- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Characterization of turbulence in a Schwarz Diamond monolith using time-resolved MRI

Daniel A. Clarke1,2,3,*, Petrik Galvosas2, and Daniel J. Holland3

  • *Contact author: daniel.clarke@chem.lu.se
PDF

Phys. Rev. Fluids 11, 104605 – Published 15 October, 2026

DOI: https://doi.org/10.1103/h7v1-p8sh

Abstract

Periodic porous structures manufactured using additive manufacturing are candidates for intensified process devices. Detailed measurements of the velocity field are required to characterize the turbulent mixing generated by flow through the channel network. We used time-resolved single-shot spiral magnetic resonance velocity (MRV) imaging to noninvasively record snapshots of the velocity field for water flowing through a Schwarz Diamond monolith. The time-resolved velocity data lend themselves to more detailed analysis than is possible with conventional time-averaged magnetic resonance  methods. The turbulent energy spectra measured with MRV confirmed that the pore network places an upper size limit on the eddies that could exist in the flow. The −5/3 power-law scaling from Kolmogorov theory was only attained at a Reynolds number around 2000, indicative of a higher transition point to a fully turbulent regime than is usually reported for random porous media. The local Reynolds stress tensor showed considerable anisotropy within the network, which affects mass transport to the channel surfaces. The time-resolved MRV method offers valuable insights into turbulence in porous media that can inform the design and operation of novel geometries.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (80)

  1. T. Van Gerven and A. Stankiewicz, Structure, energy, synergy, time–The fundamentals of process intensification, Ind. Eng. Chem. Res. 48, 2465 (2009).
  2. J. J. John, J. P. Gueneau de Mussy, S. Kuhn, and T. Van Gerven, The role of process intensification in the current business context of the chemical process industry, J. Adv. Manuf. Proc. 3, e10086 (2021).
  3. L. R. Rosseau, V. Middelkoop, H. A. Willemsen, I. Roghair, and M. van Sint Annaland, Review on additive manufacturing of catalysts and sorbents and the potential for process intensification, Front. Chem. Eng. 4, 834547 (2022).
  4. S. Feast, C. Fee, T. Huber, and D. Clarke, Printed monolith adsorption as an alternative to expanded bed adsorption for purifying M13 bacteriophage, J. Chromatogr., A 1652, 462365 (2021).
  5. T. Femmer, A. J. Kuehne, and M. Wessling, Estimation of the structure dependent performance of 3-D rapid prototyped membranes, Chem. Eng. J. 273, 438 (2015).
  6. J. Kim and D.-J. Yoo, 3D printed compact heat exchangers with mathematically defined core structures, J. Comput. Des. Eng. 7, 527 (2020).
  7. B. W. Reynolds, C. J. Fee, K. R. Morison, and D. J. Holland, Characterisation of heat transfer within 3D printed TPMS heat exchangers, Int. J. Heat Mass Transf. 212, 124264 (2023).
  8. N. C. Ellebracht, P. Roy, T. Moore, A. E. Gongora, D. I. Oyarzun, J. K. Stolaroff, and D. T. Nguyen, 3D printed triply periodic minimal surfaces as advanced structured packings for solvent-based CO2 capture, Energy Environ. Sci. 16, 1752 (2023).
  9. G. Homem Alves César Ribeiro, D. Rouzineau, and M. Meyer, Experimental evaluation of novel gas liquid contactor packing based on triply periodic minimal surfaces (TPMS), Chem. Eng. Sci. 301, 120690 (2025).
  10. W. Li, J. Ding, X. Chen, Y. Wang, X. Song, and S. Zhang, A 3D-printed CuNi alloy catalyst with a triply periodic minimal surface for the reverse water-gas shift reaction, J. Mater. Chem. A 12, 314 (2023).
  11. H. Wang, H. Tian, Q. Zhang, and L. Zhang, Novel TPMS carbon-based monolithic catalysts by three-dimensional printing for enhancement of nitrobenzene hydrogenation reaction, React. Chem. Eng. 9, 1816 (2024).
  12. S. Reid, F. Lecarpentier, D. Symons, and M. Watson, Can 3D-printed catalysts improve hydrogen peroxide thruster performance? An experimental and numerical study, Chem. Eng. Sci. 302, 120783 (2025).
  13. D. A. Clarke, F. Dolamore, C. J. Fee, P. Galvosas, and D. J. Holland, Investigation of flow through triply periodic minimal surface-structured porous media using MRI and CFD, Chem. Eng. Sci. 231, 116264 (2021).
  14. Z. Dai, D. F. Fletcher, and B. S. Haynes, Impact of tortuous geometry on laminar flow heat transfer in microchannels, Int. J. Heat Mass Transf. 83, 382 (2015).
  15. M. Coe, Z. Rahnama, B. Reynolds, and D. Holland, Numerical investigation of heat transfer enhancement by the stretching of triply periodic minimal surfaces, Int. J. Heat Mass Transf. 247, 127064 (2025).
  16. F. J. Keil, Process intensification, Rev. Chem. Eng. 34, 135 (2018).
  17. C. Hutter, C. Allemann, S. Kuhn, and P. Rudolf von Rohr, Scalar transport in a milli-scale metal foam reactor, Chem. Eng. Sci. 65, 3169 (2010).
  18. A. Muley, C. Kiser, B. Sundén, and R. K. Shah, Foam heat exchangers: A technology assessment, Heat Transf. Eng. 33, 42 (2012).
  19. D. Seguin, A. Montillet, J. Comiti, and F. Huet, Experimental characterization of flow regimes in various porous media—II: Transition to turbulent regime, Chem. Eng. Sci. 53, 3897 (1998).
  20. B. D. Wood, X. He, and S. V. Apte, Modeling turbulent flows in porous media, Annu. Rev. Fluid Mech. 52, 171 (2020).
  21. J. Bałdyga and R. Pohorecki, Turbulent micromixing in chemical reactors—a review, Chem. Eng. J. Biochem. Eng. J. 58, 183 (1995).
  22. G. Wang, F. Yang, K. Wu, Y. Ma, C. Peng, T. Liu, and L.-P. Wang, Estimation of the dissipation rate of turbulent kinetic energy: A review, Chem. Eng. Sci. 229, 116133 (2021).
  23. R. Häfeli, M. Altheimer, D. Butscher, and P. Rudolf von Rohr, PIV study of flow through porous structure using refractive index matching, Exp. Fluids 55, 1717 (2014).
  24. D. O. Kuethe, Measuring distributions of diffusivity in turbulent fluids with magnetic-resonance imaging, Phys. Rev. A 40, 4542 (1989).
  25. T.-Q. Li, J. D. Seymour, R. L. Powell, K. L. McCarthy, L. Ödberg, and M. J. McCarthy, Turbulent pipe flow studied by time-averaged NMR imaging: Measurements of velocity profile and turbulent intensity, Magn. Reson. Imag. 12, 923 (1994).
  26. B. Newling, C. C. Poirier, Y. Zhi, J. A. Rioux, A. J. Coristine, D. Roach, and B. J. Balcom, Velocity imaging of highly turbulent gas flow, Phys. Rev. Lett. 93, 154503 (2004).
  27. J. Cooper, A. York, A. Sederman, and L. Gladden, Measuring velocity and turbulent diffusivity in wall-flow filters using compressed sensing magnetic resonance, Chem. Eng. J. 377, 119690 (2019).
  28. A.-R. P. Gauthier, N. Stocek, and B. Newling, Diffusion tensor imaging of anisotropic inhomogeneous turbulent flow, Phys. Rev. E 106, 015108 (2022).
  29. C. J. Elkins, M. T. Alley, L. Saetran, and J. K. Eaton, Three-dimensional magnetic resonance velocimetry measurements of turbulence quantities in complex flow, Exp. Fluids 46, 285 (2009).
  30. S. Romig, K. John, S. Schmidt, S. Schmitter, S. Grundmann, and M. Bruschewski, Improving MRI turbulence quantification by addressing the measurement errors caused by the derivatives of the turbulent velocity field–Sequence development and in vitro validation, Magn. Reson. Imag. 117, 110333 (2025).
  31. D. Clarke, P. Galvosas, and D. J. Holland, Characterization of unsteady flow in a 3D-printed Schwarz Diamond monolith using magnetic resonance velocimetry, AIChE J. 69, e18097 (2023).
  32. D. A. Clarke, M. J. Coe, P. Galvosas, and D. J. Holland, Investigation of turbulent flow in a Schwarz Diamond monolith using MRI and CFD, Chem. Eng. J. 524, 168921 (2025).
  33. J. Hennig, A. Nauerth, and H. Friedburg, RARE imaging: A fast imaging method for clinical MR, Magn. Reson. Med. 3, 823 (1986).
  34. S. Kuczera and P. Galvosas, Advances and artefact suppression in RARE-velocimetry for flow with curved streamlines, J. Magn. Reson. 259, 135 (2015).
  35. P. Mansfield, Multi-planar image formation using NMR spin echoes, J. Phys. C 10, L55 (1977).
  36. C. B. Ahn, J. H. Kim, and Z. H. Cho, High-speed spiral-scan echo planar NMR imaging-I, IEEE Trans. Med. Imag. 5, 2 (1986).
  37. K. Kose, One-shot velocity mapping using multiple spin-echo EPI and its application to turbulent flow, J. Magn. Reson. (1969) 92, 631 (1991).
  38. J. Gatenby and J. Gore, Echo-planar-imaging studies of turbulent flow, J. Magn. Reson. Ser. A 121, 193 (1996).
  39. A. J. Sederman, M. D. Mantle, C. Buckley, and L. F. Gladden, MRI technique for measurement of velocity vectors, acceleration, and autocorrelation functions in turbulent flow, J. Magn. Reson. 166, 182 (2004).
  40. A. Penn, T. Tsuji, D. O. Brunner, C. M. Boyce, K. P. Pruessmann, and C. R. Müller, Real-time probing of granular dynamics with magnetic resonance, Sci. Adv. 3, e1701879 (2017).
  41. A. Bordbar, W. Zia, J. Birnbaum, J. Omidi, R. F. Lee, E. Lev, and C. M. Boyce, Magnetic resonance imaging of a stream of bubbles injected into liquid suspensions, Chem. Eng. J. 494, 153282 (2024).
  42. A. B. Tayler, D. J. Holland, A. J. Sederman, and L. F. Gladden, Exploring the origins of turbulence in multiphase flow using compressed sensing MRI, Phys. Rev. Lett. 108, 264505 (2012).
  43. A. B. Tayler, D. J. Holland, A. J. Sederman, and L. F. Gladden, Applications of ultra-fast MRI to high voidage bubbly flow: Measurement of bubble size distributions, interfacial area and hydrodynamics, Chem. Eng. Sci. 71, 468 (2012).
  44. M. Lu, Transition between flow regimes in porous media using magnetic resonance velocimetry: From laminar to turbulent, Ph.D. thesis, University of Cambridge, 2019.
  45. M. Sangal, M. Anikeeva, S. C. Priese, H. Mattern, J.-B. Hövener, and O. Speck, MR based magnetic susceptibility measurements of 3D printing materials at 3 Tesla, J. Magn. Reson. Open 16-17, 100138 (2023).
  46. M. R. Serial, S. Benders, P. Rotzetter, D. L. Brummerloh, J. P. Metzger, S. P. Gross, J. Nussbaum, C. R. Müller, K. P. Pruessmann, and A. Penn, Temperature distribution in a gas-solid fixed bed probed by rapid magnetic resonance imaging, Chem. Eng. Sci. 269, 118457 (2023).
  47. D. G. Nishimura, P. Irarrazabal, and C. H. Meyer, A velocity k-space analysis of flow effects in echo‐planar and spiral imaging, Magn. Reson. Med. 33, 549 (1995).
  48. A. B. Tayler, D. J. Holland, A. J. Sederman, and L. F. Gladden, Time resolved velocity measurements of unsteady systems using spiral imaging, J. Magn. Reson. 211, 1 (2011).
  49. P. Dyverfeldt, A. Sigfridsson, J.-P. E. Kvitting, and T. Ebbers, Quantification of intravoxel velocity standard deviation and turbulence intensity by generalizing phase-contrast MRI, Magn. Reson. Med. 56, 850 (2006).
  50. D. J. Holland, C. R. Müller, J. S. Dennis, L. F. Gladden, and A. J. Sederman, Spatially resolved measurement of anisotropic granular temperature in gas-fluidized beds, Powder Technol. 182, 171 (2008).
  51. G. H. Glover, Simple analytic spiral k-space algorithm, Magn. Reson. Med. 42, 412 (1999).
  52. D.-H. Kim, E. Adalsteinsson, and D. M. Spielman, Simple analytic variable density spiral design, Magn. Reson. Med. 50, 214 (2003).
  53. J. G. Pipe and N. R. Zwart, Spiral trajectory design: A flexible numerical algorithm and base analytical equations, Magn. Reson. Med. 71, 278 (2014).
  54. Y. Zur, M. Wood, and L. Neuringer, Spoiling of transverse magnetization in steady-state sequences, Magn. Reson. Med. 21, 251 (1991).
  55. J. H. Duyn, Y. Yang, J. A. Frank, and J. W. van der Veen, Simple correction method for k-space trajectory deviations in MRI, J. Magn. Reson. 132, 150 (1998).
  56. N. G. Papadakis, A. A. Wilkinson, T. A. Carpenter, and L. D. Hall, A general method for measurement of the time integral of variant magnetic field gradients: Application to 2D spiral imaging, Magn. Reson. Imag. 15, 567 (1997).
  57. M. T. Alley, G. H. Glover, and N. J. Pelc, Gradient characterization using a Fourier-transform technique, Magn. Reson. Med. 39, 581 (1998).
  58. P. Webb and A. Macovski, Rapid, fully automatic, arbitrary-volume in vivo shimming, Magn. Reson. Med. 20, 113 (1991).
  59. L.-C. Man, J. M. Pauly, and A. Macovski, Improved automatic off-resonance correction without a field map in spiral imaging, Magn. Reson. Med. 37, 906 (1997).
  60. M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, Compressed sensing MRI, IEEE Sign. Proc. Mag. 25, 72 (2008).
  61. M. Blumenthal, M. Heide, C. Holme, M. Juschitz, B. Rapp, P. Schaten, N. Scholand, J. Tamir, C Tönnes, and M. Uecker, mrirecon/bart version 0.9.00, Zenodo, 2023, doi: 10.5281/zenodo.10277939.
  62. H. Tennekes and J. L. Lumley, A First Course in Turbulence (MIT Press, Cambridge, MA, 1972).
  63. S. Liu, C. Meneveau, and J. Katz, On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet, J. Fluid Mech. 275, 83 (1994).
  64. W. Hwang and J. Eaton, Creating homogeneous and isotropic turbulence without a mean flow, Exp. Fluids 36, 444 (2004).
  65. K. V. Sharp and R. J. Adrian, PIV study of small-scale flow structure around a Rushton turbine, AIChE J. 47, 766 (2001).
  66. P. Saarenrinne and M. Piirto, Turbulent kinetic energy dissipation rate estimation from PIV velocity vector fields, Exp. Fluids 29, S300 (2000).
  67. A. Delafosse, M.-L. Collignon, M. Crine, and D. Toye, Estimation of the turbulent kinetic energy dissipation rate from 2D-PIV measurements in a vessel stirred by an axial Mixel TTP impeller, Chem. Eng. Sci. 66, 1728 (2011).
  68. S. J. Beresh, N. E. Miller, and B. Smith, Practical challenges in the calculation of turbulent viscosity from PIV data, in 2018 Aerodynamic Measurement Technology and Ground Testing Conference (American Institute of Aeronautics and Astronautics, Reston, VA, 2018), p. 2987.
  69. D. Clarke, P. Galvosas, and D. Holland, Data from: Characterization of turbulence in a Schwarz Diamond monolith using time-resolved MRI, Zenodo, 2026, doi: 10.5281/zenodo.20744281.
  70. M. Harrower and C. A. Brewer, ColorBrewer.org: An online tool for selecting colour schemes for maps, Cartograph. J. 40, 27 (2003).
  71. S. Arseneau, 3D Quiver with volumized arrows, https://www.mathworks.com/matlabcentral/fileexchange/12285-3d-quiver-with-volumized-arrows (2006), [Online; accessed August 27, 2025].
  72. Y. Shao, The fractional Ornstein-Uhlenbeck process as a representation of homogeneous Eulerian velocity turbulence, Physica D 83, 461 (1995).
  73. V. A. Patil and J. A. Liburdy, Turbulent flow characteristics in a randomly packed porous bed based on particle image velocimetry measurements, Phys. Fluids 25, 043304 (2013).
  74. D. Nield, The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface, Int. J. Heat Fluid Flow 12, 269 (1991).
  75. Y. Jin, M.-F. Uth, A. V. Kuznetsov, and H. Herwig, Numerical investigation of the possibility of macroscopic turbulence in porous media: A direct numerical simulation study, J. Fluid Mech. 766, 76 (2015).
  76. X. He, S. V. Apte, J. R. Finn, and B. D. Wood, Characteristics of turbulence in a face-centred cubic porous unit cell, J. Fluid Mech. 873, 608 (2019).
  77. H. Dillinger, C. McGrath, C. Guenthner, and S. Kozerke, Fundamentals of turbulent flow spectrum imaging, Magn. Reson. Med. 87, 1231 (2022).
  78. G. Kindlmann, Superquadric tensor glyphs, in Proceedings of the Sixth Joint Eurographics-IEEE TCVG Conference on Visualization (Eurographics Association, Goslar, Germany, 2004), pp. 147–154.
  79. S. Banerjee, R. Krahl, F. Durst, and C. Zenger, Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches, J. Turbul. 8, N32 (2007).
  80. P. B. Kingsley, Introduction to diffusion tensor imaging mathematics: Part II. Anisotropy, diffusion-weighting factors, and gradient encoding schemes, Concepts Magn. Reson. 28A, 123 (2006).

Outline

Information

Phys. Rev. Fluids 11, 104605– Published 15 October, 2026
  • Received 18 June 2026
  • Accepted 23 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation