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Register a new userA new model of turbulent relative dispersion: a self-similar telegraph equation based on persistently separating motions
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Turbulent relative dispersion is studied theoretically with a focus on the evolution of probability distribution of the relative separation of two passive particles. A finite separation speed and a finite correlation of relative velocity, which are crucial for real turbulence, are implemented to a master equation by multiple-scale consideration. A telegraph equation with scale-dependent coefficients is derived in the continuous limit. Unlike the conventional case, the telegraph equation has a similarity solution bounded by the maximum separation. The evolution is characterized by two parameters: the strength of persistency of separating motions and the coefficient of the drift term. These parameters are connected to Richardsons constant and, thus, expected to be universal. The relationship between the drift term and coherent structures is discussed for two 2-D turbulences.
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Properties of two equations describing the evolution of the probability density function (PDF) of the relative dispersion in turbulent flow are compared by investigating their solutions: the Richardson diffusion equation with the drift term and the self-similar telegraph equation derived by Ogasawara and Toh [J. Phys. Soc. Jpn. 75, 083401 (2006)]. The solution of the self-similar telegraph equation vanishes at a finite point, which represents persistent separation of a particle pair, while that of the Richardson equation extends infinitely just after the initial time. Each equation has a similarity solution, which is found to be an asymptotic solution of the initial value problem. The time lag has a dominant effect on the relaxation process into the similarity solution. The approaching time to the similarity solution can be reduced by advancing the time of the similarity solution appropriately. Batchelor scaling, a scaling law relevant to initial separation, is observed only for the telegraph case. For both models, we estimate the Richardson constant, based on their similarity solutions.
The relative dispersion process in two-dimensional free convection turbulence is investigated by direct numerical simulation. In the inertial range, the growth of relative separation, $r$, is expected as $<r^2(t)>propto t^5$ according to the Bolgiano-Obukhov scaling. The result supporting the scaling is obtained with exit-time statistics. Detailed investigation of exit-time PDF shows that the PDF is divided into two regions, the Region-I and -II, reflecting two types of separating processes: persistent expansion and random transitions between expansion and compression of relative separation. This is consistent with the physical picture of the self-similar telegraph model. In addition, a method for estimating the parameters of the model are presented. Comparing two turbulence cases, two-dimensional free convection and inverse cascade turbulence, the relation between the drift term of the model and nature of coherent structures is discussed.
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We introduce a time-dependent Eulerian-Lagrangian length-scale and an inverse locality hypothesis which explain scalings of second order one-particle Lagrangian structure functions observed in Kinematic Simulations (KS) of homogeneous isotropic turbulence. Our KS results are consistent with the physical picture that particle trajectories are more/less autocorrelated if they are smoother/rougher as a result of encountering less/more straining stagnation points, thus leading to enhanced/reduced turbulent diffusion.
We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose-Einstein condensates. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum $n(omega)$ at the zero frequency $omega$. The self-similar solution is of the second kind, and it satisfies boundary conditions corresponding to a nonzero constant spectrum (with all its derivative being zero) at $omega=0$ and a power-law asymptotic $n(omega) to omega^{-x}$ at $omega to infty ;; xin mathbb{R}^+$. Finding it amounts to solving a nonlinear eigenvalue problem, i.e. finding the value $x^*$ of the exponent $x$ for which these two boundary conditions can be satisfied simultaneously. To solve this problem we develop a new high-precision algorithm based on Chebyshev approximations and double exponential formulas for evaluating the collision integral, as well as the iterative techniques for solving the integro-differential equation for the self-similar shape function. This procedures allow to achieve a solution with accuracy $approx 4.7 %$ which is realized for $x^* approx 1.22$.
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