Influence of VLSMs in drag reduction by streamwise travelling wave of
spanwise wall velocity
Qiang Yang and Yongmann M. Chung
School of Engineering and Centre for Scientific Computing
University of Warwick
September 3, 2015
Q. Yang & Y. M. Chung (Warwick)
DR control
September 3, 2015
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Introduction
Drag reduction by streamwise travelling wave
Drag reduction can be achieved by applying panwise wall velocity in the form
w = A sin(kx x − ωt);
First explored by Quadrio et al. [2009], extensively studied by Hurst et al. [2014] with
particular emphasise on the Reynolds number effect;
Drag reduction maps at three Reynolds numbers: Reτ = 200, 400, and 800.
Drag reduction region (blue); drag increase region (yellow).
Q. Yang & Y. M. Chung (Warwick)
DR control
September 3, 2015
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Introduction
Reynolds number effect
Effect 1: at fixed control parameters, the drag reduction rate deteriorates at higher
Reynolds number (Touber and Leschziner [2012]).
Effect 2: the optimal control parameter (frequency, ω or wavenumber, κx ) moves to a
higher value at higher Reynolds number.
500 200
60
100
T+
50
1000
λ+
500
40
DR
DR
40
20
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
0
-20
5000 2000
60
0
0.05
0.1
0.15
20
Reτ=200
Reτ=400
Reτ=800
Reτ=1600
0
0.2
ω+
-20
0
0.005
0.01
0.015
0.02
κ+x
Drag reduction rate for oscillation cases (κx = 0.0, left) and stationary wave cases (ω = 0.0, right).
Q. Yang & Y. M. Chung (Warwick)
DR control
September 3, 2015
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Introduction
Graphic view of inner and outer structures
↑ Near wall vertical structures (small scales)
↑ Outer VLSMs (large/super scales)
↓ Footprints of VLSMs at y+ = 5
(no control/ oscillation/ stationary wave/ travelling wave)
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Introduction
Characteristic eddy
Near wall small streaks outer region super streaks, simultantaneously captured
using the characteristic eddy defined by Moin and Moser [1989].
80
0.6
60
y
y+
0.8
0.4
0.2
0-1
40
20
-0.5
0
0.5
1
Q. Yang & Y. M. Chung (Warwick)
-100
-50
0
50
100
z+
z
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Introduction
Role of VLSMs
Mathis et al. [2011]
u+ (y+ ) =
+
+
+
u∗ (y+ ) {1 + β u+
OL (yO , θL )} + αuOL (yO , θL )
| {z } |
{z
} |
{z
}
universal f ield
amplitudemodulation
linearsuperposition
Superimposition: VLSMs directly contribute to the near wall fluctuation though their
footprints.
Modulations: VLSMs modify the amplitude, frequency and phase of the near wall small
scale structures.
2p
Two point amplitude modulation CAM
= uL (y1 )EL (uS )(y2 ) at Reτ = 200, 400, 800 and 1600
Bernardini and Pirozzoli [2011].
Q. Yang & Y. M. Chung (Warwick)
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Result: superimposition
Streamwise wall shear stress, τw
Spectra filter size: λx ≥ 3h, λz ≥ 0.5h.
FIK (Fukagata and Kasagi [2002], Deck et al. [2014]):
Z 1
τw (kx , kz ) =
0
(1 − y)Φuv dy
Main contribution to τw at λx = h, λz = 0.5h; near wall structures have much stronger
footprints.
kx kz Φτw0 τw0
Q. Yang & Y. M. Chung (Warwick)
kx kz Φτw τw
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September 3, 2015
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Result: superimposition
Decomposed τw0
All scales
Small scales
Large scales
cu (kx , kz , y) = −Imhû∗ ∂t ûi/ kx h|û|2 i (del Álamo and Jiménez [2009])
uc
+
10
5
τw, present
τw, Jeon et. al. 1999
0
Q. Yang & Y. M. Chung (Warwick)
DR control
-1
10
0
λx
10
1
10
September 3, 2015
8 / 13
Result: superimposition
Correlation between τw0 VS τw
Small scales are easier to be damped;
Damping large scales is crucial for laminarization.
0.5
All
LS
SS
τw’/τw,0
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
τw/τw,0
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September 3, 2015
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Result: modulation
Conditioned DR maps under positive and negative VLSMs
1
overall
neg
pos
0.8
pdf
0.6
0.4
0.2
0
-2
-1
0
1
2
τw/σ
Pdf of VLSMs, CDF = 30% (Agostini and Leschziner [2014])
Wall shear stress (ω + = 0.0, κx+ = 0.004)
Q. Yang & Y. M. Chung (Warwick)
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September 3, 2015
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Result: modulation
Conditioned DR maps under positive and negative VLSMs
DR |P =
τw,0|P − τw|P
τw,0|P
× 100; DR |N =
τw,0|N − τw|N
τw,0|N
× 100
Drag increase region is narrower under positive VLSMs region;
Maximum DR is larger under positive VLSMs region.
All regions (DR)
Q. Yang & Y. M. Chung (Warwick)
Positive region (DR |P )
DR control
Negative region (DR |N )
September 3, 2015
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Result: modulation
Conditioned spectra under positive and negative VLSMs
Small scales under positive VLSMs region are more intense, shorter and convect
c↑
faster. ω ↑= uc kx = 2πu
(Ganapathisubramani et al. [2012]).
λ↓
x
↓ streamwise spectra
↑ Conditional windows
x ,ω)
|ω=ωc = 0)↓
kx − ω spectra (uc (kx ) = −ωc /kx , ∂ Φ(k
∂ω
×10-6
0.4
SS|pos+SS|neg
SS
SS|pos
SS|neg
0.3
kxΦττ
λ+x =640
0.2
0.1
0
λ+x =760
-1
0
10
10
10
1
λx
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Conclusions
Conclusions
VLSMs can be affected by streamwise travelling wave of wall motion, but the
control is less effective than the control to the small scales.
VLSMs modulate the small scales underneath it, leading the small scales to be short
in length and faster in convection velocity within positive VLSMs; the opposite in
negative VLSMs.
At higher Reynolds number, the near wall control needs to target at the positive
VLSMs footprint region.
Q. Yang & Y. M. Chung (Warwick)
DR control
September 3, 2015
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Conclusions
FFT filter size
λx ≥ 3h, λz ≥ 0.5h.
8
u+2
u+2
L
u+2
S
4
+2
+2
u , uL , uS
+2
6
2
0
0
10
10
1
2
10
y+
FFT
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HHT
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Conclusions
Characteristic eddy
Characteristic eddy (Moin and Moser [1989])
ubi (kx , y, kz ) =
1
4π 2
Z Z
ui (x, y, z)e−(ikx rx +kz rz ) drx drz
Φi j (kx , kz ) = hubi (kx , y0 , kz )ubj ∗ (kx , y, kz )i
Z
∗
d
(n) (k , y, k )dy0 = λ (n) (k , k )φ
(n) (k , y, k )
Φi j φd
x
z
x z
x
z
Z
E=
hui ui idy = ∑ λ (n)
n
ZZ
(u, v, w) =
!
d
(1)
∑ ∑ φ (kx , y, kz ) e−(ikx rx +kz rz ) dkx dkz
kx kz
Q. Yang & Y. M. Chung (Warwick)
DR control
September 3, 2015
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Conclusions
Conditioned λ2 under positive and negative VLSMs
0.3
overall
pos
neg
0.2
τw,S’/τw
0.1
0
-0.1
-0.2
-0.3
0
0.2
0.4
0.6
0.8
1
t/T
All regions
Positive region
Negative region
Q. Yang & Y. M. Chung (Warwick)
DR control
September 3, 2015
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