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Linear Hydrodynamic Stability of Fluid Flow in Curved Rectangular Ducts: Semi-Analytical Study

Published online by Cambridge University Press:  09 August 2019

H. Nowruzi ,
H. Ghassemi Open the ORCID record for H. Ghassemi [Opens in a new window]  and
S. S. Nourazar
H. Nowruzi
Affiliation:
Department of Maritime Engineering, Amirkabir University of Technology (Tehran Polytechnic)Tehran, Iran
H. Ghassemi*
Affiliation:
Department of Maritime Engineering, Amirkabir University of Technology (Tehran Polytechnic)Tehran, Iran
S. S. Nourazar
Affiliation:
Department of Mechanical EngineeringAmirkabir University of Technology (Tehran Polytechnic)Tehran, Iran
*
*Corresponding author (gasemi@aut.ac.ir)
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Abstract

In the current study, for the first time, a semi-analytical technique is used for solving eigenvalue problem arising from linear hydrodynamics stability of fluid flow through the curved rectangular ducts at different curvature ratios and aspect ratios. To this accomplishment, symmetric disturbances are assumed and the Homotopy perturbation method (HPM) is applied to solve our eigenvalue problem for curvature ratios ranging from 0.01 to 0.8 and aspect ratios ranging from 0.05 to 20. Our semi-analytical results are validated through the existing numerical and experimental data, showing good agreement. The semi-analytical results indicate that, as the curvature ratio increases the critical Dean number (Dnc) is increased and the flow becomes more stable, especially for aspect ratios lower than 1.Moreover, for all intended curvature ratios, irregular behavior in variation of Dnc is detected by an increase in the aspect ratio. So that, the Dnc is decreased when the aspect ratio increases from 0.05 up to 1 and the fluid flow becomes unstable. When the aspect ratio is increased from 1 to 5, it causes to increase the Dnc and fluid flow becomes stable. Furthermore, when the aspect ratio increases from 5 to 20, the Dnc is decreased again. In addition, Dnc and eigenvalues of critical complex wave number corresponding to Dnc for the onset of Dean flow instability is reported under different curvature ratios and aspect ratios.

Keywords

Hydrodynamic stabilityCritical Dean numberAspect ratioCurvature ratioSemi analytical method
Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 5 , October 2019 , pp. 747 - 765
Copyright
© The Society of Theoretical and Applied Mechanics 2019 

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References

REFERENCES

Fischer, A.C., Forsberg, F., Lapisa, M., Bleiker, S.J., Stemme, G., Roxhed, N. and Niklaus, F., “Integrating MEMS and ICs,” Microsystems & Nanoengineering, 1, pp. 1–16 (2015).Google Scholar
Drazin, P.G. and Reid, W.H., Hydrodynamic Stability, 2nd Edition, Cambridge University Press, Cambridge (2004).CrossRefGoogle Scholar
Winters, K.H., “A Bifurcation Study of Laminar Flow in a Curved Tube of Rectangular Cross-Section,” Journal of Fluid Mechanics, 180, pp. 343–369 (1987).CrossRefGoogle Scholar
Chandratilleke, T.T. and Nursubyakto, A., “Numerical Prediction of Secondary Flow and Convective Heat Transfer in Externally Heated Curved Rectangular Ducts,” International Journal of Thermal Sciences, 42, pp. 187–198 (2003).CrossRefGoogle Scholar
Yanase, S., Kaga, Y. and Daikai, R., “Laminar Flows through a Curved Rectangular Duct over a Wide Range of the Aspect Ratio,” Fluid Dynamics Research, 31, pp. 151–183 (2002).Google Scholar
Fellouah, H., Castelain, C., El Moctar, A.O. and Peerhossaini, H., “A Criterion for Detection of the Onset of Dean Instability in Newtonian Fluids,” European Journal of Mechanics-B/Fluids, 25, pp. 505–531 (2006).CrossRefGoogle Scholar
Boutabaa, M., Helin, L., Mompean, G. and Thais, L., “Numerical Study of Dean Vortices in Developing Newtonian and Viscoelastic Flows through a Curved Duct of Square Cross-Section,” Comptes Rendus Mecanique, 337, pp. 40–47 (2009).Google Scholar
Vinuesa, R., Noorani, A., Lozano-Durán, A., Khoury, G.K.E., Schlatter, P., Fischer, P.F. and Nagib, H.M., “Aspect Ratio Effects in Turbulent Duct Flows Studied through Direct Numerical Simulation,” Journal of Turbulence, 15, pp. 677–706 (2014).CrossRefGoogle Scholar
Tanveer, A., Hayat, T., Alsaedi, A. and Ahmad, B., “Heat Transfer Analysis for Peristalsis of MHD Carreau Fluid in a Curved Channel through Modified Darcy Law,” Journal of Mechanics, pp. 1–9 (2018).Google Scholar
Ligrani, P.M. and Niver, R.D., “Flow Visualization of Dean Vortices in a Curved Channel with 40 to 1 Aspect Ratio,” The Physics of Fluids, 31, pp. 3605–3617 (1988).Google Scholar
Yamamoto, K., Wu, X., Nozaki, K. and Hayamizu, Y., “Visualization of Taylor–Dean Flow in a Curved Duct of Square Cross-Section,” Fluid Dynamics Research, 38, pp. 1–18 (2006).CrossRefGoogle Scholar
Sugiyama, S., Hayashi, T. and Yamazaki, K., “Flow Characteristics in the Curved Rectangular Channels: Visualization of Secondary Flow,” Bulletin of JSME, 26, pp. 964–969 (1983).CrossRefGoogle Scholar
Dean, W.R., “Fluid Motion in a Curved Channel,” Proceedings of the Royal Society of London, Series A, 121, pp. 402–420 (1928)Google Scholar
Topakoglu, H.C., “Steady Laminar Flow of an Incompressible Viscous Fluid in a Curved Pipe,” Journal of Mathematics and Mechanics, 16, pp. 1231–1237 (1967).Google Scholar
Finlay, W.H. and Nandakumar, K., “Onset of Two-Dimensional Cellular Flow in Finite Curved Channels of Large Aspect Ratio,” Physics of Fluids A: Fluid Dynamics, 2, pp. 1163–1174 (1990).Google Scholar
Walowit, J., Tsao, S. and Diprima, R.C., “Stability of Flow between Arbitrarily Spaced Concentric Cylindrical Surfaces Including the Effect of a Radial Temperature Gradient,” Journal of Applied Mechanics, 31, pp. 585–593 (1964).CrossRefGoogle Scholar
Deka, R.K. and Takhar, H.S., “Hydrodynamic Stability of Viscous Flow between Curved Porous Channel with Radial Flow,” International Journal of Engineering Science, 42, pp. 953–966 (2004).CrossRefGoogle Scholar
Chen, K.T., Yarn, K.F., Chen, H.Y., Tsai, C.C., Luo, W.J. and Chen, C.N., “Aspect Ratio Effect on Laminar Flow Bifurcations in a Curved Rectangular Tube Driven by Pressure Gradients,” Journal of Mechanics, 33, pp. 831–840 (2017).CrossRefGoogle Scholar
Orszag, S.A., “Accurate Solution of the Orr–Sommerfeld Stability Equation,” Journal of Fluid Mechanics, 50, pp. 689–703 (1971).Google Scholar
Yamamoto, K., Yanase, S. and Jiang, R., “Stability of the Flow in a Helical Tube,” Fluid Dynamics Research, 22, pp. 153–170 (1998).Google Scholar
Mondal, R.N., Islam, M.Z., Islam, M.M. and Yanase, S., “Numerical Study of Unsteady Heat and Fluid Flow through a Curved Rectangular Duct of Small Aspect Ratio,” Thammasat International Journal of Science and Technology, 20, pp. 1–20 (2015).Google Scholar
Deka, R.K. and Paul, A., “Stability of Taylor–Couette and Dean Flow: A Semi-analytical Study,” Applied Mathematical Modelling, 37, pp. 1627–1637 (2013).Google Scholar
He, J.H., “A Review on Some New Recently Developed Nonlinear Analytical Techniques,” International Journal of Nonlinear Sciences and Numerical Simulation, 1, pp. 51–70 (2000).CrossRefGoogle Scholar
Batchelor, G.K., An Introduction to Fluid Dynamics, 1st Edition, Cambridge University Press, Cambridge (2000).Google Scholar
Hille, P., Vehrenkamp, R. and Schulz-Dubois, E.O., “The Development and Structure of Primary and Secondary Flow in a Curved Square Duct,” Journal of Fluid Mechanics, 151, pp. 219–241 (1985).CrossRefGoogle Scholar
Mees, P.A., Nandakumar, K. and Masliyah, J.H., “Secondary Instability of Flow in a Curved Duct of Square Cross-Section,” Journal of Fluid Mechanics, 323, pp. 387–409 (1996).Google Scholar
Huerre, P. and Monkewitz, P.A., “Local and Global Instabilities in Spatially Developing Flows,” Annual Review of Fluid Mechanics, 22, pp. 473–537 (1990).Google Scholar
Mahapatra, T.R., Dholey, S. and Gupta, A.S., “Stability of Hydromagnetic Dean Flow between Two Arbitrarily Spaced Concentric Circular Cylinders in the Presence of a Uniform Axial Magnetic Field,” Physics Letters A, 373, pp. 4338–4345 (2009).Google Scholar
Bottaro, A., “Spatially Developing Flow in Curved Duct,” Numerical Methods in Laminar and Turbulent Flow, 7, pp. 696–706 (1991).Google Scholar
Chomaz, J.M., “Absolute and Convective Instabilities in Nonlinear Systems,” Physical Review Letters, 69, pp. 1931–1934 (1992).CrossRefGoogle ScholarPubMed
Bottaro, A., “On Longitudinal Vortices in Curved Channel Flow,” Journal of Fluid Mechanics, 251, pp. 627–660 (1993).CrossRefGoogle Scholar
Maple, Maple User Manual, Version 18.02, The Waterloo Maple Inc (2014).Google Scholar
Chandratilleke, T., Nadim, N. and Narayanaswamy, R., “Analysis of Secondary Flow Characteristics and Hydrodynamic Instability in Fluid Flow through Curved Ducts,” Proceedings of the 8th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Mauritius (2011).Google Scholar
Vidal, A., Vinuesa, R., Schlatter, P. and Nagib, H.M., “Influence of Corner Geometry on the Secondary Flow in Turbulent Square Ducts,” International Journal of Heat and Fluid Flow, 67, pp. 69–78 (2017).CrossRefGoogle Scholar
Nowruzi, H., Ghassemi, H. and Nourazar, S.S., “On Hydrodynamic Stability of Dean Flow by Using Energy Gradient Methods,” Journal of Mechanics, pp. 1–14 (2018).CrossRefGoogle Scholar
Prandtl, L., Essentials of Fluid Dynamics: With Applications to Hydraulics, Aeronautics, Meteorology and Other Subjects, 1st Edition, Hafner Publishing Company, New York (1952).Google Scholar
Yanase, S., Goto, N. and Yamamoto, K., “Dual Solutions of the Flow through a Curved Tube,” Fluid Dynamics Research, 5, pp. 191–201 (1989).CrossRefGoogle Scholar
Cheng, K.C. and Mok, S.Y., “Flow Visualization Studies on Secondary Flow Patterns and Centrifugal Instability Phenomena in Curved Tubes,” Fluid Control and Measurement, 2, pp. 765–773 (1986).Google Scholar