effect of finite size rigid fiber in turbulence drag...
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Effect of finite size rigid fiber in turbulence drag reduction
Minh Do-Quang and Geert Brethouwer and Gustav Amberg and Arne Johansson.Linné FLOW Centre, KTH mechanics,
Stockholm - Sweden
Trondheim, October 24-26, 2012
2
Suspension turbulent channel flow
Simula'on: An isotropic grid, 26 million cellsWall unit Δ+ = 2
Channel high ~ 2 cm, Reτ ~ 180.Fiber length ~ 0.15÷1.5mm (L+=3.2÷24) Density ra'o = 1.2 (cellulose vs. water) Volume frac'on = 0.1÷0.4%(dilute regime)No gravity effect.
3
Two phase flow model
Turbulent flow is model by Entropy Lattice Boltzmann method.(Keating 2007)
The no-slip boundary condition on the moving particle boundary is handled by an external boundary force method(Wu & Aidun 2010)
• Including the contact and lubrication force models for fibre-fibre and fibre-wall interactions (four ways coupling).
• Fiber model “could” take into account the fiber bending, twisting by apply the bending and twisting moments on each hinge.
Fibre modelHinges
Lagrangian points
3
Two phase flow model
Case Np rρ L Vf npL3 St St+
#1 4752 1.2 24 0.11% 0.325 0.245 2.326#2 11800 1.2 9.6 0.11% 0.052 0.181 1.716#3 35640 1.2 3.2 0.11% 0.006 0.110 1.038#5 35640 1.0 3.2 0.11% 0.006 0.091 0.865
14 M. Do-Quang, G. Amberg, G. Brethouwer, A.V. Johansson
case Np ap L
+vf [%] ↵ = npL
3r⇢ St St
+
#1 4752 15 24 0.11 0.325 1.2 0.245 2.33
#2 11800 6 9.6 0.11 0.052 1.2 0.181 1.72
#3 35640 2 3.2 0.11 0.006 1.2 0.110 1.04
#4 44642 6 9.6 0.43 0.195 1.2 0.181 1.72
#5 35640 2 3.2 0.11 0.006 1.0 0.091 0.87
Table 2. Fibre parameters for the LBM simulations.
We define a fibre Stokes number based on the fibre response time, ⌧p
(see Andersson
et al. (2012)) and the bulk flow time scale ⌧f
= 1/�̇, where �̇ is a shear rate of the bulk
flow (�̇ = |u|/�). We also formulate a near-wall Stokes number using the near-wall time
scale ⌧+ = ⌫/u2⌧
:
⌧p
=2⇢
f
9⇢l
⌫
L2
ap
ln(ap
+q
a2p
� 1)q
a2p
� 1, (4.2)
St =⌧p
⌧f
, St+ =⌧p
⌧+. (4.3)
Table 2 lists the fibre parameter values for the simulations performed. It is noticed that
the bulk Stokes number St is of the order of 0.1, while the near-wall Stokes number is
always less than 3.
In order to analyze the motion of the fibres, we have recorded the fibre statistics at
every 5 time steps. Statistical quantities such as mean velocity, fluctuation intensities,
Reynolds shear stress etc., were then computed based on one million samples, equivalent
to the recording time t = 30t⇤, to make sure that the fibre distribution achieved a
statistically steady state. To analyze the carrier phase, the fluid quantities were stored
every 10 time steps. Here, the averaging was performed over the entire composite flow
field, i.e., containing the regions of fluid and immersed fibres together, since the statistical
di↵erence between the composite fluid and the fluid is very small, as shown by Uhlmann
(2008).
4.1. Turbulent flow of a fibre suspension.
The streamwise mean velocity profiles of fibres huf
i and fluid hul
i for cases #1, #2, #3
and #5, referring to the cases listed in table 2, are shown in Figure 6 as function of
• Small Stokes number• Small inertial effects because the fibres are almost
(or exactly) neutrally buoyant, rρ=1.2 or 1.0•No gravity.
5
Preferential distribution of the fibres.
Three dimensional view, fibre length L+ = 9.6, vf = 0.44%. The fibre plotted here are fibres within (y+ < 16). The dimensions of the outlined box are 1420, 64 and 396 wall units in the
x, y and z dimensions, respectively.
High speed
streakHigh speed
streak High speed
streakHigh speed
streakLow speed
streakLow speed
streak
6
Preferential distribution of the fibres.
Fiber number density distribution as a function of fluctuating streamwise velocity. The data is collected for a region (4 < y+ < 16) and a duration
time t = 30t*. Number of bins = 100.
10
20
30
40
y+
30
50
70
90
z+
−0.4
0
0.4
q [/
]0.2
0.5
0.8
e [/
]
0.1
0.5
0.9
Cb
−5
0
5
1x.1
04
−5
0
5
1y.1
04
0 50 100 150 200 250−0.8
0
0.8
1z.1
05
Time
⎨
Time history of a fibre moving in the vicinity of the wall
7
Location of the fibre
⎨Orientation
Schematic of fluid flow between streaks
Fibre stay in the high speed streak longer than in low speed streak
Fibre length L+=24
A fibre moving in the vicinity of the wall
8
ρs/ρf=1.2ρf; Reτ=180; R=0.8+, aspect ra'o=15 ; Volume frac'on = 0.11%RED color: fiber is go away from the wall; GREEN color is go to the wall.
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Fibers have higher mean velocity than the fluid near the wall
Case Np rρ L Vf npL3 St St+
#1 4752 1.2 24 0.11% 0.325 0.245 2.326#2 11800 1.2 9.6 0.11% 0.052 0.181 1.716#3 35640 1.2 3.2 0.11% 0.006 0.110 1.038#5 35640 1.0 3.2 0.11% 0.006 0.091 0.865
The higher velocity not due to the inertial
as point particle !
8
Preferential concentration
Preferential concentration due to fibre-wall interaction
12
ρs/ρf=1.2ρf; Reτ=180; R=0.8+
aspect ra'o=15 ; Volume frac'on = 0.11%
aspect ra'o=20 ; Volume frac'on = 0.44%
10
Turbulent vortices in the flow field.The second invariant Q2
Without fiber
With fibers L+=24, Volume fraction Vf = 0.11%
With fibers L+=32, Volume fraction Vf = 0.44%
100 101
10−4
10−3
10−2
10−1
100
Wavenumber (k)
E xx /
u o2
L+=32, Vf=0.44%L+=24, Vf=0.11%No fiberMoser et al. (1999)
20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
y+
−<u’v’>/u o2
No fibersL+=10, Vf=0.11%L+=24, Vf=0.44%L+=32, Vf=0.44%
101 1024
6
8
10
12
14
16
18
20
y+
<u>/u o
With fibre L+=10, Vf=0.11%With fibre L+=24, Vf=0.44%With fibre L+=32, Vf=0.44%
10
Turbulent drag reduction by rigid fibres
0 20 40 60 80 100 120 140 160 1800
0.5
1
1.5
2
2.5
3
y+
u rms
+, v
rms
+, w
rms
+
No fibresWith fibre L+=10, Vf=0.11%With fibre L+=24, Vf=0.44%With fibre L+=32, Vf=0.44%
10
Mean values of direction cosines
y+
|cos(ax)|
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y+
|cos(ay)|
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y+
|cos(az)|
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Streamwise SpanwiseWall-normal
ρs/ρf=1.2ρf; Reτ=180; R=0.8+, aspect ra'o=20 ; Volume frac'on = 0.44%
0 20 40 60 80 100 120 140 160 1800.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y+
|cos(a x)|
L+=3.2, case #3L+=9.6, case #2L+=24, case #1
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
y+
|cos(a y)|
L+=3.2, case #3L+=9.6, case #2L+=24, case #1
0 20 40 60 80 100 120 140 160 1800.1
0.2
0.3
0.4
0.5
0.6
y+
|cos(a z)|
L+=3.2, case #3L+=9.6, case #2L+=24, case #1
(C)
(B)
(A)
10
Mean values of direction cosines
11
ρs/ρf=1.2ρf; Reτ=180; R=0.8+, aspect ra'o=20 ; Volume frac'on = 0.44%
12
Conclusions
• The finite size leads to fibre-turbulence interactions that are significantly different from earlier published results for point-like particles (e.g. elongated ellipsoids smaller than the Kolmogorov scale). -An effect that becomes increasingly accentuated with fibre length is an accumulation in high-speed regions near the wall, resulting in a mean fibre velocity that is higher than the mean fluid velocity.
-There is also a resulting preferential concentration near the wall, which was shown to persist even for neutrally buoyant fibres.
• The finite length was also shown to influence the orientation of the fibres in the near-wall region, -Short fibres preferentially rotating around the spanwise direction.
-Long fibres preferentially rotating in the shear (xz) plane and reducing the turbulence intensities in the buffer region (increasing in the bulk).
Acknowledgment