Many-dimensional periodicity and turbulence of Rayleigh–Taylor and Richtmyer–Meshkov instabilities

Abstract

The work begins with a presentation of results concerning periodic Rayleigh-Taylor (RT)and Richtmyer–Meshkov (RM) dynamics. The periodic RT flows are considered in 2D and 3D cases. The periodic RM flows are considered in a 2D case. Nonstationary and steadystate statements of RT and RM problems are given. Nonstationary statements of RT and RM problems are given in the2D case. Steady-state statements of RT and RM problems are considered in the 2D and 3D cases (RT) and in the 2D case (RM). Questions about turbulence are given in the second part of the work. In the nonstationary statement, the beginning of a transition from an equilibrium close to hydrostatics and behavior of a dynamic system when it is close to the steady-state solution are considered. To study the smooth evolution of the dynamic system in the RT case (at last during an initial of order of unity t ≃ 1 stage) we have to cut of f an inverse turbulent cascade incoming from the small scales (inverse means that the typical k decreases in time during the cascade). The cascade starts from some small scale k ≫ 1 (here g = 1) and appears in the scales k ≃ 1 after time t ≃ 1 after start. We have to consider exponentially smoothed initial data to cut it off. This means that amplitudes of initial spectrum of perturbations have to be exponentially decreased in the direction of high k. The steadystates have been studied in two ways. In both of them the spectral decomposition of a velocity field and an elevation of the contact surface has been used. In one of them, corresponding to the nonstationary situation, the ordinary differential system for time dependent spectral components has been integrated. The other solutions of the algebraic equations for constant components have been studied. These results have been compared with a computer simulation that has been done by large particles and artificial compressibility methods. High-order nonlinear terms corresponding to the interaction of a wide spectrum of harmonics are considered in both ways. The spatial structure of the many dimensional periodic solutions is studied. It is known that there are two important features in the flows under consideration: the bubble envelope and the jet “phase.” Spatial structures of the 2D and 3D flows are compared in the report. It is shown that the apices of the bubbles are isolated points and the bubble envelopes in both flows are qualitatively similar; the difference has a quantitative character. On the contrary, the jet “phases” differ qualitatively in the 2D and 3D cases. In the 2D case in the plane picture, the jets are separated from each other. In this picture they are isolated fingers. In the 3D case, near the bubble envelope after some time of development following after cos kx cos qy type initial perturbation, they are not the isolated fingers. They form a structure of the intersecting walls. We see, thus, that the 2D and 3D cases are topologically different. Most powerful ejections of dense fluid are in the points where these wall jets intersect each other.