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

- 323 Downloads

### 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.

## References

- Abdulmohsin RS, Al-Dahhan MH (2017) Pressure drop and fluid flow characteristics in a packed pebble bed reactor. Nucl Technol 198(1):17–25CrossRefGoogle Scholar
- Amini N, Hassan YA (2012) An investigation of matched index of refraction technique and its application in optical measurements of fluid flow. Exp Fluids 53(6):2011–2020CrossRefGoogle Scholar
- Antohe B, Lage J (1997) A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int J Heat Mass Transf 40(13):3013–3024CrossRefMATHGoogle Scholar
- Bear J (2013) Dynamics of fluids in porous media. Courier Corporation, North ChelmsfordMATHGoogle Scholar
- Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech 25(1):539–575MathSciNetCrossRefGoogle Scholar
- Boomsma A, Bhattacharya S, Troolin D, Pothos S, Vlachos P (2016) A comparative experimental evaluation of uncertainty estimation methods for two-component PIV. Meas Sci Technol 27(9):094006CrossRefGoogle Scholar
- Calis H, Nijenhuis J, Paikert B, Dautzenberg F, Van Den Bleek C (2001) CFD modelling and experimental validation of pressure drop and flow profile in a novel structured catalytic reactor packing. Chem Eng Sci 56(4):1713–1720CrossRefGoogle Scholar
- Charonko JJ, Vlachos PP (2013) Estimation of uncertainty bounds for individual particle image velocimetry measurements from cross-correlation peak ratio. Meas Sci Technol 24(6):065301CrossRefGoogle Scholar
- Chen KK, Tu JH, Rowley CW (2012) Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J Nonlinear Scince 22(6):887–915MathSciNetCrossRefMATHGoogle Scholar
- de Lamotte A, Delafosse A, Calvo S, Toye D (2017) Analysis of PIV measurements using modal decomposition techniques, POD and DMD, to study flow structures and their dynamics within a stirred-tank reactor. Chem Eng SciGoogle Scholar
- de Lemos MJ, Pedras MH (2001) Recent mathematical models for turbulent flow in saturated rigid porous media. J Fluids Eng 123(4):935–940CrossRefGoogle Scholar
- Dixon A, Cresswell D (1986) Effective heat transfer parameters for transient packed-bed models. AIChE J 32(5):809–819CrossRefGoogle Scholar
- Du Toit C, Rousseau PG, Greyvenstein GP, Landman W (2006) A systems cfd model of a packed bed high temperature gas-cooled nuclear reactor. Int J Therm Sci 45(1):70–85CrossRefGoogle Scholar
- Dumas R (1990) Observations on the boundary layer based on measured correlations with various improvements. Near-Wall Turbul 1:437–452Google Scholar
- Eckstein A, Vlachos PP (2009a) Assessment of advanced windowing techniques for digital particle image velocimetry (DPIV). Meas Sci Technol 20(7):075402CrossRefGoogle Scholar
- Eckstein A, Vlachos PP (2009b) Digital particle image velocimetry (DPIV) robust phase correlation. Meas Sci Technol 20(5):055401CrossRefGoogle Scholar
- Eckstein AC, Charonko J, Vlachos P (2008) Phase correlation processing for DPIV measurements. Exp Fluids 45(3):485–500CrossRefGoogle Scholar
- Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48:89–94Google Scholar
- Falchi M, Romano GP (2009) Evaluation of the performance of high-speed piv compared to standard piv in a turbulent jet. Exp Fluids 47(3):509–526CrossRefGoogle Scholar
- Fick L, Merzari E, Hassan Y (2015) Direct numerical simulation of the flow through a structured pebble bed near a wall boundary. In: ASME/JSME/KSME 2015 joint fluids engineering conference, American Society of Mechanical Engineers, pp V001T03A008–V001T03A008Google Scholar
- Fick LH, Merzari E, Hassan YA (2017a) Direct numerical simulation of pebble bed flows: database development and investigation of low-frequency temporal instabilities. J Fluids Eng 139(5):051301CrossRefGoogle Scholar
- Fick LH, Merzari E, Marin O, Hassan YA (2017b) Investigation of the dynamics of incompressible flow in domains of multiple close-packed spheres. In: ASME 2017 fluids engineering division summer meeting, American Society of Mechanical Engineers, pp V01BT12A007–V01BT12A007Google Scholar
- Finn JR, Apte SV, Wood BD (2012) Characteristics of porescale vortical structures in random and arranged packed beds of spheres. In: ASME 2012 fluids engineering division summer meeting collocated with the ASME 2012 heat transfer summer conference and the ASME 2012 10th international conference on nanochannels, microchannels, and minichannels. American Society of Mechanical Engineers, pp 129–138Google Scholar
- Fort C, Fu CD, Weichselbaum NA, Bardet PM (2015) Refractive index and solubility control of para-cymene solutions for index-matched fluid–structure interaction studies. Exp Fluids 56(12):210CrossRefGoogle Scholar
- Getachew D, Minkowycz W, Lage J (2000) A modified form of the \(\kappa\)-\(\varepsilon\) model for turbulent flows of an incompressible fluid in porous media. Int J Heat Mass Transf 43(16):2909–2915CrossRefMATHGoogle Scholar
- Gunn D (1987) Axial and radial dispersion in fixed beds. Chem Eng Sci 42(2):363–373CrossRefGoogle Scholar
- Guo T, Rau MJ, Vlachos PP, Garimella SV (2017) Axisymmetric wall jet development in confined jet impingement. Phys Fluids 29(2):025102CrossRefGoogle Scholar
- Haam S, Brodkey R, Fort I, Klaboch L, Placnik M, Vanecek V (2000) Laser Doppler anemometry measurements in an index of refraction matched column in the presence of dispersed beads: part i. Int J Multiph Flow 26(9):1401–1418CrossRefMATHGoogle Scholar
- Hassan YA (2008) Large eddy simulation in pebble bed gas cooled core reactors. Nucl Eng Des 238(3):530–537CrossRefGoogle Scholar
- Hassan YA, Dominguez-Ontiveros E (2008) Flow visualization in a pebble bed reactor experiment using PIV and refractive index matching techniques. Nucl Eng Des 238(11):3080–3085CrossRefGoogle Scholar
- Higham J, Brevis W, Keylock C, Safarzadeh A (2017) Using modal decompositions to explain the sudden expansion of the mixing layer in the wake of a groyne in a shallow flow. Adv Water Resour 107:451–459CrossRefGoogle Scholar
- Higham J, Brevis W, Keylock C (2018) Implications of the selection of a particular modal decomposition technique for the analysis of shallow flows. J Hydraul Res 1–10Google Scholar
- Holmes P, Lumley JL, Berkooz G (1998) Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, CambridgeMATHGoogle Scholar
- Huang AY, Huang MY, Capart H, Chen RH (2008) Optical measurements of pore geometry and fluid velocity in a bed of irregularly packed spheres. Exp Fluids 45(2):309–321CrossRefGoogle Scholar
- Johnston W, Dybbs A, Edwards R (1975) Measurement of fluid velocity inside porous media with a laser anemometer. Phys Fluids 18(7):913–914CrossRefGoogle Scholar
- Jones B, Planchon H, Hammersley R (1973) Turbulent correlation measurements in a two-stream mixing layer. AIAA J 11(8):1146–1150CrossRefGoogle Scholar
- Kerhervé F, Fitzpatrick J (2011) Measurement and analysis of the turbulent length scales in jet flows. Exp Fluids 50(3):637–651CrossRefGoogle Scholar
- Khayamyan S, Lundström TS (2015) Interaction between the flow in two nearby pores within a porous material during transitional and turbulent flow. J Appl Fluid Mech 8(2):281–290CrossRefGoogle Scholar
- Khayamyan S, Lundström TS, Hellström JGI, Gren P, Lycksam H (2017) Measurements of transitional and turbulent flow in a randomly packed bed of spheres with particle image velocimetry. Transp Porous Media 116(1):413–431MathSciNetCrossRefGoogle Scholar
- Lumley JL (1967) The structure of inhomogeneous turbulent flows. Atmospheric turbulence and radio wave propagation, pp 166–178Google Scholar
- Masuoka T, Takatsu Y (1996) Turbulence model for flow through porous media. Int J Heat Mass Transf 39(13):2803–2809CrossRefMATHGoogle Scholar
- Mezić I (2005) Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn 41(1–3):309–325MathSciNetMATHGoogle Scholar
- Moroni M, Cushman JH (2001) Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous dispersion. ii. Experiments. Phys Fluids 13(1):81–91CrossRefMATHGoogle Scholar
- Muld TW, Efraimsson G, Henningson DS (2012) Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition. Comput Fluids 57:87–97MathSciNetCrossRefMATHGoogle Scholar
- Nakayama A, Kuwahara F (1999) A macroscopic turbulence model for flow in a porous medium. J Fluids Eng 121(2):427–433CrossRefGoogle Scholar
- Neal DR, Sciacchitano A, Smith BL, Scarano F (2015) Collaborative framework for PIV uncertainty quantification: the experimental database. Meas Sci Technol 26(7):074003CrossRefGoogle Scholar
- Nguyen T, Hassan Y (2017) Stereoscopic particle image velocimetry measurements of flow in a rod bundle with a spacer grid and mixing vanes at a low reynolds number. Int J Heat Fluid Flow 67:202–219CrossRefGoogle Scholar
- Nguyen T, Goth N, Jones P, Lee S, Vaghetto R, Hassan Y (2017) PIV measurements of turbulent flows in a 61-pin wire-wrapped hexagonal fuel bundle. Int J Heat Fluid Flow 65:47–59CrossRefGoogle Scholar
- Nguyen T, Goth N, Jones P, Vaghetto R, Hassan Y (2018) Stereoscopic PIV measurements of near-wall flow in a tightly packed rod bundle with wire spacers. Exp Thermal Fluid Sci 92:420–435CrossRefGoogle Scholar
- Northrup MA, Kulp TJ, Angel SM, Pinder GF (1993) Direct measurement of interstitial velocity field variations in a porous medium using fluorescent-particle image velocimetry. Chem Eng Sci 48(1):13–21CrossRefGoogle Scholar
- Patil VA, Liburdy JA (2012) Optical measurement uncertainties due to refractive index mismatch for flow in porous media. Exp Fluids 53(5):1453–1468CrossRefGoogle Scholar
- Patil VA, Liburdy JA (2013a) Flow characterization using piv measurements in a low aspect ratio randomly packed porous bed. Exp Fluids 54(4):1497CrossRefGoogle Scholar
- Patil VA, Liburdy JA (2013b) Turbulent flow characteristics in a randomly packed porous bed based on particle image velocimetry measurements. Phys Fluids 25(4):043304CrossRefGoogle Scholar
- Peurrung LM, Rashidi M, Kulp TJ (1995) Measurement of porous medium velocity fields and their volumetric averaging characteristics using particle tracking velocimetry. Chem Eng Sci 50(14):2243–2253CrossRefGoogle Scholar
- Raffel M, Willert CE, Wereley S, Kompenhans J (2013) Particle image velocimetry: a practical guide. Springer, BerlinGoogle Scholar
- Romano GP (1995) Analysis of two-point velocity measurements in near-wall flows. Exp Fluids 20(2):68–83CrossRefGoogle Scholar
- Rowley CW, Mezić I, Bagheri S, Schlatter P, Henningson DS (2009) Spectral analysis of nonlinear flows. J Fluid Mech 641:115–127MathSciNetCrossRefMATHGoogle Scholar
- Ruhe A (1984) Rational krylov sequence methods for eigenvalue computation. Linear Algebra Appl 58:391–405MathSciNetCrossRefMATHGoogle Scholar
- Ruiz T, Sicot C, Brizzi L, Borée J, Gervais Y (2010) Pressure/velocity coupling induced by a near wall wake. Exp Fluids 49(1):147–165CrossRefGoogle Scholar
- Saleh S, Thovert J, Adler P (1992) Measurement of two-dimensional velocity fields in porous media by particle image displacement velocimetry. Exp Fluids 12(3):210–212CrossRefGoogle Scholar
- Scheidegger AE (1974) The physics of flow through porous media. University of Toronto Press, Toronto, Tech. repGoogle Scholar
- Schmid PJ (2010) Dynamic mode decomposition of numerical and experimental data. J Fluid Mech 656:5–28MathSciNetCrossRefMATHGoogle Scholar
- Schmid PJ (2011) Application of the dynamic mode decomposition to experimental data. Exp Fluids 50(4):1123–1130CrossRefGoogle Scholar
- Sciacchitano A, Wieneke B (2016) Piv uncertainty propagation. Meas Sci Technol 27(8):084006CrossRefGoogle Scholar
- Sciacchitano A, Neal DR, Smith BL, Warner SO, Vlachos PP, Wieneke B, Scarano F (2015) Collaborative framework for PIV uncertainty quantification: comparative assessment of methods. Meas Sci Technol 26(7):074004CrossRefGoogle Scholar
- Shams A, Roelofs F, Komen E, Baglietto E (2013a) Quasi-direct numerical simulation of a pebble bed configuration. part i: Flow (velocity) field analysis. Nucl Eng Des 263:473–489CrossRefGoogle Scholar
- Shams A, Roelofs F, Komen E, Baglietto E (2013b) Quasi-direct numerical simulation of a pebble bed configuration, part-ii: Temperature field analysis. Nucl Eng Des 263:490–499CrossRefGoogle Scholar
- Sirovich L, Kirby M (1987) Low-dimensional procedure for the characterization of human faces. J Opt Soc Am 4(3):519–524CrossRefGoogle Scholar
- Spalart PR (1988) Contributions of numerical simulation data bases to the physics, modeling, and measurement of turbulence. Adv Turbul 11Google Scholar
- Stephenson J, Stewart W (1986) Optical measurements of porosity and fluid motion in packed beds. Chem Eng Sci 41(8):2161–2170CrossRefGoogle Scholar
- Timmins BH, Wilson BW, Smith BL, Vlachos PP (2012) A method for automatic estimation of instantaneous local uncertainty in particle image velocimetry measurements. Exp Fluids 53(4):1133–1147CrossRefGoogle Scholar
- Van Staden M, Janse Van Rensburg C, Viljoen C (2002) Cfd simulation of helium gas cooled pebble bed reactor. In: Proceedings of 1st Int. Conf. on heat transfer fluid mechanics and thermodynamics. Kruger Park, South AfricaGoogle Scholar
- Welch P (1967) The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans Audio Electroacoust 15(2):70–73CrossRefGoogle Scholar
- Wen CY, Fan LT et al (1975) Models for flow systems and chemical reactors. M. DekkerGoogle Scholar
- Westerweel J (1994) Efficient detection of spurious vectors in particle image velocimetry data. Exp Fluids 16(3–4):236–247CrossRefGoogle Scholar
- Wilson BM, Smith BL (2013) Uncertainty on PIV mean and fluctuating velocity due to bias and random errors. Meas Sci Technol 24(3):035302CrossRefGoogle Scholar
- Wood B, Apte S, Liburdy J, Ziazi R, He X, Finn J, Patil V (2015) A comparison of measured and modeled velocity fields for a laminar flow in a porous medium. Adv Water Resour 85:45–63CrossRefGoogle Scholar
- Yarlagadda A, Yoganathan A (1989) Experimental studies of model porous media fluid dynamics. Exp Fluids 8(1–2):59–71CrossRefGoogle Scholar
- Zhang Q, Liu Y, Wang S (2014) The identification of coherent structures using proper orthogonal decomposition and dynamic mode decomposition. J Fluids Struct 49:53–72CrossRefGoogle Scholar