Structural elements and collapse regimes in 3D flows on a slope
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TICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Structural Elements and Collapse Regimes in 3D Flows on a Slope V. P. Goncharov Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Pyzhevskii per. 3, Moscow, 109017 Russia email: [email protected] Received March 19, 2011
Abstract—The mechanisms and structural elements of an instability whose development results in the col lapse of flow fragments have been studied in the scope of the Hamilton version of the “shallow water” 3D model on a slope. The study indicated that the 3D model differs from its 2D analog in a more varied set of collapsing solutions. In particular, the solutions describing anisotropic collapse, during which the area of a collapsing fragment in contact with the slope contracts into a segment rather than a point, exist together with the solutions describing radially symmetric (isotropic) collapse. DOI: 10.1134/S1063776111090044
1. INTRODUCTION For gravity flows (as well as for any other nonlinear system), possible final regimes resulting from the development of hydrodynamic instability are of fun damental importance. Instabilities develop differently depending on the model and initial data. The Ray leigh–Taylor (RTI) and Richtmaier–Meshkov (RMI) instabilities are important for understanding instabili ties originating at an interface and play a key role in many natural processes and applications. Cavitation, inertial thermonuclear synthesis, high energy density physics, astrophysics, and geophysics are among such processes and applications; the complete list is much larger [1]. The formation of singularities during a finite time or collapse is a rather universal mechanism by which instabilities manifest themselves in nonlinear disper sion systems [2–4]. This is observed in many hydrody namic models including those related to studying the evolution of interfaces [5–8]. The main aim of this work is to generalize the solu tions describing the collapse regimes and structural elements in the flat model of gravity flow on a slope for the 3D case. According to the theory [9], in the flat model, instability and collapse can develop in two ways depending on initial conditions. Singularities at an interface are formed as (t0 – t)–1/3 and (t0 – t)–2/7 if the RMI and RTI scenarios are implemented, respectively (t and t0 are the current and collapse times, respec tively). We can anticipate that other rapider collapse scenarios exist in the 3D model, where one more addi tional dimension is included. 2. 3D MODEL OF GRAVITY FLOW ON A SLOPE We now consider a layer or a finite volume of an incompressible homogeneous fluid with a free surface, assuming that this fluid moves without vortices in the
gravitational field with acceleration g and is limited from below by a uniform slope inclined at angle ϑ with respect to the horizon. We will consider this model in the Cartesian coordinate system moving downward along a slope with acceleration gsinϑ in such a way that coordinates r = (x1, x2) are in the slope plane and the x3 axis is perpendicular
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