Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-29T00:58:58.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulent hydraulic jumps in a stratified shear flow

Published online by Cambridge University Press:  14 May 2010

S. A. THORPE
S. A. THORPE*
Affiliation:
School of Ocean Sciences, Bangor University, Menai Bridge, Anglesey LL59 5AB, UK
*
†Address for correspondence: ‘Bodfryn’, Glanrafon, Llangoed, Anglesey LL58 8PH, UK. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The conditions are examined in which stationary hydraulic jumps may occur in a continuously stratified layer of fluid of finite thickness moving over a horizontal boundary at z = 0 and beneath a deep static layer of uniform density. The velocity and density in the flowing layer are modified by turbulent mixing in the transition region. Entrainment of fluid from the overlying static layer is possible. Results are expressed in terms of a Froude number, Fr, characterizing the flow upstream of a transition. A Froude number, Fr*, is found that must be exceeded if conditions for the conservation of volume, mass and momentum fluxes across a hydraulic transition are satisfied. The condition Fr > Fr* is satisfied if the kinetic energy (KE) per unit area is greater than the potential energy per unit area, or if ∫0h [u2(z) − z2N2(z)]dz > 0, in a flow of speed u(z), in a layer of thickness h, with buoyancy frequency N(z). In the particular case (referred to as an ‘η profile’) of a flow with velocity and density that are constant if z ≤ ηh, decrease linearly if ηh ≤ z ≤ h, and in which u(z) = 0 and density is constant when z ≥ h, long linear internal waves can propagate upstream, ahead of a stationary hydraulic jump, for Fr in a range Fr* < Fr < Frc; here Frc is the largest Fr at which long waves, and wave energy, can propagate upstream in a flow with specified η. (Profiles other than the η profile exhibit similar properties.) It is concluded that whilst, in general, Fr > Fr* is a necessary condition for a hydraulic jump to occur, a more stringent condition may apply in cases where Fr* < Frc, i.e. that Fr > Frc.

Physical constraints are imposed on the form of the flow downstream of the hydraulic jump or transition that relate, for example, to its static and dynamic stability and its stability against a further hydraulic jump. A further condition is imposed that relates the rate of dissipation of turbulent KE within a transition to the loss in energy flux of the flow in passing through the transition from the upstream side to the downstream. The constraints restrict solutions for the downstream flow to those in which the flux of energy carried downstream by internal waves is negligible and in which dissipation of energy occurs within the transition region.

Although the problem is formulated in general terms, particular examples are given for η profiles, specifically when η = 0 and 0.4. The jump amplitude, the entrainment rate, the loss of energy flux and the shape of the velocity and density profiles in the flow downstream of a transition are determined when Fr > Fr* (and extending to those with Fr > Frc) in a number of extreme conditions: when the loss of energy flux in transitions is maximized, when the entrainment is maximized, when the jump amplitude is least and when loss of energy flux is maximized subject to the entrainment into the transitions being made zero. The ratio of the layer thickness downstream and upstream of a transition, the jump amplitude, is typically at least 1.4 when jumps are just possible (i.e. when Fr ~ Frc). The amplitude, entrainment and non-dimensionalized loss in energy flux increase with Fr in each of the extreme conditions, and the maximum loss in energy flux is close to that when the entrainment is greatest. The magnitude of the advective and diffusive fluxes across isopycnal surfaces, i.e. the diapycnal fluxes characterizing turbulent mixing in the transition region, also increase with Fr. Results are compared to those in which the velocity and density profiles downstream of the transition are similar to those upstream, and comparisons are also made with equivalent two-layer profiles and with a cosine-shaped profile with continuous gradients of velocity and density. If Fr is larger than a certain value (about 7 and > Frc, if η = 0.4), no solutions for flows downstream of a transition are found if there is no entrainment, implying that fluid must be entrained if a transition is to occur in flows with large Fr. Although the extreme conditions of loss of energy flux, jump amplitude or entrainment provide limits that it must satisfy, in general no unique downstream flow is found for a given flow upstream of a jump.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 654 , 10 July 2010 , pp. 305 - 350
Copyright
Copyright © Cambridge University Press 2010

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.)

References

REFERENCES

Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press, p. 482.Google Scholar
Bell, T. H. 1974 Effects of shear on the properties of internal gravity waves. Dtsch. Hydrograph Z. 27, 57–62.CrossRefGoogle Scholar
Farmer, D. & Armi, L. 1999 Stratified flow over topography: the role of small-scale entrainment and mixing in flow establishment. Proc. R. Soc. Lond. A 455, 3221–3258.CrossRefGoogle Scholar
Ferron, B., Mercier, H., Speer, K., Gargett, A. & Polzin, K. 1998 Mixing in the Romanche fracture zone. J. Phys. Oceanogr. 28, 1929–1945.2.0.CO;2>CrossRefGoogle Scholar
Hassid, S., Regev, A. & Poreh, M. 2007 Turbulent energy dissipation in density jumps. J. Fluid Mech. 572, 1–12.CrossRefGoogle Scholar
Holland, D. M., Rosales, R. R., Stefamica, D. & Tabak, E. G. 2002 Internal hydraulic jumps and mixing in two-layer flows. J. Fluid Mech. 470, 63–83.CrossRefGoogle Scholar
Hornung, H. G., Willert, C. & Turner, S. 1995 The flow field downstream of a hydraulic jump. J. Fluid Mech. 287, 299–316.CrossRefGoogle Scholar
Hoyt, J. W. & Sellin, R. H. J. 1989 Hydraulic jump as “Mixing Layer”. J. Hydraul. Engng 115, 1607–1614.CrossRefGoogle Scholar
Klemp, J. B., Rotunno, R. R. & Skamarock, W. C. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81–106.CrossRefGoogle Scholar
Koop, C. G. & Browand, F. K. 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93, 135–159.CrossRefGoogle Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press, p. 504.Google Scholar
Mackinnon, J. A., Johnston, T. M. S. & Pinkel, R. 2008 Strong transport and mixing through the Southwest Indian Ridge. Nature Geosci. 1, 755–758.CrossRefGoogle Scholar
McEwan, A. D. & Baines, P. G. 1974 Shear fronts and an experimental stratified shear flow. J. Fluid Mech. 63, 257–272.CrossRefGoogle Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10, 83–89.2.0.CO;2>CrossRefGoogle Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35, 135–167.CrossRefGoogle Scholar
Peregrine, D. H. & Svendsen, I. A. 1978 Spilling breakers, bores and hydraulic jumps. In Proceedings of the 16th International Conference on Coastal Engineering, ASCE, pp. 540–550.Google 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
Polzin, K., Speer, K. G., Toole, J. M. & Schmitt, R. W. 1996 Intense mixing of Antarctic Bottom Water in the equatorial Atlantic Ocean. Nature 380, 54–55.CrossRefGoogle Scholar
St Laurent, L. C. & Thurnherr, A. M. 2007 Intense mixing of lower thermocline water on the crest of the mid-Atlantic Ridge. Nature 448, 680–684, doi:10.1038/nature06043.CrossRefGoogle ScholarPubMed
Saffman, P. G. & Yuen, H. C. 1982 Finite-amplitude interfacial waves in the presence of a current. J. Fluid Mech. 123, 459–476.CrossRefGoogle Scholar
Simpson, J. E. 1997 Gravity Currents, 2nd edn. Cambridge University Press, p. 244.Google Scholar
Stastna, M. & Lamb, K. G. 2008 Sediment resuspension mechanisms associated with internal waves in coastal waters. J. Geophys. Res. 113, C10016, doi:10.1029/2007/JC004711.Google Scholar
Svendsen, I. B. A., Veeramony, J., Bakunin, J. & Kirby, J. T. 2000 The flow in weak turbulent hydraulic jumps. J. Fluid Mech. 418, 25–57.CrossRefGoogle Scholar
Swaters, G. E. 2009 Mixed bottom-friction-Kelvin–Helmholtz destabilization of source-driven abyssal overflows in the ocean. J. Fluid Mech. 626, 33–66.CrossRefGoogle Scholar
Thorpe, S. A. 1974 Near-resonant forcing in a shallow two-layer fluid : a model for the internal surge in Loch Ness. J. Fluid Mech. 63, 509–527.CrossRefGoogle Scholar
Thorpe, S. A. 2007 Dissipation in hydraulic transitions in flows through abyssal channels. J. Mar. Res. 65 (1), 147–168.CrossRefGoogle Scholar
Thorpe, S. A. & Kavčič, I. 2008 The circular internal hydraulic jump. J. Fluid Mech. 610, 99–129.CrossRefGoogle Scholar
Thorpe, S. A. & Liu, Z. 2009 Marginal instability? J. Phys. Oceanogr. 39, 2373–2381.CrossRefGoogle Scholar
Thorpe, S. A. & Ozen, B. 2007 Are cascading flows stable? J. Fluid Mech. 589, 411–432.CrossRefGoogle Scholar
Thorpe, S. A. & Ozen, B. 2009 Corrigendum: are cascading flows stable? J. Fluid Mech. 631, 441–442.CrossRefGoogle Scholar
Thurnherr, A. M., St. Laurent, L. C., Speer, K. G., Toole, J. M. & Ledwell, J. R. 2005 Mixing associated with sills in a canyon on the mid-ocean flank. J. Phys. Oceanogr. 35, 1370–1381.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press, p. 367.CrossRefGoogle Scholar
Wilkinson, D. L. & Wood, I. R. 1971 A rapidly varied flow phenomenon in a two-layer flow. J. Fluid Mech. 47, 241–256.CrossRefGoogle Scholar
Yeh, H. 1991 Vorticity generation mechanisms in bores. Proc. R. Soc. Lond. A 432, 215–231.Google Scholar