The three-dimensional turbulent boundary layer near a plane of symmetry

Abstract

The asymptotic structure of the three-dimensional turbulent boundary layer near a plane of symmetry is considered in the limit of large Reynolds number. A selfconsistent two-layer structure is shown to exist wherein the streamwise velocity is brought to rest through an outer defect layer and an inner wall layer in a manner similar to that in two-dimensional boundary layers. The cross-stream velocity distribution is more complex and two terms in the asymptotic expansion are required to yield a complete profile which is shown to exhibit a logarithmic region. The flow in the inner wall layer is demonstrated to be collateral to leading order; pressure gradient effects are formally of higher order but can cause the velocity profile to skew substantially near the wall at the large but finite Reynolds numbers encountered in practice. The governing set of ordinary differential equations describing a self-similar flow is derived. The calculated numerical solutions of these equations are matched asymptotically to an inner wall-layer solution and the results show trends that are consistent with experimental observations.

Near-Wall Measurements of a Three-Dimensional Turbulent Boundary Layer.

“…As might be guessed by the discussions to this point, none of the relations adequately described all of the flows. Degani, Smith and Walker (1992) performed an asymptotic study of the mean crossflow profile in a three-dimensional turbulent boundary layer with small crossflow at high Reynolds number. They found that the near-wall flow should be collateral to leading order, and that W profiles should display a logarithmic region at high Reynolds number.…”

Theoretical prediction and design for vortex generators in turbulent boundary layers

“…As a result, the governing equations reduce to a forward-marching three-dimensional vortex system, allowing relatively fast accurate computation and theoretical analysis (in 9 3, where a linearized system is obtained, in 994 and 5, which address single-mode VGs, and then in $6, which is concerned with realistic VG shapes and arrays). A representative time-mean turbulence model is used (see $9 2 and 3), namely the Cebeci-Smith one but extended into the current three-dimensional context ; other models for three-dimensional flows are considered in Chima & Yokota (1989), Vatsa & Wedan (1988), Degani, Smith & Walker (1992), Cebeci & Smith (1974). The applicability and 'workability' of this model are discussed elsewhere (Degani et al 1992;Neish & Smith 1988 for a variety of configurations.…”

“…A representative time-mean turbulence model is used (see $9 2 and 3), namely the Cebeci-Smith one but extended into the current three-dimensional context ; other models for three-dimensional flows are considered in Chima & Yokota (1989), Vatsa & Wedan (1988), Degani, Smith & Walker (1992), Cebeci & Smith (1974). The applicability and 'workability' of this model are discussed elsewhere (Degani et al 1992;Neish & Smith 1988 for a variety of configurations. In addition, however, the model is felt likely to be increasingly appropriate for the current low-profile VGs anyway, where most of the VG-generated flow effects occur at first in the logarithmic part of the boundary layer (the significance of which is addressed in the next paragraph).…”

“…Smith starting point, the incompressible regime is studied here, with the fluid density being p*, and with the oncoming boundary layer being two-dimensional, although the main application areas are more in the transonic regime with cross-flow. The wall friction velocity is denoted by u,*, which is of the order of u: (In&-' at large Re, and the oncoming boundary layer is then two-tiered (Bush & Fendell 1972;Mellor 1972;Cebeci & Smith 1974;Degani et al 1992;Neish & Smith 1988. The main findings of the research so far, then, are contained essentially in @6 and 7 below (along with the parameter groupings and comparisons in Appendix C).…”

The structure of a three-dimensional turbulent boundary layer

“…In contrast, both terms in (3.23) are necessary to describe the cross-stream velocity profile in the outer layer (see also Degani et al 1992). Using (3.22d), a composite profile is U, tan 0, (7.3) where (3.21) has been used to define an appropriate non-dimensionalizing quantity for u2.…”