Parameterization and Results of SWE for Gravity Currents Are Sensitive to the Definition of Depth
Publication: Journal of Hydraulic Engineering
Volume 147, Issue 5
Abstract
Rigorously derived shallow water equations (SWEs) are applied to results of large eddy simulation (LES) of a continuously fed gravity current in order to assess (1) sensitivity of current depth results to its definition; (2) coefficients in depth-averaged continuity and momentum equation due to the nonuniformity of density and velocity profiles; and (3) sensitivity of entrainment coefficient to definition of current depth. It is shown that using different definitions of the current depth may produce significantly different numerical results. The coefficients due to nonuniformity in the continuity equation are very close to unity, whereas the coefficients in the momentum flux and the pressure term in the momentum equation are different from unity by a margin that is very sensitive to the definition of current depth. The entrainment coefficient is more sensitive to the selected parameterization than to the definition of the current depth.
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Data Availability Statement
Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
The first author acknowledges support from the Swiss National Science Foundation (SNSF Grant No. 200021_159249). The second author acknowledges support from the Binks Trust Foundation. The authors would like to acknowledge the associate editor and the two anonymous reviewers for the constructive comments, which allowed significant improvement of the earlier version of this article.
References
Adduce, C., G. Sciortino, and S. Proietti. 2012. “Gravity currents produced by lock exchanges: Experiments and simulations with a two-layer shallow-water model with entrainment.” J. Hydraul. Eng. 138 (2): 111–121. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000484.
Altinakar, M. 1988. “Weakly depositing turbidity currents on small slopes.” Ph.D. thesis, Laboratoire de recherches hydrauliques, Ecole polytechnique fédérale de Lausanne.
Buckee, C., B. Kneller, and J. Peakall. 2001. Turbulence structure in steady, solute-driven gravity currents. New York: Wiley.
Cenedese, C., and C. Adduce. 2008. “Mixing in a density-driven current flowing down a slope in a rotating fluid.” J. Fluid Mech. 604: 369–388. https://doi.org/10.1017/S0022112008001237.
Cenedese, C., and C. Adduce. 2010. “A new parameterization for entrainment in overflows.” J. Phys. Oceanogr. 40 (8): 1835–1850. https://doi.org/10.1175/2010JPO4374.1.
Cenedese, C., R. Nokes, and J. Hyatt. 2018. “Lock-exchange gravity currents over rough bottoms.” Environ. Fluid Mech. 18 (1): 59–73. https://doi.org/10.1007/s10652-016-9501-0.
Christodoulou, G. C. 1986. “Interfacial mixing in stratified flows.” J. Hydraul. Res. 24 (2): 77–92. https://doi.org/10.1080/00221688609499323.
Chu, F. H., W. D. Pilkey, and O. H. Pilkey. 1979. “An analytical study of turbidity current steady flow.” Mar. Geol. 33 (3–4): 205–220. https://doi.org/10.1016/0025-3227(79)90081-1.
Constantinescu, G. 2014. “Les of lock-exchange compositional gravity currents: A brief review of some recent results.” Environ. Fluid Mech. 14 (2): 295–317. https://doi.org/10.1007/s10652-013-9289-0.
Ellison, T. H., and J. S. Turner. 1959. “Turbulent entrainment in stratified flows.” J. Fluid Mech. 6 (3): 423–448. https://doi.org/10.1017/S0022112059000738.
Fernando, H. J. S. 1991. “Turbulent mixing in stratified fluids.” Annu. Rev. Fluid Mech. 23 (1): 455–493. https://doi.org/10.1146/annurev.fl.23.010191.002323.
Fragoso, A. T., M. D. Patterson, and J. S. Wettlaufer. 2013. “Mixing in gravity currents.” J. Fluid Mech. 734: R2. https://doi.org/10.1017/jfm.2013.475.
García, M. H. 1993. “Hydraulic jumps in sediment-driven bottom currents.” J. Hydraul. Eng. 119 (10): 1094–1117. https://doi.org/10.1061/(ASCE)0733-9429(1993)119:10(1094).
Grundy, R. E., and J. W. Rottman. 1985. “The approach to self-similarity of the solutions of the shallow-water equations representing gravity-current releases.” J. Fluid Mech. 156: 39–53. https://doi.org/10.1017/S0022112085001975.
Hacker, J., P. Linden, and S. Dalziel. 1996. “Mixing in lock-release gravity currents.” Dyn. Atmos. Oceans 24 (1–4): 183–195. https://doi.org/10.1016/0377-0265(95)00443-2.
Hogg, A. J., M. A. Hallworth, and H. E. Huppert. 2005. “On gravity currents driven by constant fluxes of saline and particle-laden fluid in the presence of a uniform flow.” J. Fluid Mech. 539 (1): 349–385. https://doi.org/10.1017/S002211200500546X.
Holzner, M., and B. Lüthi. 2011. “Laminar superlayer at the turbulence boundary.” Phys. Rev. Lett. 106 (13): 134503. https://doi.org/10.1103/PhysRevLett.106.134503.
Huppert, H. E., and J. E. Simpson. 1980. “The slumping of gravity currents.” J. Fluid Mech. 99 (4): 785–799. https://doi.org/10.1017/S0022112080000894.
Kneller, B. C., S. J. Bennett, and W. D. McCaffrey. 1999. “Velocity structure, turbulence and fluid stresses in experimental gravity currents.” J. Geophys. Res. Oceans 104 (C3): 5381–5391. https://doi.org/10.1029/1998JC900077.
Krug, D., M. Holzner, B. Lüthi, M. Wolf, W. Kinzelbach, and A. Tsinober. 2013. “Experimental study of entrainment and interface dynamics in a gravity current.” Exp. Fluids 54: 1530. https://doi.org/10.1007/s00348-013-1530-6.
Krug, D., M. Holzner, B. Lüthi, M. Wolf, W. Kinzelbach, and A. Tsinober. 2015. “The turbulent non-turbulent interface in an inclined dense gravity current.” J. Fluid Mech. 765: 303–324. https://doi.org/10.1017/jfm.2014.738.
Legg, S. 2012. “Overflows and convectively driven flows.” Chap. 5 in Buoyancy-driven flows, 203–239. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511920196.006.
Lofquist, K. 1960. “Flow and stress near an interface between stratified liquids.” Phys. Fluids 3 (2): 158. https://doi.org/10.1063/1.1706013.
Lombardi, V., C. Adduce, G. Sciortino, and M. La Rocca. 2015. “Gravity currents flowing upslope: Laboratory experiments and shallow-water simulations.” Phys. Fluids 27 (1): 016602. https://doi.org/10.1063/1.4905305.
Manjura, N. 2016. “Numerical analysis of flow metrics and overall mixing in density current using large eddy simulation.” Ph.D. thesis, Dept. of Civil and Environmental Engineering, Univ. of Texas at San Antonio.
Marino, B. M., and L. P. Thomas. 2002. “Spreading of a gravity current over a permeable surface.” J. Hydraul. Eng. 128 (5): 527–533. https://doi.org/10.1061/(ASCE)0733-9429(2002)128:5(527).
Middleton, G. V. 1993. “Sediment deposition from turbidity currents.” Annu. Rev. Earth Planet. Sci. 21 (1): 89–114. https://doi.org/10.1146/annurev.ea.21.050193.000513.
Nogueira, H. I. S. 2013. “Experimental characterization of Unsteady gravity currents developing over smooth and rough beds.” Ph.D. thesis, Dept. of Civil Engineering, Universidade de Coimbra.
Nogueira, H. I. S., C. Adduce, E. Alves, and M. J. Franca. 2014. “Dynamics of the head of gravity currents.” Environ. Fluid Mech. 14 (2): 519–540. https://doi.org/10.1007/s10652-013-9315-2.
Ooi, S. K., G. Constantinescu, and L. Weber. 2007. “A numerical study of intrusive compositional gravity currents.” Phys. Fluids 19 (7): 076602. https://doi.org/10.1063/1.2750672.
Ooi, S. K., G. Constantinescu, and L. Weber. 2009. “Numerical simulations of lock-exchange compositional gravity current.” J. Fluid Mech. 635: 361–388. https://doi.org/10.1017/S0022112009007599.
Ottolenghi, L., C. Adduce, R. Inghilesi, V. Armenio, and F. Roman. 2016a. “Entrainment and mixing in unsteady gravity currents.” J. Hydraul. Res. 54 (5): 541–557. https://doi.org/10.1080/00221686.2016.1174961.
Ottolenghi, L., C. Adduce, R. Inghilesi, F. Roman, and V. Armenio. 2016b. “Mixing in lock-release gravity currents propagating up a slope.” Phys. Fluids 28 (5): 056604. https://doi.org/10.1063/1.4948760.
Parker, G., Y. Fukushima, and H. M. Pantin. 1986. “Selfaccelerating turbidity currents.” J. Fluid Mech. 171: 145–181. https://doi.org/10.1017/S0022112086001404.
Parker, G., M. Garcia, Y. Fukushima, and W. Yu. 1987. “Experiments on turbidity currents over an erodible bed.” J. Hydraul. Res. 25 (1): 123–147. https://doi.org/10.1080/00221688709499292.
Pierce, C. D., and P. Moin. 2004. “Progress-variable approach for large-eddy simulation of non-premixed turbulent combustion.” J. Fluid Mech. 504: 73–97. https://doi.org/10.1017/S0022112004008213.
Pokrajac, D., and G. A. Kikkert. 2011. “Radins equations for aerated shallow water flows over rough beds.” J. Hydraul. Res. 49 (5): 630–638. https://doi.org/10.1080/00221686.2011.597940.
Pokrajac, D., S. Venuleo, and M. J. Franca. 2018. “Pressure term in depth-averaged momentum equation for gravity currents with varying density.” J. Hydraul. Res. 56: 424–430. https://doi.org/10.1080/00221686.2017.1335245.
Princevac, M., H. J. S. Fernando, and C. D. Whiteman. 2005. “Turbulent entrainment into natural gravity-driven flows.” J. Fluid Mech. 533: 259–268. https://doi.org/10.1017/S0022112005004441.
Rodi, W., G. Constantinescu, and T. Stoesser. 2013. Large eddy simulation in hydraulics. IAHR monograph. Boca Raton, FL: CRC Press.
Rottman, J. W., and J. E. Simpson. 1983. “Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel.” J. Fluid Mech. 135: 95–110. https://doi.org/10.1017/S0022112083002979.
Sequeiros, O. E. 2012. “Estimating turbidity current conditions from channel morphology: A Froude number approach: Gravity flow estimation froude approach.” J. Geophys. Res. Oceans 117 (C4): C04003. https://doi.org/10.1029/2011JC007201.
Sequeiros, O. E., H. Naruse, N. Endo, M. H. Garcia, and G. Parker. 2009. “Experimental study on self-accelerating turbidity currents.” J. Geophys. Res. 114 (C5): C05025. https://doi.org/10.1029/2008JC005149.
Sequeiros, O. E., B. Spinewine, R. T. Beaubouef, T. Sun, M. H. Garcia, and G. Parker. 2010. “Characteristics of velocity and excess density profiles of saline underflows and turbidity currents flowing over a mobile bed.” J. Hydraul. Eng. 136 (7): 412–433. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000200.
Sher, D., and A. W. Woods. 2015. “Gravity currents: Entrainment, stratification and self-similarity.” J. Fluid Mech. 784: 130–162. https://doi.org/10.1017/jfm.2015.576.
Shin, J. O., S. B. Dalziel, and P. F. Linden. 2004. “Gravity currents produced by lock exchange.” J. Fluid Mech. 521: 1–34. https://doi.org/10.1017/S002211200400165X.
Stacey, M. W., and A. J. Bowen. 1988. “The vertical structure of density and turbidity currents: Theory and observations.” J. Geophys. Res. Oceans 93 (C4): 3528–3542. https://doi.org/10.1029/JC093iC04p03528.
Stagnaro, M., and M. Bolla Pittaluga. 2014. “Velocity and concentration profiles of saline and turbidity currents flowing in a straight channel under quasi-uniform conditions.” Earth Surf. Dyn. 2 (1): 167–180. https://doi.org/10.5194/esurf-2-167-2014.
Steenhauer, K., T. Tokyay, and G. Constantinescu. 2017. “Dynamics and structure of planar gravity currents propagating down an inclined surface.” Phys. Fluids 29 (3): 036604. https://doi.org/10.1063/1.4979063.
Strang, E. J., and H. J. S. Fernando. 2001. “Entrainment and mixing in stratified shear flows.” J. Fluid Mech. 428: 349–386. https://doi.org/10.1017/S0022112000002706.
Tokyay, T., and G. Constantinescu. 2015. “The effects of a submerged non-erodible triangular obstacle on bottom propagating gravity currents.” Phys. Fluids 27 (5): 056601. https://doi.org/10.1063/1.4919384.
Tokyay, T., G. Constantinescu, and E. Meiburg. 2012. “Tail structure and bed friction velocity distribution of gravity currents propagating over an array of obstacles.” J. Fluid Mech. 694: 252–291. https://doi.org/10.1017/jfm.2011.542.
Tokyay, T. E., and M. H. García. 2014. “Effect of initial excess density and discharge on constant flux gravity currents propagating on a slope.” Environ. Fluid Mech. 14 (2): 409–429. https://doi.org/10.1007/s10652-013-9317-0.
Turner, J. S. 1986. “Turbulent entrainment: The development of the entrainment assumption, and its application to geophysical flows.” J. Fluid Mech. 173: 431–471. https://doi.org/10.1017/S0022112086001222.
Ungarish, M. 2007. “Axisymmetric gravity currents at high reynolds number: On the quality of shallow-water modeling of experimental observations.” Phys. Fluids 19 (3): 036602. https://doi.org/10.1063/1.2714990.
Ungarish, M., and H. E. Huppert. 2000. “High-reynolds-number gravity currents over a porous boundary: Shallow-water solutions and box-model approximations.” J. Fluid Mech. 418: 1–23. https://doi.org/10.1017/S0022112000008302.
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Received: Feb 11, 2020
Accepted: Oct 26, 2020
Published online: Mar 12, 2021
Published in print: May 1, 2021
Discussion open until: Aug 12, 2021
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