An Eulerian-Lagrangian approach to the Navier-Stokes equations

This work presents an approach to the Navier-Stokes equations that is phrased in unbiased Eulerian coordinates, yet describes objects that have Lagrangian significance: particle paths, their dispersion and diffusion. The commutator between Lagrangian and Eulerian derivatives plays an important role in the Navier-Stokes equations: it contributes a singular perturbation to the Euler equations, in addition to the Laplacian. Bounds for the Lagrangian displacements, their first and second derivatives are obtained without assumptions. Some of these rigorous bounds can be interpreted in terms of the heuristic Richardson law of pair dispersion in turbulent flows

A formulation for the boundary-layer equations in general coordinates

This is a working paper in which a formulation is given for solving the boundary-layer equations in general body-fitted curvilinear coordinates while retaining the original Cartesian dependent variables. The solution procedure does not require that any of the coordinates be orthogonal, and much of the software developed for many Navier-Stokes schemes can be readily used. A limited number of calculations has been undertaken to validate the approach

Intrinsic character of Stokes matrices

Two germs of linear analytic differential systems $x^{k+1}Y^\prime=A(x)Y$ with a non resonant irregular singularity are analytically equivalent if and only if they have the same eigenvalues and equivalent collections of Stokes matrices. The Stokes matrices are the transition matrices between sectors on which the system is analytically equivalent to its formal normal form. Each sector contains exactly one separating ray for each pair of eigenvalues. A rotation in $S$ allows supposing that $\mathbb R^+$ lies in the intersection of two sectors. Reordering of the coordinates of $Y$ allows ordering the real parts of the eigenvalues, thus yielding triangular Stokes matrices. However, the choice of the rotation in $x$ is not canonical. In this paper we establish how the collection of Stokes matrices depends on this rotation, and hence on a chosen order of the projection of the eigenvalues on a line through the origin.Comment: 10 pages, 3 figure

Stokes Coordinates

Cage-based structures are reduced subspace deformers enabling non-isometric stretching deformations induced by clothing or muscle bulging. In this paper, we reformulate the cage-based rigging as an incompressible Stokes problem in the vorticity space. The key to our approach is a compact stencil allowing the expression of fluid-inspired high-order coordinates. Thus, our cage-based coordinates are obtained by vorticity transport as the numerical solution of the linearized Stokes equations. Then, we turn the incompressible creeping Newtonian flow into Stokes equations, and we devise a second-order compact approximation with center differencing for solving the vorticity-stream function. To the best of our knowledge, our work is the first to devise a vorticity-stream function formulation as a computational model for cage-based weighting functions. Finally, we demonstrate the effectiveness of our new techniques for a collection of cage-based shapes and applications

Eddy diffusivities of inertial particles under gravity

The large-scale/long-time transport of inertial particles of arbitrary mass density under gravity is investigated by means of a formal multiple-scale perturbative expansion in the scale-separation parametre between the carrier flow and the particle concentration field. The resulting large-scale equation for the particle concentration is determined, and is found to be diffusive with a positive-definite eddy diffusivity. The calculation of the latter tensor is reduced to the resolution of an auxiliary differential problem, consisting of a coupled set of two differential equations in a (6+1)-dimensional coordinate system (3 space coordinates plus 3 velocity coordinates plus time). Although expensive, numerical methods can be exploited to obtain the eddy diffusivity, for any desirable non-perturbative limit (e.g. arbitrary Stokes and Froude numbers). The aforementioned large-scale equation is then specialized to deal with two different relevant perturbative limits: i) vanishing of both Stokes time and sedimenting particle velocity; ii) vanishing Stokes time and finite sedimenting particle velocity. Both asymptotics lead to a greatly simplified auxiliary differential problem, now involving only space coordinates and thus easy to be tackled by standard numerical techniques. Explicit, exact expressions for the eddy diffusivities have been calculated, for both asymptotics, for the class of parallel flows, both static and time-dependent. This allows us to investigate analytically the role of gravity and inertia on the diffusion process by varying relevant features of the carrier flow, as e.g. the form of its temporal correlation function. Our results exclude a universal role played by gravity and inertia on the diffusive behaviour: regimes of both enhanced and reduced diffusion may exist, depending on the detailed structure of the carrier flow.Comment: 8 figures (12 plots), submitted to JF