Dynamics of scaled norms of vorticity for the three-dimensional Navier-Stokes and Euler equations

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 $\Omega_{m}$ for $m\geq 1$. These are scaled to form the dimensionless sequence $D_{m}= (\varpi_{0}^{-1}\Omega_{m})^{\alpha_{m}}$ where $\varpi_{0}$ is a constant frequency and $\alpha_{m} = 2m/(4m-3)$. 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 $\int_{0}^{t}D_{m}\,d\tau$. The behaviour of the $D_{m}$ is also important in the Euler case in suggesting a method by which possible singular behaviour might also be tested.Comment: To appear in the Procedia IUTAM volume of papers Topological Fluid Dynamic

Quasi-conservation laws for compressible 3D Navier-Stokes flow

We formulate the quasi-Lagrangian fluid transport dynamics of mass density $\rho$ and the projection q=\bom\cdot\nabla\rho of the vorticity \bom onto the density gradient, as determined by the 3D compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of $q$ cannot cross a level set of $\rho$. That is, in this formulation, level sets of $\rho$ (isopychnals) are impermeable to the transport of the projection $q$.Comment: 2 page note, to appear in Phys Rev

Stretching and folding processes in the 3D Euler and Navier-Stokes equations

Stretching and folding dynamics in the incompressible, stratified 3D Euler and Navier-Stokes equations are reviewed in the context of the vector \bdB = \nabla q\times\nabla\theta where q=\bom\cdot\nabla\theta. The variable $\theta$ is the temperature and \bdB satisfies \partial_{t}\bdB = \mbox{curl}\,(\bu\times\bdB). These ideas are then discussed in the context of the full compressible Navier-Stokes equations where $q$ takes the two forms q = \bom\cdot\nabla\rho and q = \bom\cdot\nabla(\ln\rho).Comment: UTAM Symposium on Understanding Common Aspects of Extreme Events in Fluid

Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics

After a review of the isentropic compressible magnetohydrodynamics (ICMHD) equations, a quaternionic framework for studying the alignment dynamics of a general fluid flow is explained and applied to the ICMHD equations.Comment: 12 pages, 2 figures, submitted to a Focus Issue of New Journal of Physics on "Magnetohydrodynamics and the Dynamo Problem" J-F Pinton, A Pouquet, E Dormy and S Cowley, editor

Conditional regularity of solutions of the three dimensional Navier-Stokes equations and implications for intermittency

Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on $L^{2m}$-norms of the vorticity, denoted by $\Omega_{m}(t)$, and particularly on $D_{m} = \Omega_{m}^{\alpha_{m}}$, where $\alpha_{m} = 2m/(4m-3)$ for $m\geq 1$. The first result, more appropriate for the unforced case, can be stated simply : if there exists an $1\leq m < \infty$ for which the integral condition is satisfied ($Z_{m}=D_{m+1}/D_{m}$) $$\int_{0}^{t}\ln (\frac{1 + Z_{m}}{c_{4,m}}) d\tau \geq 0$$ then no singularity can occur on $[0, t]$. The constant $c_{4,m} \searrow 2$ for large $m$. Secondly, for the forced case, by imposing a critical \textit{lower} bound on $\int_{0}^{t}D_{m} d\tau$, no singularity can occur in $D_{m}(t)$ for \textit{large} initial data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive $\int_{0}^{t}D_{m} d\tau$ over this critical value can be ruled out whereas other types cannot.Comment: A frequency was missing in the definition of D_{m} in (I5) v3. 11 pages, 1 figur