Finite time singularities in a class of hydrodynamic models

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form ${\cal L}\sim\int k^\alpha|{\bf v_k}|^2d^3{\bf k}$ in 3D Fourier representation, where $\alpha$ is a constant, $0<\alpha< 1$. Unlike the case $\alpha=0$ (the usual Eulerian hydrodynamics), a finite value of $\alpha$ results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel vortex filaments and an analog of the Crow instability is found at small wave-numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like $(t^*-t)^{1/(2-\alpha)}$, where $t^*$ is the singularity time.Comment: LaTeX, 17 pages, 3 eps figures. This version is close to the journal pape

On formation of a locally self-similar collapse in the incompressible Euler equations

The paper addresses the question of existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the $L^p$-condition for velocity or vorticity and for a range of scaling exponents. In particular, in $N$ dimensions if in self-similar variables $u \in L^p$ and u \sim \frac{1}{t^{\a/(1+\a)}}, then the blow-up does not occur provided \a >N/2 or -1<\a\leq N/p. This includes the $L^3$ case natural for the Navier-Stokes equations. For \a = N/2 we exclude profiles with an asymptotic power bounds of the form |y|^{-N-1+\d} \lesssim |u(y)| \lesssim |y|^{1-\d}. Homogeneous near infinity solutions are eliminated as well except when homogeneity is scaling invariant.Comment: A revised version with improved notation, proofs, etc. 19 page

Application and Comparison of Two Different DNS Algorithms for Simulating Transition to Turbulence in Taylor-Green Vortex Flow

An all-Mach number, fully implicit, non-dissipative DNS algorithm and an incompressible, dissipative DNS algorithm were applied for simulating transition to turbulence in TGV flow to assess their behavior for this flow regime. The all-Mach number solver was developed and parallelized. A method was also adopted to remove oscillating pressure corrections in time. In order to compare the behavior of the algorithms, various flow diagnostics were calculated. The results were also compared to results given in the literature. The development of the flow and the peak structures show some differences due to different dissipative and energy conserving properties of the algorithms. However the physics of TGV flow are well captured by both, even though the grid is not fully resolved