A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation
type: Item
instance of: scholarly article
title: A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation
zbMATH Open document ID: 0899.76106
published in: Computational Mechanics
publication date: 1997-09-09T00:00:00Z
review text: The authors consider the problem of hydrodynamic stability in incompressible flows around objects that move in a fluid which is initially at rest. The problem is formulated in terms of vorticity-stream function \(\zeta\). Let \(F\) be a bounded domain of a plane (solid region) with the boundary \(\Gamma\), \(\Omega\) denotes the region outside \(\Gamma\) (fluid region). If \(\vec v\) denotes the velocity of the fluid, then \(\vec v\) and \(\zeta\) satisfy the system \(\frac{\partial \zeta}{\partial t}= -u\cdot\nabla \zeta+ \frac 1{\text{Re}}\nabla^2\zeta\), \(x\in\Omega\), \(t>0\), \(\vec v(x,t)=\vec k\times\int\limits_{\Omega}\nabla G(x,y)\zeta(y,t) dy\), \(x\in\Omega\), \(t>0\) with boundary condition \(\vec n\cdot[\vec k\times\int_{\Omega}\nabla G(x,y)\zeta(y,t) dy]=\vec n\cdot\vec v|_B\), \(x\in\Gamma\). Here \(G(x,y)=\frac 1{2\pi}\ln | x-y|\) is the fundamental solution of the two-dimensional Laplace operator, \(\vec k\) is the unit normal to the plane of the motion, \(\vec n\) is the unit vector to the boundary. The authors discretize the problem, reduce it to a dynamical system and study the stability of the solution of the system as the Reynolds number (Re) is increased. A numerical result on the problem of two-dimensional flows around a circular cylinder is presented.
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76E99
zbMATH DE Number: 1058793
zbMATH Keywords: nonstationary Navier-Stokes equations
zbMATH Keywords: fundamental solution
zbMATH Keywords: Laplace operator
zbMATH Keywords: dynamical system
zbMATH Keywords: circular cylinder
author: C. Cossu
author: Luigi Morino
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: s004660050243
OpenAlex ID: W2126003299
DOI: 10.1007/S004660050243
description: scientific article; zbMATH DE number 1058793