skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Stochastic Lagrangian dynamics of vorticity. Part 2. Application to near-wall channel-flow turbulence
We use an online database of a turbulent channel-flow simulation at $$Re_\tau =1000$$ (Graham et al. J. Turbul. , vol. 17, issue 2, 2016, pp. 181–215) to determine the origin of vorticity in the near-wall buffer layer. Following an experimental study of Sheng et al. ( J. Fluid Mech. , vol. 633, 2009, pp.17–60), we identify typical ‘ejection’ and ‘sweep’ events in the buffer layer by local minima/maxima of the wall stress. In contrast to their conjecture, however, we find that vortex lifting from the wall is not a discrete event requiring $$\sim$$ 1 viscous time and $$\sim$$ 10 wall units, but is instead a distributed process over a space–time region at least $$1\sim 2$$ orders of magnitude larger in extent. To reach this conclusion, we exploit a rigorous mathematical theory of vorticity dynamics for Navier–Stokes solutions, in terms of stochastic Lagrangian flows and stochastic Cauchy invariants, conserved on average backward in time. This theory yields exact expressions for vorticity inside the flow domain in terms of vorticity at the wall, as transported by viscous diffusion and by nonlinear advection, stretching and rotation. We show that Lagrangian chaos observed in the buffer layer can be reconciled with saturated vorticity magnitude by ‘virtual reconnection’: although the Eulerian vorticity field in the viscous sublayer has a single sign of spanwise component, opposite signs of Lagrangian vorticity evolve by rotation and cancel by viscous destruction. Our analysis reveals many unifying features of classical fluids and quantum superfluids. We argue that ‘bundles’ of quantized vortices in superfluid turbulence will also exhibit stochastic Lagrangian dynamics and satisfy stochastic conservation laws resulting from particle relabelling symmetry.  more » « less
Award ID(s):
1633124
PAR ID:
10204877
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Journal of Fluid Mechanics
Volume:
901
ISSN:
0022-1120
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Journal of Fluid Mechanics)
    null (Ed.)
    Prior mathematical work of Constantin & Iyer ( Commun. Pure Appl. Maths , vol. 61, 2008, pp. 330–345; Ann. Appl. Probab. , vol. 21, 2011, pp. 1466–1492) has shown that incompressible Navier–Stokes solutions possess infinitely many stochastic Lagrangian conservation laws for vorticity, backward in time, which generalize the invariants of Cauchy ( Sciences mathématiques et physique , vol. I, 1815, pp. 33–73) for smooth Euler solutions. We reformulate this theory for the case of wall-bounded flows by appealing to the Kuz'min ( Phys. Lett. A , vol. 96, 1983, pp. 88–90)–Oseledets ( Russ. Math. Surv. , vol. 44, 1989, p. 210) representation of Navier–Stokes dynamics, in terms of the vortex-momentum density associated to a continuous distribution of infinitesimal vortex rings. The Constantin–Iyer theory provides an exact representation for vorticity at any interior point as an average over stochastic vorticity contributions transported from the wall. We point out relations of this Lagrangian formulation with the Eulerian theory of Lighthill (Boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), 1963, pp. 46–113)–Morton ( Geophys. Astrophys. Fluid Dyn. , vol. 28, 1984, pp. 277–308) for vorticity generation at solid walls, and also with a statistical result of Taylor ( Proc. R. Soc. Lond. A , vol. 135, 1932, pp. 685–702)–Huggins ( J. Low Temp. Phys. , vol. 96, 1994, pp. 317–346), which connects dissipative drag with organized cross-stream motion of vorticity and which is closely analogous to the ‘Josephson–Anderson relation’ for quantum superfluids. We elaborate a Monte Carlo numerical Lagrangian scheme to calculate the stochastic Cauchy invariants and their statistics, given the Eulerian space–time velocity field. The method is validated using an online database of a turbulent channel-flow simulation (Graham et al. , J. Turbul. , vol. 17, 2016, pp. 181–215), where conservation of the mean Cauchy invariant is verified for two selected buffer-layer events corresponding to an ‘ejection’ and a ‘sweep’. The variances of the stochastic Cauchy invariants grow exponentially backward in time, however, revealing Lagrangian chaos of the stochastic trajectories undergoing both fluid advection and viscous diffusion. 
    more » « less
  2. ; ; (, Journal of Fluid Mechanics)
    While it has long been recognized that Lagrangian drift at the ocean surface plays a critical role in the kinematics and dynamics of upper ocean processes, only recently has the contribution of wave breaking to this drift begun to be investigated through direct numerical simulations (Deike et al. ,  J. Fluid Mech. , vol. 829, 2017, pp. 364–391; Pizzo et al. ,  J. Phys. Oceanogr. , vol. 49(4), 2019, pp. 983–992). In this work, laboratory measurements of the surface Lagrangian transport due to focusing deep-water non-breaking and breaking waves are presented. It is found that wave breaking greatly enhances mass transport, compared to non-breaking focusing wave packets. These results are in agreement with the direct numerical simulations of Deike  et al. ( J. Fluid Mech. , vol. 829, 2017, pp. 364–391), and the increased transport due to breaking agrees with their scaling argument. In particular, the transport at the surface scales with $$S$$ , the linear prediction of the maximum slope at focusing, while the surface transport due to non-breaking waves scales with $$S^{2}$$ , in agreement with the classical Stokes prediction. 
    more » « less
  3. ; ; (, Journal of Fluid Mechanics)
    We investigate the effect of constant-vorticity background shear on the properties of wavetrains in deep water. Using the methodology of Fokas ( A Unified Approach to Boundary Value Problems , 2008, SIAM), we derive a higher-order nonlinear Schrödinger equation in the presence of shear and surface tension. We show that the presence of shear induces a strong coupling between the carrier wave and the mean-surface displacement. The effects of the background shear on the modulational instability of plane waves is also studied, where it is shown that shear can suppress instability, although not for all carrier wavelengths in the presence of surface tension. These results expand upon the findings of Thomas et al.  ( Phys. Fluids , vol. 24 (12), 2012, 127102). Using a modification of the generalized Lagrangian mean theory in Andrews & McIntyre ( J. Fluid Mech. , vol. 89, 1978, pp. 609–646) and approximate formulas for the velocity field in the fluid column, explicit, asymptotic approximations for the Lagrangian and Stokes drift velocities are obtained for plane-wave and Jacobi elliptic function solutions of the nonlinear Schrödinger equation. Numerical approximations to particle trajectories for these solutions are found and the Lagrangian and Stokes drift velocities corresponding to these numerical solutions corroborate the theoretical results. We show that background currents have significant effects on the mean transport properties of waves. In particular, certain combinations of background shear and carrier wave frequency lead to the disappearance of mean-surface mass transport. These results provide a possible explanation for the measurements reported in Smith ( J. Phys. Oceanogr. , vol. 36, 2006, pp. 1381–1402). Our results also provide further evidence of the viability of the modification of the Stokes drift velocity beyond the standard monochromatic approximation, such as recently proposed in Breivik et al.  ( J. Phys. Oceanogr. , vol. 44, 2014, pp. 2433–2445) in order to obtain a closer match to a range of complex ocean wave spectra. 
    more » « less
  4. ; ; (, Journal of Fluid Mechanics)
    We propose a more conservative, physically-intuitive criterion, namely, the boundary enstrophy flux ( $BEF$ ), to characterise leading-edge-type dynamic stall onset in incompressible flows. Our results are based on wall-resolved large-eddy simulations of pitching aerofoils, with fine spatial and temporal resolution around stall onset. We observe that $|BEF|$ reaches a maximum within the stall onset regime identified. By decomposing the contribution to $BEF$ from the flow field, we find that the dominant contribution arises from the laminar leading edge region, due to the combined effect of large clockwise vorticity and favourable pressure gradient. A relatively small contribution originates from the transitional/turbulent laminar separation bubble (LSB) region, due to LSB-induced counter-clockwise vorticity and adverse pressure gradient. This results in $BEF$ being nearly independent of the integration length as long as the region very close to the leading edge is included. This characteristic of $BEF$ yields a major advantage in that the effect of partial or complete inclusion of the noisy LSB region can be filtered out, without changing the $BEF$ peak location in time significantly. Next, we analytically relate $BEF$ to the net wall shear and show that its critical value ( $$=\max (|BEF|)$$ ) corresponds to the instant of maximum net shear prevailing at the wall. Finally, we have also compared $BEF$ with the leading edge suction parameter ( $LESP$ ) (Ramesh et al. , J. Fluid Mech. , vol. 751, 2014, pp. 500–538) and find that the former reaches its maximum value between $$0.3^{\circ }$$ and $$0.8^{\circ }$$ of rotation earlier. 
    more » « less
  5. ; ; (, Journal of Fluid Mechanics)
    We present high-resolution three-dimensional (3-D) direct numerical simulations of breaking waves solving for the two-phase Navier–Stokes equations. We investigate the role of the Reynolds number ( Re , wave inertia relative to viscous effects) and Bond number ( Bo , wave scale over the capillary length) on the energy, bubble and droplet statistics of strong plunging breakers. We explore the asymptotic regimes at high Re and Bo , and compare with laboratory breaking waves. Energetically, the breaking wave transitions from laminar to 3-D turbulent flow on a time scale that depends on the turbulent Re up to a limiting value $$Re_\lambda \sim 100$$ , consistent with the mixing transition in other canonical turbulent flows. We characterize the role of capillary effects on the impacting jet and ingested main cavity shape and subsequent fragmentation process, and extend the buoyant-energetic scaling from Deike et al. ( J. Fluid Mech. , vol. 801, 2016, pp. 91–129) to account for the cavity shape and its scale separation from the Hinze scale, $$r_H$$ . We confirm two regimes in the bubble size distribution, $$N(r/r_H)\propto (r/r_H)^{-10/3}$$ for $$r>r_H$$ , and $$\propto (r/r_H)^{-3/2}$$ for $$r 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email