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

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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.

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Abbreviations

y + :

Normalized wall distance

x i :

Cartesian coordinates (vector notation)

x, y, z :

Cartesian coordinates

u :

Velocity vector

u i :

Cartesian velocity components

u :

Cartesian velocity component in direction x

v :

Cartesian velocity component in direction y

w :

Cartesian velocity component in direction z

w bulk :

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

u τ :

Friction velocity

λ:

Wavelength of the coherent structures

D h :

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

a i :

Coefficients of the POD modes

S ij :

Strain tensor

τ ij :

SGS stress tensor

M :

Number of snapshots

N eq :

Number of modes used in the ODE set

M eq :

Number of oscillatory modes contained in the ODE set

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Authors and Affiliations

  1. Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology, N1-5, 2-12-1 O-Okayama, Meguro-ku, Tokyo, 152-8550, Japan

    Elia Merzari & H. Ninokata

  2. PNR-R3, Delft University of Technology, Mekelweg 15, 2629 JB, Delft, The Netherlands

    A. Mahmood & M. Rohde

Authors
  1. Elia Merzari
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  2. H. Ninokata
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  3. A. Mahmood
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  4. M. Rohde
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Correspondence to Elia Merzari.

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Merzari, E., Ninokata, H., Mahmood, A. et al. Proper orthogonal decomposition of the flow in geometries containing a narrow gap. Theor. Comput. Fluid Dyn. 23, 333–351 (2009). https://doi.org/10.1007/s00162-009-0152-3

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