Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation

Abstract

We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as 'tipping points', is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems

Continuation of probability density functions using a generalized Lyapunov approach

“…In practice, one has to provide the Jacobian matrix A of a deterministic fixed point and the matrix B representing the stochastic forcing in order to solve for the stationary covariance matrix C. In a matrix-based pseudo-arclength continuation method the Jacobian is available since a Newton-Raphson method is used [8]. The mass matrix M is readily available from the model equations.…”

“…The different methods are distinguished by whether the Jacobian matrix is explicitly computed ('matrix-based' methods) or whether matrix vector products of this matrix are used ('matrix-free' methods). The applications shown in [8] range from traditional flows and free-surface flows to magneto-hydrodynamic flows and ocean flows.…”

“…Recently, an overview [8] was given of numerical techniques for computing fixed points and periodic orbits for systems of partial differential equations (PDEs). These PDEs are discretized by any spectral, finite difference or finite element method.…”

Bifurcation Tracking for High Reynolds Number Flow Around an Airfoil

“…These have predominately been aimed at low-dimensional, enclosed flows, such as Taylor-Couette flow, of which a review was conducted by Cliffe et al [2001]. More recently continuation methods have been applied to larger fluid dynamics systems but with the main interest still being well-conditioned problems as the review by Dijkstra et al [2014] highlights. The studies by Wales [Wales, 2010;Wales et al, 2012b,a] have shown that continuation methods can be applied to high Reynolds number external flows, such as is of interest in this work.…”

Numerical computation of bifurcations in large equilibrium systems in matlab

“…[2][3][4][5] and references there. The review article [6] provides an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems.…”