Browsing by Author "Fetahu, Elona"

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Asymptotic Behaviour of Gravity Driven Free Surface Flows Resulting From Cavity Collapse
(Izmir Institute of Technology, 2020-07) Fetahu, Elona; Yılmaz, Oğuz; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
In this thesis, the gravity driven potential flows that result from cavity collapse are studied. Initially, the collapse of a vertical cylindrical cavity of circular cross sections surrounded by a liquid region is examined for two different situations. In the first one the cavity has same depth as the fluid and in the second one the cavity starts from the free surface and has less depth than the fluid. The problem is formulated by using a small parameter that represents the short duration of the stage. The first problem, as the radius and the centre of the cavity approach infinity, reduces to the classical two-dimensional dam break problem solved by Korobkin and Yilmaz (2009). The singularity of the radial velocity at the bottom circle is shown to be of logarithmic type. In the second problem, where the cavity is less deep than the fluid, the flow region is separated into two regions: the interior one, which is underneath the cylindrical cavity and above the rigid bottom, and the exterior one, which is the rest of the flow. The corresponding new problems are solved separately and then the coefficients are found by applying the matching conditions at the interface, where the fluid radial velocities and pressures coincide. On the limiting case, the problem reduces to the two-dimensional dam break flow of two immiscible fluids by Yilmaz et al. (2013a). Singularity at the bottom circle of the cavity is observed, which is of the same type as in the latter paper. Next, a third problem studies the gravity driven flow caused by the collapse of a rectangular section of a vertical plate. During the early stage, the flow is described by the velocity potential. Attention is paid to determining the velocity potential and free surface shapes. The solution follows the Fourier series method in Renzi and Dias (2013) and the boundary element method in Yilmaz et al. (2013a). Singularity is observed at the side edges and lower edge of the rectangular section. The horizontal velocity of the initially vertical free surface along the vertical line of symmetry of the rectangle is the same to the one in the two-dimensional problem Korobkin and Yilmaz (2009). The singularities observed in these problems lead to the jet formation for the initial stage. The methods applied in these computations are expected to be helpful in the analysis of gravity-driven flow free surface shapes. This thesis is a contribution towards the 3-D generalizations of dam break problems.
Initial Stages of a Three Dimensional Dam Break Flow
(Elsevier, 2022-10) Fetahu, Elona; Ivanyshyn Yaman, Olha; Yılmaz, Oğuz; 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.
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.