I recommend reading Pope, S. B., Turbulent Flows, Cambridge University Press, 2000. doi:10.1017/CBO9780511840531.

I am not a specialist of turbulent theory, and I may be wrong. Near the wall, the grad of velocity is high, so the contribution of friction is very important near the wall. Hence, it is a good parameter for dimensionless of velocity and finally to get a velocity profile U+ =f(y+).

This is the way I see that.

In the wall turbulence (that is in the turbulent BL) we can assume a balance between diffusion and convection of momentu,. That could be assumed by the equilibrium between the wall stress and the convective flux based on the characteristic velocity in the BL.

Thus, we can write tauwall = rho*(utau)2 that corresponds to assume a Reynolds number based on the wall-based variables to be O(1).

Scaling is a very useful tool in turbulence studies. If we measure dimensional properties of a turbulent flow and find a scale parameter such that scaled relationships apply to a wide range of conditions and/or flows, then we have made progress in the empirical description those flows. If that scale parameter also has `physical significance’ then it may also contribute to more general modelling of those flows.

It is known, empirically, that $u_\tau$ scales the log wind profile very well for neutral flows over flat surfaces, both in the laboratory and outdoors. There remains some debate on whether it works perfectly (Is the von Karman constant truly a constant?) but most of us just accept that it does and get on with other things. Questions come when we expand the range of flows we look at.

Log layers are also found near the bottoms of highly convective atmospheric boundary layers. Velocity statistics in such log layers do not scale on $u_\tau$. For example, the spectrum of vertical wind fluctuations collapses onto a universal curve when scaled by matching inertial subranges, which is to say when scaled on $(kz\varepsilon)^{1/3}$, not when scaled on $u_\tau$. Can the same flow have two velocity scales, one for mean velocity (a vector) and another for everything else?

Shear stress at the base of convective boundary layers can vary markedly in time, both in strength and direction. How should we then calculate $u_\tau$? Is it the magnitude of the vector-mean shear stress, which is the conventional definition, or should we use the mean magnitude of the instantaneous shear stress? There is a case for the latter.

Momentum is passed down to the surface in a series of highly organized transfers from larger to smaller eddies. But turbulent flows are dissipative systems, so organization has its cost in energy dissipation and entropy production. Taking the dissipation rate to be the energy flow that supports this organization at any level, we can use $(kz\varepsilon)^{1/3}$ as a velocity scale. Here $\varepsilon$ is the dissipation rate averaged over a time long compared with the local eddy time scale, $(z^2/\varepsilon)^{1/3}$, but short compared to the outer time scale of the flow. Averaging $(z^2/\varepsilon)^{1/3}$ over longer times gives a scale velocity that increases as the variability of $\tau$ increases, and is always greater than $u_\tau$.

So, to answer your question Daud, $u_\tau$ has been found, empirically, to work well in a range of neutral flows. Its physical significance is that it is associated with the organizational cost of transporting momentum down to the surface in cases where $\tau$ is steady in time. It has not been established in log layers where $\tau$ varies in time. There is reason to believe that it will have to be modified in such cases.

Daud Hasan , Friction velocity uτ=τw/ρuτ=τw/ρ is defined from wall shear stress so that the wall momentum flux per unit area is expressed as an equivalent velocity scale for the flow near a boundary. Physically, uτuτ measures the intensity of shear-driven momentum transfer at the wall, making it the natural scale for near‑wall turbulence dynamics independent of the outer free‑stream speed.

What it represents

Wall shear stress τwτw is the tangential stress exerted by the fluid on the surface, and recasting it as uτuτ transforms that stress into a velocity scale directly comparable to turbulent fluctuations.

In inner variables, the near‑wall momentum balance shows τw/ρτw/ρ sets the order of magnitude of viscous and Reynolds shear terms, so uτuτ emerges as the characteristic velocity of the buffer and viscous sublayers. Hence uτuτ quantifies how strongly the wall extracts or injects momentum into the flow through shear.

Why it’s useful

Using uτuτ defines the inner scaling u+=u/uτu+=u/uτ and y+=yuτ/νy+=yuτ/ν, which collapses velocity profiles across Reynolds numbers in a universal “law of the wall.”

The logarithmic law u+=1κln⁡y++Bu+=κ1lny++B employs uτuτ as its velocity scale, reflecting that near‑wall dynamics are governed by wall shear rather than outer conditions. Because uτuτ is tied to τwτw, it also anchors skin‑friction and drag correlations used for pipes, channels, and boundary layers.

Practical implications

Skin‑friction coefficient and roughness functions are naturally expressed via uτuτ, enabling consistent comparison of experiments, DNS, and models for wall‑bounded turbulence.

The same scaling informs mixing and transport estimates in environmental and engineering flows, where shear‑related dispersion often scales with a fraction of uτuτ. In short, defining friction velocity from wall shear stress provides a physically grounded yardstick for the strength of near‑wall turbulence and for nondimensionalizing wall‑bounded flows.