^{1,a)}, Sylvain Marty

^{1}and Alain Cartellier

^{1}

### Abstract

We carry out an inviscid spatial linear stabilityanalysis of a planar mixing layer, where a fast gas stream destabilizes a slower parallel liquid stream, and compare the predictions of this analysis with experimental results. We study how the value of the liquid velocity at the interface and the finite thickness of the gas jet affect the most unstable mode predicted by the inviscid analysis: in particular a zero interface velocity is considered to account for the presence in most experimental situations of a splitter splate separating the gas and the liquid. Results derived from this theory are compared with experimentally measured frequencies and growth rates: a good agreement is found between the experimental and predicted frequencies, while the experimental growth rates turn out to be much larger than expected.

I. INTRODUCTION

II. INVISCID ANALYSIS

A. Effect of a velocity deficit

B. Asymptotic analysis and influence of the liquid on the gas mode

C. Effect of a finite gas thickness

III. EXPERIMENTAL RESULTS

IV. CONCLUSION

### Key Topics

- Vortex dynamics
- 24.0
- Dispersion relations
- 17.0
- Spatial analysis
- 17.0
- Viscosity
- 12.0
- Gas liquid flows
- 11.0

## Figures

(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.

(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.

(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.

(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.

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}.

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}.

(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 }/δ_{ l } is fixed at *n* = 1, and the density ratio at *r* = 10^{−3}.

(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 }/δ_{ l } is fixed at *n* = 1, and the density ratio at *r* = 10^{−3}.

(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 }/δ_{ 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}.

(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 }/δ_{ 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}.

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).

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).

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

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

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

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

Variation of the (a) real part and (b) imaginary part of the wavenumber of the most unstable mode as a function of *H* _{ g }/δ_{ 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.

Variation of the (a) real part and (b) imaginary part of the wavenumber of the most unstable mode as a function of *H* _{ g }/δ_{ 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.

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.

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.

(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.

(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.

Measurements of Ben Rayana^{9}: (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 }/δ_{ g }: the series are collapsed.

Measurements of Ben Rayana^{9}: (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 }/δ_{ g }: the series are collapsed.

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

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

(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.

(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.

(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}.

(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}.

(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}

(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}

Comparison of data sets for the dimensionless frequency as a function of *M* ^{−1/2}: x results of Raynal^{7,12}; □ results of Ben Rayana^{9,13}; • 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).

Comparison of data sets for the dimensionless frequency as a function of *M* ^{−1/2}: x results of Raynal^{7,12}; □ results of Ben Rayana^{9,13}; • 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).

(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).

(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).

(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*/δ_{ 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.

(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*/δ_{ 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.

(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}.

(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}.

(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}.

(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}.

(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.

(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.

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

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

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