List of Table
III. Results and discussion
3.4 Turbulent structures
It has been reported that turbulence production and elevated skin-friction drag are closely associated with turbulent coherent structures (Robinson 1991; Adrian 2007). Therefore, reducing both the Reynolds shear stress and skin-friction drag is accompanied by the weakening of related turbulent vortical structures. In drag-reducing flows, previous studies have shown that the Reynolds shear stress decreases significantly in turbulent flows with polymer solutions and riblets (Wei & Willmarth 1992;
Goldstein et al. 1995). In this section, in order to provide a clear understanding of the relationship between turbulent vortical structures and DR in pipe and channel flows over SHSs, we examine modifications of turbulent structures based on an instantaneous view of vortical structures, probability density functions and the conditional averaged flow fields with an event.
Fig. 10. Skin friction drag for pipe and channel flows over SHSs calculated from the FIK identity with velocity-vorticity correlation: (a) no-slip, (b) GF=0.5, P/δ=0.39, (c) GF=0.5, P/δ=6.28 and (d) GF=0.75, P/δ=6.28.
Figs. 11 and 12 show iso-surfaces of three-dimensional vortical structures visualized using the swirling strength λci for turbulent pipe and channel flows over SHSs. The swirling strength is defined as the imaginary part of the complex conjugate eigenvalues of the local velocity gradienttensor and extracts compact vortical cores against regions of shear (Zhou et al. 1999). Gray and white regions at the wall represent no-slip and shear-free surfaces. The contour level is 20% of the maximum λci based on a no-slip flow. For no-slip flows of the pipe and channel (Figs. 11a and 12a), a number of arch- shaped (or spanwise) vortices with relatively intense strength are observed in the outer layer, and quasi- streamwise vortices are dominant close to the wall (Adrian et al. 2000). For turbulent flows over SHSs with an increase in P/δ (up to P/δ=1.56), as shown in Figs. 11(b-c) and 12(b-c), the population of spanwise vortical structures in the outer region is clearly reduced, in particular over the slip surface.
This weakening is the one of the key features of reduced-drag wall-bounded turbulent flows (Choi et al.
1993). However, as P/δ increases further (P/δ≥3.14) in Figs. 11(d) and 12(d), the vortical structures for the pipe and channel flows over SHSs become more active than those for P/δ=1.56 on the no-slip surface near the wall (y/h<0.5), although the drag for P/δ≥3.14 is slightly less than that for P/δ=1.56 in Fig. 2.
For a small P/δ (≤1.56), the vortical structures over the no-slip surface are directly affected by the shear- free surface, leading to the weakening of the vortical structures throughout the entire domain. However, as P/δ exceeds a critical value (here, P/δ>1.56), the influence of the slip surface does not significantly modify the vortical structures over the no-slip surface. Therefore, the quasi-streamwise vortices in the
-5 0 5 10
<Cf>xzsum
<Cf1>xz
<Cf2>xz<Cf3>xz<Cf4>xz<Cf6>xz
-5 0 5 10
Pipe Channel
<Cf>xzsum
<Cf1>xz
<Cf2>xz<Cf3>xz<Cf4>xz<Cf6>xz
(b)
(d) (a)
(c)
< C
f>
xz -50 5 10
<Cf>xzsum
<Cf1>xz
<Cf2>xz<Cf3>xz<Cf4>xz<Cf6>xz
(103)
(103) (103)
(103)
< C
f>
xz -50 5 10
<Cf>xzsum
<Cf1>xz
<Cf2>xz<Cf3>xz<Cf4>xz<Cf6>xz
near-wall region and spanwise-oriented vortices above the buffer layer over the no-slip surface remain dominant for a large P/δ, creating an increase in the wall-normal and spanwise turbulent stresses and the Reynolds shear stress when P/δ≥3.14 (Fig. 5). Figs. 11 (e-f) and 12(e-f) show iso-surfaces of vortical structures with an increase in GF for a fixed P/δ=6.28. As expected, the turbulent structures are increasingly weakened as GF increases.
Fig. 11. Snapshots of vortical structures in turbulent channel flows with SHSs. The contour is normalized by Uco/δ and the maximum contour of 20% is employed to depict the structures. (a) no-slip, (b) GF=0.5, P/δ=0.39, (c) GF=0.5, P/δ=1.56, (d) GF=0.5, P/δ=6.28, (e) GF=0.25, P/δ=6.28 and (f) GF=0.75, P/δ=6.28. The grey and white colors along the bottom wall indicate no-slip and shear-free surfaces, respectively. The distance between the ticks on each axis is 0.5δ. Only 40% of the streamwise domain (~10δ) is depicted.
Fig. 12. Identical to that in Fig. 11, but in turbulent pipe flows over SHSs. In (a-c), only half of the spanwise domain is shown. In (d-f), the entire spanwise domain for the pipe flows is divided into two parts for clarity.
In order to provide statistical evidence of the variation of the vortical structures based on the instantaneous flow fields in the pipe and channel flows over SHSs, the p.d.f. of the swirling strength are plotted in Fig. 13. Here, the two-dimensional (2-D) velocity gradient tensors Dx2D, Dy2D and Dz2D for the swirling strength (e.g., λx, λy and λz) are calculated to demarcate the orientation of the vortical structures (Adrian et al. 2000):
2D x
v v y z
D w w
y z
, 2Dy
u u x z
D w w
x z
and z2D
u u x y
D v v
x y
. (14)
In addition, two reference wall-normal heights (yr) for the inner and outer regions are considered. In Fig. 13, it is clear that the population of streamwise, wall-normal and spanwise vortices with respect to the strength is decreased with an increase in P/δ (P/δ≤1.56), consistent with the findings of Park et al.
(2013), who reported the weakening of the streamwise vorticity. However, for a large P/δ (P/δ≥3.14), the population of the vortices increases slightly both in the inner and outer regions. Note that relatively weak vortex components in the outer layer are observed much frequently than the no-slip cases in Figs.
13(b, d and f), most likely due to presence of less coherent and small structures, especially near the edge between the no-slip and shear-free surfaces. Consistent with the earlier observations from the instantaneous views, the statistical analysis using the p.d.f. indicates that the presence of SHSs weakens the vortical motions in all three spatial directions for a small P/δ, although the vortical motions are enhanced for a large P/δ≥3.14. Given that it has been reported that the wall-normal and spanwise vorticity are primarily created by the tilting and stretching of the mean spanwise vorticity, respectively, increased vorticity is expected near the boundary between the no-slip and shear-free surfaces, as shown in Figs. 11 and 12. However, the weakening of the streamwise mean shear (e.g., dU/dy) over the shear- free surface leads to a reduction of the mean spanwise vorticity, resulting in the suppression of the vortical structures. In Fig. 13, the population of the vortical structures in the pipe flows over SHSs is smaller than those of the channel flows in all three spatial directions, and the difference increases with an increase in P/δ, consistent with the variation of the drag (Fig. 2).
Next, we examine conditionally averaged flow fields with an event of negative streamwise velocity fluctuation (u') to identify the turbulent structures responsible for the variation of the streamwise Reynolds stresses. The conditionally averaged velocity field around a negative-u' event is defined based on a linear stochastic estimation (Tomkins & Adrian 2003):
' '
z 1
( , , ) | ( , , ) 0 ( , , )
'
i x y r r i r r
u r r r u x y z L u x y z (15) with i=1,2 and 3, and
'
1 ' 2
( , , ) ( , , )
( , , )
'
r r i x r y r z
i
r r
u x y z u x r y + r z r L
u x y z
(16)
Fig. 14 illustrates the conditionally averaged fluctuating fields for the channel flows over SHSs, when the events occur at the spanwise locations of z ref, / 2 0.25and 0.5 and at a wall-normal location yr+
=1. The conditional estimate in a regular no-slip channel flow is shown in Fig. 14(a) for comparison.
It is clear that the average structure with velocity vectors of the estimates is a large region of low-speed
Fig. 13. Probability density functions of the 2-D swirling strength (
xi τo
λ U / δ) in turbulent pipe and channel flows over SHSs for GF=0.5 (a, c, e) at yr+
=1 and (b, d, f) yr/δ=0.23. (a-b) Streamwise, (c-d) wall-normal and (e-f) spanwise components.
streaks around rz/δ=0 (dark grey), consistent with earlier findings by Lee et al. (2015). When a negative- u' event is given over the no-slip region with GF=0.5 and P/δ=0.39 (Fig. 14b), a positive-u' structure is present in the shear-free region, qualitatively distinguishable from a regular no-slip channel flow in terms of its shape. Although a regular channel flow has a zero spanwise velocity at the wall, the spanwise velocity in the channel with SHSs is finite over the shear-free region. However, because the wall-normal velocity is almost zero due to the no-slip condition at the wall, ejection and sweep events are little observed. As P/δ increases to 1.56 with GF=0.5 (Fig. 14d), the magnitude of the negative-u' structure near the event location decreases. However, a further increase of P/δ (=6.28) enhances the
z/P
<
z'
2>
+00 0.2 0.4 0.6 0.8 1
0 5 10
P/
p .d .f .(
x+
)
10-4 10-3 10-2 10-1
(e) (a)
P/
<
x'
2>
+0 0 5 10 15 20 25P/ P/
(a) (b)
P/
z/P
<
z'
2>
+00 0.2 0.4 0.6 0.8 1
0 5 10
P/
p .d .f .(
y+
)
10-4 10-3 10-2 10-1
(e) (c)
P/<
y'
2>
+0 0 5 10 15 20P/
P/
(c) (d)
P/
z/P
<
z'
2>
+00 0.2 0.4 0.6 0.8 1
0 5 10
P/
xiU
o/
p .d .f .(
z+
)
5 10 15 20 25
10-4 10-3 10-2 10-1
Pipe,P/h=0.39 Pipe,P/h=1.56 Pipe,P/h=6.28 Channel,P/h=0.39 Channel,P/h=1.56 Channel,P/h=6.28 Pipe, No_slip Channel, No_slip
(e) (e)
P/
z/P
<
z'
2>
+00 0.2 0.4 0.6 0.8 1
0 5 10
P/
xU
o/
p .d .f .(
x+
)
10-4 10-3 10-2 10-1
(e) (a)
P/
z/P
<z'2 >+0
0 0.2 0.4 0.6 0.8 1
0 5 10
P/
xiUo/
p.d.f.(z+ )
5 10 15 20 25
10-4 10-3 10-2 10-1
(e) (e)
P/
z/P
<
z'
2>
+00 0.2 0.4 0.6 0.8 1
0 5 10 15 20 25
P/ P/
ciU
0/
5 10 15 20 25
(e) (f)
P/
z/P
<
z'
2>
+00 0.2 0.4 0.6 0.8 1
0 5 10
P/
xU
o/
p .d .f .(
x+
)
10-4 10-3 10-2 10-1
(e) (a)
P/
z/P
<z'2 >+0
0 0.2 0.4 0.6 0.8 1
0 5 10
P/
xiUo/
p.d.f.(z+ )
5 10 15 20 25
10-4 10-3 10-2 10-1
(e) (e)
P/
strength of the negative-u' structure. A similar trend is also shown in Figs. 14(c), (e) and (g), when the reference position is placed at the edge between the no-slip and shear-free regions. Compared to averaged estimates with the event over the no-slip region, the velocity vectors for P/δ=1.56 and 6.28 show relatively intense counter-clockwise rotating motions over the edge, although the spanwise motion for a small P/δ (Fig. 14c) is weaker than the corresponding motion over the no-slip region (Fig. 14b).
Because the spanwise slip velocity increases continuously with an increase in P/δ irrespective of the spanwise reference position, the intense swirling motion contributes to the strong transport of the near- wall high-momentum fluid from the no-slip to shear-free regions, similar to the observation depicted in Fig. 10.
Fig. 14. Conditionally averaged fluctuating velocity fields based on a negative-u' event in the cross- stream plane. Contours show the streamwise velocity disturbance. The vectors are the in-plane velocity fluctuations, colored by the absolute magnitude of w u' '0 . (a) Regular no-slip channel, (b-c) P/δ=0.39, (d-e) P/δ=1.56 and (f-g) P/δ=6.28. (b, d, f) z ref, / 2 0.25 and (c, e, g) z ref, / 2 0.5. The wall-normal reference height is yr+=1.
Fig. 15. Identical to that in Fig. 14, but at yr/δ=0.23.
Fig. 15 displays the conditionally averaged velocity fields with a negative-u' event at yr/δ=0.23.
Contrary to the flow fields near the wall, the estimates in the outer region over the no-slip region exhibit a counter-rotating streamwise vortex pattern accompanied by ejection and sweep motions to facilitate the transport of momentum. However, the general behavior of the spanwise velocity components when P/δ varies is consistent with that in Fig. 14. In addition, the strength of the negative-u' structures near the event location is decreased when P/δ≤1.56 and increased with a large P/δ ratio. In Figs. 14 and 15, it is reasonable to conclude that the increase of the near-wall streamwise velocity fluctuations (u') in Fig. 5 in the range of P/δ≤1.56 can be attributed to an enhancement of the spanwise slip velocity, which then weakens the streamwise fluctuating structures over the entire spanwise domain. However, although the spanwise slip velocity increases further when P/δ≥3.14, the considerable influence of the turbulence activity near the wall and in the outer region for a large P/δ induces a decrease of the streamwise velocity fluctuations (u'), as shown in Fig. 5. Although not shown here, conditionally averaged flow fields with a negative-u' event for pipe flows over SHSs are similar to the finding with channel flows over SHSs.
Finally, to provide a proper explanation of the sudden growth of near-wall vortical structures when P/δ≥3.14, we examine not only the advections of the wall-normal vorticity using the spanwise velocity (vortex stretching) but also the spanwise vorticity via the wall-normal velocity (vortex transport), which play important roles in the self-sustained mechanism for near-wall streamwise vortices at low to
moderate Reynolds numbers (Jimenez & Pinelli 1999). These two vorticity flux terms are also associated with the gradient of the Reynolds shear stress (e.g., the net force term in the Navier-Stokes equations). The profiles of the two terms are shown in Fig. 16 with an increase in P/δ for GF=0.5. In Fig. 16(a), the absolute vortex stretching term increases continuously when y+<5 with an increase in P/δ (≤1.56). In addition, the wall-normal location for the minimum moves to the wall. However, the large P/δ (=6.28) reduces the vortex stretching term with y+<5. The absolute vortex stretching term continuously decreases when y+>5 with an increase in P/δ. The vortex stretching term in the pipe flows is smaller than that in the channel flows for all P/δ throughout the entire wall layer; thus, vortex stretching also increases the difference in the skin friction drag between the pipe and channel flows (Fig.
10). Fig. 16(b) shows that the vorticity transport term is reduced at all wall-normal locations as P/δ increases. However, a sudden increase in the vorticity transport terms when P/δ=6.28 is clearly observed in the near-wall region. Note that the positive and negative vorticity transport terms are mainly induced by the ejection of the low-momentum region near the wall and the lifting of hairpin vortices (Klewicki et al. 1994). The increased vorticity transport term induces an increased population of near-wall structures, as the vorticity transport is linked to the generation of near-wall structures from the advection of the mean shear by the streamwise vorticity (Jimenez & Pinelli 1999). This result indicates that the increased wall-normal and spanwise Reynolds stresses for P/δ≥3.14 in Fig. 5 can be attributed to the generation of near-wall vortices. However, because the new vortices are not directly grown onto typical u'-streaky structures near the wall, the streamwise Reynolds stress is not immediately increased for a large P/δ. Compared to the channel flows, the vorticity transport term in the pipe flows is larger for all P/δ throughout the entire wall layer, demonstrating that the influence of the advective vorticity transport on the skin friction drag in the pipe flows exceeds that in the channel flows, as shown in Fig. 10.
Fig. 16. Inner-scaled (a) vortex stretching term and (b) vortex transport term of pipe and channel flows as a function of P/δ for a fixed GF value of 0.5.
y
+< w
'
' y>
xz /U
o3
0 20 40 60 80
-0.05
0 P/ P/
y
+< v
'
' z>
xz /U
o3
0 20 40 60 80
-0.04 0
Pipe,GF=0.5,P/h=0.39 Pipe,GF=0.5,P/h=1.56 Pipe,GF=0.5,P/h=6.28 Channel,GF=0.5,P/h=0.39 Channel,GF=0.5,P/h=1.56 Channel,GF=0.5,P/h=6.28 Pipe, No_slip Channel, No_slip
P/
P/