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Scale by scale analysis of magnetoconvection with uniform wall-normal and wall-parallel magnetic fields at low magnetic Reynolds number

Jake Ineson, Aleksander Dubas, Alex Skillen

Abstract

Rayleigh-Bénard convection under an imposed inductionless magnetic field is analysed statistically from the perspective of single-point and multi-scale energy budgets. The data is obtained from direct numerical simulations with a Rayleigh number of $10^6$, a Prandtl number of $1$ and Hartmann numbers of $0$, $20$, $40$ and $80$. Wall-parallel and wall-normal magnetic fields are considered as two separate cases. The initial analysis focuses qualitatively on the influence of the magnetic field upon the coherent structures. A central contribution of this work is the interpretation of these structural modifications through magnetohydrodynamically modified turbulent kinetic energy budgets. For example, in the wall-normal case, the thinning of the thermal plumes can be attributed to the damping of the pressure-diffusion mechanisms due to the Lorentz dissipation. In the wall-parallel configuration, Joule dissipation induces a pressure-strain redistribution mechanism that preferentially transfers kinetic energy from the wall-normal velocity component to the field-perpendicular, wall-parallel velocity component but less so to the field-parallel velocity component. This description is then extended to scale-space by considering budgets relating second- and third-order structure functions. Here, the anisotropy is accounted for by analysing directional structure functions. Despite the anisotropy, the Lorentz force appears as an isotropic sink damping intermediate and large scales of motion. The result of this is a lack of transfer between scales of motion and hence a flow with suppressed small-scale turbulence. These results establish a link between qualitative observations and long-term energy balances, providing new insight into magnetoconvective turbulence and informing future modelling and theoretical approaches to such flows.

Scale by scale analysis of magnetoconvection with uniform wall-normal and wall-parallel magnetic fields at low magnetic Reynolds number

Abstract

Rayleigh-Bénard convection under an imposed inductionless magnetic field is analysed statistically from the perspective of single-point and multi-scale energy budgets. The data is obtained from direct numerical simulations with a Rayleigh number of , a Prandtl number of and Hartmann numbers of , , and . Wall-parallel and wall-normal magnetic fields are considered as two separate cases. The initial analysis focuses qualitatively on the influence of the magnetic field upon the coherent structures. A central contribution of this work is the interpretation of these structural modifications through magnetohydrodynamically modified turbulent kinetic energy budgets. For example, in the wall-normal case, the thinning of the thermal plumes can be attributed to the damping of the pressure-diffusion mechanisms due to the Lorentz dissipation. In the wall-parallel configuration, Joule dissipation induces a pressure-strain redistribution mechanism that preferentially transfers kinetic energy from the wall-normal velocity component to the field-perpendicular, wall-parallel velocity component but less so to the field-parallel velocity component. This description is then extended to scale-space by considering budgets relating second- and third-order structure functions. Here, the anisotropy is accounted for by analysing directional structure functions. Despite the anisotropy, the Lorentz force appears as an isotropic sink damping intermediate and large scales of motion. The result of this is a lack of transfer between scales of motion and hence a flow with suppressed small-scale turbulence. These results establish a link between qualitative observations and long-term energy balances, providing new insight into magnetoconvective turbulence and informing future modelling and theoretical approaches to such flows.
Paper Structure (13 sections, 47 equations, 33 figures, 3 tables)

This paper contains 13 sections, 47 equations, 33 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Governing equations and numerical method
  3. Flow snapshots and coherent structures
  4. Distribution of TKE and temperature variance in physical space
  5. Turbulent Kinetic Energy budget
  6. Temperature variance budget
  7. Multiscale analysis of TKE and temperature variance
  8. Scale by scale budgets of $\delta u^2$ and $\delta \theta^2$
  9. Isotropic scale by scale analysis of wall-normal field cases
  10. Anisotropic scale by scale analysis of wall-parallel field cases
  11. Conclusion
  12. Acknowledgements
  13. Derivation of scale-energy equation under buoyancy and Lorentz forces

Figures (33)

  • Figure 1: Simulation domain setup where the walls are held at $T_h=0.5$ and $T_c = -0.5$. Both fields along the $y$ and $x$ axis are considered in this setup.
  • Figure 2: Instantaneous temperature plots in the $y-z$ plane with a wall-normal magnetic field. (Top to bottom) cases S, D, E and F respectively. $\textbf{g} \downarrow \textbf{B} \uparrow$.
  • Figure 3: Instantaneous temperature plots in the $y-z$ plane with a wall-parallel magnetic field. (Top to bottom) cases S, D, E and F respectively. $\textbf{g} \downarrow \textbf{B} \otimes$.
  • Figure 4: Instantaneous $w$ plots in the $y-z$ plane with a wall-parallel magnetic field. (Top to bottom) cases S, A, B and C respectively. $\textbf{g} \downarrow \textbf{B} \otimes$.
  • Figure 5: Instantaneous $u$ plots in the $y-z$ plane with a wall-parallel magnetic field. (Top to bottom) cases S, A, B and C respectively. $\textbf{g} \downarrow \textbf{B} \otimes$.
  • ...and 28 more figures