Analytical Study of the Bed Roughness and Entrainment Ratio of Saline Density Currents on Hydraulic Jump

Document Type : Research Paper

Author

• Nader Barahmand

Abstract

 Density current is produced due to a density contrast with the ambient fluid. This density difference can result from dissolved solids, suspended materials, temperature, etc. Density jumps significantly influence the characteristics of the gravity currents and the ambient fluid (e.g., velocities and concentrations). In this paper, the density jump is studied analytically by considering the bed roughness and entrainment ratio. For both smooth and roughened beds, a generalized relationship was obtained for estimating the conjugate depth ratio as a function of the upstream densimetric Froude number, the entrainment ratio, and the relative roughness. Moreover, various equations for calculating the maximum possible value of the relative roughness, the minimum possible value of the prejump densimetric Froude number, and the maximum possible value of the relative roughness were proposed. The minimum possible value of the conjugate depth ratio was determined. It was found that both the conjugate depth ratio  and the length of the roller zone decrease as the bed roughness increases. Moreover, as the entrainment ratio increases, the ratio  may also decrease, or may initially increase and then decrease. Finally, a relationship was obtained for estimating the minimum possible value of the obstacle height as a function of the conjugate depth ratio.

Keywords

• Density jump
• ambient fluid
• relative roughness
• Entrainment ratio
• densimetric Froude number

20.1001.1.20086377.1396.10.33.3.1

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References

1) Baddour, R.E. and H. Abbink. 1983. Turbulent underflow in a short channel of limited depth. J. Hydraul. Eng. 109(5): 722–740.

2) Borden, Z., T. Koblitz, and E. Meiburg. 2012a. Turbulent mixing and wave radiation in non-Boussinesq internal bores. Phys. Fluids. 24: 082106.

3) Borden, Z., E. Meiburg, and G. Constantinescu. 2012b. Internal bores: An improved model via a detailed analysis of the energy budget. J. Fluid Mech. 703: 279-314.

4) Cantero, M. I. and M. H. Garcia. 2001. Sediment management in water reservoirs by jet-induced density currents. Proc., Int. Symp. Env. Hydr. Tempe, Ariz. 12-33

 

5) Carollo, F. G. and V. Ferro. 2004. “Contributo allo studio della lunghezza del risalto libero su fondo liscio e scabro.” Rivista di Ingegneria Agraria, 35: 13–20 (in Italian).

6) Carollo, F.G., V. Ferro, and V. Pampalone. 2007. Hydraulic jumps on rough beds. J. Hydraul. Eng. 133: 989-999.

7) Carollo, F.G., V. Ferro, and V. Pampalone. 2009. A new solution of classical hydraulic jump. J. Hydraul. Eng. 135: 527-531.

8) Chikita, K. and Y. Okumura. 1990. Dynamics of turbidity currents measured in Katsurasawa reservoir, Hokkido, Japan.’ J. Hydrol. 177: 323-338.        

9) Garcia, M.H. 1993. Hydraulic jumps in sediment-driven bottom currents. J. Hydraul. Eng. 119: 1094-1117.

10) Hassid, S., A. Regev, and M. Poreh. 2007. Turbulent energy dissipation in density jumps. J. Fluid Mech. 572: 1-12.

11) Hogg, A.J., and A.W. Woods. 2001. The transition from inertia-to-bottom-drag-dominated motion of turbulent gravity current. J. Fluid Mech. 449: 201-210.

12) Holland, D.M., R.R., Rosales D. Stefanica, and E.G. Tabak. 2002. Internal hydraulic jump and mixing in two-layer flows. J. Fluid Mech. 470: 63–83.

13) Huang, H., J. Imran, C. Pirmez, Q. Zhang, and G. Chen. 2009. The critical densimetric Froude number of subaqueous gravity currents can be non-unity or nonexistent. J. Sed. Res. 79: 479-485.

14) Klumpp, C.C., J. Jennifer Bountry, and B. Blair Greimann. 2003. Case studies in dam decommissioning at the Bureau of Reclamation. Proc. World Water Resources Congress. Philadelphia, Pennsylvania. 214-239.

15) Kostic, S. and G. Parker. 2006. The response of turbidity currents to a canyon-fan transition: Internal hydraulic jumps and depositional signatures. J. Hydraul. Res. 44:631-653.

16) Liu, J. and A. Tominaga. 2003. New development of sediment flushing technique. Proc. World Water Resources Cong. Philadelphia, Pennsylvania. 551-597.

17) Najafpour, N., M. Samie, B. Firoozabadi, and H. Afshin. 2014. Theoretical and experimental investigation of density jump on an inclined surface. Scientia Iranica B. 21: 1655-1665.

18) Nasrabadi, M., M.H. Omid, and J. Farhoudi. 2012. Submerged hydraulic jump with sediment-laden flow. Int. J. Sed. Res. 27: 100-111.

19) Nourmohammadi, Z., H. Afshin, and B. Firoozabadi. 2011. Experimental observation of the flow structure of turbidity currents. J. Hydraul. Res. 49: 168-177.

20) Oehy, Ch. and A. Schleiss. 2001. Numerical modeling of a turbidity current passing over an obstacle—Practical application in Lake Grimsel, Switzerland. Proc. Int. Symp. Env. Hydr. (CD-ROM), Tempe, Ariz. 11-37.

21) Oehy, Ch., and A. Schleiss. 2007. Control of turbidity currents in reservoirs by solid and permeable obstacles. J. Hydraul. Eng. 133: 637-648.

22) Qu, L. and W.K. Chow. 2012. Numerical studies on density jump in a long corridor fire. Tunel. Underground Space Tech. 32: 113-126.

23) Rayson, M.D., N.L. Jones, G.N. Ivey, and O.B. Fringer. 2011. Internal hydraulic jump formation in a deep water, continuously-stratified, unsteady channel flow. 7th Int. Symp. on Stratified Flows, Italy. 1-18.

24) Regev, A., S. Hassid, and M. Poreh. 2004. Density jumps in smoke flow along horizontal ceilings. Fire Safety J. 39: 465–479.

25) Regev, A., S. Hassid, and M. Poreh. 2006. Calculation of entrainment in density jumps. J. Environ Fluid Mech. 6: 407–424.

26) Sumner, E., J. Peakall, D. Parsons, R. Wynn, S. Darby, R. Dorrell, S. McPhail, J. Perrett, A.Webb, and D. White. 2013. First direct measurements of hydraulic jumps in an active submarine density current. Geophy. Res. Let. 40: 1-5.

27) Thrope, S.A. 2008. Dissipation in hydraulic transitions in flows through abyssal channels. J. Marine Res. 65: 147-168.

28) Thrope, S.A. 2010. Turbulent hydraulic jumps in a stratified shear flow. J. Fluid Mech. 654: 305-350.

29) Wilkinson, D.L. and I.R. Wood. 1971. A rapidly varied flow phenomenon in a two-layer flow. J. Fluid Mech. 47: 241-256.

30) Wood, I.R. and J.E. Simpson. 1984. ‘Jumps in layered miscible fluids.’ J. Fluid Mech. 140: 329–342.

31) Xi, Y.H., W.K. Chow, and J. Mao, 2015. Aerodynamics simulation on density jump in a long corridor fire. Tunnel. Underground Space Tech. 50: 23-31.

32) Xia, Q. and J. Liu. 2003. Sediment management at Naodehai reservoir. Proc. World Water Resources Congress. Philadelphia, Pennsylvania. 1-12.

33) Yih, C.S. and C.R. Guha. 1955. Hydraulic jump in a fluid system of two layers. Tellus. 7: 358-366.