I have read quite a few tutorials / watched several clips on the derivation of the Falkner-Skan boundary layer equations using similarity and then solution using RK solvers such as ode45 or odeint.
However, apart from light discussion on the meaning of $\beta$ and $m$ (being positive, negative, equal to 1 (stagnation) or equal to 0 (flat plate)), I cant seem to find (or work out) the link which would allow me to calculate the BL profiles for a known freestream pressure which varies with $x$. So, for example, I would like to be able to take a pre-calculated pressure field $P(x)$ and then apply it to the F-S equations and then the boundary layer profiles at any given $x$.
From basic F-S, we have:
$$ f''' + ff'' + \beta\left(1-f'^2\right) = 0 $$ $$ f(0) = f'(0) = 0 $$ $$ f'(\infty) = 1 $$
where $\beta = \frac{2m}{m+1}$ and $\beta$ can be related to the half-angle of a wedge.
We also know that for F-S the velocity is given by the potential flow power law relation: $$ U(x) = Cx^m $$ and that using inviscid Bernoulli, the boundary layer pressure gradient can be related to the freestream (inviscid) pressure:
$$ -\frac{\partial P}{\partial x} = \rho U(x) \frac{d U(x)}{d(x)} = \rho Cx^m mCx^{m-1} = \frac{U(x)^2 m}{x}$$
where $U(x)$ is the freestream velocity, $C$ is a velocity "coefficient" and $m$ is the dimensionless parameter above. So, as it should be, the pressure gradient $-\frac{\partial P}{\partial x}$ is related to the freestream velocity, which is given by the power law expression for potential flow.
So the question then is, if i have an arbitrary varying pressure field, can I use the above reasoning to calculate a "local" $m$ value based on the local pressure gradient and then calculate a local set of characteristic curves for the BL flow. Ultimately the reason would be to use such an approach for inverse design where, have designed a pressure field, one can calculate and then check (as a first approximation) the BL state for things like separation etc.
