Experiments in Fluids

, 54:1497 | Cite as

Flow characterization using PIV measurements in a low aspect ratio randomly packed porous bed

  • Vishal A. Patil
  • James A. Liburdy
Research Article


Low aspect ratio porous beds (bed width to bead diameter) have engineering applications such as catalytic reactors, combustors and heat exchangers. The nature of the packing within the bed and the influence of the near-wall region especially for randomly packed beds are expected to affect the velocity field and consequently the statistical characteristics of the flow. Planar PIV measurements were taken using refractive index matching at discrete locations throughout a randomly packed bed with aspect ratio of 4.67 for steady, low pore Reynolds number flows, Re pore ~ 4. Details of the measurement uncertainties as well as methods to determine local magnification and determination of the dynamic velocity range are presented. The data are analyzed using the PIV correlation averaging method with the largest velocity uncertainties arising from out-of-plane motion. Results show the correspondence with the geometric and velocity correlation functions across the bed and that the centerline of the bed shows a random-like distribution of velocity with an integral length scale on the order of one hydraulic diameter (or 0.38 bead diameters based on the porosity for this bed). The velocity variance is shown to increase by a factor of 1.8 when comparing the center plane data versus using data across the entire bed. It is shown that the large velocity variance contributes strongly to increased dispersion estimates and that based on the center plane data of the variance and integral length scales, the dispersion coefficient matches well with that measured in high aspect ratio beds using global data.


Longitudinal Dispersion Integral Length Scale Bead Geometry Laser Light Sheet Bead Diameter 
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List of symbols


Seed displacement variation within interrogation window


Cross-sectional area of bed


Constant for an optical arrangement (see Eq. 18)


Imaged bead diameter in pixel units


Seed image size in pixel units


Detector pixel size


Distance between lens center and image plane


Distance between lens center and object plane


Distance between bed wall and object plane (see Fig. 8)


Distance along optic axis of object plane at position ‘1’ from bead vertex (see Fig. 3)


Distance along optic axis of object plane at position ‘2’ from bead vertex (see Fig. 3)


Bead diameter


Hydraulic diameter \(\left( {\frac{2}{3}\frac{\phi }{{\left( {1 - \phi } \right)}}D_{\text{B}} } \right)\)


Interrogation window size


Effective dispersion coefficient in longitudinal direction


Mechanical dispersion in longitudinal direction


Longitudinal dispersion due to tracer molecular diffusion only


Molecular diffusivity of tracer


In-plane loss of seed image pairs


Out-of-plane loss of seed image pairs


Larger dimension of D I or t laser


Correlation separation distance normalized with hydraulic diameter D B


Correlation separation distance normalized with hydraulic diameter D H


Bed width


Integral length scale in longitudinal direction


Magnification of optical system


Refractive index of medium


Imaged seed number density within interrogation window of size D I


Peclet number \(\left( {\frac{{\left\langle V \right\rangle^{f} D_{\text{B}} }}{{D_{\text{M}} }}} \right)\)


Reynolds number based on V int and D H \(\left( {\frac{{V_{\text{int}} D_{\text{H}} }}{{\nu_{f} }}} \right)\)


Nominal image distance


Nominal object distance


Laser light sheet thickness


Elapsed time in second


Lagrangian integral time scale in longitudinal direction


Eulerian fluid velocity in transverse direction seen on image plane (pixels/s)


Eulerian fluid velocity in transverse direction (m/s)


Eulerian fluid velocity in longitudinal direction seen on image plane (pixels/s)


Eulerian fluid velocity in longitudinal direction (m/s)


Average bed velocity \(\left( {\frac{Q}{{A_{\text{bed}} }}} \right)\)


Average interstitial or pore velocity in bed \(\left( {\frac{{V_{Darcy} }}{\phi }} \right)\)


Lagrangian velocity of an ideal tracer


x position in bed (transverse direction perpendicular to optic axis)


y position in bed (longitudinal direction)


z position in bed (transverse direction parallel to optic axis)



Ratio of maximum transverse velocity to maximum longitudinal velocity in bed


Seed displacement in-plane transverse direction (pixels)


Seed displacement in-plane longitudinal direction (pixels)


Seed displacement out-of-plane transverse direction (pixels)


Seed displacement in-plane transverse direction (meters)


Seed displacement in-plane longitudinal direction (meters)


Seed displacement out-of-plane transverse direction (meters)


Eulerian longitudinal velocity variation within the interrogation window (pixels/s)


PIV image pair separation time


Local error in velocity at interrogation window center


Bed porosity


Kinematic viscosity of fluid phase


Spatial autocorrelation in longitudinal direction of longitudinal velocity, v


Spatial autocorrelation in longitudinal direction of bead geometry


RMS error in velocity for measurement plane


Tortuosity of bed



Object plane at position ‘1’ (see Fig. 3)


Object plane at position ‘2’ (see Fig. 3)


Error source due to in-plane seed displacement


Error source due to seed displacement gradients


Distance measured in pixel unit on image plane


Initial position of tracer


Error source due to fluid linear strain within interrogation window


Liquid phase


Measured seed displacement


Error source introduced due to uncertainty in magnification determination




Distance measured on object plane


Error source due to perspective motion of seeds


Random error due to finite-imaged seed density, N I


Error source due to refraction effects at solid–liquid interface


Error source due to fluid shear strain within interrogation window


Solid phase


Actual seed displacement




Component in in-plane transverse direction (pixels)


Component in in-plane longitudinal direction (pixels)


Component in in-plane transverse direction (meters)


Component in in-plane longitudinal direction (meters)



Variable estimated using values in fluid phase only


Average operator

max ()

Maximum of values in set


Spatial averaging of variable in fluid phase



This study was supported in part by NSF through grant 0933857 under the Particulate and Multiphase Processing Program, with program manger Dr. Ashok S. Sangani, and is gratefully acknowledged. The authors are also grateful for the helpful comments from reviewers of this paper.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Mechanical EngineeringOregon State UniversityCorvallisUSA

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