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Averaging theory for heat transfer in circular hydraulic jumps with a separation bubble

Published online by Cambridge University Press:  18 January 2024

R. Solana Gómez Open the ORCID record for R. Solana Gómez [Opens in a new window] ,
T. Bohr Open the ORCID record for T. Bohr [Opens in a new window] ,
S. Nielsen Open the ORCID record for S. Nielsen [Opens in a new window] ,
W. Rohlfs Open the ORCID record for W. Rohlfs [Opens in a new window] ,
R. Kneer Open the ORCID record for R. Kneer [Opens in a new window]  and
H. Askarizadeh Open the ORCID record for H. Askarizadeh [Opens in a new window]
R. Solana Gómez*
Affiliation:
Institute of Heat and Mass Transfer, RWTH Aachen University, Augustinerbach 6, 52056 Aachen, Germany
T. Bohr
Affiliation:
Department of Physics and Center for Fluid Dynamics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
S. Nielsen
Affiliation:
Department of Physics and Center for Fluid Dynamics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
W. Rohlfs
Affiliation:
Department of Thermal and Fluid Engineering, University of Twente, Drienerlolaan 5, 7522 NB Enschede, The Netherlands
R. Kneer
Affiliation:
Institute of Heat and Mass Transfer, RWTH Aachen University, Augustinerbach 6, 52056 Aachen, Germany
H. Askarizadeh*
Affiliation:
Institute of Heat and Mass Transfer, RWTH Aachen University, Augustinerbach 6, 52056 Aachen, Germany
*
†Email addresses for correspondence: solana.gomez@wsa.rwth-aachen.de, askarizadeh@wsa.rwth-aachen.de
†Email addresses for correspondence: solana.gomez@wsa.rwth-aachen.de, askarizadeh@wsa.rwth-aachen.de
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Abstract

Analytical investigations of heat transfer during the vertical impingement of an unsubmerged axisymmetric liquid jet on a horizontal plate have been limited to the regions ahead of the jump. This limitation is due to the complex flow physics in the jump region arising from sudden changes in the flow field. This is addressed in here by extending the averaging theory (AT) introduced by Bohr et al. (Phys. Rev. Lett., vol. 79, issue 6, 1997, pp. 1038–1041) which was further developed by Watanabe et al. (J. Fluid Mech., vol. 480, 2003, pp. 233–265), to describe the heat transfer problem in circular hydraulic jumps including separation. The applicability of the resulting theory to determine the temperature field in the jump region is evaluated using the data available in the literature and also by means of fully resolved numerical solutions. Good agreement is observed for moderate Prandtl numbers. However, for sufficiently high Prandtl numbers, deviations become notable. The reasons for the deviations according to their relevance are (i) monotonically decreasing temperature profile inherent to the AT, whereas the fully resolved numerical solutions exhibit a local maximum in the temperature profile away from the plate; and (ii) inapplicability of the concept of dividing the flow field into a region affected and a region unaffected by heat transfer according to the thermal boundary layer thickness. This concept leads to the overestimation of the temperature close to the wall and to the existence of a threshold Prandtl number, for which the thermal boundary layer thickness does not meet the free surface anymore. Around this threshold Prandtl number, the temperature field shows a discontinuous behaviour.

JFM classification

Boundary Layers: Boundary layer separation Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 979 , 25 January 2024 , A41
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Askarizadeh, H., Ahmadikia, H., Ehrenpreis, C., Kneer, R., Pishevar, A. & Rohlfs, W. 2019 Role of gravity and capillary waves in the origin of circular hydraulic jumps. Phys. Rev. Fluids 4 (11), 114002.CrossRefGoogle Scholar
Askarizadeh, H., Ahmadikia, H., Ehrenpreis, C., Kneer, R., Pishevar, A. & Rohlfs, W. 2020 Heat transfer in the hydraulic jump region of circular free-surface liquid jets. Intl J. Heat Mass Transfer 146, 118823.CrossRefGoogle Scholar
Askarizadeh, H., Ehrenpreis, C., Kneer, R. & Rohlfs, W. 2021 Assessment of the interface compression scheme in the VOF modeling of circular hydraulic jumps. Atomiz. Sprays 31 (5), 21–35.CrossRefGoogle Scholar
Bohr, T., Dimon, P. & Putkaradze, V. 1993 Shallow-water approach to the circular hydraulic jump. J. Fluid Mech. 254, 635–648.CrossRefGoogle Scholar
Bohr, T., Putkaradze, V. & Watanabe, S. 1997 Averaging theory for the structure of hydraulic jumps and separation in laminar free-surface flows. Phys. Rev. Lett. 79 (6), 1038–1041.CrossRefGoogle Scholar
Brackbill, J.U., Kothe, D.B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.CrossRefGoogle Scholar
Bush, J.W., Aristoff, J.M. & Hosoi, A.E. 2006 An experimental investigation of the stability of the circular hydraulic jump. J. Fluid Mech. 558, 33–52.CrossRefGoogle Scholar
Deshpande, S.S., Anumolu, L. & Trujillo, M.F. 2012 Evaluating the performance of the two-phase flow solver interfoam. Comput. Sci. Disc. 5 (1), 014016.CrossRefGoogle Scholar
Duchesne, A., Lebon, L. & Limat, L. 2014 Constant Froude number in a circular hydraulic jump and its implication on the jump radius selection. Europhys. Lett. 107 (5), 54002.CrossRefGoogle Scholar
Ellegaard, C., Hansen, A.E., Haaning, A. & Bohr, T. 1996 Experimental results on flow separation and transitions in the circular hydraulic jump. Phys. Scr. T67, 105–110.CrossRefGoogle Scholar
Higuera, F.J. 1994 The hydraulic jump in a viscous laminar flow. J. Fluid Mech. 274, 69–92.CrossRefGoogle Scholar
Higuera, F.J. 1997 The circular hydraulic jump. Phys. Fluids 9, 1476–1478.CrossRefGoogle Scholar
Hirt, C.W. & Nichols, B.D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39 (1), 201–225.CrossRefGoogle Scholar
Ishigai, S., Nakanishi, Sh., Mizuno, M. & Imamura, T. 1977 Heat transfer of the impinging round water jet in the interference zone of film flow along the wall. Bull. JSME 20 (139), 85–92.CrossRefGoogle Scholar
Jasak, H. 1996 Error analysis and estimation for the finite volume method with applications to fluid flows. PhD thesis, University of London.Google Scholar
von Kármán, T. 1921 Über laminare und turbulente Reibung. Z. Angew. Math. Mech. 1 (4), 233–252.CrossRefGoogle Scholar
Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S. & Zanetti, G. 1994 Modelling merging and fragmentation in multiphase flows with surfer. J. Comput. Phys. 113 (1), 134–147.CrossRefGoogle Scholar
Leveque, M. 1928 Les lois de la transmission de chaleur par convection. Annus Mines Ser 12 (13), 234–242.Google Scholar
Liu, X., Gabour, L.A. & Lienhard, J.H. 1993 Stagnation-point heat transfer during impingement of laminar liquid jets: analysis including surface tension. Trans. ASME J. Heat Transfer 115 (1), 99–105.CrossRefGoogle Scholar
Liu, X. & Lienhard, J.H. 1989 Liquid jet impingement heat transfer on a uniform flux surface. In Heat Transfer Phenomena in Radiation, Combustion, and Fires: Presented at the 1989 National Heat Transfer Conference, Philadelphia, Pennsylvania, August 6–9, 1989 (ed. R.K. Shah & and American Society of Mechanical Engineers. Heat Transfer Division), HTD (Series), 106, 523–530.Google Scholar
Liu, X. & Lienhard, J.H. 1993 The hydraulic jump in circular jet impingement and in other thin liquid films. Exp. Fluids 15, 108–116.CrossRefGoogle Scholar
Liu, X., Lienhard, J.H. & Lombara, J.S. 1991 Convective heat transfer by impingement of circular liquid jets. Trans. ASME J. Heat Transfer 113 (3), 571–582.CrossRefGoogle Scholar
Nielsen, S.M. 2015 The circular hydraulic jump. Bachelor's thesis, Technical University of Denmark.Google Scholar
Pohlhausen, K. 1921 Zur näherungsweisen Integration der Differentialgleichung der laminaren Grenzschicht. Z. Angew. Math. Mech. 1 (4), 252–290.CrossRefGoogle Scholar
Press, W.H., Teukolsky, S.A., Vetterling, W.T. & Flannery, B.P. 2007 Numerical Recipes, 3rd edn. Cambridge University Press.Google Scholar
Rohlfs, W., Ehrenpreis, C., Haustein, H.D. & Kneer, R. 2014 Influence of viscous flow relaxation time on self-similarity in free-surface jet impingement. Intl J. Heat Mass Transfer 78, 435–446.CrossRefGoogle Scholar
Rohlfs, W., Figueiredo, P. & Pischke, P. 2020 Smooth interface compression: an improved algebtraic VOF method to model flows dominated by capillary forces. Multiphase Sci. Technol. 32 (4), 259–293.CrossRefGoogle Scholar
Rohlfs, W., Pischke, P. & Scheid, B. 2017 Hydrodynamic waves in films flowing under an inclined plane. Phys. Rev. Fluids 2 (4), 044003.CrossRefGoogle Scholar
Rusche, H. 2002 Computational fluid dynamics of dispersed two-phase flows at high phase fractions. PhD thesis, Imperial College London.Google Scholar
Schlichting, H. & Gersten, K. 2006 Grenzschichttheorie, 10th edn. Springer.Google Scholar
Stevens, J.W. 1988 Measurements of local heat transfer coefficients: results for an axisymmetric, single-phase water jet impinging normally on a flat plate with uniform heat flux. PhD thesis, Brigham Young University. Department of Mechanical Engineering.Google Scholar
Sung, J., Choi, H.G. & Yoo, J.Y. 1999 Finite element simulation of thin liquid film flow and heat transfer including a hydraulic jump. Intl J. Numer. Meth. Engng 46 (1), 83–101.3.0.CO;2-D>CrossRefGoogle Scholar
Ubbink, O. 1997 Numerical prediction of two fluid systems with sharp interfaces. PhD thesis, Imperial College London.Google Scholar
de Vita, F., Lagrée, P.-Y., Chibbaro, S. & Popinet, S. 2020 Beyond shallow water: appraisal of a numerical approach to hydraulic jumps based upon the boundary layer theory. Eur. J. Mech. (B/Fluids) 79, 233–246.CrossRefGoogle Scholar
Watanabe, Sh., Putkaradze, V. & Bohr, T. 2003 Integral methods for shallow free-surface flows with separation. J. Fluid Mech. 480, 233–265.CrossRefGoogle Scholar
Watson, E.J. 1964 The radial spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20 (3), 481–499.CrossRefGoogle Scholar