A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation
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.
scientific article; zbMATH DE number 1058793