Returning to Ludwig Prandtl’s One-Equation Model

In my turbulence class this semester, I recently reviewed Prandtl’s one-equation model, which was developed over 20 years since the time of boundary theory in the early 1900s. The major paper by Ludwig Prandtl was published in the early 1940s. He presented the first one-equation turbulence model for the closure of the boundary layer equations, specifically for incompressible flow. He calibrated coefficients via channel measurements, which were carefully conducted. Therefore, the prediction of the model in terms of modeled coefficients corresponded exactly with his experiments. Many later investigators, through the early 1990s and one particular recent paper in the Royal Society published in 2023, reexamined the model and its analysis. Many have programmed this model in computational fluid dynamics, and it is the basis of many one-equation models, especially those based on a k or turbulent kinetic energy equation. However, it is generally shown that the model is not predictive for many flows without modification. Some have misattributed Prandtl’s model to a reduction of Andrei Kolmogorov’s two-equation model from 1945. Prandtl was unlikely to know about the Soviet invention during World War II, and it can be truly attributed to him. We should attribute to Ludwig Prandtl the invention of the world’s first one-equation turbulence model, even though it may be flawed, this is only because of the lack of the contemporary digital computer.