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 1. General theory for viscous, incompressible fluids
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
Award ID(s):
1633124
PAR ID:
10204876
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.)
    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
  2. ; ; (, 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
  3. ; ; (, 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
  4. ; (, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)
    null (Ed.)
    Abstract Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems. 
    more » « less
  5. ; (, Journal of Fluid Mechanics)
    A computational study of vorticity reconnection, associated with the breaking and reconnection of vortex lines, during vortex cutting by a blade is reported. A series of Navier–Stokes simulations of vortex cutting with different values of the vortex strength are described, and the different phases in the vortex cutting process are compared to those of the more traditional vortex tube reconnection process. Each of the three phases of vortex tube reconnection described by Melander & Hussain ( Phys. Fluids  A, vol. 1(4), 1989, pp. 633–635) are found to have counterparts in the vortex cutting problem, although we also point out numerous differences in the detailed mechanics by which these phases are achieved. Of particular importance in the vortex cutting process is the presence of vorticity generation from the blade surface within the reconnection region and the presence of strong vortex stretching due to the ambient flow about the blade leading edge. A simple exact Navier–Stokes solution is presented that describes the process by which incident vorticity is stretched and carried towards the surface by the ambient flow, and then interacts with and is eventually annihilated by diffusive interaction with vorticity generated at the surface. The model combines a Hiemenz straining flow, a Burgers vortex sheet and a Stokes first problem boundary layer, resulting in a nonlinear ordinary differential equation and a partial differential equation in two scaled time and distance variables that must be solved numerically. The simple model predictions exhibit qualitative agreement with the full numerical simulation results for vorticity annihilation near the leading-edge stagnation point during vortex cutting. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email