1. a H. Kyotoh, b R. Nakamura & a P. J. Baruah a a Institute of Engineering Mechanics and Systems, University of Tsukuba, Ibaraki, Japan b Third Plan Design Industrial Co., Ltd., Gifu, Japan Incipient Oscillations of a Falling Water Sheet and their Instability Mechanisms

2. (1)Hagerty and Shea(1995) found that only two types of waves are possible on a flat liquid sheet, i.e., sinuous and dilatational modes. (2) Lin(1981) showed that a viscous liquid curtain becomes unstable when the Weber number of the curtain flow exceeds 1/2. (3) Luca and Costa(1997) studied instability of a spatially developing liquid sheet by using a multiple-scale perturbation analysis. (a) Nappe oscillations of weirs and dams (e.g., Shwartz(1964), Binnie(1972) and Honma and Ogiwara(1975)). (b) Prediction and control of the sheet breakup height for the design of fountains (Casperson(1993)). (c) Stabilizing a plane liquid sheet for film coatings and paper technology (e.g., Weinstein et al.(1997) and Luca(1999)). (d) Disintegration of liquid for the mixing of fuel and gas (e.g., Lasheras and Hopfinger(2000)). Importance of study Review of research

3. (1) Experimental results obtained from “free falls”, “free falls with vibrations”, “free falls with a back wall”(without vibrations), are presented. (2) A model describing the motion of the two-dimensional sheet is developed. Shear waves observed in falling water sheet are characterized by a linear stability analysis of the Navier –Stokes equations. Present studies

4. Side view Front view 300cm 40cm Side wall Laser displacement sensor 40cm Experimental apparatus 300cm Side wall (Transparent) Nappe Oscillator Back wall 40cm Pt.1 EXPERIMENTAL STUDIES

5. High Speed Camera Images x=50cm ～ 130 cmx=130cm ～ 180 cmx=180 ～ 220 cm Water depth=3 cm ; Discharge=0.1312 m 3 /min Free fall without vibrations 40 cm

6. Wavelets on a water sheetA hole on a water sheet 20 cm30 cm Enlargement

7. The Thickness of the Sheet and Boundary Layer & Reynolds Numbers for the Water and Air Flows The Thickness of the Sheet and Boundary Layer & Reynolds Numbers for the Water and Air Flows (cm)

8. 120cm 40cm80cm 160cm200cm Frequency(Hz) Power spectrum Distance from the weir downstream A. Free Fall without Vibrations Power spectrum changes downstream （ Water depth at the weir crest:2.24cm; Discharge:0.135m 3 /min ）

9. Spectra near breaking point for various discharge （ The data at the distance=180 cm ） Spectra near breaking point for various discharge （ The data at the distance=180 cm ） 1.95cm (0.12m 3 /min) 2.46cm (0.14m 3 /min) 2.69cm (0.15m 3 /min) 3.5cm (0.19m 3 /min) Water head （ Discharge ）

10. Forced vibrations of roughly 1 mm in amplitude were applied at the water surface near the weir crest. The frequencies given are 4,6,8,10,15,20,25,30,35Hz,…. Spectrum for forced vibrations with 15 Hz A clear peak of the spectrum appeared, which has the same frequency of the vibration. Spectrum for forced vibrations with 15 Hz A clear peak of the spectrum appeared, which has the same frequency of the vibration. Water depth at the weir crest: 2.32 cm, Discharge: 0.139 m 3 /min B. Free Fall with Vibrations Frequency (Hz)

11. Figure shows the logarithm of the sheet amplitude normalized by that at x=0 cm for various forced frequencies. Response of the sheet under the forced vibrations Exponential growth of the wave amplitude along the stream Response of the sheet under the forced vibrations Exponential growth of the wave amplitude along the stream Amplification rate(4Hz ～ 20Hz) Amplification rate(25Hz ～ 55Hz) 8Hz 10Hz 15Hz 20Hz 30Hz 35Hz 40Hz 45Hz 50Hz 55Hz

12. In order to reveal the effect of the confined air behind the water sheet, the length of the back wall was changed from 40 cm to 90 cm and then to180 cm. Water depth at the weir crest: 2.78cm （ discharge 0.16m 3 /min ） 90cm 40cm Amplitude of the oscillations as a function of the distance from the weir crest Modulation and amplification of the sheet oscillation C. Free Fall with a Back Wall

13. Theory 1―― Surface tension is dominant （ Kelvin-Helmholtz instability ） Theory 2―― Forced oscillations of a water sheet confined by a water sheet and walls （ Potential flow theory ） Theory 3―― Shear instabilities of air flow ( Falkner-Skan flow; Instability of a suction flow) Pt.2 THEORETICAL DISCUSSIONS

14. Boundary conditions Theory 2. Potential flow theory

15. Long wave approximation Perturbation method Sheet thicknessSheet displacementFlow velocity

16. Governing equations Mass x- momentum y- momentum Assuming that

17. Pressure Kinematic B.C. Pressure at the free surface

18. Non-dimensionalization Parameters governing the flow

19. Forced oscillations Non-dimensional frequency Maximum amplitude of sheet

20. Weir flow Forced vibrations 5 Hz8 Hz10 Hz15 Hz h 0 =2.3 cm, u 0 =15 cm/s (Q=0.13 m 3 /min) 3m 1.5m

21. Theory 3. Shear Wave Instability Governing Equations of Water Flow Thin sheet approximation (Lowest order) Water sheet ― Long wave + Thin sheet approximations Air flow ― Boundary layer approximation for steady flow

22. Equations for Water Sheet Movements ･ 7 unknowns for 7 equations Momentum eq. Kinematic eq. Normal stress eq. Tangential stress eq.

23. Governing Equations of Air Flow Navier-Stokes equation No slip boundary conditions at the sheet surfaces ･ 4 constitutive relations for 4 equations

24. Steady Flow ―― Boundary layer approximation ―→ Falkner-Skan similarity solution y=5 cm Suction flow

25. Non-dimensionalization Governing Parameters Gradually Varying Flow Linear Stability Analysis for Gradually Varying Flow

26. Fundamental Solutions of the Orr-Sommerfelds Equation Eigen-value Relation for the Sinuous Mode

27. Neutral curve of a water sheet with uniform thickness Falkner-Skan flow Water sheet Frequency and wavelength of the most unstable mode at x=100 cm and 150 cm. cm Hz

28. Unstable waves on the water sheet were visualized by a high- speed camera. The longer wave, which leads to the sheet break-up, has a frequency of 20 Hz ~ 30 Hz and a wave-length of 30 cm ~ 50 cm. The frequency response of the sheet varies with position along the sheet and therefore with the thickness of the sheet, with the sheet resonating at higher frequencies in the thinner section of the sheet, i.e., at greater distances from the weir crest. The confined air between the walls and the water sheet causes a modulation of the sheet amplitude. CONCLUSIONS A model describing the motion of the sheet in the longitudinal and normal directions for the back-wall case was developed assuming that the flow was irrotational. This model explains the amplification of the sheet oscillations and the modulation of the amplitude of oscillation caused by the propagation of pressure fluctuations under the influence of the confined air. Incipient oscillations of the sheet observed in the free-fall experiments were characterized by using a locally uniform-flow model subjected to a linear stability analysis of the corresponding Navier–Stokes equations.