Welcome to mathematics for
engineers, the capstone course.
In this course, we'll use some of
our new math knowledge to tackle a challenging problem,
the computation of
the two-dimensional flow around the cylinder.
This capstone course consists of three weeks.
In the first week,
we'll learn the governing equations;
this is the famous Navier-Stokes equations.
We'll write these equations in two dimensions
using a scalar vorticity and a stream function.
We'll discuss the geometry of
our flow and how to define a Reynolds number.
We'll also learn how to use
the log-polar coordinate system.
After each lecture, I'll give you
some interesting problems to solve.
In the second week,
we'll start doing computations.
The steady solutions you'll compute
are stable for low Reynolds numbers.
We'll apply the finite difference method to discretize
the vorticity and stream function equations and
iterate these difference equations using the SOR method.
We'll derive boundary conditions for our flow,
then we'll plan our computational code.
Your project for this second week is to write
a MATLAB program that finds
the steady flow solution at a Reynolds number of 10.
In the third week,
we'll tackle unsteady flows.
This will lead to the numerical solution
of the Karman vortex street.
We'll assume periodic boundary conditions
in the polar angle.
The stream function equation is solved using
matrix algebra techniques and
the scalar vorticity equation is
solved using a Runge-Kutta method.
Your project for this third week is to write
a MATLAB program that can
time evolve the vorticity field.
From your solution,
you can produce a movie of the Karman street.
Thank you for joining me on
this learning adventure in mathematics for engineers,
the capstone course.
Let's get started.