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A mathematical model of turbulent heat and mass transfer in stably stratified shear flow

Published online by Cambridge University Press:  26 April 2006

G. I. Barenblatt ,
M. Bertsch ,
R. Dal Passo ,
V. M. Prostokishin  and
M. Ughi
G. I. Barenblatt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK.
M. Bertsch
Affiliation:
Dipartimento di Matematica, Universitá di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy.
R. Dal Passo
Affiliation:
Instituto per le Applicazioni del Calcolo Mauro Picone, Viale del Policlinico 137, 00161 Roma, Italy.
V. M. Prostokishin
Affiliation:
P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, 23 ul. Krasikova, Moscow 117218, Russia.
M. Ughi
Affiliation:
Instituto delle Scienze delle Costruzioni, Universitá di Trieste, Piazzale Europa 1, 34127 Trieste, Italy.
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Abstract

It is commonly assumed that heat flux and temperature diffusivity coefficients obtained in steady-state measurements can be used in the derivation of the heat conduction equation for fluid flows. Meanwhile it is also known that the steady-state heat flux as a function of temperature gradient in stably stratified turbulent shear flow is not monotone: at small values of temperature gradient the flux is increasing, whereas it is decreasing after a certain critical value of the temperature gradient. Therefore the problem of heat conduction for large values of temperature gradient becomes mathematically ill-posed, so that its solution (if it exists) is unstable.

In the present paper it is shown that a well-posed mathematical model is obtained if the finiteness of the adjustment time of the turbulence field to the variations of temperature gradient is taken into account. An evolution-type equation is obtained for the temperature distribution (a similar equation can be derived for the concentration if the stratification is due to salinity or suspended particles). The characteristic property which is obtained from a rigorous mathematical investigation is the formation of stepwise distributions of temperature and/or concentration from continuous initial distributions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 253 , August 1993 , pp. 341 - 358
Copyright
© 1993 Cambridge University Press

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References

Barenblatt, G. I., Bertsch, M., Dal Passo, R. & Ughi, M. 1993 A degenerate pseudoparabolic regularisation of a nonlinear forward backward heat equation arising in the theory of heat and mass transfer in stably stratified turbulent shear flow. SIAM J. Nonlin. Anal. (to appear).Google Scholar
Djumagazieva, S. Kh. 1983 Numerical integration of a certain partial differential equation. J. Numer. Math. Phys. 23, 839–847.Google Scholar
Felsenbaum, A. I. & Boguslavsky, S. G. 1977 On a relation between heat flux and temperature gradient in the sea. Dokl. (C. R.) USSR Acad. Sci. 235-3, 557–559.
Höllig, K. 1983 Existence of infinitely many solutions for a forward backward heat equation. Trans. Am. Math. Soc. 278, 299–316.Google Scholar
Ivey, G. N. & Imberger, G. 1991 On the nature of turbulence in stratified fluid. Part 1. The energetics of mixing. J. Phys. Oceanogr. 21, 650–658.Google Scholar
Linden, P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn 13, 3–23.Google Scholar
Mamayev, O. I. 1958 The influence of stratification on vertical turbulent mixing in the sea. Izv. (Bull.) USSR Acad. Sci. Ser. Geophys. 4, 870–875.Google Scholar
Monin, A. S. & Yaglom, A. M. 1971 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. MIT Press.
Munk, W. H. & Anderson, F. R. 1948 Notes on a theory of the thermocline. J. Mar. Res. 7, 276–295.Google Scholar
Padron, V. 1990 Sobolev regularization of some nonlinear ill-posed problems. Thesis, Department of Mathematics, University of Minnesota.
Petukhov, B. S. & Polyakov, A. F. 1989 Heat Transfer in Turbulent Mixed Convection. Hemisphere.
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid is unstable? Deep-Sea Res. 19, 79–81.Google Scholar
Posmentier, E. S. 1977 The generation of salinity finestructures by vertical diffusion. J. Phys. Oceanogr. 7, 298–300.Google Scholar
Rossby, C. G. & Montgomery, R. B. 1935 The layer of frictional influence in wind and ocean currents. Pap. Phys. Oceanogr. Met., vol. 3.Google Scholar
Ruddick, B. R., Mcdougall, T. J. & Turner, J. S. 1989 The formation of layers in a uniformly stirred density gradient. Deep-Sea Res. 36, 597–609.Google Scholar