Skip to main content

On the Lamb vector divergence in Navier–Stokes flows

  • CURTIS W. HAMMAN (a1), JOSEPH C. KLEWICKI (a2) and ROBERT M. KIRBY (a1) (a3)

The mathematical and physical properties of the Lamb vector divergence are explored. Toward this aim, the instantaneous and mean dynamics of the Lamb vector divergence are examined in several analytic and turbulent flow examples relative to its capacity to identify and characterize spatially localized motions having a distinct capacity to effect a time rate of change of momentum. In this context, the transport equation for the Lamb vector divergence is developed and shown to accurately describe the dynamical mechanisms by which adjacent high- and low-momentum fluid parcels interact to effect a time rate of change of momentum and generate forces such as drag. From this, a transport-equation-based framework is developed that captures the self-sustaining spatiotemporal interactions between coherent motions, e.g. ejections and sweeps in turbulent wall flows, as predicted by the binary source–sink distribution of the Lamb vector divergence. New insight into coherent motion development and evolution is found through the analysis of the Lamb vector divergence.

Hide All
Adrian R. J. & Moin P. 1988 Stochastic estimation of organized turbulent structure: Homogeneous shear flow. J. Fluid Mech. 190, 531559.
Alim A. 2007 A physical comprehensive definition of a vortex based on the Lamb vector. Algerian J. Appl. Fluid Mech. 1, 15.
Aubry N., Holmes P., Lumley J. L. & Stone E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115173.
Bakewell H. P. & Lumley J. L. 1967 Viscous sublayer and adjacent wall region in turbulent pipe flow. Phys. Fluids 10, 18801889.
Batchelor G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Berkooz G., Holmes P. & Lumley J. L. 1991 Intermittent dynamics in simple models of the turbulent wall layer. J. Fluid Mech. 230, 7595.
Berkooz G., Holmes P. & Lumley J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539575.
Blackburn H. M., Govardhan R. & Williamson C. H. K. 2001 A complementary numerical and physical investigation of vortex-induced vibration. J. Fluids Struct. 15, 481488.
Brasseur J. G. & Lin W. 2005 Kinematics and dynamics of small-scale vorticity and strain-rate structures in the transition from isotropic to shear turbulence. Fluid Dyn. Res. 36, 357384.
Chakraborty P., Balachandar S. & Adrian R. J. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189214.
Chambers D. H., Adrian R. J., Stewart D. S. & Sung H. J. 1988 Karhunen–Loève expansion of Burgers' model of turbulence. Phys. Fluids 31, 25732582.
Choi H., Moin P. & Kim J. 1993 Direct numerical simulation of turbulent flow over riblets. J. Fluid Mech. 255, 503539.
Crawford C. H., Marmanis H. & Karniadakis G. E. 1998 The Lamb vector and its divergence in turbulent drag reduction. In Proc. Intl Symposium on Seawater Drag Reduction, pp. 99–107.
Dombre T., Frisch U., Greene J. M., Henon M., Mehr A. & Soward A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353391.
Ferrante A. & Elghobashi S. 2004 On the physical mechanisms of drag reduction in a spatially developing turbulent boundary layer laden with microbubbles. J. Fluid Mech. 503, 345355.
Hinze J. O. 1975 Turbulence. McGraw-Hill.
Howe M. S. 1975 Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech. 71, 625673.
Jeong J. & Hussain F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.
Klewicki J. C. 1989 Velocity–vorticity correlations related to the gradients of the Reynolds stresses in parallel turbulent wall flows. Phys. Fluids A 1, 12851288.
Klewicki J. C., Fife P., Wei T. & McMurtry P. A. 2007 A physical model of the turbulent boundary layer consonant with mean momentum balance structure. Phil. Trans. R. Soc. Lond. 365, 823839.
Klewicki J. C. 1998 Connecting vortex regeneration with near-wall stress transport. In AIAA Paper 1998-2963.
Kollmann W. 2006 Critical points and manifolds of the Lamb vector field in swirling jets. Comput. Fluids 35, 746754.
Küchemann D. 1965 Report on the IUTAM symposium on concentrated vortex motions in fluids. J. Fluid Mech. 21, 120.
Lighthill M. J. 1952 On sound generated aerodynamically. Part I. Proc. R. Soc. Lond. A 211, 564587.
Lumley J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. Yaglom A. M. & Tatarsky V. I.), pp. 166178. Nauka, Moscow.
Marcu B., Meiburg E. & Newton P. K. 1994 Dynamics of heavy particles in a Burgers vortex. Phys. Fluids 7, 400410.
Marmanis H. 1998 Analogy between the Navier–Stokes equations and Maxwell's equations: Application to turbulence. Phys. Fluids 10, 14281437.
Moffatt H. K. & Tsinober A. 1992 Helicity in laminar and turbulent flow. Annu. Rev. Fluid Mech. 24, 281312.
Moffatt H. K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals. J. Fluid Mech. 159, 359378.
Moffatt H. K. 1986 a Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations. J. Fluid Mech. 166, 359378.
Moffatt H. K. 1986 b On the existence of localized rotational disturbances which propagate without change of structure in an inviscid fluid. J. Fluid Mech. 173, 289302.
Monin A. & Yaglom A. 1977 Statistical Fluid Mechanics. MIT Press.
Moser R. D., Kim J. & Mansour N. N. 1999 Direct numerical simulation of turbulent channel flow up to Reτ=590. Phys. Fluids 11, 943945.
Offen G. R. & Kline S. J. 1975 A proposed model of the bursting process in turbulent boundary layers. J. Fluid Mech. 70, 209228.
Protas B. & Wesfreid J. E. 2001 Drag force in the open-loop control of the cylinder wake in the laminar regime. Phys. Fluids 14, 810826.
Robinson S. K. 1991 Coherent motions in turbulent boundary layers. Annu. Rev. Fluid Mech. 23, 601639.
Rousseaux G., Seifer S., Steinberg V. & Wiebel A. 2007 On the Lamb vector and the hydrodynamic charge. Exps. Fluids 42, 291299.
Shiels D. & Leonard A. 2001 Investigation of a drag reduction on a circular cylinder in rotary oscillation. J. Fluid Mech. 431, 297322.
Sposito G. 1997 On steady flows with Lamb surfaces. Intl J. Engng Sci. 35, 197209.
Taylor G. I. 1935 Statistical theory of turbulence. Proc. R. Soc. Lond. 151, 421444.
Truesdell C. 1951 A form of Green's transformation. Am. J. Maths 73, 4347.
Truesdell C. 1954 The Kinematics of Vorticity. Indiana University Press.
Tsinober A. 1990 On one property of Lamb vector in isotropic turbulent flow. Phys. Fluids 2, 484486.
Tsinober A. 1998 Is concentrated vorticity that important? Eur. J. Mech. B 17, 421449.
Tsinober A. & Levich E. 1983 On the helical nature of three dimensional coherent structures in turbulent flows. Phys. Rev. Lett. A 99, 321323.
Wei T., Fife P., Klewicki J. & McMurtry P. 2005 Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows. J. Fluid Mech. 522, 303327.
Wu J.-Z., Ma H.-Y. & Zhou M.-D. 2006 Vorticity and Vortex Dynamics. Springer.
Xu J., Maxey M. R. & Karniadakis G. E. 2002 Numerical simulation of turbulent drag reduction using micro-bubbles. J. Fluid Mech. 468, 271281.
Yang Y. T., Zhang R. K., An Y. R. & Wu J. Z. 2007 Steady vortex force theory and slender-wing flow diagnosis. Acta Mechanica Sinica 23, 609619.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 73 *
Loading metrics...

Abstract views

Total abstract views: 207 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 25th November 2017. This data will be updated every 24 hours.