Use the slider to choose varying values of \( R_m \); the values of \( R_\tau \), \( n \), and \( s \) will simultaneously update.

The panel below will plot the same flow expressed in inner wall variables \( y^+ \) and \( u^+ \). The inner wall variable serves to magnify the three distinct regimes in a wall bounded turbulent flow.

Superimposed on the panel are two functions \( u^+ = y^+ \) (in red) and \( u^+ = \frac{1}{\kappa}\ln y^+ + c \) (in green) where \( \kappa = 0.41 \) and \( c = 5.5 \) are empirical constants (e.g. [10]).

The first relationship which holds in the range \( 0 < y^+ < 5 \) is called the "viscous sublayer region" and the second relationship which holds in the range \( 30 < y^+ < 100 \) is called the "logarithmic law of the wall". Beyond \( y^+ > 100 \) is the "velocity defect region" where \( \frac{u_m - \bar{u}}{u_\tau} = F\left(\eta\right) \) where \( F \) is an empirical function (e.g. [11]).