Transient energy growth of channel flow with cross-flow

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 function G(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 crossflow 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 crossflow 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 nonnormal character of the modified Orr-Sommerfeld operator tends to a high sensitivity of pseudo-spectra structures.


Introduction
The eigenvalue analysis is able to predict instability behavior for some fluid systems, such as Rayleigh-Bénard convection and Taylor-Couette flow [1]. Several theoretical papers show, for all Reynolds numbers the Couette and Poiseuille flows are unconditionally stable [2]. However, this approach does not correspond to experimental results for other problem [3,4], in which the transition to turbulence is observed at 350 370  Re for Couette flow [3]. 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. Fransson and Alfredsson [5] carried out a linear modal stability analysis of the plane Poiseuille flow with cross-flow. 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. In addition, they showed the stabilizing and the destabilizing effect of a uniform cross flow.  The governing equations that describe the flow of an incompressible Newtonian fluid are the Navier-Stokes and continuity equations. In non-dimensional form, they are written as: -Continuity equation:

Physical problem and
Boundary conditions: where V*, P*, respectively, the velocity and the pressure.
Using reference variables respectively, length, pressure, velocity and time, ( 0 U represents the maximum streamwise velocity), as follows non-dimensional variables: The basic velocity in non-dimensional form can be written as:

Linear stability analysis
For the perturbed flow, the velocity and pressure fields are expressed as the sum of a steady and a perturbation field, i.e.
The disturbance quantities u', ' v p' and are assumed periodic and of the form: The modified Orr-Somm their with equation erfeld boundary conditions are: For non-modal approach, stability is redefined in a broader sense as the response to general input variables, including initial conditions, impulsive and continuous external excitations, the p be also can seudospectra defined in other equivalent ways [7].  Figure. 2 presents the effect of the cross-flow Reynolds number on the pseudospectral boundaries and also the spectra in the ω for α= 1.0 and Re =6000. It is observed that the number of eigenvalues on each branch depends on the cross-flow Reynolds number. In addition, the sensibility of the vicinity of the intersection is dependent on the cross-flow Reynolds number. In figure. 3 exhibits the evolution of the energy growth function G(t) as a function the time for different values of Rc at α= 1.0 and Re =6000. It can be seen that; the transient energy of the optimal perturbation grows rapidly at short times when the cross-flow rate is low or strong and decline slowly at long times when the crossflow is intermediate. Figure.  Results show that the region of high maximum energy growth expanded with (Rc , Re). this result in also given in [5] using the modal stability.

Conclusion
The present work has considered the effect of a uniform cross flow (injection/ suction) on the transient energy growth of a plane Poiseuille flow is investigated. Nonmodal linear stability analysis is carried out to determine the two-dimensional optimal perturbations for maximum growth. 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 nonnormal character of the modified Orr-Sommerfeld operator tends to a high sensitivity of pseudo-spectra structures.