Componentality of velocity-derivatives in wall turbulence (and algebraic proof of Lumley's triangle)

The paper studies the statistics of velocity derivatives in wall turbulence. Componentality is studied by anisotropy invariant mapping ( AIM ) which can be applied to various physically significant positive-semidefinite (or negative-semidefinite) order-2 symmetric tensors constructed from velocity-derivatives statistics. In this respect, an alternative algebraic proof of Lumley's (1978 Adv. Appl. Mech. 18 123-176) realizability triangle is formulated including proof of the converse theorem. AIM of tensors constructed by the gradient (dissipation, vorticity-covariance), or the Hessian (destruction-of-dissipation, destruction-of-vorticity, viscous-acceleration-covariance) of the fluctuating velocities, that are significant in turbulence dynamics are then studied, using DNS data for turbulent plane channel flow, highlighting the influence of the derivative-order on near wall componentality. Particular attention is also given to the orientation of the principal axes, which is shown to be an indispensable complement to AIM .