Stagnation Flow - Part I

Now that the inviscid portion of ZFoil is working (although I already have some tweaks I want to try), it’s time to work on boundary layer code. The appropriate place to begin is, of course, the beginning, and the boundary layer begins at the leading edge, at the stagnation point where the flow is brought to a halt. Let us build such a flow using some of the tools I’ve already introduced.
Let us start with a uniform flow:
$$\overline{V}={{1}\over{2}}B$$ $$\Phi={{1}\over{2}}Bz$$
B is a real constant, and the fraction will prove to be convenient. The flow is too simple to even bother plotting, but let us apply the following transform:
$$w^2=z$$ $$\Phi={{1}\over{2}}Bw^2$$ $$\overline{V}=Bw$$

> (define (Phi w) (* 1/2 B w w))

> (define B 1)

> (plot (contours          (λ(x y)            (imag-part (Phi (make-rectangular x y))))         -3.0 3.0 0.0 1.5         #:levels '(-4.0 -2.0 -1.0 -0.5 -0.25 0.25 0.5 1.0 2.0 4.0))         #:height 200)

(object:2d-plot-snip% ...)

In z space, B is a measure of velocity, but in w space, it is a measure of velocity gradient. The stagnation point is at w = 0. Note that I have plotted the imaginary component of the complex potential \(\Phi\) ("fee"), the stream function \(\Psi\) ("psi", very close to "sigh"). When I introduce friction in my next post, we will be able to retain \(\Psi\), but \(\Phi\) will have to go, as it assumes irrotational flow. I now present an explicit form of the inviscid stream function:
$$\begin{align} \Phi&={{1}\over{2}}Bw^2\\ \Phi&={{1}\over{2}}B(x+iy)^2\\ \Phi&={{1}\over{2}}B[(x^2-y^2)+i(2xy)]\\ \Psi&=Bxy\\ \end{align}$$
Yes, the streamlines are hyperbolas. In my next post, we will deal with how to modify this stream function to account for friction.