Mathematical model for thermodynamics of an elongated reservoir

A two-dimensional (profile) mathematical model is developed for calculating hydroand thermodynamic processes in elongated reservoirs. A projection grid method is used for discretizing the model equations. The method allows one to retain some integral properties of the initial differential equations in the discrete model. Numerical modelling of flows and heat transfer in the Nurek water storage is presented. To forecast serious environmental impacts of human activities is an important problem of rational use of natural resources and environment control. Hydro technical projects, territorial redistribution of water resources, and the like are closely connected with the development of justified forecasts of hydrothermal operating conditions of water-supply systems. Methods of numerical modelling allow one to solve a number of both diagnostic and prediction problems, which are of practical interest as far as environment control and functioning of objects being designed are concerned. In this paper we describe a mathematical model for calculating nonstationary turbulent stratified flows and heat transfer in elongated reservoirs. The model is based on equations of hydroand thermodynamics which are two-dimensional with respect to the vertical plane. The model allows one to set various boundary conditions in both inlet and outlet reservoir sections, which can consist of liquid and impenetrable portions. The choice of boundary conditions depends on the particular situation and the available data. The projection grid method is used to discretize equations of the model. The method allows one to retain some integral properties of the initial differential equations in the discrete model. Numerical calculations of flows and heat transfer in the Nurek water storage are presented. 1. FORMULATION OF THE PROBLEM Nonstationary equations which are two-dimensional with respect to the vertical plane are often used in modelling hydrophysical processes in deep and rather narrow reservoirs. The equations allow one to take into account both the horizontal and vertical structures of flows and also the density stratification of water [2,4,7,17,18]. Let us introduce the rectangular system of coordinates xyz. The jc-axis is directed along the reservoir axis and the z-axis is directed vertically down. In the plane xQz let us consider the domain ß0 = {h(x) 0 and h(x) is a sufficiently smooth function describing the bottom profile and satisfying the condition ' Scientific Research Centre of Mathematical Modelling, the Kirghizstan Academy of Sciences, Bishkek 720071, Kirghizstan 516 T. V. Dunets and S. N. Sklyar