Mathematics Colloquia and Seminars

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We have created a robust numerical method for calculating fully three dimensional, time dependent non-Newtonian flows particularly where inertial forces are important. We have used our unstructured, finite-volume code to compute a wide variety of viscoelastic flows over a large range of Reynolds (Re) and Weissenberg (Wi) numbers. The method is based on a continuum implementation of the FENE-P constitutive model to describe the flow of dilute polymeric solutions, but the algorithm can be used for wide range of differential constitutive equations. This formulation thus allows for the investigation of the flow of high molecular weight polymers (L up to 100 in the FENE-P model where L is the polymer length in number of Kuhn steps) as well as high polymer relaxation times (Wi on the order of 100).

I will present the use of the code to examine the viscoelastic effects on the inertial wake in flow past bluff bodies, with the ultimate goal of examining transition to turbulence. First, full simulations were performed at Reynolds numbers of both 300 and 3900, and in each case significant stabilization of structures typically seen in Newtonian flows is seen. At Re = 300, the characteristic Newtonian mode A and mode B instabilities can be either be weakened or completely suppressed based on the polymer extensibility L – an effect which has been observed experimentally. Furthermore, at Re = 3900, even a small concentration of low extensibility polymers has the ability to stabilize the shear layer (which has transitioned for pure Newtonian flow), and revert the wake structure back to one resembling the mode B instability, a state seen in Newtonian flows at much lower Reynolds numbers.

In an effort to better quantify this stabilizing nature of viscoelasticity, a Floquet linear stability analysis was also performed for the time-periodic, viscoelastic cylinder wake. The aforementioned code was further modified in a way that allows for the solution of the linear FENE-P equations with a given perturbation spanwise wavenumber, and instability growth rates are therefore obtained for different specified perturbation forms (both amplitude and spanwise wavenumber). Results from the measurements of these perturbation growth rates and their comparison with the full simulations will be presented and discussed.