Fluid Deformation in Random Unsteady Flow

Abstract

Fluid deformation controls myriad processes including mixing and dispersion, development of stress in complex fluids, droplet breakup and emulsification, fluid-structure interaction, chemical reactions and biological activity. We develop a simple stochastic model for fluid deformation in random unsteady flows such as homogeneous isotropic turbulence. We show that although the Lagrangian velocity process is non-Markovian and non-Fickian, temporal decorrelation due to the unsteady nature of the flow renders the evolution of the Lagrangian velocity gradient tensor to be Fickian. Application of a coordinate transform renders the velocity gradient tensor upper triangular, eliminating vortical rotation and decoupling principal stretches from shear deformations, leading to a stochastic model of fluid deformation as a simple Brownian process. We develop closed-form expressions for the evolution of the Cauchy-Green tensor and show that the finite-time Lyapunov exponents are Gaussian distribution. Application of this model to DNS calculations of forced isotropic turbulence at Taylor-scale Reynolds number Re$_λ\approx 433$, confirms the underlying model assumptions and provides excellent agreement with theoretical results.

Figures (2)

• Figure 1: (a) Temporal and (b) spatial autocorrelation functions for the velocity magnitude from numerical simulations of isotropic turbulence. Dashed lines represent fitted respective temporal and spatial autocorrelation functions $R_t(t)=\exp(-t/t_d)$, $R_s(s)=\exp(-s.s_d)$, where $t_d=0.844$ [s], $s_d=0.439$ [m]. PDF of (c) Eulerian velocity with fit of Nakagami distribution (dashed line). (d) Lagrangian correlation functions for diagonal $\epsilon^\prime_{ii}$ velocity gradient components. PDFs of the (e) diagonal $\epsilon^\prime_{ii}$ and (f) off-diagonal $\epsilon^\prime_{ii}$ components of the velocity gradient tensor. Dashed lines represent fits of the Laplace distribution to these velocity gradient PDFs.
• Figure 2: (a) Evolution of (a) ensemble mean $\langle\xi_{ii}(t)\rangle_N$ and (b) ensemble variance $\langle\xi_{ii}(t)^2\rangle-\langle\xi_{ii}(t)\rangle_N^2$ of log-stretches (solid blue lines) over $10^3$ trajectories and respective analytic solutions $\lambda_i t$, $\sigma_{ii}^2 t$ (dashed black lines) for $i=1:3$. (c) Convergence of $A_{ij}(\mathbf{X},t)$ to steady value $a_{ij}(\mathbf{X})$ for 30 sample trajectories. (d) Decay of $m_{ij}(\mathbf{X},t)$ toward zero for 30 sample trajectories. (e) Convergence of components $C_{ij}(\mathbf{X},t)/F_{11}(\mathbf{X},t)^2$ to a constant value for 30 sample trajectories. (f) Evolution of FTLE $\lambda(\mathbf{X},t)$ (light blue lines) over 30 sample trajectories and ensemble averaged FTLE $\langle\lambda(\mathbf{X},t)\rangle_N$ with Lagrangian time $t$ (black dashed line) toward Lyapunov exponent $\lambda_\infty$ (solid black line). The analytic expression (\ref{['eqn:FTLEmean']}) for $\langle\lambda(\mathbf{X},t)\rangle_N$ (solid gray line) is different to the numerical solution at short times as convergence to the central limit theorem is still developing.)