This assignment involves simulating air flow over an airplane wing. Air can be modeled as a fluid and the simplest model of fluid flow is Laplace's equation
where the velocity potential is related to the flow velocity (air speed) by
Note that if we solve for then we can easily find the flow velocity. In fact, the function has no physical interpretation other than being used indirectly for finding the flow velocities.
This equation makes several simplifying assumptions about fluid flow. In particular it assumes that the fluid is incompressible (i.e. the pressure is constant), that there is no viscosity (i.e. no friction) and that the fluid cannot rotate to form vortices. These assumptions are reasonable for simulating air flow at low speeds (no turbulence) and if you don't care about drag.
The goal of this homework is to simulate fluid flow under these assumptions over an object, in particular an airplane wing. To do this we need to solve Laplace's equation over a region. We also need to set the boundary conditions to specify the ``wind speed'' (we will effectively be simulating a wind tunnel). Consider the following picture:
To simulate airflow throughout the region we will use the following boundary conditions. First we assume that the object itself is impenetrable so that the air flow at the surface of the wing has no component perpendicular to the surface ( where is the unit vector normal to the surface). Second we will assume that air is blowing in from the left, and assume that the component of flow entering on the left is constant. We assume that there is no flow through the top or bottom (as if it were a wind tunnel). Finally we will set the along the right boundary. We do this so that we lock down the value of at that surface. I'll explain the motivation for this in class.