The equations that govern fluid mechanics are a system of nonlinear partial differential equations. It is only possible to find analytical solutions in very special cases. In general, the most powerful tools at our disposal are computational methods. Computational methods can be applied to virtually any model of interest. We use high-order numerical methods to study the evolution of fluid flows in a variety of different models. Also, we compute the stability of flows by solving large eigenvalue problems. A lot of the code we use is generated in house but we also use other existing software when it is appropriate for the investigation in question.