Turbulence In The Shell Model Of Passive Scalar Singularity

Since the 1980s,studies of passive scalar have been one of hotspots in turbulence research.Experiments and numerical simulations have provided lots of new phenomena and concepts,which put an impetus to reinterpret the classical turbulent theories and modify application models at present.Our studies on passive scalar compose of three parts:1,The power law of the structure function is an important property of the turbulence.In this thesis,a shell-model version of passive scalar problem is introduced which is inspired by the model of K.Ohkitani and M.Yakhot.As in the original problem,the prescribed random velocity field is Gaussian andδcorrelated in time.Deterministic differential equations are regarded as nonlinear Langevin equation.Then,the Fokker-Planck equations of PDF for passive scalars are obtained and solved numerically.In energy input range(n＜5,n is shell number.),the probability distribution function(PDF) of passive scalars are near the Gaussian distribution.In inertial range(5≤n≤16) and dissipation range(n≥17),the probability distribution function(PDF) of passive scalars has obvious intermittence. And the scaling power of passive scalar is anomalous.The results of numerical simulations are compared with experimental measurements.2,We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of passive scalars of turbulence.Different to the original problem, the distribution function of the prescribed random velocity field is the multi-dimensional normal and delta-correlated in time.Here,our random velocity field is spatially correlative.For comparison,we also given the result obtained by the Gaussian random velocity field without spatial correlation The anomalous scaling exponents H(p) of passive scalar advected by two kinds of random velocity above are determined for structure function up to p = 15 by numerical simulations of the random shell model with Runge-Kutta methods to solve the stochastic differential equations. We observed that the H(p)'s obtained by the multi-dimension normal distribution random velocity are more anomalous than those obtained by the independent Gaussian random velocity.3,We consider temporal-spatial correlations of the turbulence's velocity here. We simply analyze the correlation of the velocity on the time-space serial.With numerical simulation on GOY shell model,we show the correlation function is exponential decay function.The advecting velocity field is regarded as colored noise field,which is spatially and temporally correlative.For comparison,we also given the scaling exponents of passive scalars obtained by the Gaussian random velocity field, the multi-dimensional normal velocity field and SL velocity field.We observed that anomalous RSH scaling exponents H(p)/H(2) of passive scalar obtained by colored noise field are more anomalous than those obtained by other three velocity fields.