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1

An approximate method for estimation of fluid motion in a lake containing islands

Klonowska-Prosnak M. E., Prosnak W. J., Cześnik P. P.

Archives of Hydro-Engineering and Environmental Mechanics

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2005

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Vol. 52, nr 4

335-349

EN

The physical phenomenon considered in the paper deals with flow in a lake, which contains one or several islands. The flow is generated by rivers, their inlets and outlets being distributed on the shoreline of the lake. The problem to be solved consists in determination of the velocity field - in the domain bounded by the shorelines of the lake and the islands. In fact, it is attempted to arrive solely at an estimation of the field. Consequently, a rather simple physical model of the phenomenon, as well as its simple mathematical description has been applied. In particular, plane, irrotational and steady flow of ideal liquid has been introduced, the inlets and outlets of the rivers being simulated by sources and sinks. Hence, the problem reduces to determination of a complex function, representing the complex velocity field, which satisfies the impermeability condition on all contours representing the shorelines, the field being generated by the singularities already mentioned. Unfortunately, the so formulated problem is "overconditioned" or "too stiff", what means that the impermeability condition on the outer contour cannot be satisfied. Nevertheless, we arrived at a simple method for circumventing this obstacle, the payoff consisting in some modification of this contour. We had this particular modification in mind, applying the word "approximate" in the title of the paper. The paper contains results - in the form of streamline patterns - for lakes containing from 1 to 3 islands. In the relevant figures the distances between the given and the modified exterior contours can be seen distinctly - allowing the reader to draw conclusions, whether the errors due to the modifications are admissible or not. Of course, it depends anyway on the point of view of the user of the results.

2

A method for determination of flow at the plane, horizontal interface separating streams of two different fluids

Prosnak W. J., Cześnik P. P.

Archives of Hydro-Engineering and Environmental Mechanics

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2005

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Vol. 52, nr 2

89-110

EN

The paper deals with the theoretical investigation of the phenomenon, which consists in generation - by wind - of the movement of water, the bodies of air and of water being separated by a horizontal, plane interface, which is identical with the free surface of water. A set of assumptions defining a physical model of the phenomenon, borrowed from Lock (1951), is introduced, as well as the mathematical description of this model. It reduces to a composite ordinary differential problem, containing two non-linear equations of the third order, which have to satisfy some boundary conditions. A novel method of solution of the differential problem just mentioned is presented in the paper. In the method use is made of exact formulae for coefficients of series representing the solution. The method seems to be competitive with the one given in the paper by Lock (1951).

3

On Plane, Steady, Creeping Flow, Generated Around an Arbitrary Cylinder by a Translating Flat Plate

Klonowska-Prosnak M.E., Prosnak W.J., Cześnik P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2003

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Vol. 51, nr 2

113-144

EN

Plane, steady, creeping flow around an arbitrary cylinder situated in vicinity of a flat plate is considered, the flow being generated by translation of this plate along itself with constant speed, perpendicular to generatrices. The stream function of the flow satisfies the biharmonic equation, so that all properties of the flow - such as velocity and pressure fields as well as fields of other stress tensor components - are expressed in terms of the Goursat functions. Hence, the problem of determination of the flow reduces to determination of these functions. The two-step approach is applied to the solution of this problem. The first step consists in conformal mapping of the original domain of solution onto an annulus - by means of a suitable set of mapping functions. The second step consists in development of the two Goursat functions in Laurent series extended by two logarithmic terms each. Unknown coefficients of the series have to satisfy a system of linear algebraic equations, following from boundary conditions. The system is arrived at by means of the pseudospectral method. The so obtained velocity field is applied to generation of streamline patterns. Such a pattern is presented in the paper, and compared with an analogous one, corresponding to potential flow.

4

Conformal mapping of the Gulf of Gdańsk onto a canonical domain

Cześnik P. P., Prosnak W. J.

Oceanologia

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2002

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No. 44 (2)

179-207

EN

The paper deals with the conformal mapping of a plane, finite, simply connected domain, representing the southern segment of the Gulf of Gdansk, 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 & Czesnik (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 Gdansk 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.

5

Profil w pobliżu ziemi

Prosnak W. J.

Prace Instytutu Lotnictwa

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2001

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Nr 4 (167)

62-68

PL

Przedmiotem pracy jest zagadnienie opływu profilu płaskim niestacjonarnym strumieniem cieczy doskonałej, jednorodnym w nieskończoności, w sytuacji gdy obszar w którym występuje zjawisko jest półpłaszczyzną, a profil znajduje się w pobliżu ograniczającej ją prostej. Podstawowy pomysł wyznaczenia takiego opływu polega na uzupełnieniu półpłaszczyzny do "pełnej" płaszczyzny - poprzez wprowadzenie symetrycznej odbitki wspomnianej półpłaszczyzny względem ograniczącej ją prostej. Innymi słowy, sprowadza się perwotne zagadnienie, dotyczące opływu profilu pojedynczego - do zagadnienia pomocniczego, dotyczącego opływu symetrycznej pary profilów. Rozwiązanie drugiego z tych zagadnień polega na wyznaczeniu dwu funkcji holomorficznych, któremają wspólną "standardową" postać, a nieznane współczynniki. Pierwsza z funkcji konforemnie odwzorowuje domknięte zewnętrze pary profilów na takież zewnętrze pary okręgów. Wyznaczenie jej współczynników, a także środków i promieni okręgów, jest niełatwym zagadnieniem nieliniowym. Druga z tych funkcji jest potencjałem zespolonym przepływu pomocniczego, a jej współczynniki wyznacza się z liniowego układu równań. W obu przypadkach cytuje się programy komputerowe umożliwiające wyznaczanie współczynników. Przedstawiona teoria obejmuje zarówno profil jednoelementowy, jak i wieloelementowy, o dowolnej liczbie urządzeń hipernośnych. Opisane ujęcie teoretyczne problemu wymienionego w tytule może być traktowane jako daleko idące uogólnienie wyników Bondera [1], które dotyczy zadania odwrotnego: kształt jednoelementowego profilu nie jest w tej teorii z góry znany. Na zakończenie wspomina się o możliwości uwzględnienia lepkości płynu i przedstawia rezultat, dotyczący pełzającego opływu zadanego profilu.

EN

The problem of determination of plane, steady flow of inviscid, and incompressible fluid about an airfoil is considered in the paper - in case, when the domainof solution is represented by the half - plane; the airfoil is situated in vicinity of the straight line bounding the domain, and the flowtends with the distance from the airfoil to uniform stream, parallel to the just mentioned line. The fundamental idea for solving such a problem consists in extension of the domain to the "full" plane - by introduction of its image, symmetric with respect to the said straight line. In other words, the original problem dealing with flow about just one airfoil is reduced to an auxiliary one, dealing with flow about a symmetric couple of airfoils. Solution to the auxiliary problem is sought in form of two holomorphic functions, which have the same "standard" form, but different - unknown - coefficients. The first one of these functions transforms conformally the closed exterior of the couple of airfoils onto such an exterior of a couple of circles. determination of its coefficients together with radii and complex centres of the circles - represents a nonlinear, and rather not too easy a task. The second function represents complex potential of the auxialiary flow, and its coefficient follow from a linear set of algebraic equations. Computer programs enabling the potential user solution of both these tasks are referred to. In fact, they can be applied not only to single element airfoils but also to the multielement ones. Such a theoretical approach to the considered problem can be treated as a far reaching generalisation of the Bonders paper [1] which, however, deals solely with a corresponding inverse problem, the shape of the single element airfoil not being given in advance. In conclusion, also the probability of taking into account tthe viscosity of the fluid is mentioned, and a result concerning the creeping flow around a given airfoil is presented.

6

Transformation of the Vistula Lagoon onto a canonical domain

Prosnak W. J., Cześnik P. P.

Oceanologia

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2001

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No. 43 (2)

169-200

EN

The paper deals with the conformal mapping of finite, plane, simply connected domains, representing oceans, lakes, estuaries, bays, lagoons, and other natural water bodies of this kind. As a rule, they are bounded by geometrically complex shorelines. The partial differential problems investigated in Oceanology and posed in such domains have turned out to be difficult to solve for at least three reasons. They follow on from the mathematical properties of the differential equations governing such problems, from the just-mentioned geometrical complexity of the domains of solution, and from the sensitivity of the solutions to boundary conditions. In view of the last reason the contours admitted as boundaries of the domains of the solution ought to be as close to the real shorelines as possible. The obviously inaccurate approximation of the shorelines by "staircases", which appears rather often (cf. Catewicz & Jankowski 1983, Lin & Chandler-Wilde 1996) as a consequence of applying finite difference methods to the solution of the partial differential problems, raises serious doubts from the point of view of Numerical Fluid Mechanics. It is recalled in the paper that such inaccuracies are not unavoidable: that complicated plane domains can be transformed accurately by means of properly applied conformal mapping onto regular, canonical domains - in particular, onto discs or squares. Such a transformation is demonstrated on the rather difficult example of the Vistula Lagoon. The transformation begins with the decomposition of the domain into five plane subdomains, each one of which is eventually transformed onto a disc. Every such result is arrived at quite independently of the remaining subdomains, by means of a set of properly selected consecutive mappings. Hence, the final canonical domain consists in this case of a system of five discs which, however, within the framework of this differential problem, have to be treated as interconnected. The interconnections involve images of four segments of straight lines, separating the original subdomains. The transformations and the resulting canonical domain presented in the paper are intended to be applied to the solution of certain hydrodynamical problems concerning the Vistula Lagoon, which will be published elsewhere.

7

Algorithms for the exact solution of two matrix equations occurring in the driven cavity problem

Klonowska M. E., Prosnak W. J.

Oceanologia

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1999

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No. 41 (1)

3-23

EN

It is shown that two particular systems of linear equations, derived in an earlier paper by Prosnak and Kosma (1991), can be solved in an exact time- and storage-saving manner. First of all, by the proper elimination of unknowns, each system can be reduced to a smaller one containing only half of the unknowns. In the first case, the matrix of coefficients of the so reduced system turns out to be tridiagonal, its elements consisting of square submatrices. Moreover, the reduced system can be split into two independent ones. In the second case, the matrix of the reduced system can be presented as the product of two triangular ones, each one being partitioned in square submatrices. Corresponding algorithms and computer programs have been developed in order to investigate whether some economy in storage and computing time is really attainable. Affirmative conclusions are drawn from the results of computations. This means that the new method of solving problems governed by the Navier-Stokes equations, presented in the cited paper, can be applied in a more effective manner.