Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-25T06:32:37.780Z Has data issue: false hasContentIssue false
  • English
  • Français

On Hydrodynamic Stability of Dean Flow by Using Energy Gradient Methods

Published online by Cambridge University Press:  25 January 2018

H. Nowruzi ,
H. Ghassemi  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 Engineering Amirkabir University of Technology (Tehran Polytechnic)Tehran, Iran
*
*Corresponding author (gasemi@aut.ac.ir)
Get access

Abstract

In the present paper, we investigate the hydrodynamic instability of Dean flow under different Dean numbers ranging from 1 to 2500, curvature ratios from 0.0001 up to 1000 and temperatures ranging from 273.15 K to 373.15 K. To study of fluid flow instability, analytical velocity profiles under intended conditions and energy gradient function K in the energy gradient method are evaluated. The results of present study show that, as the curvature ratio increases the flow becomes more stable. Moreover, no regular and significant effects on the energy gradient function K were achieved by increasing of temperatures. We found that, the origin of instability in the entire flow field is located on the inner wall of the parallel curved walls, especially for larger curvature ratios. We also reported the critical value of the energy gradient function K for the onset of instability corresponding to the critical Dean number.

Keywords

Hydrodynamic stabilityDean flowAnalytical methodEnergy gradient method
Type
Research Article
Information
Journal of Mechanics , Volume 35 , Issue 3 , June 2019 , pp. 405 - 418
Copyright
© The Society of Theoretical and Applied Mechanics 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Drazin, P. G. and Reid, W. H., Hydrodynamic Stability, 2nd Edition, Cambridge University Press, Cambridge (2004).Google 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
Joseph, B., Smith, E. P. and Adler, R. J., “Numerical Treatment of Laminar Flow in Helically Coiled Tubes of Square Cross Section: Part I. Stationary Helically Coiled Tubes,” AIChE Journal, 21, pp. 965–974 (1975).Google Scholar
Cheng, K. C., Lin, R. C. and Ou, J. W., “Fully Developed Laminar Flow in Curved Rectangular Channels,” Journal of Fluid Engineering, 98, pp. 41–48 (1976).Google Scholar
Ghia, K. N. and Sokhey, J. S., “Laminar Incompressible Viscous Flow in Curved Ducts of Regular Cross-Sections,” Journal of Fluid Engineering, 99, pp. 640–648 (1977).Google Scholar
Cheng, K. C. and Yuen, F. P., “Flow Visualization Experiments on Secondary Flow Patterns in an Isothermally Heated Curved Pipe,” Journal of Heat Transfer, 109, pp. 55–61 (1987).Google Scholar
Masliyah, J. H., “On Laminar Flow in Curved Semicircular Ducts,” Journal Fluid Mechanics, 99, pp. 469–479 (1980).Google Scholar
Dennis, S. C. R., “Dual Solutions for Steady Laminar Flow through a Curved Tube,” The Quarterly Journal of Mechanics and Applied Mathematics, 35, pp. 305–324 (1982).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 Fluid Mechanics, 151, pp. 219–241 (1985).Google Scholar
Li, Y., Wang, X., Yuan, S. and Tan, S. K., “Flow Development in Curved Rectangular Ducts with Continuously Varying Curvature,” Experimental Thermal and Fluid Science, 75, pp. 1–15 (2016).Google Scholar
Reid, W. H., “On the Stability of Viscous Flow in a Curved Channel,” Proceedings of the Royal Society of London, Series A, 244, pp. 186–198 (1958).Google Scholar
Reid, W. H. and Harris, D. L., “Some Further Results on the Bénard Problem,” Physics of Fluids, 1, pp. 102–110 (1958).Google Scholar
Bahl, S. K., “The Effect of Radial Temperature Gradient on the Stability of a Viscous Flow between Two Rotating Coaxial Cylinders,” Journal of Applied Mechanics, 39, pp. 593–595 (1972).Google Scholar
Chandrasekar, S., Hydrodynamic and Hydromagnetic Stability, Oxford University Press, London (1961).Google Scholar
Zhiming, L. and Yulu, L., “A Calculation Method for Fully Developed Flows in Curved Rectangular Tubes,” Applied Mathematics and Mechanics, 18, pp. 315–320 (1997).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).Google Scholar
Deka, R. K. and Takhar, H. S., “Hydrodynamic Stability of Viscous Flow between Curved Porous Channels with Radial Flow,” International Journal of Engineering Science, 42, pp. 953–966 (2004).Google Scholar
Orszag, S. A., “Accurate Solution of the Orr- Sommerfeld Stability Equation,” Journal 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
Helal, M. N. A., Ghosh, B. P. and Mondal, R. N., “Numerical Simulation of Two-Dimensional Laminar Flow and Heat Transfer through a Rotating Curved Square Channel,” American Journal of Fluid Dynamics, 6, pp.1–10 (2016).Google Scholar
Yanase, S., Kaga, Y. and Daikai, R., “Laminar Flows through a Curved Rectangular Curved Duct over a Wide Range of Aspect Ratio,” Fluid Dynamics Research, 31, pp. 151–183 (2002).Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. and Thomas, Jr. A., Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin (2012).Google Scholar
Deka, R. K. and Paul, A., “Stability of Taylor– Couette and Dean Flow: A Semi-Analytical Study,” Applied Mathematical Modeling, 37, pp. 1627–1637 (2013).Google Scholar
Dou, H. S., “Energy Gradient Theory of Hydrodynamic Instability,” The Third International Conference on Nonlinear Science, Singapore (2004).Google Scholar
Dou, H. S., “Mechanism of Flow Instability and Transition to Turbulence,” International Journal of Non-Linear Mechanics, 41, pp. 512–517 (2006).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
Dou, H. S., “Three Important Theorems for Fluid Dynamics,” arXiv Preprint Physics/0610082 (2006).Google Scholar
Dou, H. S., Khoo, B .C. and Yeo, K. S., “Energy Loss Distribution in the Plane Couette Flow and the Taylor–Couette Flow between Concentric Rotating Cylinders,” International Journal of Thermal Sciences, 46, pp. 262–275 (2007).Google Scholar
Farahbakhsh, I., Nourazar, S. S., Ghassemi, H., Dou, H. S. and Nazari-Golshan, A., “On the Instability of Plane Poiseuille Flow of Two Immiscible Fluids Using the Energy Gradient Theory,” Journal of Mechanics, 30, pp. 299–305 (2014).Google Scholar
Nowruzi, H., Nourazar, S. S. and Ghassemi, H., “On the Instability of Two Dimensional Backward-Facing Step Flow Using Energy Gradient Method,” Journal of Applied Fluid Mechanics, 11, pp. 241–256 (2018).Google Scholar
Topakoglu, H. C., “Steady Laminar Flows of an Incompressible Viscous Fluid in Curved Pipes,” Journal of Mathematics and Mechanics, 16, pp. 1321–1337 (1967).Google Scholar
Siggers, J. H. and Waters, S. L., “Steady Flows in Pipes with Finite Curvature,” Physics of Fluids, 17, 077102 (2005).Google Scholar
Dou, H. S. and Ben, A. Q., “Simulation and Instability Investigation of the Flow around a Cylinder between Two Parallel Walls,” Journal of Thermal Science, 24, pp. 140–148 (2015).Google Scholar
Hossain, M. A., Munir, M. S. and Rees, D. A. S., “Flow of Viscous Incompressible Fluid with Temperature Dependent Viscosity and Thermal Conductivity Past a Permeable Wedge with Uniform Surface Heat Flux,” International Journal of Thermal Sciences, 39, pp. 635–644 (2000).Google Scholar
Wang, C., Liu, S., Wu, J. and Li, Z., “Effects of Temperature-Dependent Viscosity on Fluid Flow and Heat Transfer in a Helical Rectangular Duct with a Finite Pitch,” Brazilian Journal of Chemical Engineering, 31, pp. 787–797 (2014).Google Scholar
Kafoussias, N. G. and Williams, E. W., “The Effect of Temperature-Dependent Viscosity on Free-Forced Convective Laminar Boundary Layer Flow Past a Vertical Isothermal Flat Plate,” Acta Mechanica, 110, pp. 123–137 (1995).Google Scholar
Speight, J. G., Lange’s Handbook of Chemistry, 17th Edition, McGraw-Hill, New York (2005).Google Scholar