Near-wall Lamb vector and its temporal–spatial evolution in the viscous sublayer of wall-bounded flows

Abstract

This paper gives the relation between the Lamb vector and fundamental surface quantities in the vicinity of a no-slip flat wall using the Taylor-series expansion solution of the Navier–Stokes equations for incompressible viscous flow. The wall-normal component of the Lamb vector is dominated by the boundary enstrophy at the first order and the boundary enstrophy flux at the second order. The tangential Lamb vector is contributed by the terms related to both the boundary vorticity divergence and the skin friction divergence. Then, the derived relation is validated in the three-dimensional nonorthogonal Hiemenz flow (an attachment line flow) and a single-phase turbulent channel flow simulated using the lattice Boltzmann method. For this Hiemenz flow where the boundary vorticity divergence vanishes, the skin friction divergence-related term dominates the distribution of the near-wall tangential Lamb vector. In the turbulent channel flow, both the skin friction divergence and boundary vorticity divergence-related terms have significant contributions to the streamwise component of the Lamb vector, which are associated with the strong wall-normal velocity events (SWNVEs) in the viscous sublayer. In contrast, the boundary vorticity divergence-related terms have the main contribution to the spanwise component of the Lamb vector. Furthermore, the temporal–spatial evolution of the kinetic energy of the Lamb vector (half of the inner product of the Lamb vector denoted by J) is studied. In the turbulent channel flow, the SWNVEs are the main contributors to the temporal–spatial evolution of J in the viscous sublayer. This evolution is dominated by the viscous dissipation effect due to the gradient of the Lamb vector and the coupling effect between a viscous source term and the Lamb vector. The relations presented in this paper could be useful in understanding the physical mechanisms of the initial formation and evolution of the Lamb vector in the viscous sublayer of wall-bounded turbulence.