Receptivity and sensitivity analysis of Jeffery-Hamel flow

In local stability framework, receptivity and sensitivity analysis for Jeffery-Hamel (JH) flow for converging 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 as a 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 in transition process. A mathematical model as a harmonically driven input-output system is formulated (through resolvent norm) to quantify the amplification of energy and identifying the resonant external frequencies of the system. Sensitivity analysis is also mapped by resolvent norm by highlighting the most sensitive eigenvalues in the pseudospectrum 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 is utilized. The wall normal direction were discretized at Chebyshev collocation points in order to achieve higher accuracy. We have studied three different cases for near critical Reynolds number values. In 2D diverging JH flow case (at wavenumbers k(alpha) = 1.66, k(z) = 0) with near critical Reynolds no = 250 and diverging angle alpha = 1 degrees, the resonant peak is observed at omega = 0.7653. For 3D diverging JH flow case (at wavenumbers k(alpha) = 0, k(z) = 1) with Reynolds no = 250 and diverging angle alpha = 1 degrees, the resonance occurs at omega = 0.0102, having comparatively higher peak. Whereas for converging JH flow (alpha = -0.005 degrees), Re = 9000, k(alpha) = 0 and k(z) = 2, the eigen-frequency omega = 0.0102 resonates with that of external frequency, with even higher magnitude as compared with that of both diverging JH flow cases. The JH flows are accompanied with single resonant peak, as compared with that of plane Poiseuille flow. This qualitatively links the inherited better stability of the JH flows as compared to the plane Poiseuille flow. These resonances could induce the starting step leading to transition in the flow.