Abstract
The effect of a uniform cross flow (injection/ suction) on the transient energy growth of a plane Poiseuille flow is investigated. Non-modal linear stability analysis is carried out to determine the two-dimensional optimal perturbations for maximum growth. The linearized Navier-Stockes equations are reduced to a modified Orr Sommerfeld equation that is solved numerically using a Chebychev collocation spectral method. Our study is focused on the response to external excitations and initial conditions by examining the energy growth functionG(t)and the pseudo-spectrum. Results show that, the transient energy of the optimal perturbation grows rapidly at short times and decline slowly at long times when the cross-flow rate is low or strong. In addition, the maximum energy growth is very pronounced in low injection rate than that of the strong one. For the intermediate cross-flow rate, the transient energy growth of the perturbation, is only possible at the long times with a very high-energy gain. Analysis of the pseudo-spectrum show that the non-normal character of the modified Orr-Sommerfeld operator tends to a high sensitivity of pseudo-spectra structures.
Highlights
The eigenvalue analysis is able to predict instability behavior for some fluid systems, such as RayleighBénard convection and Taylor-Couette flow [1]
The gap between the eigenvalue analysis and experiments leads to the emergence of a new theory called: theory of non-modal stability [6]
This, we motivate to reproduce the results of Fransson and Alfredsson [5] by using the non-modal approach, in which, we focus on the response to initial conditions by examining the pseudo-spectra structures and the transient energy growths
Summary
The eigenvalue analysis is able to predict instability behavior for some fluid systems, such as RayleighBénard convection and Taylor-Couette flow [1]. For all Reynolds numbers the Couette and Poiseuille flows are unconditionally stable [2]. This approach does not correspond to experimental results for other problem [3,4], in which the transition to turbulence is observed at 350 Re 370 for Couette flow [3]. The authors made corrections to the problems discussed in [3,4] and they proved that the stability of this problem depends on the choice of the velocity scale. They showed the stabilizing and the destabilizing effect of a uniform cross flow
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