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Self-regularization in turbulence from the Kolmogorov 4/5-law and alignment
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 more »
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Publication Date:
NSF-PAR ID:
10327734
Journal Name:
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume:
380
Issue:
2226
ISSN:
1364-503X
3. Transition from laminar to turbulent flow occurring over a smooth surface is a particularly important route to chaos in fluid dynamics. It often occurs via sporadic inception of spatially localized patches (spots) of turbulence that grow and merge downstream to become the fully turbulent boundary layer. A long-standing question has been whether these incipient spots already contain properties of high-Reynolds-number, developed turbulence. In this study, the question is posed for geometric scaling properties of the interface separating turbulence within the spots from the outer flow. For high-Reynolds-number turbulence, such interfaces are known to display fractal scaling laws with a dimension$D≈7/3$, where the 1/3 excess exponent above 2 (smooth surfaces) follows from Kolmogorov scaling of velocity fluctuations. The data used in this study are from a direct numerical simulation, and the spot boundaries (interfaces) are determined by using an unsupervised machine-learning method that can identify such interfaces without setting arbitrary thresholds. Wide separation between small and large scales during transition is provided by the large range of spot volumes, enabling accurate measurements of the volume–area fractal scaling exponent. Measurements show a dimension of$D=2.36±0.03$over almost 5 decades of spot volume, i.e., trends fully consistent with high-Reynolds-number turbulence. Additional observations pertainingmore »
4. We experimentally investigate the rise velocity of finite-sized bubbles in turbulence with a high energy dissipation rate of $\unicode[STIX]{x1D716}\gtrsim 0.5~\text{m}^{2}~\text{s}^{-3}$ . In contrast to a 30–40 % reduction in rise velocity previously reported in weak turbulence (the Weber number ( $We$ ) is much smaller than the Eötvös number ( $Eo$ ); $We\ll 1 5. Dimensional analysis suggests that the dissipation length scale ($\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 TKEmore »