I have the following two equations:
\begin{align} \frac{\lambda}{u_L} = 3.2693\left(\frac{\nu}{\epsilon}\right)^{1/2 }\end{align}
\begin{align} u_L = \left(\epsilon \lambda\right)^{1/3}\end{align}
I have tried to solve for $\lambda$ as follows:
\begin{align} \frac{\lambda}{\left(\epsilon \lambda\right)^{1/3}} = 3.2693\left(\frac{\nu}{\epsilon}\right)^{1/2}\end{align}
combining variables on the LHS: \begin{align} \frac{{\lambda}^{2/3}}{\epsilon^{1/3}} = 3.2693\left(\frac{\nu}{\epsilon}\right)^{1/2}\end{align}
Multiplying both sides by ${\epsilon}^{1/3}$
\begin{align} {\lambda}^{2/3} = 3.2693\left(\frac{\nu^3}{\epsilon^3}\right)^{1/6}\left({\epsilon^2}\right)^{1/6}\end{align}
This simplifies to: \begin{align} {\lambda}^{2/3} = 3.2693\left(\frac{\nu^3}{\epsilon}\right)^{1/6}\end{align}
Raising both sides to the 3/2 power yields: \begin{align} {\lambda} = 3.2693^{3/2}\left(\frac{\nu^3}{\epsilon}\right)^{3/12}\end{align}
Finally, \begin{align} {\lambda} = 5.911\left(\frac{\nu^3}{\epsilon}\right)^{1/4}\end{align}
However, when I use this equation to calculate the value of $\lambda$ from the data, I'm not getting values equal to an alternative equation for calculating $\lambda$, which alternative equation I am certain is correct.
Does anyone see an error; I can't seem to find one.