Stability of plane Couette flow with constant wall transpiration

Plane Couette flow with constant wall transpiration, i.e., constant blowing from below and constant suction from above, is a solution of the Navier-Stokes equations in terms of exponentials. It is characterized by two Reynolds numbers Re and ${\text{Re}}_V$, where Re parametrizes the moving wall and ${\text{Re}}_V$ describes the influence of the transpiration. For this flow the modified Orr-Sommerfeld equation admits one of the very few exact solutions in terms of hypergeometric functions. Based on this, we solve the stability problem, though for the main part focus on temporal stability. For small wall-transpiration rates up to ${\text{Re}}_V\simeq 6.71$, the flow remains unconditionally stable, analogous to the classical Couette flow for arbitrary Reynolds numbers Re. By further increasing ${\text{Re}}_V$ an instability sets in and a minimum critical Reynolds number of $\text{Re}=668350.491$ is reached at ${\text{Re}}_V=9.799$. Hence, around this point, the destabilizing effect of blowing outweighs the stabilizing effect of suction. By further increasing the transpiration rate beyond this point, the corresponding critical Reynolds number Re increases again and continues to grow. This limiting case is accompanied by the development of a strong boundary-layer-like velocity profile near the upper wall and the flow transitions to the asymptotic suction boundary layer (ASBL). Thus, the present analysis comprises the whole range from classical Couette flows extended by transpiration to the ASBL, which is known to have a strongly stabilizing effect.

Figure 1

Dimensionless velocity profile of PCFT with wall-transpiration rates from ${\text{Re}}_V=0$ (pure PCF) to ${\text{Re}}_V=50$.

Figure 2

Dimensionless velocity profile of the ASBL with the displacement thickness ${\delta }_1$ and wall-normal constant suction $V_0$.

Figure 3

(a)–(c) Different perspectives, projections, and parameter ranges of the neutral stability surface of PCFT as parametrized by the flow Reynolds number Re, the transpiration Reynolds number ${\text{Re}}_V$, and the streamwise wave number $\alpha$. (d) Neutral stability curve at ${\text{Re}}_V={\text{Re}}_{V,\text{cr}}$ together with contours of growth rates ${\omega }_i>0$.

Figure 4

Temporal and spatial spectra of PCFT at the globally critical values of ${\text{Re}}_{\text{cr}}=668350.491$ and ${\text{Re}}_{V,\text{cr}}=9.799$ for (a) streamwise wave number ${\alpha }_{\text{cr}}=1.320$ and (b) frequency ${\omega }_{\text{cr}}=1.123$.

Figure 5

Temporal spectra are displayed at fixed ${\text{Re}}_{\text{cr}}=668350.491, {\alpha }_{\text{cr}}=1.320$, and varying ${\text{Re}}_V$ for (a) ${\text{Re}}_V=5$, (b) ${\text{Re}}_V=15$, and (c) ${\text{Re}}_V=20$.

Figure 6

Plots of the (a) $u$ and (b) $v$ eigenfunctions parametrized by ${\text{Re}}_V$ at fixed $\text{Re}={\text{Re}}_{\text{cr}}$ and $\alpha ={\alpha }_{\text{cr}}$. The eigenfunctions are respectively normalized by the maxima of the $u$ components and were plotted employing the exact solution (6).