Bounding curves in the energy,enstrophy-plane are derived for the 3D Navier-Stokes equations under an assumption on coherence of the vorticity direction. The analysis in the critical case where the direction is Hölder continuous with exponent $r=1/2$ results in a curve with extraordinarily large maximal enstrophy (exponential in Grashof), in marked contrast to the subcritical case, $r>1/2$ (algebraic in Grashof).
Existence of 2D enstrophy cascade in a suitable mathematical setting, and under suitable conditions compatible with 2D turbulence phenomenology, is known both in the Fourier and in the physical scales. The goal of this paper is to show that the same geometric condition preventing the formation of singularities - 1/2-Hölder coherence of the vorticity direction - coupled with a suitable condition on a modified Kraichnan scale, and under a certain modulation assumption on evolution of the vorticity, leads to existence of 3D enstrophy cascade in physical scales of the flow.
We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in ${\mathbb R}^2$. We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstrophy balance.
Working directly from the 3D magnetohydrodynamical equations and entirely in physical scales we formulate a scenario wherein the enstrophy flux exhibits cascade-like properties. In particular we show the inertially-driven transport of current and vorticity enstrophy is from larger to smaller scale structures and this inter-scale transfer is local and occurs at a nearly constant rate. This process is reminiscent of the direct cascades exhibited by certain ideal invariants in turbulent plasmas. Our results are consistent with the physically and numerically supported picture that current and vort…
It is not yet known if the global attractor of the space periodic 2D Navier-Stokes equations contains nonstationary solutions $u(x,t)$ such that their energy and enstrophy per unit mass are constant for every $t \in (-\infty, \infty)$. The study of the properties of such solutions was initiated in \cite{CMM13}, where, due to the hypothetical existence of such solutions, they were called "ghost solutions". In this work, we introduce and study geometric structures shared by all ghost solutions. This study led us to consider a subclass of ghost solutions for which those geometric structures have …
This study provides sufficient conditions for the temporal monotonic decay of enstrophy for two-dimensional perturbations traveling in the incompressible, viscous, plane Poiseuille and Couette flows. Extension of J. L. Synge's procedure (1938) to the initial-value problem allowed us to find the region of the wavenumber-Reynolds number map where the enstrophy of any initial disturbance cannot grow. This region is wider than the kinetic energy's one. We also show that the parameters space is split in two regions with clearly distinct propagation and dispersion properties.
Enstrophy is an averaged measure of fluid vorticity. This quantity is particularly important in {\em rotating} geophysical flows. We investigate the dynamical evolution of enstrophy for large-scale quasi-geostrophic flows under random wind forcing. We obtain upper bounds on the enstrophy, as well as results establishing its Hölder continuity and describing the small-time asymptotics.
We study well-posedness of a velocity-vorticity formulation of the Navier--Stokes equations, supplemented with no-slip velocity boundary conditions, a no-penetration vorticity boundary condition, along with a natural vorticity boundary condition depending on a pressure functional. In the stationary case we prove existence and uniqueness of a suitable weak solution to the system under a small data condition. The topic of the paper is driven by recent developments of vorticity based numerical methods for the Navier--Stokes equations.
In this article we derive Biot-Savar law for exterior 2D domain. No-slip condition in terms of vorticity is expressed in integral form. For non-stationary fluid dynamics equations no-slip condition generates affine invariant manifold and no-slip integral relations on vorticity can be transferred to Robin-type boundary condition.
We propose two different proofs of the fact that Oseen's vortex is the unique solution of the two-dimensional Navier-Stokes equation with a Dirac mass as initial vorticity. The first argument, due to C.E. Wayne and the second author, is based on an entropy estimate for the vorticity equation in self-similar variables. The second proof is new and relies on symmetrization techniques for parabolic equations.
We show that any solution of the two-dimensional Navier-Stokes equation whose vorticity distribution is uniformly bounded in $L^1(R^2)$ for positive times is entirely determined by the trace of the vorticity at $t = 0$, which is a finite measure. When combined with previous existence results by Cottet, by Giga, Miyakawa, and Osada, and by Kato, this uniqueness property implies that the Cauchy problem for the vorticity equation in $R^2$ is globally well-posed in the space of finite measures. In particular, this provides an example of a situation where the Navier-Stokes equation is well-posed fo…
We develop a new approach for regularity estimates, especially vorticity estimates, of solutions of the three-dimensional Navier-Stokes equations with periodic initial data, by exploiting carefully formulated linearized vorticity equations. An appealing feature of the linearized vorticity equations is the inheritance of the divergence-free property of solutions, so that it can intrinsically be employed to construct and estimate solutions of the Navier-Stokes equations. New regularity estimates of strong solutions of the three-dimensional Navier-Stokes equations are obtained by deriving new exp…
We study the two-dimensional Navier-Stokes system on a flat cylinder with the usual Dirichlet boundary conditions for the velocity field u. We formulate the problem as an infinite system of ODE's for the natural Fourier components of the vorticity, and the boundary conditions are taken into account by adding a vorticity production at the boundary. We prove equivalence to the original Navier-Stokes system and show that the decay of the Fourier modes is exponential for any positive time in the periodic direction, but it is only power-like in the other direction.
Both experimental and numerical studies of fluid motion indicate that initially localized regions of vorticity tend to evolve into isolated vortices and that these vortices then serve as organizing centers for the flow. In this paper we prove that in two dimensions localized regions of vorticity do evolve toward a vortex. More precisely we prove that any solution of the two-dimensional Navier-Stokes equation whose initial vorticity distribution is integrable converges to an explicit self-similar solution called ``Oseen's vortex''. This implies that the Oseen vortices are dynamically stable for…
Invariant parameterization schemes for the eddy-vorticity flux in the barotropic vorticity equation on the beta-plane are constructed and then applied to turbulence modeling. This construction is realized by the exhaustive description of differential invariants for the maximal Lie invariance pseudogroup of this equation using the method of moving frames, which includes finding functional bases of differential invariants of arbitrary order, a minimal generating set of differential invariants and a basis of operators of invariant differentiation in an explicit form. Special attention is paid to …
We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain in the plane with a C^2-boundary. Navier boundary conditions can be expressed in the form w = (2 K - A) v . T and v . n = 0 on the boundary, where v is the velocity, w the vorticity, n a unit normal vector, T a unit tangent vector, and A is a bounded measurable function on the boundary. Such solutions have been considered for simply connected domains by Clopeau, Mikelić, and Robert under the assumption that the initial vorticity is bounded, and by Lopes Filho, Nussenzveig Lopes, and Planas …
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure $p$ can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolut…
WE PRESENT THE RANDOM REPRESENTATIONS FOR THE NAVIER-STOKES VORTICITY EQUATIONS FOR AN INCOMPRESSIBLE FLUID IN A SMOOTH MANIFOLD WITH BOUNDARY AND REFLECTING BOUNDARY CONDITIONS FOR THE VORTICITY. WE SPECIALIZE OUR CONSTRUCTIONS TO R(n-1)xR+. WE EXTEND THESE CONSTRUCTIONS TO GIVE THE RANDOM REPRESENTATIONS FOR THE KINEMATIC DYNAMO PROBLEM OF MAGNETOHYDRODYNAMICS. WE CARRY OUT THESE INTEGRATIONS THROUGH THE APPLICATION OF THE METHODS OF STOCHASTIC DIFFERENTIAL GEOMETRY, I.E. THE GAUGE THEORY OF DIFFUSION PROCESSES ON SMOOTH MANIFOLDS.
A series of numerical experiments is suggested for the three-dimensional Navier-Stokes and Euler equations on a periodic domain based on a set of $L^{2m}$-norms of vorticity $Ω_{m}$ for $m\geq 1$. These are scaled to form the dimensionless sequence $D_{m}= (\varpi_{0}^{-1}Ω_{m})^{α_{m}}$ where $\varpi_{0}$ is a constant frequency and $α_{m} = 2m/(4m-3)$. A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of $D_{m}$ and $D_{m+1}$ while another is furnished by imposing a critical lower bound on $\int_{0}^{t}D_{m}\,dτ$. The behaviour of the $D_{…
As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analogue (enstrophy) in the inviscid case. With a large-scale forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency and self-similarity in the dynamical system structure.
Assuming that initial velocity and initial vorticity are bounded in the plane, we show that on a sufficiently short time interval the unique solutions of the Navier-Stokes equations converge uniformly to the unique solution of the Euler equations as viscosity approaches zero. We also establish a rate of convergence.
The issue of intermittency in numerical solutions of the 3D Navier-Stokes equations on a periodic box $[0,\,L]^{3}$ is addressed through four sets of numerical simulations that calculate a new set of variables defined by $D_{m}(t) = \left(\varpi_{0}^{-1}Ω_{m}\right)^{α_{m}}$ for $1 \leq m \leq \infty$ where $α_{m}= \frac{2m}{4m-3}$ and $\left[Ω_{m}(t)\right]^{2m} = L^{-3}\I |\bom|^{2m}dV$ with $\varpi_{0} = νL^{-2}$. All four simulations unexpectedly show that the $D_{m}$ are ordered for $m = 1\,,...,\,9$ such that $D_{m+1} < D_{m}$. Moreover, the $D_{m}$ squeeze together such that $D_{m+1}/D_…
We derive the vorticity equation for an incompressible fluid on a 2-dimensional surface with arbitrary topology embedded in 3-dimensional Euclidean space by using a tailored Clebsch parametrization of the flow. In the inviscid limit, we identify conserved surface energy and enstrophy, and obtain the corresponding noncanonical Hamiltonian structure. We then discuss the formulation of the diffusion operator on the surface by examining two alternatives. In the first case, we follow the standard approach for the Navier-Stokes equations on a Riemannian manifold and calculate the diffusion operator …
A new variational principle - extremizing the fixed frame kinetic energy under constant relative enstrophy - for a coupled barotropic flow - rotating solid sphere system is introduced with the following consequences. In particular, angular momentum is transfered between the fluid and the solid sphere through a modelled torque mechanism. The fluid's angular momentum is therefore not fixed but only bounded by the relative enstrophy, as is required of any model that supports super-rotation. The main results are: At any rate of spin $Ω$ and relative enstrophy, the unique global energy maximizer fo…
We establish local balance equations for smooth functions of the vorticity in the DiPerna-Majda weak solutions of 2D incompressible Euler, analogous to the balance proved by Duchon and Robert for kinetic energy in 3D. The anomalous term or defect distribution therein corresponds to the ``enstrophy cascade'' of 2D turbulence. It is used to define a rather natural notion of ``dissipative Euler solution'' in 2D. However, we show that the DiPerna-Majda solutions with vorticity in $L^p$ for $p>2$ are conservative and have zero defect. Instead, we must seek an alternative approach to dissipative sol…
Concentration of Stationary Measures onto Large Scales in the Fast-Advection Limit of the Stochastically Forced Two-Dimensional Navier-Stokes Equations
