Is there any specific relation between Re_b (bulk/ Mean Reynolds number) and Re_\tau (friction Reynolds Number). In few of the literature and experimental work I have gone through researchers give approximate values of these Reynolds numbers in their papers. I want to know, are these approximate values based in experimental correlations or any specific relations between these values?
One of the example of literature review by Schule & Flack i have added in the query as a snapshot for reference.
The Reynolds numbers based on the friction velocity [=sqrt(Thau/rho)] and the bulk (or freestream velocity, V) are indirectly linked through the Darcy friction factor, Cf=Thau/(0,5*rho*V^2). Cf is often expressed as a function of Re (based on V), hence also the friction velocity can be expressed as a function of V.
Reb = Vb L/ ni =( utauL/ni)(Vb/utau)= Retau V+.
if the equations are made non dimensional by using utau as reference velocity you have just a scaling by the non dimensional velocity V+.
Hi, to my knowledge one of the best correlations relating Re_b to Re_tau is the one proposed by Dean, Cf = 0.073 Re^−1/4 Article Reynolds Number Dependence of Skin Friction and Other Bulk F...
with C f= tau_w/(1/2 rho U_b^2) = 2 (u_tau/U_b)^2 = 2 (Re_tau/Re)^2
More recently, other relations for Cf have been proposed, for instance by Zanoun et al. (2003 and 2009)
Hi, sorry for asking here but the question may be relevant. Is there any state of the art review paper on Reynolds number definition(s) in various types of flows?
Maciej: The short answer to your question is no (as far as i know). The main challenge is defining a proper length scale in Re, and the sensible solution is to use some hydraulic diameter for internal flows, and the local position for external flows (flat plates, aerofoils, tubes etc). The main thing is that Re in some way should relate to the boundary layer thickness, which can be difficult (This is why the Nu and Cf for developing flows must be corrected with a d/L-expression for internal flows). Some attempts have been made at using the momentum thickness as length scale, which has showed some success in estimating the transition to turbulence.
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