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Statistical characterisation of turbulence for an unsteady gravity current

Published online by Cambridge University Press:  20 August 2020

Joë Pelmard Open the ORCID record for Joë Pelmard [Opens in a new window] ,
S. Norris  and
H. Friedrich Open the ORCID record for H. Friedrich [Opens in a new window]
Joë Pelmard*
Affiliation:
Department of Civil and Environmental Engineering, University of Auckland, 20 Symonds Street, Auckland1010, New Zealand
S. Norris
Affiliation:
Department of Mechanical Engineering, University of Auckland, 20 Symonds Street, Auckland1010, New Zealand
H. Friedrich
Affiliation:
Department of Civil and Environmental Engineering, University of Auckland, 20 Symonds Street, Auckland1010, New Zealand
*
†Email address for correspondence: jpel442@aucklanduni.ac.nz
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Abstract

The present study aims to provide a statistical analysis of turbulence in the mixing layer of a lock-exchange gravity current propagating over a 2 % slope based on large eddy simulation using a Boussinesq code. The statistics are calculated from the ensemble and spanwise averaging of 200 simulations for two time steps corresponding to the initial constant velocity slumping phase and the decelerating inertial phase. The overall energy balance and structure of the mixing layer are weakly influenced by the propagation time following the lock release. Thereby, streamwise dominated turbulence is produced by the positive buoyancy flux and subsequently converted into averaged flow through energy backscatter in the nose, whereas the current's interface takes the structure of a stratified mixing layer unstable to Kelvin–Helmholtz instabilities in the rear of the head. The dependency of the current head/body structure on the evolution of the turbulence kinetic energy (TKE) along the mixing layer is also investigated. The transition from the head to the body is associated with a peak of TKE and the flux Richardson number exceeding the stability criterion $Ri_f = 0.2$. It is furthermore observed that the turbulence intensity in all three spatial directions stabilises to satisfy $\langle u'u' \rangle = 2 \langle v'v' \rangle = 2 \langle w'w' \rangle$, where $u', v' \ \text{and} \ w'$ are respectively the streamwise, spanwise and vertical turbulent perturbations of velocity. Finally, a region of statistically stationary TKE is identified once the gradient Richardson number plateaus to a value dependent on the current's propagation approximately 5.5 lock heights backward from the front, where the depth-averaged TKE budget reduces to the balance between the contributions due to shear (P), buoyancy (B) and viscous dissipation $(\varepsilon)$ as $\langle P \rangle _d + \langle B \rangle _d - \langle \varepsilon \rangle _d \approx 0$.

JFM classification

Geophysical and Geological Flows: Gravity currents Turbulent Flows: Shear layer turbulence Turbulent Flows: Stratified turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 901 , 25 October 2020 , A7
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Balaras, E., Piomelli, U. & Wallace, J. M. 2001 Self-similar states in turbulent mixing layers. J. Fluid Mech. 446, 1–24.CrossRefGoogle Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31 (2), 209–248.CrossRefGoogle Scholar
Bhaganagar, K. 2017 Role of head of turbulent 3-D density currents in mixing during slumping regime. Phys. Fluids 29 (2), 020703.CrossRefGoogle Scholar
Bonnecaze, R. T., Huppert, H. E. & Lister, J. R. 1993 Particle-driven gravity currents. J. Fluid Mech. 250, 339–369.CrossRefGoogle Scholar
Bonometti, T. & Balachandar, S. 2008 Effect of Schmidt number on the structure and propagation of density currents. Theor. Comput. Fluid Dyn. 22 (5), 341–361.CrossRefGoogle Scholar
Britter, R. E. 1974 An experiment on turbulence in a density stratified fluid. PhD thesis, Monash University.Google Scholar
Cantero, M. I., Balachandar, S., Garcia, M. H. & Bock, D. 2008 Turbulent structures in planar gravity currents and their influence on the flow dynamics. J. Geophys. Res. 113 (C8).CrossRefGoogle Scholar
Cantero, M. I., Balachandar, S. & Parker, G. 2009 Direct numerical simulation of stratification effects in a sediment-laden turbulent channel flow. J. Turbul. 10 (27), 1–28.CrossRefGoogle Scholar
Cantero, M. I., Lee, J. R., Balachandar, S. & Garcia, M. H. 2007 On the front velocity of gravity currents. J. Fluid Mech. 586, 1–39.CrossRefGoogle Scholar
Cenedese, C. & Adduce, C. 2010 A new parameterization for entrainment in overflows. J. Phys. Oceanogr. 40 (8), 1835–1850.CrossRefGoogle Scholar
Chang, K. C. & Lee, K. H. 2017 Determination of mixing length in turbulent mixing layer on basis of vorticity field. Int. J. Heat Fluid Flow 66, 121–126.CrossRefGoogle Scholar
Chang, K. C. & Li, C. T. 2011 Redefining mixing length in turbulent mixing layer in terms of shear-induced vorticity. J. Fluid Sci. Technol. 6 (4), 662–673.CrossRefGoogle Scholar
Eggenhuisen, J. T. & McCaffrey, W. D. 2012 The vertical turbulence structure of experimental turbidity currents encountering basal obstructions: implications for vertical suspended sediment distribution in non-equilibrium currents. Sedimentology 59 (3), 1101–1120.CrossRefGoogle Scholar
Ellison, T. H. 1957 Turbulent transport of heat and momentum from an infinite rough plane. J. Fluid Mech. 2 (5), 456–466.CrossRefGoogle Scholar
Ellison, T. H. & Turner, J. S. 1959 Turbulent entrainment in stratified flows. J. Fluid Mech. 6 (3), 423–448.CrossRefGoogle Scholar
Fragoso, A. T., Patterson, M. D. & Wettlaufer, J. S. 2013 Mixing in gravity currents. J. Fluid Mech. 734, R2.CrossRefGoogle Scholar
Garcia, M. H. 1994 Depositional turbidity currents laden with poorly sorted sediment. J. Hydraul. Engng ASCE 120 (11), 1240–1263.CrossRefGoogle Scholar
Gray, T. E., Alexander, J. & Leeder, M. R. 2006 Longitudinal flow evolution and turbulence structure of dynamically similar, sustained, saline density and turbidity currents. J. Geophys. Res. 111 (C8).CrossRefGoogle Scholar
Hacker, J., Linden, P. F. & Dalziel, S. B. 1996 Mixing in lock-release gravity currents. Dyn. Atmos. Oceans 24 (1–4), 183–195.CrossRefGoogle Scholar
Hallworth, M. A., Huppert, H. E., Phillips, J. C. & Sparks, R. S. J. 1996 Entrainment into two-dimensional and axisymmetric turbulent gravity currents. J. Fluid Mech. 308, 289–311.CrossRefGoogle Scholar
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
Inghilesi, R., Adduce, C., Lombardi, V., Roman, F. & Armenio, V. 2018 Axisymmetric three-dimensional gravity currents generated by lock exchange. J. Fluid Mech. 851, 507–544.CrossRefGoogle Scholar
Jovanovic, J. 2013 The Statistical Dynamics of Turbulence. Springer Science & Business Media.Google Scholar
Keulegan, G. H. 1957 An experimental study of the motion of saline water from locks into fresh water channels. NBS Report 5168. US National Bureau of Standards.Google Scholar
Kneller, B. C., Bennett, S. J. & McCaffrey, W. D. 1999 Velocity structure, turbulence and fluid stresses in experimental gravity currents. J. Geophys. Res. 104 (C3), 5381–5391.CrossRefGoogle Scholar
Komar, P. D. 1972 Relative significance of head and body spill from a channelized turbidity current. Geol. Soc. Am. Bull. 83 (4), 1151–1156.CrossRefGoogle Scholar
Krug, D., Holzner, M., Luthi, B., Wolf, M., Kinzelbach, W. & Tsinober, A. 2013 Experimental study of entrainment and interface dynamics in a gravity current. Exp. Fluids 54 (5), 1530.CrossRefGoogle Scholar
Krug, D., Holzner, M., Luthi, B., Wolf, M., Kinzelbach, W. & Tsinober, A. 2015 The turbulent/non-turbulent interface in an inclined dense gravity current. J. Fluid Mech. 765, 303–324.CrossRefGoogle Scholar
Kyrousi, F., Leonardi, A., Roman, F., Armenio, V., Zanello, F., Zordan, J., Juez, C. & Falcomer, L. 2018 Large eddy simulations of sediment entrainment induced by a lock-exchange gravity current. Adv. Water Resour. 114, 102–118.CrossRefGoogle Scholar
Lombardi, V., Adduce, C. & La Rocca, M. 2018 Unconfined lock-exchange gravity currents with variable lock width: laboratory experiments and shallow-water simulations. J. Hydraul. Res. 56 (3), 399–411.CrossRefGoogle Scholar
Lumley, J. L. & Newman, G. R. 1977 The return to isotropy of homogeneous turbulence. J. Fluid Mech. 82 (1), 161–178.CrossRefGoogle Scholar
Marino, B. M., Thomas, L. P. & Linden, P. F. 2005 The front condition for gravity currents. J. Fluid Mech. 536, 49–78.CrossRefGoogle Scholar
Meiburg, E. & Kneller, B. 2010 Turbidity currents and their deposits. Annu. Rev. Fluid Mech. 42, 135–156.CrossRefGoogle Scholar
Meiburg, E., Radhakrishnan, S. & Nasr-Azadani, M. M. 2015 Modeling gravity and turbidity currents: computational approaches and challenges. Appl. Mech. Rev. 67 (4), 040802.CrossRefGoogle Scholar
Mellor, G. L. & Yamada, T. 1982 Development of a turbulence closure-model for geophysical fluid problems. Rev. Geophys. 20 (4), 851–875.CrossRefGoogle Scholar
Middleton, G. V. 1966 Experiments on density and turbidity currents: I. Motion of the head. Can. J. Earth Sci. 3 (4), 523–546.CrossRefGoogle Scholar
Nasr-Azadani, M. M. & Meiburg, E. 2011 Turbins: an immersed boundary, Navier–Stokes code for the simulation of gravity and turbidity currents interacting with complex topographies. Comput. Fluids 45 (1), 14–28.CrossRefGoogle Scholar
Necker, F., Härtel, C., Kleiser, L. & Meiburg, E. 2002 High-resolution simulations of particle-driven gravity currents. Intl J. Multiphase Flow 28 (2), 279–300.CrossRefGoogle Scholar
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
Nogueira, H. I. S., Adduce, C., Alves, E. & Franca, M. J. 2014 Dynamics of the head of gravity currents. Environ. Fluid Mech. 14 (2), 519–540.CrossRefGoogle Scholar
Norris, S. E. 2000 A parallel Navier–Stokes solver for natural convection and free surface flow. PhD thesis, University of Sydney.Google Scholar
Odier, P., Chen, J. & Ecke, R. E. 2014 Entrainment and mixing in a laboratory model of oceanic overflow. J. Fluid Mech. 746, 498–535.CrossRefGoogle Scholar
Ooi, S. K., Constantinescu, G. & Weber, L. 2009 Numerical simulations of lock-exchange compositional gravity current. J. Fluid Mech. 635, 361–388.CrossRefGoogle Scholar
Osborn, T. R. 1980 Estimates of the local-rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10 (1), 83–89.2.0.CO;2>CrossRefGoogle Scholar
Ottolenghi, L., Adduce, C., Inghilesi, R., Armenio, V. & Roman, F. 2016 a Entrainment and mixing in unsteady gravity currents. J. Hydraul. Res. 54 (5), 541–557.CrossRefGoogle Scholar
Ottolenghi, L., Adduce, C., Inghilesi, R., Roman, F. & Armenio, V. 2016 b Mixing in lock-release gravity currents propagating up a slope. Phys. Fluids 28 (5), 056604.CrossRefGoogle Scholar
Ottolenghi, L., Adduce, C., Roman, E. & Armenio, V. 2017 a Analysis of the flow in gravity currents propagating up a slope. Ocean Model. 115, 1–13.CrossRefGoogle Scholar
Ottolenghi, L., Cenedese, C. & Adduce, C. 2017 b Entrainment in a dense current flowing down a rough sloping bottom in a rotating fluid. J. Phys. Oceanogr. 47 (3), 485–498.CrossRefGoogle Scholar
Ottolenghi, L., Prestininzi, P., Montessori, A., Adduce, C. & La Rocca, M. 2018 Lattice Boltzmann simulations of gravity currents. Eur. J. Mech. B/Fluids 67, 125–136.CrossRefGoogle Scholar
Özgökmen, T. M., Iliescu, T. & Fischer, P. F. 2009 Reynolds number dependence of mixing in a lock-exchange system from direct numerical and large eddy simulations. Ocean Model. 30 (2-3), 190–206.CrossRefGoogle Scholar
Pacanowski, R. C. & Philander, S. G. H. 1981 Parameterization of vertical mixing in numerical models of tropical oceans. J. Phys. Oceanogr. 11 (11), 1443–1451.2.0.CO;2>CrossRefGoogle Scholar
Pelmard, J., Norris, S. E. & Friedrich, H. 2018 LES grid resolution requirements for the modelling of gravity currents. Comput. Fluids 174, 256–270.CrossRefGoogle Scholar
Pérez-Díaz, B., Castanedo, S., Palomar, P., Henno, F. & Wood, M. 2018 a Modeling nonconfined density currents using 3D hydrodynamic models. J. Hydraul. Engng 145 (3), 04018088.CrossRefGoogle Scholar
Pérez-Díaz, B., Palomar, P., Castanedo, S. & Alvarez, A. 2018 b PIV-PLIF characterization of nonconfined saline density currents under different flow conditions. J. Hydraul. Engng 144 (9), 04018063.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Reeuwijk, Van, Holzner, M., Caulfield, M. & P., C. 2019 Mixing and entrainment are suppressed in inclined gravity currents. J. Fluid Mech. 873, 786–815.CrossRefGoogle Scholar
Rohr, J. J., Itsweire, E. C., Helland, K. N. & Van Atta, C. W. 1988 Growth and decay of turbulence in a stably stratified shear flow. J. Fluid Mech. 195, 77–111.CrossRefGoogle Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–110.CrossRefGoogle Scholar
Sher, D. & Woods, A. W. 2015 Gravity currents: entrainment, stratification and self-similarity. J. Fluid Mech. 784, 130–162.CrossRefGoogle Scholar
Shin, J. O., Dalziel, S. B. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1–34.CrossRefGoogle Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weath. Rev. 91 (3), 99–164.2.3.CO;2>CrossRefGoogle Scholar
Steenhauer, K., Tokyay, T. & Constantinescu, G. 2017 Dynamics and structure of planar gravity currents propagating down an inclined surface. Phys. Fluids 29 (3), 036604.CrossRefGoogle Scholar
Strang, E. J. & Fernando, H. J. S. 2001 Entrainment and mixing in stratified shear flows. J. Fluid Mech. 428, 349–386.CrossRefGoogle Scholar
Tavoularis, S. 1985 Asymptotic laws for transversely homogeneous turbulent shear flows. Phys. Fluids 28 (3), 999–1001.CrossRefGoogle Scholar
Thomas, L. P., Dalziel, S. B. & Marino, B. M. 2003 The structure of the head of an inertial gravity current determined by particle-tracking velocimetry. Exp. Fluids 34 (6), 708–716.CrossRefGoogle Scholar
Thorpe, S. A. 1968 A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32 (4), 693–704.CrossRefGoogle Scholar
Tokyay, T., Constantinescu, G. & Meiburg, E. 2012 Tail structure and bed friction velocity distribution of gravity currents propagating over an array of obstacles. J. Fluid Mech. 694, 252–291.CrossRefGoogle Scholar
Tokyay, T., Constantinescu, G. & Meiburg, E. 2014 Lock-exchange gravity currents with a low volume of release propagating over an array of obstacles. J. Geophys. Res. 119 (5), 2752–2768.CrossRefGoogle Scholar
Townsend, A. A. 1958 The effects of radiative transfer on turbulent flow of a stratified fluid. J. Fluid Mech. 4 (4), 361–375.CrossRefGoogle Scholar
Townsend, A. A. 1980 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
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
Vreman, B., Geurts, B. & Kuerten, H. 1997 Large-eddy simulation of the turbulent mixing layer. J. Fluid Mech. 339, 357–390.CrossRefGoogle Scholar
Wilson, R. I., Friedrich, H. & Stevens, C. 2017 Turbulent entrainment in sediment-laden flows interacting with an obstacle. Phys. Fluids 29 (3), 036603.CrossRefGoogle Scholar
Wilson, R. I., Friedrich, H. & Stevens, C. 2018 a Flow structure of unconfined turbidity currents interacting with an obstacle. Environ. Fluid Mech. 18 (6), 1571–1594.CrossRefGoogle Scholar
Wilson, R. I., Friedrich, H. & Stevens, C. 2018 b Image thresholding process for combining photometry with intrusive flow instruments. J. Hydraul. Res. 56 (2), 282–290.CrossRefGoogle Scholar
Yang, W. B., Zhang, H. Q., Chan, C. K. & Lin, W. Y. 2004 Large eddy simulation of mixing layer. J. Comput. Appl. Math. 163 (1), 311–318.CrossRefGoogle Scholar
Zhao, L., He, Z., Lv, Y., Lin, Y. T., Hu, P. & Pähtz, T. 2018 Front velocity and front location of lock-exchange gravity currents descending a slope in a linearly stratified environment. J. Hydraul. Engng 144 (11), 04018068.CrossRefGoogle Scholar