Scalar Mixing inScalar Mixing inTurbulent, ConfinedTurbulent, ConfinedAxisymmetric Co-flowsAxisymmetric Co-flows

C.N. Markides & E. MastorakosC.N. Markides & E. MastorakosHopkinson Laboratory, Department of EngineeringHopkinson Laboratory, Department of Engineering

Monday, 6Monday, 6thth of of February, 2006February, 2006

The Turbulent, The Turbulent, ConfinedConfinedAxisymmetric Co-flowAxisymmetric Co-flow11 Axisymmetric flow configuration (r,z)Axisymmetric flow configuration (r,z)

Dimensions:Dimensions:– Quartz tube inner Quartz tube inner Ø D=Ø D=33.96mm33.96mm– Injector outer Injector outer ØØ d doo=2.975mm=2.975mm– Injector inner Injector inner ØØ d=2.248mm and 1.027mm d=2.248mm and 1.027mm– Domain (laser sheet) height 60mmDomain (laser sheet) height 60mm

Co-flow air preheated to 200Co-flow air preheated to 200±1±1ooCC

Upstream (63mm from injector nozzle) perforated Upstream (63mm from injector nozzle) perforated grid (M=3mm circular holes, 44% solidity) grid (M=3mm circular holes, 44% solidity) enhances co-flow turbulence levelenhances co-flow turbulence level

Injected stream:Injected stream:– Begins as nitrogen-diluted fuelBegins as nitrogen-diluted fuel– YYC2H2C2H2=0.73 C=0.73 C22HH22/N/N22 or Y or YH2H2=0.14 H=0.14 H22/N/N22

– Passes though seeder that introduces 20% Passes though seeder that introduces 20% acetone b.v.acetone b.v.

For co-flow define:For co-flow define:– ReRecoco = U = UcocoM/M/νν, or, Re, or, Returbturb = u = u''cocoLLturbturb//νν– ReRecoco range: 395-950; ***Re range: 395-950; ***Returbturb≈Re≈Recoco/10***/10***

For injected stream define:For injected stream define:– υυinjinj=U=Uinjinj/U/Ucoco

– υυinjinj range: 1.1-4.9; ***Not a free jet*** range: 1.1-4.9; ***Not a free jet***– δδinjinj= = ρρinjinj//ρρcoco

– δδinjinj=1.2 =1.2 forfor CC22HH22/N/N22/acetone/acetone– δδinjinj=0.8=0.8 for H for H22/N/N22/acetone/acetone

1. See Markides and Mastorakos (2005) in Proc. Combust. Inst.; Markides (2006) Ph.D. 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 Thesis; Markides, De Paola and Mastorakos (2006) shortly in Exp. Therm. Fluid Sci. for details.details.

Grid

Injector

Quartz Tube

Air from MFC

Uco, Tco

Acetone Seeded Fuel

Laser Sheet

Fuel Injection Uinj, Tinj

z

r

1/19

Planar Laser-Induced Planar Laser-Induced Fluorescence Fluorescence Measurements (Brief Measurements (Brief OverviewOverview22)) Each ‘Run’ corresponds to a set of fixed UEach ‘Run’ corresponds to a set of fixed Ucoco, U, Uinjinj and Y and Yfuelfuel conditions (i.e. Re conditions (i.e. Recoco/Re/Returbturb, ,

υυinjinj and and δδfuelfuel))

The raw measurements are near-instantaneous (integrated over 0.4The raw measurements are near-instantaneous (integrated over 0.4μμs) images of s) images of size (height x width) 1280 x 480 pixelssize (height x width) 1280 x 480 pixels

For each ‘Run’ we generated 200 images at 10Hz (every 0.1s)For each ‘Run’ we generated 200 images at 10Hz (every 0.1s)

Images are 2-dimensional planar measurements:Images are 2-dimensional planar measurements:– Smallest lengthscale in the flow is Kolmogorov (Smallest lengthscale in the flow is Kolmogorov (ηηKK) and was measured at ) and was measured at 0.2-0.3mm0.2-0.3mm– Laser-sheet thickness (spatial resolution) ≈ Laser-sheet thickness (spatial resolution) ≈ 0.100.10±0.03±0.03mmmm– Measured intensity at any image pixel is the spatial average over a square region of length Measured intensity at any image pixel is the spatial average over a square region of length

0.050-0.055mm0.050-0.055mm at that point in the flow at that point in the flow– Ensemble spatial resolution is Ensemble spatial resolution is 0.09mm0.09mm– Ability to transfer contrast information quantified by Modulation Transfer Function (MTF). Ability to transfer contrast information quantified by Modulation Transfer Function (MTF).

Investigation revealed ability to resolve 70-80% of spatial detail with Investigation revealed ability to resolve 70-80% of spatial detail with 4-5 pixels or 0.3mm4-5 pixels or 0.3mm– Local intensity proportional to local volumetric/molar concentration of acetone vapour, so Local intensity proportional to local volumetric/molar concentration of acetone vapour, so

that:that:

Convert to mass-based mixture fraction by:Convert to mass-based mixture fraction by:

Two-dimensional scalar dissipation (the Greek one - Two-dimensional scalar dissipation (the Greek one - χχ) was calculated by:) was calculated by:

2. See Markides and Mastorakos (2006) in Chem. Eng. Sci.; Markides (2006) Ph.D. Thesis 2. See Markides and Mastorakos (2006) in Chem. Eng. Sci.; Markides (2006) Ph.D. Thesis for details.for details.

0,0

,~

zrn

zrn

111 ])~

1(1[ inj

22

2 2zr

DD

***But before this was done the images of ξ were filtered and denoised***

2/19

Quantifying Quantifying Measurement ResolutionMeasurement Resolution

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log 1

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mal

ized

Pd

f) (

-)

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log 1

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mal

ized

Pd

f) (

-)

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K

= [3

Ltu

rb/u

'3 ]1/4 ( m

)

z = 1 mmz = 2 mmz = 22 mmz = 42 mm

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Pdf

(-)

Black

White

3/19

Obtaining the instantaneous Obtaining the instantaneous χχ22DD field from the instantaneous field from the instantaneous ξ ξ field (I)field (I)

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4/19

Obtaining the instantaneous Obtaining the instantaneous χχ22DD field from the instantaneous field from the instantaneous ξ ξ field (II)field (II)

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5/19

Obtaining the instantaneous Obtaining the instantaneous χχ22DD field from the instantaneous field from the instantaneous ξ ξ field (III)field (III)

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χ2D

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(-

) and

/2

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)

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6/19

Obtaining the instantaneous Obtaining the instantaneous χχ22DD field from the instantaneous field from the instantaneous ξ ξ field (IV)field (IV) At this stage have considered the squares of the spatial gradients of At this stage have considered the squares of the spatial gradients of ξξ, ,

((∂∂ξξ/∂r)/∂r)22 and and ((∂∂ξξ/∂z)/∂z)22; we also have the spatial gradients of ; we also have the spatial gradients of ξξ'', , ((∂∂ξξ''/∂r)/∂r)22 and and ((∂∂ξξ''/∂z)/∂z)22 Finally, need molecular diffusivity (D):Finally, need molecular diffusivity (D):

– Consider ternary (acetone-’1’, nitrogen-’2’ and fuel-’3’) diffusion Consider ternary (acetone-’1’, nitrogen-’2’ and fuel-’3’) diffusion coefficients from binary diffusion coefficients (Dcoefficients from binary diffusion coefficients (D1111, D, D1212, D, D1313, D, D2222, D, D2323, D, D33)33)

– Simplify by assuming that DSimplify by assuming that D1212≈D≈D1313 or X or X11<<1 so that D<<1 so that D1212TT<<D<<D1111

TT and and

calculate Dcalculate D1,mix1,mix at each point in the flow, given that we already have the at each point in the flow, given that we already have the

instantaneous Xinstantaneous X11 (from (from ξ)ξ)

23/13/1

11

75.1

2

0143.0

BABA

AB

MMp

TD

13

3

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2,1

1

D

X

D

XD mix

112

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1311

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123 XX

DD

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T

7/19

Calculating Mixing Calculating Mixing Quantities (I)Quantities (I) For each ‘Run’ and at each pixel representing For each ‘Run’ and at each pixel representing

a location in physical space (r,z) loop through a location in physical space (r,z) loop through 200 images:200 images:– Of each version of Of each version of ξξ (raw and all stages of (raw and all stages of

processing) to obtain the mean and variance of processing) to obtain the mean and variance of each version of each version of ξξ

– Of each version of corresponding (Of each version of corresponding (∂∂ξξ//∂∂r)r)22 and and ((∂∂ξξ//∂∂z)z)22 to obtain the mean and variance of each to obtain the mean and variance of each version of (version of (∂∂ξξ//∂∂r)r)22 and ( and (∂∂ξξ//∂∂z)z)22

– Of each version of corresponding (Of each version of corresponding (∂∂ξξ''//∂∂r)r)22 and and ((∂∂ξξ''//∂∂z)z)22 to obtain the mean and variance of each to obtain the mean and variance of each version of (version of (∂∂ξξ''//∂∂r)r)22 and ( and (∂∂ξξ''//∂∂z)z)22

– Evaluate spatial 2-point autocorrelation matrices Evaluate spatial 2-point autocorrelation matrices along centreline and at left/right half-widthsalong centreline and at left/right half-widths

– Compile radial volume-averaged quantities (i.e. at Compile radial volume-averaged quantities (i.e. at one z group all r data together)one z group all r data together)

8/19

Calculating Mixing Calculating Mixing Quantities (II)Quantities (II) For each ‘Run’ and at each pixel representing a location in physical space For each ‘Run’ and at each pixel representing a location in physical space

(r,z) consider a window of size 1x1 (or 2x10) (r,z) consider a window of size 1x1 (or 2x10) ηηKK containing 40 (or 720) containing 40 (or 720) pixels and loop through 200 images:pixels and loop through 200 images:– At 15 axial locations from 1mm to 60mm in steps of 4mmAt 15 axial locations from 1mm to 60mm in steps of 4mm– At 5 radial locations with r=0, At 5 radial locations with r=0, ±±d/2, d/2, ±±dd– Calculate the local mean, variance, skewness and kurtosis of all versions of all Calculate the local mean, variance, skewness and kurtosis of all versions of all

variables (variables (ξξ and and χχ2D2D))

– Calculate the local mean, variance, skewness and kurtosis of the logarithm of Calculate the local mean, variance, skewness and kurtosis of the logarithm of all versions of all versions of χχ2D2D

– Compile pdfs of all versions of all variables (Compile pdfs of all versions of all variables (ξξ and and χχ2D2D) each composed of 90 ) each composed of 90 and 30 points respectively spanning the min-max range (from about 140,000 and 30 points respectively spanning the min-max range (from about 140,000 data points)data points)

– Separate the Separate the χχ2D2D data into 30 groups between the min-max range of data into 30 groups between the min-max range of ξξ by by considering the corresponding values of considering the corresponding values of ξξ ( (χχ2D2D||ξξ))

– Calculate the local mean, variance, skewness and kurtosis of Calculate the local mean, variance, skewness and kurtosis of χχ2D2D||ξξ and ln( and ln(χχ2D2D||ξξ))

– Compile pdfs of all versions of Compile pdfs of all versions of χχ2D2D||ξξ:: unless number of data points in the local pdf is less than 300unless number of data points in the local pdf is less than 300 each composed of 30 points respectively spanning the min-max rangeeach composed of 30 points respectively spanning the min-max range from about 2,000-10,000 data pointsfrom about 2,000-10,000 data points

9/19

Mean Mean ξξ (I) (I)

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– All are equal velocity cases (UAll are equal velocity cases (U injinj≈U≈Ucoco; υ; υinjinj=1.0±0.2) with =1.0±0.2) with the 2.248mm injector and varying Rethe 2.248mm injector and varying Recoco/Re/Returbturb

Right:Right:– Jet cases Jet cases with the 2.248mm injector with the 2.248mm injector ((υυinjinj=3 and =3 and

4)4)

Not affected by image processingNot affected by image processing

10/19

Mean Mean ξξ (II) (II)

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/ (

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11/19

Variance of Variance of ξξ (I) (I)

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12/19

Left:Left:– Equal velocity caseEqual velocity case

Right:Right:– Jet caseJet case

Affected by image processingAffected by image processing

Variance of Variance of ξξ (II) (II)

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'

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'2 /

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Run 1: Re=405, =1.1Run 2: Re=425, =1.2Run 4: Re=530, =1.1Run 5: Re=560, =1.1Run 8: Re=560, =2.4Run 10: Re=745, =2.4Run 11: Re=895, =4.3Run 12: Re=895, =5.0

13/19

Mean Mean χχ22DD (I) (I)

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Mean Mean χχ22DD (II) (II)

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(Non-strict) Isotropy in (Non-strict) Isotropy in χχ2D2D

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Mean Scalar Mean Scalar Dissipation Modelling Dissipation Modelling and Cand CDD (I) (I) At each location in physical space where we would like to At each location in physical space where we would like to

evaluate:evaluate:

Firstly we need to recover the mean full 3-dimensional Firstly we need to recover the mean full 3-dimensional χχ from the from the

mean mean χχ2D2D (along the centreline by symmetry the mean gradients (along the centreline by symmetry the mean gradients

squared in the radial and azimuthal direction are equal)squared in the radial and azimuthal direction are equal)

– Also examined isotropy of the two componentsAlso examined isotropy of the two components

We also need knowledge of the turbulent timescale (k/We also need knowledge of the turbulent timescale (k/εε) where k ) where k

is the turbulence kinetic energy and is the turbulence kinetic energy and εε the mean turbulence the mean turbulence

dissipationdissipation

– Use (k/Use (k/εε)/(L)/(Lturbturb/u/u'')≈)≈1.71.7±±0.2 0.2 Pope (2000)Pope (2000)

k

CD 2

17/19

Mean Scalar Mean Scalar Dissipation Modelling Dissipation Modelling and Cand CDD (II) (II)

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U/Uco

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18/19

Centrelin

e

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Mean Scalar Mean Scalar Dissipation Modelling Dissipation Modelling and Cand CDD (III) (III)

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τturb/τturb(z/d=35)

Scalar Mixing inScalar Mixing inTurbulent, ConfinedTurbulent, ConfinedAxisymmetric Co-flowsAxisymmetric Co-flows

C.N. Markides & E. MastorakosC.N. Markides & E. MastorakosHopkinson Laboratory, Department of EngineeringHopkinson Laboratory, Department of Engineering

Monday, 6Monday, 6thth of of February, 2006February, 2006