAM meeting in Alexandria Virginia, June 1980 and at the Second International Congress on Numerical Methods for Eygineering, G.A.M.N.I. 2, December 1980, Chatenay-Malabry, Paris, France

Permanent address: Gulf Science and Technology, P.O. Drawer 2038, Pittsburgh, PA, U.S.A. They showed two examples each of states for m = 3 and 4 and one trans-lating V-state, all valid to three significant figures. R. T. Pierrehumbert 6 applied an ad-hoc iterative method 7 to calculate the additional members of the set of uniformly translating V-states. Saffman and Szeto 8 used a Newton-Raphson procedure to determine properties of a pair of

corotating vortex regions of like shape and area and they used the same 9 procedure in to determine properties of a stationary single periodic array of FAVR's,that is a structured "free- shear" layer. Saffman and Schatzman 10 obtained stationary solutions consisting of two staggered parallel rows of identically shaped and oppositely signed FAVR's that is, the finite- area analog of the von Karman "wake".

University of Pittsburgh Page 1 Dec 31, 1980

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STATIONARY SOLUTIONS OF THE EULER-EQUATIONS IN TWO DIMENSIONS, SINGLY-AND DOUBLY-CONNECTED V-

STATES

In this paper we present further examples and detailed properties of isolated rotating V-states (Sec. III)...