On the Stability of Magnetohydrodynamic Shear Flows: Characterization of Critical Pressure-Velocity-Magnetic-Field Interactions
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Date
2018-10-25
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Abstract
Plasma shear flows are abundant in nature and frequently encountered in engineering applications.
The stability characteristics of plasma shear flows are of much fundamental interest. Shear
flows are susceptible to various algebraic and modal instabilities, i.e., velocity perturbations grow
as a polynomial or an exponential function of time. It is well known in literature that background
magnetic field applied along the flow direction and compressibility have a stabilizing influence on
these shear instabilities. In this dissertation, a systematic investigation of the stabilization mechanisms
is performed. This dissertation consists of three studies, each addressing a different type of
free shear layer: Study 1 - homogeneously sheared flows in the incompressible regime; Study 2 - inhomogeneously sheared mixing layers in the incompressible regime; Study 3 - inhomogeneously
sheared planar jets in the compressible regime. The common theme of all the studies investigate
the nature of pressure-velocity-magnetic-field interactions that influence stabilizing mechanisms.
For the case of homogeneous shear investigated in the first study, velocity perturbations in the
absence of the magnetic field are susceptible to algebraic instability, i.e., kinetic energy contained
in the perturbations (k) grows as, k ~ O(t^n). The stabilizing influence of magnetic field strength
and perturbation orientation (β) on the instability is characterized using linear analysis and direct
numerical simulations. Linear analysis indicates that the perturbation growth is dependent on the
parameter, RvA ≡ VvAk/S, where, VvA, k and S are the Alfvén wave speed, initial wavenumber and
mean flow shear, respectively. Analytical solutions for various perturbation energies at extreme
RvA regimes – RvA» 1 and RvA « 1 – are derived and compared to numerical simulations. The
behavior of perturbations at different RvA regimes and β values is also explicated using numerical
simulations.
In the second study, a tangent hyperbolic profile is chosen for the mean velocity field. Owing
to the presence of an inflection point in the profile, the flow field is subjected to Kelvin-Helmholtz
(KH) instability leading to exponential growth of perturbations, i.e., k ~ O(e^t). In the absence
of any magnetic field (hydrodynamic limit), the precursor vortices form and roll up into a primary
vortex. The primary vortex further entrains fluid leading to the onset of nonlinear asymptotic stage
and formation of secondary vortex bands. We investigate the linear and nonlinear effects of magnetic
field on this three-stage evolution of KH instability. Flow field features such as circulation,
gauge pressure and perturbation energies are utilized to delineate the parameter space into strong,
weak and intermediate magnetic-field stabilization regimes. The mechanisms of magnetic field
stabilization in each of the three regimes is investigated using direct numerical simulations.
In the third study, the evolution of pressure-, kinetic- and magnetic-perturbation energies for
the case of compressible magnetohydrodynamic (MHD) planar jets is investigated. A streamwise
background magnetic field is again applied. The change in the nature of interactions between
velocity and magnetic fields due to compressibility is established using linear analysis. Numerical
simulations of single mode and random, isotropic initial perturbations are performed to examine
these new agencies of exchange and their subsequent effect on the overall stability of the flow field.
The findings of this dissertation are expected to further our understanding of various compressible
and magnetic field mechanisms and their roles in perturbation evolution. This will aid in the
development of closure models for MHD shear flows which could be used for designing efficient
plasma propulsion engines and flow control devices.
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Keywords
Magnetohydrodynamics, Shear flows