Spanwise pairing of finite-amplitude longitudinal vortex rolls in inclined free-convection boundary layers

Abstract

Buoyancy-driven flow on a heated inclined plate can become unstable to static longitudinal roil instability at a critical distance, measured by $\tilde{R}_{\rm c}$, from the leading edge. Experiments in water by Sparrow & Husar (1969) indicate that these rolls undergo a second transition further downstream such that adjacent rolls merge and their spanwise wavelength is doubled. We study this secondary bifurcation phenomenon here with a set of model equations by first constructing the full eigenspectrum and eigenfunctions with a Chebyshev–Tau spectral method and then deriving the pertinent amplitude equations. By stipulating that the dimensional cross-stream wavelength of the rolls remains constant beyond $\tilde{R}_{\rm c}$, which is consistent with experimental observation, we show that the finite-amplitude primary rolls are destabilized by the ½ subharmonic mode at another critical distance $\tilde{R}_{\frac{1}{2}}$ from the edge. This ½ mode is shown to have an asymmetric spatial phase shift of ½π relative to the original 1 mode of the primary rolls, thus explaining the unique dislocation of tracer streaks after the rolls coalesce in the experiments. Also consistent with experimental observation is the theoretical result that the merged rolls are annihilated downstream by a saddle-node bifurcation before further wavclength doubling can occur. Simple amplitude criteria and critical distances from the leading edge for the various transitions are derived and compared to experimental values.