紊动流场中悬浮颗粒分布的随机理论

THE DISTRIBUTION OF SOLID PARTICLES SUSPENDED IN A TURBULENT FLOW: A STOCHASTIC APPROACH

通过分析固体颗粒在紊动流场中的随机运动,建立了二维流场中垂直于时均流动的方向上颗粒随机位移的概率密度分布函数所满足的方程。由该方程解出的分布函数在一定条件下即相当于颗粒浓度分布函数。运用这一方法研究了[1]、[2]中报道的壁面附近颗粒浓度降低的现象。

The random motion of solid particles suspended in two-dimentional turbulent flow is considered in this paper. Mean values of partical velocity and displacement in a direction normal to the mean streamlines of the flow are calculated and it is found out that particle velocity vp can be decomposed into a mean velocity (vp) and a velocity fluctuation vp - (vp) where (vp) is equal to the settling velocity of the same particle in tranquil fluid. A Langevin random differential equation for particle displacement Yp is developed, from which a Fokker-Planck equation for the probability function p(y, t) is derived on the basis of the theory of Markovian process. Thus the distribution of p(y, t) is interrelated to the random motion of the particle. The lift effect to which a particle will be subject in the vicinity of the wall is taken into account and a corresponding Fokker-Planck equation is developed. Analytic solution of this equation shows that the probability density p(y, t) describing particle displacement has a maximum value at y = H where the perpendicular component of the resulting lift force precisely balances the particle gravity. Interpretation of experimental observations reported in the literature is given using this theory.