Stationary and Non-Stationary Periodic Flows Mathematical Modelling using Various Vortex Viscosity Models
Authors: Protsenko E. A. , Protsenko S. V. 
Published: January 2024

Introduction. Mathematical modelling of currents is an urgent research topic in the ﬁeld of hydrodynamics and oceanography. Despite ongoing research in the ﬁeld of developing accurate and eﬃcient numerical methods for solving Navier-Stokes equations that take into account vortex viscosity, the problems of accurate prediction and control of turbulence remain unresolved. The inﬂuence of nonlinear eﬀects in vortex viscosity models on the accuracy of forecasts and their applicability to various ﬂow conditions also remains relevant. The aim of the study is to study the inﬂuence of linearized and quadratic bottom friction and two turbulence models on the numerical solution of stationary and non-stationary periodic ﬂows. Special emphasis is placed on comparing numerical results with analytical solutions within the framework of using various models of bottom friction.

Materials and Methods. The computational models used in this study are based on a simpliﬁed two-dimensional wave model and full three-dimensional Navier-Stokes equations. The classical model of shallow water motion and the 2D model without taking into account dynamic changes in the geometry of the reservoir surface are derived from a system of equations for a spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics of a shallow reservoir. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear eﬀects – nonlinearities caused by higher-order terms in the equations of motion, i. e. terms of advective acceleration and friction, and nonlinear eﬀects caused by geometric nonlinearities, this is due, for example, to diﬀerent water depths and reservoir widths, which will be important when modelling a real sea.

Results. The results of modeling stationary and non-stationary periodic ﬂows in a schematized rectangular basin using linearized bottom friction are presented. The inﬂuence of linearization on the numerical solution is investigated in comparison with analytical proﬁles using models calculating bottom friction in a quadratic formulation. In combination with quadratic bottom friction, two turbulence models are studied: the constant vortex viscosity and the Prandtl mixing length model. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modeling and analytical solutions averaged in depth.

Discussion and Conclusion. New approaches to modelling and studying ﬂows with variable vortex viscosity are proposed, including analysis of the inﬂuence of linearization and the use of various turbulence models. For the linearized and quadratic formulations of bottom friction, it is proved that the numerical results for the case of stationary ﬂow show great similarity with analytical solutions, since the surface height is much less than the water depth and advection can be neglected. The numerical results for the unsteady ﬂow also show a good agreement with the theory. Unlike analytical solutions, numerical modelling has minor deviations in the long run. The study of ﬂows, within the framework of using various turbulence models, will make it possible to take into account the inﬂuence of nonlinear eﬀects in vortex viscosity models on the accuracy of forecasts and their applicability to various ﬂow conditions. The results obtained make it possible to better understand and describe the physical processes occurring in shallow waters. This opens up new possibilities for applying mathematical modelling to predict and analyze the impact of human activities on the marine environment and to solve other problems in the ﬁeld of oceanology and geophysics.