On the evolution of the turbulent spot in a laminar boundary layer with a favourable pressure gradient

Abstract

The evolution of a turbulent spot in an accelerating laminar boundary-layer flow was investigated. The type of boundary layer chosen for this experiment resembles in every respect the flow in the vicinity of a stagnation point theoretically described by Falkner and Skan. The rate of growth of the spot was significantly inhibited by the favourable pressure gradient in all three directions. It became much shorter and narrower in comparison with a similar spot generated in a Blasius boundary layer at comparable distances from its origin and comparable Reynolds numbers. The celerities of its boundaries did not scale with the local free-stream velocity as they do in the absence of a pressure gradient. Dimensional analysis was used to identify and correlate the independent variables determining the size, the convection speed, and the relative rate of growth of this spotThe familiar arrowhead shape of the spot gave way to a rounded triangular shape with the trailing interface being straight and perpendicular to the direction of streaming. The familiar Tollmien-Schlichting wave packet was not observed in this pressure gradient because the surrounding boundary layer was very stable at the Re considered. Since the arrowhead shape of the spot is associated with the breakdown of the waves within the packet it cannot occur below the critical Re. The relative size of the ‘calmed region’ following the spot also diminished; however, one could only speculate as to the origin of this region.

Strong interaction of a turbulent spot with a shock-induced separation bubble

“…With favorable pressure gradient the laminar velocity profile is fuller leading to a reduced ⌬u m . Hence the growth rate of the spot is reduced, consistent with Katz et al 10 ͑ii͒ In an adverse pressure gradient flow, following the same argument, ⌬u increases due to the deceleration of the laminar base flow and an enhanced spot spreading can be obtained, consistent with was reduced during spot merging into a full-turbulent boundary layer downstream. In this case the surrounding fluid u l ͑y͒ changes to turbulent and ⌬u m reduces, ultimately to zero, leaving normal turbulent diffusion, rather than enhanced lateral growth as the main mechanism.…”

“…This has been confirmed in later work, including Katz et al 10 for a favorable pressure gradient boundary layer and Seifert and Wygnanski, 7 who found a doubling of spot growth rate in a strong adverse pressure gradient flow. Measurements based on surface heat transfer Zhong et al 11 also showed a significant increase in spot lateral spreading rates with increasing adverse pressure gradients, although the growth angles measured in this way are smaller than those based on velocity measurements.…”

On the propagation of naturally-occurring turbulent spots

“…The first such study was conducted by Wygnanski [8] who determined that, in a favourable pressure gradient, neither the leading nor the trailing-edge convection rate of the turbulent spot scaled directly with the freestream velocity. This result was later expanded upon by Katz et al [9], who studied a turbulent spot developing in a boundary layer accelerating in a pressure gradient which Could be described by the Falkner-Skan parameter fl = 1. They showed that, under these conditions, the leading-and trailing-edge convection velocities of the spot increased as the square root of distance from the spot origin, while the freestream velocity increased linearly with the same length.…”

Numerical Investigation of Laminar-Turbulent Transition in a Cone Boundary Layer at Mach 6

“…For incompressible flow with zero-pressure-gradient, the lateral spreading angle is approximately 10 degrees (Schubauer & Klebanoff, 1956). Many other characteristics of turbulent spots in incompressible (low-speed) flows have been well documented in the literature (see, Wygnanski et al, 1979;Katz et al, 1982;Gad-el Hak et al, 1981;Glezer et al, 1989;Seifert & Wygnanski, 1995). The lateral spreading or growth of the turbulent spot strongly depends on the Mach number.…”