The Lagrangian Deformation Structure of Three-Dimensional Steady Flow

TL;DR

The paper tackles the problem of predicting Lagrangian deformation from Eulerian flow features in steady 3D flows. It introduces a Protean coordinate transform that aligns with streamlines and renders the velocity gradient upper triangular, enabling closed-form expressions for the deformation gradient $\mathbf{F}'$ and revealing how helicity and topological constraints shape deformation. Applied to ABC, Kraichnan, dual streamfunction, and random potential flows, the method shows that deformation structure in Protean coordinates is largely simple and often Gaussian in random flows, with FTLEs closely linked to principal components of $\boldsymbol\epsilon'$, and that zero-helicity flows obey sub-exponential growth consistent with Lamb surfaces. This framework lays the groundwork for stochastic models of Lagrangian deformation, extending 2D CTRW approaches to 3D by deriving deformation kernels from Eulerian statistics and enabling prediction of deformation PDFs and FTLEs from flow data.

Abstract

Fluid deformation and strain history are central to wide range of fluid mechanical phenomena ranging from fluid mixing and particle transport to stress development in complex fluids and the formation of Lagrangian coherent structures (LCSs). To understand and model these processes it is necessary to quantify Lagrangian deformation in terms of Eulerian flow properties, currently an open problem. To elucidate this link we develop a Protean (streamline) coordinate transform for steady three-dimensional (3D) flows which renders both the velocity gradient and deformation gradient upper triangular. This frame not only simplifies computation of fluid deformation metrics such as finite-time Lyapunov exponents (FTLEs) and elucidates the deformation structure of the flow, but moreover explicitly recovers kinematic and topological constraints upon deformation such as those related to helicity density and the Poincaré-Bendixson theorem. We apply this transform to several classes of steady 3D flow, including helical and non-helical, compressible and incompressible flows, and find random flows exhibit remarkably simple (Gaussian) deformation structure, and so are fully characterised in terms of a small number of parameters. As such this technique provides the basis for the development of stochastic models of fluid deformation in random flows which adhere to the kinematic constraints inherent to various flow classes.

Figures (15)

• Figure 1: (a) Convergence of the transverse orientation angle $\alpha(t)$ for different initial conditions $\alpha(0)=\alpha_0$ (thin black lines) toward the inertial manifold $\mathcal{M}(t)=\alpha(t;\alpha_{0,\infty})$ (thick gray line) for the ABC flow. Note the correspondence between $\mathcal{M}(t)$ and the transverse angle $\alpha^\star(t)$ associated with minimum (gray dashed line) divergence $\partial g/\partial\alpha(\alpha,t)$ responsible for creation of the inertial manifold. The maximum divergence is also shown (black dashed line). (b) Convergence of the initial angle associated with the approximate inertial manifold $\alpha_{0,\tau}$ toward the infinite time limit $\alpha_{0,\infty}$.
• Figure 2: (a) Typical particle trajectory in the ergodic region of the Arnol'd-Beltrami-Childress flow, (b) contour plot of velocity magnitude distribution in $x_3=0$ plane.
• Figure 3: (a) Relative growth of the length $|\mathbf{l}(t)|$ of an infinitesimal material line along a single trajectory in the ABC flow (\ref{['eqn:ABC']}) calculated by (grey) particle tracking and (black) from $\mathbf{l}(t)=\mathbf{F}^\prime(t)\cdot\mathbf{l}(0)$. (b) Solution of the orientation angle $\alpha(t)$ along the inertial manifold $\mathcal{M}$. (c) Determinant error for the (grey) Cartesian $\mathbf{F}(t)$ and (black) Protean $\mathbf{F}^\prime(t)$ deformation tensors. (d) Convergence of the principal stretching exponent $\lambda(t,\mathbf{X})$ to the FTLE $\mu(t)$.
• Figure 4: Errors between Cartesian and Protean deformation tensors for time steps $\Delta t=10^{-3}$ (black), $10^{-2}$ (dark grey), $10^{-1}$ (medium grey), $10^0$ (light grey) in terms of (a) determinant of the deformation tensor $\mathbf{F}(t)$, (b) eigenvalues of right Cauchy-Green tensor $\mathbf{C}$, (c) norm of error between deformation tensors, (d) error in material line length $l(t)$.
• Figure 5: (a) Distribution of diagonal reoriented rate of strain components $\boldsymbol\epsilon^\prime_{ii}(t)$ for the ABC flow, dark grey $\epsilon^\prime_{11}$, medium grey $\epsilon^\prime_{22}$, light grey $\epsilon^\prime_{33}$, (b) distribution of off-diagonal reoriented rate of strain components $\boldsymbol\epsilon^\prime_{ij}(t)$ for the ABC flow, dark grey $\epsilon^\prime_{12}$, medium grey $\epsilon^\prime_{13}$, light grey $\epsilon^\prime_{23}$.