The objective of this work is to determine how the addition of small amounts of polymer affects the sequence of hydrodynamic instabilities leading to turbulence in free shear layers. The FENE-P constitutive equation is chosen to describe the fluid rheology, and a Hartley transform-based pseudo-spectral method is applied to the coupled, nonlinear system of partial differential equations governing the fluid vorticity and the FENE-P dumbbell configuration. Both two- and three-dimensional numerical simulations are performed using a time-developing mixing layer model. In the 2D simulations, the roll-up instability is studied for values of the elasticity number, E (the ratio of the fluid relaxation time to the characteristic time for vorticity diffusion), up to 0 (1), and over a range of the maximum dumbbell extensibility, b. For sufficiently small values of E and b, a viscoelastic quasi-steady state analogous to the Newtonian one is found, with no significant changes to the streamlines. For E = 0 (1) and sufficiently large b, the fundamental is unable to dominate the flow, and the roll-up process is inhibited. The underlying mechanism involves the generation of higher harmonics by large polymer stress gradients which serve as a vorticity source/sink. The higher harmonics then resist the tendency of the fundamental to concentrate the vorticity into a single core region. In the 3D simulations, spanwise perturbations are introduced to the 2D quasi-steady states, and exponential growth rates for the resulting secondary instability are computed. For low values of E and b, the viscoelastic quasi-steady states are more unstable to 3D perturbations than the Newtonian quasi-steady states, while for large enough values of E and b, they may become less unstable. The primary mechanism responsible for the differences in the Newtonian and viscoelastic 3D growth rates is a distortion to the shape of the 2D vortices. These observations indicate that for large enough values of E and b, the addition of polymers may inhibit the instabilities leading to turbulence in free shear layers, and suggest a plausible mechanism through which polymers reduce small scale turbulence in turbulent flows.
Bibliographical noteFunding Information:
We thank Dr Jalel Azaiez for helpful discussions regarding the 2D simulations. This research was supported by a grant from the United States Department of Energy, Office of Basic Energy Sciences. Satish Kumar also received support through a Graduate Fellowship from the National Science Foundation and a Kodak Fellowship from the Eastman Kodak Company.
Copyright 2004 Elsevier Science B.V., Amsterdam. All rights reserved.
- Free shear layers
- Mixing layers
- Numerical simulation
- Viscoelastic fluid mechanics