Direct numerical simulation of coflowing rough and smooth turbulent channel flows

Fully developed turbulent flows through ribbed channels have been simulated using direct numerical simulation (DNS). Square ribs were arranged transversely to the flow on one of the channel walls, and based on their spanwise extent, resulted in two configurations—two-dimensional (2D) configuration resulting from full-span-width ribs and a three-dimensional (3D) configuration, where the ribs extend only up to half the span-width of the channel, leaving the other half smooth. The 3D configuration thus produced a unique problem of coflowing rough and smooth turbulent channel flows. A striking phenomenon has been observed of the secondary roll cells, exhibiting a strong updraft in the smooth half of the channel. The mean velocity profile from the smooth half surprisingly possesses a linear region of constant slope at the channel core. Comparisons were also drawn with DNS of a smooth channel at the same friction Reynolds number 400 and it was found that the roll cells on the smooth half not only affect the bulk flow negatively but also attenuate the turbulence significantly. In spite of having a higher bulk velocity, the rough half of the 3D configuration is more turbulent than the 2D configuration; this has been attributed to the momentum transfer from the smooth half to the rough half. Statistical turbulence quantities, and the production rates of turbulent kinetic energy and enstrophy have been used to arrive at the inferences. In essence, the roll cells buffer the variations between the rough and the smooth halves in the 3D configuration. The instantaneous streamwise velocity fluctuations in the 3D configuration displayed a wavelike form.

Figure 1

Computational domain of (a) RC and (b) RSC, where $k=0.1h$ and pitch $\lambda =0.8h$; $L_x=12.8h, L_y=6.4h$.

Figure 2

Two-point correlations of the streamwise velocity fluctuations $u$ from RSC in (a) streamwise and (b) spanwise directions at $z/h=1.0$.

Figure 3

Grid resolution for RC and RSC, in terms of Grötzbach's criterion [see Eq. (3)]; the variation has been shown along the wall-normal direction.

Figure 4

Profiles of (a), (c) mean streamwise velocity $\langle U\rangle$; (b), (d) RMS of velocity fluctuations, computed using MGLET. Top panel: Profiles from smooth channel DNS at ${\text{Re}}_{\tau }=400$, normalized with the friction velocity $u_{\tau }$ and compared with those from Abe et al.  [17] at ${\text{Re}}_{\tau }=395$; bottom panel: Profiles from ribbed channel DNS at ${\text{Re}}_{\tau }=260$, normalized with the maximum streamwise velocity $U_{\text{max}}$ and compared with those from Burattini et al.  [6] at the same Reynolds number. In panels (c) and (d), $z/H=0$ corresponds to the plane of rib-crests. All the data have been averaged in $x$ and the normalized profiles of $u_{\text{rms}}$ and $v_{\text{rms}}$ are shifted, in panel (b) by 1.0 and 0.5 and in panel (d) by 0.1 and 0.05, respectively.

Figure 5

Contour plots on the $(y,z)$ plane, of (a) mean streamwise velocity $\langle U\rangle /U_m$; (b) $\sqrt{{\langle V\rangle }^2+{\langle W\rangle }^2}/\langle U\rangle$; (c) mean spanwise velocity $\langle V\rangle /U_m$. Contours in panels (a) and (b) are superimposed with the mean secondary streamlines. $U_m$ denotes the bulk mean velocity.

Figure 6

Mean streamwise velocity normalized with (a) bulk velocity, $U_m$; (c) friction velocity of rough wall, $u_{{\tau }_r}$ from the corresponding flows. (b) Total shear-stress normalized with $u_{\tau }$. $z_{\tau }$ in panel (c) denotes the zero-crossing of the shear stress profiles in panel (b). The red dashed line in panel (a) shows slope of the central linear region in the profile from RSC at $y/h=4.8$.

Figure 7

Contours of instantaneous skin friction coefficient $C_f=\mu \frac{dU}{dz}/\left(\frac{1}{2}\rho u_{\tau }^2\right)$ on the bottom walls of (a) RSC and (b) RC; top walls of (c) RSC and (d) RC; (e) one of the walls of SC.

Figure 8

RMS of velocity fluctuations normalized with the scales $u_{{\tau }_r}$ and $z_{\tau }$, from RSC, RC, and SC.

Figure 9

The variation of nonzero components of the anisotropy tensor, $b_{ij}$ from RSC, RC, and SC.

Figure 10

Instantaneous turbulent kinetic energy normalized with $u_{\tau }^2$ in the $(x,z)$ plane from (a) RSC-smooth $(y/h=4.8)$; (b) SC $(y/h=1.6)$; (c) RSC-rough $(y/h=1.6)$; (d) RC $(y/h=3.2)$.

Figure 11

Variation of (a) production rate $P_k^{+}$ and (b) dissipation rate ${\epsilon }^{+}$, of turbulent kinetic energy in RSC, RC, and SC. The data have been normalized with $u_{{\tau }_r}^4/\nu$.

Figure 12

Instantaneous spanwise vorticity, normalized with $u_{\tau }^2/\nu , {\langle {\omega }_y\rangle }^{+}$ in the $(x,z)$ plane from (a) RSC-smooth $(y/h=4.8)$; (b) SC $(y/h=1.6)$; (c) RSC-rough $(y/h=1.6)$; (d) RC $(y/h=3.2)$.

Figure 13

Profiles of $\left|\langle \omega \rangle \right|=\sqrt{{\langle {\omega }_x\rangle }^2+{\langle {\omega }_y\rangle }^2+{\langle {\omega }_z\rangle }^2}$, normalized with $u_{{\tau }_r}^2/\nu , {\left|\langle \omega \rangle \right|}^{+}$ on a logarithmic scale.

Figure 14

Streamwise vorticity fluctuation, normalized with $u_{\tau }^2/\nu , {{\omega }_x^{\prime }}^{+}$ in the $(y,z)$ plane at a midrib location, at $x/h=1.55$ from (a) RSC and (b) RC.

Figure 15

Wall-normal vorticity fluctuation normalized with $u_{\tau }^2/\nu , {{\omega }_z^{\prime }}^{+}$ in the $(x,y)$ plane at (a) $z/h=0.1$ and (b) $z/h=0.2$ from RSC. The data has been presented for only half the span and one-fourth of the streamwise extent.

Figure 16

Isosurfaces of ${\lambda }_2=-0.14$ colored with $z^{+}$, extending up to $z/h=0.5$, from (a) RSC and (b) RC.

Figure 17

Isosurfaces of ${\lambda }_2=-0.02$ colored with $z^{+}$, extending up to $z/h=0.5$ from (a) RSC-smooth and (b) SC.

Figure 18

Isosurfaces of positive enstrophy production rate, ${\omega }_i{S}_{ij}{\omega }_j^{+}={\omega }_i{S}_{ij}{\omega }_j/{\left(\frac{u_{{\tau }_r}^2}{\nu }\right)}^3$ colored with $z^{+}$, in the proximity of the ribs in (a) RSC, ${\omega }_i{S}_{ij}{\omega }_j^{+}=0.04$; (b) RC, ${\omega }_i{S}_{ij}{\omega }_j^{+}=0.02$. The domain has been shown only for one-fourth of the extent in $x$.

Figure 19

Isosurfaces of negative enstrophy production rate, ${\omega }_i{S}_{ij}{\omega }_j^{+}={\omega }_i{S}_{ij}{\omega }_j/{\left(\frac{u_{{\tau }_r}^2}{\nu }\right)}^3$ colored with $z^{+}$, in the proximity of the ribs in (a) RSC, ${\omega }_i{S}_{ij}{\omega }_j^{+}=-9.74\times {10}^{-3}$; (b) RC, ${\omega }_i{S}_{ij}{\omega }_j^{+}=-3.35\times {10}^{-3}$. The domain has been shown only for one-fourth of the extent in $x$.

Figure 20

Variation of the streamwise velocity fluctuation $u$ in RSC. (a) Time traces of $u$ obtained from the spanwise points at the midchannel $z/h=1$ at a midrib location; (b), (c) Contours of $u$ in $(y,z)$ planes obtained from instants (I) and (II), respectively; (d), (e) Contours of $u$ in $(x,y)$ planes at $z/h=0.5$ obtained from instants (I) and (II), respectively. The streamlines in panels (b) and (c) were obtained by averaging instantaneous data in $(y,z)$ planes cutting the ribs.

Figure 21

Spanwise variation of the pitch-averaged mean streamwise velocity $\langle U\rangle$, at the midrib location at different wall-normal locations $z/h$. The velocity data is normalized by the bulk velocity $U_m$.

Figure 22

Time traces of streamwise velocity fluctuation $u$ along the $x$ direction at $z/h=1.0$ in (a) RSC-rough $(y/h=1.6)$; (b) RSC-smooth $(y/h=4.8)$; (c) RC $(y/h=3.2)$; (d) SC $(y/h=1.6)$. In the case of RC and RSC, the data has been presented up to only half the extent in $x$.