Wall Bounded Flows and a General Proof of the Validity of the Universal Logarithmic Law of the Wall

Abstract

Abstract The logarithmic law of the wall is usually derived for the flat plate assuming stationary, two-dimensional fully developed flow with no external pressure gradient. The Prandtl mixing length model for the turbulence is applied, which assumes homogeneous turbulence and two empirical constants, and the logarithmic wall law is derived. It is than stated in the textbooks that it is universally valid without a proof. As a justification experimental evidence is shown. First this proof will be shown in detail. Than a more general approach based on similarity considerations is made to show the universal validity of the logarithmic law of the wall. Starting from the Navier-Stokes equation a general non dimensional form of this equation is derived showing its dependency from four non-dimensional numbers, the Strouhal, Euler, Reynolds and the Froude number. Then wall bounded laminar flows are analyzed by dimensional analysis. The laminar boundary length and time scales are derived and used to non-dimensionalize the Navier-Stokes equation. With this specific non-dimensionalization for the laminar boundary layer a more specific non dimensional Navier-Stokes equation is derived. Then the high Reynolds limit is taken with considerations of orders of magnitude and the boundary layer equations are derived. Finally, for turbulent near wall flows a dimensional analysis is made and the corresponding near wall non-dimensional velocities and coordinates y+ and u+ are derived from the Buckingham-Π theorem. Using these variables to non-dimensionalize the Navier-Stokes equations in the near wall turbulent region the third author Malcherek showed that the so derived non-dimensional Navier-Stokes equations do not depend on any non-dimensional number and has a unique solution. Hence, the logarithmic law of the wall must be universally valid, without any simplification, any turbulence model, empirical constant or further assumptions. In such a way the students do not have to believe anymore in the universality of the logarithmic law of the wall based on empirical evidence only, now this fact has been proven by the third author Malcherek and the larger context has been elaborated by all authors for an advanced teaching of wall bounded flows.