Magnetohydrodynamic Turbulence: numerical simulation and study of the transition between two-dimensional and three-dimensional turbulence, with and without walls

• Applicant: Professor Dr. Alban Pothérat
• Subject Area: Fluid Mechanics
• Term: from 2005 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 13029236

• Final Report Year: 2010

Final Report Abstract

Looking back, this project has led to considerable achievements: a new type of spectral methods that relies on the actual physical properties of the flow has been developed. Its application to MHD turbulence in a 3D spatially periodic domain and to simple channel flow problems has not only demonstrated its cost-effectiveness in terms of CPU cost, but also proved that it could eliminate the problem of having to mesh fine boundary layers, a drastic limitation for traditional methods. We have also laid the mathematical foundations and algorithm for the implementation of this method to virtually any problem of fluid mechanics. A full numerical implementation of spectral methods based on the least dissipative modes is now mostly a question of programming and shall be the next natural step towards DNS of MHD channel flows at High Re and high Ha. An application to fund a 3 year project around this theme has been submitted to the Leverhulme Trust. In the longer run, this method can potentially be applied to the simulation of any complex flow in arbitrary geometry. This project was however not exclusively of algorithmic nature and the physical properties of our methods have allowed considerable progress in the study of MHD turbulence itself: the establishment of Kolmogorov-like laws for the size of the dissipative scales in terms of a generalised ”anisotropic wavenumber” λ opens a way to a new theoretical approach to these flows: like the Kolmogorov laws, the traditional laws of turbulence for the Power spectral density and energy transfer too can be expected to have an MHD counterpart expressed in terms of λ. This too is a promising direction for future theoretical investigation, in MHD and beyond since the λ-sequence can be found for many other types of flows. Clarifying the mechanisms of the transition between two and three dimensional MHD turbulence is still an on-going task. Now that the sensitivity to initial flow conditions and the central role of partial vortex pairing at low Ha have been singled out, further simulations are needed to understand the spectacular ”re-stabilisation” mechanism that gives birth to the steady inverted Y-shape 3D vortices we discovered experimentally and numerically. This shall be one of the aims of the proposal we submitted to the Leverhulme Trust. Numerical work towards this aim is currently being undertaken at the lowest values of Ha with help of the powerful Finite Volume Code we have developed in the last part of this project.

Publications

• Attractor dimension and determining modes in Low-Rm MHD Turbulence. Imperial College, London, UK 2007
  A. Pothérat
  
• DNS of MHD Turbulence Based on the Least Dissipative Modes. In: ICIAM conference Zurich, Switzerland 2007
  V. Dymkou and A. Pothérat
  
• DNS of MHD turbulence based on the least dissipative modes. In: International workshop on numerical simulations in MHD flows Karlsruhe, Germany 2007
  V. Dymkou and A. Pothérat
  
• DNS of MHD turbulence based on the least dissipative modes. In: Meeting of the P17 COST action ”MHD fundamentals” Coventry, UK 2007
  A. Pothérat and V. Dymkou
  
• DNS of MHD turbulence based on the least dissipative modes. In: Annual UKMHD meeting Saleford, UK 2008
  V. Dymkou and A. Pothérat
  
• DNS of MHD Turbulence Based on the Least Dissipative Modes. Proc. Appl. Math. Mech. 7 (1), 4140011–4140012 (2008)
  V. Dymkou and A. Pothérat
  
• Spectral Methods for Low-Rm MHD Turbulence. In: PAMIR conference pages 393–397 Ghiens, France 2008
  V. Dymkou and A. Pothérat
  
• Transition between two and three-dimensional MHD flows. In: 3rd international conference on Bifurcations and Instabilities in Fluid Dynamics Nottingham, UK 2009
  A. Pothérat, R. Klein and V. Dymkou
  
• Spectral Methods Based on the Least Dissipative Modes for Wall-Bounded MHD Turbulence. J. Theor. Comp. Fluid Mech. 23 (6), 535–555 (2009)
  V. Dymkou and A. Pothérat
  
• Transition between 2D and 3D MHD turbulence. In: Meeting of the P17 COST action Grenoble, France 2009
  R. Klein, A. Pothérat and V. Dymkou
  
• Numerical simulations of MHD flows in a duct past obstacles in a duct under externally applied magnetic field. PhD thesis, Coventry University, UK 2010
  V. Dousset
  
• Transition between 2D and 3D MHD flows. In: Annual UKMHD meeting Leeds, UK 2010
  R. Klein, A. Pothérat and V. Dymkou
  
• Transition between two and three-dimensional wall-bounded MHD flows. In: EUROMECH conference Bad Reichenhall, Germany 2010
  A. Pothérat, R. Klein and V. Dymkou