Alternate Turbulent Wall Scalings

Wall turbulent fluctuating properties, such as pressure, Reynolds stresses, velocities, wall shear stress, are most often made nondimensional using the friction velocity (u_tau²) Sometimes this leads to quantities that increase logarithmically with Reynolds number.   The fluctuating wall pressure, the fluctuating wall shear stress or vorticity, and the peak if <uu> are known to behave this way; for example, the wall shear stress fluctuations, <t₀t₀>⁺ ~ A ln (Re_tau) + B.  Sometimes it is desirable to have an order-one quantity as the Reynolds number becomes large.  A new scaling of the dependent variable will accomplish this.  Theoretically, all wall layers have a friction velocity law of the form U / u_tau = (1/k) ln(Re_tau) + C.  This is a companion result of the logarithmic mean velocity overlap laws. The ln(Re_tau) term can be eliminated between these expressions.   Rearrangement introduces U / u_tau into the dependent scaling so that one u_tau is canceled and a mixed velocity (U u_tau)^1/2 occurs. <t₀t₀>⁺u_tau/U  ~ D +E  u_tau /U.  Now the expression is of order one and a modifying term with gauge function u_tau/U(Re_tau) occurs.