Conformal mapping of the Gulf of Gdańsk onto a canonical domain

• Autorzy: Czesnik P.P. , Prosnak W.J.
• Języki publikacji: EN

Abstrakty

EN

The paper deals with the conformal mapping of a plane, finite, simply connected domain, representing the southern segment of the Gulf of Gdańsk, enclosed from the North by a parallel, tangential to Cape Rozewie. The segment contains the Hel Peninsula. The method of double decomposition, presented in Prosnak & Cześnik (2001), is applied to the transformation of such an original domain onto a canonical one, which consists of two separate unit discs. The first decomposition concerns the domain which is split into two adjacent subdomains by means of a segment of a straight line. The second decomposition involves two holomorphic functions, each one mapping a subdomain onto a separate disc. The decomposition consists in replacing the function by a sequence of simple ones, so that the mapping is performed step-wise. Each sequence starts with the Schwarz-Christoffel function, the last step consisting in an inversion that transforms an infinite circular domain onto a disc. The data for the problem is contained in the Annex, and is represented by two sets of rectangular coordinates defining consecutive discrete points of the contours bounding the subdomains. The solution to the problem consists of:
– two sets of functions, consecutively transforming each of the subdomains;
– the numerical values of the parameters of these functions;
– a set of figures illustrating the consecutive transformations. The accuracy of the first, and of the penultimate transformation are given, because only in the case of these two functions do the unknown coefficients have to be determined by means of a suitable iterative process. The coefficients of all the remaining functions are evaluated from exact formulae. It should be recalled that the depth of the Gulf of Gdańsk varies considerably – from a few to 110 metres – the gradients of the bottom being rather large. Therefore, the domain of the solution for any mathematical problem describing the hydrodynamical phenomena occurring in the Gulf is usually taken to be three-dimensional. Nevertheless, the paper deals with the transformation of the free surface of the Gulf, assumed as plane. It should be emphasized that this assumption does not mean that the whole domain of the solution has to be regarded as plane.

Słowa kluczowe

EN

Schwarz-Christoffel function; solution; conformal mapping; shoreline; Gdansk Gulf; inversion

• Wydawca: -
• Czasopismo: Oceanologia
• Rocznik: 2002
• Tom: 44
• Numer: 2
• Opis fizyczny: p.179-207,fig.,ref.

Twórcy

autor

Czesnik P.P.

• Polish Academy of Sciences, Powstancow Warszawy 55, 81-712 Sopot, Poland

autor

Prosnak W.J.

Bibliografia

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• Driscoll T.A., 1996, A MATLAB toolbox for Schwarz-Christoffel mapping, ACM Trans. Math. Soft., 22, 168–186, (cited after Driscoll T.A., Schwarz-Christoffel Toolbox User’s Guide available over the Web at http:/www.math.udel.edu/∼driscoll/SC/).
• Druet Cz., Jankowski A., 1983, Characteristics of hydrodynamic states and processes in the Baltic Sea induced by a uniform wind field, [in:] Fundamentals of economics in the marine environment, Stud. i Mater. Oceanol., 40, 129–202, (in Polish).
• Prosnak W. J., Cześnik P.P., 2001, Transformation of the Vistula Lagoon onto a canonical domain, Oceanologia, 43 (2), 169–200.
• Prosnak W. J., Klonowska M.E., 1996, Introduction to numerical fluid mechanics. Part B: conformal mapping, Ossolineum, Wrocław, 998 pp., (in Polish).