The development of computational fluid dynamics is not only restricted to the computer simulation of fluid flow phenomena. The fact that fluid dynamics was a very attractive research problem for computational scientist does not make the numerical methods of prime usage for fluid dynamics simulations. However, there was a complete link between the development of numerical methods for PDEs and the simulation of fluid flow problems.
Since the 1940s, analytical solutions to most fluid dynamics problems, especially those arising in aerodynamics, were readily available for simplified or idealized situations. However, it was soon realized that a wide range of problems still need to be solved especially with the increasing demands of the industry. This was an incentive for the inception of asymptotic/semi-analytical methods. These methods included perturbation techniques (asymptotics) and scale similarity analysis which found appreciable applications for viscous flow problems and inviscid compressible flow.
Numerical methods, on the other hand, were known since the time of Newton in the 1700s. Methods for the solution of ODEs or PDEs were conceptually conceived, but only on paper. With the absence of the personal computer, there was no way for the application of these techniques.
See this blog post with some information regarding the important people contributing to Fluid Mechanics.
To be continued - waiting for permissions to use a couple of references