Large-scale structure in skin friction drag reduction
by a spanwise wall oscillation
Qiang Yang, Yongmann M. Chung
School of Engineering and Centre for Scientific Computing
University of Warwick, Coventry, CV4 7AL, U.K.
Spanwise wall oscillation has been shown to be an efficient way to reduce turbulent skin friction Cf ,
as much as 40% drag reduction (DR) at Reτ ≈ 200 [1]. However, the maximum DR tends to reduce
as Reynolds number increases [2, 3]. This Reynolds number effect could be due to the very large scale
motions (VLSMs) [4] in the outer region at high Reynolds numbers [2]. The aim of current work is to
measure quantitatively the effect of VLSMs on Cf in the spanwise wall oscillation control.
Figure 1(a) shows the optimal oscillation frequency moves from ω + = 0.06 at Reτ = 200 to ω + = 0.08
at Reτ = 800, while the maximum DR is decreased with the Reynolds number. The DR was split into a
contribution from inner region and one from outer region for no conctrol cases and wall oscillation cases
at ω + = 0.06, shown in figure 1(b). The inner and outer regions are separated by the peak location of the
turbulent shear stress profile, −uv. It shows that, when Reynolds number is increased, the contribution
to DR from the outer region becomes more important.
The VLSMs are indicated by a spectra ridge in figure 2(a). The dominate peak at λ+
x ≈ 1000 and
+
λz ≈ 100, corresponds to the low speed streaks and quasi-streamwise vortices. When Reynolds number
is increased from Reτ = 800 to 1600, the outer peak at lower wavenumbers contains more energy. The
role of the outer peak (VLSMs) is more important at even higher Reynolds number turbulence. Figure
2(b) shows the premultiplied energy density spectra, kx Φuu , at Reτ = 800 for the wall oscillation, with
ω + = 0.06 and 0.03. When wall oscillation is applied, the inner peak moves towards higher wavenumber
and also a higher wall normal location. The energy of the inner peak is much weaker, especially for
ω + = 0.06 case. The energy of the outer peak tends to be less reduced. The uv spectra are split into
two parts, i.e., small scale (λx < 3h and λz < 0.5h) and VLSMs (λx ≥ 3h and λz ≥ 0.5h). By using
the FIK identity [5], contributions from small and large scales to skin friction, Cf , are quantified: a
R1
laminar contribution 6/Rem ; a small scale turbulence contribution 6 0 (1 − y)(−uv)S dy and a large scale
R1
turbulence contribution 6 0 (1 − y)(−uv)L dy. These data are shown in table 1. As can be seen, the
contribution from the VLSMs becomes more important as the Reynolds number increases. In the wall
oscillation case, the contribution to DR from the large scales is significant.
Table 1: Contributions to skin friction, Cf .
flow type
no control (NO)
no control (NO)
wall oscillation (WO)
Reτ
800
1600
800
laminar
7%
4%
10% (0% ↓)
small scales
57%
57%
54% (31% ↓ )
large scales
36%
39%
36% (25% ↓ )
References
[1] W. J. Jung, N. Mangiavacchi, and R. Akhavan. Suppression of turbulence in wall-bounded flows by
high-frequency spanwise oscillations. Physics of Fluids A, 4(8):1605–1607, 1992.
1
500 200
60
100
T+
50
1
0.8
C f / C f0
DR
40
20
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
0
-20
(a)
0
0.05
0.1
ω+
0.15
0.6
0.4
Outer
Inner
Laminar
0.2
0
0.2
(b)
500
1000
1500 2000
Reτ
Figure 1: (a) Shift of optimal oscillation frequency; (b) Cf contributions from laminar component, inner
and outer regions for wall oscillation with ω + = 0.06 (solid lines), and no control cases (dashed lines).
(a)
(b)
Figure 2: (a) Premultiplied uu energy density spectra, kx kz Φuu for Reτ = 800 (lines) and 1600 (shaded)
at y + ' 15; (b) premultiplied energy density spectra, kx Φuu , comparison among no control case (shaded),
ω + = 0.06 oscillation case (red lines) and ω + = 0.03 oscillation case (yellow lines) at Reτ = 800.
[2] E. Touber and M. A. Leschziner. Near-wall streak modification by spanwise oscillatory wall motion
and drag-reduction mechanisms. Journal of Fluid Mechanics, 693:150–200, 2012.
[3] E. Hurst, Q. Yang, and Y. M. Chung. The effect of Reynolds number on turbulent drag reduction
by streamwise travelling waves. Journal of Fluid Mechanics, under review, 2014.
[4] N. Hutchins and I. Marusic. Evidence of very long meandering features in the logarithmic region of
turbulent boundary layers. Journal of Fluid Mechanics, 579:1–28, 2007.
[5] K. Fukagata, K. Iwamoto, and N. Kasagi. Contribution of Reynolds stress distribution to the skin
friction in wall-bounded flows. Physics of Fluids, 14(11):L73–L76, 2002.
2