Curvature Law of the Wall for Swirling Axial Flows in Rotating Machinery

A new law of the wall accounting for curvature effects in swirling axial flows is derived. The influence of the curvature on the turbulence mixing lengths in both axial and tangential directions is examined theoretically using the Reynolds stress transport equations. For equilibrium flows with weak curvature, identical mixing lengths are derived for the axial and tangential directions. Additionally, the effect of finite local curvature and shear stress ratio on the near-wall velocities is systematically explored. It is found that the curvature effect in swirling axial flows is suppressed by a factor of $1∕(1+σw2)$ compared to that in curved channel flows, where $σw$ is the ratio of the axial to swirl shear stress. For a given curvature radius, the maximum velocity deviation occurs when the axial-to-swirl shear stress ratio is zero. Finally, the performance of the new curvature law is evaluated by implementing it as a wall function in a well-established CFD code. The new wall function provides improved agreement for swirl velocity distributions inside labyrinth cavities in comparison with existing experimental laser Doppler anemometry measurements.