An official website of the United States government
A defining feature of three-dimensional hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon—anomalous dissipation—is sometimes called the ‘zeroth law of turbulence’ as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions S p ( ℓ ) := ⟨ | δ ℓ u | p ⟩ exhibit persistent power-law scaling in the inertial range, namely S p ( ℓ ) ∼ | ℓ | ζ p for exponents ζ p > 0 over an ever increasing (with Reynolds) range of scales. This behaviour indicates that the velocity field retains some fractional differentiability uniformly in the Reynolds number. The Kolmogorov 1941 theory of turbulence predicts that ζ p = p / 3 for all p and Onsager’s 1949 theory establishes the requirement that ζ p ≤ p / 3 for p ≥ 3 for consistency with the zeroth law. Empirically, ζ 2 ⪆ 2 / 3 and ζ 3 ⪅ 1 , suggesting that turbulent Navier–Stokes solutions approximate dissipative weak solutions of the Euler equations possessing (nearly) the minimal degree of singularity required to sustain anomalous dissipation. In this note, we adopt an experimentally supported hypothesis on the anti-alignment of velocity increments with their separation vectors and demonstrate that the inertial dissipation provides a regularization mechanism via the Kolmogorov 4/5-law. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
more »
« less
The Onsager theory of wall-bounded turbulence and Taylor’s momentum anomaly
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
- Award ID(s):
- 1738918
- PAR ID:
- 10338053
- Date Published:
- Journal Name:
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 380
- Issue:
- 2218
- ISSN:
- 1364-503X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Recent works on wall-bounded flows have corroborated the coexistence of wall-attached eddies, whose statistical features are predicted through Townsend's attached-eddy hypothesis (AEH), and very-large-scale motions (VLSMs). Furthermore, it has been shown that the presence of wall-attached eddies within the logarithmic layer is linked to the appearance of an inverse-power-law region in the streamwise velocity energy spectra, upon significant separation between outer and viscous scales. In this work, a near-neutral atmospheric surface layer is probed with wind light detection and ranging to investigate the contributions to the streamwise velocity energy associated with wall-attached eddies and VLSMs for a very-high-Reynolds-number boundary layer. Energy and linear coherence spectra (LCS) of the streamwise velocity are interrogated to identify the spectral boundaries associated with eddies of different typologies. Inspired by the AEH, an analytical model for the LCS associated with wall-attached eddies is formulated. The experimental results show that the identification of the wall-attached-eddy energy contribution through the analysis of the energy spectra leads to an underestimate of the associated spectral range, maximum height attained and turbulence intensity. This feature is due to the overlap of the energy associated with VLSMs obscuring the inverse-power-law region. The LCS analysis estimates wall-attached eddies with a streamwise/wall-normal ratio of about 14.3 attaining a height of about 30 % of the outer scale of turbulence.more » « less
-
The dissolution of minute concentration of polymers in wall-bounded flows is well-known for its unparalleled ability to reduce turbulent friction drag. Another phenomenon, elasto-inertial turbulence (EIT), has been far less studied even though elastic instabilities have already been observed in dilute polymer solutions before the discovery of polymer drag reduction. EIT is a chaotic state driven by polymer dynamics that is observed across many orders of magnitude in Reynolds number. It involves energy transfer from small elastic scales to large flow scales. The investigation of the mechanisms of EIT offers the possibility to better understand other complex phenomena such as elastic turbulence and maximum drag reduction. In this review, we survey recent research efforts that are advancing the understanding of the dynamics of EIT. We highlight the fundamental differences between EIT and Newtonian/inertial turbulence from the perspective of experiments, numerical simulations, instabilities, and coherent structures. Finally, we discuss the possible links between EIT and elastic turbulence and polymer drag reduction, as well as the remaining challenges in unraveling the self-sustaining mechanism of EIT.more » « less
-
null (Ed.)Dimensional analysis suggests that the dissipation length scale ( $$\ell _{{\it\epsilon}}=u_{\star }^{3}/{\it\epsilon}$$ ) is the appropriate scale for the shear-production range of the second-order streamwise structure function in neutrally stratified turbulent shear flows near solid boundaries, including smooth- and rough-wall boundary layers and shear layers above canopies (e.g. crops, forests and cities). These flows have two major characteristics in common: (i) a single velocity scale, i.e. the friction velocity ( $$u_{\star }$$ ) and (ii) the presence of large eddies that scale with an external length scale much larger than the local integral length scale. No assumptions are made about the local integral scale, which is shown to be proportional to $$\ell _{{\it\epsilon}}$$ for the scaling analysis to be consistent with Kolmogorov’s result for the inertial subrange. Here $${\it\epsilon}$$ is the rate of dissipation of turbulent kinetic energy (TKE) that represents the rate of energy cascade in the inertial subrange. The scaling yields a log-law dependence of the second-order streamwise structure function on ( $$r/\ell _{{\it\epsilon}}$$ ), where $$r$$ is the streamwise spatial separation. This scaling law is confirmed by large-eddy simulation (LES) results in the roughness sublayer above a model canopy, where the imbalance between local production and dissipation of TKE is much greater than in the inertial layer of wall turbulence and the local integral scale is affected by two external length scales. Parameters estimated for the log-law dependence on ( $$r/\ell _{{\it\epsilon}}$$ ) are in reasonable agreement with those reported for the inertial layer of wall turbulence. This leads to two important conclusions. Firstly, the validity of the $$\ell _{{\it\epsilon}}$$ -scaling is extended to shear flows with a much greater imbalance between production and dissipation, indicating possible universality of the shear-production range in flows near solid boundaries. Secondly, from a modelling perspective, $$\ell _{{\it\epsilon}}$$ is the appropriate scale to characterize turbulence in shear flows with multiple externally imposed length scales.more » « less
-
Abstract Coral reef roughness produces turbulent boundary layers and bottom stresses that are important for reef metabolism monitoring and reef circulation modeling. However, there is some uncertainty as to whether field methods for estimating bottom stress are applicable in shallow canopy environments as found on coral reefs. Friction velocities () and drag coefficients () were estimated using five independent methods and compared across 14 sites on a shallow forereef (2–9 m deep) in Palau with large and spatially variable coral roughness elements (0.4–1 m tall). The methods included the following: (a) momentum balance closure, (b) log‐fitting to velocity profiles, (c) Reynolds stresses, (d) turbulence dissipation, and (e) roughness characterization from digital elevation models (DEMs). Both velocity profiles and point turbulence measurements indicated good agreement with log‐layer scaling, suggesting that measurements were taken within a well‐developed turbulent boundary layer and that canopy effects were minimal. However, estimated from the DEMs, momentum budget and log‐profile fitting were consistently larger than those estimated from direct turbulence measurements. Near‐bed Reynolds stresses only contributed about 1/3 of the total bottom stress and drag produced by the reef. Thus, effects of topographical heterogeneity that induce mean velocity fluxes, dispersive stresses, and form drag are expected to be important. This decoupling of total drag and local turbulence implies that both rates of mass transfer as well as values of fluxes inferred from concentration measurements may be proportional to smaller, turbulence‐derived values of rather than to those based on larger‐scale flow structure.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rsta.2021.0079
