Session W12: General Fluid Dynamics: Theory (10:00am - 10:45am CST)

Maximum Amplification of Enstrophy in 3D Navier-Stokes Flows

This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes equation --- as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no finite a priori bounds available for the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. To quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy $\mathcal{E}_0$ are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time $T$. Such problems are solved computationally using a large-scale adjoint-based gradient approach. By solving these problems for a broad range of values of $T$ and $\mathcal{E}_0$, we demonstrate that the maximum growth of enstrophy appears finite and scales in proportion to $\mathcal{E}_0^{3/2}$. Thus, in the worst-case scenario the enstrophy remains bounded for all times and there is no evidence for formation of singularity in finite time.   

Asymptotic Convergence to a Full Nonlinear Solution

A perturbation technique is used to investigate the nonlinear effects and asymptotic convergence to the full nonlinear solution in a flow that propagates omnidirectional waves in a modified set of Euler equations. The physical dissipative mechanisms considered within the differential system are viscosity (momentum diffusion) and heat conduction (energy diffusion). This asymptotic convergence is used to predict a lower bound calculated by the perturbation truncation error in the differential system.   

An atomistic model for the Navier slip condition

The behavior of a fluid at the interface with a solid boundary is affected, to a large extent, by the potential landscape imposed on the fluid by the solid. In previous work [Wang and Hadjiconstantinou, {\it Phys. Rev. Fluids}, 2, 094201, 2017] we have shown how this potential gives rise and determines the layering observed at the fluid-solid interface. In this presentation we discuss how fluid slip at the interface with a solid boundary can be modeled as forced Brownian motion in a periodic potential landscape. The resulting model goes beyond simple transition-state-theory approaches and uses well-defined atomistic parameters to capture the salient features of the slip process in both the linear and non-linear forcing regimes, yielding excellent agreement with MD simulation results, as well as previous modeling results. An explicit expression for the Navier slip coefficient in terms of molecular-level system parameters is derived.   

Normal modes with Boundary Dynamics

Consider infinitesimal wave perturbations to a resting stable fluid equilibrium and suppose that there are restoring forces at the boundary as well as in the interior. Examples include a stratified Boussinesq fluid with a free-surface or a quasigeostrophic fluid with bottom topography. In such problems, the resulting eigenvalue problem for the normal modes will include the eigenvalue parameter in the boundary conditions. This leads to normal modes with fundamentally new properties: 1) The modes form a basis in a larger function space than $L^2$ and may be discontinuous at the boundaries. Any function with a finite-jump discontinuity at an active boundary may be represented by the modes. 2) The modes are orthonormal with respect to an indefinite inner product that contains contributions from Dirac delta ``functions''. Some modes may have a negative square norm. 3) Distinct modes may have an identical number of internal zeros. Assuming the eigenfunctions are ordered by their eigenvalues, the number of internal zeros for the eigenfunctions may take forms such as $1,0,0,1,2,3,\dots$ 4) Such problems include a $\delta$-sheet formulation analogous to the Bretherton (1966) interpretation of boundary buoyancy gradients as infinitely thin sheets of potential vorticity.   

Small-scale averaging coarse-grains passive scalar turbulence

Capturing the multi-scale dynamics of turbulent mixing remains a theoretical and computational challenge. Therefore, many practical applications require a coarse-grained description, which treats the small scales effectively. Here, we address this challenge at the example of a stochastic, one-dimensional Kraichnan model for passive scalar mixing. We propose that effective large-scale equations can be obtained by ensemble-averaging over the small-scale velocity fluctuations. We show that this procedure leads to an effective diffusivity reminiscent of phenomenological eddy viscosity models. Additionally, we establish an exact filtering procedure that maps second-order statistics of the fully resolved passive scalar field to the one obtained by small-scale averaging. Combined with fully resolved simulations, we show that small-scale averaging also captures higher-order large-scale statistics of passive scalar fields.   

Anomalous diffusion process detected in macroscale Taylor dispersion

Anomalous dispersion process is described by a non-linear relationship (usually a power law one) between the second centered spatial moment of a solute under transport and time. In this work, we show that anomalous spreading regimes can be seen in the transient upscaled balance equation describing Taylor dispersion assuming that the appropriate transient dispersion coefficient is adopted. While most of the previous work on anomalous behavior for Taylor dispersion has focused on directly solving the microscale transport equations in free space; here we illustrate that the anomalous dispersion is also evident within the averaged equations themselves. Similar to the microscale equations, three distinct dispersive regimes can be observed in the averaged equations: (i) early time diffusive spreading, (ii) anomalous / ballistic dispersive spreading, and (iii) classical Taylor dispersion. Dispersion of solutes shows significant superdiffusive behavior for $Pe > 10$, and ballistic-type spreading is observed at early times for $Pe < 1000$. The observations about the spreading regimes for the averaged equation are corroborated by averaging the results of microscale numerical simulations.