## Effect of Schmidt number on the structure and propagation of density currents.(English)Zbl 1178.76115

Summary: The results of a numerical study of two- and three-dimensional Boussinesq density currents are described. They are aimed at exploring the role of the Schmidt number on the structure and dynamics of density driven currents. Two complementary approaches are used, namely a spectral method and a finite-volume interface capturing method. They allow for the first time to describe density currents in the whole range of Schmidt number $$1 \leq Sc \leq \infty$$ and Reynolds number $$10^{2} \leq Re \leq 10^{4}$$. The present results confirm that the Schmidt number only weakly influences the structure and dynamics of density currents provided the Reynolds number of the flow is large, say of $$O(10^{4})$$ or more. On the contrary low- to moderate-$$Re$$ density currents are dependant on $$Sc$$ as the structure of the mixing region and the front velocities are modified by diffusion effects. The scaling of the characteristic density thickness of the interface has been confirmed to behave as $$(ScRe)^{ - 1/2}$$. Three-dimensional simulations suggest that the patterns of lobes and clefts are independent of $$Sc$$. In contrast the Schmidt number is found to affect dramatically (1) the shape of the current head as a depression is observed at high-$$Sc$$, (2) the formation of vortex structures generated by Kelvin-Helmholtz instabilities. A criterion is proposed for the stability of the interface along the body of the current based on the estimate of a bulk Richardson number. This criterion, derived for currents of arbitrary density ratio, is in agreement with present computed results as well as available experimental and numerical data.

### MSC:

 76D05 Navier-Stokes equations for incompressible viscous fluids 76M22 Spectral methods applied to problems in fluid mechanics 76M12 Finite volume methods applied to problems in fluid mechanics
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### References:

 [1] Allen J.: Principles of Physical Sedimentology. George Allen and Unwin Ltd., London (1985) [2] Benjamin T.B.: Density urrents and related phenomena. J. Fluid Mech. 31, 209–248 (1968) · Zbl 0169.28503 [3] Birman V., Martin J.E., Meiburg E.: The non-Boussinesq lock–exchange problem. Part 2. High-resolution simulations. J. Fluid Mech. 537, 125–144 (2005) · Zbl 1138.76345 [4] Birman V.K., Battandier B.A., Meiburg E., Linden P.F.: Lock–exchange flows in sloping channels. J. Fluid Mech. 577, 53–77 (2007) · Zbl 1178.76114 [5] Bonometti T., Magnaudet J.: Transition from spherical cap to toroidal bubbles. Phys. Fluids 18, 052102 (2006) [6] Bonometti T., Magnaudet J.: An interface-capturing method for incompressible two-phase flows. Validation and application to bubble dynamics. Int. J. Multiph. Flow 33, 109–133 (2007) [7] Brenner H.: Kinematics of volume transport. Physica A 349, 11–59 (2005) [8] Cantero M., Balachandar S., Garcia M., Ferry J.: Direct numerical simulations of planar and cyindrical density currents. J. Appl. Mech. 73, 923–930 (2006) · Zbl 1111.74338 [9] Cantero M.I., Lee J.R., Balachandar S., Garcia M.H.: On the front velocity of gravity currents. J. Fluid Mech. 586, 1–39 (2007) · Zbl 1178.76135 [10] Canuto C., Hussaini M., Quarteroni A., Zang T.: Spectral Methods in Fluid Dynamics. Springer, Heidelberg (1988) · Zbl 0658.76001 [11] Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford (1961) · Zbl 0142.44103 [12] Cortese T., Balachandar S.: High performance spectral simulation of turbulent flows in massively parallel machines with distributed memory. Int. J. Supercomput. Ap. 9, 187–204 (1995) [13] Daly B., Pracht W.: Numerical study of density-current surges. Phys. Fluids 11, 15–30 (1968) · Zbl 0153.57103 [14] Drazin P.G., Reid W.H.: Hydrodynamic Stability, 2nd edn. Cambridge University Press, Cambridge (1981) [15] Fay J.: The spreads of oil slicks on a calm sea. In: Hoult, D.P.(eds) Oils in the Sea, pp. 53–63. Plenum Press, New yorkm (1969) [16] Grant G.B., Jagger S.F., Lea C.J.: Fires in tunnels. Philos. Trans. R. Soc. Lond. A 356, 2873–296 (1998) [17] Gröbelbauer H.P., Fanneløp T.K., Britter R.E.: The propagation of intrusion fronts of high density ratio. J. Fluid Mech. 250, 669–687 (1993) [18] Härtel C., Meiburg E., Necker F.: Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front spedd for slip and no-slip boundaries. J. Fluid Mech. 418, 189–212 (2000a) · Zbl 0985.76042 [19] Härtel C., Carlsson F., Thunblom M.: Analysis and direct numerical simulation of the flow at a gravity-current head. Part 2. The lobe-and-cleft instability. J. Fluid Mech. 418, 213–229 (2000b) · Zbl 1103.76337 [20] Hoult D.: Oil spreading in the sea. Annu. Rev. Fluid Mech. 4, 341–368 (1972) [21] Huppert H.: The propagation of two-dimensional and axisymmetric viscous density currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58 (1982) [22] Huppert H.E.: Density currents: a personnel perspective. J. Fluid Mech. 554, 299–322 (2006) · Zbl 1090.76022 [23] Huppert H., Simpson J.: The slumping of gravity currents. J. Fluid Mech. 99, 785–799 (1980) [24] Joseph D., Renardy Y.: Fundamentals of Two Fluids Dynamics. Part II. Springer, Heidelberg (1992) [25] Klemp J.B., Rotunno R., Skamarock W.C.: On the dynamics of density currents in a channel. J. Fluid Mech. 269, 169–198 (1994) · Zbl 0800.76076 [26] Lowe R.J., Rottman J.W., Linden P.F.: The non-Boussinesq lock–exchange problem. Part 1. Theory and experiments. J. Fluid Mech. 537, 101–124 (2005) · Zbl 1138.76332 [27] Marino B., Thomas L., Linden P.: The front condition for density currents. J. Fluid Mech. 536, 49–78 (2005) · Zbl 1077.76020 [28] Necker F., Härtel C., Kleiser L., Meiburg E.: Mixing and dissipation in particle-driven density currents. J. Fluid Mech. 545, 339–372 (2005) · Zbl 1085.76559 [29] Ozgökmen T., Fischer P., Duan J., Iliescu T.: Three-dimensional turbulent bottom density currents from a high-order nonhydrostatic spectral element model. J. Phys. Oceanogr. 34, 2006–2026 (2004) [30] Pawlak G., Armi L.: Mixing and entrainment in developing stratified currents. J. Fluid Mech. 424, 45–73 (2000) · Zbl 0958.76510 [31] Ritter A.: Die fortplanzung der wasserwellen. Z. Verein. Deutsch. Ing. 36, 947–954 (1892) [32] Rottman J., Simpson J.: Density currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–110 (1983) [33] Schmidt W.: Zur Mechanik der boen. Z. Meteorol. 28, 355–362 (1911) [34] Shin J., Dalziel S., Linden P.: Density currents produced by lock exchange. J. Fluid Mech. 521, 1–34 (2004) · Zbl 1065.76037 [35] Simpson J.: Effect of the lower boundary on the head of a gravity current. J. Fluid Mech. 53, 759–768 (1972) [36] Simpson J.: Density Currents, 2nd edn. Cambridge University Press, Cambridge (1997) [37] Thorpe S.A.: A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32, 693–704 (1968) [38] von Karman T.: The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615–683 (1940) · JFM 66.0985.04 [39] Zalesak S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31, 335–362 (1979) · Zbl 0416.76002
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