Application of Stochastic Theories and Three-Dimensional Particle Tracking Velocity (3D-PTV) Experiments to Study Anomalous Dispersion

为模拟多孔介质而发展的Cushman模型通常认为污染物的弥散通量与浓度梯度成正比，其扩散系数为常数或随时间变化。这些模型是对多孔介质中传输的粗略近似，在观测尺度上具有不断演变的非均质性。据说，如果色散张量是常数，多孔介质的行为是菲克的，如果张量是时间相关的，它是准菲克的，如果通量是卷积的，它是卷积菲克的。更一般形式的弥散通量是可能的，在任何情况下，如果弥散通量和浓度梯度之间没有恒定的比例系数，则称弥散通量为反常。所提出的效应的一个主要目的是利用现有的混合过程模型和三维粒子跟踪速度(3D-PTV)来研究这些理论对于各种类型的非均质性的精度。此外，利用全中间散射函数和有限大小李雅普诺夫指数等非线性动力学概念对这些模型进行了推广。具体的实验目标是：(I)构建匹配的折射率、非均匀、多孔基质的流体混合物的序列；(Ii)使用3D-PTV重建拉格朗日粒子轨迹；(Iii)使用轨迹来确定均方位移、速度分布、速度关联(单粒子和多粒子)函数、经典弥散张量、自部分和完全中间散射函数、广义波矢量和频率相关的弥散张量以及有限大小的Lyapunov指数；(Iv)在实验室尺度上研究甘油中的浮力驱动的空气流动。具体的理论目标是：(I)利用实验数据检验现有非均匀介质输运模型的充分性；(Ii)发展有限尺寸Lyapunov指数与非均匀介质中色散之间的关系；(Iii)发展依赖于多粒子关联函数、完全中间散射函数和有限尺寸Lyapunov指数的多孔性介质中的色散理论；以及(Iv)用实验数据检验新理论。

0003878CushmanModel developed to simulate in porous media often consider the dispersive flux of the contaminant species to be proportional to the concentration gradient via a constant, or time-dependent dispersion coefficient. These models are a crude approximation for transport in porous media with evolving scales of heterogeneity on the scale of observation. It is said that a porous medium behaves in a fickian fashion if the dispersion tensor is constant, it is quasi-fickian if the tensor is time dependent, and it is convolution fickian if the flux is a convolution. More general forms of the dispersive flux are possible, and in any case, dispersive fluxes are called anomalous if there is no constant coefficient of proportionality between the dispersive flux and the gradient of concentration.A main purpose of the proposed effect is to use existing models of the mixing process in conjunction with three-dimensional particle tracking velocity (3D-PTV) to study the accuracy of these theories for various types of heterogeneity. In addition, it is proposed to extend these models by using the full intermediate scattering function and concepts from nonlinear dynamics such as finite-size Lyapunov exponents. The specific experimental objectives are: (i) to construct a sequence of matched index, heterogeneous, porous-matrix fluid mixtures; (ii) to use 3D-PTV to reconstruct lagrangian particle trajectories; (iii) to use the trajectories to determine mean square displacements, velocity distributions velocity correlation (single and multiparticle) functions, classical dispersion tensors, self-part and full intermediate scattering functions, generalized wave-vector and frequency dependent dispersion tensors, and finite-size Lyapunov exponents; (iv) to investigate buoyancy driven flow of air in glycerol in matched index formations, both homogeneous and heterogeneous on the lab scale. The specific theoretical objectives are: (i) to examine the adequacy of existing models of transport in heterogeneous media using experimental data: (ii) to develop the relationship between the finite-size Lyapunov exponents and dispersion in heterogeneous media; (iii) to develop a theory of dispersion in porous media with evolving heterogeneity which relies upon multiparticle correlation functions, the full intermediate scattering function, and the finite-size Lyapunov exponents; and (iv) to test the new theory with data obtained experimentally.