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Mathematical formulation of mode-to-mode energy transfers and energy fluxes in compressible turbulence

Dhananjay Singh*, Harshit Tiwari†, and Lekha Sharma‡

Mahendra K. Verma§

  • *Contact author: sdhanjay@iitk.ac.in
  • †Contact author: tharshit@iitk.ac.in
  • ‡Contact author: lekhas@iitk.ac.in
  • §Contact author: mkv@iitk.ac.in
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Phys. Rev. Fluids 10, 114609 – Published 14 November, 2025

DOI: https://doi.org/10.1103/nmly-ql47

Abstract

Understanding compressible turbulence is critical for modeling atmospheric, astrophysical, and engineering flows. However, compressible turbulence poses a more significant challenge than incompressible turbulence. We present a novel mathematical framework to compute mode-to-mode energy transfer rates and energy fluxes for compressible flows. The formalism captures detailed energy conservation within triads and allows decomposition of transfers into rotational, compressive, and mixed components, providing a clear picture of energy exchange among velocity and internal energy modes. We also establish analogies with incompressible hydrodynamic and magnetohydrodynamic flows, highlighting the framework's universality in studying energy transfers.

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References (54)

  1. S. K. Lele, Compressibility effects on turbulence, Annu. Rev. Fluid Mech. 26, 211 (1994).
  2. B. W. Carroll and D. A. Ostlie, An Introduction to Modern Astrophysics, 2nd ed. (Cambridge University Press, Cambridge, 2017).
  3. H. C. Spruit, Å. Nordlund, and A. M. Title, Solar convection, Annu. Rev. Astron. Astrophys. 28, 263 (1990).
  4. G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics (Cambridge University Press, Cambridge, 2006).
  5. A. Fujisawa, Review of plasma turbulence experiments, Proc. Jpn. Acad., Ser. B 97, 103 (2021).
  6. T. von Kármán, Compressibility effects in aerodynamics, J. Spacecr. Rockets 40, 992 (2003).
  7. M. Lesieur, Turbulence in Fluids (Springer-Verlag, Dordrecht, 2008).
  8. A. N. Kolmogorov, Dissipation of energy in locally isotropic turbulence, Dokl. Acad. Nauk. SSSR 32, 16 (1941).
  9. A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Acad. Nauk. SSSR 30, 301 (1941).
  10. R. H. Kraichnan, The structure of isotropic turbulence at very high Reynolds numbers, J. Fluid Mech. 5, 497 (1959).
  11. G. Dar, M. K. Verma, and V. Eswaran, Energy transfer in two-dimensional magnetohydrodynamic turbulence: Formalism and numerical results, Phys. D 157, 207 (2001).
  12. M. K. Verma, Statistical theory of magnetohydrodynamic turbulence: Recent results, Phys. Rep. 401, 229 (2004).
  13. R. H. Kraichnan, Inertial-range transfer in two-and three-dimensional turbulence, J. Fluid Mech. 47, 525 (1971).
  14. J. A. Domaradzki and R. S. Rogallo, Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence, Phys. Fluids A 2, 413 (1990).
  15. Y. Zhou, Degrees of locality of energy transfer in the inertial range, Phys. Fluids A 5, 1092 (1993).
  16. Y. Zhou, Interacting scales and energy transfer in isotropic turbulence, Phys. Fluids A 5, 2511 (1993).
  17. M. K. Verma, A. Ayyer, O. Debliquy, S. Kumar, and A. V. Chandra, Local shell-to-shell energy transfer via nonlocal interactions in fluid turbulence, Pramana 65, 297 (2005).
  18. Y. Zhou, Turbulence theories and statistical closure approaches, Phys. Rep. 935, 1 (2021).
  19. Y. Zhou, Hydrodynamic Instabilities and Turbulence: Rayleigh–Taylor, Richtmyer–Meshkov, and Kelvin–Helmholtz Mixing (Cambridge University Press, Cambridge, 2024).
  20. S. Kida and S. A. Orszag, Energy and spectral dynamics in forced compressible turbulence, J. Sci. Comput. 5, 85 (1990).
  21. S. Kida and S. A. Orszag, Energy and spectral dynamics in decaying compressible turbulence, J. Sci. Comput. 7, 1 (1992).
  22. J. Wang, Y. Yang, Y. Shi, Z. Xiao, X. T. He, and S. Chen, Cascade of kinetic energy in three-dimensional compressible turbulence, Phys. Rev. Lett. 110, 214505 (2013).
  23. W. Schmidt and P. Grete, Kinetic and internal energy transfer in implicit large-eddy simulations of forced compressible turbulence, Phys. Rev. E 100, 043116 (2019).
  24. Y. Sakurai and T. Ishihara, Direct numerical simulations of compressible turbulence in a periodic box: Effect of isothermal assumptions on turbulence statistics, Phys. Fluids 36, 085152 (2024).
  25. S. Jagannathan and D. A. Donzis, Reynolds and Mach number scaling in solenoidally-forced compressible turbulence using high-resolution direct numerical simulations, J. Fluid Mech. 789, 669 (2016).
  26. S. Sarkar, The pressure–dilatation correlation in compressible flows, Phys. Fluids A 4, 2674 (1992).
  27. H. Miura and S. Kida, Acoustic energy exchange in compressible turbulence, Phys. Fluids 7, 1732 (1995).
  28. D. S. Praturi and S. S. Girimaji, Effect of pressure-dilatation on energy spectrum evolution in compressible turbulence, Phys. Fluids 31, 055114 (2019).
  29. J. P. John and D. A. Donzis, Strong evidence for universality in homogeneous compressible turbulence, Phys. Fluids 36, 106121 (2024).
  30. G. P. Zank and W. H. Matthaeus, The equations of nearly incompressible fluids. I - hydrodynamics, turbulence, and waves, Phys. Fluids A 3, 69 (1991).
  31. M. K. Verma, Energy Transfers in Fluid Flows: Multiscale and Spectral Perspectives (Cambridge University Press, Cambridge, 2019).
  32. P. Sagaut and C. Cambon, Homogeneous Turbulence Dynamics, 2nd ed. (Cambridge University Press, Cambridge, 2018).
  33. J. Wang, M. Wan, S. Chen, and S. Chen, Kinetic energy transfer in compressible isotropic turbulence, J. Fluid Mech. 841, 581 (2018).
  34. C. Luo, L. Fang, J. Fang, H. Xu, A. Pumir, and P.-F. Yang, The contribution of dilatational motion to energy flux in homogeneous compressible turbulence, J. Fluid Mech. 1018, A7 (2025).
  35. J. P. Graham, R. Cameron, and M. Schüssler, Turbulent small-scale dynamo action in solar surface simulations, Astrophys. J. 714, 1606 (2010).
  36. Y. Yang, M. Wan, W. H. Matthaeus, and S. Chen, Energy budget in decaying compressible MHD turbulence, J. Fluid Mech. 916, A4 (2021).
  37. H. Aluie, Compressible turbulence: The cascade and its locality, Phys. Rev. Lett. 106, 174502 (2011).
  38. H. Aluie, Scale decomposition in compressible turbulence, Phys. D 247, 54 (2013).
  39. G. Falkovich, I. Fouxon, and Y. Oz, New relations for correlation functions in Navier–Stokes turbulence, J. Fluid Mech. 644, 465 (2010).
  40. S. Banerjee and S. Galtier, A Kolmogorov-like exact relation for compressible polytropic turbulence, J. Fluid Mech. 742, 230 (2014).
  41. A. G. Kritsuk, R. Wagner, and M. L. Norman, Energy cascade and scaling in supersonic isothermal turbulence, J. Fluid Mech. 729, R1 (2013).
  42. P. Grete, B. W. O'shea, K. Beckwith, W. Schmidt, and A. Christlieb, Energy transfer in compressible magnetohydrodynamic turbulence, Phys. Plasmas 24, 092311 (2017).
  43. P. K. Kundu, I. M. Cohen, and D. R. Dowling, Fluid Mechanics, 6th ed. (Academic Press, San Diego, 2015).
  44. O. Debliquy, M. K. Verma, and D. Carati, Energy fluxes and shell-to-shell transfers in three-dimensional decaying magnetohydrodynamic turbulence, Phys. Plasmas 12, 042309 (2005).
  45. M. K. Verma, Variable energy flux in turbulence, J. Phys. A: Math. Theor. 55, 013002 (2022).
  46. M. K. Verma, M. K. Sharma, S. Chatterjee, and S. Alam, Variable energy fluxes and exact relations in magnetohydrodynamics turbulence, Fluids 6, 225 (2021).
  47. C.-S. Yih, A transformation for non-homentropic flows, with an application to large-amplitude motion in the atmosphere, J. Fluid Mech. 9, 68 (1960).
  48. J. L. Rodriguez Azara and G. Emanuel, Compressible rotational flows generated by the substitution principle, Phys. Fluids 31, 1058 (1988).
  49. Y. Zhou, Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I, Phys. Rep. 720-722, 1 (2017).
  50. Y. Zhou, J. D. Sadler, and O. A. Hurricane, Instabilities and mixing in inertial confinement fusion, Annu. Rev. Fluid Mech. 57, 197 (2025).
  51. H. Tiwari, L. Sharma, and M. K. Verma, Compressible turbulent convection at very high Rayleigh numbers, Int. J. Heat Mass Transfer 242, 126821 (2025).
  52. P. Wesseling, Principles of Computational Fluid Dynamics, 1st ed. (Springer-Verlag, Berlin, Heidelberg, 2000).
  53. C. Ouyang, S. He, Q. Xu, Y. Luo, and W. Zhang, A McCormack-TVD finite difference method to simulate the mass flow in mountainous terrain with variable computational domain, Comput. Geosci. 52, 1 (2013).
  54. D. Singh, H. Tiwari, L. Sharma, and M. K. Verma, Data for Mathematical formulation of mode-to-mode energy transfers and energy fluxes in compressible turbulence [Data set], Zenodo, (2025) https://doi.org/10.5281/zenodo.17403535.

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Phys. Rev. Fluids 10, 114609– Published 14 November, 2025
  • Received 4 May 2025
  • Accepted 14 October 2025
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