Long Eddies in Sheared Flows

Math. Models Methods Appl. Sci.

2004

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.

Section: Introductionmentioning

confidence: 90%

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

Teshukov

Russo

Math. Models Methods Appl. Sci.

2004

“…The approach developed in studies [5] of planeparallel flows with a critical layer was used to obtain a class of exact solutions describing two-dimensional flows with return zones.…”

Section: Steady-state Rotationally Symmetric Solutionsmentioning

confidence: 99%

Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow

2008

J Appl Mech Tech Phy

This paper considers nonlinear equations describing the propagation of long waves in two-dimensional shear flow of a heavy ideal incompressible fluid with a free boundary. A nine-dimensional group of transformations admitted by the equations of motion is found by symmetry methods. Twodimensional subgroups are used to find simpler integrodifferential submodels which define classes of exact solutions, some of which are integrated. New steady-state and unsteady rotationally symmetric solutions with a nontrivial velocity distribution along the depth are obtained.Introduction. Approximate models of shallow-water theory are used to model wave processes in fluids and to describe large-scale motions in the atmosphere and ocean. A mathematical foundation for the classical (depth-averaged) shallow-water approximation was given by Ovsyannikov [1]. The long-wave model taking into account velocity shear along the depth, especially in the two-dimensional case has been studied to a lesser extent. The nonlinear equations of rotational shallow water for plane-parallel motions were studied in [2-7], etc., where infinite series of conservation laws were found, classes of exact solutions were constructed, and conditions for the generalized hyperbolicity and well-posedness of the Cauchy problem were formulated. Teshukov [8,9] studied the shallow-water equations for two-dimensional shear flow, established the existence of simple waves, constructed an extension of Prandtl-Meyer waves, and formulated conditions for the generalized hyperbolicity of the steady-state equations.In the present work, a theoretical group analysis of the two-dimensional shallow-water equations for shear flows was performed. The 9-dimensional group of the admitted transformations was found. It was established that the Lie algebra of operators L 9 corresponding to these transformations is isomorphic to the Lie algebra of the admitted operators for the equations of two-dimensional isentropic motion of a polytropic gas with an adiabatic exponent γ = 2, for which the optimal system subalgebras [10] is known. New classes of exact solutions were constructed using an optimal system of subalgebras that allows a classification of submodels. Steady-state rotationally symmetric solutions describing motion with zones of return flow were obtained. Stable unsteady shear solutions describing the spread (collapse) of a parabolic cavity were found.1. Mathematical Model and Admitted Transformations. The solutions of the system of differential equations

“…The characteristic roots c = c 1 < u 0 and c = c 2 > u 1 of the characteristic equation χ (c) = 0 define speeds of the long-surface waves traveling to the left and right of the stream. In cases when the solution has singularities, the integral in the formula (5) can become convergent at c = u 0 (and/or c = u 1 ). When the characteristic root disappears, the wave perturbations (in corresponding direction) propagate along with the stream and do not outrun it.…”

Section: Perturbation Dissipating Class Of Flowsmentioning

confidence: 99%

“…Nonlinear equations of the long-wave theory were a subject of research in a number of papers. Infinite series of conservation laws were found in [3,4], flows with critical layer were considered in [5], families of exact solutions were constructed and interpreted in [6,7]. Papers [8,9] and some others deal with study of infinite hydrodynamic momentum chains, resulting from Benney's equations.…”

Section: Introductionmentioning

confidence: 99%

Is Landau Damping Possible in a Shear Fluid Flow?

Khe

2013

Stud Appl Math

Asymptotic analysis for small long-wave perturbations of a given stationary shear flow of an ideal fluid with free boundary as t → ∞ is performed. It is shown that small disturbances of the flow are attracted to periodic solution in the case where the governing equations are hyperbolic on the main shear flow solution. A class of shear flows for which Landau damping is realizable, is described. Analytical results obtained are validated by numerical calculations.