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Stationary and nonstationary energy cascades in homogeneous ferrofluid turbulence

Sukhdev Mouraya*, Nandita Pan, and Supratik Banerjee

  • *Contact author: sukhdevmourya@gmail.com
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Phys. Rev. Fluids 9, 094604 – Published 13 September, 2024

DOI: https://doi.org/10.1103/PhysRevFluids.9.094604

Abstract

The nonlinear transfer rate of the total energy (transfer rate of kinetic energy + transfer rate due to the work done by the magnetization) for an incompressible turbulent ferrofluid system is studied under the assumption of statistical homogeneity. Using the formalism of the two-point correlators, an exact relation connecting the second-order statistical moments to the average energy injection rate is derived for the scale-to-scale transfer of the total energy. We validate the universality of the exact relation through direct numerical simulations for stationary and nonstationary cascade regimes. For a weak external magnetic field, both kinetic and the total energy cascade with nearly the same cascade rate. A stationary cascade regime is achieved, and hence a good agreement between the exact energy transfer rate and the average energy injection is found. Due to the rapid alignment of the ferrofluid particles in the presence of strong external fields, the turbulence dynamics becomes nonstationary. Interestingly, there too, both kinetic and the total energy exhibit inertial range cascades but with different cascade rates which can be explained using the nonstationary form of our derived exact relation.

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References (56)

  1. A. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Akademiia Nauk SSSR Doklady 30, 301 (1941).
  2. T. de Karman and L. Howarth, On the statistical theory of isotropic turbulence, Proc. R. Soc. London Ser. A 164, 192 (1938).
  3. H. Politano and A. Pouquet, von Kármán–Howarth equation for magnetohydrodynamics and its consequences on third-order longitudinal structure and correlation functions, Phys. Rev. E 57, R21 (1998).
  4. H. Politano and A. Pouquet, Dynamical length scales for turbulent magnetized flows, Geophys. Res. Lett. 25, 273 (1998).
  5. H. Politano, T. Gomez, and A. Pouquet, von Kármán–Howarth relationship for helical magnetohydrodynamic flows, Phys. Rev. E 68, 026315 (2003).
  6. S. Galtier, von Kármán–Howarth equations for Hall magnetohydrodynamic flows, Phys. Rev. E 77, 015302(R) (2008).
  7. A. S. Monin and A. M. Yaglom., Statistical Fluid Mechanics: Mechanics of Turbulence (MIT Press, Cambridge, MA, 1975), Vol. 2.
  8. R. A. Antonia, M. Ould-Rouis, F. Anselmet, and Y. Zhu, Analogy between predictions of Kolmogorov and Yaglom, J. Fluid Mech. 332, 395 (1997).
  9. J. J. Podesta, Laws for third-order moments in homogeneous anisotropic incompressible magnetohydrodynamic turbulence, J. Fluid Mech. 609, 171 (2008).
  10. P. Augier, S. Galtier, and P. Billant, Kolmogorov laws for stratified turbulence, J. Fluid Mech. 709, 659 (2012).
  11. N. Pan and S. Banerjee, Exact relations for energy transfer in simple and active binary fluid turbulence, Phys. Rev. E 106, 025104 (2022).
  12. S. Galtier and S. Banerjee, Exact relation for correlation functions in compressible isothermal turbulence, Phys. Rev. Lett. 107, 134501 (2011).
  13. S. Banerjee and S. Galtier, Exact relation with two-point correlation functions and phenomenological approach for compressible magnetohydrodynamic turbulence, Phys. Rev. E 87, 013019 (2013).
  14. S. Banerjee and S. Galtier, A Kolmogorov-like exact relation for compressible polytropic turbulence, J. Fluid Mech. 742, 230 (2014).
  15. S. Banerjee and A. G. Kritsuk, Exact relations for energy transfer in self-gravitating isothermal turbulence, Phys. Rev. E 96, 053116 (2017).
  16. S. Banerjee and S. Galtier, An alternative formulation for exact scaling relations in hydrodynamic and magnetohydrodynamic turbulence, J. Phys. A: Math. Theor. 50, 015501 (2017).
  17. S. Banerjee and S. Galtier, Chiral exact relations for helicities in Hall magnetohydrodynamic turbulence, Phys. Rev. E 93, 033120 (2016).
  18. N. Andrés and S. Banerjee, Statistics of incompressible hydrodynamic turbulence: An alternative approach, Phys. Rev. Fluids 4, 024603 (2019).
  19. S. Banerjee and A. G. Kritsuk, Energy transfer in compressible magnetohydrodynamic turbulence for isothermal self-gravitating fluids, Phys. Rev. E 97, 023107 (2018).
  20. S. Banerjee and N. Andrés, Scale-to-scale energy transfer rate in compressible two-fluid plasma turbulence, Phys. Rev. E 101, 043212 (2020).
  21. M. I. Shliomis, Effective viscosity of magnetic suspensions, Sov. Phys. JETP 34, 1291 (1972).
  22. A. Zebib, Thermal convection in a magnetic fluid, J. Fluid Mech. 321, 121 (1996).
  23. R. Rosensweig, Ferrohydrodynamics (Dover Publications Inc., Mineola, New York, USA, 1997).
  24. P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Phys. Rev. E 70, 026313 (2004).
  25. M. I. Shliomis and K. I. Morozov, Negative viscosity of ferrofluid under alternating magnetic field, Phys. Fluids 6, 2855 (1994).
  26. J.-C. Bacri, R. Perzynski, M. I. Shliomis, and G. I. Burde, “Negative-viscosity” effect in a magnetic fluid, Phys. Rev. Lett. 75, 2128 (1995).
  27. B. A. Finlayson, Spin-up of ferrofluids: The impact of the spin viscosity and the Langevin function, Phys. Fluids 25, 073101 (2013).
  28. J. P. Huang, Z. W. Wang, and C. Holm, Computer simulations of the structure of colloidal ferrofluids, Phys. Rev. E 71, 061203 (2005).
  29. M. H. Buschmann, Critical review of heat transfer experiments in ferrohydrodynamic pipe flow utilising ferronanofluids, Int. J. Therm. Sci. 157, 106426 (2020).
  30. R. Adrian and J. Buckmaster, Suppression of turbulence in magnetically stabilized ferroliquids, Int. J. Eng. Sci. 19, 303 (1981).
  31. K. R. Schumacher, I. Sellien, G. S. Knoke, T. Cadar, and B. A. Finlayson, Experiment and simulation of laminar and turbulent ferrofluid pipe flow in an oscillating magnetic field, Phys. Rev. E 67, 026308 (2003).
  32. K. R. Schumacher, J. J. Riley, and B. A. Finlayson, Homogeneous turbulence in ferrofluids with a steady magnetic field, J. Fluid Mech. 599, 1 (2008).
  33. K. R. Schumacher, J. J. Riley, and B. A. Finlayson, Effects of an oscillating magnetic field on homogeneous ferrofluid turbulence, Phys. Rev. E 81, 016317 (2010).
  34. K. R. Schumacher, J. J. Riley, and B. A. Finlayson, Turbulence in ferrofluids in channel flow with steady and oscillating magnetic fields, Phys. Rev. E 83, 016307 (2011).
  35. S. Altmeyer, Y. Do, and Y. C. Lai, Transition to turbulence in Taylor-Couette ferrofluidic flow, Sci. Rep. 5, 10781 (2015).
  36. S. Galtier, S. V. Nazarenko, A. C. Newell, and A. Pouquet, A weak turbulence theory for incompressible magnetohydrodynamics, J. Plasma Phys. 63, 447 (2000).
  37. F. Boyer and E. Falcon, Wave turbulence on the surface of a ferrofluid in a magnetic field, Phys. Rev. Lett. 101, 244502 (2008).
  38. S. Dorbolo and E. Falcon, Wave turbulence on the surface of a ferrofluid in a horizontal magnetic field, Phys. Rev. E 83, 046303 (2011).
  39. E. A. Kochurin, Numerical simulation of the wave turbulence on the surface of a ferrofluid in a horizontal magnetic field, J. Magn. Magn. Mater. 503, 166607 (2020).
  40. K. Raj, B. Moskowitz, and R. Casciari, Advances in ferrofluid technology, J. Magn. Magn. Mater. 149, 174 (1995).
  41. S. M. Snyder, T. Cadar, and B. A. Finlayson, Finite element model of magnetoconvection of a ferrofluid, J. Magn. Magn. Mater. 262, 269 (2003).
  42. L. Sha, Y. Ju, H. Zhang, and J. Wang, Experimental investigation on the convective heat transfer of Fe3O4/water nanofluids under constant magnetic field, Appl. Therm. Eng. 113, 566 (2017).
  43. M. Kole and S. Khandekar, Engineering applications of ferrofluids: A review, J. Magn. Magn. Mater. 537, 168222 (2021).
  44. B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech. 40, 753 (1970).
  45. B. U. Felderhof and H. J. Kroh, Hydrodynamics of magnetic and dielectric fluids in interaction with the electromagnetic field, J. Chem. Phys. 110, 7403 (1999).
  46. S. Mouraya and S. Banerjee, Determination of energy flux rate in homogeneous ferrohydrodynamic turbulence using two-point statistics, Phys. Rev. E 100, 053105 (2019).
  47. M. Mortensen and H. P. Langtangen, High performance Python for direct numerical simulations of turbulent flows, Comput. Phys. Commun. 203, 53 (2016).
  48. T. Gotoh, D. Fukayama, and T. Nakano, Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation, Phys. Fluids 14, 1065 (2002).
  49. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
  50. M. A. Taylor, S. Kurien, and G. L. Eyink, Recovering isotropic statistics in turbulence simulations: The Kolmogorov 4/5th law, Phys. Rev. E 68, 026310 (2003).
  51. R. Ferrand, Multi-scale compressible turbulence in astrophysical plasmas viewed through theoretical, numerical and observational methods, Earth and Planetary Astrophysics, Université Paris-Saclay, 2021.
  52. L. Danaila, F. Anselmet, T. Zhou, and R. A. Antonia, A generalization of Yaglom's equation which accounts for the large-scale forcing in heated decaying turbulence, J. Fluid Mech. 391, 359 (1999).
  53. S. Banerjee, A. Halder, and N. Pan, Universal turbulent relaxation of fluids and plasmas by the principle of vanishing nonlinear transfers, Phys. Rev. E 107, L043201 (2023).
  54. N. Pan, S. Banerjee, and A. Halder, Universal relaxation of turbulent binary fluids, Commun. Phys. 7, 4 (2024).
  55. S. Banerjee and A. Halder, Fundamental units of triadic interactions in Hall magnetohydrodynamic turbulence: How far can we go? Phys. Plasmas 31, 062301 (2024).
  56. S. Mouraya and S. Banerjee, Temperature evolution equation of a compressible turbulent ferrofluid, Phys. Fluids 35, 015120 (2023).

Outline

Information

Phys. Rev. Fluids 9, 094604– Published 13 September, 2024
  • Received 7 May 2024
  • Accepted 29 August 2024
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