On the Singular Nature of Turbulent Boundary Layers

Abstract

An estimate is derived for the rate at which, with increasing Reynolds number, the vorticity in turbulent boundary layers is confined to a diminishing fraction of the overall flow domain. For laminar boundary layers this rate is reflected in the self-similar coordinate stretching that determines the Reynolds number scalings for boundary layer growth and skin friction. For the turbulent boundary layer this rate is shown to also derive from an underlying similarity structure. An accounting of the magnitude ordering of terms in the mean dynamical equation for the turbulent boundary layer reveals a four layer structure. This structure forms during the transitional regime, persists for all subsequent Reynolds numbers, and provides a framework for describing the evolution of the boundary layer vorticity and momentum fields. Multiscale analyses that exploit the four layer ordering reveal that two kinds of self-similarities are formally admitted. With increasing Reynolds number, these are shown to be associated with two kinds of scale-separation between the motions characteristic of the velocity and vorticity fields. One pertains to the near-wall spatial confinement of the vorticity field owing to vorticity stretching, and the other pertains to the advective transport of decreasingly smaller scale vortical motions over a domain that approaches the overall flow width as the Reynolds number becomes large. The scalings associated with the self-similar structure indicate that slightly greater than 50% of the total vorticity content is, with increasing Reynolds number, confined to a near-wall layer of diminishing thickness, with the remainder attributable to a domain that approaches the total layer thickness. Within the larger domain at least 50% of the vorticity is concentrated in narrow vortical fissures that also decrease in relative scale with Reynolds number. Spanwise vorticity measurements in laboratory boundary layers and the atmospheric surface layer are shown to be in agreement with the theoretical predictions, but also provide evidence that at sufficiently high Reynolds number, the vortical fissures develop an intermittent internal structure. Collectively, these results indicate that, on average at any given instant, at least 75% of the boundary layer circulation (per unit length) is confined to a region that diminishes like

as the boundary layer Reynolds number, δ ⁺= δ uτ/ν →∞.