Keywords:-
Article Content:-
Abstract
An investigation into some possible behaviours of rising plumes with a quadratic dependence relation assumption was made with the consideration that the plumes arises from a virtual sources. Our results showed that both forced and pure plumes provides buoyancy flux at the source. Meanwhile, the lazy plume rises with a significant volume flux and finite temperature with zero volume flux for both the forced plume and pure at the source. The forced plume also possesses some significant upward momentum flux and zero momentum flux for the pure plume. Determination of zero-buoyancy height and the fountain-top height are key when it comes to rising plumes. If the ambient water depth into consideration is less than the zero-buoyancy height then the warm water will definitely spread outwards as surface gravity current; otherwise, it will form a fountain.Our results showed that the upward momentum flux attains its pick at; and changes sign in the buoyancy force at this point while the momentum flux tends to zero with a finite final volume flux
Qf from its maximum rise height. Meanwhile, as M0 approaches the value , so the Qf and the Q0 tends to the value. It is worth stating also that the fountain-top height will be of great significant if the focus is on erosion of an ice cover. We therefore suggest maintaining a low source Froude number which will result to minimising impact on the lake floor or on the surface. Though, some level of carefulness is needed whenever we intend to make use of the present results to foretell the behaviour of real plumes. In particular, if laboratory experiments are required either to examine the theory or to model larger-scale flows in the environment. Lastly, with the entrainment model, it is required that the plume will be fully turbulent, a condition usually obtained with a Reynolds number Re > 2000. This can be likened to the power station warm discharge which will definitely be fully turbulent with it large volume flux. These are basic conditions on the validity of the assumptions. There are some limitations as stated in the conclusion. But then, the overall numerical results here are very good as they present us with more insight into rising plume with the quadratic dependence relation assumption.
References:-
References
Bloomfield, L. J. and Kerr, R. C. (2000). A theoretical model of a turbulent fountain J. Fluid Mech 424 Pp. 197 - 216.
Caulfield, C-C.P. and Woods, A.W. (1995). Plumes with non-monotonic mixing behaviour Geophys. Astrophys. Fluid Dynamics 79 Pp. 173 - 199.
Fischer, H. B., List, E. J., Koh, R. C.Y., Imberger, J. and Brooks, N. H. (1979). Mixing in Inland and Coastal Waters. Academic Press
George, M. A. and Osaisai, F. E.(2023). Flow Parameter Dependence of Laminar Plumes with Buoyancy Reversal from a line Source Current Trends in Engineering Science (CTES) 3(4) 1036.
George, M. A., Osaisai, F. E. and Dienagha, N. (2023). Laminar Plumes from a Line Source and Their Possible Flow Behaviours Asian Research Journal of Mathematics 19(4) 15 - 30.
George, M. A. and Kay, A. (2022). Numerical Simulations of the Cabbeling Phenomenon in Surface Gravity Currents in Cold Fresh Water Journal of Scientific Research & Reports 28(1)Pp. 68 - 85.
George, M. A. and Osaisai, F. E. (2022). Density Current Simulations In Cold Fresh Water And Its Cabbeling phenomenon: A Comparative Analysis With Given Experimental Results Current Journal of Applied Science and Technology 41(29) Pp. 37 - 52.
George, M. A. and Kay, A. (2017). Warm discharges in cold fresh water: 2. Numerical simulation of laminar line plumes Environ. Fluid Mech.doi: http://dx.doi.org/10.1007/s10652-016-9468-x.
George, M. A. and Kay, A. (2017). Numerical simulation of a line plume impinging on a ceiling in cold fresh water International Journal of Heat and Mass Transfer 108 Pp. 1364 - 1373.
Hoglund, B. and Spigarelli, S. A. (1972). Studies of the sinking plume phenomenon Proc. 15th Conf. Great Lakes Res Pp. 614 - 624.
Hunt, G. R. and Kaye, N. B. (2005). Lazy plumes. J. Fluid Mech., 533 Pp. 329 - 338.
Kay, A. (2007). Warm discharges in cold fresh water. Part 1. Line plumes in a uniform ambient. Journal of Fluid Mechanics, 574, Pp. 239 - 271. doi:10.1017/S0022112006004101
Macqueen, J. F. (1979) Turbulence and cooling water discharges from power stations. In: C. J. Harris (ed.) Mathematical Modelling of Turbulent Diffusion in the Environment Pp. 379 - 437.
Morton, B. R. (1959). Forced plumes. J. Fluid Mech., 5 Pp. 151 - 163.
Morton, B. R., Taylor, G. I. and Turner, J.S. (1956). Turbulent gravitational convection from maintained and instantaneous sources Proc. R. Soc. Lond. A 234(29) Pp. 1 - 23.
Ricou, P. F. and Spalding, B D. (1961). Measurement of entrainment by axisymmetrical turbulent jets. Journal of Fluid Mechanics, 11(1), Pp. 21 - 32.
Turner, J. S. (1973). Buoyancy Effects in Fluids. Cambridge University Press
Turner, J. S (1966). Jets and plumes with negative and reversing buoyancy. J. Fluid Mech., 26 Pp. 779 - 792.
Wiiest, A., Brooks, N. H. and Imboden, D. M. (1992). Bubble plume modeling for lake restoration. Water Resources Res. 28 Pp. 3235 - 3250.