arXiv subjects

Darryl D. Holm

Publications and source records attributed to Darryl D. Holm.

155 records · Page 9Linked to original sources

arXiv2001

Renormalized HVBK dynamics for Superfluid Helium Turbulence

We review the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations for superfluid Helium turbulence and discuss their implications for recent measurements of superfluid turbulence decay. A new Hamiltonian formulation of these equations renormalizes the vortex line velocity to incorporate finite temperature effects. These effects also renormalize the coupling constant in the mutual friction force between the superfluid and normal fluid components by a factor of rho_s / ρ(the superfluid mass fraction) but they leave the vortex line tension unaffected. Thus, the original HVBK form is recovered at zero temperature and its mutual friction coefficients are renormalized at nonzero temperature. The HVBK equations keep their form and no new parameters are added. However, a temperature dependent trade-off does arise between the mutual friction coupling and the vortex line tension. The renormalized HVBK equations obtained via this new Hamiltonian approach imply a dynamical equation for the space-integrated vortex tangle length, which is the quantity measured by second sound attenuation experiments in superfluid turbulence. A Taylor-Proudman theorem also emerges for the superfluid vortices that shows the steady vortex line velocity becomes columnar under rapid rotation.

nlin.CD↗

arXiv2001

Euler-Poincaré Dynamics of Perfect Complex Fluids

Lagrangian reduction by stages is used to derive the Euler-Poincaré equations for the nondissipative coupled motion and micromotion of complex fluids. We mainly treat perfect complex fluids (PCFs) whose order parameters are continuous material variables. These order parameters may be regarded geometrically either as objects in a vector space, or as coset spaces of Lie symmetry groups with respect to subgroups that leave these objects invariant. Examples include liquid crystals, superfluids, Yang-Mills magnetofluids and spin-glasses. A Lie-Poisson Hamiltonian formulation of the dynamics for perfect complex fluids is obtained by Legendre transforming the Euler-Poincare formulation. These dynamics are also derived by using the Clebsch approach. In the Hamiltonian and Lagrangian formulations of perfect complex fluid dynamics Lie algebras containing two-cocycles arise as a characteristic feature. After discussing these geometrical formulations of the dynamics of perfect complex fluids, we give an example of how to introduce defects into the order parameter as imperfections (e.g., vortices) that carry their own momentum. The defects may move relative to the Lagrangian fluid material and thereby produce additional reactive forces and stresses.

nlin.CD↗

arXiv2001

Variational Principles for Lagrangian Averaged Fluid Dynamics

The Lagrangian average (LA) of the ideal fluid equations preserves their transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its convection of potential vorticity and its conservation of helicity. Lagrangian averaging also preserves the Euler-Poincaré (EP) variational framework that implies the LA fluid equations. This is expressed in the Lagrangian-averaged Euler-Poincaré (LAEP) theorem proven here and illustrated for the Lagrangian average Euler (LAE) equations.

nlin.CD↗

arXiv2001

Averaged Lagrangians and the mean dynamical effects of fluctuations in continuum mechanics

We begin by placing the Generalized Lagrangian Mean (GLM) equations for a compressible adiabatic fluid into the Euler-Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then derive a set of approximate small amplitude GLM equations (glm) that express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The new glm EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or glm) fluid theory with a Taylor hypothesis closure (THC). The glm equations lead to generalizations of the Euler-alpha models to include compressibility and magnetic fields.

nlin.CD↗

arXiv1999

The Euler-Poincare Equations in Geophysical Fluid Dynamics

Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: (1) Euler-Poincaré equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained; (2) an abstract Kelvin-Noether theorem is derived for such systems. By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincaré form and discuss their corresponding Kelvin-Noether theorems and potential vorticity conservation laws.

chao-dyn↗

arXiv1999

A Nonlinear Analysis of the Averaged Euler Equations

This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter $α$; one interpretation is that they are obtained by ensemble averaging the Euler equations in Lagrangian representation over rapid fluctuations whose amplitudes are of order $α$. The particle flows associated with these equations are shown to be geodesics on a suitable group of volume preserving diffeomorphisms, just as with the Euler equations themselves (according to Arnold's theorem), but with respect to a right invariant $H^1$ metric instead of the $L^2$ metric. The equations are also equivalent to those for a certain second grade fluid. Additional properties of the Euler equations, such as smoothness of the geodesic spray (the Ebin-Marsden theorem) are also shown to hold. Using this nonlinear analysis framework, the limit of zero viscosity for the corresponding viscous equations is shown to be a regular limit, {\it even in the presence of boundaries}.

chao-dyn↗

arXiv1999

Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid motion

We formulate equations for the slow time dynamics of fluid motion that self consistently account for the effects of the variability upon the mean. The time-average effects of the fluctuations introduce nonlinear dispersion that acts to spatially smooth the transport velocity of the mean flow relative to its circulation or momentum velocity, by the inversion of a Helmholtz operator whose length scale corresponds to the covariance of the fluctuations.

chao-dyn↗

arXiv1999

Direct Numerical Simulations of the Navier-Stokes Alpha Model

We explore the utility of the recently proposed alpha equations in providing a subgrid model for fluid turbulence. Our principal results are comparisons of direct numerical simulations of fluid turbulence using several values of the parameter alpha, including the limiting case where the Navier-Stokes equations are recovered. Our studies show that the large scale features, including statistics and structures, are preserved by the alpha models, even at coarser resolutions where the fine scales are not fully resolved. We also describe the differences that appear in simulations. We provide a summary of the principal features of the alpha equations, and offer some explanation of the effectiveness of these equations used as a subgrid model for three-dimensional fluid turbulence.

chao-dyn↗

arXiv1998

The Camassa-Holm equations as a closure model for turbulent channel flow

We propose the viscous Camassa-Holm equations as a closure approximation for the Reynolds-averaged equations of the incompressible Navier-Stokes fluid. This approximation is tested on turbulent channel flows with steady mean. Analytical solutions for the mean velocity and the Reynolds shear stress across the entire channel are obtained, showing good agreement with experimental measurements and direct numerical simulations. As Reynolds number varies, these analytical mean velocity profiles form a family of curves whose envelopes are shown to have either power law, or logarithmic behavior, depending on the choice of drag law.

chao-dyn↗

arXiv1998

Hamilton's principle for quasigeostrophic motion

We show that the equation of quasigeostrophic (QG) potential vorticity conservation in geophysical fluid dynamics follows from Hamilton's principle for stationary variations of an action for geodesic motion in the f-plane case or its prolongation in the beta-plane case. This implies a new momentum equation and an associated Kelvin circulation theorem for QG motion. We treat the barotropic and two-layer baroclinic cases, as well as the continuously stratified case.

chao-dyn↗

arXiv1993

An integrable shallow water equation with peaked solitons

We derive a new completely integrable dispersive shallow water equation that is biHamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak.

patt-sol↗