Listen to Euler describing his equations: Everything that the theory of fluids contains is embodied in the two equations I have formulated. It is not the laws of mechanics that we lack in order to pursue this research, only the analysis which has not been sufficiently developed for this purpose. We have to wait until the age of the computer to solve the equations.
Does computational science represent a paradigm shift in the sense that questions and answers are radically changed? Let us make a case study of the fluid dynamics of slighty viscous incompressible flow of basic importance in aero/hydrodynamics.
Euler formulated the basic equations expressing conservation of mass and momentum as the Euler Equations in 1752, and conjectured that all of slightly viscous incompressible flow is “embodied” in the equations so that theoretical fluid dynamics thus reduces to solving his equations. This grand Euler paradigm was however just a dream since solving the Euler equations in general was impossible and particular potential solutions showed to be unphysical as expressed in d’Alembert’s paradox of zero drag of a potential solutions at variance with observation.
But Euler’s dream can come true today, as demonstrated in Computational Turbulent Incompressible Flow: It is possible to computationally solve of the Euler equations using millions of mesh points and obtain results in close correspondence with observation in large generality.
Solutions turn out to be partly turbulent, but with large scale structures which can be captured conceptually: For example, the important area of bluff body flow can be described as potential solutions modified by turbulent 3d rotational separation.
Is this a paradigm shift in the sense of Kuhn, as a shift from “normal science” of “puzzle solving”? Yes, it is because the normal science of slightly viscous incompressible flow has consisted of a set of analytical solutions of the Euler equations for special cases, which has filled books since the time of Euler into present time.
Since turbulence is beyond analytical description, all these “puzzle solutions” have missed essential physics which has caused an split between theory and practice in fluid dynamics into pseudo-science.
When now the Euler equations can be solved computationally in general, the “puzzle solutions” comprising “normal science” no longer have any role to play, and a shift of paradigm is awaiting: Stay tuned! A first catch is a New Theory of Flight.