UDC 532.592; 517.958
V. M. TeshukovThe classical shallow-water equations describing the propagation of long waves in flow without a shear of the horizontal velocity along the vertical coincide with the equations describing the isentropic motion of a polytropic gas for a polytropic exponent γ = 2 (in the theory of fluid wave motion, this fact is called the gas-dynamic analogy). A new mathematical model of long-wave theory is derived that describes shear free-boundary fluid flows. It is shown that in the case of one-dimensional motion, the equations of the new model coincide with the equations describing nonisentropic gas motion with a special choice of the equation of state, and in the multidimensional case, the new system of longwave equations differs significantly from the gas motion model. In the general case, it is established that the system of equations derived is a hyperbolic system. The velocities of propagation of wave perturbations are found.
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.
In this paper we derive an approximate multi-dimensional model of dispersive waves propagating in a two-layer fluid with free surface. This model is a "two-layer" generalization of the Green-Naghdi model. Our derivation is based on Hamilton's principle. From the Lagrangian for the full-water problem we obtain an approximate Lagrangian with accuracy O(ε 2 ), where ε is the small parameter representing the ratio of a typical vertical scale to a typical horizontal scale. This approach allows us to derive governing equations in a compact and symmetric form. Important properties of the model are revealed. In particular, we introduce the notion of generalized vorticity and derive analogues of integrals of motion, such as Bernoulli integrals, which are well known in ideal Fluid Mechanics. Conservation laws for the total momentum and total energy are also obtained.
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