Lester Daniel R, Dentz Marco, Le Borgne Tanguy, de Barros Felipe P J
School of Engineering, RMIT University, 3000 Melbourne, Australia.
Spanish National Research Council (IDAEA-CSIC), 08034 Barcelona, Spain.
J Fluid Mech. 2018 Nov;855:770-803. doi: 10.1017/jfm.2018.654. Epub 2018 Sep 19.
The deformation of elementary fluid volumes by velocity gradients is a key process for scalar mixing, chemical reactions and biological processes in flows. Whilst fluid deformation in unsteady, turbulent flow has gained much attention over the past half century, deformation in steady random flows with complex structure - such as flow through heterogeneous porous media - has received significantly less attention. In contrast to turbulent flow, the steady nature of these flows constrains fluid deformation to be anisotropic with respect to the fluid velocity, with significant implications for e.g. longitudinal and transverse mixing and dispersion. In this study we derive an coupled continuous time random walk (CTRW) model of fluid deformation in random steady three-dimensional flow that is based upon a streamline coordinate transform which renders the velocity gradient and fluid deformation tensors upper-triangular. We apply this coupled CTRW model to several model flows and find these exhibit a remarkably simple deformation structure in the streamline coordinate frame, facilitating solution of the stochastic deformation tensor components. These results show that the evolution of longitudinal and transverse fluid deformation for chaotic flows is governed by both the Lyapunov exponent and power-law exponent of the velocity PDF at small velocities, whereas algebraic deformation in non-chaotic flows arises from the intermittency of shear events following similar dynamics as that for steady two-dimensional flow.
基本流体微团因速度梯度而产生的变形是流动中标量混合、化学反应和生物过程的关键过程。虽然非定常湍流中的流体变形在过去半个世纪中受到了广泛关注，但具有复杂结构的定常随机流动（如通过非均质多孔介质的流动）中的变形受到的关注要少得多。与湍流不同，这些流动的定常特性使得流体变形相对于流体速度呈各向异性，这对纵向和横向混合及扩散等有重大影响。在本研究中，我们基于流线坐标变换推导了一个随机定常三维流动中流体变形的耦合连续时间随机游走（CTRW）模型，该变换使速度梯度和流体变形张量成为上三角矩阵。我们将这个耦合CTRW模型应用于几个模型流动，发现它们在流线坐标系中呈现出非常简单的变形结构，便于求解随机变形张量分量。这些结果表明，混沌流动中纵向和横向流体变形的演化由小速度下速度概率密度函数的李雅普诺夫指数和幂律指数共同控制，而非混沌流动中的代数变形则源于剪切事件的间歇性，其动力学与定常二维流动相似。
