Mathematics for Engineers: The Capstone Course

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  • Governing Equations
    • This week we learn the governing equations for the flow around a cylinder. We begin with the Navier-Stokes equations and the continuity equation, and derive a pair of coupled equations for the stream function and scalar vorticity. These equations are nondimensionalized and contain only a single dimensionless parameter called the Reynolds number. The governing equations are then simplified using log-polar coordinates.
  • Steady Flows
    • In this week, we formulate the computational fluid dynamics problem of the steady flow around a cylinder. We introduce the finite difference method and derive iteration equations. Boundary conditions are derived and the outline of a MATLAB program is discussed. Students will write a program to compute the stream function at a Reynolds number of ten.
  • Unsteady Flows
    • In this week, we formulate the computational fluid dynamics problem of the unsteady flow around a cylinder. We introduce periodic boundary conditions in the polar angle and show how to solve for the stream function using matrix methods. The solution for the scalar vorticity will use a MATLAB ode integrator. Students will write code to compute the time-dependent scalar vorticity at a Reynolds number of sixty.

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