Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.

Singularities of Euler Flow? Not Out of the Blue!

Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.(A version of the paper with higher-quality figures is available at this http URL)

Singularities of Euler flow? Not out of the blue!

The three‐dimensional (3‐D) time evolution of a high‐symmetry initial condition [J. Phys. Soc. Jpn. 54, 2132 (1985)] is simulated using a Fourier pseudospectral method for Re=1/ν=500, 1000, 2000, and 5000 with an effective resolution of 10243 collocation points (1713 independent modes, maximum wave number kmax=340). It is found that much before the peak enstrophy is reached, there is a short interval when the local quantities increase sharply. It is also found that during this interval, six vortex dipoles (at the origin) and three dipoles (at the π/2 corner) collapse toward two separate vorticity null points at the opposite corners of the domain in a nearly self‐similar fashion. The coherent vortices break up afterward, followed by a sharp decrease in local quantities. The singularity analysis shows that, within the limits of the resolution, the maximum vorticity scales approximately as (T−Tc)−1, shortly before the breakup. However, the increase in peak vorticity stops at a certain time, possibly due to viscous dissipation effects. The temporal evolution of the width of the analyticity strip shows that δ approaches zero at a rate faster than exponential, but reaches a minimum value and starts to increase. This suggests that the solution remains uniformly analytic, as is the case in the viscous Burgers equation.

Direct numerical simulation of transition to turbulence from a high-symmetry initial condition

Discrete group theory is applied to certain well-known flows with discrete symmetries. A flow is associated with particular discrete point or space group if flow components possess the same symmetries as the irreducible representations of the group. The flows examined are or have been considered candidates for finite-time blowup. The properties of the groups are related to the characteristics of the flows.

Discrete groups, symmetric flows and hydrodynamic blowup

We briefly review our previous work, high-Reynolds number simulations of a high-symmetry flow, and the evidence found therein for a singularity in incompressible, inviscid flow. We show the physical picture of a self-similar collapse of a 12-vortex structure during transition. We find that vortex reconnection plays an important role in the alteration of the topology of vortex lines and in the development of a turbulent flow. We also present an analysis of the behavior of third-order velocity derivatives at the origin, the center of the collapse.

On a possible Euler singularity during transition in a high-symmetry flow

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

R. B. Pelz

Author Tools

Chat about Author

Papers

On a possible Euler singularity during transition in a high-symmetry flow