I'm trying to solve two-dimensional potential flow over airfoils with the finite element method, using the stream function formulation ($\Delta\psi = 0$, $u = -\partial\psi/\partial y$, $v = \partial\psi/\partial x$) and superposition to account for and enforce the Kutta condition (zero velocity) at the trailing edge. Following the approach of de Vries we have [1]
\begin{equation} \psi = \psi_1 + b*\psi_2 + c*\psi_3 \end{equation}
where
\begin{equation} \psi_1: \Delta\psi = 0,\quad \psi_{\Gamma_2} = -U_\infty y,\quad \psi_{\Gamma_1} = 0\end{equation} \begin{equation} \psi_2: \Delta\psi = 0,\quad \psi_{\Gamma_2} = 1,\quad \psi_{\Gamma_1} = 0\end{equation} \begin{equation} \psi_3: \Delta\psi = 0,\quad \psi_{\Gamma_2} = 0,\quad \psi_{\Gamma_1} = 1\end{equation}
and $\Gamma_2$ is the external boundary of the domain with horizontal velocity $U_\infty$, and $\Gamma_1$ the boundary of the airfoil. Using superposition, and the fact that $u_T=v_T=0$ at the point of the trailing edge $T$, we should be able to determine the constants $b$ and $c$ with (equations 16a and 16b in [1])
\begin{equation} u_T = u_{1T} + bu_{2T} + cu_{3T} = 0\end{equation} \begin{equation} v_T = v_{1T} + bv_{2T} + cv_{3T} = 0\end{equation}
All this seems resonable until one considers the fact that $\psi_2$ and $\psi_3$ will be linearly dependent $(\psi_3 = 1-\psi_2)$ so the system $\{u_T,v_T\}$ is not determined. Is there something I'm missing here or have misunderstood? Clearly the author has used this approach successfully, but I just can't see how it can work in this case.
[1] G. de Vries and D. H. Norrie, "The Application of the Finite-Element Technique to Potential Flow Problems", J. Appl. Mech 38(4), 798-802, 1971, DOI: 10.1115/1.3408957
