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The application of bees algorithm in finding the neutral stability curve for plane Poiseuille flow

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Abstract

This paper investigates the application of the bees algorithm (BA) in finding the neutral stability curve of the Orr-Sommerfeld equation for basic flows between parallel plates. To demonstrate the performance of the proposed method, determination of neutral stability curve for plane Poiseuille flow is considered as a case study. Some minor modifications have been applied to BA in order to minimize the computational cost. To better assess the effectiveness of the bees algorithm, its results were compared with those of the modified genetic algorithm (MGA). The obtained results confirm the superiority of BA over MGA in terms of the computational cost.

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References

  1. Orr WM (1907) The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid. Proc R Ir Acad A 27:69–138

    Google Scholar 

  2. Sommerfeld A (1908) Ein Beitrag zur hydrodynamischen Erkelaerung der turbulenten Fluessigkeitsbewegungen. In: Proceedings of the fourth congress of mathematicians 3, pp 16–124

  3. Thomas LH (1953) The stability of plane Poiseuille flow. Phys Rev 91:780–783

    Article  ADS  MATH  Google Scholar 

  4. Gary J, Helgason R (1970) A matrix method for ordinary differential eigenvalue problems. J Comput Phys 5(2):169–187

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Dolph CL, Lewis DC (1958) On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseuille flow. Q Appl Math 16:97–110

    MathSciNet  MATH  Google Scholar 

  6. Orszag SA (1971) Accurate solution of the Orr-Sommerfeld stability equation. J Fluid Mech 50:689–703

    Article  ADS  MATH  Google Scholar 

  7. Nachtsheim PR (1964) An initial value method for the numerical treatment of the Orr-Sommerfeld equation for the case of plane Poiseuille flow. Technical Note D-2414, NASA

  8. Huang WZ, Sloan DM (1994) The pseudospectral method for solving differential eigenvalue problems. J Comput Phys 111(2):399–409

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Welfert BD (1997) Generation of pseudospectral differentiation matrices I. SIAM J Numer Anal 34:1640–1657

    Article  MathSciNet  MATH  Google Scholar 

  10. Weideman JAC, Reddy SC (2000) A MATLAB differentiation matrix suite. ACM Trans Math Softw 26(4):465–519

    Article  MathSciNet  Google Scholar 

  11. McBain GD, Armfield SW (2004) Natural convection in a vertical slot: Accurate solution of the linear stability equations. ANZIAM J 45(E):92–105

    MathSciNet  Google Scholar 

  12. McBain GD, Chubb TH, Armfield SW (2009) Numerical solution of the Orr–Sommerfeld equation using the viscous Green’s function and split Gaussian quadrature. J Comput Appl Math 224:397–404

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Amini Kafiabad H, Sadeghy K (2010) On the use of genetic algorithm for finding the neutral instability curve in plane Poiseuille flow. Int J Non-Linear Mech 45:691–698

    Article  Google Scholar 

  14. Pham DT, Ghanbarzadeh A, Koc E, Otri S, Rahim S, Zaidi M (2005) The bees algorithm. Technical note, Manufacturing Engineering Centre, Cardiff University, UK

  15. Lahmann JR, Plum M (2004) A computer-assisted instability proof for the Orr-Sommerfeld equation with Blasius profile. Z Angew Math Mech 84(3):188–204

    Article  MathSciNet  MATH  Google Scholar 

  16. Tammisola O, Lundell F, Schlatter P, Wehrfritz A, Söderberg LD (2011) Global linear and nonlinear stability of viscous confined plane wakes with co-flow. J Fluid Mech 675:397–434

    Article  ADS  MATH  Google Scholar 

  17. Boyd JP (2000) Chebyshev and Fourier spectral methods, 2nd edn. Dover, New York

    Google Scholar 

  18. Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral methods in fluid dynamics. Springer, Berlin

    Book  MATH  Google Scholar 

  19. Fornberg B (1996) A practical guide to pseudospectral methods. Cambridge University Press, New York

    MATH  Google Scholar 

  20. Baltensperger R, Berrut J (1999) The errors in calculating the pseudospectral differentiation matrices for Chebyshev-Gauss-Lobatto points. Comput Math Appl 37(1):41–48

    Article  MathSciNet  MATH  Google Scholar 

  21. Bayliss A, Class A, Matkowsky BJ (1995) Roundoff error in computing derivatives using the Chebyshev differentiation matrix. J Comput Phys 116(2):380–383

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Pham DT, Ghanbarzadeh A, Koc E, Otri S, Rahim S, Zaidi M (2006) The bees algorithm—a novel tool for complex optimisation problems. In: Proceeding of the 2nd international virtual conference on intelligent production machines and systems. Elsevier, Oxford

    Google Scholar 

  23. Van Den Braembussche RA (2011) Tuning of optimization strategies. NATO RTO-EN-AVT.167

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Authors and Affiliations

  1. Mechanical Engineering Department, Shahid Chamran University, Ahvaz, 61355, Iran

    S. S. Bahrainian, Z. Mehrdoost & A. Ghanbarzadeh

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  1. S. S. Bahrainian
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  2. Z. Mehrdoost
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  3. A. Ghanbarzadeh
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Correspondence to Z. Mehrdoost.

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Bahrainian, S.S., Mehrdoost, Z. & Ghanbarzadeh, A. The application of bees algorithm in finding the neutral stability curve for plane Poiseuille flow. Meccanica 48, 2255–2261 (2013). https://doi.org/10.1007/s11012-013-9742-7

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  • DOI: https://doi.org/10.1007/s11012-013-9742-7

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