Solving Single Phase Fluid Flow Instability Equations Using Chebyshev Tau- QZ Polynomial

Document Type : Research Paper


1 Professor, Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

2 Assistant professor, Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

3 PhD candidate, Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran


In this article the instability of single phase flow in a circular pipe from laminar to turbulence regime has been investigated. To this end, after finding boundary conditions and equation related to instability of flow in cylindrical coordination system, which is called eigenvalue Orr Sommerfeld equation, the solution method for these equation has been investigated. In this article Chebyshev polynomial Tau-QZ algorithm has been selected for the solution technique to solve the Orr Sommerfeld equation because in this method some of complex terms in the instability equation in cylindrical coordination will be appeared. After finding Orr Sommerfeld parameters related to Chebyshev polynomial Tau-QZ algorithm the solution have been done for Re=5000 and Re=1000, then the results had been compared with the results of valid references where other methods had been used in them. It have been observed that the use of Chebyshev Tau-QZ algorithm has higher accuracy concerning the results and it also has a higher accurate technique to solve the Orr Sommerfeld instability equations in cylindrical coordination system.


Main Subjects

[1]           Davidson, P., 2015, Turbulence: an introduction for scientists and engineers. Oxford University Press, USA.
[2]           Schmid, Peter J., and Dan S. Henningson, 2012, Stability and transition in shear flows, Vol. 142. Springer     Science & Business Media.
[3]           Dou, Hua-Shu., 2006, Mechanism of flow instability and transition to turbulence, International Journal of Non-Linear Mechanics 41.4: 512-517.
[4]           Dou, Hua-Shu., 2006, Physics of flow instability and turbulent transition in shear flows, arxiv preprint physics/0607004.
[5]           Mullin,T., Experimental Studies of Transition to Turbulence in a Pipe, Annual Review of Fluid Mechanics,Vol. 43:1-24.
[6]           Kerswell, R. R., and O. R. Tutty, 2007, Recurrence of travelling waves in transitional pipe flow, Journal of Fluid Mechanics 584: 69-102.  
[7]           Eckhardt, Bruno, et al., 2007, Turbulence transition in pipe flow, Annu. Rev. Fluid Mech. 39: 447-468.
[8]           Fox, R. W., McDonald, A. T., Pritchard, P. J.,2011, Introduction to Fluid Mechanics, John Wiley & Sons Press.
[9]           Mellibovsky, Fernando, et al., 2009, Transition in localized pipe flow turbulence, Physical review letters 103.5: 054502.
[10]        Drazin, P. G. and W. H. Reid, 1981, Hydrodynamic stability, Cambridge university press Cambridge.
[11]         Sexl, T. , 1927, Zur stabilitätsfrage der Poiseuilleschen und Couetteschen strömung." Annalen der Physik 388(14): 835-848.
[12]         Dongarra, J., et al., 1996, Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems, Applied Numerical Mathematics 22(4): 399-434.
[13]         Gardner, D. R., et al., 1989, A modified tau spectral method that eliminates spurious eigenvalues, Journal of Computational Physics 80(1): 137-167.
[14]        Fox, L., 1962,Chebyshev methods for ordinary differential equations, The Computer Journal 4(4): 318-331.
[15]         Davey, A. and P. Drazin, 1969, The stability of Poiseuille flow in a pipe, Journal of Fluid Mechanics 36(2): 209-218.
Volume 50, Issue 1
June 2019
Pages 135-139
  • Receive Date: 26 February 2018
  • Revise Date: 24 April 2018
  • Accept Date: 28 April 2018