Receptivity and sensitivity analysis of Jeffery-Hamel flow

Abstract

In local stability framework, receptivity and sensitivity analysis for Jeffery-Hamel (JH) flow forconverging and diverging angles are presented here. The frequencies are pointed out, upon which the internal eigen-frequency of the system resonates with that of external forcing frequencies. This resonance is often characterized asa starting step of disturbance growth of internal disturbances influenced from external environment disturbance.Identifying and avoiding such frequencies in external disturbance environment apriori, can certainly help delay intransition process. A mathematical model as a harmonically driven input-output system is formulated (throughresolvent norm) to quantify the amplification of energy and identifying the resonant external frequencies of thesystem. Sensitivity analysis is also mapped by resolvent norm by highlighting the most sensitive eigenvalues in thepseudospectrum of the system. Numerical simulation is done for small angles of converging and diverging JH flow,for which parallel flow assumptions are also valid. For numerical discretization, Chebyshev spectral method isutilized. The wall normal direction were discretized at Chebyshev collocation points in order to achieve higheraccuracy. We have studied three different cases for near critical Reynolds number values. In 2D diverging JH flowcase (at wavenumberskx¼1:66,kz¼0) with near critical Reynolds no = 250 and diverging anglea¼1 , theresonant peak is observed atx¼0:7653. For 3D diverging JH flow case (at wavenumberskx¼0,kz¼1) withReynolds no = 250 and diverging anglea¼1 , the resonance occurs atx¼0:0102, having comparatively higherpeak. Whereas for converging JH flow (a¼ 0:005 ), Re¼9000,kx¼0 andkz¼2, the eigen-frequencyx¼0:0102 resonates with that of external frequency, with even higher magnitude as compared with that of bothdiverging JH flow cases. The JH flows are accompanied with single resonant peak, as compared with that of planePoiseuille flow. This qualitatively links the inherited better stability of the JH flows as compared to the planePoiseuille flow. These resonances could induce the starting step leading to transition in the flow.