Secondary flows in turbulent boundary layers over longitudinal surface roughness

Direct numerical simulations of turbulent boundary layers over longitudinal surface roughness are performed to investigate the impact of the surface roughness on the mean flow characteristics related to counter-rotating large-scale secondary flows. By systematically changing the two parameters of the pitch (P) and width (S) for roughness elements in the ranges of $0.57\le P/\delta \le 2.39$ and $0.15\le S/\delta \le 1.12$, where δ is the boundary layer thickness, we find that the size of the secondary flow in each case is mostly determined by the value of P − S, i.e., the valley width, over the ridge-type roughness. However, the strength of the secondary flows on the cross-stream plane relative to the flow is increased when the value of P increases or when the value of S decreases. In addition to the secondary flows, additional tertiary and quaternary flows are observed both above the roughness crest and in the valley as the values of P and S increase further. Based on an analysis using the turbulent kinetic energy transport equation, it is shown that the secondary flow over the ridge-type roughness is both driven and sustained by the anisotropy of turbulence, consistent with previous observations of a turbulent boundary layer over strip-type roughness [Anderson et al., J. Fluid Mech. 768, 316 (2015)]. Careful inspection of the turbulent kinetic energy budget reveals that the opposite rotational sense of the secondary flow between the ridge- and strip-type roughness elements is primarily attributed to the local imbalance of energy budget created by the strong turbulent transport term over the ridge-type roughness. The active transport of the kinetic energy over the ridge-type roughness is closely associated with the upward deflection of spanwise motions in the valley, mostly due to the roughness edge.

Figure 1

Schematic of a turbulent boundary layer with longitudinal surface roughness. Roughness elements with height H and width S are periodically arranged in the spanwise direction with pitch P. A step change from a smooth to a rough surface occurs at $80{\theta }_{in}$ downstream from the inlet, which is defined as $x/{\theta }_{in}=0$.

Figure 2

Variations of the spanwise-averaged (a) skin-friction drag ($C_f$), (b) form the drag ($P_d$) and (c) friction velocity ($U_{\tau }/U_{\infty }$) along the downstream direction.

Figure 3

Variations of the spanwise-averaged (a) boundary layer thickness ($\delta /{\theta }_{in}$), (b) displacement thickness (${\delta }^{*}/{\theta }_{in}$), (c) momentum thickness ($\theta /{\theta }_{in}$) and (d) shape parameter (H) along the downstream direction.

Figure 4

Variations of the spanwise-averaged mean velocity defect profiles along the downstream direction for P12S3.

Figure 5

Variation of the turbulent Reynolds stresses along the streamwise direction for P12S3. The line symbols are identical to those in Fig. 4.

Figure 6

Streamwise locations required to achieve new equilibrium states after a step change with increases of P and S, normalized by the inlet momentum thickness (${\theta }_{in}$).

Figure 7

(a) Isosurfaces of the phase-averaged mean-signed swirling strength ($\langle {\mathrm{\Lambda }}_{ci}\rangle {\theta }_{in}/U_{\infty }$) for P12S3, showing the formation of a secondary flow throughout the entire turbulent boundary layer with a longitudinal roughness element. Red and blue represent positive and negative values, respectively, and the contour levels are 10% of the maximum and minimum values at $x/{\theta }_{in}=250$. The aspect ratio in each direction is (4:1:1). In (b–e), subplots of $\langle {\mathrm{\Lambda }}_{ci}\rangle \delta /U_{\infty }$ in (a) are drawn on the $yz$ planes with vectors constructed using the phased-averaged mean wall-normal ($\langle v\rangle /U_{\infty }$) and spanwise velocities ($\langle w\rangle /U_{\infty }$): (b) $x/{\theta }_{in}=16$, (c) 50, (d) 100 and (e) 300.

Figure 8

Contours of the phase-averaged mean-signed swirling strength ($\langle {\mathrm{\Lambda }}_{ci}\rangle \delta /U_{\infty }$) with vectors constructed using the phased-averaged mean wall-normal ($\langle v\rangle /U_{\infty }$) and spanwise velocities ($\langle w\rangle /U_{\infty }$) as P and S vary: (a) P12S3 ($P/\delta =0.58,S/\delta =0.14$), (b) P24S3 ($P/\delta =1.18,S/\delta =0.15$), (c) P12S6 ($P/\delta =0.57,S/\delta =0.27$), (d) P24S6 ($P/\delta =1.18,S/\delta =0.29$) and (e) P24S12 ($P/\delta =1.17,S/\delta =0.59$). On the right in each figure, the phase-averaged mean spanwise velocity $(\langle w\rangle /U_{\infty })$ is drawn, with line contours of $\langle w\rangle /U_{\infty }=0.001$ (solid line) and −0.001 (dashed line) included. The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$) for each flow and the vectors are uniformly distributed for clarity.

Figure 9

Identical to those in Fig. 8 but for (a) P48S3 ($P/\delta =2.39$, $S/\delta =0.15$) and (b) P48S24 ($P/\delta =2.39$, $S/\delta =1.12$). The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$).

Figure 10

Contours of the phase-averaged mean streamwise velocity ($\langle u\rangle /U_{\infty }$) with vectors constructed using the phased-averaged mean wall-normal ($\langle v\rangle /U_{\infty }$) and spanwise velocities ($\langle w\rangle /U_{\infty }$) while varying P and S: (a) P12S3, (b) P12S6, (c) P24S3, (d) P24S6 and (e) P24S12. The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$) for each flow and the vectors are uniformly distributed for clarity.

Figure 11

Contours of the wall-normal gradient of phase-averaged mean streamwise velocity $\left(\partial \langle u\rangle /\partial y\right)$ normalized by $U_{\infty }/{\theta }_{in}$ as P and S vary: (a) P12S3, (b) P12S6, (c) P24S3, (d) P24S6 and (e) P24S12. The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$). In each figure, the spanwise variation for the wall-normal gradient of the phase-averaged mean streamwise velocity at the wall ${\left(\partial \langle u\rangle /\partial y\right)}_w$ is included (green line). The label for ${\left(\partial \langle u\rangle /\partial y\right)}_w$ is plotted in the right-hand side of the figures.

Figure 12

Contours of the Reynolds stresses for P12S3 normalized by $U_{\infty }^2$: (a) $\langle u^{\prime }{u}^{\prime }\rangle$, (b) $\langle v^{\prime }{v}^{\prime }\rangle$, (c) $\langle w^{\prime }{w}^{\prime }\rangle$, (d) $\langle u^{\prime }{v}^{\prime }\rangle$, (e) $\langle u^{\prime }{w}^{\prime }\rangle$ and (f) $\langle v^{\prime }{w}^{\prime }\rangle$. The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$).

Figure 13

Contours of budget terms for the turbulent kinetic energy: (a, b) production term, $P_k$, (c, d) dissipation term,$-{\varepsilon }_k$, (e, f) turbulent transport term, $T_k$, (g, h) viscous diffusion term, $D_k$, and (i, j) velocity pressure-gradient term, ${\Pi }_k$. (a, c, e, g, i) P12S3 and (b, d, f, g, h) P24S12. The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$) for each flow.

Figure 14

Contours of (a, b) the sum of the turbulent kinetic energy production and dissipation terms, $P_k-{\varepsilon }_k$, (c, d) the sum of all turbulent kinetic energy budget terms, $P_k-{\varepsilon }_k+T_k+D_k+{\Pi }_k$ and (e, f) the sum of the turbulent transport, viscous diffusion and velocity pressure-gradient terms, $T_k+D_k+{\Pi }_k$. (a, c, e) P12S3 and (b, d, f) P24S12. The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$).

Figure 15

Contours of the turbulent transport term: (a) $-1/2\left(\partial \langle u^{\prime }{u}^{\prime }{v}^{\prime }\rangle /\partial y+\partial \langle v^{\prime }{v}^{\prime }{v}^{\prime }\rangle /\partial y+\partial \langle w^{\prime }{w}^{\prime }{v}^{\prime }\rangle /\partial y\right)$, (b) $-1/2\left(\partial \langle u^{\prime }{u}^{\prime }{w}^{\prime }\rangle /\partial z+\partial \langle v^{\prime }{v}^{\prime }{w}^{\prime }\rangle /\partial z+\partial \langle w^{\prime }{w}^{\prime }{w}^{\prime }\rangle /\partial z\right)$, (c) $-1/2\partial \langle u^{\prime }{u}^{\prime }{v}^{\prime }\rangle /\partial y$, (d) $-1/2\partial \langle u^{\prime }{u}^{\prime }{w}^{\prime }\rangle /\partial z$, (e) $\langle u^{\prime }{u}^{\prime }{v}^{\prime }\rangle$ and (f) $\langle u^{\prime }{u}^{\prime }{w}^{\prime }\rangle$, normalized by $U_{\infty }^3/{\theta }_{in}$ for P12S3. The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$).

Figure 16

Contours of (a, b) the turbulent kinetic energy and (c, d) the sum of the convective terms ($C_{2,k}+C_{3,k}$). (a, c) P12S3 and (b, d) P24S12. The data are extracted in the equilibrium region ($x/{\theta }_{in}=512$).