List of Table
III. Results and discussion
3.5 Mean secondary flow
in channel and pipe flows with GF=0.5 and P/δ=0.39. The grey and white colors on the bottom wall indicate no-slip and shear-free surfaces, respectively. An identical contour legend is used in (b-d) and (e-f). In Figs. 17(b) and 18(b), the tke production associated with <u'v'> exceeds the other four terms.
Because the contours of Pk k Tk Dk k are locally imbalanced in Figs. 17(b) and 18(b), advective velocities for secondary flows are induced in the flows to preserve the conservation of energy.
The sum of the right-hand side of Eq. (18) is similar to sum of the tke advection terms, C2,k and C3,k. In addition, C2,k shows a positive value over the no-slip region, indicating that Pk k Tk Dk k 0 and that C3,k is close to zero over the no-slip region, suggesting that C2,k is similar to the sum of the right-hand side of Eq. (18) over the no-slip region. Because the wall-normal gradients of the tke contour in Figs. 17(a) and 18(a) show positive values near the wall within the no-slip region, the phase-averaged wall-normal velocity must be positive (Figs. 17e and 18e) based on the contour of <v>∂<k>/∂y>0 in Figs. 17(c) and 18(c). In addition, C3,k near the wall shows a negative value for all edges between the no-slip and shear-free regions. Because the spanwise gradients of the tke contour show a positive value from no-slip to shear-free regions and a negative value from shear-free to no-slip regions near the wall, the corresponding phase-averaged spanwise velocities must be negative and positive along the spanwise direction (Figs. 17f and 18f) based on the contour of <w>∂<k>/∂z<0 in Figs. 17(d) and 18(d). In order to preserve the mean flow continuity, the wall-normal and spanwise velocities must be balanced near the wall, and the sweep event in the shear-free region must be balanced by a lateral outflow near the wall, leading to the generation of a secondary flow.
Images similar to those in Figs. 18 and 19 but for a larger P/δ (=6.28) are presented in Figs. 19 and 20. Although the vectors of
v w,
for a small P/δ show pairs of counter-rotating streamwise vortices which extend roughly one pitch length along the wall-normal direction (Figs. 17 and 18), the secondary flow for P/δ=6.28 is shown to cover the entire pipe radius or channel height. In addition, it is clear that the strength of the secondary flow for the pipe flows in Figs. (17)-(20) is larger than that of the channel flows for all P/δ due to the higher spanwise slip velocity. The most interesting observation in the figures is that although the vectors for a small P/δ show downward motion over the shear-free surface and upward motion over the no-slip surface, those when P/δ=6.28 indicate the existence of a secondary flow for which the rotational direction is opposite. Specifically, upwelling arises above the shear-free surface, although relatively weak downwelling motion is observed over the no-slip surface. Though not shown here, secondary flows with varying GF (=0.25, 0.75) when P/δ=6.28 show upwelling motion above the shear-free region close to the interface, similar to findings by Türk et al. (2014) and Stroh et al. (2016).In Figs. 19(a) and 20(a), compared to the tke contours for a small P/δ, a distinctive spatial feature for a large P/δ is sign change of the wall-normal gradient for the tke contour. Because the gradients of the tke contours show negative values near the wall within the center of the shear-free region, the phase- averaged wall-normal velocity must be positive (Figs. 19e and 20e) based on the negative contour of
Fig. 17. Contours of (a) the turbulent kinetic energy (k+), (b) the sum of PkkTkDk k, (c) v k
y
, (d) k
w z
, (e) v and (f) w in turbulent channel flows over SHSs on the yz plane for GF=0.5 and P/δ=0.39. The phase-averaged velocity vector of
v w,
is superimposed to reveal the role of the anisotropy of the Reynolds stress tensor in generating a secondary flow. The grey and white colors along the bottom wall indicate no-slip and shear-free surfaces, respectively. The color legends in (b-d) and (e-f) are identical.<v>∂<k>/∂y<0 shown in Figs. 19(c) and 20(c). In addition, C3,k near the wall shows a negative value for all edges between no-slip and shear-free regions. Because the spanwise gradients of the tke contour show negative values from the no-slip to shear-free regions and positive values from the shear-free to no-slip regions near the wall based on the contour of <w>∂<k>/∂z<0 shown in Figs. 19(d) and 20(d), the corresponding phase-averaged spanwise velocity must be positive and negative along the spanwise direction (Figs. 19f and 20f). Thus, contrary to the secondary flow with the sweep event in the shear- free region for a small P/δ, the wall-normal and spanwise velocities for a large P/δ create ejecting
Fig. 18. Identical to Fig. 17 but for a pipe flow on the rθ plane with GF=0.5 and P/δ=0.39.
motion over the shear-free region. Hinze (1967) suggested that the advective velocities transport fluid with the lowest tke to the fluid with the highest tke in the region of Pk k TkDk k 0 (the no- slip region for a small P/δ) to achieve optimal equilibrium of the energy, with the fluid with the highest tke transported to the fluid with the lowest tke in the region of Pk k Tk Dk k 0 (shear-free region for large P/δ). This result is consistent with our observation of the conditionally averaged flow fields in Figs. 14 and 15, where the strength of the spanwise velocity over the edge exceeds the
corresponding velocity component with the opposite sign over the no-slip surface with an increase in P/δ. The present observation demonstrates that the driving mechanism responsible for the secondary flow in motion is Prandtl’s secondary flow of the second kind caused by the local imbalance of the tke budget, which necessitates secondary advective velocities.
Fig. 19Identical to that in Fig. 17 but with GF=0.5 and P/δ=6.28.
Fig. 20. Identical to that in Fig. 18 but with GF=0.5 and P/δ=6.28. The color legends for (a) and (b-d) are shown on the left and right sides on the top while that for (e-f) is depicted on the bottom.