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We focus on the wake function F(Y), where Y=y/δ: the excess velocity above the logarithmic profile in wall-bounded turbulence. It is first measured using direct numerical simulation data of turbulent channel flow at Reτ≃5200, which has a distinguishable overlap layer. The function Q(Y)=YF′(Y) is seen to be significant only beyond Yc≃0.16, and also linear (with a slope α≃1.15) up to Y≃0.5. Our simplest Q(Y) model is then a linear ramp function that starts at the measured intercept. Our Q(Y) model is furthermore extended up to the channel center, by subtracting a quadratic term which satisfies the boundary condition at Y=1. The analytical integration of Q(Y) provides the model for the wake function with offset Yc, and which covers the full range; it is seen to fit very well the DNS data. Our Q(Y) model is also compared to the “extended law of the wall” model without offset. Furthermore, and to comply with some recent literature, a version of our model is also developed with some small slope α0≪α contribution to Q(Y) within the overlap layer. The DNS of a zero pressure gradient boundary layer at Reτ≃2300 is considered next: the amplitude of the wake function is much larger than in channel flow (α≃7.3) and the intercept is smaller (Yc≃0.11). The obtained F(Y) model also compares very well with the DNS data over the full range. Lastly, we also revisit the Coles wake function model and we show that adding the offset improves it significantly.
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Winckelmans, G., & Duponcheel, M. (2021). Modeling the law of the wake using an offset from the wall. Physical Review Fluids, 6(6), 64606. https://doi.org/10.1103/PhysRevFluids.6.064606 (Original work published 2021)