Karman constant and accurate mean flow prediction in a turbulent pipe

Zhen-Su She, Xi Chen, You Wu, Fazle Hussain

Published 2011-12-29, updated 2011-12-30Version 2

The Karman constant \kappa - widely used in atmospheric science and engineering turbulence modelling, and proposed by Prandtl in 1925 and von Karman in 1930 to describe the mean velocity of a turbulent wall-bounded flow - leads to a logarithmic profile in an overlap region near the wall. For over eighty years, its value was believed to be ~0.41. But more recently, many argue that it is not a constant, because of measured variations in different flows and at different Reynolds numbers (Re). Here, a multi-layer analytic theory is shown to lead to a re-interpretation of \kappa as a global constant for both the overlap region and outer flow, and to yield a new method for its measurement. The newly determined value is 0.45 for both channel and pipe. It is shown that this new \kappa, together with other wall constants, yields a 99% accuracy in the prediction of mean velocity data at all points in high Re (up to 40 million) pipe flow. The theory also describes finite Re effect, and discovers a transition at the friction Re (i.e. Re_\tau)=5000. An accurate model for the prediction of turbulent transport in canonical pipe and channel flows is achieved here, and we propose the model to be valid for a wide class of turbulent flows.