High Reynolds Number Boundary Layers Subjected to
Various Pressure-Gradients, the Kármán “Coefficient”
and the Log Law Prevails
Hassan M. Nagib
Department of Mechanical, Materials and Aerospace Engineering,
Illinois Institute of Technology (I I T), Chicago, IL 60616-3793, USA ;
Mean velocity distributions in the overlap region, over the range of Reynolds numbers 10,000 < Req < 70,000, under five different pressure-gradient conditions are accurately described by a log law. The pressure-gradient conditions include adverse, zero, favorable, strongly favorable, and a complex distribution, with parameters b ranging from - 0.15 to + 0.25. The wall-shear stress was measured using oil-film interferometry, and hot-wire sensors were used to measure velocity profiles. Parameters of the logarithmic overlap region developed from these higher Reynolds number boundary layers continue to be consistent with our recent findings and to remain independent of Reynolds number. The best estimate of the log-law parameters from the zero-pressure gradient boundary layers is k = 0.384, B = 4.127. However, the Kármán “coefficient” (k) is found to vary considerably for the non-equilibrium boundary layers under the various pressure gradients. The results clearly demonstrate that the power-law relation does not represent the overlap region of high Reynolds number boundary layers. A slightly modified version of the almost century old Prandtl-Kármán skin friction relation, as well as the four decades old Coles-Fernholz logarithmic relation using the appropriate k value of 0.384, provide exceptional agreement with the recent data from KTH and IIT for zero pressure gradient conditions. Some comments and recent results on transitional surface roughness effects may also be presented.