HYDRODYNAMICS AND ACOUSTICS

2019 ◊ Volume 1 (91) ◊ Issue 4 ◊ p. 469-483

O. G. Stetsenko^*

^* Institute of Hydromechanics of NAS of Ukraine, Kyiv, Ukraine

A velocity potential of unsteady motion of a system of two sources in a rectangular channel

Gidrodin. akust. 2019, 1(4):469-483

TEXT LANGUAGE: Ukrainian

ABSTRACT

The paper deals with the problem of determining the velocity potential generated by a non-stationary motion of a system of two sources with similar time-variable intensities located under the free surface of a rectangular cross-section channel symmetrically to its vertical median plane. The overall solution is the sum of two potentials. The first one is the sum of potentials of the sources represented as a product of a source intensity and Green function for Laplace equation in the infinite medium, the said potentials including the sources being a reflection in respect of the moving system relative to a channel bottom and the infinite source system being a reflection of the moving system relative the channel side walls in addition to the mowing system itself. This approach allows the fulfilling of boundary conditions for this potential on the bottom and the side walls. By integral representation for a single source and Poisson formulae for the above-mentioned infinite source system, this component is a Fourier cosine series in respect of a transverse coordinate. The involved coefficients comprise an integral representation by longitudinal coordinate for a function being determined by the known characteristics of the mowing sources and the sources being the reflection of the above moving sources relative to the bottom. The second potential is calculated to satisfy the boundary conditions on the free surface. It is responsible for the influence of the free surface on the solution behavior. A field of surface waves generated by the moving source system is described. Like the first potential, it has the form of a Fourier cosine series with coefficients that include the unknown image function. To determine the latter, an initial-boundary problem was formulated, with the fulfillment of boundary conditions on the bottom and free surface. The solution for the second potential also has two components. The first component obtained in analytical form represents the system of two periodic motions that occurred in the hydrodynamic field corresponding to the motion of the mentioned source system. The Froude number and wave spectrum corresponding to each mode determine the period of these time oscillations. The second component is a double integral, by the time and longitudinal wave number. It depends on the behavior of a non-stationary velocity variation, the intensity of the moving sources, and the Froude number. This term represents the contribution of the inertial forces generated by the added mass.

KEY WORDS

ship hydrodynamics, velocity potential, rectangular channel, Froude number, unsteadiness

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