Hostname: page-component-669899f699-tpknm Total loading time: 0 Render date: 2025-04-24T18:57:43.377Z Has data issue: false hasContentIssue false
  • English
  • Français

Internal gravity wave radiation from a stratified turbulent wake

Published online by Cambridge University Press:  11 February 2020

K. L. Rowe Open the ORCID record for K. L. Rowe [Opens in a new window] ,
P. J. Diamessis  and
Q. Zhou Open the ORCID record for Q. Zhou [Opens in a new window]
K. L. Rowe*
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY14853, USA
P. J. Diamessis
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY14853, USA
Q. Zhou
Affiliation:
Department of Civil Engineering, University of Calgary, Calgary, Alberta, T2N 2N4, Canada
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The near-field energetics and directional properties of internal gravity waves (IGWs) radiated from the turbulent wake of a sphere towed through a linearly stratified fluid are investigated using a series numerical experiments. Simulations have been performed for an initial Reynolds number $Re\equiv UD/\unicode[STIX]{x1D708}\in \{5\times 10^{3},10^{5},4\times 10^{5}\}$ and internal Froude number $Fr\equiv 2U/ND\in \{4,16,64\}$, defined using body-based scales – $D$, the sphere diameter; $U$, the tow speed; and $N$, the Brunt–Väisälä frequency. Snapshots of temporally evolving wake flow fields are sampled over the full wake evolution. The energy extracted from the wake through internal wave radiation is quantified by computing the total wave power emitted through a wake-following elliptic cylinder. The total time-integrated wave energy radiated is found to increase with $Re$ and decrease with $Fr$. The peak radiated wave power occurs at early times, near to the onset of buoyancy control, and is found to be approximately unchanged in magnitude as $Re$ is increased. For the two higher $Re$ considered, at late times, IGWs continue to be emitted – accounting for a distinct increase in total radiated wave energy. The most powerful IGWs are radiated out of the wake at a wide range of angles for $Nt<10$, at $20^{\circ }{-}70^{\circ }$ to the horizontal for $10\leqslant Nt\leqslant 25$, and nearly parallel to the horizontal late in the non-equilibrium regime of wake evolution. Internal wave radiation is found to be an important sink for wake kinetic energy after $Nt=10$, suggesting wave radiation cannot be neglected when modelling stratified turbulent wakes in geophysical and ocean engineering applications.

JFM classification

Wakes/Jets: Wakes Geophysical and Geological Flows: Internal waves Turbulent Flows: Stratified turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , A25
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abdilghanie, A. M.2010 A numerical investigation of turbulence-driven and forced generation of internal gravity waves in stratified mid-water. PhD thesis, Cornell University.Google Scholar
Abdilghanie, A. M. & Diamessis, P. J. 2013 The internal gravity wave field emitted by a stably stratified turbulent wake. J. Fluid Mech. 720, 104–139.CrossRefGoogle Scholar
Abkar, M. & Porte-Agel, F. 2015 Influence of atmospheric stability on wind-turbine wakes: a large-eddy simulation study. Phys. Fluids 27, 035104.CrossRefGoogle Scholar
Abramowitz, M. & Stegun., I. A. 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 9. Dover.Google Scholar
Allaerts, D. & Meyers, J. N. 2018 Gravity waves and wind-farm efficiency in neutral and stable conditions. Boundary-Layer Meteorol. 166 (2), 269–299.CrossRefGoogle ScholarPubMed
Augier, P., Chomaz, J.-M. & Billant, P. 2012 Spectral analysis of the transition to turbulence from a dipole in stratified fluid. J. Fluid Mech. 713, 86–108.CrossRefGoogle Scholar
Bonneton, P., Chomaz, J. M. & Hopfinger, E. J. 1993 Internal waves produced by the turbulent wake of a sphere moving horizontally in a stratified fluid. J. Fluid Mech. 254, 23–40.CrossRefGoogle Scholar
Boyd, J. P. 2012 Numerical, perturbative and Chebyshev inversion of the incomplete elliptic integral of the second kind. Appl. Maths Comput. 218 (13), 7005–7013.CrossRefGoogle Scholar
Brandt, A. & Rottier, J. R. 2015 The internal wavefield generated by a towed sphere at low Froude number. J. Fluid Mech. 769, 103–129.CrossRefGoogle Scholar
Brucker, K. A. & Sarkar, S. 2010 A comparative study of self-propelled and towed wakes in a stratified fluid. J. Fluid Mech. 652, 373–404.CrossRefGoogle Scholar
de Bruyn Kops, S. M. & Riley, J. J. 2019 The effects of stable stratification on the decay of initially isotropic homogeneous turbulence. J. Fluid Mech. 860, 787–821.CrossRefGoogle Scholar
Bühler, O. 2014 Waves and Mean Flows. Cambridge University Press.CrossRefGoogle Scholar
Dallard, T. & Spedding, G. R. 1993 2D wavelet transforms: generalisation of the Hardy space and application to experimental studies. Eur. J. Mech. (B/Fluids) 12, 107–134.Google Scholar
Diamessis, P. J., Domaradzki, J. A. & Hesthaven, J. S. 2005 A spectral multidomain penalty method model for the simulation of high Reynolds number localized incompressible stratified turbulence. J. Comput. Phys. 202 (1), 298–322.CrossRefGoogle Scholar
Diamessis, P. J., Spedding, G. R. & Domaradzki, J. A. 2011 Similarity scaling and vorticity structure in high-Reynolds-number stably stratified turbulent wakes. J. Fluid Mech. 671, 52–95.CrossRefGoogle Scholar
Dommermuth, D. G., Rottman, J. W., Innis, G. E. & Novikov, E. A. 2002 Numerical simulation of the wake of a towed sphere in a weakly stratified fluid. J. Fluid Mech. 473, 83–101.CrossRefGoogle Scholar
Gibson, C. H., Nabatov, V. & Ozmidov, R. 1993 Measurements of turbulence and fossil turbulence near Ampere seamount. Dyn. Atmos. Oceans 19, 175–204.CrossRefGoogle Scholar
Gourlay, M. J., Arendt, S. C., Fritts, D. C. & Werne, J. 2001 Numerical modeling of initially turbulent wakes with net momentum. Phys. Fluids 13 (12), 3783–3802.CrossRefGoogle Scholar
Karniadakis, G. E., Israeli, M. & Orszag, S. 1991 High-order splitting methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 97 (2), 414–443.CrossRefGoogle Scholar
Lamb, K. G. 2007 Energy and pseudoenergy flux in the internal wave field generated by tidal flow over topography. Cont. Shelf Res. 27 (9), 1208–1232.CrossRefGoogle Scholar
Lighthill, J. 2001 Waves in Fluids. Cambridge University Press.Google Scholar
Lin, J.-T. & Pao, Y.-H. 1979 Wakes in stratified fluids. Annu. Rev. Fluid Mech. 11 (1), 317–338.CrossRefGoogle Scholar
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.CrossRefGoogle Scholar
Maffioli, A., Davidson, P. A., Dalziel, S. B. & Swaminathan, N. 2014 The evolution of a stratified turbulent cloud. J. Fluid Mech. 739, 229–253.CrossRefGoogle Scholar
Meunier, P., Diamessis, P. J. & Spedding, G. R. 2006 Self-preservation in stratified momentum wakes. Phys. Fluids 18, 106601.CrossRefGoogle Scholar
Olbers, D., Willebrand, J. & Eden, C. 2012 Ocean Dynamics. Springer Science & Business Media.CrossRefGoogle Scholar
Orszag, S. A. & Pao, Y. H. 1975 Numerical computation of turbulent shear flows. Adv. Geophys. 18 (1), 225–236.CrossRefGoogle Scholar
Pao, H. P., Lai, R. Y. & Schemm, C. E.1982 Vortex trails in stratified fluids. Tech. Rep. 3(1). Johns Hopkins Applied Physics Laboratory Technical Digest.Google Scholar
Pawlak, G., MacCready, P., Edwards, K. A. & McCabe, R. 2003 Observations on the evolution of tidal vorticity at a stratified deep water headland. Geophys. Res. Lett. 30 (24), 2234.CrossRefGoogle Scholar
Perfect, B., Kumar, N. & Riley, J. J. 2018 Vortex structures in the wake of an idealized seamount in rotating, stratified flow. Geophys. Res. Lett. 45 (17), 9098–9105.CrossRefGoogle Scholar
Pham, H. T., Sarkar, S. & Brucker, K. A. 2009 Dynamics of a stratified shear layer above a region of uniform stratification. J. Fluid Mech. 630, 191–223.CrossRefGoogle Scholar
Plougonven, R. & Zeitlin, V. 2002 Internal gravity wave emission from a pancake vortex: an example of wave–vortex interaction in strongly stratified flows. Phys. Fluids 14 (3), 1259–1268.CrossRefGoogle Scholar
Redford, J. A., Lund, T. S. & Coleman, G. N. 2015 A numerical study of a weakly stratified turbulent wake. J. Fluid Mech. 776, 568–609.CrossRefGoogle Scholar
Riley, J. J. & de Bruyn Kops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15, 2047.CrossRefGoogle Scholar
Riley, J. J. & Lindborg, E. 2012 Recent progress in stratified turbulence. In Ten Chapters in Turbulence (ed. Davidson, P. A., Kaneda, Y. & Sreenivasan, K. R.), pp. 269–317. Cambridge University Press.CrossRefGoogle Scholar
Rotunno, R., Grubisic, V. & Smolarkiewicz, P. K. 1999 Vorticity and potential vorticity in mountain wakes. J. Atmos. Sci. 56 (16), 2796–2810.2.0.CO;2>CrossRefGoogle Scholar
Shete, K. P. & de Bruyn Kops, S. M. 2020 Area of scalar isosurfaces in homogeneous isotropic turbulence as a function of Reynolds and Schmidt numbers. J. Fluid Mech. 883, A38.CrossRefGoogle Scholar
Spedding, G. R. 1997 The evolution of initially turbulent bluff-body wakes at high internal froude number. J. Fluid Mech. 337, 283–301.CrossRefGoogle Scholar
Spedding, G. R. 2014 Wake signature detection. Annu. Rev. Fluid Mech. 46, 273–302.CrossRefGoogle Scholar
Spedding, G. R., Browand, F. K., Bell, R. & Chen, J. 2000 Internal waves from intermediate, or late-wake vortices. In Stratified Flows I Proceedings of the 5th International Symposium on Stratied Flows, Vancouver, Canada, pp. 113–118. UBC.Google Scholar
Spedding, G. R., Browand, F. K. & Fincham, A. M. 1996 Turbulence, similarity scaling and vortex geometry in the wake of a towed sphere in a stably stratified fluid. J. Fluid Mech. 314, 53–103.CrossRefGoogle Scholar
Spiegel, E. A. & Veronis, G. 1960 On the boussinesq approximation for a compressible fluid. Astrophys. J. 131, 442.CrossRefGoogle Scholar
Sutherland, B. R. 2011 Internal Gravity Waves. Cambridge University Press.Google Scholar
Taylor, J. R. & Sarkar, S. 2007 Internal gravity waves generated by a turbulent bottom Ekman layer. J. Fluid Mech. 590, 331–354.CrossRefGoogle Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.CrossRefGoogle Scholar
Voisin, B. 1995 Internal wave generation by turbulent wakes. In Mixing in Geophysical Flows, pp. 291–301. CIMNE.Google Scholar
Watanabe, T., Riley, J. J., de Bruyn Kops, S. M., Diamessis, P. J. & Zhou, Q. 2016 Turbulent/non-turbulent interfaces in wakes in stably stratified fluids. J. Fluid Mech. 797, R1.CrossRefGoogle Scholar
Zhou, Q.2015 Far-field evolution of turbulence-emitted internal waves and Reynolds number effects on a localized stratified turbulent flow. PhD thesis, Cornell University, Ithaca, New York.Google Scholar
Zhou, Q. & Diamessis, P. J. 2016 Surface manifestation of internal waves emitted by submerged localized stratified turbulence. J. Fluid Mech. 798, 505–539.CrossRefGoogle Scholar
Zhou, Q. & Diamessis, P. J. 2019 Large-scale characteristics of stratified wake turbulence at varying Reynolds number. Phys. Rev. Fluids 4, 084802.CrossRefGoogle Scholar