Scalar Mixing in Turbulent, Confined Axi-symmetric Co

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Scalar Mixing in
Turbulent, Confined
Axisymmetric Co-flows
C.N. Markides & E. Mastorakos
Hopkinson Laboratory, Department of Engineering
Monday, 6th of February, 2006
1/19
The Turbulent, Confined
Axisymmetric Co-flow1

Axisymmetric flow configuration (r,z)

Dimensions:
–
–
–
–
Quartz tube inner Ø D=33.96mm
Injector outer Ø do=2.975mm
Injector inner Ø d=2.248mm and 1.027mm
Domain (laser sheet) height 60mm

Co-flow air preheated to

Upstream (63mm from injector nozzle) perforated grid
(M=3mm circular holes, 44% solidity) enhances co-flow
turbulence level

Begins as nitrogen-diluted fuel
YC2H2=0.73 C2H2/N2 or YH2=0.14 H2/N2
Passes though seeder that introduces 20% acetone b.v.
Fuel Injection
z
r
Uco, Tco
Injector
Quartz Tube
For co-flow define:
–
–

Uinj, Tinj
Injected stream:
–
–
–

Laser Sheet
200±1oC
Reco = UcoM/ν, or, Returb = u'coLturb/ν
Reco range: 395-950; ***Returb≈Reco/10***
For injected stream define:
–
–
–
–
–
Acetone Seeded
Fuel
Air from MFC
υinj=Uinj/Uco
υinj range: 1.1-4.9; ***Not a free jet***
δinj= ρinj/ρco
δinj=1.2 for C2H2/N2/acetone
δinj=0.8 for H2/N2/acetone
1. See Markides and Mastorakos (2005) in Proc. Combust. Inst.; Markides (2006) Ph.D. Thesis;
Markides, De Paola and Mastorakos (2006) shortly in Exp. Therm. Fluid Sci. for details.
Grid
2/19
Planar Laser-Induced Fluorescence
Measurements (Brief Overview2)

Each ‘Run’ corresponds to a set of fixed Uco, Uinj and Yfuel conditions (i.e. Reco/Returb, υinj
and δfuel)

The raw measurements are near-instantaneous (integrated over 0.4μs) images of size
(height x width) 1280 x 480 pixels

For each ‘Run’ we generated 200 images at 10Hz (every 0.1s)

Images are 2-dimensional planar measurements:
–
–
–
–
–
–
Smallest lengthscale in the flow is Kolmogorov (ηK) and was measured at 0.2-0.3mm
Laser-sheet thickness (spatial resolution) ≈ 0.10±0.03mm
Measured intensity at any image pixel is the spatial average over a square region of length 0.0500.055mm at that point in the flow
Ensemble spatial resolution is 0.09mm
Ability to transfer contrast information quantified by Modulation Transfer Function (MTF).
Investigation revealed ability to resolve 70-80% of spatial detail with 4-5 pixels or 0.3mm
Local intensity proportional to local volumetric/molar concentration of acetone vapour, so that:
~
 

nr , z 
nr  0, z  0
Convert to mass-based mixture fraction by:
~
  [1  (1   1 ) inj 1 ]1

Two-dimensional scalar dissipation (the Greek one - χ) was calculated by:
 2D
   2    2 
 2 D       ***But before this was done the images of ξ were filtered and denoised***
 r   z  
2. See Markides and Mastorakos (2006) in Chem. Eng. Sci.; Markides (2006) Ph.D. Thesis for details.
3/19
Quantifying Measurement
Resolution
(m)
1
3 1/4
0.6
3
0.4
0.2
25
8
6
4
3
Resolution (pixels)
x 10
Black
0
-1
z = 1 mm
z = 2 mm
z = 22 mm
z = 42 mm
-0.5
0
2r/D (-)
0.5
1
6
4
White
2
0
0
50
100
150 200
Intensity/Counts (-)
250
-1
-2
-3
10
Pdf (-)
8
log (Normalized Pdf) (-)
10
-1
10
log (Normalized Pdf) (-)
100
0
4
-3
-10
200
2
0
-2
300
K = [ Lturb/u' ]
MTF
0.8
400
300
-5
0
5
10
Normalized Pixel Intensity/Counts (-)
-4
-5
-80
-40
0
40
80
Normalized Pixel Intensity/Counts (-)
4/19
Obtaining the instantaneous χ2D field
from the instantaneous ξ field (I)
ξ
100
100
100
90
90
90
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
10 20 30 40 50 60 70 80 90 100
χ2D
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
100
100
100
90
90
90
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
5/19
Obtaining the instantaneous χ2D field
from the instantaneous ξ field (II)
ξ
100
100
100
90
90
90
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
10 20 30 40 50 60 70 80 90 100
χ2D
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
100
100
100
90
90
90
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50 60 70 80 90 100
6/19
Obtaining the instantaneous χ2D field
from the instantaneous ξ field (III)
90
90
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
10 20 30 40 50 60 70 80 90 100
χ2D
1
0.8
 
2
0.6
 
3
0.4
 
2
0.2
0
0
10 20 30 40 50 60 70 80 90 100
100
100
100
90
90
90
80
80
80
70
70
70
60
60
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
10 20 30 40 50 60 70 80 90 100
 
1
-6
100
 (-) and /2D (1x10 /s)
ξ
100
10 20 30 40 50 60 70 80 90 100
20
40
60
z(r=0) (pixels)
80
100
10 20 30 40 50 60 70 80 90 100
7/19
Obtaining the instantaneous χ2D field
from the instantaneous ξ field (IV)

At this stage have considered the squares of the spatial gradients of ξ, (∂ξ/∂r)2
and (∂ξ/∂z)2; we also have the spatial gradients of ξ', (∂ξ'/∂r)2 and (∂ξ'/∂z)2

Finally, need molecular diffusivity (D):
DAB
–
0.0143T 1.75



2
  1A/ 3   1B/ 3
p
1
1 
 MA  MB 


2
Consider ternary (acetone-’1’, nitrogen-’2’ and fuel-’3’) diffusion coefficients from
binary diffusion coefficients (D11, D12, D13, D22, D23, D33)
D11T 
D13
D

X 2  13  1
 D12 
1
X 1 D23  1  X 1 
D12
–
Simplify by assuming that D12≈D13 or X1<<1 so that D12T<<D11T and calculate D1,mix at
each point in the flow, given that we already have the instantaneous X1 (from ξ)
D1,mix 
1
X2 X3

D12 D13
8/19
Calculating Mixing
Quantities (I)

For each ‘Run’ and at each pixel representing a
location in physical space (r,z) loop through 200
images:
– Of each version of ξ (raw and all stages of processing) to
obtain the mean and variance of each version of ξ
– Of each version of corresponding (∂ξ/∂r)2 and (∂ξ/∂z)2 to
obtain the mean and variance of each version of (∂ξ/∂r)2
and (∂ξ/∂z)2
– Of each version of corresponding (∂ξ'/∂r)2 and (∂ξ'/∂z)2 to
obtain the mean and variance of each version of (∂ξ'/∂r)2
and (∂ξ'/∂z)2
– Evaluate spatial 2-point autocorrelation matrices along
centreline and at left/right half-widths
– Compile radial volume-averaged quantities (i.e. at one z
group all r data together)
9/19
Calculating Mixing
Quantities (II)

For each ‘Run’ and at each pixel representing a location in physical space (r,z)
consider a window of size 1x1 (or 2x10) ηK containing 40 (or 720) pixels and loop
through 200 images:
–
–
–
–
–
–
–
–
At 15 axial locations from 1mm to 60mm in steps of 4mm
At 5 radial locations with r=0, ±d/2, ±d
Calculate the local mean, variance, skewness and kurtosis of all versions of all variables (ξ
and χ2D)
Calculate the local mean, variance, skewness and kurtosis of the logarithm of all versions
of χ2D
Compile pdfs of all versions of all variables (ξ and χ2D) each composed of 90 and 30 points
respectively spanning the min-max range (from about 140,000 data points)
Separate the χ2D data into 30 groups between the min-max range of ξ by considering the
corresponding values of ξ (χ2D|ξ)
Calculate the local mean, variance, skewness and kurtosis of χ2D|ξ and ln(χ2D|ξ)
Compile pdfs of all versions of χ2D|ξ:



unless number of data points in the local pdf is less than 300
each composed of 30 points respectively spanning the min-max range
from about 2,000-10,000 data points
10/19
Mean ξ (I)
20
20
0.8
0.3
0.2
0
-2 0 2
r/d (-)
0.8
Not affected by image processing
8
8
8
8
8
6
6
6
6
6
6
10
4
4
z/d (-)
0.2
z/d (-)
4
z/d (-)
4
10
0.8
z/d (-)
z/d (-)
8
4
0.4
0.4
4
0
r/d (-)
2
0
r/d (-)
0.6
2
0.8
0.8
2
0
-2
0
r/d (-)
0.2
0.4 .6
0
0
-2
0
-2
2
0.2
2
0.8
2
0.2
0
r/d (-)
0.8
0.4
0.2
2
0.8
0.2 0.4 .6
0
0
r/d (-)
0
-2
2
0.6
6
0.
0.
6
0.6
0.4
0.2
0
-2
2
0.4
2
0.2
z/d (-)
10
10
0.1
0.8
10
10
0.2
0
-2 0 2
r/d (-)
0.6
0.4
Jet cases with the 2.248mm injector (υinj=3 and 4)
0.2
0.6
0.4

0.4
5
5
Right:
–
0.2
0.5
0.2
10
0.4
0.2

All are equal velocity cases (Uinj≈Uco; υinj=1.0±0.2) with the
2.248mm injector and varying Reco/Returb
10
0.4
–
0.6
z/d (-)
Below:
z/d (-)

0.7
15
15
2
0
-2
0
r/d (-)
2
11/19
Mean ξ (II)
1
1
   (-)
   (-)
z/d=1
0.5
0
0
5
10
15
z/d (-)
20
25
0
z/d=6
-1
0
r/d (-)
1
2
10
z/d=5
 / (r=0) (-)
10
   (-)
z/d=4
z/d=5
0
-2
0
-2
-2
10
z/d=1
z/d=4, 5 and 6
-1
-4
10 -2
10
0.5
0
10
z/d (-)
10
2
10
0
0.05
0.1
2
2
(r/d) /(z/d) (-)
0.15
12/19
Variance of ξ (I)
Left:
–
Right:
–
Affected by image processing
10
10
10
0.02
8
8
8
0.02
0
-2
2
0.1
0.0
08.1
0
r/d (-)
2
0
-2
z/d (-)
0.0
6
0.02
z/d (-)
0.0
4
0.02
z/d (-)
2
4
0.04
6
4
2
0.02
0
r/d (-)
0.
04
0.04
0.02
2
6
0.04
0
r/d (-)
4
0.02
0.040.06 8
0.0
0
-2
0.02
2
6 0.04
0.0
0.0
8
1
.
0
0.08
0.1
6
0.02
4
8
0.02
0.02
6
0.02
2
0
-2
0.04
00
.0.0
24
10
0.02

Jet case
0.04

Equal velocity case
z/d (-)

0.02
0
r/d (-)
2
13/19
Variance of ξ (II)
5
0.1
 ' /   (-)
Run 1: Re=405,=1.1
Run 5: Re=560,=1.1
Run 8: Re=560,=2.4
Run 10: Re=745,=2.4
2
2
 '  (-)
0.15
0.05
0
0
5
10
z/d (-)
15
20
4
3
Run 1: Re=405, =1.1
Run 2: Re=425, =1.2
Run 4: Re=530, =1.1
Run 5: Re=560, =1.1
Run 8: Re=560, =2.4
Run 10: Re=745, =2.4
Run 11: Re=895, =4.3
Run 12: Re=895, =5.0
2
1
0
0
5
10
15
z/d (-)
20
25
14/19
Mean χ2D (I)

Equal velocity case

Significantly affected by image processing
5
5
5
60
4
10
4
4
70
1
0
-1
0
r/d (-)
20
2
30
1
0
-1
40
20
10
20
0
r/d (-)
40
20 30
60
10
3
1
40
30
0
-1
1
40
60
2
30 0
4
1
40
2
3
z/d (-)
20
10
20
z/d (-)
20
10
3
20
z/d (-)
10
50
30
40
0
r/d (-)
10
1
15/19
Mean χ2D (II)
80
300
 2D  (1/s)
60
 2D  (1/s)
Run 1: Re=405,  =1.1
Run 5: Re=560,  =1.1
Run 8: Re=560,  =2.4
Run 10: Re=745,  =2.4
200
40
100
20
0
0
5
10
15
z/d (-)
20
25
0
-1
-0.5
0
r/d (-)
0.5
1
16/19
(Non-strict) Isotropy in χ2D
2
2
10
10
 (1/s)
0
radial
10
-1
10
-2
10
-3
10 -3
-2
-1
0
1
10
10
10
10
10
 axial  (1/s)
0
10


radial
 (1/s)
1
10
-2
10
2
10 -2
10
0
10
 axial  (1/s)
10
2
17/19
Mean Scalar Dissipation
Modelling and CD (I)

At each location in physical space where we would like to evaluate:
CD 


 2
k
 
 
Firstly we need to recover the mean full 3-dimensional χ from the
mean χ2D (along the centreline by symmetry the mean gradients
squared in the radial and azimuthal direction are equal)
– Also examined isotropy of the two components

We also need knowledge of the turbulent timescale (k/ε) where k is
the turbulence kinetic energy and ε the mean turbulence dissipation
– Use (k/ε)/(Lturb/u')≈1.7±0.2 Pope (2000)
18/19
Mean Scalar Dissipation
Modelling and CD (II)
U/Uco
2
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0
20
0.4
0.14
0.35
0.12
0.3
Centreline
(z+63mm)/d
22
24
26
28
30
32
34
36
38
40
Centreline
0.1
0.25
0.08
0.2
u'/Uco
0.06
0.15
0.04
0.1
0.02
0.05
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0
20
(z+63mm)/d
22
24
26
28
30
32
34
36
38
40
19/19
Mean Scalar Dissipation
Modelling and CD (III)
1.8
10
τturb/τturb(z/d=35)
8
3D /  '  .(k/)
2
1.6
2
1.4
1.2
6
4
1
0.8
2
0.6
0.4
0.2
0
20
22
24
26
28
30
32
34
36
38
40
0
0
0.5
 /(k/ ) (-)
res
1
Scalar Mixing in
Turbulent, Confined
Axisymmetric Co-flows
C.N. Markides & E. Mastorakos
Hopkinson Laboratory, Department of Engineering
Monday, 6th of February, 2006

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