Hydrodynamic stability study in a curved square duct by using the energy gradient method

  • Hashem NowruziEmail author
  • Hassan Ghassemi
  • S. Salman Nourazar
Technical Paper


Curved non-circular ducts have remarkable applications in the macro and micro scales engineering. Stability of flow through the curved ducts is a challenging topic in the context of fluid mechanics. In the current paper, hydrodynamic stability of fully developed 3D steady flow of incompressible fluid through the 180° curved square duct is investigated via energy gradient method. To this accomplishment, different Dean numbers (De) ranging from 19.60 to 1181.25 at curvature ratio 6.45 are considered. To investigate of flow stability, we analyzed the distributions of velocity, total pressure and control parameter of energy gradient function K. Obtained results indicated an appropriate agreement via both experimental and CFD data. Results of our investigation show that the maximum of energy gradient function Kmax is decreased by a reduction in the Dean number. In addition, we found that the origin of Dean hydrodynamic instability (i.e., position of the Kmax) is placed in the radial position between the center of the duct and outer curvature wall at azimuthally angular position θ < 72°. Results of Kmax and its cylindrical coordinates corresponding to the onset of Dean flow instability under various Dean numbers are reported.


Hydrodynamic stability Curved duct Dean number Energy gradient theory 

List of symbols

Latin letters


Computational fluid dynamics


Reynolds number


Dean number


Finite volume method


Root mean square error


Aspect ratio


Energy gradient function

\(K_{\rm{max} }\)

Maximum value of \(K\)


Value of \(K_{\rm{max} }\) for instability


Velocity vector


Averaged of stream-wise velocity


Hydraulic diameter


Curvature radius


Duct’s width


Duct’s height


Total mechanical energy


Static pressure of flow field


Total energy lost


Direction of transverse coordinate


Direction of stream-wise coordinate


Amplitude of the velocity disturbance


Amplitude of the disturbance distance

Greek letters




Dynamic viscosity


Kinematic viscosity


Curvature ratio


Frequency of the disturbance


Size of grid element



CFD computations presented in this paper have been performed on the parallel machines of the high-performance computing research center (HPCRC) of Amirkabir University of Technology (AUT); their supports are gratefully acknowledged. We thank Dr. Hua-Shu Dou from Zhejiang Sci-Tech University for his helpful comments and the referees for comments that greatly improved the manuscript.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Hashem Nowruzi
    • 1
    Email author
  • Hassan Ghassemi
    • 1
  • S. Salman Nourazar
    • 2
  1. 1.Department of Maritime EngineeringAmirkabir University of Technology (Tehran Polytechnic)TehranIran
  2. 2.Mechanical Engineering DepartmentAmirkabir University of Technology (Tehran Polytechnic)TehranIran

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