Mathematical modeling of 3D current flows for narrow shallow water bodies of complicated forms

. This article is devoted to the modeling of three-dimensional currents for narrow shallow water systems like Kerch straight. Model, which is presented in this article, is based on previously constructed 3D discrete model which has used cell filling function and rectangular uniform grids. The effect of rising free surface function has been detected in narrowest part of straight in numerical modelling. The proposed discrete models remain stable at depth differences tens of times, which is an important factor for coastal systems. Also this approach may be applied for wave evolution prediction in narrow straits of complicated bottom relief and coastal line.


Introduction
The article deals with the mathematical modeling of three-dimensional currents for narrow shallow water bodies of complex shape. The assumed effect of the free surface elevation function in the narrowest part of the computational domain was confirmed by numerical simulation.
To describe wave processes in shallow water bodies, a system of Navier-Stokes equations is used, which includes three equations of motion in areas with dynamically changing geometry of the computational domain. Discrete models remain stable, in contrast to known ones, with significant depth differences, which is an important factor for coastal systems. Software has been developed that makes it possible to predict three-dimensional currents in narrow places of shallow water bodies of complex shape, such as the Azov Sea, including the Kerch Strait, where it is possible to increase the amplitude of the free surface rise function due to the narrowing and difference in depths.
-equations of motion (Navier-Stokes): -continuity equation: is the velocity vector of the water flow of a shallow water body;  is the density of the aquatic environment; p is the hydrodynamic pressure; g is the gravitational acceleration; , are coefficients of turbulent exchange in the horizontal and vertical directions; n is the normal vector to the surface describing the boundary of the computational domain.
the surface of the structure: where  is the intensity of evaporation of a liquid, , w is the wind velocity relative to water, ρa is the atmosphere density, s Cd is the dimensionless surface resistance coefficient, which depends on wind speed [4].
Let us set the tangential stress vector for the bottom taking into account the movement of water as follows:  accordingly. The method of correction to pressure was used to solve the hydrodynamic problem. The variant of this method in the case of a variable density will take the form [6][7][8]: We introduce the coefficients 0 q , 1 q , 2 q , 3 q , 4 q , 5 q , 6 q , describing VOF of regions The    account the VOF, can be written in the following form [11,12]: Similarly, approximations for the remaining coordinate directions will be recorded. The error in approximating the mathematical model is equal to The conservation of the flow at the discrete level of the developed hydrodynamic model is proved, as well as the absence of non-conservative dissipative terms obtained as a result of discretization of the system of equations. A sufficient condition for the stability [13] and monotony of the developed model is determined on the basis of the maximum principle [14], with constraints on the step with respect to the spatial coordinates: Discrete analogs of the system of equations (5) are solved by an adaptive modified alternating-triangular method of variational type [15].
Results of numerical experiments based on hydrodynamic model In places with «bottleneck» geometry, frequent occurrence is the free surface rise in the narrowest part of the computational domain.
To simulate 3D current flows for narrow shallow water bodies, the bottom geometry of the «bottleneck» type was used with depths as close as possible to the real geometry of the bottom of the Kerch Strait (maximum depth 18 m from the Black Sea).
The source of disturbances is set at a given distance from the coastline. The modeling area is 50 by 50 km, the depth is 18 m; the peak rises 2 m above sea level.
When digitizing the surface relief, it is assumed that the average level is zero. The distances between the grid nodes are 10 cm vertically, 500 m horizontally, that is, 500 × 500 m.
A modern software package has been created, adapted for modeling 3D current flows for narrow shallow water bodies, used in a wide range of parameters for calculating the velocity and pressure fields of the water environment, as well as for assessing the impact on the coast.  The developed software package allows experiments with both simplified geometry (bottleneck type) and geometry close to the real strait. Visualization of the results shows that a free surface elevation does indeed appear on the simplified geometry, and the results obtained for the simplified geometry are consistent with the results for the bottom geometry close to the real strait. The developed numerical algorithms for solving model problems and implementing a complex of programs can be used to study 3D current flows for narrow shallow water bodies of complicated forms, such as the Azov and White Seas, the North Caspian, as well as to find the velocity and pressure fields of the aquatic environment. Figure 3, 4 show the results of the forecast of hydrodynamic processes in the flow of liquid in the strait, taking into account the simplified and real geometry of the bottom, the figure shows the vector field of the velocity of the water medium (cut from the XOZ plane) at the same time. Figure 5 shows the pressure fields for the simplified and original bottom geometry.

Conclusion
The article describes mathematical modeling of 3D current flows for narrow shallow water bodies of complicated forms. This paper presents a continuous three-dimensional mathematical model of the hydrodynamics of shallow water bodies, its discretization. When constructing discrete mathematical models of hydrodynamics, the filling of control cells is taken into account, which makes it possible to increase the accuracy of the solution in a complex region of the investigated area by improving the approximation of the boundary. The article discusses the approximation of the problem of calculating the velocity field at an intermediate time layer in terms of spatial variables. The software package allows you to calculate the free surface elevation function, provided that discrete models are stable with dozens of times depth differences, which is an important element for coastal systems.