A class of exact Navier–Stokes solutions for homogeneous flat-plate boundary layers and their linear stability

Abstract

We introduce a new boundary layer formalism on the basis of which a class of exact solutions to the Navier-Stokes equations is derived. These solutions describe laminar boundary layer flows past a flat plate under the assumption of one homogeneous direction, such as the classical swept Hiemenz boundary layer (SHBL), the asymptotic suction boundary layer (ASBL) and the oblique impingement boundary layer. The linear stability of these new solutions is investigated, uncovering new results for the SHBL and the ASBL. Previously, each of these flows had been described with its own formalism and coordinate system, such that the solutions could not be transformed into each other. Using a new compound formalism, we are able to show that the ASBL is the physical limit of the SHBL with wall suction when the chordwise velocity component vanishes while the homogeneous sweep velocity is maintained. A corresponding non-dimensionalization is proposed, which allows conversion of the new Reynolds number definition to the classical ones. Linear stability analysis for the new class of solutions reveals a compound neutral surface which contains the classical neutral curves of the SHBL and the ASBL. It is shown that the linearly most unstable Görtler-Hämmerlin modes of the SHBL smoothly transform into Tollmien-Schlichting modes as the chordwise velocity vanishes. These results are useful for transition prediction of the attachment-line instability, especially concerning the use of suction to stabilize boundary layers of swept-wing aircraft.

Secondary instability and subcritical transition of the leading-edge boundary layer

“…With κ denoting its non-dimensional strength, wall suction is known to increase both the linear critical Reynolds number Re crit (κ) (Hall et al 1984) and the transition Reynolds number Re tr (κ) (Spalart 1988;Arnal et al 1997). This behaviour was linked to the linear stability theory of a broader class of homogeneous flat-plate boundary layers by John, Obrist & Kleiser (2012, 2014a. In particular, the SHBL with suction turns into the highly stable asymptotic effective in leading to transition, as demonstrated for Tollmien-Schlichting (TS) eigenmodes interacting with cross-flow disturbances (Bippes 1989;Meyer & Kleiser 1989;Wintergerste & Kleiser 1995;Bippes 1999;Wintergerste 2002), for travelling cross-flow vortices (Wassermann & Kloker 2003) or for two streaks interacting with each other (Brandt & de Lange 2008).…”

Secondary Instability and Subcritical Transition in the Leading-Edge Boundary Layer

“…With κ denoting its non-dimensional strength, wall suction is known to increase both the linear critical Reynolds number Re crit (κ) (Hall et al 1984) and the transition Reynolds number Re tr (κ) (Spalart 1988;Arnal et al 1997). This behaviour was linked to the linear stability theory of a broader class of homogeneous flat-plate boundary layers by John, Obrist & Kleiser (2012, 2014a. In particular, the SHBL with suction turns into the highly stable asymptotic suction boundary layer (ASBL) in the limit of vanishing chordwise flow, e.g.…”