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Internal waves generated by a stratified wake: experiment and theory

  • P. Meunier (a1) (a2), S. Le Dizès (a2), L. Redekopp (a1) and G. R. Spedding (a1)


This paper presents experimental and theoretical results on the internal waves emitted by a bluff body moving horizontally in a linearly stratified fluid. Three different bluff bodies (a sphere, a spheroid and a cylinder) have been used in order to study the effect of the shape of the bluff body, although most of the results are obtained for the sphere. Two types of internal waves have been observed experimentally: large wavelength lee waves generated by the bluff body itself and small wavelength coherent wake waves generated by the turbulent wake. First, the lee waves are separated from the wake waves by averaging the experimental measurements in the frame moving with the bluff body. The velocity amplitude of the lee waves scales as the inverse of the Froude number $F=2U_{B}/(ND)$ for $F>2$ (where $U_{B}$ is the towing velocity, $D$ the diameter and $N$ the buoyancy frequency). This scaling proves that the internal waves are related to the drag of the bluff body which is due to the separation of the flow behind the bluff body. This separation is usually not taken into account in the classical models which assume that the flow is dipolar. The drag can be modelled as a point force in the Navier–Stokes equations, which gives a correct prediction of the structure and the amplitude of the lee waves. Second, the wake waves have been separated from the lee waves by averaging the velocity fields in the frame moving at the phase velocity of the waves. The phase velocity and the wavelength scale as $F^{-2/3}$ and $F^{1/3}$ respectively which correspond to the velocity and distance between same sign vortices of the von Kármán vortex street. A simplified model is derived for the internal waves emitted by the double row of moving point vortices of the von Kármán street. The amplitude of the wake waves is measured experimentally and seems to depend on the Reynolds number.


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Abdilghanie, A. M. & Diamessis, P. J. 2013 The internal gravity wave field emitted by a stably stratified turbulent wake. J. Fluid Mech. 720, 104139.
Baines, P. G. 1987 Upstream blocking and airflow over mountains. Annu. Rev. Fluid Mech. 19 (1), 7595.
Blevins, R. 1984 Applied Fluid Dynamics Handbook. Van Nostrand Reinhold Company.
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, 2340.
Boyer, D., Davies, P., Fernando, H. & Zhang, X. 1989 Linearly stratified flow past a horizontal circular cylinder. Phil. Trans. R. Soc. Lond. A 328 (1601), 501528.
Brandt, A. & Rottier, J. 2015 The internal wavefield generated by a towed sphere at low Froude number. J. Fluid Mech. 769, 103129.
Chomaz, J. M., Bonneton, P., Butet, A. & Hopfinger, E. J. 1993a Vertical diffusion in the far wake of a sphere moving in a stratified fluid. Phys. Fluids A 5, 27992806.
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, 5295.
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, 83101.
Dupont, P. & Voisin, B. 1996 Internal waves generated by a translating and oscillating sphere. Dyn. Atmos. Oceans 23 (1), 289298.
Fincham, A. M. & Spedding, G. R. 1997 Low-cost high-resolution DPIV for turbulent flows. Exp. Fluids 23, 449462.
Gilreath, H. E. & Brandt, A. 1985 Experiments on the generation of internal waves in a stratified fluid. AIAA J. 23, 693700.
Gourlay, M. J., Arendt, S. C., Fritts, D. C. & Werne, J. 2001 Numerical modeling of initially turbulent wakes with net momentum. Phys. Fluids 13, 37833802.
Hopfinger, E., Flor, J.-B., Chomaz, J.-M. & Bonneton, P. 1991 Internal waves generated by a moving sphere and its wake in a stratified fluid. Exp. Fluids 11 (4), 255261.
Lighthill, M. J. 1967 On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids. J. Fluid Mech. 27, 725752.
Lin, J. T. & Pao, Y. H. 1979 Wakes in stratified fluids: a review. Annu. Rev. Fluid Mech. 11, 317338.
Lin, Q., Boyer, D. & Fernando, H. 1993 Internal waves generated by the turbulent wake of a sphere. Exp. Fluids 15 (2), 147154.
Meunier, P. 2012 Stratified wake of a tilted cylinder. Part 2. Lee internal waves. J. Fluid Mech. 699, 198215.
Meunier, P., Diamessis, P. J. & Spedding, G. R. 2006 Self-preservation in stratified momentum wakes. Phys. Fluids 18 (10), 106601.
Meunier, P. & Spedding, G. R. 2004 A loss of memory in stratified momentum wakes. Phys. Fluids 16 (2), 298305.
Milder, M.1974 Internal waves radiated by a moving source. Vol. I. Tech. Rep. No. 782-262, National Technical Information Service.
Miles, J. W. 1968 Lee waves in a stratified flow. Part 2. Semi-circular obstacle. J. Fluid Mech. 33 (4), 803814.
Miles, J. W. 1971 Internal waves generated by a horizontally moving source. Geophys. Astrophys. Fluid Dyn. 2 (1), 6387.
Munroe, J. R. & Sutherland, B. R. 2014 Internal wave energy radiated from a turbulent mixed layer. Phys. Fluids 26 (9), 096604.
Pal, A., Sarkar, S., Posa, A. & Balaras, E. 2017 Direct numerical simulation of stratified flow past a sphere at a subcritical Reynolds number of 3700 and moderate Froude number. J. Fluid Mech. 826, 531.
Redford, J., Lund, T. & Coleman, G. 2015 A numerical study of a weakly stratified turbulent wake. J. Fluid Mech. 776, 568609.
Riley, J. R. & Lelong, M. P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32, 613.
Robey, H. F. 1997 The generation of internal waves by a towed sphere and its wake in a thermocline. Phys. Fluids 9 (11), 33533367.
Scase, M. & Dalziel, S. 2004 Internal wave fields and drag generated by a translating body in a stratified fluid. J. Fluid Mech. 498, 289313.
Scase, M. & Dalziel, S. 2006 Internal wave fields generated by a translating body in a stratified fluid: an experimental comparison. J. Fluid Mech. 564, 305331.
Spedding, G. R. 1997 The evolution of initially turbulent bluff-body wakes at high internal Froude number. J. Fluid Mech. 337, 283301.
Spedding, G. R. 2002 The streamwise spacing of adjacent coherent structures in stratified wakes. Phys. Fluids 14 (11), 38203828.
Spedding, G. R. 2014 Wake signature detection. Annu. Rev. Fluid Mech. 46, 273302.
Spedding, G. R., Browand, F. K., Bell, R. & Chen, J. 2000 Internal waves from intermediate, or late-wake vortices. In Stratified Flows I, Proc. 5th Int. Symp. on Stratified Flows, Vancouver Canada, UBC (ed. Lawrence, G. A., Pieters, R. & Yonemitsu, N.), pp. 113118. University of British Columbia.
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, 53103.
de Stadler, M. B., Sarkar, S. & Brucker, K. A. 2010 Effect of the Prandtl number on a stratified turbulent wake. Phys. Fluids 22 (9), 095102.
Stevenson, T. & Thomas, N. 1969 Two-dimensional internal waves generated by a travelling oscillating cylinder. J. Fluid Mech. 36 (03), 505511.
Stevenson, T. N. 1968 Some two-dimensional internal waves in a stratified fluid. J. Fluid Mech. 33, 715720.
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.
Thorpe, S. 2016 Layers and internal waves in uniformly stratified fluids stirred by vertical grids. J. Fluid Mech. 793, 380413.
Thorpe, S. A. 2005 The Turbulent Ocean. Cambridge University Press.
Vasholz, D. P. 2011 Stratified wakes, the high Froude number approximation, and potential flow. Theor. Comput. Fluid Dyn. 25 (6), 357379.
Voisin, B. 1991 Internal wave generation in uniformly stratified fluids. Part 1. Green’s function and point sources. J. Fluid Mech. 231, 439480.
Voisin, B. 1994 Internal wave generation in uniformly stratified fluids. Part 2. Moving point sources. J. Fluid Mech. 261, 333374.
Voisin, B. 2007 Lee waves from a sphere in a stratified flow. J. Fluid Mech. 574, 273315.
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.
Zhou, Q. & Diamessis, P. J. 2016 Surface manifestation of internal waves emitted by submerged localized stratified turbulence. J. Fluid Mech. 798, 505539.
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