Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-24T06:10:14.320Z Has data issue: false hasContentIssue false

Multi-mode models of flow and of solute dispersion in shallow water. Part 1. General derivation

Published online by Cambridge University Press:  26 April 2006

Ronald Smith
Affiliation:
Department of Mathematical Sciences, Loughborough University of Technology, LE11 3TU, UK

Abstract

Instead of considering just the vertically averaged current and the vertically averaged concentration, a multi-mode model is derived in which more of the vertical structure can be computed directly rather than being lumped into a dispersion coefficient. Test cases, of laminar flows, are used to quantify the accuracy of the lowest non-trivial truncation (two modes) in replicating both the flow and the dispersion process.

Type
Research Article
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. Roy. Soc. Lond. A 235, 6777.Google Scholar
Blumberg, A. F. & Mellor, G. L. 1987 A description of a three-dimensional coastal ocean circulation model. In Three-dimensional Coastal Ocean Models (ed. N. S. Heaps), 116. AGU.
Bugliarello, G. & Jackson, E. D. 1964 Random walk study of convective diffusion. Engng Mech. Div. ASCE 90, 4977.Google Scholar
Chatwin, P. C. 1970 The approach to normality of the concentration distribution of a solute in solvent flowing along a pipe. J. Fluid Mech. 43, 321352.Google Scholar
Chickwendu, S. C. 1986 Calculation of longitudinal shear dispersivity using an N-zone model as N →∞. J. Fluid Mech. 167, 1930.Google Scholar
Davies, A. M. 1987 Spectral models in continental shelf sea oceanography. Three-dimensional Coastal Ocean Models (ed. N. S. Heaps), pp. 71106. AGU.
Elder, J. W. 1959 The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5, 544560.Google Scholar
Falconer, R. A. 1976 Mathematical modelling of jet-forced circulation in reservoirs and harbours. Ph.D Thesis, Imperial College, London
Fischer, H. B. 1969 The effects of bends on dispersion in streams. Water Resources Res. 5, 496506.Google Scholar
Fischer, H. B. 1978 On the tensor form of the bulk dispersion coefficient in a bounded skewed shear flow. J. Geophys. Res. 83, 23732375.Google Scholar
Gill, W. N. & Sankarasubramanian, R. 1970 Exact analysis of unsteady convective diffusion. Proc. Roy. Soc. Lond. A 316, 341350.Google Scholar
Heaps, N. S. 1972 On the numerical solution of the three-dimensional hydrodynamical equations for tides and storm surges. Mem. Soc. Sci. Liege.(6), 2, 143180.Google Scholar
Hutton, A. G., Smith, R. M. & Hickmott, S. 1987 The computation of turbulent flows of industrial complexity by the finite element method – progress and prospects. Finite Element Fluids 7, 289309.Google Scholar
Koutitas, C. & Koutita, M. G. 1986 A comparative study of three mathematical models for windgenerated circulation in coastal areas. Coastal Engng 10, 127138.Google Scholar
Lamb, H. 1945 Hydrodynamics. Cambridge University Press.
Phillips, N. A. 1957 A coordinate system having some special advantages for numerical forecasting. J. Met. 14, 184186.Google Scholar
Prych, E. A. 1970 Effect of density differences on lateral mixing in open-channel flow. Keck Lab. Hydraul. Water Res. Calif. Inst. Tech. Rep. KH-R-21.
Saffman, P. G. 1962 The effect of wind shear on horizontal spread from an instantaneous ground source. Q. J. R. Met. Soc. 88, 382393.Google Scholar
Smith, R. 1977 Coriolis, curvature and buoyancy effects upon dispersion in a narrow channel. Hydrodynamics of Estuaries and Fjords (ed. J. C. J. Nihoul), pp. 217231. Elsevier.
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. Lond. A 219, 186203.Google Scholar