Mathematical and Computational Engineering Stream

Origin of hydrodynamical instability in rotating shear flows

Faculty: Banibrata Mukhopadhyay (Physics), O. N. Ramesh (Aerospace Eng.)

Summary:

Origin of hydrodyamical instability and subsequent turbulence in shear flows is a centuty old problem, particularly the ones which are Rayleigh stable. For example, in laboratory, plane Couette flow becomes turbulent at Reynolds number (Re) as low as 350, but based on linear stability analysis it is always stable. Similar issues are there in Astrohysical sites. Astrophysical accretion disk, which is a rotating shear flow, is understood to be unstable and tubulent based on observed data, but Rayleigh stable. Although in presence of magnetic fields, particularly when the underlying flow is ionized, instability can be argued, e.g. by Magneto-Rotational Instability, many accreting systems are reasonably cold and neutral in charge, and laboratory flows are in general nonmagnetic. How to resolve this mismatch between results in experiments/observations and theory?

One of the ideas floating around in last few decades is that based on non-normality. The plane Couette and Poiseuille flows as well as accreion disk flow are non-normal. That means, underlying perturbation modes are not linearly independent. Therefore, though they are decaying with time eventually, hence linearly stable, at an intermediate time, depending on suitable initial conditions, they may produce a huge energy growth depending on Re: e.g. for plane Couette flow the peak growth Gr ~ Re2, for the Keplerian disk Gr ~ Re2/3. The huge Gr at large Re (which is definitely the case for astrophysical flows) argues for emergence of nonlinearity and turbulence. However, often the question arises against sustanance of such a plausible turbulence. To encounter it, another related plausibility is the forced flow, e.g. the force emerged from thermal fluctuation, fluid-grain interactions etc. This force could not only lead to the modified energy dispersion relation but in the first place affect the background equilibrium flow.

The plan of the proposed project is to use above mentioned ideas and techniques to enlighten the true origin of hydrodynamic instability in laboratory shear flows and astrophysical flows, which are otherwise linearly stable. As in presence of magnetic field instability may kick in, one question is if pure hydrodynamics could produce perturbation growth rate same as that due to magnetic field. The proposed Ph.D. project will aim to set up first a simplified model of forced Navier-Stokes equation and its linear and nonlinear solutions, without and with magnetic fields.

As to the astrophysical accretion disk, the aim would be to model a small part of the disk and its stability analysis. Research at the Aerospace Engineering (AE), IISc, led by Prof. O. N. Ramesh has focussed on many aspects of linear stability in the traditional local sense (where a parallel flow assumotion is made for the base flow) and a more general global stability framework where such a parallel flow assumption is not made. This has brought important connections between various stability theories such as modal local and non-local theories and the global stability theories for a variety of flow configurations such as a boundary layer (of both separated and attached flows), wall jet and plane jets.

Research at the Department of Physics (PH), IISc, led by Prof. Banibrata Mukhopadhyay, aims at understanding origin of pure hydrodynamic instability and turbulence, as seen in experiments or observed data, in apparently Rayleigh stable flow. Over the years, he has used severeal techniques towards this mission and partly resolved the issues, but more to follow. The Ph.D. student should have a good mathematical and preferably computational backgrounds. Knowledge of fluid dynamics and numerical solutions of PDEs will be an advantage. The student will have access to the HPC machines in SERC, IISc.

Selected publications:

  1. Ghosh, S., and Mukhopadhyay, B., Forced linear shear flows with rotation: rotating Couette-Poiseuille flow, its stability and astrophysical implications – The Astrophysical Journal 922, 161, 2021; arXiv:2107.04012.
  2. Ghosh, S., and Mukhopadhyay, B., Origin of hydrodynamic instability from noise: from laboratory flow to accretion disk – Physical Review Fluids 6, 013903, 2021; arXiv:2012.13417.
  3. Mukhopadhyay, B., Mathew, R. and Raha, S., Growing pseudo-eigenmodes and positive logarithmic norms in rotating shear flows – New Journal of Physics 13, 023029, 2011; arXiv:1101.4608.
  4. Mukhopadhyay, B., Afshordi, N. and Narayan, R., Bypass to turbulence in hydrodynamic accretion disks: An eigenvalue analysis – The Astrophysical Journal 629, 383, 2005; astro-ph/0412193.
  5. Diwan, S. and Ramesh, O. N., Relevance of local parallel theory to the linear stability of laminar separation bubbles, Journal of Fluid Mechanics 698, 468, 2012.
  6. Varghese, J. and Ramesh, O. N., Linear Instability of a plane wall jet, To be submitted to Physical Review Fluids.