Famed physicist and aerodynamicist Dr. Theodore von Karman made significant contributions to the study of fluid dynamics. Turbulent flow refers to a gas which is moving at such a speed, that it ceases to flow uniformly (laminar flow), and breaks up into turbulence. Turbulent flow is bad for aircraft, because it induces buffeting 9uneven forces on the aircraft). My question: did Dr. von Karman solve the turbulence problem mathematically? Or is turbulent flow and example of chaotic behavior, and cannot be solved in closed form?
My understanding of the state of the art in understanding turbulence is that turbulent flow is chaotic and generally has no closed form solutions. I think people use the Navier-Stokes equations and, often, use a “turbulence model” such as the Kappa Epsilon or the Spalart-Almaraz or any of a half dozen other popular ones. In these models new field variables represent turbulent energy, and the viscosity of the fluid is made fictitious in a way that depends on the turbulent energy. This changes the shear stress transport across streamlines but allows the modeled flow to be nonturbulent. The idea is to have fictitious viscosity in fictitious laminar flow that transports shear stress the same way that the true viscosity and real, but unmodelable, turbulent flow do.
That this can be done is called the Bousinesque Hypothesis, from which the approaches start.
Napier gave a very good answer, and I won’t waste time by repeating him. But maybe a can add a couple points.
The bottom line is that, no, we don’t have closed-form solutions for turbulence modeling. It is chaotic behavior, the effects of which can only be approximated. Different turbulence models used in CFD (computational fluid dynamics) emphasize the modeling accuracy of different aspects of the flow. If you care mostly about heat transfer, you’d use a particular model. If you care about mixing and mass transport, you might use a different model.
There’s no reason the Navier-Stokes equations in their pure form couldn’t model turbulence to any arbitrary level of precision if you had the computational resources. The issue isn’t with the equations; they’re quite theoretically sound and complete. The problem arises from not having good initial conditions to start the solution. The N-S equations are full of partial derivatives that can be highly sensitive to input errors. So to model something even close to accurately would require having detailed understanding of the flow before you even start the solution. That’s not practical, and so the “shortcut” provided by turbulence models is used.