The present study is concerned with the stability of a flow of viscous conducting liquid driven by a pressure gradient between two parallel walls in the presence of a transverse magnetic field, which is investigated using a Chebyshev collocation method. This magnetohydrodynamic counterpart of the classic plane Poiseuille flow is generally known as Hartmann flow. Although the magnetic field has a strong stabilizing effect, the turbulence is known to set in this flow similarly to its hydrodynamic counterpart well below the threshold predicted by the linear stability theory. Such a nonlinear transition to turbulence is thought to be mediated by unstable equilibrium flow states which may exist in addition to the base flow. Firstly, the weakly nonlinear stability analysis carried out in this study shows that Hartmann flow is subcritically unstable to small finite-amplitude disturbances regardless of the magnetic field strength. Secondly, two-dimensional nonlinear travelling wave states are found to exist in Hartmann flow at substantially subcritical Reynolds numbers starting from Ren = 2939 without the magnetic field and from Ren ∼ 6.50 × 103Ha in a sufficiently strong magnetic field defined by the Hartmann number Ha. Although the latter value is by a factor of seven lower than the linear stability threshold Rel ∼ 4.83×104Ha and by almost a factor of two lower than the value predicted by the mean-field (monoharmonic) approximation, it is still more than an order of magnitude higher than the experimentally observed value for the onset of turbulence in this flow. Three-dimensional disturbances are expected to bifurcate from these two-dimensional travelling waves or infinity and to extend to significantly lower Reynolds numbers. The by-product of this study are two developments of numerical techniques for linear and weakly nonlinear stability analysis. Firstly, a simple technique for avoiding spurious eigenvalues is developed for the solution of the Orr-Sommerfeld equation. Secondly, an efficient numerical method for evaluating Landau coefficients which describe small amplitude states in the vicinity of the linear stability threshold is introduced. The method differs from the standard approach by applying the solvability condition to the discretised rather than the continuous problem.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:683891 |
| Date | January 2013 |
| Creators | Hagan, J. |
| Publisher | Coventry University |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://curve.coventry.ac.uk/open/items/cc5976b0-419c-4944-a2ff-3af446a03d05/1 |
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