Exploring symmetry plane conditions in numerical Euler solutions (original) (raw)

Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

Journal of Nonlinear Science, 2006

We study the interplay between the local geometric properties and the nonblowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition [15] [Phys. Fluids 5 (1993), 1725]. We use a pseudo-spectral method with resolution up to 1536×1024×3072 to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than doubly exponential in time, up to t = 19, beyond the singularity time t = 18.7 predicted by Kerr's computations [15], [18]. The velocity, the enstrophy, and the enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.

The Growth of Vorticity Moments in the Euler Equations

Procedia IUTAM, 2013

A new rescaling of the vorticity moments and their growth terms is used to characterise the evolution of anti-parallel vortices governed by the 3D Euler equations. To suppress unphysical instabilities, the initial condition uses a balanced profile for the initial magnitude of vorticity along with a new algorithm for the initial vorticity direction. The new analysis uses a new adaptation to the Euler equations of a rescaling of the vorticity moments developed for Navier-Stokes analysis. All rescaled moments grow in time, with the lower-order moments bounding the higher-order moments from above, consistent with new results from several Navier-Stokes calculations. Furthermore, if, as an inviscid flow evolves, this ordering is assumed to hold, then a singular upper bound on the growth of these moments can be used to provide a prediction of power law growth to compare against. There is a significant period where the growth of the highest moments converges to these singular bounds, demonstrating a tie between the strongest nonlinear growth and how the rescaled vorticity moments are ordered. The logarithmic growth of all the moments are calculated directly and the estimated singular times for the different Dm converge to a common value for the simulation in the best domain.

Development of high vorticity structures in incompressible 3D Euler equations

Physics of Fluids, 2015

We perform the systematic numerical study of high vorticity structures that develop in the 3D incompressible Euler equations from generic large-scale initial conditions. We observe that a multitude of high vorticity structures appear in the form of thin vorticity sheets (pancakes). Our analysis reveals the self-similarity of the pancakes evolution, which is governed by two different exponents e −t/T and e t/Tω describing compression in the transverse direction and the vorticity growth respectively, with the universal ratio T /T ω ≈ 2/3. We relate development of these structures to the gradual formation of the Kolmogorov energy spectrum E k ∝ k −5/3 , which we observe in a fully inviscid system. With the spectral analysis we demonstrate that the energy transfer to small scales is performed through the pancake structures, which accumulate in the Kolmogorov interval of scales and evolve according to the scaling law ω max ∝ −2/3 for the local vorticity maximums ω max and the transverse pancake scales .

Vortex events in Euler and Navier–Stokes simulations with smooth initial conditions

We present high-resolution numerical simulations of the Euler and Navier-Stokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the Navier-Stokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time (t s ) with scaling |u| ∞ ∼ (t s − t) −1/2 , |ω| ∞ ∼ (t s − t) −1 . This blow-up is associated with the formation of a k −3 spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward t s , the total enstrophy is observed to increase at a slower rate, Ω ∼ (t s − t) −3/4 , than would naively be expected given the behaviour of the maximum vorticity, ω ∞ ∼ (t s − t) −1 . This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various Re, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid t s . In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching k −5/3 . The simulations show that the peak value of the enstrophy scales as Re 3/2 , which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of Re, supporting the validity of Kolmogorov's law of finite energy dissipation. At later times the kinetic energy shows a t −1.2 decay for all Re, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to t s , large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a k −5/3 range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space.

Velocity and scaling of collapsing Euler vortices

2006

New analysis of the scaling structure of a numerical solution of the Euler equations finds that initially anti-parallel vortex tubes collapse into two wings whose cross-sections can be described using two length scales ρ and R. The first ρ ∼ (T − t) for the leading edge and the distance between the position of peak vorticity and the dividing plane. The second R ∼ (T −t) 1/2 describes the extent of the wings and the distance of the peak in vortical velocity sup x |v| from the peak in vorticity. All measures of singular growth within the inner region give the same singular time. This includes a blowup in the peak of vortical or axial velocity going as (T − t) −1/2 at a distance R from the position of ω ∞ . Outside this self-similar region, energy, enstrophy, circulation and helicity accumulate. Twisting of vortex lines consistent with vortex line length growing to infinity is observed in the outer region. Vorticity in the intermediate zone between the inner and outer regions, while no longer growing at the singular rate, could be the major source of the strain interactions that drive the flow.

Vorticity alignment results for the three-dimensional Euler and Navier - Stokes equations

Nonlinearity, 1997

We address the problem in Navier-Stokes isotropic turbulence of why the vorticity accumulates on thin sets such as quasi-one-dimensional tubes and quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst, Kerstein, Kerr and Gibbon, who observed that the vorticity vector ω aligns with the intermediate eigenvector of the strain-matrix S, we study this problem in the context of both the three-dimensional Euler and Navier-Stokes equations using the variables α =ξ · Sξ and χ =ξ × Sξ whereξ = ω/ω. This introduces the dynamic angle φ(x, t) = arctan( χ α ), which lies between ω and Sω. For the Euler equations a closed set of differential equations for α and χ is derived in terms of the Hessian matrix of the pressure P = {p ,ij }. For the Navier-Stokes equations, the Burgers vortex and shear-layer solutions turn out to be the Lagrangian fixed-point solutions of the equivalent (α, χ) equations with a corresponding angle φ = 0. Under certain assumptions for more general flows it is shown that there is an attracting fixed point of the (α, χ) equations which corresponds to positive vortex stretching and for which the cosine of the corresponding angle is close to unity. This indicates that near alignment is an attracting state of the system and is consistent with the formation of Burgers-like structures.

Extreme vortex states and the growth of enstrophy in three-dimensional incompressible flows

Journal of Fluid Mechanics, 2017

In this investigation we study extreme vortex states defined as incompressible velocity fields with prescribed enstrophy mathcalE0{\mathcal{E}}_{0}mathcalE0 which maximize the instantaneous rate of growth of enstrophy textdmathcalE/textdt\text{d}{\mathcal{E}}/\text{d}ttextdmathcalE/textdt . We provide an analytic characterization of these extreme vortex states in the limit of vanishing enstrophy mathcalE0{\mathcal{E}}_{0}mathcalE0 and, in particular, show that the Taylor–Green vortex is in fact a local maximizer of textdmathcalE/textdt\text{d}{\mathcal{E}}/\text{d}ttextdmathcalE/textdt in this limit. For finite values of enstrophy, the extreme vortex states are computed numerically by solving a constrained variational optimization problem using a suitable gradient method. In combination with a continuation approach, this allows us to construct an entire family of maximizing vortex states parameterized by their enstrophy. We also confirm the findings of the seminal study by Lu & Doering (Indiana Univ. Math. J., vol. 57, 2008, pp. 2693–2727) that these extreme vortex states saturate (up to ...

Geometric Properties and Nonblowup of 3D Incompressible Euler Flow

Communications in Partial Differential Equations, 2005

By exploring a local geometric property of the vorticity field along a vortex filament, we establish a sharp relationship between the geometric properties of the vorticity field and the maximum vortex stretching. This new understanding leads to an improved result of the global existence of the 3D Euler equation under mild assumptions.

Dynamics of Scaled Norms of Vorticity for the Three-dimensional Navier-Stokes and Euler Equations

Procedia IUTAM

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 ≥ 1. These are scaled to form the dimensionless sequence D m = (ϖ −1 0 Ω m) α m where ϖ 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 t 0 D m dτ. 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.