Onset of two-dimensional cellular flow in finite curved channels of large aspect ratio

Abstract

The development of streamwise-oriented, symmetric, two-dimensional vortices in curved channels of large aspect ratios is studied near the threshold of the first centrifugal instability. The nonlinear equations of motion governing the two-dimensional, stationary flows are solved numerically over a range of parameter values. The dynamical parameter is the normalized streamwise pressure gradient defined as ε=[(∂p/∂x)−(∂p/∂x)c]/(∂p/∂x)c, where (∂p/∂x)c is the critical value for the infinite geometry. The development of interior cells and the selection of the wavelength as ε is gradually increased through zero is quite different from that observed in Taylor vortex flow. For pressure gradients of 0.5% and 1.0% (ε=0.005, 0.01) above critical, the interior cells begin to grow spontaneously and are strongest in the middle of the channel. Unlike the interior cells in Taylor vortex flow, they are only weakly coupled to the end cells. The end cells (or Ekman vortices) are also found to be anomalously long. As ε is increased further to 0.04 and 0.07, the amplitude and wavelength of the interior cells are more uniform. There is, however, a complex interaction between the Ekman vortices and their neighboring interior cells, often resulting in the formation of additional cells in that region. Next, a simple Ginzburg–Landau (GL) model is tested for weakly nonlinear, two-dimensional vortices. The coefficient in the steady form of this equation is evaluated for a wide parameter range using high accuracy calculations of infinite aspect ratio neutral stability curves. (When suitably normalized, neutral stability curves are found to vary only a little with radius ratio.) For large aspect ratio curved channels, predictions from the model are compared with results from numerical simulation. The variation with ε of vortex amplitude near the center of the duct is correctly predicted by the Ginzburg–Landau equation. For given ε, however, agreement of the spanwise variation of vortex amplitude and spacing between the numerical simulation and the model is not obtained. The development of consistent amplitude equations and boundary conditions that link the interior flow to the boundary is expected to be a challenging task.