Turbulent plane Couette flow with a roughened wall

In this direct numerical simulation (DNS) study, a fully developed turbulent plane Couette flow (pCf), where the bottom wall is roughened and the top smooth wall moves with a constant velocity $U_w$, is reported. For roughening, roughness elements in the form of square ribs of height $r=0.2h$ (where $h$ is channel half-height) are mounted only on the bottom wall with streamwise pitch separations of $5r$ and $10r$. The Reynolds number, based on half the velocity difference between the walls and $h$, is set to 1300. Upon roughening a wall, roughness would generate vortical structures near the vicinity of the roughness elements and enhance turbulence locally. We report that in the current problem, turbulence kinetic energy (TKE) is enhanced near both the rough and the smooth walls. The major factor aiding this process is the blockage effect offered by the rib roughness, which enhances the skin friction on both walls, and therefore the average friction Reynolds number is increased by 25% and 33% in the cases with pitches $5r$ and $10r$, respectively. In addition, in each of the rough pCf cases, streamwise variations in the mean flow, Reynolds stresses and the TKE budget terms are confined within the channel centerline, thereby indicating streamwise homogeneity above the channel centerline.

Figure 1

Schematic diagram of the rough plane Couette flow, where the bottom wall is stationary and the top wall moves in the streamwise direction with a constant velocity $U_w$. Square ribs are added to the stationary wall with a streamwise pitch separation of $s/r=5$ and $s/r=10$.

Figure 2

Two-point correlation of the velocity fluctuations at $z/h=1$ (center-plane) from the rough pCf cases (d-type roughness and k-type roughness). (a) Streamwise correlation; (b) spanwise correlation. Note that only half the domain length is shown here.

Figure 3

(a) Ratio of the mean grid width to the Kolmogorov length scale $\eta$ for the rough pCf cases plotted at midcavity (I) and midrib (II) locations and compared with those of the smooth pCf. The dash-dotted vertical line indicates the location of the roughness crest ($z/h=0.2$). (b) Profiles of root-mean-square velocities shown up to $z/h=1$ for smooth pCf: —, present smooth pCf; $\square$, Bech et al. [16]; $\circ$, Hu et al. [42]; $\mathrm{\Delta }$, Holstad et al. [20].

Figure 4

Instantaneous turbulent kinetic energy $\frac{1}{2}(u^{\prime 2}+v^{\prime 2}+w^{\prime 2})$ contours in an $xy$-plane for d-type roughness (c, d) and k-type roughness (e, f) compared with those of smooth pCf (a, b). Contours near the rough wall at $z/h=0.25$ (c, e) and smooth wall at $z/h=1.9$ (d, f). Contour values are normalized by $U_w^2$.

Figure 5

Instantaneous turbulent kinetic energy for d-type roughness (b) and k-type roughness (c) compared with those of smooth pCf (a). Contours shown in an $xz$-plane at the midspan location $y/h=8.4$. Contour values are normalized by $U_w^2$. The flow is from left to right and for the purpose of clarity only half of the streamwise domain is shown.

Figure 6

Instantaneous negative ${\lambda }_2$ vortices normalized by $\frac{U_w^4}{{\nu }^2}$ ($-9\times {10}^{-8}$), colored with nondimensional wall-normal distance $z/h$. The color varies from blue (near wall) to red (near channel centerline). (a, b) Smooth pCf; (c, d) d-type roughness; (e, f) k-type roughness. (a, c, e) Lower half of the domain shown from the bottom stationary wall ($z/h=0$) to the channel centerline ($z/h=1$), (b, d, f) upper half of the domain shown from the channel centerline ($z/h=1$) to the top moving wall ($z/h=2$).

Figure 7

Instantaneous skin-friction coefficient ($C_f$) for (a) smooth pCf; (b) d-type roughness; (c) k-type roughness. Contours shown in $xy$-plane at the smooth moving wall ($z/h=2$).

Figure 8

Mean streamlines between two consecutive ribs. (a) d-type roughness; (b) k-type roughness. Here $x$ is the streamwise direction and $z$ is the wall-normal direction. Flow is from left to right. $s$ refers to pitch separation.

Figure 9

Profiles of (a) mean streamwise velocity and (b) mean velocity slope for d-type roughness and k-type roughness cases plotted at midcavity (I) and midrib (II) locations. Results are compared with those of smooth pCf.

Figure 10

Contours of mean wall-normal velocity ($\overline{w}$) for (a) d-type roughness; (b) k-type roughness.

Figure 11

Mean secondary flow streamlines superimposed with the mean wall-normal velocity contours in the background: (a) smooth pCf, (b) d-type roughness, (c) k-type roughness. The contours are shown in $yz$-plane at the midcavity (I) location. Here $y$ is the spanwise direction.

Figure 12

Profiles of Reynolds normal stresses (a–c) and Reynolds shear stress (d) for the d-type roughness and k-type roughness plotted across the channel at midcavity (I) and midrib (II) locations and compared with those of the smooth pCf. Reynolds stresses are normalized by $u_{\tau }^2$ where $u_{\tau }$ is the global wall friction velocity of each case. The dash-dotted and dotted vertical lines indicate the rib crest and the upper limit of the roughness sublayer ($z=5r$), respectively.

Figure 13

Profiles of the turbulent kinetic energy ($k$) for the d-type roughness and k-type roughness, plotted at the midcavity (I) location and compared with the smooth pCf. The data are normalized by (a) $U_w^2$ (absolute scaling) and (b) $u_{\tau }^2$ (local scaling). The dash-dotted vertical line indicates the location of the rib crests.

Figure 14

Contours of turbulent kinetic energy ($k$) normalized by $u_{\tau }^2$ where $u_{\tau }$ is the global wall friction velocity of each case. (a) d-type roughness and (b) k-type roughness.

Figure 15

Contours of Reynolds normal stresses normalized by $u_{\tau }^2$ where $u_{\tau }$ is the global wall friction velocity of each case. (a, b) $\overline{u^{\prime }{u}^{\prime }}$, (c, d) $\overline{v^{\prime }{v}^{\prime }}$, and (e, f) $\overline{w^{\prime }{w}^{\prime }}$. (a, c, e) d-type roughness and (b, d, f) k-type roughness.

Figure 16

Contours of (a, b) production rate ($P$), (c, d) dissipation rate ($\varepsilon$), and (e, f) advection rate ($C$) normalized by $\frac{u_{\tau }^4}{\nu }$ where $u_{\tau }$ is the global wall friction velocity of each case. (a, c, e) d-type roughness and (b, d, f) k-type roughness. The dotted lines are negative values.

Figure 17

Contours of (a, b) turbulent diffusion rate ($T$), (c, d) viscous diffusion rate ($V$), and (e, f) pressure diffusion rate ($G$) normalized by $\frac{u_{\tau }^4}{\nu }$ where $u_{\tau }$ is the global wall friction velocity of each case. (a, c, e) d-type roughness and (b, d, f) k-type roughness. The dotted lines are negative values.

Figure 18

Contours of secondary production rate $-\overline{u^{\prime }{u}^{\prime }}\frac{d\overline{u}}{dx}$ normalized by $\frac{u_{\tau }^4}{\nu }$ where $u_{\tau }$ is the global wall friction velocity of each case. (a) d-type roughness and (b) k-type roughness. The dotted lines are negative values.

Figure 19

Ratio of production rate to the dissipation rate for d-type roughness and k-type roughness plotted at the midcavity (I) location and compared with those with the smooth pCf.