I am studying some Viscous Flow lecture notes, and I am looking at the following extract.
- "Consider the steady two-dimensional flow of a stream of viscous fluid with far-field velocity $U{\bf i}$ past an obstacle of typical dimension $L$, with ${\rm Re} = UL/\nu \gg 1$. We might expect a thin boundary layer around the body of thickness of ${\cal O}(L/{\rm Re}^{1/2})$ in which viscous effects are important."
- "The envisaged boundary-layer scaling is then $y = Y /{\rm Re}^{1/2}$, where $Y = {\cal O}(1)$ as ${\rm Re} \to\infty$, so that locally $Y = 0$ looks like a flat plate."
- "In the boundary layer the dimensional streamfunction $$\psi \sim \frac{LU}{{\rm Re}^{1/2}} \Psi$$ as ${\rm Re}\to\infty$, where the dimensionless streamfunction $\Psi(x,Y)$ satisfies the boundary layer equation $$\dfrac{\partial\Psi}{\partial Y}\dfrac{\partial^{2}\Psi}{\partial x\partial Y}-\dfrac{\partial\Psi}{\partial x}\dfrac{\partial^{2}\Psi}{\partial Y^{2}}=-\dfrac{\partial p}{\partial x}+\dfrac{\partial^{3}\Psi}{\partial Y^{3}}.$$
I have tried starting with the Navier-Stokes equation and deriving the streamfunction formulation of it. I have tried nondimensionalising and making the rescalings suggested, but I am just not seeing where the boundary-layer thickness and rescaling of $y$ come from. Could somebody please shed some light on this?