Linear instability of channel flow with microgroove-type anisotropic superhydrophobic walls

We study the temporal linear instability of channel flow subject to a tensorial slip boundary condition that models the slip effect induced by microgroove-type superhydrophobic surfaces. The microgrooves are not necessarily aligned with the driving pressure gradient. Pralits et al. [Phys. Rev. Fluids 2, 013901 (2017)] investigated the same problem and reported that a proper tilt angle of the microgrooves about the driving pressure gradient can reduce the critical Reynolds number and that the flow with a single superhydrophobic wall is much more unstable or less stable than that with two superhydrophobic walls. In contrast, we show that the lowest critical Reynolds number is always realized with two superhydrophobic walls, and we obtain critical Reynolds numbers significantly lower than that reported. Besides, we show that the critical Reynolds number can be further reduced by increasing the anisotropy in the slip length. As the tilt angle changes, there appears to be a strong correlation between the strength of the instability and the magnitude of the cross-flow component of the base flow incurred by the tilt angle. In case the tilt angles of the microgrooves differ on the two walls, the critical Reynolds number increases as the difference in the tilt angles increases, i.e., two superhydrophobic walls with parallel microgrooves give the lowest critical Reynolds number. The results are informative for designing the microgroove-type wall texture to introduce instability at low Reynolds number channel flow, which may be of interest for enhancing mixing or heat transfer in small flow systems where turbulence cannot be triggered.

Figure 1

The effect of the tilt angle $\theta$ on the base flow. (a) The maximum of $W\left(y\right)$ over $y$. Note that for the one-SH-wall case, $W\left(y\right)$ maximizes at the slippery wall at $y=-1$. (b) The streamwise velocity at the bottom slippery wall $U(y=-1)$ and at the channel center $U(y=0)$. The slip parameters are ${\lambda }^{\parallel }=0.155$ and ${\lambda }^{\perp }={\lambda }^{\parallel }/2$.

Figure 2

The effects of the tilt angle on the distribution of eigenvalue ${\omega }_i$ in the wave number plane. Slip lengths are set to be ${\lambda }^{\parallel }=0.155$ and ${\lambda }^{\perp }={\lambda }^{\parallel }/2$. (a) One-SH-wall with $\theta =0$ and $\mathrm{Re}=10000$. (b) Two-SH-wall with $\theta =0$ and $\mathrm{Re}=3\times {10}^5$. (c) One-SH-wall with $\theta =\pi /4$ and $\mathrm{Re}=1000$. (d) Two-SH-wall with $\theta =\pi /4$ and $\mathrm{Re}=1000$. The bold line encloses the linearly unstable region in the wave-number plane.

Figure 3

The critical Reynolds number as a function of the tilt angle $\theta$ for the two-SH-wall channel. (a) ${\lambda }^{\parallel }=0.05$, (b) ${\lambda }^{\parallel }=0.155$. Our calculations are shown as red squares and the results of Ref. [19] are shown as blue circles for comparison.

Figure 4

(a), (b) The critical Reynolds number as a function of the tilt angle $\theta$ for the one-SH-wall channel. Our calculations are shown as green triangles and the results of Ref. [19] are shown as blue circles for comparison. (c) The critical Reynolds number for two-SH-wall and one-SH-wall cases in the present work are compared.

Figure 5

(a) The most unstable wave numbers at the critical Reynolds number as functions of tilt angle $\theta$ for two-SH-wall and one-SH-wall cases. (b) A close-up at small $\theta$.

Figure 6

The flow structure of the leading eigenmode for (a), (b) the one-SH-wall and (c), (d) two-SH-wall cases with $\theta =0$ at the respective critical Reynolds number. In panels (a) and (b), the bottom wall is slippery, the Reynolds number is $\mathrm{Re}=7417$, and the wave numbers are $(\alpha ,\beta )=(0.69,0)$. In panels (c) and (d), the Reynolds number is $\mathrm{Re}=2.3\times {10}^5$ and the wave numbers are $(\alpha ,\beta )=(0.19,-0.65)$. In the top row, color shows the spanwise vorticity in the x-y plane at $z=0$ and vectors show the inplane velocities. In the bottom row, color shows the spanwise vorticity on the bottom wall. The magnitudes of the shown quantities are arbitrary.

Figure 7

The flow structure of the leading eigenmode for (a), (b) the one-SH-wall and (c) (d) two-SH-wall cases with $\theta =\pi /4$ at the respective critical Reynolds number. In panels (a) and (b), the bottom wall is slippery, the Reynolds number is $\mathrm{Re}=486$, and the critical wave numbers are $(\alpha ,\beta )=(0.04,-1.23)$. In panels (c) and (d), the Reynolds number is $\mathrm{Re}=429$ and the critical wave numbers are $(\alpha ,\beta )=(0.12,-1.63)$. The color shows the streamwise velocity in the top row and wall-normal velocity in the bottom row, and vectors show the in-plane velocities. The magnitudes of the shown quantities are arbitrary.

Figure 8

The phase speeds of the leading eigenmode for the ${\lambda }^{\parallel }=0.155$ case. The corresponding Reynolds numbers and wave numbers are shown in Figs. 4 and 5. (a) Streamwise phase speed. (b) Spanwise phase speed.

Figure 9

The critical Reynolds number as a function of ${\lambda }^{\parallel }$ with $\theta =0$ and $\theta =\pi /4$: (a) one-SH-wall case, (b) two-SH-Wall case. The results of Ref. [19] are also plotted for comparison.

Figure 10

The critical (a) streamwise and (b) spanwise wave numbers as functions of ${\lambda }^{\parallel }$ with $\theta =0$ and $\theta =\pi /4$.

Figure 11

The (a) streamwise and (b) spanwise phase speeds of the most unstable eigenmodes as functions of ${\lambda }^{\parallel }$ with $\theta =0$ and $\theta =\pi /4$.

Figure 12

(a) The critical Reynolds number as a function of the tilt angle $\theta$ for the one-SH-wall (dashed lines) and two-SH-wall (solid lines) with ${\lambda }^{\parallel }=0.1$ and ${\lambda }^{\parallel }/{\lambda }^{\perp }=2$, $10/3$, and 10. (b) The minimum critical Reynolds number over $\theta$ as a function of the slip length ratio ${\lambda }^{\parallel }/{\lambda }^{\perp }$.

Figure 13

(a) The growth rate $\gamma$ of the most unstable mode for the two-SH-wall case with different tilt angle $\theta$ on the two walls. Slip lengths are ${\lambda }^{\parallel }=0.155$ and ${\lambda }^{\perp }={\lambda }^{\parallel }/2$. The angle is fixed at $\theta ={51}^{\circ }$ on the top wall and is varied on the bottom wall. The Reynolds numbers is fixed at $\mathrm{Re}=419$. (b) The base flow profiles $U$ and $W$ in case of $\theta ={51}^{\circ }$ on top wall and $\theta =-{51}^{\circ }$ on the bottom wall. Slip lengths are the same as in panel (a).

Figure 14

(a)–(c) The eigenspectra calculated using the $\mathit{u}-p$ formulation (circles) and $v-\eta$ formulation (crosses). The real and imaginary parts of the eigenvalues are denoted ${\omega }_r$ and ${\omega }_i$, respectively. The parameters are detailed in Table 2. Panels (a)–(c) correspond to Cases (1)–(3), respectively. In panel (d), using the DNS formulation, the time series of the kinetic energy ($KE$) of small perturbations in the three cases are shown. The $KE$ is normalized by its value at $t=0$. The exponential growth rate can be calculated by Eqs. (B1).

Figure 15

The neutral curve (bold line) in the $\alpha \text{−}Re$ plane for $\beta =-1.76$. The slip lengths are ${\lambda }^{\parallel }=0.05$ and ${\lambda }^{\perp }=0.025$ with $\theta =\pi /4$. The eigenvalue ${\omega }_i$ is plotted as the colormap.