On the Effect of Small Steps on Pre-transitional Streamwise Elongated Streaks

Abstract

It is generally agreed that in the process of transition from laminar to turbulent flow in boundary layers at moderately high freestream turbulence, the streamwise elongated streaks play an important role. These streaks may evolve inside boundary layer flows over flat plates or curved surfaces when the amplitudes of freestream disturbances exceed a certain level, or as a result of the imbalance between centrifugal effects and radial pressure gradients, respectively. In this paper, we study the effect of small surface steps on the transient growth associated with streamwise elongated streaks. The question that we aim at answering here is how do such surface imperfections impact the onset of secondary instabilities? These short-scale instabilities are thought to be the precursor to streak breakdown and there continues to remain many unanswered questions about their basic properties. To excite the streaks, we introduce a spanwise periodic distribution of localized surface roughness elements followed by a portion of concave surface in the downstream. Two step configurations of forward and backward facing types, and three 'streak strengths' (quantified by the difference between the maximum and minimum of the high-and low-speed regions, respectively) are considered. We numerically solve the boundary region equations subject to upstream boundary conditions as derived in Goldstein et al. 11,12 The generalized Rayleigh pressure equation is solved as an eigenvalue problem to determine the growth rates associated with the secondary instabilities. It is found that as the height of the step is increased beyond 0.5 of boundary layer displacement thickness, the growth rate reduces at the peak streamwise wavenumber of the instability. It appears that this reduction is greater for forward step configuration for all considered cases of 'streak strengths'.