Travelling waves in a straight square duct

Isothermal, incompressible flow in a straight duct with square cross-section is known to be linearly stable [1]. Direct numerical simulation, on the other hand, has revealed that turbulence in this geometry is self-sustained above a Reynolds number value of approximately 1100, based on the bulk velocity and the duct half-width [2]. Numerous non-linear equilibrium solutions have already been identified in plane Couette, plane Poiseuille and pipe flows [3, 4, 5], and their role in the transition process as well as their relevance to the statistics of turbulent flow have been investigated [6, 7, 8]. No non-linear travelling-wave solutions for the flow through a square duct have been published to date. In the specific case of square duct flow, it can be anticipated that travelling wave solutions will help to shed further light on the origin of mean secondary flow, whose appearance has been linked to the near-wall coherent structures [2]. Here we will present results obtained by applying an iterative solution strategy to the steady Navier-Stokes equations in a moving frame of reference. In the absence of a “natural” primary bifurcation point, we resort to the method proposed by Waleffe [9], where streamwise vortices are artificially added to the base flow and forced against viscous decay, leading to streaks, which are in turn linearly unstable, feeding back into the original vortices. The non-linear solution is then continued back to the original problem, i.e. the unforced flow.