The law of the wake: 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 of the law of the wall in wall-bounded turbulence at high Reynolds number. 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 first part is then a linear ramp function that starts at Y_c. Our model is furthermore 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 our 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 our 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 the ramp function for Q(Y) is measured as α ≈ 7.3) and the measured intercept is smaller (Y_c ≈ 0.11). The finally obtained F(Y) model is again seen to reproduce very well with the DNS data. 

To comply with some recent literature, a version of our model is also developed where an added linear contribution (of slope α₀ much smaller than α) is added to Q(Y) within the overlap layer. 

Lastly, we also revisit the wake function model of Coles, and we show that adding the offset improves it significantly.