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Degani et al. (1993) observe that despite much effort, models to predict the mean- flow behaviour in three-dimensional turbulent boundary layers remain unsatisfac- tory. This is especially pertinent to perturbed, pressure-driven flows given their relative complexity. The Squire-Winter-Hawthorn (SWH) relationship however, has been shown repeatedly to provide a reasonable description of the outer layer velocity profile even when subjected to an adverse streamwise pressure gradient (Coleman et al., 2000). The relationship describes the inviscid skewing of the boundary-layer vortex lines away from the (perpendicular) mean-flow direction. The crossflow ve- locity predicted using the SWH relationship reads:

W

Qe = (

1− U

Qe)

The behaviour of the inner layer is more complicated. Using asymptotic anal- ysis, Degani et al. (1993) investigated the effects of pressure gradients and Reynolds number for equilibrium pressure-driven three-dimensional flows, finding the cross- flow velocity profile dependent on both. Examples of this are the duct-bend flows of Schwarz and Bradshaw (1994) and Flack and Johnston (1998). Despite the sim- ilarity in their test section geometries, the skewing at the wall (γe−γw) is greater in the latter for a given value ofγe. The only significant difference between the two flows tested appears to be their respective Reynolds numbers.

Turbulent Behaviour

It might be expected that the addition of mean three-dimensionality into a two- dimensional turbulent boundary layer would have a destabilising effect on the flow, supplying additional energy to the turbulence. A distinguishing feature of three- dimensional boundary layers however, is that the opposite is often the case, with many turbulent quantities indicating a more stable flow. This is best illustrated by

the structural parametera1, calculated as the ratio of the turbulent shear stress to

two times the turbulent kinetic energy:

a1 = τt 2k = √ u′_v′+_v′_w′ u′2+_v′2+_w′2 (2.2)

This statistic contains much information about the flow and can be thought of as the ‘efficiency’ with which the turbulent motions can extract energy from the mean shear and turn it into shear stress. For a parallel turbulent boundary layer,a1 is typically

0.15 between 0.1<y/δ<0.8, goes to zero at the wall and drops to low values in the outer regions. Broadly, for three-dimensional turbulent boundary layers, increasing the skew angleγ=tan−1(

W/U) leads to a larger reduction in the streamwise shear

stress (−u′v′) and a

agreement though, about the magnitude of the reduction in a1 and in pressure-

driven flows the outer layer of boundary layer seems equally likely to see a rise in the streamwise shear stress.

The typical response is illustrated by the swept-wing flows of Bradshaw and Pontikos (1985), Baskaran et al. (1990) and Itoh and Kobayashi (2000) with all three

measuring a reduction in a1 in excess of 30% as the skew angle increases. A drop

in the turbulent kinetic energy and a rapid decline in the streamwise shear stress are also observed. The cross-stream shear stress (v′w′) rises, which is expected, but at a much slower rate compared to the decline in the streamwise component. Moin et al. (1990), Coleman et al. (1996) and Kannepalli and Piomelli (2000) investigate shear-driven three-dimensional boundary layers in channel flows numerically using DNS or LES. Moin et al. (1990) simulate a spanwise acceleration of the channel wall velocity, and Coleman et al. (1996) and Kannepalli and Piomelli (2000) investigate a sudden, constant motion of the wall. The transient and equilibrium behaviour of Moin et al. (1990) therefore differs, but all three come to similar conclusions with

the typical decline in turbulent shear stress, kinetic energy anda1 near the wall, as

well as an increase in the dissipation rate.

A further distinguishing feature of three-dimensional turbulent boundary layers is their anisotropy, as demonstrated by the difference seen in experiments between the shear stress vector direction:

θτ =tan−1(−

v′w′

u′_v′) (2.3)

and that of the strain-rate vector:

θg =tan−1(

∂W

∂y /

∂U

∂y) (2.4)

The use of a scalar isotropic eddy viscosity as a turbulence model, is then, clearly inappropriate for such boundary layers. The divergence between these two angles is a consistent trend in almost all three-dimensional turbulent boundary layer studies

and does not appear to be influenced severely by streamwise velocity gradients. Measurements between studies largely agree with regard to the outer layer, where the shear stress lags behind the velocity gradient direction and the magnitude of the difference depends on the degree through which the flow has turned. Near the wall where available measurements are few, there is greater disagreement. Flack

and Johnston (1998) for example, show thatθg leadsθτ for most of the outer layer

in their 30° bent duct, with a maximum magnitude of 14°. For the inner layer, a

region of collateral flow is evident between 20<y+<_{50 and nearer the wall, the shear} stress vector is found to lead the velocity gradient vector. Regions whereθτ leadsθg are few in the literature and are difficult to explain. A further example is provided by Moin et al. (1990).

Johnston and Flack (1996) imagine that the divergence between the direc- tions of shear-stress and velocity gradient vectors is caused by the ‘contamination’ of the newer downstream turbulence with the ‘history’ of the gradually decaying upstream flow. An illustration of this is the difference between perturbed and equi- librium three-dimensional flows. The rotating-disk boundary layer investigated by Littell and Eaton (1994) for example, is close to equilibrium so there is almost no di-

vergence betweenθgandθτ and the turbulence appears close to isotropy. In a similar

investigation on an enclosed rotating disk, Itoh et al. (1992) show that the station- ary side comes closer to equilibrium than the rotating-disk side and demonstrates

less deviation betweenθτ and θg. A further example is provided by Kannepalli and

Piomelli (2000). The section of spanwise moving wall in their channel-flow simula- tion was long enough to see the formation of an equilibrium collateral region near the wall. This is accompanied by a realigning and recovery of the turbulence as

equilibrium is approached, and the divergence betweenθg andθτ gradually declines;