Polynomial SOS in fluid dynamics: a review with a look ahead

S. Chernyshenko, P. J. Goulart, D. Huang and A. Papachristodoulou

@article{CGHP:2014,
  author = {S. Chernyshenko and P. J. Goulart and D. Huang and A. Papachristodoulou},
  title = {Polynomial SOS in fluid dynamics: a review with a look ahead},
  journal = {Philisophical Transactions of the Royal Society A},
  year = {2014},
  volume = {372},
  number = {2020},
  url = {http://dx.doi.org/10.1098/rsta.2013.0350},
  doi = {10.1098/rsta.2013.0350}
}

The first part of this paper reviews the application of the sum-of-squares-of-polynomials technique to the problem of global stability of fluid flows. It describes the known approaches and the latest results, in particular, obtaining for a version of the rotating Couette flow a better stability range than the range given by the classic energy stability method. The second part of this paper describes new results and ideas, including a new method of obtaining bounds for time-averaged flow parameters illustrated with a model problem and a method of obtaining approximate bounds that are insensitive to unstable steady states and periodic orbits. It is proposed to use the bound on the energy dissipation rate as the cost functional in the design of flow control aimed at reducing turbulent drag.