Turbulent skin-friction drag reduction by travelling waves induced
by spanwise Lorentz force
Qiang Yang and Yongmann M. Chung
School of Engineering and Centre for Scientific Computing
University of Warwick
August 26, 2015
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
1 / 16
Introduction
Introduction
Spanwise wall oscillation can have as much as 40% drag reduction (DR) at
Reτ = 200 (Jung et al. [1992]).
Spanwise oscillation generated by spanwise Lorentz force (Berger et al. [2000]).
Spanwise travelling wave generated by spanwise Lorentz force (Du and Karniadakis
[2000]).
Streamwise travelling wave generated by spanwise Lorentz force (Huang and Li
[2010]).
Spanwise travelling wave by plasma actuator force (Choi et al. [2011]).
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
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Methodology
Navier-Stokes solver
Incompressible Navier-Stokes equations:
∂ ui ∂ ui u j
∂p
1 ∂ 2 ui
+
=
+
+ fi ,
∂t
∂xj
∂ xi Re ∂ xi2
f1 = 0, f2 = 0, f3 = Ae−y/∆ sin(κx x + κz z − ωt).
∆ - penetration depth; κx , κz - wavenumber; ω - angular frequency.
Finite volume code (fully implicit
fractional step method);
Geometry and coordinate
2DECOMP&FFT library for
parallelisation;
Standard channel flow: periodic
boundary in streamwise (x) and
spanwise (z) directions; no-slip
boundary on top and bottom walls (y).
Initial field for control: a fully
developed turbulent flow at Reτ = 200.
Constant mass flow rate.
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
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Result
Body force distribution
oscillation OC ( f3 = Ae−y/∆ sin(−ωt));
forward/backward streamwise travelling wave FST,BST ( f3 = Ae−y/∆ sin(κx x ± ωt));
spanwise travelling wave ST ( f3 = Ae−y/∆ sin(κz z − ωt));
oblique travelling wave OT ( f3 = Ae−y/∆ sin(κx x + κz z − ωt)).
FST
Q. Yang & Y. M. Chung (Warwick)
BST
ST
Drag reduction by spanwise Lorentz force
OT
August 26, 2015
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Result
Validation
DR validation for spanwise oscillation Lorentz force with A = 0.5.
∆+ = 10
T + = 100
T + = 50 T + = 100
∆+ = 5
∆+ = 10
DR (Berger et al. [2000])
15
25
26
25
DR (Present)
16
25
19
25
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
∆+ = 20
23
24
August 26, 2015
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Result
DR map
(a)
(b)
Drag reduction map for (a) streamwise travelling wave and (b) spanwise travelling wave.
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
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Result
Oblique travelling wave
√
Instantaneous velocity magnitude u2 + v2 + w2 at y+ = 5 for different travelling wave
◦
◦
◦
◦
angles: θ = 0 , 15 , 30 , 45 , 60◦ , 75◦ , 90◦ , 105◦ , 120◦ , 135◦ , 150◦ , 180◦ .
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
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Result
Oblique travelling wave
1
0.15
0.9
θ
ui,rms
C f /C f,0
0.1
0.8
0.7
θ
0.05
θ
0.6
0.5
0
50
100
150
200
250
300
0
10
t
(a)
0
1
2
10
10
y+
(b)
×10-3
2.5
u1 = u cos θ − w sin θ
2
-u1u2(1-y)
2
2
2
2
u02
1 = (u −U ) cos θ + (w −
2
2
W ) sin θ − (uw −UW ) sin(2θ )
u01 u02 = uv cos θ − vw sin θ
1.5
θ
1
0.5
(a) Wall shear stress time history,
(b) urms and (c) −uv statistics
profiles.
Q. Yang & Y. M. Chung (Warwick)
0
0
(c)
Drag reduction by spanwise Lorentz force
0.2
0.4
0.6
0.8
1
y/h
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Result
Convection time scale
T = λ /(Uc − ω/κ) → T = λ /(Uc cos(θ ) − ω/κ).
80
40
OC
FST
ST
BST
60
y+
DR
20
∆+=10,OC
∆+=5,OC
+
∆ =10,oblique
∆+=10,stationary
∆+=5,stationary
0
-20
0
0.05
0.1
ω+
0.15
40
20
0
-0.4
-0.2
0
0.2
0.4
W0
(a)
(b)
(a) Drag reduction comparison (Uc+ = 5 (DR scaled by 1.1 for ∆+ = 10 and 0.7 for
∆+ = 5) for stationary wave; Uc+ = 8 for oblique wave) and (b) generalised Stokes layer.
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
9 / 16
Result
Streaks behaviour
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)
Drag reduction by spanwise Lorentz force
August 26, 2015
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Result
Streaks behaviour
High- (red) and low-speed (green) streaks during one oscillation period (phase 1-16 (left
to right, top to bottom)).
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
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Result
Vertical structure behaviour
Ensemble averaged λ2 structures (Jeong et al. [1997])
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
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Result
Vertical structure behaviour
Vertical structures based on λ2 were conditioned on ωx0 .
Positive (yellow) and negative (red) λ2 structure during one oscillation period.
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
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Result
Vertical structure behaviour
Vertical structures based on λ2 were conditioned on ωx0 .
(a)
(b)
(c)
(d)
(e)
(f)
FST
ST
BST
Positive (yellow) and negative (red) λ2 structure in the first half wave length (a,c,e) and
second half wave length (b,d,f).
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
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Result
Spectra change
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
OC
FST
ST
BST
One-dimensional pre-multiplied energy spectra kx kz Φuu and kx kz Φww over the half channel
height.
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
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Conclusion
Conclusion
Travelling wave in streamwise and spanwise by spanwise Lorentz force can achieve
drag reduction.
Backward travelling wave is more efficient than wave travel in other directions.
Streamwise travelling wave can achieve higher drag reduction than oscillation, but
spanwise travelling wave always has less drag reduction than oscillation.
A time scale T = λ /(Uc cos(θ ) − ω/κ) is important to divide the drag reduction
map regions, even for oblique travelling wave.
The periodic behaviour of streaks and vertical structures were shown by POD and
conditional average.
Spectra analysis suggests the break-up of near wall streaks in the drag reduction
cases.
Q. Yang & Y. M. Chung (Warwick)
Drag reduction by spanwise Lorentz force
August 26, 2015
16 / 16