In this article, we propose a modified FxLMS (Filtered least mean square algorithm) fluid flow control model of a linear convectively unstable disturbance over a flat plate in one dimensional sense. In many flow situations, we need to suppress the flow instabilities as it causes unnecessary drag and noise due toenhancement of perturbation amplitude, whereas in few other flow situations, we deliberately need the flow perturbations in an enhanced desired permissible set level to obtain a desired mixing or controlled enhancement of heat transfer through the fluids. We propose a one dimensional modified FxLMS flow control model through which we can achieve controlled attenuation as well as enhancement in the amplitude of fluid perturbation with single additional gain in the control model. This method is widely used in noise control and signal processing areas. We adopt the similar modeling approach to confront the problem. One dimensional linearized Kuramoto– Sivashinsky equation model (KS equation) is used to model the fluid flow. We make use of model-free adaptive method to further apply it for fluids flows. A structurally modified adaptive algorithm incorporating FxLMS is designed and tested to get the desired set level of fluctuations in the fluid flow. In this control model, with a setvalue of additional gain, we achieve cancellation, attenuation, enhancement, and neutralization of perturbation amplitude downstream of the fluid flow in one dimensional sense. With the fast Fourier transformation (FFT) analysis, we have also observed that, the modified FxLMS fluid flow adaptive control model attenuates/enhances the set of troubling frequencies accordingly.

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 eigenfrequencyof 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 pseudospectrumof 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_x = 1:66, k_z = 0) with near critical Reynolds no = 250 and diverging angle α = 1°, theresonant peak is observed at ω = 0:7653. For 3D diverging JH flow case (at wavenumbers k_x = 0, k_z = 1) with Reynolds no = 250 and diverging angle α = 1°, the resonance occurs at ω = 0:0102, having comparatively higher peak. Whereas for converging JH flow (α = -0:005°), Re = 9000, k_x = 0 and k_z = 2, the eigen-frequency ω = 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.