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Register a new userFinite-Time Euler singularities: A Lagrangian perspective
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We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate for a finite-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, we present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high resolution adaptively refined numerical simulations and inject Lagrangian tracer particles to monitor geometric properties of vortex line segments. We then verify the assumptions made by analytical non-blowup criteria introduced by Deng et. al [Commun. PDE 31 (2006)] connecting vortex line geometry (curvature, spreading) to velocity increase to rule out singular behavior.
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There are two main strategies for improving the projection-based reduced order model (ROM) accuracy: (i) improving the ROM, i.e., adding new terms to the standard ROM; and (ii) improving the ROM basis, i.e., constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.
The Green Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler Poincare theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models.
We investigate the formation of singularities in the incompressible Navier-Stokes equations in $dgeq 2$ dimensions with a fractional Laplacian $| abla |^alpha$. We derive analytically a sufficient but not necessary condition for solutions to remain always smooth and show that finite time singularities cannot form for $alphageq alpha_c= 1+d/2$. Moreover, initial singularities become unstable for $alpha>alpha_c$.
We review the continuous symmetry approach and apply it to find the solution, via the construction of constants of motion and infinitesimal symmetries, of the 3D Euler fluid equations in several instances of interest, without recourse to Noethers theorem. We show that the vorticity field is a symmetry of the flow and therefore one can construct a Lie algebra of symmetries if the flow admits another symmetry. For steady Euler flows this leads directly to the distinction of (non-)Beltrami flows: an example is given where the topology of the spatial manifold determines whether the flow admits extra symmetries. Next, we study the stagnation-point-type exact solution of the 3D Euler fluid equations introduced by Gibbon et al. (Physica D, vol.132, 1999, pp.497-510) along with a one-parameter generalisation of it introduced by Mulungye et al. (J. Fluid Mech., vol.771, 2015, pp.468-502). Applying the symmetry approach to these models allows for the explicit integration of the fields along pathlines, revealing a fine structure of blowup for the vorticity, its stretching rate, and the back-to-labels map, depending on the value of the free parameter and on the initial conditions. Finally, we produce explicit blowup exponents and prefactors for a generic type of initial conditions.
New aspects of turbulence are uncovered if one considers flow motion from the perspective of a fluid particle (known as the Lagrangian approach) rather than in terms of a velocity field (the Eulerian viewpoint). Using a new experimental technique, based on the scattering of ultrasounds, we have obtained a direct measurement of particle velocities, resolved at all scales, in a fully turbulent flow. It enables us to approach intermittency in turbulence from a dynamical point of view and to analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the walk have random uncorrelated directions but a magnitude that is extremely long-range correlated in time. Theoretically, we study a Langevin equation that incorporates these features and we show that the resulting dynamics accounts for the observed one- and two-point statistical properties of the Lagrangian velocity fluctuations. Our approach connects the intermittent statistical nature of turbulence to the dynamics of the flow.
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