Von Kármán’s Similarity Hypothesis and Its Application to Turbulent Boundary Layer Theory

Von Kármán’s similarity hypothesis, that the turbulent fluctuations in two-dimensional parallel mean flow are similar at all points in the flow, is extended to two-dimensional turbulent boundary layer flow. The Boussinesq hypothesis which introduced the eddy viscosity for turbulent mean flow and Prandtl’s mixing length model for the eddy viscosity for two-dimensional parallel mean flow are briefly reviewed. The derivation of mixing length for two-dimensional parallel mean flow based on von Kármán’s similarity hypothesis is outlined. The physical interpretation of von Kármán’s mixing length given by Betz is provided alongside a German-to-English translation of Betz’s original paper. The extension of von Kármán’s mixing length to two-dimensional turbulent boundary layer flow is presented by defining characteristic scales for the mean flow in the boundary layer and for the fluctuations, and by imposing the boundary layer approximation. Finally, finding von Kármán-like mixing lengths using dimensional analysis is presented.