Skip to main content
SpringerLink
Log in

Direct numerical simulation of turbulent flows in eccentric pipes

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

A numerical algorithm was developed for solving the incompressible Navier-Stokes equations in curvilinear orthogonal coordinates. The algorithm is based on a central-difference discretization in space and on a third-order accurate semi-implicit Runge-Kutta scheme for time integration. The discrete equations inherit some properties of the original differential equations, in particular, the neutrality of the convective terms and the pressure gradient in the kinetic energy production. The method was applied to the direct numerical simulation of turbulent flows between two eccentric cylinders. Numerical computations were performed at Re = 4000 (where the Reynolds number Re was defined in terms of the mean velocity and the hydraulic diameter). It was found that two types of flow develop depending on the geometric parameters. In the flow of one type, turbulent fluctuations were observed over the entire cross section of the pipe, including the narrowest gap, where the local Reynolds number was only about 500. The flow of the other type was divided into turbulent and laminar regions (in the wide and narrow parts of the gap, respectively).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Moin and K. Mahesh, “Direct Numerical Simulation: A Tool in Turbulence Research,” Ann. Rev. Fluid Mech. 3, 539–578 (1998).

    MathSciNet  Google Scholar 

  2. J. Kim, P. Moin, and R. Moser, “Turbulence Statistics in Fully Developed Channel Flow at Low Reynolds Number,” J. Fluid Mech. 177, 133–166 (1987).

    Google Scholar 

  3. P. R. Spalart, “Direct Simulation of a Turbulent Boundary Layer up to Re = 1410,” J. Fluid Mech. 187, 61–98 (1988).

    MATH  Google Scholar 

  4. J. G. M. Eggels, F. Unger, M. H. Weiss, et al., “Fully Developed Turbulent Pipe Flow: A Comparison between Direct Numerical Simulation and Experiment,” J. Fluid Mech. 268, 175–209 (1994).

    Google Scholar 

  5. N. V. Nikitin, “Direct Numerical Simulation of Three-Dimensional Turbulent Flows in Circular Pipes,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 6, 14–26 (1994).

  6. N. V. Nikitin, “Statistical Characteristics of Near-Wall Turbulence,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 32–43 (1996).

  7. A. O. Demuren and W. Rodi, “Calculation of Turbulence-Driven Secondary Motion in Noncircular Ducts,” J. Fluid Mech. 140, 189–222 (1984).

    Google Scholar 

  8. S. Gavrilakis, “Numerical Simulation of Low-Reynolds-Number Turbulent Flow through a Straight Square Duct,” J. Fluid Mech. 244, 101–129 (1992).

    Google Scholar 

  9. A. Huser and S. Biringen, “Direct Numerical Simulation of Turbulent Flow in a Square Duct,” J. Fluid Mech. 257, 65–95 (1993).

    Google Scholar 

  10. N. V. Nikitin, “Numerical Simulation of Turbulent Flows in a Pipe of Square Cross Section,” Dokl. Akad. Nauk 353, 338–342 (1997) [Dokl. Phys. 42, 158–162 (1997)].

    MATH  Google Scholar 

  11. N. Nikitin and A. Yakhot, “Direct Numerical Simulation of Turbulent Flow in Elliptical Ducts,” J. Fluid Mech. 532, 141–164 (2005).

    Article  Google Scholar 

  12. J. Jimenez and A. Pinelli, “The Autonomous Cycle of Near-Wall Turbulence,” J. Fluid Mech. 389, 335–359 (1999).

    MathSciNet  Google Scholar 

  13. J. Kim and P. Moin, “Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations,” J. Comput. Phys. 59, 308–323 (1985).

    Article  MathSciNet  Google Scholar 

  14. M. M. Rai and P. Moin, “Direct Simulations of Turbulent Flow Using Finite-Difference Schemes,” J. Comput. Phys. 96, 15–53 (1991).

    Google Scholar 

  15. P. R. Spalart, R. D. Moser, and M. Rogers, “Spectral Methods for the Navier-Stokes Equations with One Infinite and Two Periodic Directions,” J. Comput. Phys. 96, 297–324 (1991).

    Article  MathSciNet  Google Scholar 

  16. R. Verzicco and P. Orlandi, “A Finite-Difference Scheme for the Three-Dimensional Incompressible Flows in Cylindrical Coordinates,” J. Comput. Phys. 123, 402–414 (1996).

    Article  MathSciNet  Google Scholar 

  17. N. Nikitin, “Third-Order-Accurate Semi-Implicit Runge-Kutta Scheme for Incompressible Navier-Stokes Equations,” Int. J. Numer. Methods Fluids (2005).

  18. N. V. Nikitin, “Spectral Finite-Difference Method for Incompressible Turbulent Flows in Pipes and Channels,” Zh. Vychisl. Mat. Mat. Fiz. 34, 909–925 (1994).

    MATH  MathSciNet  Google Scholar 

  19. D. L. Brown, R. Cortez, and M. L. Minion, “Accurate Projection Methods for the Incompressible Navier-Stokes Equations,” J. Comput. Phys. 168, 464–499 (2001).

    Article  MathSciNet  Google Scholar 

  20. J. K. Dukowicz and A. S. Dvinsky, “Approximate Factorization as a High Order Splitting for the Implicit Incompressible Flow Equations,” J. Comput. Phys. 102, 336–347 (1992).

    Article  MathSciNet  Google Scholar 

  21. J. B. Perot, “An Analysis of the Fractional Step Method,” J. Comput. Phys. 108, 51–58 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  22. N. V. Nikitin, “Spatial Approach to Numerical Simulation of Turbulence in Pipes,” Dokl. Akad. Nauk 343, 767–770 (1995).

    MATH  Google Scholar 

  23. N. V. Nikitin, “Numerical Analysis of Laminar-Turbulent Transition in a Circular Pipe under Periodic Inlet Perturbations,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 2, 42–55 (2001).

  24. J. Jimenez and P. Moin, “The Minimal Flow Unit in Near-Wall Turbulence,” J. Fluid Mech. 225, 213–240 (1991).

    Google Scholar 

  25. F. H. Harlow and J. E. Welsh, “Numerical Calculation of Time-Dependent Viscous Incompressible Flow with Free Surface,” Phys. Fluids 8, 2182–2189 (1965).

    Article  Google Scholar 

  26. A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations (Nauka, Moscow, 1978; Birkhäuser, Basel, 1989).

    Google Scholar 

  27. P. N. Swarztrauber, “A Direct Method for the Discrete Solution of Separable Elliptic Equations,” SIAM J. Numer. Anal. 11, 1136–1150 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  28. H. Choi and P. Moin, “Effects of the Computational Time Step on Numerical Solutions of Turbulent Flow,” J. Comput. Phys. 113, 1–4 (1994).

    Article  Google Scholar 

  29. F. E. Ham, F. S. Lien, and A. B. Strong, “A Fully Conservative Second-Order Finite Difference Scheme for Incompressible Flow on Nonuniform Grids,” J. Comput. Phys. 177, 117–133 (2002).

    Article  Google Scholar 

  30. N. V. Nikitin, Extended Abstract of Doctoral Dissertation in Mathematics and Physics (Mosk. Gos. Univ., Moscow, 1996).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanics, Moscow State University, Michurinskii pr. 1, Moscow, 119192, Russia

    N. V. Nikitin

Authors
  1. N. V. Nikitin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © N.V. Nikitin, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 3, pp. 509–526.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nikitin, N.V. Direct numerical simulation of turbulent flows in eccentric pipes. Comput. Math. and Math. Phys. 46, 489–504 (2006). https://doi.org/10.1134/S0965542506030158

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542506030158

Keywords

Search

Navigation