Evidence of quasi equilibrium in pressure-gradient turbulent boundary layers

Two sets of measurements utilizing hot-wire anemometry and oil film interferometry for flat-plate turbulent boundary layers, exposed to various controlled adverse and favorable pressure gradients, are used to evaluate history effects of the imposed and varying freestream gradients. The results are from the NDF wind tunnel at ILLINOIS TECH (IIT) and the MTL wind tunnel at KTH, over the range $800 < Re_\tau < 22,000$ (where $Re_{\tau}$ is the friction Reynolds number). The streamwise pressure-gradient parameter $\beta \equiv (-\ell/\tau_{w}) \cdot (\partial P_{e}/\partial x)$ varied between $-2 < \beta < 7$, where $\ell$ is an outer length scale for boundary layers equivalent to the half height of channel flow and the radius of pipe flow, and is estimated for each boundary-layer profile. Extracting from each profile the three parameters of the overlap region, following the recent work of \cite{mon23} that led to an overlap region of combined logarithmic and linear parts, we find minimum history effects in the overlap region. Thus, the overlap region in this range of pressure-gradient boundary layers appears to be in ``quasi equilibrium".