Large scale structures in skin friction drag reduction by spanwise wall
oscillation
Qiang Yang and Yongmann M. Chung
School of Engineering and Centre for Scientific Computing
University of Warwick
September 22, 2014
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
1 / 21
Introduction
Drag reduction by spanwise wall oscillation
Spanwise wall oscillation can have as much as 40% drag reduction (DR) at
Reτ = 200 (Jung et al. [1992]);
Cf
,
C f ,0
are the skin friction for control and no-control flows, respectively.
DR = 1 −
where C f and C f ,0
Simple formula (no need for feedback):
w = A sin(ωt) (or w = A sin( 2π
T t)),
where w is spanwise wall velocity; A is the spanwise peak wall velocity; ω is the wall
oscillation frequency ( T is the wall oscillation period).
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
2 / 21
Introduction
Simulation set up
Cases
Reτ
Rem
Lx × Ly × Lz
Nx × Ny × Nz
∆x+
∆y+
∆z+
CH200
CH400
CH800
CH800L
CH1600
200
400
800
800
1600
3150
7000
15700
15700
34500
16h × 2h × 6h
16h × 2h × 6h
12h × 2h × 4h
24h × 2h × 8h
12h × 2h × 4h
320 × 140 × 240
640 × 240 × 480
960 × 384 × 640
3840 × 384 × 1280
1920 × 800 × 1280
10.0
10.0
10.0
5.0
10.0
0.4 ∼ 6.0
0.4 ∼ 7.2
0.4 ∼ 9.7
0.4 ∼ 9.7
0.4 ∼ 9.2
5.0
5.0
5.0
5.0
5.0
Computational domain sizes and grid resolutions for DNS.
Using an in-house second-order accuracy FVM code in both space and time (Hurst
et al. [2014]).
CH800 (236 millions points) was simulated in local cluster Minerva at Warwick
University.
CH1600 (∼ 2 billion points) was simulated on HECToR (and Archer).
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
3 / 21
Introduction
Reynolds number effect for spanwise wall oscillation
1
0.9
100
T+
50
40
0.8
DR
C f / C f,0
500 200
60
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
20
0.7
0.6
0.5
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
0
0
500
1000
t
1500
2000
-20
0
0.05
+
0.1
0.15
0.2
ω+
Initial response of skin friction C f for T + = 100 at four Reynolds numbers (left) and DR
changes with oscillation frequencies and Reynolds numbers (right); .
As Reynolds number increases from Reτ = 200 to 1600:
Maximum DR decreases;
Optimal oscillation frequency shifts towards a higher value.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
4 / 21
Introduction
Large scale structures in no control flows
32
1.5
++
rms0
rms0
uwv rms
2
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
1
1
0.5
00 -1-1
10
10
0
10 0
10
1
10 1
10
+
+
yy
2
10 2
10
3
10 3
10
0
urms plots (left) and 2D spectra kx kz Φuu at y+ = 15 (right).
The logarithmic region keeps developing to form an outer peak in urms profile.
A spectra handle (Hoyas and Jiménez [2006]) in premultiplied spectra kx kz Φuu
becomes clearer at Reτ = 800 (lines) and Reτ = 1600 (shaded).
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
5 / 21
Introduction
Large scale structures in no control flows
32
1.5
++
rms0
rms0
uwv rms
2
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
1
1
0.5
00 -1-1
10
10
0
10 0
10
1
10 1
10
+
+
yy
2
10 2
10
3
10 3
10
0
urms plots (left) and 2D spectra kx kz Φuu at y+ = 15 (right).
The logarithmic region keeps developing to form an outer peak in urms profile.
A spectra handle (Hoyas and Jiménez [2006]) in premultiplied spectra kx kz Φuu
becomes clearer at Reτ = 800 (lines) and Reτ = 1600 (shaded).
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
6 / 21
Introduction
Large scale structures in no control flows
1D premultiplied spectra kx Φuu (left) and kz Φuu (right).
Inner peak scales well in wall units, with λx+ ≈ 1000 and λz+ ≈ 100.
The outer site penetrates deeply into the near wall region.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
7 / 21
Introduction
Large scale structures in no control flows
1D premultiplied spectra kx Φuu (left) and kz Φuu (right).
Inner peak scales well in wall units, with λx+ ≈ 1000 and λz+ ≈ 100.
The outer site penetrates deeply into the near wall region.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
8 / 21
Introduction
Large scale structures in no control flows
1D premultiplied spectra kx Φuu (left) and kz Φuu (right).
Inner peak scales well in wall units, with λx+ ≈ 1000 and λz+ ≈ 100.
The outer site penetrates deeply into the near wall region.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
9 / 21
Visualisation
Visualisation for CH1600
Visualisation for channel flow at Reτ = 1600 with and without spanwise wall
oscillation (movie).
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
10 / 21
Spectra & decomposition
Spectra change (kx kz Φuu )
No control cases
2D spectra kx kz Φuu at y+ = 15 (left) and 1D spectra kx Φuu (right).
The inner peak shifts towards smaller scales by wall oscillation.
Energy in the large scales is also modified by wall oscillation.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
11 / 21
Spectra & decomposition
Spectra change (kx kz Φuu )
Controlled at T + = 100
2D spectra kx kz Φuu at y+ = 15 (left) and 1D spectra kx Φuu (right).
The inner peak shifts towards smaller streamwise scales by wall oscillation.
Energy in the large scales is also modified by wall oscillation.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
12 / 21
Spectra & decomposition
Inner/Outer decomposition
1
1
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
0.6
0.8
(1-y)(1-uv0)
-uv0
0.8
0.4
0.2
0.6
0.4
0.2
0
0
0
0.2
0.4
y/h
0.6
0.8
1
0
0.2
0.4
y/h
0.6
0.8
1
Turbulent shear stress distribution, −uv0 (left) and the wall distance weighted form,
(1 − y)(−uv0 ) (right).
Contribution to C f is considered by using the formula from Fukagata et al. [2002]:
Cf =
6
+6
Re
Z 1
(1 − y)(−uv)dy
0
Q. Yang & Y. M. Chung (Warwick)
=
6
+6
Re
|
Z yp
0
(1 − y)(−uv)dy + 6
{z
} |
inner
Large structures in spanwise wall oscillation
Z 1
yp
(1 − y)(−uv)dy
{z
}
outer
September 22, 2014
13 / 21
Spectra & decomposition
Inner/Outer decomposition
1
1
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
0.6
0.8
(1-y)(1-uv0)
-uv0
0.8
0.4
0.2
0.6
0.4
0.2
0
0
0
0.2
0.4
y/h
0.6
0.8
1
0
0.2
0.4
y/h
0.6
0.8
1
Turbulent shear stress distribution, −uv0 (left) and the wall distance weighted form,
(1 − y)(−uv0 ) (right).
Contribution to C f is considered by using the formula from Fukagata et al. [2002]:
Cf =
6
+6
Re
Z 1
(1 − y)(−uv)dy
0
Q. Yang & Y. M. Chung (Warwick)
=
6
+6
Re
|
Z yp
0
(1 − y)(−uv)dy + 6
{z
} |
inner
Large structures in spanwise wall oscillation
Z 1
yp
(1 − y)(−uv)dy
{z
}
outer
September 22, 2014
14 / 21
Spectra & decomposition
Inner/Outer decomposition
1
40
30
0.6
DR
C f / C f0
0.8
0.4
Outer
Inner
Laminar
0.2
0
500
1000
1500 2000
20
Outer
Inner
Total
10
0
Reτ
500
1000
1500 2000
Reτ
Contribution to skin friction, C f , from turbulent shear stress in the inner region and outer
region. (. . . no control; — T + = 100).
As Reynolds number increases:
Laminar component to C f deceases;
Inner contribution for controlled cases remains almost constant;
C f reduction in the outer region is very large and remains almost constant.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
15 / 21
Spectra & decomposition
Spectra decomposition (kx kz Φuv )
2
Reτ=800 Reτ=1600 LevelkxkzΦuv/uτ
2
1
no control:
+
T =100:
0.0025
0.1
0.005
Reτ=800 Reτ=1600
λz
+
-uv, (-uv)L, (-uv)S
103
102
no control:
T+=100:
0.002
0.0015
0.001
0.0005
0
0
0.2
0.4
0.6
0.8
1
y/h
1
10
2
10
10
3
4
10
λ+x
2D spectra kx kz Φuv at y+ = 15 (left) and decomposed turbulent shear stress, −uv (right).
Spectra filter size λx /h = 3 and λz /h = 0.5.
Decomposed large scales, −uvL , for no control cases are scaled well in outer units.
Significant turbulent shear stress reduction comes from the large scales, −uvL .
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
16 / 21
Spectra & decomposition
Spectra decomposition (kx kz Φuv )
2
Reτ=800 Reτ=1600 LevelkxkzΦuv/uτ
2
1
no control:
+
T =100:
0.0025
0.1
0.005
Reτ=800 Reτ=1600
λz
+
-uv, (-uv)L, (-uv)S
103
102
no control:
T+=100:
0.002
0.0015
0.001
0.0005
0
0
0.2
0.4
0.6
0.8
y/h
1
10
2
10
10
3
4
10
λ+x
2D spectra kx kz Φuv at y+ = 15 (left) and decomposed turbulent shear stress, −uv (right).
Spectra filter size λx /h = 3 and λz /h = 0.5.
Decomposed large scales, −uvL , for no control cases are scaled well in outer units.
Significant turbulent shear stress reduction comes from the large scales, −uvL .
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
17 / 21
Spectra & decomposition
Spectra decomposition (kx kz Φuv )
2
Reτ=800 Reτ=1600 LevelkxkzΦuv/uτ
2
1
no control:
+
T =100:
0.0025
0.1
0.005
Reτ=800 Reτ=1600
λz
+
-uv, (-uv)L, (-uv)S
103
102
no control:
T+=100:
0.002
0.0015
0.001
0.0005
0
0
0.2
0.4
0.6
0.8
1
y/h
1
10
2
10
10
3
4
10
λ+x
2D spectra kx kz Φuv at y+ = 15 (left) and decomposed turbulent shear stress, −uv (right).
Spectra filter size λx /h = 3 and λz /h = 0.5.
Decomposed large scales, −uvL , for no control cases are scaled well in outer units.
Significant turbulent shear stress reduction comes from the large scales, −uvL .
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
18 / 21
Conclusion
Conclusions
Cases
CH800 (NO)
CH1600 (NO)
CH800 (WO)
Type
no control
no control
T + = 100
laminar
7%
4%
10% (0% ↓)
small scales
57%
57%
54% (31% ↓ )
large scales
36%
39%
36% (25% ↓ )
Contribution to skin friction, C f , from laminar, small scales and large scales.
The effect of large scales becomes important as Reynolds number increases.
Wall oscillation can also significantly weaken the large scales, even though those
structures are far away from the wall.
The interaction between the small and large scales may explain the Reynolds number
effect in spanwise wall oscillation.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
19 / 21
Conclusion
Acknowledgements
Thank You
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
20 / 21
References
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.
M. Quadrio, P. Ricco, and C. Viotti. Streamwise-travelling waves of spanwise wall
velocity for turbulent drag reduction. Journal of Fluid Mechanics, 627:161–178, 2009.
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.
K. C. Kim and R. J. Adrian. Very large-scale motion in the outer layer. Physics of Fluids,
11(2):417422, 1999.
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.
S. Hoyas and J. Jiménez. Scaling of the velocity fluctuations in turbulent channels up to
Reτ = 2003. Physics of Fluids, 18:011702, 2006.
R. Mathis, N. Hutchins, and I. Marusic. Large-scale amplitude modulation of the
small-scale structures in turbulent boundary layer. Journal of Fluid Mechanics, 628:
311–337, 2009.
B. Ganapathisubramani, N. Hutchins, J. P. Monty, D. Chung, and I. Marusic. Amplitude
and frequency modulation in wall turbulence. Journal of Fluid Mechanics, 712:61–91,
2012.
M. Bernardini and S. Pirozzoli. Inner/outer layer interactions in turbulent boundary
layers: A refined measure for the large-scale amplitude modulation mechanism. Physics
of Fluids, 23(6):061701, 2011.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
20 / 21
Conclusion
L. Agostini and M. A. Leschziner. On the influence of outer large-scale structures on
near-wall turbulence in channel flow. Physics of Fluids, 26:075107, 2014.
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.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
20 / 21
Conclusion
Density spectra
covariance of Fourier coefficients:
Rbi, j (κ,t) = hubi (κ,t)b
u j (κ 0 ,t)i = hubi (κ,t)b
u j (−κ,t)i = hubi (κ,t)b
u∗j (κ,t)i, κ ∈ [−∞, ∞].
(1)
one-sided energy spectra:
Ei j (κ,t) ≡ Ei j (κx , κz ,t)
= Ri j (κx , κz ,t) + Ri j (−κx , κz ,t) + Ri j (−κx , −κz ,t) + Ri j (κx , −κz ,t)
= ubi (κx , κz ,t)b
u∗j (κx , κz ,t) + ubi (−κx , κz ,t)b
u∗j (−κx , κz ,t)
(2)
+ ubi (−κx , −κz ,t)b
u∗j (−κx , −κz ,t) + ubi (κx , −κz ,t)b
u∗j (κx , −κz ,t)
= ubi (κx , κz ,t)b
u j (−κx , −κz ,t) + ubi (−κx , κz ,t)b
u j (κx , −κz ,t)
+ ubi (−κx , −κz ,t)b
u j (κx , κz ,t) + ubi (κx , −κz ,t)b
u j (−κx , κz ,t).
one-sided energy density spectra:
Φi, j (κ,t) = Ei, j (κ,t)/dκ.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
(3)
September 22, 2014
20 / 21
Conclusion
1D spectra validation
2.5
103
2
1.5
Φuu/(uτν)
10
kxΦuu/u2τ
1
10-1
10-3
1
-5
10
0.5
10-7
-9
10
-3
10
-2
10
10
0
-1
-3
-2
10
10
+
8
6
6
u2
∫Φuudk/u2τ
-1
kx
8
4
2
0
10
+
kx
4
2
-3
10
-2
10
10
-1
0
-1
10
0
10
+
kx
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
10
y
1
10
2
+
September 22, 2014
20 / 21
Conclusion
2D spectra validation
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
20 / 21
Conclusion
Spectra size
Table: Contribution to C f : spectra filter size test.
Cases
(1)λx ≥ 3h and λz ≥ 0.5h :
Reτ = 800(NC)
Reτ = 800(Osci)
(2)λx ≥ 3h:
Reτ = 800(NC)
Reτ = 800(Osci)
(3)λz ≥ 0.5h:
Reτ = 800(NC)
Reτ = 800(Osci)
(4)λx ≥ h:
Reτ = 800(NC)
Reτ = 800(Osci)
(5)λz ≥ h:
Reτ = 800(NC)
Reτ = 800(Osci)
Q. Yang & Y. M. Chung (Warwick)
laminar
turbulence (S)
turbulence (L)
7.3%
10.0% (0% ↓)
57.2%
53.8% (30.6% ↓ )
35.5%
36.2% (25.1% ↓ )
7.3%
10.0% (0% ↓)
44.0%
43.5% (27.3% ↓ )
48.7%
46.5% (29.6% ↓ )
7.3%
10.0% (0% ↓)
38.1%
30.8% (30.8% ↓ )
54.6%
54.2% (26.9% ↓ )
7.3%
10.0% (0% ↓)
27.3%
26.1% (28.0% ↓ )
65.4%
63.3% (28.7% ↓ )
7.3%
10.0% (0% ↓)
56.6%
53.5% (30.4% ↓ )
36.1%
36.5% (25.6% ↓ )
Large structures in spanwise wall oscillation
September 22, 2014
20 / 21
Conclusion
Spectra decomposition
Accumulative energy uu for Reτ = 800 and Reτ = 1600.
Spectra filter size: λx /h = 3 and λz /h = 0.5 (dash dot lines).
0
100
10
Level∫Φuudk
Level∫Φuudk
0.9
0.6
0.3
3
2
1
0.9
0.6
0.3
y
10-1
y
10-1
3
2
1
10-2
10
-2
-3
10
-3
10
-1
0
10
10
10
1
λx /h
-1
10
10
0
λz /h
Around 30% uu energy is contained in large scales near the wall.
More than 50% uu energy is contained in large scales in the outer region.
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
20 / 21
Conclusion
Spectra decomposed field
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
20 / 21
Conclusion
Location for Inner/Outer decomposition
1
40
30
0.6
DR
C f / C f0
0.8
0.4
Outer
Inner
Laminar
0.2
0
500
1000
1500 2000
20
Outer
Inner
Total
10
0
Reτ
500
1000
1500 2000
Reτ
Contribution to skin friction, C f from turbulent shear stress in the inner region and outer
region. (. . . no control; — T + = 100).
Q. Yang & Y. M. Chung (Warwick)
Large structures in spanwise wall oscillation
September 22, 2014
21 / 21