First-order moment equations of hybrid second-order moment model are obtained by Euler method, while second-order moment equations are deduced by Lagrange equations. Equations for particle fraction and momentum are provided firstly. A Lagrange model with mean Langevin equations is obtained and Reynold stress equation is deduced, so that hybrid second-order moment model is closed without additional approximate assumptions. Wall-jet-flow loaded with solid particles is simulated. It shows that the model is effective.

In one-dimensional nonlinear system, linear parts of acoustic wave meet wave equation and nonlinear part is assumed similar to that of nonlinear Schrödinger equation. Transmission coefficients of two incident acoustic waves from opposite directions are calculated respectively. It is shown that this model has unidirectional penetrability for acoustic wave within certain frequency and amplitude, and expresses behaviors of diode. Increasing nonlinear intensity leads to a remarkable enhancement of unidirectional penetrability.

A Langevin equation of fluctuating velocity of fluid ‘seen’ by particle is simplified in Minier's model.Autocorrelation of turbulence fluctuating velocity is obtained,which has non-isotropic property.Corresponding PDF model of particle is obtained and a simplified form is used to predict turbulent particle dispersion in grid-generating turbulence flow.PDF model is solved analytically and solutions are compared with experimental results of Wells and Stock.

With Langevin equation of particle motion,PDF equations in two levels are obtained in which the fluid fluctuation velocity is expanded in dimension and one-dimensional colored noise is transformed into two-dimensional white noise.They are two-phase PDF transport equation of turbulence and particle-phase PDF transport equation.With particle motion equations and assumption of Gaussian distribution,a closure problem of PDF equation is solved and a second-order-moment model is obtained.It is simplified into an algebra stress model [ASM].A finite analytic numerical method for convective diffusion equation is applied to solve the two-order moment model of two-phase flows.A wall-jet-flow loaded with solid particles is studied in the ASM by finite analytic numerical method.Numerical results are compared with experimental results.

A method is presented to deal with high-dimensional scalar PDF(probability density function) transport equation for two-phase flow in phase space (X, V), in which the PDF transport equation is reduced to velocity phase space.The particle velocity PDF f^v(V) is solved analytically and the particle position PDF is obtained statistically with Monte Carlo techniques. It shows that the method is better than the particle Reynolds stress trajectory model(PRST Model) in simulating a turbulent wall jet loaded with solid particles.

Based on a colored noise model with general Gaussian process,by functional derivation and small correlation time expansion,high-dimensional colored noises are calculated approximately and an effective Fokker-Planck equation(FPE) is derived.The effective FPE is applied to obtain probability density function transport equations of particles in two-phase flows and a second order moment model of two-phase flows.It is simplified into an algebra stress model(ASM).The finite analytic numerical method used in convective diffusion equation is applied to the second-order moment model.A wall-jet-flow loaded with solid particles is calculated with ASM and finite analytic numerical method.Numerical results are compared with experimental results.

A Lagrangian equation of v'_giv'_pi on the particles trajectory is obtained with stochastic process theory, and then Lagrangian equation of particle Reynolds stresses is exactly set up. Wang Stock experiment on heavy particle dispersion considered scale ratio and flow decay is simulated by both this method and Zhang's model^[1]. Simulation results show the new method is better than Zhang's model. A backward-facing step two-phase flow is calculated by the new model and the results show good agreement with the experiment results.