Control of optimal growth of instabilities in Jeffery-Hamel flow

Ravi Kant,Narayanan Vinod

doi:10.1063/1.5087432

Abstract

In this article, we present an active control of optimal growth of instabilities in Jeffery-Hamel (JH) flow through the standard wall transpiration control technique. The motivation behind this work is to delay the transition process in converging/diverging JH flows with small angles. We formulate the governing equations and the control model for the JH flow with the parallel flow approximations. In the non-modal stability framework, we have constructed a state space model which incorporates control actuation as periodic suction/blowing of fluid through walls (wall transpiration). The variational method is used to compute the optimal growth of the system. An optimal feedback control gain is obtained (assuming the knowledge of the full state) through linear quadratic regulator (LQR) and feed-backed to the system to suppress the maximum amplification of optimal growth of the instabilities in the flow. We found that the optimally amplified growth in both converging as well as diverging JH flow is reduced after applying controlled wall transpiration. The present study could be the starting step towards the control of non-parallel global instabilities.

Highlights

Jeffery Hamel (JH) flows are defined as the flow between two non-parallel walls with a source or a sink at the point of intersection of the walls
We have numerically investigated the optimal growth of disturbances in the JH flow at different Reynolds numbers and converging/diverging angles
We identify the most unstable wavenumber pair at a particular Reynolds number, the controller is designed to actuate the signal associated with the same wavenumber pair

Summary

Introduction

Jeffery Hamel (JH) flows are defined as the flow between two non-parallel walls with a source or a sink at the point of intersection of the walls It has gained attention in the past decades due to the wide range of applications of JH flows, which include aerospace, chemical, civil, environmental, mechanical, industrial and bio-mechanical engineering.[1,2] Such application include heat transfer and cold drawing applications, extrusion of molten materials through converging dies, control of liquid metal flows, crystal growth, design of medical diagnostic devices and pressure driven transport through converging/diverging channels.[2,3] It is desirable that flow should not undergo quick transition otherwise it may lead to unnecessary flow conditions. In order to overcome this, flow control is required.

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