Vol. 55 No. 1 (2023): ITU ARI, 55(1), 2023
ITU ARI-B Engineering Sciences

Lumped Vortex Element Flying Over Free Water Surface

Sakir Bal
Istanbul Technical University, Dept. of Naval Architecture and Marine Engineering
Cover image of ITU ARI

Published 04/24/2023


  • Lumped vortex element,
  • wave deformation,
  • lift coefficient,
  • Froude number

How to Cite

Bal, Sakir. 2023. “Lumped Vortex Element Flying Over Free Water Surface”. ITU ARI Bulletin of Istanbul Technical University 55 (1):1-6. https://ari.itu.edu.tr/index.php/ituari/article/view/69.


In this study, the lift coefficients (circulation) of two-dimensional flat-plate flying with a constant speed over a free surface have been calculated by a closed-form (analytical) solution. The effects of very high speed have also been included in the calculations. The flat-plate has been modeled by a lumped vortex element under the conditions of potential flow theory. While the kinematic boundary condition (zero normal velocity condition) is satisfied at three-quarter chord length of flat-plate, linearized and combined (kinematic and dynamic) condition has been applied on the free water surface. The total velocity potential has then been calculated by the method of images. Kutta condition is satisfied automatically at the trailing edge by this lumped vortex element. The wave elevations on the free surface have also been calculated in a closed-form solution. First, the lift coefficient by the present analytical solution have been validated with those of another numerical method for NACA0004 foil section. Later, the effects of Froude number, clearance (vertical distance) of flat-plate from calm free water surface, and the angle of attack on the results (namely lift coefficients and free surface deformations) have been discussed in a detailed manner. It has been found that the lift coefficient varies significantly with Froude number particularly for lower clearance values. An increase in Froude number causes also an increase both in wave-length and in wave-height on the free surface. On the other hand, a decrease in clearance (means a closer distance to free surface) causes an increase in wave-height but not in wave-length. A similar finding is noted for angle of attack. An increase in angle of attack causes an increase in loading as well as in wave-height but not in wave-length.


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