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Experimental and analytical study of the shear instability of a gas-liquid mixing layer
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10.1063/1.3642640
/content/aip/journal/pof2/23/9/10.1063/1.3642640
http://aip.metastore.ingenta.com/content/aip/journal/pof2/23/9/10.1063/1.3642640
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(a) Velocity profile with two vorticity thicknesses; (b) Velocity profile with a gas vorticity thickness only; and (c) Profile exhibiting a velocity deficit at the interface.

Image of FIG. 2.
FIG. 2.

(a) Dimensionless wavenumber k r and (b) growth rate k i obtained by stability analysis, as a function of dimensionless frequency ωδ g /U g . The solid curve is obtained with a base flow having a full velocity deficit at the interface (α i  = 0), and the dotted curve with an interface velocity equal to the liquid velocity (α i  = α l ). Density ratio r = 10−3, velocity ratio α l  = 10−2 and vorticity thickness ratio n = 1.

Image of FIG. 3.
FIG. 3.

Dimensionless wavenumber k r and growth rate k i obtained by stability analysis, as a function of dimensionless frequency ωδ g /U g , for ratios U l /U g  = 5.10−4, 10−3, 2.10−3, 5.10−3, 10−2, 2.10−2, 4.10−2, 6.10−2, and 9.10−2 (from left to right). α i  = 0, n = 1, and r = 10−3.

Image of FIG. 4.
FIG. 4.

(a) Variations of the dimensionless frequency and (b) growth rate of the most unstable mode as a function of the liquid to gas velocity ratio α l . Solid curve: velocity deficit at the interface; dotted curve: no velocity deficit. The vorticity thickness ratio n = δ g l is fixed at n = 1, and the density ratio at r = 10−3.

Image of FIG. 5.
FIG. 5.

(a) Dimensionless wavenumber k r and (b) growth rate k i as a function of dimensionless frequency ωδ g /U g , for gas to liquid vorticity thickness ratio n = δ g l  = 0.15, 0.2, 0.5, 1, and 10 (from left to right). The liquid to gas velocity ratio is fixed α l  = 10−2 and the density ratio r = 10−3.

Image of FIG. 6.
FIG. 6.

Frequency of the most unstable mode found when the full dispersion relation is solved numerically for spatial solutions (solid line), and predicted by the asymptotic expansion (dotted line), when the velocity ratio α l is varied. The base velocity profile is taken to have a full velocity deficit at the interface (α i  = 0).

Image of FIG. 7.
FIG. 7.

Velocity profile used for the analysis: the gas jet has a finite thickness H g .

Image of FIG. 8.
FIG. 8.

Growth rate as a function of frequency, for different thicknesses of the gas stream H g : solid line H g g  = 125; dotted line H g g  = 25; dashed line H g g  = 10; dash-dotted line H g g  = 2.5; other parameters are r = 10−3, α l  = α i  = 10−2, and n = 1.

Image of FIG. 9.
FIG. 9.

Variation of the (a) real part and (b) imaginary part of the wavenumber of the most unstable mode as a function of H g g , for different density ratio r; □: r = 10−4; ×: r = 3.10−4; o: r = 10−3; *: r = 5.10−3; +: r = 10−2. Other parameters are α l  = α i  = 10−2 and n = 1.

Image of FIG. 10.
FIG. 10.

Variation of the (a) real part and (b) imaginary part of the wavenumber of the most unstable mode as a function of H g /λ, for different density ratio r; □: r = 10−4; ×: r = 3.10−4; o: r = 10−3; *: r = 5.10−3; +: r = 10−2. Other parameters are α l  = α i  = 10−2 and n = 1.

Image of FIG. 11.
FIG. 11.

(a) Variation of the frequency of the most unstable mode with H g /λ, for different density ratio r; □: r = 10−4; ×: r = 3.10−4; o: r = 10−3; *: r = 5.10−3; +: r = 10−2. Other parameters are α l  = α i  = 10−2 and n = 1. (b) Variation of the growth rate with H g /λ, same parameters except α i  = 0.

Image of FIG. 12.
FIG. 12.

Measurements of Ben Rayana9: (a) frequency of the Kelvin-Helmholtz instability as a function of the gas velocity, for different thicknesses H g . (b) Same data plotted as a function of U g g : the series are collapsed.

Image of FIG. 13.
FIG. 13.

(Color online) Sketch of the experimental set-up.

Image of FIG. 14.
FIG. 14.

(Color online) (a) Example of a spectrum of the position of the interface, for U g  = 22 m s−1: solid line U l  = 0.26 m s−1, dashed line U l  = 0.5 m s−1. The insert, on a larger scale, shows harmonics for U g  = 12 m s−1 and U l  = 0.26 m s−1; (b) Downstream variation of the spectrum of the amplitude of the instability: U g  = 12 m s−1, U l  = 0.26 m s−1. The spectrum is computed every 2δ g , up to x = 65δ g : the amplitude of the maximum increases with downstream distance.

Image of FIG. 15.
FIG. 15.

(Color online) Experimental frequency as a function of the liquid velocity U l , for different U g : ○: U g  = 12 m s−1; □: U g  = 17 m s−1; ×: U g  = 22 m s−1; •: U g  = 27 m s−1.

Image of FIG. 16.
FIG. 16.

(Color online) Dimensionless frequency as a function of M −1/2. The dotted line is (asymptotic prediction for profile of Fig. 1(b) and the solid line is (asymptotic prediction for profile of Fig. 1(c). Symbols correspond to different values of U g , ○: U g  = 12 m s−1; □: U g  = 17 m s−1; ×: U g  = 22 m s−1; •: U g  = 27 m s−1

Image of FIG. 17.
FIG. 17.

Comparison of data sets for the dimensionless frequency as a function of M −1/2: x results of Raynal7,12; □ results of Ben Rayana9,13; • results of the present study. The dotted line is (asymptotic prediction for profile of Fig. 1(b) and the solid line is (asymptotic prediction for profile of Fig. 1(c).

Image of FIG. 18.
FIG. 18.

(Color online) (a) Ratio of the experimental frequency and the predicted frequency, as a function of M. Symbols correspond to different values of U g , ○: U g  = 12 m s−1; □: U g  = 17 m s−1; ×: U g  = 22 m s−1; *: U g  = 27 m s−1. (b) Same plot with the data of Raynal (crosses) and Ben Rayana (diamonds).

Image of FIG. 19.
FIG. 19.

(Color online) (a) Illustration of the growth rate measurement (U g =12 m/s, U l  = 0.37 m/s): a histogram of the interface positions at a given downstream position is made; the amplitude of the instability is deduced from the width of the histogram (dashed line); (b)Variation of the dimensionless amplitude A g of the waves, as a function of downstream distance. From right to left, U g  = 12, 17, 22, and 27 m s−1. The region of exponential growth (enhanced by the dashed line) is drastically reduced as U g is increased.

Image of FIG. 20.
FIG. 20.

(Color online) (a) Dimensionless measured growth rate as a function of M, ∘: U g  = 12 m s−1; □: U g  = 17 m s−1; x: U g  = 22 m s−1; •: U g  = 27 m s−1; (b) Dimensionless measured growth rate as a function of U g ; *: U l  = 0.26 m s−1; ⋄: U l  = 0.31 m s−1; ▿: U l  = 0.37 m s−1; ▵: U l  = 0.50 m s−1; : U l  = 0.76 m s−1; : U l  = 0.95 m s−1.

Image of FIG. 21.
FIG. 21.

(Color online) Ratio of the experimental and predicted spatial growth rate, as a function of U g . Symbols correspond to different values of U l , *: U l  = 0.26 m s−1; ⋄: U l  = 0.31 m s−1; ▿: U l  = 0.37 m s−1; ▵: U l  = 0.50 m s−1; : U l  = 0.76 m s−1; : U l  = 0.95 m s−1.

Image of FIG. 22.
FIG. 22.

(Color online) PIV visualization of the gas flow around a wave, for an annular gas flow (H g  = 1.5 mm) around a liquid jet (radius R = 4 mm), U l  = 0.5 m s−1, U g  = 20 m s−1. The white dashed line enhances the limit of the liquid jet downstream the wave: the gas jet is detached from the liquid.

Image of FIG. 23.
FIG. 23.

(Color online) Ratio of the experimental and predicted frequency as a function of the experimental growth rate.

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/content/aip/journal/pof2/23/9/10.1063/1.3642640
2011-09-29
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Experimental and analytical study of the shear instability of a gas-liquid mixing layer
http://aip.metastore.ingenta.com/content/aip/journal/pof2/23/9/10.1063/1.3642640
10.1063/1.3642640
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