The Boussinesq approximation [4] yields an approximation for the non dimensional turbulent eddy viscosity

\[ \bar{v} := \frac{v_t}{v} = \frac{n\left( 1 - \alpha\right)\left(1 - \left( 1 - \eta\right)^{2n-2}\right)} {\alpha + n\left( 1 - \alpha\right)\left( 1 - \eta\right)\left(1 - \eta\right)^{2n - 2}} \tag{9} \]

The turbulent eddy viscosity may be interpreted as a `local' turbulence Reynolds number. A low value is associated with \( \bar{v} < 1 \) where viscous stresses dominate turbulent stresses, and a high value where turbulent stresses are dominant. The region of rapid change in \bar{v} can be identified as the logarithmic region. The levelling of \( \bar{v} \) towards the centre of the channel supports the constant eddy viscosity assumption for the outer region [4]. The value for \( \eta \) for which \( \bar{v} = 1 \) is given by

\[ \eta_{level} = 1 - \left(\frac{1}{2}\left( 1 - \frac{n - s}{n\left( s - 1\right)} \right)\right)^{\frac{1}{2n - 2}} \tag{10} \]

For the value of \( R_m \) obtain the corresponding value of \( \eta_{level} \) in inner wall variable \( y^+ \). For example, if \( R_m = 12300 \) , \( y^+_{level} = 11.467 \) which is a good estimate of the extent of the laminar sublayer. However, a Taylor series expansion of Equation (9) about \( \eta = 0 \) indicates that as \( y\rightarrow 0 \), \( \bar{v} \rightarrow y \) which is in conflict with the expected cubic behaviour [5]. This discrepancy is due to the isotropy inherent in the Boussinesq approximation.