Enstrophy change of the Reynolds-Orr solution in channel flow

The plane Poiseuille flow is one of the elementary flow configurations. Although its laminar-turbulent transition mechanism has been investigated intensively in the last century, the significant difference in the critical Reynolds number between the experiments and the theory lacks a clear explanation. In this paper, an attempt is made to reduce this gap by analyzing the solution of the Reynolds-Orr equation. Recent published results have shown that the usage of enstrophy (the volume integral of the squared vorticity) instead of the kinetic energy as the norm of perturbations predicts higher Reynolds numbers in the two-dimensional case. In addition, other research show has shown an improvement of the original Reynolds-Orr energy equation using the weighted norm in a tilted coordinate system. In this paper the enstrophy is used in three dimensions combined with the tilted coordinate system approach. The zero-enstrophy-growth constraint is applied to the classical Reynolds-Orr equation, and then the solution is further refined in the tilted coordinate system. The results are compared to direct numerical simulations published previously.

Figure 1

(a) Schematic drawing of the domain and the base flow. (b) Schematic drawing of the perturbation in the original ($x,y,z$) and the tilted ($x^{\prime },y,z^{\prime }$) coordinate system. The wall-normal direction ($y$) is unchanged. The wave number of the perturbation points into $z^{\prime }$.

Figure 2

(a) The Reynolds number (original RO equation) and (b) the temporal growth rate of the enstrophy as a function of the streamwise ($\alpha$) and spanwise ($\beta$) wave numbers for the most critical perturbation according to the RO equation. The solid magenta line represents the zero growth of enstrophy. The black dashed line shows the perturbations where the angle between the wave-number vector and the streamwise direction is ${45}^{\circ }$. The most critical perturbation with zero enstrophy growth was found at $\alpha =1.32$, $\beta =1.78$, and ${\mathrm{Re}}_{\mathrm{crit}}=57.3$ according to the original RO equation and at $\alpha =1.13$, $\beta =1.38$, and ${\mathrm{Re}}_{\mathrm{crit}}^m=140.8$ according to the method of Falsaperla et al. [6].

Figure 3

The terms of the enstrophy change in the integrand of Eq. (10) normalized by the enstrophy in the case of the critical perturbations according to the Reynolds-Orr energy theory. (a) First and second terms: $1/s{\int }_{\mathcal{V}}-{\omega }_i{u}_j\frac{\partial {\mathrm{\Omega }}_i}{\partial x_j}d\mathcal{V}=1/s{\int }_{\mathcal{V}}{\omega }_i{\mathrm{\Omega }}_j\frac{\partial u_i}{\partial x_j}d\mathcal{V}$. (b) Fourth term: $1/s{\int }_{\mathcal{V}}{\omega }_i{\omega }_j\frac{\partial U_i}{\partial x_j}d\mathcal{V}$. (c) Fifth term: $1/s{\int }_{\mathcal{V}}\frac{1}{\mathrm{Re}}{\omega }_i\frac{{\partial }^2{\omega }_i}{\partial x_j^2}d\mathcal{V}$. In panel (b), the dashed line represents the zero enstrophy growth.

Figure 4

The critical Reynolds number as a function of the wavelength in the case of perturbations with zero change of enstrophy. The continuous line is the classical solution and the dashed line is the improved solution, according to Eq. (A1). The two crosses represent the minima.

Figure 5

The critical perturbation field according to the original theory, in which case the enstrophy change is zero. From left to right: Streamwise component, wall-normal component, and spanwise component. The blue curves are the real part and the red curves are the imaginary part of the eigenfunction.

Figure 6

The components of the critical perturbation field according to the original theory at $z=0$ ($\mathrm{Re}=57.3$, $\alpha =1.32$, $\beta =1.78$): (a) the streamwise component, (b) the wall-normal component, and (c) the spanwise component.