^{1}, Maurice Rossi

^{2}and Thomas Boeck

^{1}

### Abstract

In the framework of linear stability theory, we analyze how a liquid-gas mixing layer is affected by several parameters: viscosity ratio, density ratio, and several length scales. These scales reflect the presence of a velocity defect induced by the wake behind the splitter plate and the presence of boundary layers which develop ahead of the plate trailing edge. Incorporating such effects, we compute the various temporal and spatial instability modes and identify their driving instability mechanism based on their Reynolds number dependence, spatial structure, and energy budget. It is examined how the velocity defect modifies the temporal and the spatial stability properties. In addition, the transition from convective to absolute instability occurs at lower velocity contrast between gas and liquid free streams when a defect is present. This transition is also promoted by surface tension. Compared to inviscid stability computations, our spatial stability analysis displays a better agreement with measured growth rates obtained in two recent air-water experiments.

T.B. is grateful to Stéphane Zaleski for introducing him to the problem of mixing layer instability in two-phase flows, and for valuable discussions. The authors also acknowledge discussions with Jean-Philippe Matas and Alain Cartellier, and would like to thank them for providing their experimental data. Philip Yecko is acknowledged for useful advice on the numerical method. The work of T.O. and T.B. has been supported by the Deutsche Forschungsgemeinschaft (Emmy-Noether Grant No. Bo 1668/2).

I. INTRODUCTION

II. VISCOUS BASIC VELOCITY PROFILE

III. INSTABILITY PROBLEM

IV. TEMPORAL STABILITY PROBLEM

A. Dependence on the velocity distribution

1. Mode I

2. Mode III

3. Mode II

4. The dominant mode

B. Influence of density ratio

V. SPATIAL STABILITY PROBLEM

A. Dependence on the velocity distribution

1. Unstable modes

2. Dominant mode

B. Influence of density ratio

VI. TRANSITION FROM CONVECTIVE TO ABSOLUTE INSTABILITY

VII. COMPARISON WITH EXPERIMENTAL RESULTS

VIII. DISCUSSION AND CONCLUSIONS

### Key Topics

- Reynolds stress modeling
- 61.0
- Flow instabilities
- 55.0
- Viscosity
- 46.0
- Spatial analysis
- 35.0
- Gas liquid interfaces
- 21.0

##### B01F3/00

## Figures

Viscous basic velocity profiles for δ1 = 1, different values of δ0 and two velocity ratios: (a) M = 0.5, (b) M = 0.9. The viscosity ratio is m = 0.012 (approximatively air-water case).

Viscous basic velocity profiles for δ1 = 1, different values of δ0 and two velocity ratios: (a) M = 0.5, (b) M = 0.9. The viscosity ratio is m = 0.012 (approximatively air-water case).

Location of (a) the minimum y m , and (b) the inflection points y i of the velocity profile as a function of δ0. The two curves correspond to velocity ratios M = 0.5 and M = 0.9 with parameters m = 0.012 (approximatively air-water case), δ1 = 1.

Location of (a) the minimum y m , and (b) the inflection points y i of the velocity profile as a function of δ0. The two curves correspond to velocity ratios M = 0.5 and M = 0.9 with parameters m = 0.012 (approximatively air-water case), δ1 = 1.

Rescaled growth rates ω i /M of the viscous temporal instability modes for the Stokes solution δ0 = 1: (a) Re = 59 280; (b) Re = 29 640. Case r = 0.0012, m = 0.012. Note that for the Stokes solution, ω i is proportional to M.

Rescaled growth rates ω i /M of the viscous temporal instability modes for the Stokes solution δ0 = 1: (a) Re = 59 280; (b) Re = 29 640. Case r = 0.0012, m = 0.012. Note that for the Stokes solution, ω i is proportional to M.

Temporal growth rates ω i as a function of the wavenumber α for M = 0.5. Each picture corresponds to different width δ0 and Reynolds number. The other parameters are r = 0.0012, m = 0.012. Note that mode III is not amplified for case (a) (Re = 2000, δ0 = 1).

Temporal growth rates ω i as a function of the wavenumber α for M = 0.5. Each picture corresponds to different width δ0 and Reynolds number. The other parameters are r = 0.0012, m = 0.012. Note that mode III is not amplified for case (a) (Re = 2000, δ0 = 1).

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

Energy contributions as a function of streamwise wavenumber for mode I. The parameters are M = 0.9, r = 0.0012, m = 0.012, Re = 10 000, S = 0. Defect width is (a) δ0 = 1 and (b) δ0 = 0.1.

Energy contributions as a function of streamwise wavenumber for mode I. The parameters are M = 0.9, r = 0.0012, m = 0.012, Re = 10 000, S = 0. Defect width is (a) δ0 = 1 and (b) δ0 = 0.1.

Temporal eigenfunctions of mode I at α = 1 (real and imaginary parts as well as absolute value, normalized by ϕ(0) = 1) for (a) δ0 = 1, (b) δ0 = 0.9, (c) δ0 = 0.5, (d) δ0 = 0.1. Case with Re = 2000, M = 0.5, r = 0.0012, m = 0.012.

Temporal eigenfunctions of mode I at α = 1 (real and imaginary parts as well as absolute value, normalized by ϕ(0) = 1) for (a) δ0 = 1, (b) δ0 = 0.9, (c) δ0 = 0.5, (d) δ0 = 0.1. Case with Re = 2000, M = 0.5, r = 0.0012, m = 0.012.

Rescaled maximum temporal growth rate (a) and corresponding wavenumber (b) for mode I as a function of δ0 with M = 0.5, M = 0.9, and Re = 2000, Re = 10 000. The remaining parameters are r = 0.0012, m = 0.012, and S = 0.

Rescaled maximum temporal growth rate (a) and corresponding wavenumber (b) for mode I as a function of δ0 with M = 0.5, M = 0.9, and Re = 2000, Re = 10 000. The remaining parameters are r = 0.0012, m = 0.012, and S = 0.

Mode III: temporal eigenfunctions at α = 0.3 with: (a) δ0 = 0.1; (b) δ0 = 1. Case Re = 10 000, M = 0.5, r = 0.0012, m = 0.012.

Mode III: temporal eigenfunctions at α = 0.3 with: (a) δ0 = 0.1; (b) δ0 = 1. Case Re = 10 000, M = 0.5, r = 0.0012, m = 0.012.

Reynolds number dependence of mode III: (a) δ0 = 1 and M = 0.5; (b) δ0 = 1 and M = 0.9; (c) δ0 = 0.1 and M = 0.5; (d) δ0 = 0.1 and M = 0.9. Case with r = 0.0012, m = 0.012.

Reynolds number dependence of mode III: (a) δ0 = 1 and M = 0.5; (b) δ0 = 1 and M = 0.9; (c) δ0 = 0.1 and M = 0.5; (d) δ0 = 0.1 and M = 0.9. Case with r = 0.0012, m = 0.012.

Energy contributions as a function of streamwise wavenumber for mode III. The parameters are M = 0.5, r = 0.0012, m = 0.012, Re = 10 000, S = 0. Defect width is (a) δ0 = 1 and (b) δ0 = 0.1.

Energy contributions as a function of streamwise wavenumber for mode III. The parameters are M = 0.5, r = 0.0012, m = 0.012, Re = 10 000, S = 0. Defect width is (a) δ0 = 1 and (b) δ0 = 0.1.

Mode II: Temporal eigenfunctions at α = 10: (a) δ0 = 0.1, (b) δ0 = 1. Case M = 0.5, Re = 2000, r = 0.0012, m = 0.012.

Mode II: Temporal eigenfunctions at α = 10: (a) δ0 = 0.1, (b) δ0 = 1. Case M = 0.5, Re = 2000, r = 0.0012, m = 0.012.

Mode II: growth rates as a function of wavenumber α for different velocity ratios M and width δ0. Case r = 0.0012, m = 0.012, and (a) M = 0.5, δ0 = 1, (b) M = 0.9, δ0 = 1, (c) M = 0.5, δ0 = 0.1, (d) M = 0.9, δ0 = 0.1.

Mode II: growth rates as a function of wavenumber α for different velocity ratios M and width δ0. Case r = 0.0012, m = 0.012, and (a) M = 0.5, δ0 = 1, (b) M = 0.9, δ0 = 1, (c) M = 0.5, δ0 = 0.1, (d) M = 0.9, δ0 = 0.1.

Mode II: wavenumber αmax of maximum temporal growthrate as a function of Re for various defect widths δ0. (a) M = 0.5 and (b) M = 0.9. Case r = 0.0012, m = 0.012.

Mode II: wavenumber αmax of maximum temporal growthrate as a function of Re for various defect widths δ0. (a) M = 0.5 and (b) M = 0.9. Case r = 0.0012, m = 0.012.

Energy contributions as a function of streamwise wavenumber for mode II. The parameters are M = 0.9, r = 0.0012, m = 0.012, Re = 10 000, S = 0. Defect width is (a) δ0 = 1, (b) δ0 = 0.1.

Energy contributions as a function of streamwise wavenumber for mode II. The parameters are M = 0.9, r = 0.0012, m = 0.012, Re = 10 000, S = 0. Defect width is (a) δ0 = 1, (b) δ0 = 0.1.

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

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

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

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

Maximum growth rates (a) and corresponding wavenumbers αmax (b) as function of density ratio r for Re = 1000. Case M = 0.5, m = 0.01.

Maximum growth rates (a) and corresponding wavenumbers αmax (b) as function of density ratio r for Re = 1000. Case M = 0.5, m = 0.01.

Spatial growth rates α i as a function of real wavenumbers α r for different defect widths δ0 and velocity ratios M. Case r = 0.0012, m = 0.012. Note the presence of confinement branches in the case M = 0.9 and δ0 = 0.1.

Spatial growth rates α i as a function of real wavenumbers α r for different defect widths δ0 and velocity ratios M. Case r = 0.0012, m = 0.012. Note the presence of confinement branches in the case M = 0.9 and δ0 = 0.1.

Mode I: maximum growth rate (a) and corresponding wavenumber (b) as a function of the velocity defect δ0. Case r = 0.0012, m = 0.012. For M = 0.5 all curves coincide for the same four values of Re shown for M = 0.9.

Mode I: maximum growth rate (a) and corresponding wavenumber (b) as a function of the velocity defect δ0. Case r = 0.0012, m = 0.012. For M = 0.5 all curves coincide for the same four values of Re shown for M = 0.9.

Mode II: Inverse maximum growth rate as a function of the velocity defect δ0 for (a) velocity ratio M = 0.5; (b) M = 0.9. Case r = 0.0012, m = 0.012.

Mode II: Inverse maximum growth rate as a function of the velocity defect δ0 for (a) velocity ratio M = 0.5; (b) M = 0.9. Case r = 0.0012, m = 0.012.

Mode II: Dependence of frequency on the real part of the wavenumber from temporal and spatial analysis for M = 0.9, Re = 10 000, r = 0.0012, m = 0.012. Results of spatial calculation (full lines), temporal calculation (star symbols), and estimate α r U 0 (square symbols) coincide.

Mode II: Dependence of frequency on the real part of the wavenumber from temporal and spatial analysis for M = 0.9, Re = 10 000, r = 0.0012, m = 0.012. Results of spatial calculation (full lines), temporal calculation (star symbols), and estimate α r U 0 (square symbols) coincide.

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

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

Maximum spatial growth rates ((a) and (c)) and corresponding wave numbers ((b) and (d)) as function of density ratio r. Case M = 0.9 ((a) and (b)) and case M = 0.5 ((c), and (d)). Viscosity ratio m = 0.01 and Reynolds number Re = 10 000. The abrupt ending in some of the curves with M = 0.9 is due to a change from convective to absolute instability. Quadruple precision was required for δ0 = 1.0 and r < 10−4.

Maximum spatial growth rates ((a) and (c)) and corresponding wave numbers ((b) and (d)) as function of density ratio r. Case M = 0.9 ((a) and (b)) and case M = 0.5 ((c), and (d)). Viscosity ratio m = 0.01 and Reynolds number Re = 10 000. The abrupt ending in some of the curves with M = 0.9 is due to a change from convective to absolute instability. Quadruple precision was required for δ0 = 1.0 and r < 10−4.

Two instances of absolute instability: the generalized spatial branches reconnect before the imaginary part ω i reaches zero. (a) Case M = 0.95, δ0 = 0.5; (b) case M = 0.92, δ0 = 0.1. Parameters r = 0.0012, m = 0.018, δ1 = 1, Re = 700, S = 1. Confinement branches appear close to the imaginary axis.

Two instances of absolute instability: the generalized spatial branches reconnect before the imaginary part ω i reaches zero. (a) Case M = 0.95, δ0 = 0.5; (b) case M = 0.92, δ0 = 0.1. Parameters r = 0.0012, m = 0.018, δ1 = 1, Re = 700, S = 1. Confinement branches appear close to the imaginary axis.

Curves in parameter plane (δ0,M) delimitating the convective/absolute transition. (a) Reynolds number Re = 700, (b) Re = 10 000. Case r = 0.0012, m = 0.018, δ1 = 1, and S = 0 or S = 1.

Curves in parameter plane (δ0,M) delimitating the convective/absolute transition. (a) Reynolds number Re = 700, (b) Re = 10 000. Case r = 0.0012, m = 0.018, δ1 = 1, and S = 0 or S = 1.

Pinch point location as a function of δ0: (a) ; (b) . Transition corresponding to the cases of Fig. 26 .

Pinch point location as a function of δ0: (a) ; (b) . Transition corresponding to the cases of Fig. 26 .

Unstable temporal branches for (a) case B2 from Table I and (b) C3 from Table II computed with δ0 = 0.1.

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

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

Numerical and experimental growth rates for the axisymmetric nozzle as a function of the velocity ratio M for parameter combinations C1−C4 from Table I . Theoretical inviscid spatial growth rates are computed with Eq. (44) .

Numerical and experimental frequencies for the planar nozzle as a function of the velocity ratio M 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) .

Numerical and experimental frequencies for the planar nozzle as a function of the velocity ratio M 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) .

Numerical and experimental growth rates for the planar nozzle as a function of velocity ratio M 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. 43 .

Numerical and experimental growth rates for the planar nozzle as a function of velocity ratio M 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. 43 .

Comparison between viscous results and the experimental cases A2−A4 from Table II assuming a gas boundary layer thickness χδ2 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 et al. 43

Comparison between viscous results and the experimental cases A2−A4 from Table II assuming a gas boundary layer thickness χδ2 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 et al. 43

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