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Internal bores: an improved model via a detailed analysis of the energy budget

Published online by Cambridge University Press:  13 June 2012

Zachary Borden ,
Eckart Meiburg  and
George Constantinescu
Zachary Borden
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Eckart Meiburg*
Affiliation:
Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
George Constantinescu
Affiliation:
Department of Civil and Environmental Engineering, University of Iowa, Iowa City, IA 52242, USA
*
†Email address for correspondence: [email protected]
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Abstract

Internal bores, or internal hydraulic jumps, arise in many atmospheric and oceanographic phenomena. The classic single-layer hydraulic jump model accurately predicts the bore height and propagation velocity when the difference between the densities of the expanding and contracting layers is large (i.e. water and air), but fails in the Boussinesq limit. A two-layer model, which conserves mass separately in each layer and momentum globally is more accurate in the Boussinesq limit, but it requires for closure an assumption about the loss of energy across a bore. It is widely believed that bounds on the bore speed can be found by restricting the energy loss entirely to one of the two layers, but under some circumstances, both bounds overpredict the propagation speed. A front velocity slower than both bounds implies that, somehow, the expanding layer is gaining energy. We directly examine the flux of energy within internal bores using two- and three-dimensional direct numerical simulations and find that although there is a global loss of energy across a bore, a transfer of energy from the contracting to the expanding layer causes a net energy gain in the expanding layer. The energy transfer is largely the result of turbulent mixing at the interface. Within the parameter regime investigated, the effect of mixing is much larger than non-hydrostatic and viscous effects, both of which are neglected in the two-layer analytical models. Based on our results, we propose an improved two-layer model that provides an accurate propagation velocity as a function of the geometrical parameters, the Reynolds number, and the Schmidt number.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents
Type
Papers
Information
Journal of Fluid Mechanics , Volume 703 , 25 July 2012 , pp. 279 - 314
Copyright
Copyright © Cambridge University Press 2012

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References

1. Baines, P. G. 1984 A unified description of two-layer flow over topography. J. Fluid Mech. 146, 127–167.CrossRefGoogle Scholar
2. Baines, P. G. 1995 Topographic Effects in Stratified Fluids. Cambridge University Press.Google Scholar
3. Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
4. Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224 (1159), 448–460.Google Scholar
5. Bonometti, T. & Balachandar, S. 2008 Effect of Schmidt number on the structure and propagation of density currents. Theor. Comput. Fluid Dyn. 22, 341–361.CrossRefGoogle Scholar
6. Chu, V. H. & Baddour, R. E. 1977 Surges, waves and mixing in two-layer density stratified flow. In Proc. 17th Congress of the International Association of Hydraulic Research, Baden-Baden, Germany, pp. 303–310. IAHR.Google Scholar
7. Clarke, R. H., Smith, R. K. & Reid, D. G. 1981 The morning glory of the Gulf of Carpentaria: an atmospheric undular bore. Mon. Weath. Rev. 109, 1726–1750.2.0.CO;2>CrossRefGoogle Scholar
8. Cummins, P. F. 1995 Numerical simulations of upstream bores and solitons in a two-layer flow past an obstacle. J. Phys. Oceanogr. 25, 1504–1515.2.0.CO;2>CrossRefGoogle Scholar
9. Fletcher, C. A. J. 1991 Computational Techniques for Fluid Dynamics, vol. 2, 2nd edn. Springer.Google Scholar
10. Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids 3 (7), 1760–1765.CrossRefGoogle Scholar
11. Gonzalez-Juez, E. & Meiburg, E. 2009 Shallow-water analysis of gravity-current flows past isolated obstacles. J. Fluid Mech. 635, 415–438.CrossRefGoogle Scholar
12. Gonzalez-Juez, E., Meiburg, E. & Constantinescu, G. 2009 Gravity currents impinging on bottom-mounted square cylinders: flow fields and associated forces. J. Fluid Mech. 631, 65–102.CrossRefGoogle Scholar
13. Gonzalez-Juez, E., Meiburg, E., Tokyay, T. & Constantinescu, G. 2010 Gravity current flow past a circular cylinder: forces, wall shear stresses and implications for scour. J. Fluid Mech. 649, 69–102.CrossRefGoogle Scholar
14. Haase, S. P. & Smith, R. K. 1984 Morning glory wave clouds in Oklahoma: a case study. Mon. Weather Rev. 112, 2078–2089.2.0.CO;2>CrossRefGoogle Scholar
15. Härtel, C., Meiburg, E. & Necker, F. 2000 Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front speed for slip and no-slip boundaries. J. Fluid Mech. 418, 189–212.CrossRefGoogle Scholar
16. Horn, D. A., Imberger, J. & Ivey, G. N. 2001 The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech. 434, 181–207.CrossRefGoogle Scholar
17. Hosegood, P. & van Haren, H. 2004 Near-bed solibores over the continental slope in the Faeroe–Shetland channel. Deep-Sea Res. II 51 (25–26), 2943–2971.Google Scholar
18. Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799.CrossRefGoogle Scholar
19. Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1994 On the dynamics of gravity currents in a channel. J. Fluid Mech. 269, 169–198.CrossRefGoogle Scholar
20. Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81–106.CrossRefGoogle Scholar
21. Koop, C. G. & Browand, F. K. 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93 (1), 135–179.CrossRefGoogle Scholar
22. Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
23. Leichter, J. J., Wing, S. R., Miller, S. L. & Denny, M. W. 1996 Pulsed delivery of subthermocline water to Conch Reef (Florida Keys) by internal tidal bores. Limnol. Oceanogr. 41 (7), 1490–1501.CrossRefGoogle Scholar
24. Li, M. & Cummins, P. F. 1998 A note on hydraulic theory of internal bores. Dyn. Atmos. Oceans 28, 1–7.CrossRefGoogle Scholar
25. Lilly, D. K. 1992 A proposed modification of the Germano-subgrid-scale closure method. Phys. Fluids A 4 (3), 633–635.CrossRefGoogle Scholar
26. Morozov, E. G., Trulsen, K., Velarde, M. G. & Vlasenko, V. I. 2002 Internal tides in the straight of gibraltar. J. Phys. Ocenaogr. 32, 3193–3206.2.0.CO;2>CrossRefGoogle Scholar
27. Necker, F., Härtel, C., Kleiser, L. & Meiburg, E. 2005 Mixing and dissipation in particle-driven gravity currents. J. Fluid Mech. 545, 339–372.CrossRefGoogle Scholar
28. Ooi, S. K., Constantinescu, S. G. & Weber, L. 2007 Numerical simulations of lock-exchange compositional gravity currents. J. Fluid Mech. 635, 361–388.CrossRefGoogle Scholar
29. Panton, R. L. 2005 Incompressible Flow, 3rd edn. John Wiley.Google Scholar
30. Pierce, C. D. 2001 Progress-variable approach for large eddy simulation of turbulent combustion. PhD thesis, Stanford University.Google Scholar
31. Pierce, C. D. & Moin, P. 2004 Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion. J. Fluid Mech. 504, 73–97.CrossRefGoogle Scholar
32. Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
33. Simpson, J. E. 1997 Gravity Currents, 2nd edn. Cambridge University Press.Google Scholar
34. Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
35. Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431–471.CrossRefGoogle Scholar
36. Ungarish, M. 2008 Energy balances and front speed conditions of two-layer models for gravity currents produced by lock release. Acta Mech. 201, 63–81.CrossRefGoogle Scholar
37. Ungarish, M. 2009 Energy balances for gravity currents with a jump at the interface produced by lock release. Acta Mechanica 211, 1–21.CrossRefGoogle Scholar
38. Ungarish, M. & Huppert, H. E. 2006 Energy balances for propagating gravity currents: homogeneous and stratified ambients. J. Fluid Mech. 565, 363–380.CrossRefGoogle Scholar
39. Ungarish, M. & Huppert, H. E. 2008 Energy balances for axisymmetric gravity currents in homogeneous and linearly stratified ambients. J. Fluid Mech. 616, 303–326.CrossRefGoogle Scholar
40. Wakimoto, R. M. & Kingsmill, D. E. 1995 Structure of an atmospheric undular bore generated from colliding boundaries during CaPE. Mon. Weather Rev. 123, 1374–1393.2.0.CO;2>CrossRefGoogle Scholar
41. Winant, C. D. & Browand, F. K. 1974 Vortex pairing: the mechanism of turbulent mixing-layer growth at moderate Reynolds number. J. Fluid Mech. 63 (2), 237–255.CrossRefGoogle Scholar
42. Wood, I. R. & Simpson, J. E. 1984 Jumps in layered miscible fluids. J. Fluid Mech. 140, 215–231.CrossRefGoogle Scholar
43. Yih, C. S. & Guha, C. R. 1955 Hydraulic jump in a fluid system of two layers. Tellus 7, 358–366.CrossRefGoogle Scholar
44. Zhou, J., Dupuy, B., Bertozzi, A. L. & Hosoi, A. E. 2005 Theory for shock dynamics in particle–laden thin films. Phys. Rev. Lett. 94, 117803.CrossRefGoogle ScholarPubMed