M.E. here (I took Machine Design I and II in undergrad). Machine design gets into the effects of what the aero loads actually do. A.E.'s are typically given a set of thermal and fluid boundary conditions (density, fluid properties, object velocity, the angle of attack, etc.) and compute a set of air pressure loading conditions on a surface. M.E.'s then typically take those mapped aero pressure loads (often with temperatures mapped on an external surface as well) along with acceleration loads and see what the effects are from those conditions. Those effects are seen as the material strain from which the FEA programs calculate stress. Then ultimately margins of safety are calculated based on the mechanical material properties (aka allowables).
Machine design typically discusses the basic beams, columns, plates, etc and then gets into more advanced things: bolts, gears, clutches, pulleys etc. Honestly, it may not be all that applicable to an A.E. who wants to do CFD. If you're just taking it as an elective, go for it, but if there's something more applicable to what you want to do such as an advanced comp flow or CFD class--I'd recommend taking that instead. Like you said, understanding the effects of what aero and thermal loading conditions can do to turbines may help you to provide better insight and design assistance into turbofan and turbojet aircraft engines, etc. As I've mentioned, typically it would be the M.E.'s analyzing the actual turbine blades, shafts etc. for high-stress conditions from the information the A.E.'s provide. Another area that I can't stress enough is coding for an A.E.; many of our A.E.'s are VERY well-versed in MATLAB, Fortran, and Python and I would recommend taking as many classes in that area as possible. Good luck!