COMPUTATION OF THE TWO-DIMENSIONAL FLOW DEVELOPING IN THE INTERNAL LAKE OF KIS-BALATON

Abstract

The hydraulical phenomena in nature are three-dimensional (3D) and varying in time. In most practical cases it is sufficient to consider those phenomena as two-dimensional (2D) in the horizontal plane and varying in time. These phenomena occur primarily 'where the accelera- tion along the vertical direction is negligible compared to the gravitational one. Disregarding the variation of hydraulical parameters along the vertical direction we obtain a homogeneous horizontal flow. This phenomenon is described by the relatively simple Reynolds equations. In most practical cases the variation of velocity along the vertical direction has to be taken into account. The integral equations in which the depth average is taken into account are approaching the three-dimensional phenomena rather well. In our study we performed the result of our several years' research in the field of mathe- matical modelling of two-dimensional hydrodynamical and transport processes. The mathematical background was introduced in paragraph 1. We showed how the two- dimensional equations (7)-(9) can be achieved from Reynolds equations, valid for the time averaged mean hydraulic characteristics at a certain point of a three-dimensional turbulent flow. After deriving a closed set of equations, which can be solved, we introduced the numerical solution method of the implicit finite differences in four steps and in alternative direction. We showed the calibration of the model to prove the accuracy of the results in paragraph 2. That is why we compared the results of our model with laboratory measurements. We simulated the phenomenon forming at the tailbay of a hydropower station and compared the flow patterns of the mathematical and physical models (Figs 2 and 3). We proved, that our model and its computer program are suitable for the computation of flows that may be as- sumed as two-dimensional in the horizontal plane. In paragraph 3 the application of the model for the Internal Lake of Kis-Balaton is shown. We computed the near steady flow pattern, which forms when both the water intake and outlet was 10 m 8/s (Fig. 4). Sensitivity tests were performed for both the velocity coefficient, C (Fig. 5) and the eddy viscosity, v. We established that neither the determination of C nor that of v needs field measure- ments. Satisfactory accuracy can be achieved if one uses the values and formulae based on former experiences, laboratory experiments or data found in some publications for the estima- tion of C and v. In paragraph 4 we briefly performed the application of the transport-diffusion equation for the Internal Lake of Kis-Balaton. With the help of an example it is shown, that we can compute the concentration of pollutants in a two dimensional space, varying in time (Figs 6 and 7).