Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2025-01-01T08:53:05.156Z Has data issue: false hasContentIssue false
  • English
  • Français

Alignment statistics of rods with the Lagrangian stretching direction in a channel flow

Published online by Cambridge University Press:  25 August 2020

Z. Cui Open the ORCID record for Z. Cui [Opens in a new window] ,
A. Dubey Open the ORCID record for A. Dubey [Opens in a new window] ,
L. Zhao Open the ORCID record for L. Zhao [Opens in a new window]  and
B. Mehlig Open the ORCID record for B. Mehlig [Opens in a new window]
Z. Cui
Affiliation:
AML, Department of Engineering Mechanics, Tsinghua University, 100084Beijing, PR China
A. Dubey
Affiliation:
Department of Physics, Gothenburg University, SE-41296Gothenburg, Sweden
L. Zhao*
Affiliation:
AML, Department of Engineering Mechanics, Tsinghua University, 100084Beijing, PR China
B. Mehlig
Affiliation:
Department of Physics, Gothenburg University, SE-41296Gothenburg, Sweden
*
†Email address for correspondence: zhaolihao@tsinghua.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In homogeneous isotropic turbulence, slender rods are known to align with the Lagrangian stretching direction. However, how the degree of alignment depends on the aspect ratio of the rod is not understood. Moreover, particle-laden flows are often anisotropic and inhomogeneous. Therefore we study the alignment of rods with the Lagrangian stretching direction in a channel flow, which is approximately homogeneous and isotropic near the centre but inhomogeneous and anisotropic near the walls. Our main question is how the distribution of relative angles between a rod and the Lagrangian stretching direction depends on the aspect ratio of the rod and upon the distance of the rod from the channel wall. We find that this distribution exhibits two regimes: a plateau at small angles corresponding to random uncorrelated motion, and power-law tails due to large excursions. We find that slender rods near the channel centre align better with the Lagrangian stretching direction compared with those near the channel wall. These observations are explained in terms of simple statistical models based on Jeffery's equation, qualitatively near the channel centre and quantitatively near the channel wall. Lastly we discuss the consequences of our results for the distribution of relative angles between the orientations of nearby rods (Zhao et al., Phys. Rev. Fluids, vol. 4, 2019, 054602).

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 901 , 25 October 2020 , A16
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andersson, H. I., Zhao, L. & Variano, E. A. 2015 On the anisotropic vorticity in turbulent channel flows. J. Fluids Engng 137, 084503.CrossRefGoogle Scholar
Balkovsky, E. & Fouxon, A. 1999 Universal long-time properties of lagrangian statistics in the batchelor regime and their application to the passive scalar problem. Phys. Rev. E 60, 4164–4174.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1952 The effect of homogeneous turbulence on material lines and surfaces. Proc. R. Soc. Lond. A 213 (1114), 349–366.Google Scholar
Bezuglyy, V., Mehlig, B. & Wilkinson, M. 2010 Poincaré indices of rheoscopic visualisations. Europhys. Lett. 89, 34003.CrossRefGoogle Scholar
Borgnino, M., Gustavsson, K., De Lillo, F., Boffetta, G., Cencini, M. & Mehlig, B. 2019 Alignment of nonspherical active particles in chaotic flows. Phys. Rev. Lett. 123, 138003.CrossRefGoogle ScholarPubMed
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284–304.CrossRefGoogle Scholar
Byron, M., Einarsson, J., Gustavsson, K., Voth, G., Mehlig, B. & Variano, E. 2015 Shape-dependence of particle rotation in isotropic turbulence. Phys. Fluids 27, 035101.CrossRefGoogle Scholar
Carlsson, A., Söderberg, L. D. & Lundell, F. 2010 Fibre orientation measurements near a headbox wall. Nord. Pulp Pap. Res. J. 25, 204–212.CrossRefGoogle Scholar
Challabotla, N. R., Zhao, L. & Andersson, H. I. 2015 Shape effects on dynamics of inertia-free spheroids in wall turbulence. Phys. Fluids 27, 061703.CrossRefGoogle Scholar
Chertkov, M., Kolokolov, I., Lebedev, V. & Turitsyn, K. 2005 Polymer statistics in a random flow with mean shear. J. Fluid Mech. 531, 251–260.CrossRefGoogle Scholar
Chevillard, L. & Meneveau, C. 2013 Orientation dynamics of small, triaxial-ellipsoidal particles in isotropic turbulence. J. Fluid Mech. 737, 571.CrossRefGoogle Scholar
Dehkharghani, A., Waisbord, N., Dunkel, J. & Guasto, J. S. 2019 Bacterial scattering in microfluidic crystal flows reveals giant active Taylor–Aris dispersion. Proc. Natl Acad. Sci. 116, 11119–11124.CrossRefGoogle ScholarPubMed
Deutsch, J. M. 1994 Probability distributions for multicomponent systems with multiplicative noise. Physica A 208 (3), 445–461.CrossRefGoogle Scholar
Drummond, I. T. & Münch, W. 1990 Turbulent stretching of line and surface elements. J. Fluid Mech. 215, 45–59.CrossRefGoogle Scholar
Dubey, A., Meibohm, J., Gustavsson, K. & Mehlig, B. 2018 Fractal dimensions and trajectory crossings in correlated random walks. Phys. Rev. E 98, 062117.CrossRefGoogle Scholar
Duncan, K., Mehlig, B., Östlund, S. & Wilkinson, M. 2005 Clustering by mixing flows. Phys. Rev. Lett. 95, 240602.CrossRefGoogle ScholarPubMed
Einarsson, J., Angilella, J. R. & Mehlig, B. 2014 Orientational dynamics of weakly inertial axisymmetric particles in steady viscous flows. Physica D 278–279, 79–85.CrossRefGoogle Scholar
Einarsson, J., Candelier, F., Lundell, F., Angilella, J. R. & Mehlig, B. 2015 Effect of weak fluid inertia upon Jeffery orbits. Phys. Rev. E 91, 041002(R).CrossRefGoogle ScholarPubMed
Einarsson, J., Mihiretie, B. M., Laas, A., Ankardal, S., Angilella, J. R., Hanstorp, D. & Mehlig, B. 2016 Tumbling of asymmetric microrods in a microchannel flow. Phys. Fluids 28, 013302.CrossRefGoogle Scholar
Fries, J., Einarsson, J. & Mehlig, B. 2017 Angular dynamics of small crystals in viscous flow. Phys. Rev. Fluids 2, 014302.CrossRefGoogle Scholar
Fries, J., Kumar, M. V., Mihiretie, B. M., Hanstorp, D. & Mehlig, B. 2018 Spinning and tumbling of micron-sized triangles in a micro-channel shear flow. Phys. Fluids 30, 033304.CrossRefGoogle Scholar
Girimaji, S. S. & Pope, S. B. 1990 Material-element deformation in isotropic turbulence. J. Fluid Mech. 220, 427–458.CrossRefGoogle Scholar
Guala, M., Lüthi, B., Liberzon, A., Tsinober, A. & Kinzelbach, W. 2005 On the evolution of material lines and vorticity in homogeneous turbulence. J. Fluid Mech. 533, 339–359.CrossRefGoogle Scholar
Gustavsson, K., Berglund, F., Jonsson, P. R. & Mehlig, B. 2016 Preferential sampling and small-scale clustering of gyrotactic microswimmers in turbulence. Phys. Rev. Lett. 116, 108104.CrossRefGoogle ScholarPubMed
Gustavsson, K., Einarsson, J. & Mehlig, B. 2014 Tumbling of small axisymmetric particles in random and turbulent flows. Phys. Rev. Lett. 112, 014501.CrossRefGoogle ScholarPubMed
Gustavsson, K. & Mehlig, B. 2016 Statistical models for spatial patterns of heavy particles in turbulence. Adv. Phys. 65, 1–57.CrossRefGoogle Scholar
Gustavsson, K., Mehlig, B. & Wilkinson, M. 2015 Analysis of the correlation dimension for inertial particles. Phys. Fluids 27 (7), 73305.CrossRefGoogle Scholar
Hejazi, B., Mehlig, B. & Voth, G. A. 2017 Emergent scar lines in chaotic advection of passive directors. Phys. Rev. Fluids 2, 124501.CrossRefGoogle Scholar
Hinch, E. J. & Leal, L. G. 1972 The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles. J. Fluid Mech. 52, 683–712.CrossRefGoogle Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Johnson, P. L., Hamilton, S. S., Burns, R. & Meneveau, C. 2017 Analysis of geometrical and statistical features of lagrangian stretching in turbulent channel flow using a database task-parallel particle tracking algorithm. Phys. Rev. Fluids 2, 014605.CrossRefGoogle Scholar
Kesten, H. 1973 Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
Lundell, F., Söderberg, D. & Alfredsson, H. 2011 Fluid mechanics of papermaking. Annu. Rev. Fluid Mech. 43, 195–217.CrossRefGoogle Scholar
Mansour, N. N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15–44.CrossRefGoogle Scholar
Marchioli, C., Fantoni, M. & Soldati, A. 2010 Orientation, distribution, and deposition of elongated, inertial fibers in turbulent channel flow. Phys. Fluids 22, 033301.CrossRefGoogle Scholar
Marchioli, C. & Soldati, A. 2002 Mechanisms for particle transfer and segregation in a turbulent boundary layer. J. Fluid Mech. 468, 283–315.CrossRefGoogle Scholar
Meibohm, J., Pistone, L., Gustavsson, K. & Mehlig, B. 2017 Relative velocities in bidisperse turbulent suspensions. Phys. Rev. E 96, 061102.CrossRefGoogle ScholarPubMed
Mortensen, P. H., Andersson, H. I., Gillissen, J. J. J. & Boersma, B. J. 2008 Dynamics of prolate ellipsoidal particles in a turbulent channel flow. Phys. Fluids 20, 093302.CrossRefGoogle Scholar
Ni, R., Ouelette, N. T. & Voth, G. A. 2014 Alignment of vorticity and rods with Lagrangian fluid stretching in turbulence. J. Fluid Mech. 743, R3.CrossRefGoogle Scholar
Parsa, S., Calzavarini, E., Toschi, F. & Voth, G. A. 2012 Rotation rate of rods in turbulent fluid flow. Phys. Rev. Lett. 109, 134501.CrossRefGoogle ScholarPubMed
Parsa, S., Guasto, J. S., Kishore, M., Ouellette, N. T., Gollub, J. P. & Voth, G. A. 2011 Rotation and alignment of rods in two-dimensional chaotic flow. Phys. Fluids 23, 043302.CrossRefGoogle Scholar
Pumir, A. 2017 Structure of the velocity gradient tensor in turbulent shear flows. Phys. Rev. Fluids 2, 074602.CrossRefGoogle Scholar
Pumir, A. & Wilkinson, M. 2011 Orientation statistics of small particles in turbulence. New J. Phys. 13, 093030.CrossRefGoogle Scholar
Rosén, T., Einarsson, J., Nordmark, A., Aidun, C. K., Lundell, F. & Mehlig, B. 2015 Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers. Phys. Rev. E 92, 063022.CrossRefGoogle ScholarPubMed
She, Z.-S., Jackson, E., Orszag, S. A., Hunt, J. C. R., Phillips, O. M. & Williams, D. 1991 Structure and dynamics of homogeneous turbulence: models and simulations. Proc. R. Soc. Lond. A 434 (1890), 101–124.Google Scholar
Subramanian, G. & Koch, D. L. 2005 Inertial effects on fibre motion in simple shear flow. J. Fluid Mech. 535, 383–414.CrossRefGoogle Scholar
Turitsyn, K. S. 2007 Polymer dynamics in chaotic flows with a strong shear component. J. Exp. Theor. Phys. 105, 655–664.CrossRefGoogle Scholar
Voth, G. & Soldati, A. 2017 Anisotropic particles in turbulence. Annu. Rev. Fluid Mech. 49, 249–276.CrossRefGoogle Scholar
Wilkinson, M., Bezuglyy, V. & Mehlig, B. 2009 Fingerprints of random flows? Phys. Fluids 21, 043304.CrossRefGoogle Scholar
Wilkinson, M., Bezuglyy, V. & Mehlig, B. 2011 Emergent order in rheoscopic swirls. J. Fluid Mech. 667, 158.CrossRefGoogle Scholar
Xu, H., Pumir, A. & Bodenschatz, E. 2011 The pirouette effect in turbulent flows. Nat. Phys. 7, 709.CrossRefGoogle Scholar
Zhao, L. & Andersson, H. I. 2016 Why spheroids orient preferentially in near-wall turbulence. J. Fluid Mech. 807, 221–234.CrossRefGoogle Scholar
Zhao, L., Challabotla, N. R., Andersson, H. I. & Variano, E. A. 2015 Rotation of nonspherical particles in turbulent channel flow. Phys. Rev. Lett. 115, 244501.CrossRefGoogle ScholarPubMed
Zhao, L., Gustavsson, K., Ni, R., Kramel, S., Voth, G., Andersson, H. I. & Mehlig, B. 2019 Passive directors in turbulence. Phys. Rev. Fluids 4, 054602.CrossRefGoogle Scholar