On the identification of entrance hydrodynamic region in case of laminar flow of Newtonian fluid in horizontal annular channel

A. V. Ryazhskikh

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 9 (2017) no. 3, pp. 34-40

Résumé

In the frameworks of physical linearization of the Navier–Stokes equations in a cylindrical coordinates system on the one-way axial force-feed laminar flow of Newtonian fluid, a mathematical model of the flow development in the entrance region of a horizontal annular channel is formulated. The unknown constant gradient of pressure along the channel is connected with the equation of continuity written in an integral form of stability of liquid flow in any cross section of a channel. Use of the one-way integral Laplace transformation along the longitudinal coordinate allowed to obtain an analytical expression of the local hydrodynamic field at the entrance region and determine pressure losses coincided with the classic data. Analysis of the characteristic structure of the hydrodynamic field of dimensionless velocities at the entrance region showed that for small values of the ratio of the radii of the inner and outer coaxial cylindrical tubes constituting the annular channel, asymmetry of the longitudinal velocity profile is observed with a shift of the maximum value towards the surface of a coaxial cylinder of smaller radius, and an increase in the Reynolds number practically linearly increases the length of the hydrodynamic entrance region. Assumption about the absence of drift of the radial coordinate of the maximum velocity in the entrance hydrodynamic region, limited to the so-called "regular" regime, made it possible to identify the length of the entrance hydrodynamic region in the annular channel by the completed expression in an explicit form that correlates with the classical estimates obtained as a result of computational experiments. It is noted that when the ratio of the radii of the inner and outer coaxial cylinders approaches zero or infinity (corresponding to particular cases of a circular tube and a flat channel), the known results for the lengths of the entrance hydrodynamic regions are obtained. Difference between velocity values calculated by the proposed model and experimental values in the region adjacent to the entry section is explained by the fact that kinetic energy of the liquid flow is not accounted for by leveling the pressure inhomogeneity along the channel cross-section. Nonetheless, it is shown that it does not have a significant influence on the length of hydrodynamic entrance region.

Détail

Mots-clés : annular channel
Keywords: Newtonian fluid, length of the entrance hydrodynamic region.

Bibliographie

[1] M. Massoud, Engineering termofluids: thermodynamics, fluid mechanics and heat transfer, Springer-Verlag, Berlin–Heidelberg, 2005, 1120 pp. | DOI

[2] Latif M. Jiji, Heat convection, Springer-Verlag, Berlin–Heidelberg, 2009, 543 pp. | DOI

[3] H.S. Heaton, W.C. Reynolds, W.M. Kays, “Heat transfer in annular passages-simultaneous development of velocity and temperature fields in laminar flow”, Int. J. Heat and Mass Transfer, 7:7 (1964), 763–781 | DOI | Zbl

[4] Kays W. M., Convective heat and mass transfer, McGraw-Hill Science, 1993, 480 pp.

[5] F. Durst, S. Ray, B. Ünsal, O.A. Bayoumi, “The development lengths of laminar pipe and channel flows”, Journal of Fluids Engineering, 127:6 (2005), 1154–1160 | DOI

[6] A.A. Shaker, “A numerical study of low Reynolds number incompressible flow of entrance and disturbed regions of concentric circular pipes”, J. of Engineering and Development, 16:2 (2012), 16–33 http://www.iasj.net/iasj?func=fulltext&aId=67504

[7] “Hydrodynamic entrance length for high-viscosity Newtonian fluid flow in an annular channel”, A.V. Ryazhskikh, S.V. Ryabov, 86:2 (2013), 396–401 | DOI

[8] R.J. Poole, “Development length requirements for fully developed laminar flow in concentric annuli”, Journal of Fluids Engineering, 132:6 (2010), 501–504 | DOI

[9] Bird R. B., Stewart W. E., Lightfoot E. N., Transport phenomena, John Wiley, New York, 2007, 928 pp.

[10] Doetsch G., Anleitung zum praktischen gebrauch der Laplace-transformation und der Ztransformation, R. Oldenbourg Verlag, Munchen–Wien, 1967, 256 pp. | MR

[11] Janke E., Emde F., Losch F., Tafeln hoherer Funktionen, B.G.Teubner Verlagsgesellschaft, Stuttgart, 1960, 318 pp.

[12] Ditkin V. A., Prudnikov A. P., Reference book on operational calculus, Vysshaya shkola, M., 1965, 465 pp. (in Russ.)

[13] Belyaev N. M., Ryadno A. A., Heat conductivity theory techniques, In two parts, v. 1, Vysshaya shkola, M., 1982, 327 pp. (in Russ.)

[14] B.G. Korenev, Bessel functions and their applications, Taylor Francis, NY, 2002, 288 pp. | MR | Zbl

[15] Idel'chik I. E., Reference book on hydraulic resistance, Mashinostroenie Publ., M., 1992, 672 pp. (in Russ.)

[16] Slezkin N. A., Dynamics of viscous incompressible liquid, GITTL publ., M., 1955, 519 pp. (in Russ.)

[17] S.T. McComas, “Hydrodynamic entrance length for ducts of arbitrary cross section”, J. Basic Eng., 89:4 (1967), 847–850 | DOI