The Karman constant is inversely proportional to the number of stagnation points at the upper edge of the buffer layer and is therefore not a universal constant
Seminar Room 1, Newton Institute
By an adaptation of the Rice theorem to three-dimensional incompressible vector fields we show that the average distance between turbulent velocity stagnation points at a certain distance from the wall is proportional to the Taylor microscale at that distance. Then, by using this result in conjunction with the balance between kinetic energy dissipation and production in the log-layer we calculate the Karman constant as a function of other constants. We show that the Karman constant is inversely proportional to the number of turbulent velocity stagnation points at the edge of the log-layer that is closest to the wall.
We perform three different Direct Numerical Simulations (DNS) of fully developed turbulent channel flows to test our formula. In two of these DNS the flow is forced at the wall in two different ways and in the remaining third simulation it is not. The proportionality between the inverse Karman constant and the number $C_s$ of turbulent velocity stagnation points at the upper edge of the buffer layer is observed in all three simulations even though the values of the Karman constant and $C_s$ are different in each one of these simulations. Our formula is therefore able to predict and explain why the Karman constant is not a universal constant.