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# Turbulent Eddy Viscosity

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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.

$$R_\tau =$$ ,
$$n =$$ ,
$$s =$$
$$\eta_{level} =$$