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1

Modeling of Waves Propagating in Water with a Crushed Ice Layer on the Free Surface

Szmidt K.

Archives of Hydro-Engineering and Environmental Mechanics

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2017

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Vol. 64, nr 2

87--99

EN

A transformation of gravitational waves in fluid of constant depth with a crushed ice layer floating on the free fluid surface is considered. The propagating waves undergo a slight damping along their path of propagation. The main goal of the study is to construct an approximate descriptive model of this phenomenon.With regard to small displacements of the free surface, a viscous type model of damping is considered, which corresponds to a continuous distribution of dash-pots at the free surface of the fluid. A constant parameter of the dampers is assumed in advance as known parameter of damping. This parameter may be obtained by means of experiments in a laboratory flume.

2

Added Mass of Fluid and Fundamental Frequencies of a Horizontal Elastic Circular Plate Vibrating in Fluid of Constant Depth

Szmidt K., Hedzielski B.

Archives of Hydro-Engineering and Environmental Mechanics

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2017

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Vol. 64, nr 3-4

163--186

EN

The paper deals with free vibrations of a horizontal thin elastic circular plate submerged in an infinite layer of fluid of constant depth. The motion of the plate is accompanied by the fluid motion, and thus, the pressure load on this plate results from displacements of the plate in time. The plate and fluid motions depend on boundary conditions, and, in particular, the pressure load depends on the gap between the plate and the fluid bottom. In theoretical description of this phenomenon, we deal with a coupled problem of hydrodynamics in which the plate and fluid motions are coupled through boundary conditions at the plate surfaces. This coupling leads to the so-called co-vibrating (added) mass of fluid, which significantly changes the fundamental frequencies (eigenfrequencies) of the plate. In formulation of the problem, a linear theory of small deflections of the plate is employed. At the same time, one assumes the potential fluid motion with the potential function satisfying Laplace’s equation within the fluid domain and appropriate boundary conditions at fluid boundaries. In order to solve the problem, the infinite fluid domain is divided into sub-domains of simple geometry, and the solution of problem equations is constructed separately for each of these domains. Numerical experiments have been conducted to illustrate the formulation developed in this paper.

3

Vibrations of a Horizontal Elastic Band Plate Submerged in Fluid of Constant Depth

Szmidt K., Hedzielski B.

Archives of Hydro-Engineering and Environmental Mechanics

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2016

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Vol. 63, nr 2-3

191--213

EN

The paper deals with free and forced vibrations of a horizontal thin elastic plate submerged in an infinite layer of fluid of constant depth. In free vibrations, the pressure load on the plate results from assumed displacements of the plate. In forced vibrations, the fluid pressure is mainly induced by water waves arriving at the plate. In both cases, we have a coupled problem of hydrodynamics in which the plate and fluid motions are coupled through boundary conditions at the plate surface. At the same time, the pressure load on the plate depends on the gap between the plate and the fluid bottom. The motion of the plate is accompanied by the fluid motion. This leads to the so-called co-vibrating mass of fluid, which strongly changes the eigenfrequencies of the plate. In formulation of this problem, a linear theory of small deflections of the plate is employed. In order to calculate the fluid pressure, a solution of Laplace’s equation is constructed in the doubly connected infinite fluid domain. To this end, this infinite domain is divided into sub-domains of simple geometry, and the solution of the problem equation is constructed separately for each of these domains. Numerical experiments are conducted to illustrate the formulation developed in this paper.

4

Transformation of long waves in a canal of variable section

Szmidt K., Hedzielski B.

Archives of Hydro-Engineering and Environmental Mechanics

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2016

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Vol. 63, nr 1

3--18

EN

The paper deals with long water waves propagating in a straight canal of constant depth and variable section. In the formulation of this problem, a simplified, one-dimensional model is considered that is based on the assumption of a “columnar” fluid motion. To this end, a system of material coordinates is employed as independent variables in the description of this phe- nomenon. The main attention is focused on transient solutions corresponding to a fluid motion starting from rest. With respect to the initial value problem considered, we confine our attention to a finite domain fluid motion induced by a piston-type generator placed at the beginning of the canal. For a finite elapse of time, measured from the starting point, the solution in the finite fluid area mimics a solution within an infinite domain, inherent for wave propagation problems. The main goal of our investigations is to describe the evolution of the free surface (the wave height) at the smallest section of the canal. Numerical examples are provided to illustrate the model formulation developed in this paper. The accuracy of this approximate description is assessed by comparing its results with data obtained in hydraulic experiments performed in a laboratory flume.

5

Nonlinear Interactions between Gravity Waves in Water of Constant Depth

Szmidt K., Hedzielski B.

Archives of Hydro-Engineering and Environmental Mechanics

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2015

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Vol. 62, nr 1-2

3--25

EN

The paper deals with interactions between water waves propagating in fluid of constant depth. In formulation of this problem, a nonlinear character of these interactions is taken into account. In particular, in order to simplify a solution to nonlinear boundary conditions at the free surface, a system of material coordinates is employed as independent variables in the description of the phenomenon. The main attention is focused on the transient solutions corresponding to fluid motion starting from rest. With respect to the initial value problem considered, we confine our attention to a finite fluid domain. For a finite elapse of time, measured from the starting point, the solution in a finite fluid area mimics a solution within an infinite domain, inherent for wave propagation problems. Because of the complicated structure of equations describing nonlinear waves, an approximate formulation is considered, which is based on a power series expansion of dependent variables with respect to a small parameter. Such a solution is assumed to be accurate in describing the main features of the phenomenon. Numerical experiments are conducted to illustrate the approximate formulation developed in this paper.

6

Obciążenia hydrodynamiczne rurociągów instalowanych w strefie brzegowej morza

Szmidt K., Hedzielski B.

Inżynieria Morska i Geotechnika

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2014

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nr 2

101--112

PL

Podstawowe aspekty wyznaczania obciążeń hydrodynamicznych rurociągów instalowanych w strefie brzegowej morza. Ograniczenie dyskusji do sztywnych, głównie stalowych rurociągów. Analiza wpływu prześwitu pomiędzy dnem i osią rurociągu na wielkość obciążeń oraz wpływu przepływu cieczy wewnątrz rurociągu na podstawowe częstości drgań własnych rurociągu. Określenie obciążenia hydrodynamicznego, wywołanego falowaniem morza na podstawie danych otrzymanych z pomiarów falowania w rejonie Zatoki Gdańskiej za pomocą boi pomiarowej.

EN

The problem of a description of hydrodynamic loads acting on pipes installed in a coastal sea zone. Confining the discussion to stiff, mainly steel pipes, which may be considered as one dimensional beam structures. Analysis of the influence of the gap between the pipe and the sea floor on the resultant pressure forces and the influence of the fluid velocity within the pipe on the basic free vibrations frequencies of the pipe. Calculation of the extreme wave induced forces acting on the pipe by means of data recorded by a floating measurement buoy, anchored in the Gdańsk Bay.

7

On the SPH Approximations in ModelingWaterWaves

Szmidt K.

Archives of Hydro-Engineering and Environmental Mechanics

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2013

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Vol. 60, nr 1-4

63--86

EN

This paper presents an examination of approximation aspects of the Smoothed Particle Hydrodynamics (SPH) in modeling the water wave phenomenon. Close attention is paid on consistency of the SPH formulation and its relation with a correction technique applied to improve the method accuracy. The considerations are confined to flow fields within finite domains with a free surface and fixed solid boundaries with free slip boundary conditions. In spite of a wide application of the SPH method in fluid mechanics, the appropriate modeling of the boundaries is still not clear. For solid straight line boundaries, a natural way is to use additional (virtual, ghost) particles outside the boundary and take into account mirror reflection of associated field variables. Such a method leads to good results, except for a vicinity of solid horizontal bottoms where, because of the SPH approximations in the description of pressure, a stratification of the fluid material particles may occur. In order to illustrate the last phenomenon, some numerical tests have been made. These numerical experiments show that the solid fluid bottom attracts the material particles and thus, to prevent these particles from penetration into the bottom, a mutual exchange of positions of real and ghost particles has been used in a computation procedure.

8

Boussinesq-type Equations for Long Waves in Water of Variable Depth

Szmidt K.

Archives of Hydro-Engineering and Environmental Mechanics

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2011

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Vol. 58, nr 1-4

3-21

EN

The paper deals with the problem of the transformation of long gravitational waves propagating in water of variable depth. The main attention of the paper is focused on the derivation of equations describing this phenomenon. These equations are derived under the assumption that the non-viscous fluid is incompressible and rotation free, and that the fluid velocity components may be expressed in the form of the power series expansions with respect to the water depth. This procedure makes it possible to transform the original two-dimensional problem into a one-dimensional one, in which all unknown variables depend on time and a horizontal coordinate. The partial differential equations derived correspond to the conservation of mass and momentum. The solution of these equations is constructed by the finite difference method and an approximate discrete integration in the time domain. In order to estimate the accuracy of this formulation, theoretical results obtained for a specific problem were compared with experimental measurements carried out in a laboratory flume. The comparison shows that the proposed theoretical formulation is an accurate description of long waves propagating in water of variable depth.

9

Finite Difference Analysis of Surface Wave Scattering by Underwater Rectangular Obstacles

Szmidt K.

Archives of Hydro-Engineering and Environmental Mechanics

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2010

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Vol. 57, nr 3-4

179-198

EN

This paper deals with the problem of the scattering of surface water waves by underwater obstacles. The main goal of the investigations is to estimate the efficiency of such structures in protecting sea shelf zones from open sea waves. A useful measure of the protection is the ratio of the square of the amplitude of the transmitted wave to the square of the amplitude of the arriving wave. The problem is formulated in terms of the finite difference method. It is shown that the discrete approach to the problem leads to eigenvalue problems for two matrices resulting from the discrete description. As compared to analytical formulation, the discrete method may be convenient in application to unsteady problems and obstacles of complicated geometry.

10

On the Description of Long Water Waves in Material Variables

Szmidt K., Hedzielski B.

Archives of Hydro-Engineering and Environmental Mechanics

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2009

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Vol. 56, nr 1-2

63-83

EN

Shallow water equations formulated in material variables are presented in this paper. In the model considered, a three-dimensional physical problem is substituted by a two-dimensional one describing a transformation of long waves in water of variable depth. The latter is obtained by means of the assumption that a vertical column of water particles remains vertical during the entire motion of the fluid. Under the assumption of small, continuous variation of the water depth, the equations for gravity waves are derived through Hamilton's principle formulated in terms of the material coordinates. This formulation ensures the conservation of mechanical energy. The approximation depends on the wave parameters as well as on the bed bathymetry. The latter may influence a solution of the model decisively; thus, one should be careful in applying the description to complicated geometries of fluid domains encountered in engineering practice.

11

On the Transformation of Long Gravitational Waves in a Region of Variable Water Depth: a Comparison of Theory and Experiment

Szmidt K.

Archives of Hydro-Engineering and Environmental Mechanics

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2007

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Vol. 54, nr 2

137-158

EN

The paper describes investigations on transformation of long gravitational waves in water of variable depth with reflection of the waves from a shelf barrier. In the model considered, a long water wave arrives from an area of constant water depth to an area of constant, smaller water depth, where it reflects at a vertical wall. The analysis is confined to a finite fluid domain, relevant to experimental investigations in a laboratory flume. In theoretical analysis of the phenomenon, we follow a non-linear shallow water approximation to the problem considered. The fundamental equations of fluid motion are derived with the help of a standard variational procedure in a material system of coordinates. The equations proved to be a reasonable approximation to a description of the long waves propagating in fluid with small variation of its depth. In the discussed case of reflection of such waves from a vertical barrier, however, the motion of the fluid is more complicated and therefore the long water wave theory does not deliver as good results as in the case of pure propagation of the waves. The primary objective of this paper is thus to compare the theoretical solution proposed with data obtained in experiments, and to answer the question about accuracy and applicability of the theoretical model in the description of the problem investigated.

12

Professor Piotr Wilde (1931-2006)

Szmidt K.

Archives of Hydro-Engineering and Environmental Mechanics

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2006

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Vol. 53, nr 1

3-5

13

A note on discrete descriptions of water flows in material variables

Szmidt K.

Archives of Hydro-Engineering and Environmental Mechanics

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2005

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Vol. 52, nr 2

163-175

EN

The paper describes the problem of discrete formulation of plane fluid flows in material description. The investigation is confined to chosen cases of stationary potential and vortex motion of an incompressible inviscid fluid within circular domains with perfect boundaries. The paths of fluid particles are obtained by numerical integration of momentum equations within a discrete time space. Brownian type random disturbances are attached to the displacement field obtained by the integration. It has been shown, that the discrete formulation may lead to solutions in which a small distance between two material points may grow to a relatively large value after a finite elapse of time. The last feature of the procedure may be a serious drawback of the discrete formulation in the material variables.

14

Initial boundary value problems for vortex motion of an ideal fluid in bounded domains

Szmidt K.

Archives of Hydro-Engineering and Environmental Mechanics

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2001

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Vol. 48, nr 1

19-33

EN

The paper deals with the problem of vortex motion of an incompressible perfect fluid in bounded domains. The research is confined to chosen cases of steady velocity fields within rectangular, circular and elliptic regions with rigid boundaries. The solution to the initial-value problem of the fluid flow for the assumed velocity fields is the primary object of this paper. It is demonstrated that individual particles of the fluid have their own periods of motion and thus, one should be careful in describing such problems by means of discrete methods, especially in the Lagrangian variables. The problem discussed has its origin in numerical analysis of water waves by means of the finite difference or the finite element method.