^{1}, S. Amiroudine

^{2}, D. Chatain

^{1}, T. Lyubimova

^{3}and D. Beysens

^{1,4}

### Abstract

Under the absence of gravity forces, the interaction of vibration with a thermal boundary layer (TBL) can lead to a rich variety of dynamics in a supercritical fluid (SCF). When subjected to vibration, a SCF can display different kinds of instabilities for different relative directions of the TBL and vibration. Rayleigh vibrational instability is formed when the vibration direction is tangential to the TBL. When the direction of vibration is perpendicular to the TBL, instabilities of parametric nature can develop. Two-dimensional finite volume numerical analysis of supercritical H2 filled in a square cell under vibration is carried out. The vibrational amplitudes range from 0.05 to 5 times the side of the cell and frequencies vary between 2.78 Hz and 25 Hz. Three different thermal boundary conditions (isothermal walls, adiabatic vertical/isothermal horizontal walls, and adiabatic horizontal/isothermal vertical walls) have been considered with various temperature proximities to the critical point (10 mK, 100 mK, and 1 K). The results evidence Rayleigh vibrational and parametric instabilities in a thermal field. It is for the first time that the latter type of instability is observed in the thermal field under such conditions. Additionally, the role of the cell corners is highlighted (a “corner” instability is observed). These instabilities are analyzed and quantified. In particular, the stability domains have been plotted.

The authors gratefully acknowledge financial support from the CNES (France) and the Government of Perm Region (Russia). The authors also thank the IDRIS Computing Center (France).

I. INTRODUCTION

II. DEFINITION OF THE PROBLEM AND THE GEOMETRY

III. GOVERNING EQUATIONS

IV. NUMERICAL SCHEME

V. RESULTS

A. Corner instability

B. Parametric instability

C. Rayleigh-vibrational instability

VI. CONCLUSION

### Key Topics

- Critical point phenomena
- 28.0
- Supercritical fluids
- 20.0
- Rotating flows
- 14.0
- Navier Stokes equations
- 13.0
- Viscosity
- 12.0

##### F28

## Figures

Three different computational configurations: (a) isothermal conditions on all 4 walls, (b) isothermal conditions on vertical walls and adiabatic conditions on horizontal walls, and (c) isothermal conditions on horizontal walls and adiabatic conditions on vertical walls.

Three different computational configurations: (a) isothermal conditions on all 4 walls, (b) isothermal conditions on vertical walls and adiabatic conditions on horizontal walls, and (c) isothermal conditions on horizontal walls and adiabatic conditions on vertical walls.

Evolution of the thermal field for the configuration of Fig. 1(a) (4 isothermal walls) with f ′ = 5.56 Hz, a′ = 20 mm, ΔT ′ = 1 K, and δT ′ = 100 mK. Different types of instabilities are evidenced: (1) corner instability; (2)–(6) parametric instability on the vertical walls; and (4)–(6) Rayleigh vibrational instability on the horizontal walls.

Evolution of the thermal field for the configuration of Fig. 1(a) (4 isothermal walls) with f ′ = 5.56 Hz, a′ = 20 mm, ΔT ′ = 1 K, and δT ′ = 100 mK. Different types of instabilities are evidenced: (1) corner instability; (2)–(6) parametric instability on the vertical walls; and (4)–(6) Rayleigh vibrational instability on the horizontal walls.

Evolution of the corner instability at each period of vibration. Configuration of Fig. 1(c) (adiabatic vertical walls, isothermal horizontal walls): f ′ = 2.78 Hz (period 0.36 s), a′ = 20 mm, ΔT ′ = 1 K with a temperature quench of 100 mK. (a) and (f) 1.44 s, (b) and (g) 1.8 s, (c) and (h) 2.16 s, (d) and (i) 2.52 s, and (e) and (j) 2.88 s, respectively. Top row: temperature fields; bottom row: corresponding stream lines. PV: primary vortex and SV: secondary vortex.

Evolution of the corner instability at each period of vibration. Configuration of Fig. 1(c) (adiabatic vertical walls, isothermal horizontal walls): f ′ = 2.78 Hz (period 0.36 s), a′ = 20 mm, ΔT ′ = 1 K with a temperature quench of 100 mK. (a) and (f) 1.44 s, (b) and (g) 1.8 s, (c) and (h) 2.16 s, (d) and (i) 2.52 s, and (e) and (j) 2.88 s, respectively. Top row: temperature fields; bottom row: corresponding stream lines. PV: primary vortex and SV: secondary vortex.

Stability curves for the corner instability for ΔT ′ = 10 mK with a quench of 1 mK and for ΔT ′ = 100 mK with a quench of 10 mK. The fluid is unstable above the curve and stable below it.

Stability curves for the corner instability for ΔT ′ = 10 mK with a quench of 1 mK and for ΔT ′ = 100 mK with a quench of 10 mK. The fluid is unstable above the curve and stable below it.

Parametric instability in Fig. 1(b) configuration (adiabatic horizontal walls, isothermal vertical walls) for the case f ′ = 2.78 Hz (period 0.36 s), a′ = 20 mm, ΔT ′ = 100 mK, and δT ′ = 10 mK in 6 consecutive half periods of vibration. (a) and (g) 14.58 s, (b) and (h) 14.76 s, (c) and (i) 14.94 s, (d) and (j) 15.12 s, (e) and (k) 15.3 s, and (f) and (l) 15.48 s Upper row: temperature fields; lower row: corresponding streamlines.

Parametric instability in Fig. 1(b) configuration (adiabatic horizontal walls, isothermal vertical walls) for the case f ′ = 2.78 Hz (period 0.36 s), a′ = 20 mm, ΔT ′ = 100 mK, and δT ′ = 10 mK in 6 consecutive half periods of vibration. (a) and (g) 14.58 s, (b) and (h) 14.76 s, (c) and (i) 14.94 s, (d) and (j) 15.12 s, (e) and (k) 15.3 s, and (f) and (l) 15.48 s Upper row: temperature fields; lower row: corresponding streamlines.

(a)–(d) Temperature field of the parametric instability in Fig. 1(b) configuration (adiabatic horizontal walls, isothermal vertical walls) for the case f ′ = 2.78 Hz (period 0.36 s), a′ = 20 mm, ΔT ′ = 100 mK, and δT ′ = 10 mK in 4 consecutive time periods at (a) 14.4 s, (b) 14.76 s, (c) 15.12 s, and (d) 15.48 s, respectively, plotted for the same vibration phase. (e)–(h) Temperature field of the parametric instability for a′ = 20 mm, ΔT ′ = 100 mK, δT ′ = 10 mK for frequencies: (e) 2.78 Hz, (f) 5.56 Hz, (g) 8.33 Hz, and (h) 16.66 Hz.

(a)–(d) Temperature field of the parametric instability in Fig. 1(b) configuration (adiabatic horizontal walls, isothermal vertical walls) for the case f ′ = 2.78 Hz (period 0.36 s), a′ = 20 mm, ΔT ′ = 100 mK, and δT ′ = 10 mK in 4 consecutive time periods at (a) 14.4 s, (b) 14.76 s, (c) 15.12 s, and (d) 15.48 s, respectively, plotted for the same vibration phase. (e)–(h) Temperature field of the parametric instability for a′ = 20 mm, ΔT ′ = 100 mK, δT ′ = 10 mK for frequencies: (e) 2.78 Hz, (f) 5.56 Hz, (g) 8.33 Hz, and (h) 16.66 Hz.

(a) Reduced wavelength (Λ) of the parametric instability versus reduced acceleration (Γ) for the cases ΔT ′ = 10 mK, 100 mK, and 1000 mK with corresponding quenches of δT ′ = 1 mK, 10 mK, and 100 mK, respectively (log-log plot). Line: best fit (see text). (b) Reduced wavelength (Λ) of the parametric instability versus critical point proximity (ΔT ′) for f ′ = 2.78 Hz, 5.56 Hz, and 8.33 Hz and a′ = 15 mm (log-log plot).

(a) Reduced wavelength (Λ) of the parametric instability versus reduced acceleration (Γ) for the cases ΔT ′ = 10 mK, 100 mK, and 1000 mK with corresponding quenches of δT ′ = 1 mK, 10 mK, and 100 mK, respectively (log-log plot). Line: best fit (see text). (b) Reduced wavelength (Λ) of the parametric instability versus critical point proximity (ΔT ′) for f ′ = 2.78 Hz, 5.56 Hz, and 8.33 Hz and a′ = 15 mm (log-log plot).

ω′2(ρ′1 + ρ′2) vs k′3 plot for ΔT ′ = 10 mK, 100 mK, and 1000 mK.

ω′2(ρ′1 + ρ′2) vs k′3 plot for ΔT ′ = 10 mK, 100 mK, and 1000 mK.

Stability curves for the parametric instability showing the critical amplitude (a′ cr ) vs frequency (f ′) for (a) ΔT ′ = 100 mK and 10 mK with quenches of δT ′ = 10 mK and 1 mK, respectively, and (b) ΔT ′ = 100 mK with quenches of δT ′ = 10 mK and 5 mK,

Stability curves for the parametric instability showing the critical amplitude (a′ cr ) vs frequency (f ′) for (a) ΔT ′ = 100 mK and 10 mK with quenches of δT ′ = 10 mK and 1 mK, respectively, and (b) ΔT ′ = 100 mK with quenches of δT ′ = 10 mK and 5 mK,

Critical vibrational acceleration Γ cr vs frequency (f ′) of vibration for the parametric instability for (a) ΔT ′ = 100 mK and 10 mK with quenches of δT ′ = 10 mK and 1 mK, respectively, and (b) ΔT ′ = 100 mK with quenches of δT ′ = 10 mK and 5 mK.

Critical vibrational acceleration Γ cr vs frequency (f ′) of vibration for the parametric instability for (a) ΔT ′ = 100 mK and 10 mK with quenches of δT ′ = 10 mK and 1 mK, respectively, and (b) ΔT ′ = 100 mK with quenches of δT ′ = 10 mK and 5 mK.

Critical Rayleigh vibrational number Ravc for two proximities to the critical point: (a) vs frequency and (b) vs amplitude.

Critical Rayleigh vibrational number Ravc for two proximities to the critical point: (a) vs frequency and (b) vs amplitude.

Critical Rayleigh vibrational number Rav c vs critical point proximity ΔT ′ for f ′ = 2.78 Hz, 5.56 Hz, and 8.33 Hz and a′ = 10 mm. The solid lines are the corresponding variations in CO2 from Ref. 11 with slopes −0.83 for ΔT ′ < 0.1 K and −0.31 for ΔT ′ > 0.1 K.

Critical Rayleigh vibrational number Rav c vs critical point proximity ΔT ′ for f ′ = 2.78 Hz, 5.56 Hz, and 8.33 Hz and a′ = 10 mm. The solid lines are the corresponding variations in CO2 from Ref. 11 with slopes −0.83 for ΔT ′ < 0.1 K and −0.31 for ΔT ′ > 0.1 K.

(a) Stability curve (amplitude a′ cr vs frequency f ′) for the Rayleigh vibrational instability for ΔT ′ = 10 mK and 100 mK with quenches δT ′ = 1 mK and 10 mK, respectively. (b) Critical velocity (a′f ′) cr vs frequency plot for ΔT ′ = 10 mK and 100 mK with quenches δT ′ = 1 mK and 10 mK, respectively.

(a) Stability curve (amplitude a′ cr vs frequency f ′) for the Rayleigh vibrational instability for ΔT ′ = 10 mK and 100 mK with quenches δT ′ = 1 mK and 10 mK, respectively. (b) Critical velocity (a′f ′) cr vs frequency plot for ΔT ′ = 10 mK and 100 mK with quenches δT ′ = 1 mK and 10 mK, respectively.

Stability domain for the three types of instabilities: corner (solid line with dots), parametric instability (solid line with boxes), and Rayleigh vibrational instability (solid line with circles) for (a) ΔT ′ = 0.1 K and (b) ΔT ′ = 0.01 K.

Stability domain for the three types of instabilities: corner (solid line with dots), parametric instability (solid line with boxes), and Rayleigh vibrational instability (solid line with circles) for (a) ΔT ′ = 0.1 K and (b) ΔT ′ = 0.01 K.

## Tables

Thermo-physical data for n-H2 with .

Thermo-physical data for n-H2 with .

Orders of magnitude of various time scales involved in the problem for two typical critical point proximities.

Orders of magnitude of various time scales involved in the problem for two typical critical point proximities.

Comparison of the size of the cavity with the length a′β′ p δT ′ for a′ = 30 mm.

Comparison of the size of the cavity with the length a′β′ p δT ′ for a′ = 30 mm.

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