Influence of VLSMs in drag reduction by streamwise
travelling wave of spanwise wall velocity
Qiang Yang, Yongmann M. Chung
School of Engineering and Centre for Scientific Computing
University of Warwick, Coventry, CV4 7AL, U.K.
Streamwise travelling wave by spanwise wall velocity was discovered to be a successful control method
for turbulent skin-friction drag reduction [Quadrio et al., 2009]. This control method is an extended form
from spanwise wall oscillation, with the spanwise wall velocity changing in a sinusoidal form in both
time and space, i.e., W = A sin(κx x − ωt), where A is the velocity magnitude, κx is the streamwise
wavenumber and ω is the oscillation frequency. However, similar to spanwise wall oscillation [Touber
and Leschziner, 2012], streamwise travelling wave of spanwise wall velocity also suffers from deterioration
of drag reduction at higher Reynolds number. Hurst et al. [2014] showed that the scaling parameter
α (DR ∼ Re−α
τ ) depended on the control parameters ω and κx , and at the worst case it could reach
α ≈ 0.4. This Reynolds number effect was found to be linked with the newly found very large scale
motions (VLSMs) at high Reynolds numbers [Touber and Leschziner, 2012]. Indeed, the streamwise
travelling wave can significant modify the near-wall cycle by the generalised Stokes layer, but how it
affects the VLSMs in the outer region is still not clear. Jiménez [2012] showed evidence that the small
scales lived in the local environment defined by the large scales. The small scales are modulated by the
large scales above them [Hutchins and Marusic, 2007]. Here, we investigate the wall shear stress of 54
streamwise travelling wave cases at Reτ = 800 to show the local effect from VLSMs.
Figure 1 shows the instantaneous streamwise velocity contour at 8 different wall normal locations for no
control case. The VLSMs are identified with a sharp spectra filter, with the size of λx ≥ 3h, λz ≥ 0.5h.
The zero contour line of the VLSMs is superimposed into each field, and clearly we can see that it
distinguishes well the positive region (bright colour) and negative region (dark colour). As the wall
paralleled plane moves further away from the wall, the low- and high-speed streaks become longer and
wider. The proper orthogonal decomposition (POD) was used to identify the characteristic eddy in yz
plane [Moin and Moser, 1989], and the eddy with two different scales are clearly separated, as shown in
figure 2. In the outer region, the super low- and high- speed streaks (VLSMs) have a spanwise separation
of λ ≈ 0.5h (measured from the top of logarithmic region, y/h = 0.2, see also figure 1(e)). A close view
of the near-wall region (figure 2(b)) show that the small low- and high- speed streaks have a spanwise
separation of λ+ ≈ 50. The super streaks affect the small streaks by footprint and amplitude modulation
[Marusic et al., 2010]. In figure 3, we conditioned the drag reduction map of streamwise travelling wave
based on positive and negative VLSMs regions. To make the condition effect more obvious, the whole wall
surface is divided into three regions based on the p.d.f. of VLSMs, as done by Agostini and Leschziner
[2014], where the probabilities of streamwise velocity fluctuation falls into the positive, negative and gray
(between positive and negative) regions are equally to be 1/3. The wall shear stress in positive and
negative VLSMs regions for each streamwise travelling wave case are sampled and compared with that
of the no control case, and two new drag reduction maps (figure 3(b)(c)) are created for the positive and
negative VLSMs, respectively. The two new DR maps look very similar to the one computed based all the
whole wall region, but the boundary between drag reduction and drag increase regions are significantly
changed. More drag reduction is observed under positive VLSMs than that under negative ones, and
this makes the drag increase region (brighter colour) narrower under positive VLSMs regions and wider
under negative VLSMs regions.
References
M. Quadrio, P. Ricco, and C. Viotti. Journal of Fluid Mechanics, 627:161–178, 2009.
E. Touber and M. A. Leschziner. Journal of Fluid Mechanics, 693:150–200, 2012.
E. Hurst, Q. Yang, and Y. M. Chung. Journal of Fluid Mechanics, 759:28–55, 2014.
J. Jiménez. Annual Review of Fluid Mechanics, 44:27–45, 2012.
N. Hutchins and I. Marusic. Philosophical Transactions of the Royal Society A, 365:647–664, 2007.
L. Agostini and M. A. Leschziner. Physics of Fluids, 26:075107, 2014.
P. Moin and R. D. Moser. Journal of Fluid Mechanics, 200:471–509, 1989.
I. Marusic, R. Mathis, and N. Hutchins. Science, 329:193–196, 2010.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 1: Instantaneous streamwise velocity contour at 8 different wall normal locations, i.e., y + =
5, 15, 30, 60, 150, 245, 400, 560. Contour lines indicate the zero level of the VLSMs. Domain is 12h in
length and 4h in width, and flow goes from bottom to top.
80
0.6
60
y
y+
0.8
0.4
0.2
0-1
40
20
-0.5
0
0.5
1
-100
-50
0
50
100
z+
(b)
z
(a)
Figure 2: The characteristic eddy in yz plane from: (a) global view for the whole half channel height and
(b) zoomed in view of the near-wall region. Negative contour lines are dashed.
(a)
(b)
(c)
Figure 3: Drag reduction map for (a) all the wall region; (b) positive VLSMs region only and (c) negative
VLSMs region only.