We investigate computationally whether a class of self-similar solutions of the Navier-Stokes equations in infinite channels driven by accelerating or decelerating walls, arises anywhere in a channel restricted to finite length. The self-similar solutions satisfy a nonlinear partial differential equation involving time and the vertical coordinate and have been studied previously. Admissible self-similar solutions include stable and unstable steady branches, time-periodic branches emerging from Hopf bifurcations, as well as chaotic solutions following a period-doubling Feigenbaum cascade. The objective here is to explore whether such solutions emerge when restricted to finite, although long, channels. The problem is addressed numerically using fast algorithms to solve the Navier-Stokes equations. It is established that all branches of the self-similar solutions (including time-periodic ones) can occur in finite channels provided the Reynolds numbers are not too high (approximately 500 for accelerating and 33 for decelerating wall flows, respectively). As the Reynolds number increases it becomes more difficult to recover the self-similar branches with general inflow conditions, and this has been overcome by utilizing the self-similar solutions as inflow Dirichlet conditions. At sufficiently high Reynolds numbers, the self-similar inflow conditions are incapable of producing the exact solution in the interior, but instead new steady or time-periodic states emerge which depend on the Reynolds number and the channel length. It is established numerically by solving the linearized Navier-Stokes equations that such phenomena are due to a spatial instability of the self-similar states to perturbations which are not of self-similar form. The results indicate that caution must be exercised in drawing conclusions from a stability theory restricted to self-similar perturbations and examples are given where such analysis is erroneous. In the case of decelerating wall flows the self-similar solutions are recovered on finite domains but for a much smaller range of Reynolds numbers as compared to accelerating wall flows. The results also show that time-periodic self-similar states are possible in finite channels (the Navier- Stokes equations are solved subject to the spatiotemporal self-similar solution at inflow) for accelerating wall flows but not for decelerating ones. Of special interest are computed aperiodic states of the Navier-Stokes equations in the case of accelerating walls and relatively high Reynolds numbers. These solutions behave intermittently and bounce between long-lived states that are congruent to the self-similar solutions to non-self-similar aperiodic ones. This is a complicated phenomenon due to the dimensionality and nonlinear nature of the system but the results indicate that the self-similar states can be strong attractors in some regions of the phase space.
Bibliographical noteFunding Information:
L.E. acknowledges graduate student support from the National Science Foundation and NJIT. D.T.P. was supported in part by the National Science Foundation under Grant No. DMS-0707339.