An unstructured spectral element method for hydrodynamic instability and bifurcation analysis

A high order unstructured spectral element method with a domain decomposition Stokes solver is presented for the hydrodynamic instability and bifurcation analysis. A Jacobian-Free Inexact-Newton-Krylov algorithm with a Stokes time-stepping preconditioning technique is introduced for the steady-state solution of incompressible flow. A matrix-free arc-length approach with Householder transformation is used for the numerical continuation near a turning point. An Arnoldi method is utilized to calculate the leading eigenvalues and their corresponding eigenvectors for the big system of linearized incompressible Navier-Stokes equations, which are responsible for initiating the hydrodynamic instability. The new method can do the steady and unsteady simulations in the similar way without time-splitting divergence error, it do not form the Jacobian matrix, which can reduce the memory allocation, decrease the computation cost, and speed up the convergence rate. The symmetric-breaking Hopf and Pitchfork bifurcations are considered in the flow passed a circular cylinder between two parallel plates. An antisymmetric sinusoidal velocity driven cavity problem is considered and the stable and unstable patterns are analyzed by checking the leading eigenvalues of their steady states. Besides the stable patterns of steady symmetric and steady asymmetric solutions£a new pair of unsteady asymmetric solutions are found depending on the different initial conditions.A high order unstructured spectral element method with a domain decomposition Stokes solver is presented for the hydrodynamic instability and bifurcation analysis. A Jacobian-Free Inexact-Newton-Krylov algorithm with a Stokes time-stepping preconditioning technique is introduced for the steady-state solution of incompressible flow. A matrix-free arc-length approach with Householder transformation is used for the numerical continuation near a turning point. An Arnoldi method is utilized to calculate the leading eigenvalues and their corresponding eigenvectors for the big system of linearized incompressible Navier-Stokes equations, which are responsible for initiating the hydrodynamic instability. The new method can do the steady and unsteady simulations in the similar way without time-splitting divergence error, it do not form the Jacobian matrix, which can reduce the memory allocation, decrease the computation cost, and speed up the convergence rate. The symmetric-breaking Hopf and Pitchfork bifurcations ...