Thermally enhanced gravity driven flows. (English) Zbl 1288.76014
Summary: Gravity currents play a major role in both natural and human-made settings. The driving buoyancy forces for these flows are due to density differences which may arise as a result of compositional differences (e.g. salinity), the presence of suspended material in the flow as in the case of turbidity currents, temperature differences, or a combination of these mechanisms. This article reports on a study of surface gravity currents moving horizontally over a slightly denser ambient fluid when these surface layers are subjected to an incoming heat flux which acts to enhance (or erode) the existing stratification. A general equation of state is adopted to connect the density of the upper layer to its changing temperature. A two-layer hydraulic theory is developed and conditions for its validity carefully specified. Both analytical and numerical analyses are carried out in order to examine the salient features of these flows.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 35L65 | Hyperbolic conservation laws |
References:
| [1] | Antar, N.; Moodie, T. B., Weakly nonhydrostatic effects in compositionally-driven gravity flows, Stud. Appl. Math., 111, 239-267 (2003) · Zbl 1141.76347 |
| [2] | Benjamin, T. B., Gravity currents and related phenomena, J. Fluid Mech., 31, 209-248 (1968) · Zbl 0169.28503 |
| [3] | J.R. Bennett, Thermally driven lake currents during the spring and fall transition periods, Proceedings of the 14th Conference on Great Lakes Research, Ann Arbor, Michigan, 1971, pp. 535-544.; J.R. Bennett, Thermally driven lake currents during the spring and fall transition periods, Proceedings of the 14th Conference on Great Lakes Research, Ann Arbor, Michigan, 1971, pp. 535-544. |
| [4] | Bonnecaze, R. T.; Huppert, H. E.; Lister, J. R., Particle-driven gravity currents, J. Fluid Mech., 250, 339-369 (1993) |
| [5] | D’Alessio, S. J.D.; Moodie, T. B.; Pascal, J. P.; Swaters, G. E., Gravity currents produced by sudden release of a fixed volume of heavy fluid, Stud. Appl. Math., 96, 359-385 (1996) · Zbl 0852.76093 |
| [6] | Eklund, H., Freshwatertemperature of maximum density calculated from compressibility, Science, 142, 1457-1458 (1963) |
| [7] | G.H. Elliott, A mathematical study of the thermal bar, In: Proceedings of the 14th Conference on Great Lakes Research, Ann Arbor, Michigan, 1971, pp. 545-554.; G.H. Elliott, A mathematical study of the thermal bar, In: Proceedings of the 14th Conference on Great Lakes Research, Ann Arbor, Michigan, 1971, pp. 545-554. |
| [8] | Fannelop, T. K.; Waldman, G. D., Dynamics of oil slicks, AIAA J., 10, 506-510 (1972) |
| [9] | Farrow, D. E., An asymptotic model for the hydrodynamics of the thermal bar, J. Fluid Mech., 289, 129-140 (1995) · Zbl 0843.76077 |
| [10] | Farrow, D. E., A numerical model of the hydrodymanics of the thermal bar, J. Fluid Mech., 303, 279-295 (1995) · Zbl 0854.76085 |
| [11] | Farrow, D. E., A model of the thermal bar in the rotating frame including vertically non-uniform heating, Environ. Fluid Mech., 2, 197-218 (2002) |
| [12] | Holland, P. R.; Kay, A.; Botte, V., A numerical study of the dynamics of the riverine thermal bar in a deep lake, Environ. Fluid Mech., 1, 311-332 (2001) |
| [13] | Hoult, D. P., Oil spreading on the sea, Ann. Rev. Fluid Mech., 4, 341-368 (1972) |
| [14] | D.W. Hubbard, J.D. Spain, The structure of the early spring thermal bar in Lake Superior, Proceedings of the 16th Conference on Great Lakes Research, Ann Arbor, Michigan, 1973, pp. 735-742.; D.W. Hubbard, J.D. Spain, The structure of the early spring thermal bar in Lake Superior, Proceedings of the 16th Conference on Great Lakes Research, Ann Arbor, Michigan, 1973, pp. 735-742. |
| [15] | Kundu, P. K., Fluid Mechanics (1990), Academic Press: Academic Press London · Zbl 0780.76001 |
| [16] | Lapidus, A., A detached shock calculation by second-order finite differences, J. Comput. Phys., 2, 154-177 (1967) · Zbl 0152.44804 |
| [17] | Lax, P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (1973), SIAM: SIAM Philadelphia, PA · Zbl 0268.35062 |
| [18] | Le Veque, R., Numerical Methods for Conservation Laws (1992), Birkhäuser: Birkhäuser Basel · Zbl 0847.65053 |
| [19] | Linden, P. F., The fluid mechanics of natural ventilation, Ann. Rev. Fluid Mech., 31, 201-238 (1999) |
| [20] | Malm, J.; Zilitinkevich, S., Temperature distribution and current system in a convectively mixed lake, Bound. Layer Meteor., 71, 219-234 (1994) |
| [21] | Moodie, T. B.; Pascal, J. P., Non-hydraulic effects in particle-driven gravity currents in deep surroundings, Stud. Appl. Math., 107, 217-251 (2001) · Zbl 1152.86306 |
| [22] | Moodie, T. B.; Pascal, J. P.; Bowman, J. C., Modeling sediment deposition patterns arising from suddenly released fixed-volume turbulent suspensions, Stud. Appl. Math., 105, 333-359 (2000) · Zbl 1136.76707 |
| [23] | Moodie, T. B.; Pascal, J. P.; Swaters, G. E., Sediment transport and deposition from a two-layer fluid model of gravity currents on sloping bottoms, Stud. Appl. Math., 100, 215-244 (1998) · Zbl 1003.86001 |
| [24] | Rottman, J. W.; Simpson, J. E., Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel, J. Fluid Mech., 135, 95-110 (1983) |
| [25] | Sweby, P. K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21, 995-1011 (1984) · Zbl 0565.65048 |
| [26] | von Kármán, T., The engineer grapples with nonlinear problems, Bull. Amer. Math. Soc., 46, 615-683 (1940) · JFM 66.0985.04 |
| [27] | Yu, H.; Liu, Y., A second-order accurate, component-wise TVD scheme for nonlinear, hyperbolic conservation laws, J. Comput. Phys., 173, 1-16 (2001) · Zbl 0987.65084 |
| [28] | Zilitinkevich, S. S.; Kreiman, K. D.; Terzhevik, A. Y., The thermal bar, J. Fluid Mech., 236, 27-42 (1992) · Zbl 0746.76079 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
