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Proper orthogonal decomposition of the flow in geometries containing a narrow gap

  • Elia Merzari
  • H. Ninokata
  • A. Mahmood
  • M. Rohde
Original Article

Abstract

Geometries containing a narrow gap are characterized by strong quasi-periodical flow oscillations in the narrow gap region. The above mentioned phenomena are of inherently unstable nature and, even if no conclusive theoretical study on the subject has been published, the evidence shown to this point suggests that the oscillations are connected to interactions between eddy structures of turbulent flows on opposite sides of the gap. These coherent structures travel in the direction of homogeneous turbulence, in a fashion that strongly recalls a vortex street. Analogous behaviours have been observed for arrays of arbitrarily shaped channels, within certain range of the geometric parameters. A modelling for these phenomena is at least problematic to achieve since they are turbulence driven. This work aims to address the use of Proper Orthogonal Decomposition (POD) to reduce the Navier–Stokes equations to a set of ordinary differential equations and better understand the dynamics underlying these oscillations. Both experimental and numerical data are used to carry out the POD.

Keywords

Eccentric channel POD Low-dimensional models 

List of symbols

y+

Normalized wall distance

xi

Cartesian coordinates (vector notation)

x, y, z

Cartesian coordinates

u

Velocity vector

ui

Cartesian velocity components

u

Cartesian velocity component in direction x

v

Cartesian velocity component in direction y

w

Cartesian velocity component in direction z

wbulk

Bulk velocity

\({\langle f \rangle}\)

Ensemble averaging operator on function f

f

Fluctuation over the ensemble average \({f=f-\langle f \rangle}\)

ρ

Density

υ

Kinematic viscosity

Δ

Filter width, mesh size

uτ

Friction velocity

λ

Wavelength of the coherent structures

Dh

Hydraulic diameter

Re

Bulk Reynolds number Re = D h w bulk/ν

g

Inner to outer diameter

e

Eccentricity e = d/D h

d

Distance between the cylinder axis for the eccentric channel

f

Frequency

T

Period

k

Wavenumber

m

Quantum number

L

Length of the domain in the streamwise direction

ai

Coefficients of the POD modes

Sij

Strain tensor

τij

SGS stress tensor

M

Number of snapshots

Neq

Number of modes used in the ODE set

Meq

Number of oscillatory modes contained in the ODE set

Mathematics Subject Classification (2000)

76D99 76D05 76D06 

References

  1. 1.
    Hooper J.D., Rehme K.: Large-scale structural effect in developed turbulent flows through closely-spaced rod arrays. J. Fluid Mech. 145, 305–337 (1984)CrossRefGoogle Scholar
  2. 2.
    Meyer L., Rehme K.: Large-scale turbulence phenomena in compound rectangular channels. Exp. Therm. Fluid Sci. 8, 286–304 (1994)CrossRefGoogle Scholar
  3. 3.
    Merzari E., Ninokata H., Baglietto E.: numerical simulation of the flow in tight-lattice fuel bundles. Nucl. Eng. Des. 238, 1703 (2008)CrossRefGoogle Scholar
  4. 4.
    Merzari E., Wang S., Ninokata H., Theofilis V.: Biglobal linear stability analysis for the flow in eccentric annular channels and a related geometry. Phys. Fluids 20, 114104 (2008). doi: 10.1063/1.3005864 CrossRefGoogle Scholar
  5. 5.
    Gosset A., Tavoularis S.: Laminar flow instability in a rectangular channel with a cylindrical core. Phys. Fluids 18, 044108 (2006)CrossRefGoogle Scholar
  6. 6.
    Guellouz M.S., Tavoularis S.: The structure of turbulent flow in a rectangular channel containing a cylindrical rod—Part I: Reynolds averaged experiments. Exp. Therm. Fluid Sci. 23, 59 (2000)CrossRefGoogle Scholar
  7. 7.
    Lexmond, A., Mudde, M., Van der Haagen M.: Visualization of the vortex street and characterization of the cross flow in the gap between two subchannels. In: Proceedings of 11th Nureth. Avignone, Paper 122 (2005)Google Scholar
  8. 8.
    Aubry N., Holmes P., Lumley J.L., Stone E.: The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Sanghi S., Aubry N.: Low dimensional models for the structure and dynamics in near wall turbulence. J. Fluid Mech. 247, 455–488 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lumley J.L.: Stochastic Tools in Turbulence. Academic Press, New York (1970)zbMATHGoogle Scholar
  11. 11.
    Berkooz G., Holmes P., Lumley J.L.: The proper orthogonal decomposition in the analisys of turbulent flow. Ann. Rev. Fluid Mech. 25, 539–575 (1993)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Sirovich L.: Turbulence and the dynamics of coherent structures, Part I: Coherent structures. Q. Appl. Math. XLV, 561–571 (1987)MathSciNetGoogle Scholar
  13. 13.
    Aubry N., Guyonnet R., Lima R.: Spatiotemporal analysis of complex signals: theory and applications. J. Stat. Phys. 64, 683–739 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Merzari E., Ninokata H.: Anisotropic turbulence and coherent structures in eccentric annular channels. Flow Turbul. Combust. 82, 93–120 (2009). doi: 10.1007/s10494-008-9170-2 CrossRefGoogle Scholar
  15. 15.
    Deardorff J.W.: A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech. 41, 453–480 (1970)zbMATHCrossRefGoogle Scholar
  16. 16.
    Aubry N.: On the hidden beauty of the proper orthogonal decomposition. Theor. Comput. Fluid Dyn. 2, 339–352 (1991)zbMATHCrossRefGoogle Scholar
  17. 17.
    Aubry N., Guyonnet R., Lima R.: Spatio-temporal symmetries and bifurcations via bi-orthogonal decompositions. J. Nonlinear Sci. 2, 183–215 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zang Y., Street R.L., Koseff J.R.: A non-staggered grid, fractional step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. J. Comput. Phys. 114, 1 (1994)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Jordan S.A.: Dynamic subgrid-scale modeling for large-eddy simulations in complex topologies. J. Fluids Eng. 123(3), 619–627 (2001)CrossRefGoogle Scholar
  20. 20.
    Kasagi, N., Horiuti, K., Miyake, Y., Miyauchi, T., Nagano, Y.: Establishment of the direct numerical simulation data bases of turbulent transport phenomena. Available on the Website of Tokyo University. http://www.thtlab.t.u-tokyo.ac.jp/
  21. 21.
    Nikitin N.V.: Direct numerical simulation of turbulent flows in eccentric pipes. Comput. Math. Math. Phys. 46, 509–526 (2006)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Omurtag A., Sirovich L.: On low-dimensional modeling of channel turbulence. Theor. Comput. Fluid Dyn. 13, 115–127 (1999)zbMATHCrossRefGoogle Scholar
  23. 23.
    Merzari, E., Ninokata, H.: Toward a dynamical system approach for the understanding of turbulent flow pulsation between subchannels. In: Proceedings of the 12th Nureth. Pittsburgh, USA (2007)Google Scholar
  24. 24.
    Cazemier, W.: Proper orthogonal decomposition and low dimensional models for turbulent flow. PhD thesis, University of Groningen, Groningen, The Netherlands (1997). ISBN 90-367-0682-3Google Scholar
  25. 25.
    Shampine L.F., Reichelt M.W.: The MATLAB ODE suite. SIAM J. Sci. Comput. 18, 1–22 (1997)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Elia Merzari
    • 1
  • H. Ninokata
    • 1
  • A. Mahmood
    • 2
  • M. Rohde
    • 2
  1. 1.Research Laboratory for Nuclear ReactorsTokyo Institute of TechnologyMeguro-ku, TokyoJapan
  2. 2.PNR-R3, Delft University of TechnologyDelftThe Netherlands

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