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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)’.
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Resolvent-based predictions for turbulent flow over anisotropic permeable substrates
Recent simulations indicate that streamwise-preferential porous materials have the potential to reduce drag in wall-bounded turbulent flows (Gómez-de-Segura & García-Mayoral, J. Fluid Mech. , vol. 875, 2019, pp. 124–172). This paper extends the resolvent formulation to study the effect of such anisotropic permeable substrates on turbulent channel flow. Under the resolvent formulation, the Fourier-transformed Navier–Stokes equations are interpreted as a linear forcing–response system. The nonlinear terms are considered the endogenous forcing in the system that gives rise to a velocity and pressure response. A gain-based decomposition of the forcing–response transfer function – the resolvent operator – identifies response modes (resolvent modes) that are known to reproduce important structural and statistical features of wall-bounded turbulent flows. The effect of permeable substrates is introduced in this framework using the volume-averaged Navier–Stokes equations and a generalized form of Darcy's law. Substrates with high streamwise permeability and low spanwise permeability are found to suppress the forcing–response gain for the resolvent mode that serves as a surrogate for the energetic near-wall cycle. This reduction in mode gain is shown to be consistent with the drag reduction trends predicted by theory and observed in numerical simulations. Simulation results indicate that drag reduction is limited by the emergence of spanwise rollers resembling Kelvin–Helmholtz vortices beyond a threshold value of wall-normal permeability. The resolvent framework also predicts the conditions in which such energetic spanwise-coherent rollers emerge. These findings suggest that a limited set of resolvent modes can serve as the building blocks for computationally efficient models that enable the design and optimization of permeable substrates for passive turbulence control.
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- Award ID(s):
- 1943105
- PAR ID:
- 10309747
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 913
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2020.1169
