Non-modal stability analysis of magneto-hydrodynamic flow in a single pipe

A complete understanding of the stability of fluid flows under varying magnetic field profiles is imperative for achieving control of plasma and operating fluids in the blankets of future fusion reactors. In this context, the primary objective of this study is to investigate the influence of varying magnetic profiles on the flow regime of a generic fluid, which is representative of both thermonuclear plasma and conductive fluids within a nuclear fusion reactor. To this aim in this work non-modal stability theory is adopted to perform stability analysis of a magneto-hydrodynamic (MHD) flow in an infinite circular pipe in order to study the effects of the magnetic field on the fluid dynamics of the pipe flow. In particular, the effects on the general stability of two magnetic field profiles are compared with the reference case of a pipe Poiseuille flow without magnetic field. Firstly, the classic modal stability technique is employed to study asymptotical stability. Then, non-modal stability analysis is applied to magneto-hydrodynamic pipe flow to study the system's response for a finite time immediately after a perturbation. Fourier–Chebyshev Petrov–Galerkin spectral method is used to compute the eigenvalues and pseudospectra of the weak formulation associated with the linearised system. Investigations on the dependence of spectra and transient growths on the specific magnetic profiles are conducted for different values of perturbation wave numbers. The obtained results show that in general the magnetic field has an effect of stabilization on the system, which depends on the specific magnetic profile considered. In addition, the non-modal stability analysis reveals that the inclusion of the magnetic field mitigates the effects of perturbations also in the short term, a phenomenon that cannot be seen using only modal stability analysis.