Coles "wake function" revisited and modeled using an offset from the wall

We examine the wake function F(Y) with Y = y/δ: the excess velocity above the logarithmic profile in wall-bounded turbulence at high Reynolds number, as proposed by Coles. It is first measured using DNS data of channel flow at Re_τ ≈ 5200, which has a distinguishable overlap layer with 1/κ ≈ 2.61. The "generating function" Q(Y) = Y F'(Y) is seen to be significant only beyond Y_c ≈ 0.16, and also linear (with a slope α ≈ 1.15) up to Y ≈ 0.5 . Our Q(Y) model for that part is taken as a linear ramp function that starts at Y_c. The model is also extended for the second part: by subtracting a quadratic term which satisfies the boundary condition at Y=1. The analytical integration of the Q(Y) model finally provides the complete F(Y) wake function model with offset Y_c. It is seen to fit very well the DNS data over the full range. The first part of the model is also usefully compared to the "extended law of the wall model" of Bernardini et al. (a model without offset).

The DNS data of a ZPG boundary layer at Re_τ ≈ 2300 are considered next. The amplitude of the wake function is then much larger than in channel flow. The slope of Q(Y) in its linear part is much larger (α ≈ 7.3) and the intercept is smaller (Y_c ≈ 0.11). The finally obtained F(Y) model is seen to also reproduce very well the DNS data.

To comply with some recent literature, we also present a version of the Q(Y) model that incorporates a small linear contribution (of slope α₀ much smaller than α) within the overlap layer.

Lastly, we also revisit the simple wake function model that was proposed by Coles, and we show that adding the offset improves it significantly.