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We investigate the purely spatial Lagrangian coordinate transformation from the Lagrangian to the basic Eulerian frame. We demonstrate three techniques for extracting the relativistic displacement field from a given solution in the…
We investigate the weakly nonlinear dynamics of transient gravity waves at infinite depth under the influence of a shear current varying linearly with depth. An analytical solution is permitted via integration of the Euler equations.…
Adjoint-based shape optimization most often relies on Eulerian flow field formulations. However, since Lagrangian particle methods are the natural choice for solving sedimentation problems in oceanography, extensions to the Lagrangian…
We present a numerical method of analyzing possibly singular incompressible 3D Euler flows using massively parallel high-resolution adaptively refined numerical simulations up to 8192^3 mesh points. Geometrical properties of Lagrangian…
We present a local existence result for the three dimensional incompressible Euler equations. The solution is constructed using a formulation of the equations as an active vector system in Eulerian coordinates. The formulation employs the…
The incompressible three-dimensional Euler equations develop very thin pancake-like regions of increasing vorticity. These regions evolve with the scaling $\omega_{max}\sim\ell^{-2/3}$ between the vorticity maximum and the pancake…
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…
Experimentalists now measure intense rotations of Lagrangian particles in turbulent flows by tracking their trajectories and Lagrangian-average velocity gradients at high Reynolds numbers. This paper formulates the dynamics of an…
We develop a general theory of transport barriers for three-dimensional unsteady flows with arbitrary time-dependence. The barriers are obtained as two-dimensional Lagrangian Coherent Structures (LCSs) that create locally maximal…
A new geometrically conservative arbitrary Lagrangian-Eulerian (ALE) formulation is presented for the moving boundary problems in the swirl-free cylindrical coordinates. The governing equations are multiplied with the radial distance and…
This paper develops the theory of affine Euler-Poincar\'e and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids,…
As a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial setting, Lagrangian hydrodynamics is characterized by moving meshes, advection-dominated solutions, and moving shock fronts with sharp…
A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template…
We present an Eulerian vortex method based on the theory of flow maps to simulate the complex vortical motions of incompressible fluids. Central to our method is the novel incorporation of the flow-map transport equations for line elements,…
In this paper, we study both convergence and bounded variation properties of a new fully discrete conservative Lagrangian--Eulerian scheme to the entropy solution in the sense of Kruzhkov (scalar case) by using a weak asymptotic analysis.…
In this paper, we present a novel second-order accurate Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on moving nonconforming polygonal grids, in order to avoid the typical mesh distortion caused by shear flows in Lagrangian-type…
We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow…
In this paper we derive a representation of the deterministic 3-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for…
This paper presents robust discontinuous Galerkin methods for the incompressible Navier-Stokes equations on moving meshes. High-order accurate arbitrary Lagrangian-Eulerian formulations are proposed in a unified framework for both…
We identify materially defined regions in unsteady two-dimensional flows that combine finite-time contraction with elevated accumulated intrinsic rotation along trajectories, which we term \emph{Lagrangian rotating contracting structures}…