Calculating the dimensional wall-normal coordinate for a self-similar compressible boundary layer using Levy-Lees transformation

Question

How can I convert my self-similar boundary layer solution that is a function of the nondimensional wall-normal coordinate $\eta$ to be a function of dimensional $y$? For instance, if I determine from my boundary layer solution that $\delta_{99}$ occurs at $\eta = 5$, for a given set of dimensional parameters how do I determine the corresponding physical coordinate? This is very straight forward for Blasius flow but is not as obvious to me for compressible flow. The definition of $\eta$ is shown below. The integral dependence on $y$ is what is throwing me off.

$$\eta = \frac{U_e}{\sqrt{\int_0^x\rho_eU_e\mu_edx}}\int_0^y\rho dy$$

Answer 1

Just write the transformation in its differential form and then rearrange.

$$ d\eta = \frac{U_e}{\sqrt{\int_0^x\rho_e U_e \mu_e dx}}\rho dy $$

Rearranging and integrating yields:

$$ y = \frac{\sqrt{\int_0^x\rho_e U_e \mu_e dx}}{U_e}\int_0^\eta \frac{1}{\rho}d\eta $$