This talk will present some new results on the development of algebraically growing disturbances in mixed forced-free convection boundary layer flows. Such disturbances have been conjectured as playing an important role in transition to turbulence in a wide variety of fluid flows. In the present case they are intimately linked with the development of streamwise grid-scale oscillations that arise in the numerical solution of the boundary- layer equations. Methods for the suppression of such instabilities will also be discussed.