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# Reynolds Stress

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From equation (5), the Reynolds stress is:

$-r = \frac{n\left( s - 1\right)}{s\left( n - 1\right)}\left(\left( 1-\eta\right)^{2n-1} -\left( 1-\eta\right)\right) \tag{7}$

It is easy to show that $$r$$ attains a maximum at $$\eta^*$$ given by

$\eta^* = 1 - \left(\frac{1}{2n-1}\right)^{\frac{1}{2\left( n - 1\right)}} \tag{8}$

As in the previous page, choose a value of $$R_m$$ and get the corresponding values of $$R_\tau , s , n$$. The Reynolds stress $$r \left(\eta\right)$$ will be plotted on the right.

Also note the values of $$\eta^*$$ and $$y^+$$. For example, if $$R_m = 12300$$, then $$\eta^* = 0.1082$$ and $$y^+ = 58.28$$, i.e. in the log-wall region. Notice that $$\left( 1 - \eta \right)$$ is an asymptote (in green) and that for larger values of $$R_m$$ the boundary layer near $$\eta = 0$$ becomes thinner.

$$R_\tau =$$ ,
$$n =$$ ,
$$s =$$
$$\eta^* =$$ ,
$$y^+ =$$