Fast computation of optimal disturbances for duct flows with a given accuracy

A. V. Boiko ; Yu. M. Nechepurenko ; M. Sadkane

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 11, pp. 2017-2027

Résumé

This work is devoted to the numerical analysis of small flow disturbances, i.e. velocity and pressure deviations from the steady state, in ducts of constant cross sections. The main emphasis is put on the disturbances causing the most kinetic energy density growth, the so-called optimal disturbances, whose knowledge is important in laminar-turbulent transition and robust flow control investigations. Numerically, this amounts to computing the maximum amplification of the 2-norm of a matrix exponential $\exp\{tS\}$ for a square matrix $S$ at $t\geq0$. To speed up the computations, we propose a new algorithm based on low-rank approximations of the matrix exponential and prove that it computes the desired amplification with a given accuracy. We discuss its implementation and demonstrate its efficiency by means of numerical experiments with a duct of square cross section.