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The influence of shear on mixing across density interfaces

Published online by Cambridge University Press:  29 March 2006

Robert R. Long
Robert R. Long
Affiliation:
Department of Mechanics and Materials Science, The Johns Hopkins University, Baltimore, Maryland 21218
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Abstract

This paper discusses critically observations of mixing processes across density interfaces in laboratory experiments and inferences that have been made and can be made from these observations. Fluxes of heat or salt and entrainment velocities have been found to depend on negative powers of an overall Richardson number Ri* based on the buoyancy jump across the interface, the depth of the homogeneous layer and the intensity of the turbulence near the source. When the Reynolds and Péclet numbers are large, the fluxes or entrainment velocities appear to be proportional to the minus one and minus three-halves powers of Ri* for flows with and without mean shear respectively, and this difference has caused speculation about the accuracy of the experimental data and about the reasons for the two laws if the difference is real. In the present discussion, we accept the accuracy of the two laws and attribute the higher entrainment rate for shear flows to the decrease of r.m.s. velocities near the interface with increasing Ri* in the case of zero shear. A plausible argument yields the unifying result that the entrainment rates in both cases are proportional to Ri−1, where Ri is a Richardson number based on the buoyancy jump and velocities and lengths characteristic of the turbulence near the interface. It is suggested that the $Ri^{-\frac{3}{2}}$ behaviour inferred by Turner is based on an erroneous interpretation of experimental data.

In the course of the discussion, it is shown that the drag coefficient in flow of a stratified fluid over a rough surface is independent of the Richardson number (or density jump across the interface or inversion) and depends only on the ratio of the roughness length to the depth of the homogeneous layer. This has obvious implications for problems of parameterizing the momentum flux near the ground in the atmosphere.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 70 , Issue 2 , 29 July 1975 , pp. 305 - 320
Copyright
© 1975 Cambridge University Press

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References

Arya, S. P. S. & Plate, E. J. 1969 Modeling of the stably stratified atmospheric boundary layer. J. Atmos. Sci. 26, 656–665.Google Scholar
Boussinesq, J. 1903 Théorie Analytique de la Chaleur, vol. 2. Paris: Gauthier-Villars.
Bowden, K. F. 1960 Turbulence in the Sea (ed. M. N. Hill), pp. 802–825.
Brush, L. M. 1970 Artificial mixing of stratified fluids formed by salt and heat in a laboratory reservoir. New Jersey Water Resources Res. Inst. Res. Project, B-024.Google Scholar
Crapper, P. F. 1973 An experimental study of mixing across density interfaces. Ph.D. thesis, University of Cambridge.
Crapper, P. F. & Linden, P. F. 1974 The structure of turbulent density interfaces. J. Fluid Mech. 65, 45–63.Google Scholar
Cromwell, T. 1960 Pycnoclines created by mixing in an aquarium tank. J. Mar. Res. 18, 73–82.Google Scholar
Ellison, T. H. & Turner, J. S. 1960 Mixing of dense fluid in a turbulent pipe flow. Parts 1 and 2. J. Fluid Mech. 8, 514–544.Google Scholar
Izakson, A. A. 1937 On the formula for velocity distributions near walls. Tech. Phys. U.S.S.R. 4, 27–37.Google Scholar
Kato, H. & Phillips, O. M. 1969 On the penetration of a turbulent layer into a stratified fluid. J. Fluid Mech. 37, 643–655.Google Scholar
Kitaigorodskii, S. A. 1960 On the computation of the thickness of the wind-mixing layer in the ocean. Bull. Acad. Sci. U.S.S.R., Geophys. Ser. 3, 284–287.Google Scholar
Kullenberg, G. 1971 Vertical diffusion in shallow waters. Tellus, 23, 129–135.Google Scholar
Linden, P. F. 1973 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60, 467–480.Google Scholar
Long, R. R. 1970 A theory of turbulence in stratified fluids. J. Fluid Mech. 42, 349–365.Google Scholar
Long, R. R. 1972 Some aspects of turbulence in stratified fluids. Appl. Mech. Rev. pp. 1297–1301.Google Scholar
Long, R. R. 1973 Some properties of horizontally homogeneous, statistically steady turbulence in a stratified fluid. Boundary-Layer Met. 5, 139–157.Google Scholar
Millikan, C. B. 1938 A critical discussion of turbulent flow in channels and circular tubes. Proc. 5th Int. Congr. Appl. Mech. pp. 386–392.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics: Mechanics of Turbulence. M.I.T. Press.
Moore, M. J. & Long, R. R. 1971 An experimental investigation of turbulent stratified shearing flow. J. Fluid Mech. 49, 635–655.Google Scholar
Robertson, J. M. 1959 On turbulent plane Couette flow. Proc. 6th Ann. Conf. Fluid Mech., pp. 169–179.Google Scholar
Rouse, H. & Dodu, J. 1955 Turbulent diffusion across a density discontinuity. Houille Blanche, 10, 405–410.Google Scholar
Schlichting, H. 1955 Boundary Layer Theory, pp. 408, 411. McGraw-Hill.
Spiegel, E. A. & Veronis, G. 1960 On the Boussinesq approximation for a compressible fluid. Astrophys. J. 131, 441–447.Google Scholar
Sundaram, T. R. 1973 A theoretical model for the seasonal thermal cycle of deep temperate lakes. Proc. 16th Conf. Great Lakes Res., pp. 1009–1025.Google Scholar
Thorpe, S. A. 1973 Turbulence in stably stratified fluids: a review of laboratory experiments. Boundary-Layer Met. 5, 95–119.Google Scholar
Turner, J. S. 1968 The influence of molecular-diffusivity on turbulent entrainment across a density interface. J. Fluid Mech. 33, 639–656.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids, chap. 9. Cambridge University Press.
Wolanski, E. 1972 Turbulent entrainment across stable density-stratified liquids and suspensions. Ph.D. thesis, The Johns Hopkins University.
Wu, J. 1973 Wind-induced turbulent entrainment across a stable density interface. J. Fluid Mech. 61, 275–287.Google Scholar