Lagrangian velocity-gradient evolution: Closure modeling conditioned on local streamline geometry.

We develop a stochastic diffusion model for the evolution of velocity gradient tensor (A$_{\mathrm{ij}}\equiv \partial $u$_{\mathrm{i}}$/$\partial $x$_{\mathrm{j}})$ following a fluid particle in isotropic incompressible turbulent flow. Two primary challenges toward accurate modeling of turbulence velocity-gradient dynamics are (i) intermittent nature of A$_{\mathrm{ij}}$ and (ii) non-locality of pressure and viscous terms in its evolution equation. To overcome these difficulties, we factorize A$_{\mathrm{ij}}$ into its intermittent magnitude (A$\equiv \surd $(A$_{\mathrm{kl}}$A$_{\mathrm{kl}})$, streamline scale) and normalized velocity gradient tensor (b$_{\mathrm{ij}}\equiv $A$_{\mathrm{ij}}$/A) which fully determines the local streamline shape. It is first shown that the evolution of magnitude A does not require any additional closures, once b$_{\mathrm{ij}}$ equation is suitably modeled. Then, the evolution of mathematically bounded tensor b$_{\mathrm{ij}}$ is modeled using a stochastic differential equation, and the non-local pressure and viscous terms are modeled with closures conditioned on local streamline shapes. The key advantages of this approach are:(i) relative ease of modeling the bounded tensor b$_{\mathrm{ij}}$ and (ii) amenability of conditioning nonlocal processes upon local streamline shape. The model accurately captures one-time statistics of isotropic turbulence, including high-order moments of A$_{\mathrm{ij}}$, streamline-shape distribution and vorticity-strain rate alignment.