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Viscous instability of a sheared liquid-gas interface: Dependence on fluid properties and basic velocity profile
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10.1063/1.4792311
/content/aip/journal/pof2/25/3/10.1063/1.4792311
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/3/10.1063/1.4792311

Figures

Image of FIG. 1.
FIG. 1.

Viscous basic velocity profiles for δ = 1, different values of δ and two velocity ratios: (a) = 0.5, (b) = 0.9. The viscosity ratio is = 0.012 (approximatively air-water case).

Image of FIG. 2.
FIG. 2.

Location of (a) the minimum , and (b) the inflection points of the velocity profile as a function of δ. The two curves correspond to velocity ratios = 0.5 and = 0.9 with parameters = 0.012 (approximatively air-water case), δ = 1.

Image of FIG. 3.
FIG. 3.

Rescaled growth rates ω/ of the viscous temporal instability modes for the Stokes solution δ = 1: (a) = 59 280; (b) = 29 640. Case = 0.0012, = 0.012. Note that for the Stokes solution, ω is proportional to .

Image of FIG. 4.
FIG. 4.

Temporal growth rates ω as a function of the wavenumber α for = 0.5. Each picture corresponds to different width δ and Reynolds number. The other parameters are = 0.0012, = 0.012. Note that mode III is not amplified for case (a) ( = 2000, δ = 1).

Image of FIG. 5.
FIG. 5.

Idem Figure 4 but for = 0.9. Note that mode III is not amplified for = 2000.

Image of FIG. 6.
FIG. 6.

Energy contributions as a function of streamwise wavenumber for mode I. The parameters are = 0.9, = 0.0012, = 0.012, = 10 000, = 0. Defect width is (a) δ = 1 and (b) δ = 0.1.

Image of FIG. 7.
FIG. 7.

Temporal eigenfunctions of mode I at α = 1 (real and imaginary parts as well as absolute value, normalized by ϕ(0) = 1) for (a) δ = 1, (b) δ = 0.9, (c) δ = 0.5, (d) δ = 0.1. Case with = 2000, = 0.5, = 0.0012, = 0.012.

Image of FIG. 8.
FIG. 8.

Rescaled maximum temporal growth rate (a) and corresponding wavenumber (b) for mode I as a function of δ with = 0.5, = 0.9, and = 2000, = 10 000. The remaining parameters are = 0.0012, = 0.012, and = 0.

Image of FIG. 9.
FIG. 9.

Mode III: temporal eigenfunctions at α = 0.3 with: (a) δ = 0.1; (b) δ = 1. Case = 10 000, = 0.5, = 0.0012, = 0.012.

Image of FIG. 10.
FIG. 10.

Reynolds number dependence of mode III: (a) δ = 1 and = 0.5; (b) δ = 1 and = 0.9; (c) δ = 0.1 and = 0.5; (d) δ = 0.1 and = 0.9. Case with = 0.0012, = 0.012.

Image of FIG. 11.
FIG. 11.

Energy contributions as a function of streamwise wavenumber for mode III. The parameters are = 0.5, = 0.0012, = 0.012, = 10 000, = 0. Defect width is (a) δ = 1 and (b) δ = 0.1.

Image of FIG. 12.
FIG. 12.

Mode II: Temporal eigenfunctions at α = 10: (a) δ = 0.1, (b) δ = 1. Case = 0.5, = 2000, = 0.0012, = 0.012.

Image of FIG. 13.
FIG. 13.

Mode II: growth rates as a function of wavenumber α for different velocity ratios and width δ. Case = 0.0012, = 0.012, and (a) = 0.5, δ = 1, (b) = 0.9, δ = 1, (c) = 0.5, δ = 0.1, (d) = 0.9, δ = 0.1.

Image of FIG. 14.
FIG. 14.

Mode II: wavenumber α of maximum temporal growthrate as a function of for various defect widths δ. (a) = 0.5 and (b) = 0.9. Case = 0.0012, = 0.012.

Image of FIG. 15.
FIG. 15.

Energy contributions as a function of streamwise wavenumber for mode II. The parameters are = 0.9, = 0.0012, = 0.012, = 10 000, = 0. Defect width is (a) δ = 1, (b) δ = 0.1.

Image of FIG. 16.
FIG. 16.

Maximum temporal growth rate as function of the velocity defect for modes I and II: (a) = 0.9, and (b) = 0.5.

Image of FIG. 17.
FIG. 17.

Maximum growth rates ((a) and (c)) and corresponding wavenumbers α ((b) and (d)) as function of density ratio for Reynolds numbers = 1000 ((a) and (b)) and = 10 000 ((c) and (d)). Case = 0.9, = 0.01.

Image of FIG. 18.
FIG. 18.

Maximum growth rates (a) and corresponding wavenumbers α (b) as function of density ratio for = 1000. Case = 0.5, = 0.01.

Image of FIG. 19.
FIG. 19.

Spatial growth rates α as a function of real wavenumbers α for different defect widths δ and velocity ratios . Case = 0.0012, = 0.012. Note the presence of confinement branches in the case = 0.9 and δ = 0.1.

Image of FIG. 20.
FIG. 20.

Mode I: maximum growth rate (a) and corresponding wavenumber (b) as a function of the velocity defect δ. Case = 0.0012, = 0.012. For = 0.5 all curves coincide for the same four values of shown for = 0.9.

Image of FIG. 21.
FIG. 21.

Mode II: Inverse maximum growth rate as a function of the velocity defect δ for (a) velocity ratio = 0.5; (b) = 0.9. Case = 0.0012, = 0.012.

Image of FIG. 22.
FIG. 22.

Mode II: Dependence of frequency on the real part of the wavenumber from temporal and spatial analysis for = 0.9, = 10 000, = 0.0012, = 0.012. Results of spatial calculation (full lines), temporal calculation (star symbols), and estimate α (square symbols) coincide.

Image of FIG. 23.
FIG. 23.

Maximum spatial growth rate as function of the velocity defect for modes I and II: (a) = 0.5 and (b) = 0.9.

Image of FIG. 24.
FIG. 24.

Maximum spatial growth rates ((a) and (c)) and corresponding wave numbers ((b) and (d)) as function of density ratio . Case = 0.9 ((a) and (b)) and case = 0.5 ((c), and (d)). Viscosity ratio = 0.01 and Reynolds number = 10 000. The abrupt ending in some of the curves with = 0.9 is due to a change from convective to absolute instability. Quadruple precision was required for δ = 1.0 and < 10.

Image of FIG. 25.
FIG. 25.

Two instances of absolute instability: the generalized spatial branches reconnect before the imaginary part ω reaches zero. (a) Case = 0.95, δ = 0.5; (b) case = 0.92, δ = 0.1. Parameters = 0.0012, = 0.018, δ = 1, = 700, = 1. Confinement branches appear close to the imaginary axis.

Image of FIG. 26.
FIG. 26.

Curves in parameter plane (δ,) delimitating the convective/absolute transition. (a) Reynolds number = 700, (b) = 10 000. Case = 0.0012, = 0.018, δ = 1, and = 0 or = 1.

Image of FIG. 27.
FIG. 27.

Pinch point location as a function of δ: (a) ; (b) . Transition corresponding to the cases of Fig. 26 .

Image of FIG. 28.
FIG. 28.

Unstable temporal branches for (a) case B2 from Table I and (b) C3 from Table II computed with δ = 0.1.

Image of FIG. 29.
FIG. 29.

Numerical and experimental frequencies for the axisymmetric nozzle as a function of the velocity ratio for parameter combinations from Table I : (a) A1−A4, (b) B1−B5. Theoretical inviscid frequencies are computed with Eqs. (44) and (45) .

Image of FIG. 30.
FIG. 30.

Numerical and experimental growth rates for the axisymmetric nozzle as a function of the velocity ratio for parameter combinations 1−4 from Table I . Theoretical inviscid spatial growth rates are computed with Eq. (44) .

Image of FIG. 31.
FIG. 31.

Numerical and experimental frequencies for the planar nozzle as a function of the velocity ratio for parameter combinations from Table II : (a) A1−A6, (b) B1−B6, (c) C1−C6, (d) D1−D6. Theoretical inviscid frequencies are computed with Eqs. (44) and (45) .

Image of FIG. 32.
FIG. 32.

Numerical and experimental growth rates for the planar nozzle as a function of velocity ratio for parameter combinations from Table II : (a) A1−A6, (b) B1−B6, (c) C1−C6, (d) D1−D6. The left column shows the maximum growth rates from viscous stability theory, and the right column the largest growth rate corresponding to the experimental value of the frequency. Theoretical inviscid growth rates have been obtained from Eq. (44) and from Fig. 21 in Ref. .

Image of FIG. 33.
FIG. 33.

Comparison between viscous results and the experimental cases A2−A4 from Table II assuming a gas boundary layer thickness χδ and different values of χ. The results for χ = 1 correspond to those in Figures 31(a) and 32(a) . The dashed lines correspond to the inviscid results by Matas

Tables

Generic image for table
Table I.

Parameters for the axisymmetric nozzle experiments performed by Marmottant and Villermaux.

Generic image for table
Table II.

Parameters for the planar nozzle experiments performed by Matas

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/content/aip/journal/pof2/25/3/10.1063/1.4792311
2013-03-07
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Viscous instability of a sheared liquid-gas interface: Dependence on fluid properties and basic velocity profile
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/3/10.1063/1.4792311
10.1063/1.4792311
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