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Analytic study of developing flows in a tube laden with non-evaporating and evaporating drops via a modified linearization of the two-phase momentum equations

Published online by Cambridge University Press:  30 April 2008

S. KHOSID  and
Y. TAMBOUR
S. KHOSID
Affiliation:
Department of Applied Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel
Y. TAMBOUR*
Affiliation:
Department of Aerospace Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel
*
‡Author to whom correspondence should be addressed: [email protected].
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Abstract

A novel modification of the classical Langhaar linearization of the mutually coupled momentum equations for developing two-phase flows in circular ducts is presented. This modification enables us to treat: (i) flows developing from spatially periodic initial velocity distributions without the presence of droplets, and (ii) two-phase flows in which monosize, non-evaporating and evaporating droplets suspended in a developing gas flow of an initially uniform velocity distribution exchange momentum with the host-gas flow. New solutions are presented for the downstream evolution in the velocity profiles which develop from spatially periodic initial velocity distributions that eventually reach the fully developed Poiseuille velocity profile. These solutions are validated by employing known numerical procedures, providing strong support for the physical underpinnings of the present modified linearization. New solutions are also presented for the evolution in drop velocities and vapour spatial distributions for evaporating droplets suspended in an initially uniform velocity profile of the host gas. Asymptotic solutions are presented for the flow region which lies very close to the inlet of the tube, where the relative velocity between the droplets and the host gas is high, and thus the velocity fields of the two phases are mutually coupled. These solutions provide new explicit formulae for the droplet velocity field as a function of the initial conditions and droplet diameter (relative to the tube diameter) for non-evaporating drops, and also as a function of evaporation rate for evaporating drops.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 603 , 25 May 2008 , pp. 245 - 270
Copyright
Copyright © Cambridge University Press 2008

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Footnotes

†

Present address: RAFAEL, Armament Development Authority, Israel.

References

REFERENCES

Anidjar, F., Greenberg, J. B. & Tambour, Y. 1996 On the solution of a class of polydisperse spray problems. Eur. J. Appl. Maths. 7, 75–96.CrossRefGoogle Scholar
Asmolov, E. S. 1999 The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number. J. Fluid Mech. 381, 63–87.CrossRefGoogle Scholar
Berestycki, H. & Sivashinsky, G. I. 1991 Flame extinction by periodic flow field. SIAM J. Appl. Maths 55, 344–350.CrossRefGoogle Scholar
Borghi, R. 1996 Background on droplets and sprays. In Combustion and Turbulence in Two-phase Flows. Lecture series 1996-02. Von Karman Institute of Fluid Dynamics.Google Scholar
Campbell, W. D. & Slattery, J. C. 1963 Flow in the entrance of a tube. Trans. ASME: J. Basic Engng 85, 41–46.CrossRefGoogle Scholar
Candel, S., Lacas, F., Darabiha, N. & Rolon, C. 1999 Group combustion in spray flames. Multiphase Sci. Tech. 11, 1–18.CrossRefGoogle Scholar
Chigier, N. & Gemci, T. 2002 Problems and challenges in the development of micro-propulsion devices. In Proc. 42nd Israel Conf. on Aerospace Sciences, Technion-Israel Institute of Technology, Haifa, Israel.Google Scholar
Greenberg, J. B. & Cohen, R. 1992 Spatial distortion of spray diffusion flames. Atomization Spray 2, 275–293.CrossRefGoogle Scholar
Heaton, H. S., Reynolds, W. C. & Kays, W. M. 1964 Heat transfer in annular passages: simultaneous development of velocity and temperature fields in laminar flow. Intl J. Heat Mass Transfer 7, 763–781.CrossRefGoogle Scholar
Hornbeck, R. W. 1964 Laminar flow in the entrance region of a pipe. Appl. Sci. Res. A 13, 224–232.CrossRefGoogle Scholar
Katoshevski, D. & Tambour, Y. 1993 A theoretical study of polydisperse liquid-sprays in a free shear-layer flow. Phys. Fluids A 5, 3085–3098.CrossRefGoogle Scholar
Katoshevski, D. & Tambour, Y. 1995 Sprays in radially spreading shear-layer flows. Phys. Fluids 7, 530–538.CrossRefGoogle Scholar
Khosid, S. & Tambour, Y. 1993 New closed-form analytical solutions of the discrete coagulation equation with simultaneous evaporation, and their use for validation of sectional solutions. Atomization Spray 3, 223–248.Google Scholar
Langhaar, H. L. 1942 Steady flow in the transition length of a straight tube. J. Appl. Mech. 9, A55–A58.CrossRefGoogle Scholar
Laurent, F. 2006 Numerical analysis of Eulerian multi-fluid models in the context of kinetic formulations for dilute evaporating sprays. ESAIM – Math. Model. Numer. Anal. 40, 431–468.CrossRefGoogle Scholar
Laurent, F. & Massot, M. 2001 Multi-fluid modelling of laminar polydisperse spray flame: origin, assumptions and comparison of sectional and sampling methods. Combust. Theor. Model. 5, 537–572.CrossRefGoogle Scholar
Laurent, F., Santoro, V., Noskov, M., Smooke, M. D., Gomez, A. & Massot, M. 2004 Accurate treatment of size distribution effects in polydisperse spray diffusion flames: multi-fluid modeling, computations and experiments. Combust. Theor. Model. 8, 385–412.CrossRefGoogle Scholar
Leboissetier, A., Okong'o, N. & Bellan, J. 2005 Consistent large-eddy simulation of temporal mixing layer laden with evaporating drops. Part 2. A posteriori modelling. J. Fluid Mech. 523, 37–78.CrossRefGoogle Scholar
Le Clercq, P. C. & Bellan, J. 2005 Direct numerical simulation of gaseous mixing layers laden with multicomponent-liquid drops: liquid-specific effects. J. Fluid Mech. 533, 57–94.CrossRefGoogle Scholar
Liu, J. 1974 Flow of a Bingham fluid in the entrance region of an annular tube. Master's thesis, University of Wisconsin, Milwaukee.Google Scholar
Marble, F. E. 1963 Dynamics of a gas containing small solid particles. Proc. 5th AGARD Combustion and Propulsion Colloquium (1962), pp. 175–215.Google Scholar
Matas, J.-P., Glezer, , Guazzelli, É. & Morris, J. F. 2004 a Trains of particles in finite-Reynolds-number pipe flow. Phys. Fluids 16, 4192–4195.CrossRefGoogle Scholar
Matas, J.-P., Morris, J. F. & Guazzelli, É. 2004 b Inertial migration of rigid spherical particles in Poiseuille flow. J. Fluid Mech. 515, 171–195.CrossRefGoogle Scholar
Meng, H., Hsiao, G. C., Yang, V. & Shuen, J. S. 2005 Transport and dynamics of liquid oxygen droplets in supercritical hydrogen streams. J. Fluid Mech. 527, 115–139.CrossRefGoogle Scholar
Miller, R. S. & Bellan, J. 1999 Direct numerical simulation of a confined three-dimensional gas mixing layer with one evaporating hydrocarbon-droplet-laden stream. J. Fluid Mech. 384, 293–338.CrossRefGoogle Scholar
Nakamura, M., Akamatsu, F., Kurose, R. & Katsuki, M. 2005 Combustion mechanism of liquid fuel spray in a gaseous flame. Phys. Fluids 17, 123301.CrossRefGoogle Scholar
Nekhamkina, O. A. & Rotinjan, M. A. 1978 Analysis of heat transfer in turbulent flow of multicomponent gases in tubes. J. Engng Phys. 33, 678–686 (in Russian).Google Scholar
Patankar, S. V. & Spalding, D. B. 1967 A finite-difference procedure for solving the equations of the two-dimensional boundary layer. Intl J. Heat Mass Transfer 33, 678–686.Google Scholar
Réveillon, J. & Vervisch, L. 2005 Analysis of weakly turbulent dilute-spray flames and spray combustion regimes. J. Fluid Mech. 537, 317–347.CrossRefGoogle Scholar
Saffman, P. G. 1962 On the stability of laminar flow of a dusty gas. J. Fluid Mech. 13, 120–128.CrossRefGoogle Scholar
Schonberg, J. A. & Hinch, E. J. 1989 Inertial migration of a sphere in Poiseuille flow. J. Fluid Mech. 203, 517–524.CrossRefGoogle Scholar
Segré, G. & Silberberg, A. 1962 Behaviour of macroscopic rigid spheres in Poiseuille flow. J. Fluid Mech. 14, 115–157.CrossRefGoogle Scholar
Shah, R. K. & London, A. L. 1978 Laminar Flow Forced Convection in Ducts. Academic.Google Scholar
Shumway, R. W. & McEligot, D. M. 1971 Heated laminar gas flow in annuli with temperature-dependent transport properties. Nucl. Sci. Engng 46, 394–399.CrossRefGoogle Scholar
Silverman, I., Greenberg, J. B. & Tambour, Y. 1991 Asymptotic analysis of a premixed polydisperse spray flame. SIAM J. Appl. Maths 51, 1284–1303.CrossRefGoogle Scholar
Sivashinsky, G. I. 1988 Cascade-renormalization theory of turbulent flame speed. Combust. Sci. Technol. 62, 77–96.CrossRefGoogle Scholar
Soo, S. L. 1990 Multiphase Fluid Dynamics. Science Press, Beijing, China.Google Scholar
Sparrow, E. M., Lin, S. H. & Lundgren, T. S. 1964 Flow development in the hydrodynamic entrance region of tubes and ducts. Phys. Fluids 7, 338–347.CrossRefGoogle Scholar
Tambour, Y. 1984 Vaporization of polydisperse fuel sprays in a laminar boundary layer flow: a sectional approach. Combust. Flame 58, 103–114.CrossRefGoogle Scholar
Tambour, Y. 1985 A lagrangian sectional approach for simulating droplet size distribution of vaporizing fuel sprays in a turbulent jet. Combust. Flame 60, 15–28.CrossRefGoogle Scholar
Tambour, Y. 1995 Structure of multisize sprays and its effect on spray flame properties. In Mechanics and Combustion of Droplets and Sprays (ed. Chiu, H. H. & Chigier, N.), pp. 210–234. Begell House.Google Scholar
Tambour, Y. & Katoshevski, D. 1994 Similarity analysis of spray diffusion flames in a unidirectional shear-layer flow. In Twenty-Fifth Int. Symp. on Combustion, vol. XXV, pp. 381–388.Google Scholar
Tambour, Y. & Katoshevski, D. 1995 Asymptotic analysis of droplet coalescence effects on spray diffusion flames in a unidirectional shear-layer flow. Atomization Spray 5, 357–386.CrossRefGoogle Scholar
Tambour, Y. & Zehavi, S. 1993 Derivation of near-field sectional equations for the dynamics of polydisperse spray flows: an analysis of the relaxation zone behind a normal shock wave. Combust. Flame 95, 383–409.CrossRefGoogle Scholar
Ulrichson, D. L. & Schmitz, R. A. 1965 Laminar-flow heat transfer in the entrance region of circular tubes. Intl J. Heat Mass Transfer 8, 253–258.CrossRefGoogle Scholar
Varanasi, K. K., Clack, H. L. & Miller, R. S. 2004 On preferential diffusion of binary component liquid droplets evaporating in a two-phase mixing layer. Intl J. Multiphase Flow 30, 1235–1257.CrossRefGoogle Scholar
Ward-Smith, A. J. 1980 Internal Fluid Flow: the Fluid Dynamics of Flow in Pipes and Ducts. Clarendon.Google Scholar
Wolfram, S. 1991 Mathematica: A System for Doing Mathematics by Computer. Addison-Wesley.Google Scholar
Yu, K. M., Sung, C. J. & Law, C. K. 1994 Some aspects of the freely propagating premixed flame in a spatially periodic flow field. Combust. Flame 97, 375–383.CrossRefGoogle Scholar