Analytical and Numerical Solutions of the Shallow Water Equations for 2d Rotational Flows

Abstract

In this paper we consider a system of integrodifferential equations describing, in a long wave approximation, plane-parallel rotational motions of an ideal incompressible liquid with a free surface. Using special identities, we calculate integral Riemann invariants which are conserved along the trajectories and characteristics. A new class of solutions for this system is found. The description of these special solutions simplifies essentially since in this case the integrodifferential system reduces to a hyperbolic system of two differential equations admitting formulation in the Riemann invariants. The solutions describing propagation of simple waves are obtained in an analytical form. Numerical simulations of rotational flows are performed using both the system describing the special class of the solutions and shallow water equations for rotational flows. In order to describe discontinuous rotational flows, the equations of motion are written in a special conservation form and jump conditions are derived. A non-oscillatory, second-order central scheme is used in the calculations. Some illustrative solutions are presented for smooth and discontinuous rotational flows.

The Russo–Smereka kinetic equation: Conservation laws, reductions and numerical solutions

“…Note that Eqs. (9) are similar to conservation laws of the shallow water equations for shear flows [12,13]. As a rule, in the case of infinitedimensional systems to substantiate the correctness of the choice of conservation laws is difficult.…”

Construction of exact solutions of certain equations of hyperbolic type containing a discontinuity propagating along a non-homogeneous background

“…Важной составляющей этого процесса является этап верификации вычислительного алгоритма, при этом особый интерес представляет поведение численных решений в областях, содержащих сильные и слабые разрывы. Много работ посвящены построению аналитического решения для сложных задач, имеющих прикладное значение [5][6][7][8][9][10][11]. Однако в данных работах рассматриваются либо кусочно-постоянные начальные данные [5][6][7][8][9], либо решения, исключающие разрывы [10,11].…”

“…Много работ посвящены построению аналитического решения для сложных задач, имеющих прикладное значение [5][6][7][8][9][10][11]. Однако в данных работах рассматриваются либо кусочно-постоянные начальные данные [5][6][7][8][9], либо решения, исключающие разрывы [10,11].…”

Interaction of shear flows of an ideal incompressible fluid in a channel

“…The first equation is the local conservation law for the relative momentum, the second equation is the local mass conservation law, and the last equation is the closing one. The first two equations of system (7) coincide with the conservation laws proposed in [6,7] to describe the discontinuous solutions of vortex shallow-water equations that model the propagation of long-wave perturbations in a free-boundary fluid layer. The conditions at the discontinuity corresponding to the conservation laws (7) are written as…”

“…To determine the discontinuous solutions of the model of shear fluid flow in a channel, we use the system of conservation laws (7). The first equation is the local conservation law for the relative momentum, the second equation is the local mass conservation law, and the last equation is the closing one.…”