Spectral methods for the viscoelastic time‐dependent flow equations with applications to Taylor–Couette flow

The time evolution of finite amplitude axisymmetric perturbations (Taylor cells) to the purely azimuthal, viscoelastic, cylindrical Couette flow was numerically simulated. Two time integration numerical methods were developed, both based on a pseudospectral spatial approximation of the variables, efficiently implemented using fast Poisson solvers and optimal filtering routines. The first method, applicable for finite Re numbers, is based on a time‐splitting integration with the divergence‐free condition enforced through an influence matrix technique. The second one, is based on a semi‐implicit time integration of the constitutive equation with both the continuity and the momentum equations enforced as constraints. Stability results for an upper convected Maxwell fluid were obtained for the supercritical bifurcations, either steady or time‐periodic, developed after the onset of instabilities in the primary flow. At small elasticity values, ϵ ≡ De / Re , the time integration of finite amplitude disturbances confirms the stability of the single branch of steady Taylor cells. At intermediate ϵ values the rotating wave family of time‐periodic solutions developed at the onset of instability is stable, whereas the standing wave is found to be unstable. At high ϵ values, and in particular for the limit of creeping flow (ϵ = ∞), the present study shows that the rotating wave family is unstable and the standing (radial) wave is stable, in agreement with previous finite‐element investigations. It is thus shown that spectral techniques provide a robust and computationally efficient method for the simulation of complex, non‐linear, time‐dependent viscoelastic flows.