Drag for wall-bounded flows is directly related to the spatial flux of spanwise vorticity outward from the wall. In turbulent flows a key contribution to this wall-normal flux arises from nonlinear advection and stretching of vorticity, interpretable as a cascade. We study this process using numerical simulation data of turbulent channel flow at friction Reynolds number
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. The net transfer from the wall of spanwise vorticity created by downstream pressure drop is due to two large opposing fluxes, one which is ‘down-gradient’ or outward from the wall, where most vorticity concentrates, and the other which is ‘up-gradient’ or toward the wall and acting against strong viscous diffusion in the near-wall region. We present evidence that the up-gradient/down-gradient transport occurs by a mechanism of correlated inflow/outflow and spanwise vortex stretching/contraction that was proposed by Lighthill. This mechanism is essentially Lagrangian, but we explicate its relation to the Eulerian anti-symmetric vorticity flux tensor. As evidence for the mechanism, we study (i) statistical correlations of the wall-normal velocity and of wall-normal flux of spanwise vorticity, (ii) vorticity flux cospectra identifying eddies involved in nonlinear vorticity transport in the two opposing directions and (iii) visualizations of coherent vortex structures which contribute to the transport. The ‘D-type’ vortices contributing to down-gradient transport in the log layer are found to be attached, hairpin-type vortices. However, the ‘U-type’ vortices contributing to up-gradient transport are detached, wall-parallel, pancake-shaped vortices with strong spanwise vorticity, as expected by Lighthill's mechanism. We discuss modifications to the attached eddy model and implications for turbulent drag reduction.$Re_\tau =1000$ Free, publicly-accessible full text available November 10, 2024 -
We discuss the Onsager theory of wall-bounded turbulence, analysing the momentum dissipation anomaly hypothesized by Taylor. Turbulent drag laws observed with both smooth and rough walls imply ultraviolet divergences of velocity gradients. These are eliminated by a coarse-graining operation, filtering out small-scale eddies and windowing out near-wall eddies, thus introducing two arbitrary regularization length-scales. The regularized equations for resolved eddies correspond to the weak formulation of the Navier–Stokes equation and contain, in addition to the usual turbulent stress, also an inertial drag force modelling momentum exchange with unresolved near-wall eddies. Using an Onsager-type argument based on the principle of renormalization group invariance, we derive an upper bound on wall friction by a function of Reynolds number determined by the modulus of continuity of the velocity at the wall. Our main result is a deterministic version of Prandtl’s relation between the Blasius − 1 / 4 drag law and the 1/7 power-law profile of the mean streamwise velocity. At higher Reynolds, the von Kármán–Prandtl drag law requires instead a slow logarithmic approach of velocity to zero at the wall. We discuss briefly also the large-eddy simulation of wall-bounded flows and use of iterative renormalization group methods to establish universal statistics in the inertial sublayer. This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.more » « less
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A displacement thickness based inflow generation method, for simulation of a developing turbulent boundary layer, is proposed. Following existing rescaling/recycling methods, velocities from a plane sufficiently downstream of the inlet are recycled back and used as the inflow after re-scaling based on inner and outer length-scales. The inner length-scale is based on the viscous length-scale (for smooth walls) or surface specific scales (for rough walls). Prior recycling methods for smooth and rough boundary layers typically use d99 as the outer length-scale. Since d99 is a threshold based quantity, it is strongly dependent on the mean velocity profile and can have large undesired fluctuations, particularly if the profile shape is atypical or unsteady. Here, we propose the use of profile integrated quantities such as the displacement thickness (d1) to obtain a ‘surrogate’ for d99 in order to mitigate the adverse effects of having to determine the outer scale from a point-wise measurement of the mean velocity profile. The outer length- scale at the downstream plane is determined based on the local displacement thickness and higher-order moments of the integrated velocity profile. The inlet displacement thick- ness is fixed at a desired value and the outer length-scale at the inlet is determined through an iterative method. The use of high-order moments of the velocity profile is tested a- priori on DNS data for a developing boundary layer. Also, an initial application to LES over a surface with roughness elements is presented.more » « less