Let's say I'm interested in fluid dynamics, specifically in fluid-structure interactions -- and I want to get into modeling, simulations and experiments. I am a mathematics student by training, having taken yearlong courses in introductory analysis, linear algebra, complex variables, and a semester of probability theory, stochastic calculus and fluid dynamics. I don't have much training in physics or engineering other than some decent research at a lab at our dept (our math department is well-regarded, especially in the area of fluid dynamics).

I feel I most naturally fit into a PhD program in the computational sciences / engineering space, given my math background and lack of physics and engineering background.

However, I also feel that joining a more traditional mechanical / aerospace engineering department as a PhD student would perhaps allow me to work "deeper" ... and be more of a domain expert, picking up the necessary mathematics and computational methods and modeling skills as needed. I think I much prefer this strategy, letting the fluid dynamics applications drive my graduate work in mathematics and computations.

How can I decide whether to be a computational scientist in a computational science PhD program or in a more traditional mechanical / aerospace engineering program and pick up the computation skills as needed?

(Let's assume that I can get into both computational science programs and mechanical / aerospace engineering programs ...)