Temporal behaviour of global perturbations in compressible axisymmetric flows with free boundaries

Abstract

The dynamics of small global perturbations in the form of a linear combination of a finite number of non-axisymmetric eigenmodes is studied in the two-dimensional approximation. The background flow is assumed to be an axisymmetric perfect fluid with adiabatic index γ = 5/3 rotating with a power law angular velocity distribution Ω ∝ r −q , 1.5 < q < 2.0, confined by free boundaries in the radial direction. The substantial transient growth of acoustic energy of optimized perturbations is discovered. An optimal energy growth G is calculated numerically for a variety of parameters. Its value depends essentially on the perturbation azimuthal wavenumber m and increases for higher values of m. The closer the rotation profile to the Keplerian law, the larger growth factors can be obtained but over a longer time. The highest acoustic energy increase found numerically is of order ∼ 10 2 over ∼ 6 typical Keplerian periods. Slow neutral eigenmodes with corotation radius beyond the outer boundary mostly contribute to the transient growth. The revealed linear temporal behaviour of perturbations may play an important role in angular momentum transfer in toroidal flows near compact relativistic objects.

A study of the transient dynamics of perturbations in Keplerian discs using a variational approach

“…This is done without referring to modal solutions what can be quite an involved task, especially in complex flows. For example, Zhuravlev & Shakura (2009) and Razdoburdin & Zhuravlev (2012) treated the optimals in the form of finite linear combinations of neutral acoustic modes in a quasi-Keplerian torus. The obtained optimal perturbation corresponds to a wave packet localised initially in the vicinity of the outer boundary of torus and moving towards the inner boundary.…”

“…This can be done, for example, by means of the singular value decomposition of the relevant matrix exponential projected onto the orthonormal basis, look Schmid & Henningson (2001). Butler & Farrell (1992) use this method to study the optimal transient growth in classical Couette, Poiseuille and Blasius boundary layer flows, Mukhopadhyay et al (2005) use it to investigate Keplerian flow in local approximation and Zhuravlev & Shakura (2009) and, subsequently, Razdoburdin & Zhuravlev (2012) apply it to study the global optimal growth in quasi-Keplerian torus with free boundaries. However, this strategy entails all technical difficulties related to calculation of modes and eigen-frequencies.…”

Global transient dynamics of three-dimensional hydrodynamical disturbances in a thin viscous accretion disk

“…In lieu of this, Ionnaou & Kakouris [8] reported in their study that stochastic forcing was found to lead to persistent activity with angular momentum transported outward. More recently, Zhuravlev & Shakura [31] applied optimal perturbation strategy to two-dimensional sub-Keplerian toroidal configurations. They found that optimal perturbations giving rise to substantial TG are composed of certain combinations of non-axisymmetric eigenmodes.…”

Optimal growth of small perturbations in thin gaseous disks

“…Matching (14) for the expansion of the WKBJ solution (11) in the vicinity of x 1 and x 2 gives the zero phase ϕ 0 = −nπ/2 and the dispersion relation…”