Secondary flow in turbulent ducts with increasing aspect ratio

Direct numerical simulations of turbulent duct flows with aspect ratios 1, 3, 5, 7, 10, and 14.4 at a center-plane friction Reynolds number ${\text{Re}}_{\tau ,c}\simeq 180$, and aspect ratios 1 and 3 at ${\text{Re}}_{\tau ,c}\simeq 360$, were carried out with the spectral-element code nek5000. The aim of these simulations is to gain insight into the kinematics and dynamics of Prandtl's secondary flow of the second kind and its impact on the flow physics of wall-bounded turbulence. The secondary flow is characterized in terms of the cross-plane component of the mean kinetic energy, and its variation in the spanwise direction of the flow. Our results show that averaging times of around 3000 convective time units (based on duct half-height $h$) are required to reach a converged state of the secondary flow, which extends up to a spanwise distance of around $\simeq 5h$ measured from the side walls. We also show that if the duct is not wide enough to accommodate the whole extent of the secondary flow, then its structure is modified as reflected through a different spanwise distribution of energy. Another confirmation of the extent of the secondary flow is the decay rate of kinetic energy of any remnant secondary motions for $z_c/h>5$ (where $z_c$ is the spanwise distance from the corner) in aspect ratios 7, 10, and 14.4, which exhibits a decreasing level of energy with increasing averaging time $t_a$, and in its rapid rate of decay given by $\sim t_a^{-1}$. This is the same rate of decay observed in a spanwise-periodic channel simulation, which suggests that at the core, the kinetic energy of the secondary flow integrated over the cross-sectional area, ${\langle K\rangle }_{yz}$, behaves as a random variable with zero mean, with rate of decay consistent with central limit theorem. Long-time averages of statistics in a region of rectangular ducts extending about the width of a well-designed channel simulation (i.e., extending about $\simeq 3h$ on each side of the center plane) indicate that ducts or experimental facilities with aspect ratios larger than 10 may, if properly designed, exhibit good agreement with results obtained from spanwise-periodic channel computations.

Figure 1

Schematic view of the three-dimensional effects present in fully developed turbulent duct flows. The blue regions represent the side-wall boundary layers and the dashed blue arrows show that they accelerate the flow in the duct center plane. Red and green arrows indicate the flow direction in the secondary vortices on the horizontal and vertical walls, respectively.

Figure 2

Instantaneous streamwise velocity field in the $\mathrm{AR}=10, {\text{Re}}_{\tau ,c}\simeq 180$ case at 2000 convective time units (normalized with $U_b$ and $h$) from the beginning of the simulation. Green and orange represent minimum and maximum velocities in the field, respectively. Flow is from lower left to upper right, and walls have been made transparent for clarity.

Figure 3

Streamlines of secondary mean flow $\mathrm{\Psi }$, where the coordinate $z_c$ is defined with its origin at the duct corner. (a) Square ducts at $\_\_\_\_\_$${\text{Re}}_{\tau ,c}\simeq 180$ and ${\text{Re}}_{\tau ,c}\simeq 360$; (b) $\mathrm{AR}=3$ ducts at $\_\_\_\_\_$${\text{Re}}_{\tau ,c}\simeq 180$, ${\text{Re}}_{\tau ,c}\simeq 360$ and $\_\_\_\_\_$ shows the first contour (with value $3.9\times {10}^{-4}$) from the square duct at ${\text{Re}}_{\tau ,c}\simeq 360$ for comparison; (c) ${\text{Re}}_{\tau ,c}\simeq 180$ ducts with $\_\_\_\_\_$$\mathrm{AR}=1$, $\mathrm{AR}=3$, $\_\_\_\_\_$$\mathrm{AR}=5$, and $\mathrm{AR}=10$. A total of 10 contours are shown in the square duct cases, whereas 15 are shown in the wider ducts, all of them with increments of $3.9\times {10}^{-4}$. The velocity and length scales are, respectively, $U_b$ and $h$.

Figure 4

Kinetic energy of the secondary flow $K$ for the (top) $\mathrm{AR}=3$ and (bottom) $\mathrm{AR}=10$ cases at ${\text{Re}}_{\tau ,c}\simeq 180$, together with the windows used to calculate ${\langle K\rangle }_{yz}$. Fields obtained after averaging in the streamwise direction, time, and over the four duct quadrants.

Figure 5

Kinetic energy of the secondary flow averaged over the first window ${\langle K\rangle }_{yz1}$ as a function of averaging time, for all the aspect ratios at ${\text{Re}}_{\tau ,c}\simeq 180$.

Figure 6

Kinetic energy of the secondary flow averaged over the second and third windows as a function of averaging time. Cases as in Fig. 5, excluding the square duct.

Figure 7

Kinetic energy of the secondary flow averaged over windows larger than or equal wo window 6, as a function of averaging time, for $\mathrm{AR}=7$, 10, and 14.4 at ${\text{Re}}_{\tau ,c}\simeq 180$. The same quantity is shown averaged over the cross-sectional area in a spanwise-periodic channel, together with the observed rate of decay ${\langle K\rangle }_{yz}\sim t_a^{-1}$.

Figure 8

Kinetic energy of the secondary flow averaged over the fourth and fifth windows as a function of averaging time. Cases as in Fig. 5, excluding the $\mathrm{AR}=1$ and 3 ducts, and rate of decay ${\langle K\rangle }_{yz}\sim t_a^{-1}$ shown for comparison purposes.

Figure 9

Spanwise variation of the kinetic energy of the secondary flow averaged over the wall-normal direction ${\langle K\rangle }_y$, for all the duct cases at ${\text{Re}}_{\tau ,c}\simeq 180$. The left panel shows the region close to the corner, whereas in the right panel we focus on the aspect ratios larger or equal than 7 and show a wider spanwise extent.

Figure 10

Spanwise variation of the kinetic energy of the secondary flow averaged over the wall-normal direction ${\langle K\rangle }_y$, for the $\mathrm{AR}=1$ and 3 cases at ${\text{Re}}_{\tau ,c}\simeq 180$ and 360. In the left panel, the $z_c$ coordinate is scaled in outer units, whereas in the right one inner scaling is considered.

Figure 11

(a) Inner-scaled mean flow and (b) Reynolds-stress tensor components at the center plane of various ducts with ${\text{Re}}_{\tau ,c}\simeq 180$, compared with DNS of channel flow at ${\text{Re}}_{\tau }=180$ [42]. TKE budget of ducts at ${\text{Re}}_{\tau ,c}\simeq 180$ with (c) $\mathrm{AR}=10$ and (d) $\mathrm{AR}=14.4$, compared with same reference channel flow data [42]. TKE budget terms: $\_\_\_\_\_$ production, dissipation, turbulent transport, viscous diffusion, and $\_\_\_\_\_$ velocity-pressure-gradient correlation.

Figure 12

Spanwise distributions of [(a), (b)] wall-shear stress normalized with center-plane value and [(c), (d)] streamwise and [(e), (f)] spanwise fluctuating wall-shear stress. The variable $z_c^{+}$ is defined with its origin at the corner and is formed with $u_{\tau ,c}$. The cases shown in the left panels correspond to $\_\_\_\_\_$$\mathrm{AR}=1$, $\mathrm{AR}=3$, $\mathrm{AR}=5$, $\mathrm{AR}=7$, $\_\_\_\_\_$$\mathrm{AR}=10$, and $\_\_\_\_\_$$\mathrm{AR}=14.4$, all of them at ${\text{Re}}_{\tau ,c}\simeq 180$, and the vertical $\_\_\_\_$ indicates the position where $z_c=5h$. The cases shown in the right panels are $\_\_\_\_\_$$\mathrm{AR}=1$ and $\mathrm{AR}=3$ at ${\text{Re}}_{\tau ,c}\simeq 180$ and $\mathrm{AR}=1$ and $\mathrm{AR}=3$ at ${\text{Re}}_{\tau ,c}\simeq 360$. In all the panels, the horizontal $\_\_\_\_$ represents corresponding channel flow values at the same ${\text{Re}}_{\tau }$ [42, 47].