# Time-resolved PIV measurements in a low-aspect ratio facility of randomly packed spheres and flow analysis using modal decomposition

### Abstract

This work experimentally investigated the flow characteristics in a facility with randomly packed spheres at a low-aspect ratio of 4.4. Velocity fields in the near-wall region and in the pores between spheres were obtained by employing the matched-index-of-refraction (MIR) and time-resolved particle image velocimetry (TR-PIV) techniques for Reynolds numbers of 340, 520, and 720. From the obtained TR-PIV velocity vector fields, flow characteristics including first- and second-order statistics, such as mean velocity, root-mean-square fluctuating velocity, and Reynolds stress profiles, were computed. The effects of the wall enclosure and Reynolds numbers on the flow patterns were investigated by comparing the computed flow statistics and evaluating two-point cross correlations of the velocities measured adjacent to and far from the wall. Comparisons of the mean velocities, root-mean-square fluctuating velocities, and Reynolds stress component showed an increase in flow mixing and turbulent intensity in the gaps between spheres in the packed bed. The re-circulation-region sizes, however, were found to be independent from an increase in Reynolds numbers. Finally, flow modal decompositions, such as the proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), were applied to the vorticity fields extracted from sub-regions located near and far from the wall to reveal the most dominant POD and DMD flow structures.

### Graphical abstract

## List of symbols

- \(\langle .\rangle\)
Time-averaged operator

- \(\varvec{\omega }\)
Vorticity (s\(^{-1}\))

- \(\epsilon _{\text {N}}\)
Absolute difference between the statistics

- \(\eta\)
Separation length (mm)

- \(\mu _{\text {pcy}}\)
Dynamic viscosity of

*p*-cymene (Pa \(\times\) s)- \(\nu _{\text {pcy}}\)
Kinematic viscosity of

*p*-cymene (m\(^2\)/s)- \(\rho _{\text {pcy}}\)
Density of

*p*-cymene (kg/m\(^{3}\))- \(\tau\)
Time delay (s)

- \(R_{uu0},R_{vv0}\)
Velocity–velocity spatial cross-correlation coefficients

- \(R_{uu},R_{vv}\)
Velocity–velocity spatial–temporal cross-correlation coefficients

*Re*Reynolds number

- \(\epsilon _{\text {b}}\)
Porosity

*D*Bed diameter (mm)

- \(d_{\text {h}}\)
Effective hydraulic diameter (mm)

- \(d_{\text {p}}\)
Sphere diameter (mm)

*f*Sampling frequency (Hz)

- \(L_x,L_y\)
Integral length scales (mm)

*N*Number of samples

*U*,*V*Horizontal and vertical time-averaged velocities (m/s)

- \(u^{\prime },v^{\prime }\)
Horizontal and vertical fluctuating velocities (m/s)

- \(U_{\text {m}}\)
Mean flow velocity (m/s)

- \(V_{g}\)
Superficial velocity (m/s)

- \(V_{\text {int}}\)
Interstitial velocity (m/s)

*x*,*y*Horizontal (traversal), vertical (axial) directions

## Notes

### Acknowledgements

This research is financially supported by the U.S. Department of Energy, NEAMS project and under a contract DE-NE0008550.

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