从单颗粒受力到群体运动特征的推移质研究

发布时间：2018-12-15 08:11

【摘要】：推移质运动的随机性一直是泥沙研究的难点和热点，这种运动的随机性归根结底是由颗粒受力的不确定性导致的。颗粒受力的不确定性使得在中小尺度上，颗粒运动与否，运动的速度、单步步长、单步时间和停时等特征均具有随机性，而中小尺度上这些变量的随机性导致了颗粒群体在大的时空尺度上的扩散。本文主要针对颗粒以滚动和滑动为主要运动形式的推移质运动。通过统计力学的方法，从单颗粒受力出发，将颗粒受力分解为确定性部分和随机部分，从而建立了描述单颗粒动力学的郎之万方程，并由此推导得到了描述颗粒速度概率密度函数（Probability density function, PDF）的福克—普朗克方程，从而将单颗粒受力与颗粒群体速度的统计特征结合了起来。在此基础上，将停时引入到郎之万方程中，模拟了大量颗粒间歇性的运动过程，探讨了颗粒群体在大的时空尺度上体现的对流和扩散特征。并分析了沙质床面和卵石夹沙床面的动态演化特征。福克—普朗克方程的定态解析解给出的速度PDF与实验资料符合较好，均表现出类指数分布（正向速度与负向速度的PDF均按指数衰减），该解从理论上解释了这种类指数分布是由单颗粒受力引起的，即库伦摩擦力和随机力导致了这种类指数分布，而沿水流方向的确定性的时均力使得分布产生了偏态。通过郎之万方程直接模拟单颗粒速度序列，提取单步运动过程，得到了单步步长和单步时间的关系，与实验资料符合较好，均表现出单步步长与单步时间的5/3幂律关系，体现了单步步长和单步时间是相关联的。郎之万方程虽然较准确地描述了运动颗粒的动力学特性，但难以描述颗粒静止时的受力状态，导致这种困难的主要原因是摩擦力在速度为0时的多值性。因此，在速度为0时引入不同分布的停时，模拟了颗粒在大的时空尺度上的正常/奇异对流和扩散特征。结果表明，对于均匀颗粒，由于颗粒速度是窄尾分布，即便单步步长是长尾分布，，也不一定产生超扩散，扩散特性由停时分布的尾部特征决定，不同分布的停时可导致欠扩散、超扩散和正常扩散。忽略单步时间不影响颗粒的对流特性，但影响颗粒的扩散特性。对于非均匀颗粒群体，因为不再满足随机变量的同分布假定，因此由颗粒非均匀性产生的长尾分布将导致与均匀颗粒的长尾分布完全不同的对流和扩散特征，在大的时空尺度上，非均匀颗粒群体总是出现弹道运动，这对应的不是奇异扩散，而是确定性的沿程分选过程。
[Abstract]:The randomness of bed load motion is always a difficult point and hot spot in sediment research. The randomness of this motion is caused by the uncertainty of particle force in the final analysis. The uncertainty of particle force makes the characteristics of particle motion, velocity, single step length, single step time and stop time are all random in small and medium scale, such as moving or not, moving speed, single step length, single step time and stopping time, etc. The randomness of these variables leads to the diffusion of the particle population on the large scale of time and space. This paper focuses on the bedload motion in which rolling and sliding are the main motion forms of particles. By means of statistical mechanics, starting from the single particle force, the particle force is decomposed into deterministic part and random part, and the Langzhiwan equation describing the dynamics of single particle is established. The Fokk-Planck equation describing the probability density function of particle velocity (Probability density function, PDF) is derived, and the statistical characteristics of single particle force and particle population velocity are combined. On this basis, the stopping time is introduced into the Langzhiwan equation, the intermittent motion process of a large number of particles is simulated, and the convection and diffusion characteristics of the particle population on a large space-time scale are discussed. The dynamic evolution characteristics of sand bed and gravel bed are analyzed. The steady state analytical solution of Fokk-Planck equation shows that the velocity PDF is in good agreement with the experimental data, and both show exponential distribution (the PDF of forward velocity and negative velocity decay exponentially). The solution theoretically explains that this kind of exponential distribution is caused by a single particle force, that is, Coulomb friction and random force lead to this kind of exponential distribution, and the deterministic time-averaged force along the direction of water flow causes the distribution to be skewed. By directly simulating single particle velocity series by Lang Zhiwan equation, the single step motion process is extracted, and the relationship between single step length and single step time is obtained, which is in good agreement with experimental data, and shows the 5 / 3 power law relation between single step length and single step time. It shows that the single step length and the single step time are related. Although the Langzhiwan equation accurately describes the dynamic characteristics of moving particles, it is difficult to describe the mechanical state of particles when they are still. The main reason for this difficulty is the multivalue of friction force when the velocity is zero. Therefore, the normal / singular convection and diffusion characteristics of particles on a large space-time scale are simulated when different distributions are introduced when the velocity is zero. The results show that, for homogeneous particles, the velocity of particles is narrow tail distribution, even if the single step size is long tail distribution, it is not necessarily superdiffusion. The diffusion characteristics are determined by the tail characteristics of the stopping time distribution, and the stop time of different distributions can lead to underdiffusion. Superdiffusion and normal diffusion. Ignoring the single step time does not affect the convection characteristics of the particles, but affects the diffusion characteristics of the particles. For the heterogeneous particle population, because the assumption of the same distribution of random variables is no longer satisfied, the long tail distribution generated by the particle heterogeneity will lead to convection and diffusion characteristics completely different from the long tail distribution of uniform particles. In large space-time scale, there is always ballistic motion in non-uniform particle population, which corresponds to not singular diffusion, but deterministic separation along the path.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TV142.2

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