ЧИСЛЕННОЕ КОНФОРМНОЕ ОТОБРАЖЕНИЕ В ДВУМЕРНОЙ ГИДРОДИНАМИКЕ И СМЕЖНЫЕ ПРОБЛЕМЫ ЭЛЕКТРОДИНАМИКИ И ТЕОРИИ УПРУГОСТИ Б.И. Рабинович и Ю.В. Тюрин Электронная версия доклада В.И. Прохоренко

Доклад является представлением одноименной монографии Б.И. Рабиновича и Ю.В. Тюрина, выходящей в ИКИ на английском языке в этом году. Описывается оригинальный численный алгоритм конформного отображения (RT-алгоритм), основанный на двух процедурах (R-процедуре и T-процедуре). Решение проблемы, достигаемое этим методом, охватывает произвольные односвязные и двусвязные области с кусочно-гладкими контурами. RT-алгоритм используется для построения ортогональных сеток и для решения широкого спектра внешних и внутренних двумерных задач гидродинамики, а также некоторых смежных задач электродинамики и теории упругости. В качестве основных инструментов для решения краевых задач на преобразованной области (единичный круг, круговое кольцо, область, ограниченная прямоугольником) применяются методы рядов Тейлора и Лорана, Ритца, конечных элементов, сопряженных вихрей. Все эти методы реализуются в виде программного обеспечения на PC, а результаты представляются средствами машинной графики. Для проверки получаемых численных результатов используются, когда это возможно, аналитические решения, решения, полученные другими численными методами, а также экспериментальные результаты. ЧИСЛЕННОЕ КОНФОРМНОЕ ОТОБРАЖЕНИЕ В ДВУМЕРНОЙ ГИДРОДИНАМИКЕ И СМЕЖНЫЕ ПРОБЛЕМЫ ЭЛЕКТРОДИНАМИКИ И ТЕОРИИ УПРУГОСТИ 2

NUMERICAL METHODS IN FLUID MECHANICS Boris I. Rabinovich and Yuri V. Tyurin Numerical Conformal Mapping in Two-Dimensional Hydrodynamics SPACE RESEARCH INSTITUTE RUSSIAN ACADEMY OF SCIENCES

Numerical Conformal Mapping in Two-Dimensional Hydrodynamics & Related Problems of Electrodynamics and Elasticity Theory by Boris I. Rabinovich and Yuri V. Tyurin English text edited by Djosef Cherniawsky, PhD, Program implementation of the RT-algorithm, the numerical solutions and their graphical representation has been developed by Yuri Tyurin, PhD, Rudolf Ashkinazy, Alexander Leviant, PhD, Arcadi Livshits, PhD, Eugeni Sokolin PhD 4

Acknowledgements The authors take this opportunity to express deep gratitude to their colleagues mentioned above for creatively purposefully carrying out this complex task. Their role can scarcely be exaggerated. The authors thanks Alexander Rabinovich for the great assistance in composing references on the numerical methods for conformal mapping, as well as Boris Kruglov and Yuri Shilov for their help in translation the manuscript into English and Josef Cherniawsky - for his productive comments on the manuscript. The authors also would like to thank Natalia Komarova for her careful editing the camera-ready version of the book. 5

Chapter 1 Lavrentiev Variational Principle and Auxiliary Transformations Lavrentiev variational principle and representation of mapping functions by Taylor and Laurent series. Construction of the image of variation of an arbitrary contour on a unit circle. Preliminary analytical transforms. General formulation of linear boundary- value problems for simply and doubly connected domains. 6

Chapter 2 RT-Algorithm for Conformal Mapping of an Arbitrary Domain onto the Unit Circle and of a Doubly Connected Domain onto the Annulus General idea and informal description of the RT- algorithm. Mapping of a quasi-circular domain onto a circle (R-procedure). Mapping of an arbitrary simply connected domain onto the unit circle and of a doubly connected domain onto the annulus (T-procedure). Convergence of the RT-algorithm. Methodological examples. 7

Construction on the unit circle of function variations characterising contour deviation from a circle Interior problemExterior problem 8

RT-algorithm. Conformal map example Double connected domain. Inverse mapping Joukowski transformRT- algorithm a 9

Chapter 3 Exterior problems of Hydrodynamics Two-dimensional flow problems: Complex potential and formulas of Blasius-Chaplygin and Kutta-Joukowski. Electrostatic and magnetostatic analogies. Complex potentials. Examples of electric and magnetic fields. Unsteady two- dimensional motion of a contour without circulation. Solution of a boundary-value problem for complex potentials. Circulatory flow past two contours. Flow about a contour near a rectilinear boundary. A wing in motion near the ground. General problem on the flow around two contours. Vortices in the Jovian atmosphere. Slatted wing. Generalization to multiply connected domains. Flow about a lattice made of a single row of contours. Airfoil cascade. Flow around a contour with two sharp edges in the presence of two free vortices. Conjugate-vortex method. 10

Aerodynamic problems 11

Potential Flow with two Vortices around the Arc Hydrodynamic problem 12

Magnetostatic problem - conformal mapping RT- algorithm Linear fractal transform 13

Chapter 4 Interior problems of Hydrodynamics Stokes-Joukowski problem. Interior boundary-value problems of fluid dynamics for moving cavities. Two- dimensional boundary-value problems. Complex velocity potential and associated moments of inertia of liquid. The Saint-Venant problem as an analogue of the two-dimensional Stokes-Joukowski problem. Examples of solving to the Stokes-Joukowski and Saint-Venant problems. The axisymmetric cavity. Prediction of circulatory flows of liquid in closed domains. An example of calculating of the associated moment of inertia in the presence of local vortex regions. 14

Fluid stream in the rotating tank or the torsion of an elliptical bar A 15

Rotation of the circular tank with two inner ribs 16

Without Vortices With Vortices Rotation of the circular tank with two inner ribs Stream lines 17

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Torsion of the beam of cross-shape section 19

Torsion of the shaft with sectarian cut 20

Torsion of the castellated hollow shaft 21

RT- algorithmJoukowski transform 22

RT- algorithm Joukowski transform 23

Joukowski transform RT- algorithm algorithm 24

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Sequential phase of the construction of conformal mapping and the grid conformally equivalent to a polar grid with a cut 27

Torsion of the parallelogram bar RT- algorithm a 28

Torsion of the parallelogram bar 29

Chapter 5 Liquid Sloshing in Cavities Boundary-value problems of fluid dynamics in cavities and equations of disturbed motion of the body-liquid system. Natural oscillations of a liquid within vertical cylindrical cavities. Model problems. Oscillations of liquid in cavities in the form of bodies of revolution. Natural oscillations of liquid in horizontal cylindrical cavities. Shallow water approximation. Numerical experiments. Certain generalisations 30

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Simply connected domain with inner cut Inverse mapping i 32

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Simply connected domain with inner cut Inverse mapping i 34

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Seiches: The Caspian sea - conformal mapping Simply connected domain with inner cut Inverse mapping i 36

Seiches: The Caspian sea - normal modes 37

Conclusions The Notation Table 1 The methods used for solution of the boundary-value problems 38

Table 2 Typical accuracy characteristics of numerical computations using the RT-algorithm of conformal mapping 39

NUMERICAL METHODS IN FLUID MECHANICS SPACE RESEARCH INSTITUTE RUSSIAN ACADEMY OF SCIENCES

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