TY - JOUR

T1 - On linear stability of shear flows of an ideal stratified fluid

T2 - 4th International Conference on Supercomputer Technologies of Mathematical Modelling, SCTeMM 2019

AU - Gavrilieva, A. A.

AU - Gubarev, Yu G.

PY - 2019/12/13

Y1 - 2019/12/13

N2 - The results, that obtained by the spectral method with use of integral relations for the problem of linear stability of steady-state shear plane-parallel flows of an inviscid stratified incompressible fluid in the gravity field with respect to plane perturbations in the Boussinesq approximation and without it, are specified, complemented and developed by the most powerful analytical method of the modern mathematical theory of hydrodynamic stability-the second (or direct) Lyapunov method. In both case, the new analytical method made it possible to prove that given steady-state flows of stratified fluid are absolutely unstable in theoretical sense with respect to small plane perturbations and to obtain the sufficient conditions for practical linear instability of considered flows. The illustrative analytical examples of given steady-state flows and small plane perturbations as normal waves imposed on them are constructed. Using the asymptotic method, it is proved that constructed perturbations grow in time irrespective of the fact whether the Miles-Howard and the Miles-type theorems are valid or not.

AB - The results, that obtained by the spectral method with use of integral relations for the problem of linear stability of steady-state shear plane-parallel flows of an inviscid stratified incompressible fluid in the gravity field with respect to plane perturbations in the Boussinesq approximation and without it, are specified, complemented and developed by the most powerful analytical method of the modern mathematical theory of hydrodynamic stability-the second (or direct) Lyapunov method. In both case, the new analytical method made it possible to prove that given steady-state flows of stratified fluid are absolutely unstable in theoretical sense with respect to small plane perturbations and to obtain the sufficient conditions for practical linear instability of considered flows. The illustrative analytical examples of given steady-state flows and small plane perturbations as normal waves imposed on them are constructed. Using the asymptotic method, it is proved that constructed perturbations grow in time irrespective of the fact whether the Miles-Howard and the Miles-type theorems are valid or not.

UR - http://www.scopus.com/inward/record.url?scp=85078193667&partnerID=8YFLogxK

U2 - 10.1088/1742-6596/1392/1/012006

DO - 10.1088/1742-6596/1392/1/012006

M3 - Conference article

AN - SCOPUS:85078193667

VL - 1392

JO - Journal of Physics: Conference Series

JF - Journal of Physics: Conference Series

SN - 1742-6588

IS - 1

M1 - 012006

Y2 - 19 June 2019 through 21 June 2019

ER -