Hydrodynamic stability in undulated channels

Nektar++ has been used to study various forms of hydrodynamic instabilities arising is a flow through a channel with corrugated walls. Stability analysis has been performed using direct numerical simulation and tracking growth, or attenuation of the unstable modes.

Streamwise vorticity component, illustrating stationary vortices resulting from saturation of the unstable mode.
Streamwise vorticity component, illustrating stationary vortices resulting from saturation of the unstable mode.

We ware able to show that, in the case of the large scale wall undulations applied parallel to the streamwise direction (lines of constant wall elevation are perpendicular to the flow) for a range of geometries, the centrifugal instability, caused by the streamline curvature, dominates the stability of the flow, preventing the onset of travelling waves. Our findings have been presented in a JFM paper1.

Particle trajectories in the saturated flow field
Particle trajectories in the saturated flow field

By changing the flow direction, such that lines of constant wall elevation become parallel to the flow, the flow becomes susceptible to destabilization at a much lower values of the Reynolds number (~60). Interestingly such configuration offers some drug reduction in relation to the plain Poiseuille flow. With the use of direct numerical simulation we ware able to show that the nonstationary flow, resulting from nonlinear saturation of the unstable mode retains drag reducing property for a range of over the critical values of the Reynolds number. We have reported our findings here2.

Four vortices formed as a result of the saturation of the unstable mode. Isosurfaces of the instantaneous Q-cryterion.
Four vortices formed as a result of the saturation of the unstable mode. Isosurfaces of the instantaneous Q-cryterion.
Variation of the critical Reynolds number Re_{cr} (solid lines) and corresponding phase speed v_{p}=\frac{\sigma_{r}}{\beta_{cr}} (dashed lines) as functions of channel corrugation wavenumber \alpha and amplitude S. Lines of constant flow rate related to the smooth channel reference (dotted lines) determine drag reducing zone, on the left of the \frac{Q}{Q_{r}}=1 line and drag increasing zone is on the right.

References:

1
Gepner, S. W., and Floryan, J. M., “Flow dynamics and enhanced mixing in a converging–diverging channel,” Journal of Fluid Mechanics, vol. 807, Oct. 2016, pp. 167–204. [Source]
2
Yadav, N., Gepner, S. W., and Szumbarski, J., “Instability in a channel with grooves parallel to the flow,” Physics of Fluids, vol. 29, Aug. 2017, p. 084104. [Source]