Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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Start date: 2020Institution Author: search.filters.institutionAuthor.Yılmaz, Oğuz

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Now showing 1 - 5 of 5
  • Article
    Citation - WoS: 1
    Citation - Scopus: 1
    Initial Stages of Gravity-Driven Flow of Two Fluids of Equal Depth
    (American Institute of Physics, 2023) Korobkin, Alexander; Yılmaz, Oğuz; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    Short-time behavior of gravity-driven free surface flow of two fluids of equal depth and different densities is studied. Initially, the fluids are at rest and separated with a vertical rigid plate of negligible thickness. Then, the plate disappears suddenly and a gravity-driven flow of the fluids starts. The flow in an early stage is described by the potential theory. The initial flow in the leading order is described by a linear problem, which is solved by the Fourier series method. The motions of the interface between the fluids and their free surfaces are investigated. The singular behaviors of the velocity field at the bottom point, where the interface meets the rigid bottom, and the top point, where the interface meets both free surfaces, are analyzed in detail. The flow velocity is shown to be log-singular at the bottom point. The leading-order inner asymptotic solution is constructed in a small vicinity of this point. It is shown that the flow close to the bottom point is self-similar. The motion of the interface is independent of any parameters, including the density ratio, of the problem in specially stretched variables. In the limiting case of negligible density of one of the fluids, the results of the classical dam break problem are recovered. The Lagrangian representation is employed to capture the behavior of the interface and the free surfaces at the top, where the fluid interface meets the free surfaces. The shapes of the free surfaces and the interface in the leading order computed by using the Lagrangian variables show a jump discontinuity of the free surface near the top point where the free surfaces and the interface meet. Inner region formulation is derived near the top point.
  • Article
    Citation - WoS: 1
    Citation - Scopus: 1
    Initial Stages of a Three Dimensional Dam Break Flow
    (Elsevier, 2022) Fetahu, Elona; Ivanyshyn Yaman, Olha; Yılmaz, Oğuz; Ivanyshyn Yaman, Olha; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    Short time behavior of a three dimensional, gravity-driven free surface flow is studied analytically and numerically. Initially the fluid is at rest, held by a vertical wall. A rectangular section of the wall suddenly disappears and the gravity driven three-dimensional flow starts. In order to describe the flow in the early stage, the potential theory is employed. Viscous effects are ignored for small times. The leading order problem is solved by using the Fourier series method and an integral equation method. Local analysis of the flow field close to the side edges of the rectangular section reveals a square root singularity. The flow velocity is also log-singular at the bottom edge of the rectangular section. In the limiting case, as the width of the rectangular section approaches infinity, the results of the classical two-dimensional dam break flow are recovered. Three dimensional effects become important closer to the side edges of the rectangular section.
  • Article
    Citation - WoS: 2
    Citation - Scopus: 2
    An Iterative Method for Interaction of Hydro-Elastic Waves With Several Vertical Cylinders of Circular Cross-Sections
    (MDPI, 2022) Dişibüyük, Nazile Buğurcan; Yılmaz, Oğuz; Korobkin, A. A.; Khabakhpasheva, Tatyana; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    The problem of ice loads acting on multiple vertical cylinders of circular cross-sections frozen in an ice cover of infinite extent is studied. The loads are caused by a flexural-gravity wave propagating in the ice cover towards the rigid bottom-mounted cylinders. This is a three-dimensional linearized problem of hydroelasticity with finite water depth. The flow under the ice is potential and incompressible. The problem is solved by the vertical mode method combined with an iterative method. The velocity potential is written with respect to each cylinder and is expanded into the Fourier series. The algorithm of the problem solving is reduced to calculations of the Fourier coefficients of the velocity potential. Numerical results for the forces acting on four circular cylinders are presented for different ice thicknesses, incident wave angles and cylinder spacing. The obtained wave forces are compared with the results by others. Good agreement is reported.
  • Article
    Citation - WoS: 1
    Citation - Scopus: 1
    A Three Dimensional Dam Break Flow: Small Time Behavior
    (Elsevier, 2021) Fetahu, Elona; Yılmaz, Oğuz; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    Small time behavior of gravity driven free surface flows resulting from the collapse of a cavity is studied. Initially there is a rigid vertical cylinder of circular cross section starting from the free surface of a liquid and ending at the rigid bottom. The cylinder disappears suddenly and gravity driven flow of the fluid starts. The flow in early stage is described by the potential theory. Attention is paid to the singular behavior of the velocity field at the intersection line between the bottom and the free surface of the cavity. The leading order linear problem is solved by the Fourier series method. The flow velocity is log-singular at the intersection line. In the limiting case where the radius and the center of the cavity approach infinity, the problem is reduced to the classical two dimensional dam break problem where the fluid is initially on one side of a vertical wall (dry bed case). The flow resulting from cavity collapse is a three dimensional dam break flow. It is concluded that the three dimensional effects are important when the radius of the cavity is small compared with its depth and that the local flow near the intersection line of the cavity is governed only by the hydrostatic pressure.
  • Article
    Citation - WoS: 7
    Citation - Scopus: 8
    Diffraction of Flexural-Gravity Waves by a Vertical Cylinder of Non-Circular Cross Section
    (Elsevier Ltd., 2020) Dişibüyük, Nazile Buğurcan; Yılmaz, Oğuz; Yılmaz, Oğuz; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
    The linear three-dimensional problem of flexural-gravity wave (hydro-elastic wave) diffraction by a vertical cylinder of an arbitrary smooth cross section is studied using an asymptotic approach combined with the vertical mode method for water of finite depth. The surface of the water is covered by an infinite, continuous elastic ice plate. The rigid cylinder extends from the sea bottom to the ice surface. The ice plate is frozen to the cylinder. The ice deflection is described by the equation of a thin elastic plate of constant thickness with clamped edge conditions at the cylinder. The flow under the ice is described by the linear theory of potential flows. The coupled problem of wave diffraction is solved in two steps. First, the problem is solved without evanescent waves similar to the problem of water waves diffracted by a vertical cylinder. This solution does not satisfy the edge conditions. Second, a radiation problem with a prescribed motion of the ice plate edge is solved by the vertical mode method. The sum of these two solutions solve the original problem. Both solutions are obtained by an asymptotic method with a small parameter quantifying a small deviation of the cylinder cross section from a circular one. Third-order asymptotic solutions are obtained by solving a set of two-dimensional boundary problems for Helmholtz equations in the exterior of a circle. Strains along the edge, where the ice plate is frozen to the cylinder, are investigated for nearly square and elliptic cross sections of the vertical cylinders depending on the characteristics of ice and incident wave. The strains are shown to be highest in the places of high curvatures of the cross sections. The derived asymptotic formulae can be used in design of vertical columns in ice. They directly relate the strains in ice plate to the shape of the column. © 2020 Elsevier Ltd