Effect of plate permeability on nonlinear stability of the asymptotic suction boundary layer

The nonlinear stability of the asymptotic suction boundary layer is studied numerically, searching for finite-amplitude solutions that bifurcate from the laminar flow state. By changing the boundary conditions for disturbances at the plate from the classical no-slip condition to more physically sound ones, the stability characteristics of the flow may change radically, both for the linearized as well as the nonlinear problem. The wall boundary condition takes into account the permeability $\widehat{K}$ of the plate; for very low permeability, it is acceptable to impose the classical boundary condition ($\widehat{K}=0$). This leads to a Reynolds number of approximately ${\text{Re}}_c=54400$ for the onset of linearly unstable waves, and close to ${\text{Re}}_g=3200$ for the emergence of nonlinear solutions [F. A. Milinazzo and P. G. Saffman, J. Fluid Mech. 160, 281 (1985); J. H. M. Fransson, Ph.D. thesis, Royal Institute of Technology, KTH, Sweden, 2003]. However, for larger values of the plate's permeability, the lower limit for the existence of linear and nonlinear solutions shifts to significantly lower Reynolds numbers. For the largest permeability studied here, the limit values of the Reynolds numbers reduce down to ${\text{Re}}_c=796$ and ${\text{Re}}_g=294$. For all cases studied, the solutions bifurcate subcritically toward lower Re, and this leads to the conjecture that they may be involved in the very first stages of a transition scenario similar to the classical route of the Blasius boundary layer initiated by Tollmien-Schlichting (TS) waves. The stability of these nonlinear solutions is also investigated, showing a low-frequency main unstable mode whose growth rate decreases with increasing permeability and with the Reynolds number, following a power law ${\text{Re}}^{-\rho }$, where the value of $\rho$ depends on the permeability coefficient $\widehat{K}$. The nonlinear dynamics of the flow in the vicinity of the computed finite-amplitude solutions is finally investigated by direct numerical simulations, providing a viable scenario for subcritical transition due to TS waves.

Figure 1

A schematic of the plate showing the main idea behind the imposed boundary conditions at $y=0$. Variables in capital letters refer to the base flow.

Figure 2

The energy $E$ of the wavy part of the nonlinear solutions, see Eq. (8), mapped out in Re for a streamwise wave number of $\alpha =0.154$ and eight values of $a$. The classical wall condition $a=0$ corresponds to a bifurcation at ${\text{Re}}_c\approx 54000$ for ${\alpha }_c=0.1555$ [16, 36]. With increasing $a, {\text{Re}}_c$ is reduced. This is also the case for the nonlinear lower limit, cf. Table 1. Proof of convergence is shown by the individual symbols; for $a<0.02$ we use as a base resolution $(NX,NY)=(14,140)$ and confirmed with (18,160). The $a=0.02$ solution is mapped out using a truncation of (10,160) and confirmed with (14,200), while the $a=0.08$, 0.12 and 0.20 solutions are represented by a resolution of (14,200) and confirmed with (18,220).

Figure 3

Perturbation energy $E$ of the wavy part mapped out in $\alpha$ for $a=0$ (22,100), $3.85\times {10}^{-4}$ (26,120), $5\times {10}^{-3}$ (18,160), 0.01 (18,160), and 0.02 (14,200) at $\text{Re}=3178$ (see the legends). The values in the parentheses after each value of $a$ refer to the truncation $(NX,NY)$ used. The selected Re lies in the neighborhood of the ${\text{Re}}_g$ of the $a=0$ solution as evidenced by its narrow envelope in $\alpha$. Table 1 shows the values of ${\text{Re}}_g$ for all $a$'s presented in the plot. The inset shows the energy for $a=0.20$ at $\text{Re}=305$ and 320 using a truncation of (14,180) and 3178 using (12,240). The lower values of Re shown in the smaller plot are close to the limit ${\text{Re}}_g$ for $a=0.20$ given in Table 1.

Figure 4

Energy growth extracted from a linearized three-dimensional DNS initialized with several nonlinear Tollmien-Schlichting waves. The solutions selected are the upper branch ones with $\alpha =0.20$ and $\text{Re}=3178$ (see Fig. 3). The inset shows the growth rate $\sigma$ for several values of $a$ ranging from 0 to $0.2$, whereas the main plot shows the time evolution of the three-dimensional energy for $a=0$, 0.01, 0.05, 0.1, and 0.2.

Figure 5

Variation with Re of the growth rate $\sigma$ of the unstable mode for the upper branch TWs with $\alpha =0.20, a=0$ (blue), and $a=0.01$ (green). The scaling law is represented by a dashed line.

Figure 6

Colored isocontours of the streamwise instantaneous velocity on the plane $z=2\pi$ and isosurfaces of the streamwise (red and green) and the spanwise (black and white) perturbation to the TW extracted at $t=2000$ from a linearized DNS initialized by the upper branch solution with $\alpha =0.20, \text{Re}=3178, a=0$ (left), and $a=0.01$ (right). The gray isocontours in the right frame represent the wall-normal perturbation at the wall.

Figure 7

Colored isocontours of the streamwise instantaneous velocity on the plane $z=2\pi$ and isosurfaces of the streamwise (light blue) and the wall-normal (black and white) perturbation to the base flow extracted at $t=300$, 350, 400, and 500 (from left top to right bottom) from a DNS initialized by the upper branch solution with $\alpha =0.20, \text{Re}=3178$, and $a=0$.

Figure 8

Flow trajectory plotted on an $E_v-E_w$ projection of the phase space, extracted from a DNS initialized with the upper branch solutions with $\alpha =0.20, \text{Re}=3178, a=0.0$ [red (gray) line], and $a=0.1$ (black line). The red dots indicate the state at $t=300$, 400, and 500 (for increasing dot size), whereas the black ones indicate the state at $t=450$, 550, and 650 (for increasing dot size).

Figure 9

Colored isocontours of the streamwise instantaneous velocity on the plane $z=2\pi$ and isosurfaces of the streamwise (light blue) and the wall-normal (black and white) perturbation to the base flow extracted at $t=450$, 500, 550, and 650 (from left top to right bottom) from a DNS initialized by the upper branch solution with $\alpha =0.20, \text{Re}=3178$, and $a=0.1$. The gray isocontours at $y=0$ represent the wall-normal perturbation at the wall.