Nonlinear stability results for plane Couette and Poiseuille flows

We prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less than ${\mathrm{Re}}_{\mathrm{Orr}}(2\pi /(\lambda sin\theta ))/sin\theta$ when a perturbation is a tilted perturbation in the direction $x^{\prime }$ which forms an angle $\theta \in (0,\pi /2]$ with the direction $\mathbf{i}$ of the basic motion and does not depend on $x^{\prime }$. ${\mathrm{Re}}_{\mathrm{Orr}}$ is the critical Orr-Reynolds number for spanwise perturbations which is computed for wave number $2\pi /(\lambda sin\theta )$, with $\lambda$ being any positive wavelength. By taking the minimum with respect to $\lambda$, we obtain the critical energy Reynolds number for a fixed inclination angle and any wavelength: for plane Couette flow, it is ${\mathrm{Re}}_{\mathrm{Orr}}=44.3/sin\theta$, and for plane Poiseuille flow, it is ${\mathrm{Re}}_{\mathrm{Orr}}=87.6/sin\theta$ (in particular, for $\theta =\pi /2$ we have the classical values ${\mathrm{Re}}_{\mathrm{Orr}}=44.3$ for plane Couette flow and ${\mathrm{Re}}_{\mathrm{Orr}}=87.6$ for plane Poiseuille flow). Here the nondimensional interval between the planes bounding the channel is $[-1,1]$. In particular, these results improve those obtained by Joseph, who found for streamwise perturbations a critical nonlinear value of 20.65 in the plane Couette case, and those obtained by Joseph and Carmi who found the value 49.55 for plane Poiseuille flow for streamwise perturbations. If we fix some wavelengths from the experimental data and the numerical simulations, the critical Reynolds numbers that we obtain are in a very good agreement both with the the experiments and the numerical simulation. These results partially solve the Couette-Sommerfeld paradox.

Figure 1

Plane Couette energy Orr-Reynolds number $\overline{R}=R_c$ as a function of the wave number $a$, for Eq. (10) with nonslip boundary conditions. The absolute minimum is $R_c=44.3$ and it is achieved when the wave number $a_c=1.9$. Here the channel has gap 2 and $z\in [-1,1]$.

Figure 2

Plane Poiseuille energy Orr-Reynolds number $\overline{R}=R_c$ as function of the wave number $a$, for Eq. (10) with nonslip boundary conditions. The absolute minimum is $R_c=87.6$ and it is achieved when the wave number $a_c=2.09$. Here the channel has gap 2 and $z\in [-1,1]$.

Figure 3

Laminar flows in a horizontal channel. The direction of motion is that of the $x$ axis. The direction of the $x^{\prime }$ axis forms an angle $\theta$ with the direction of the $x$ axis.

Figure 4

Couette energy Orr-Reynolds numbers given by formula (25): we plot the five critical values reported in Sec. 4 from (i) to (v), as indicated here by the different points. We have also added the continuous curve that for fixed ratio $\frac{a}{sin\theta }=\frac{2\pi }{\lambda sin\theta }$, gives ${\mathrm{Re}}_{\mathrm{Orr}}\left(\frac{2\pi }{\lambda sin\theta }\right)$. On the curve, we have reported our corresponding critical Reynolds values.

Figure 5

Comparison of our energy Orr-Reynolds numbers $\overline{R}=R_c$ as a function of the wavelength $\lambda$ and the values obtained by [16], formula (4.2), $\mathrm{Re}=\frac{\pi \lambda }{sin{24}^{\circ }}$. The dashed line represents $R_c=\frac{\pi \lambda }{sin{24}^{\circ }}$; the continuous line represents our critical Reynolds values $\overline{R}=R_c$ as a function of $\lambda$.