aH. Kyotoh, bR. Nakamura & aP. J. Baruaha

　 aInstitute of Engineering Mechanics and Systems,

　　　　　 University of Tsukuba, Ibaraki, Japan　 bThird Plan Design Industrial Co., Ltd., Gifu, Japan

Incipient Oscillations of a Falling Water Sheetand their Instability Mechanisms

(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 W

eber number of the curtain flow exceeds 1/2.

(3) Luca and Costa(1997) studied instability of a spatially developing liquid shee

t 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 studyImportance of study

Review of researchReview of research

(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 st

ability analysis of the Navier –Stokes equations.

Present studiesPresent studies

Side view Front view

300c

m

40cm

Side wall

Laser displacement sensor

40cm

Experimental apparatusExperimental apparatus

300c

m

Side wallSide wall(Transparent)

Nappe

Oscillator

Back wall

40cm

Pt.1 EXPERIMENTAL STUDIESPt.1 EXPERIMENTAL STUDIES

High Speed Camera ImagesHigh Speed Camera Images

x=50cm ～ 130 cm

x=130cm ～ 180 cm x=180 ～ 220 cm

Water depth=3 cm ; Discharge=0.1312 m3/min

Free fall without vibrations

40 cm 40 cm 40 cm

Wavelets on a water sheet A hole on a water sheet

20 cm 30 cm

Enlargement

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

0 50 100 150 200Distance from the weir crestHcmL

0

0.2

0.4

0.6

0.8

1

Thickness of the sheet and the boundary layer

Boundary layerSheet

Thickness

0 50 100 150 200Distance from the weir crestHcmL

0

500

1000

1500

2000

2500

3000

3500

Reynolds number

Air:ReaWater:Rew

Reynolds number

(cm)

0.00E+00

5.00E- 10

1.00E- 09

1.50E- 09

2.00E- 09

0 20 40 60

120cm

0.00E+00

2.00E- 10

4.00E- 10

6.00E- 10

8.00E- 10

0 20 40 60

40cm

0.00E+00

2.00E- 104.00E- 10

6.00E- 10

8.00E- 101.00E- 09

1.20E- 09

0 20 40 60

80cm

0.00E+00

1.00E- 09

2.00E- 09

3.00E- 09

4.00E- 09

5.00E- 09

0 20 40 60

160cm

0.00E+002.00E- 084.00E- 086.00E- 088.00E- 081.00E- 071.20E- 071.40E- 07

0 20 40 60

200cm

Frequency(Hz)Pow

er s

pect

rum Distance from

the weir downstream

A. Free Fall without Vibrations

　 Power spectrum changes downstream（ Water depth at the weir crest:2.24cm;

Discharge:0.135m3/min ）

0.00E+00

2.00E- 09

4.00E- 09

6.00E- 09

8.00E- 09

1.00E- 08

1.20E- 08

1.40E- 08

1.60E- 08

0 20 40 60(Hz)周波数

パワースペクトル

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 ）

0.00E+00

2.00E- 08

4.00E- 08

6.00E- 08

8.00E- 08

1.00E- 07

1.20E- 07

1.40E- 07

0 20 40 60(Hz)周波数

パワースペクトル

0.00E+00

4.00E- 08

8.00E- 08

1.20E- 07

1.60E- 07

0 20 40 60(Hz)周波数

パワースペクトル

0.00E+001.00E- 092.00E- 093.00E- 094.00E- 095.00E- 096.00E- 097.00E- 098.00E- 099.00E- 091.00E- 08

0 20 40 60(Hz)周波数

パワースペクトル

1.95cm(0.12m3/min)

2.46cm(0.14m3/min)

2.69cm(0.15m3/min)

3.5cm(0.19m3/min)

Water head（ Discharge ）

0. 0E+00

2. 0E-05

4. 0E-05

6. 0E-05

0 10 20 30 40 50

Hz周波数（ ）

パワースペクトル 160cm

below theweir crest

0. 0E+00

5. 0E- 08

1. 0E- 07

1. 5E- 07

2. 0E- 07

0 10 20 30 40 50Hz周波数（ ）

パワースペクトル

80cmbelow theweir crest

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 m3/min

B. Free Fall with VibrationsB. Free Fall with Vibrations

Frequency (Hz) Frequency (Hz)

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 vibrationsExponential growth of the wave amplitude along the stream

Response of the sheet under the forced vibrationsExponential growth of the wave amplitude along the stream

Amplification rate(4Hz ～ 20Hz) Amplification rate(25Hz ～ 55Hz)

1.00E-02

1.00E+00

1.00E+02

1.00E+04

1.00E+06

0 50 100 150 200 250

Distance from the weir(cm)

4Hz6Hz8Hz10Hz15Hz20Hz

1.00E-02

1.00E+00

1.00E+02

1.00E+04

1.00E+06

0 50 100 150 200 250

Distance from the weir(cm)

25Hz30Hz35Hz40Hz45Hz50Hz55Hz

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 0.16m3/min ）

0

0.004

0.008

0.012

0.016

0 50 100 150 200 250Distance from weir crest downstream(cm)

Root

mean s

quare

(m)

180cm90cm40cm

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 WallC. Free Fall with a Back Wall

Theory 1―― Surface tension is dominant（ Kelvin-Helmholtz instability ）

max

3

2

6646.0

,1

,)1(2

fffor

fffor

V

V

S

Vf

cr

w

awa

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 DISCUSSIONSPt.2 THEORETICAL DISCUSSIONS

0 50 100 150 200Distance from the weir crestHcmL

0

50000

100000

150000

200000

250000

300000

350000Kelvin- Helmholtz theory

Frequency: fmax

Weber number

021

,022

w

IIII

I

Pgx

yxt

Boundary conditions

npy

yxxtPTP IISI

,,

,0,0

Theory 2. Potential flow theory

sY

mY

LY

hny py

y

x

1)/(/)/( yOxO

2

10

10

10

,),(),(

,),,(),,(

,),(),(

txPtxPP

tyxtyx

txtx

SSS

III

00000

0 2

1,

2

1, npmnps

I YYx

U

Long wave approximation

Perturbation method

Sheet thickness Sheet displacementFlow velocity

Governing equations ,00

ss YU

xt

Y

,00

0 gx

UU

t

Uaww

011

2

2

0

,

222

Ux

Y

t

YVPP

x

YTVUY

xVY

t

mmSrS

mswsw

Mass

x- momentum

y- momentum

11

2

22

0

2

0

00

02

2

224

22

SrS

msw

msw

msw

msw

PPx

YTUY

xt

YUY

x

Y

x

UU

t

UY

t

YY

0t

YsAssuming that

Pressure

yyxxxn

,1

2/12

Kinematic B.C.

Pressure at the free surface

np

hLn

n

hxn

hxn

xxGr

xdtxt

xxGrtx

n

h

S

,

,tanh

coscos1,

,,,2,0

01

np

xdtxt

xxGrtxPh

aS

,

,,,2,0 2

02

1

Non-dimensionalization

hg

uu

hgh

uyq

h

gtt

h

xx e

neee

nnn

0

00

0

000

2,

2,,

1

0 2

22

0

2

22

0

22

02

2

,,

222

nInnI

n

mnInrnen

n

mnen

nn

mnen

n

m

n

m

dxtxt

YhxhxGrux

x

Yux

xt

Yux

x

Y

t

Y

Parameters governing the flow

2/3

0

0

002,

22

h

h

hgh

uyq

q

s eeen

n

2/1

0

0

00

2

1

2

h

h

hg

uu ee

ne

210/1,22100/1,1000/1 00 nee

w

a uh

hs

0 2

3 2

2 5 2

3 7 2

4

1

2

3

4

Forced oscillationsForced oscillations

0 0.2 0.4 0.6 0.8 1-2

-1

0

1

2

0 0.2 0.4 0.6 0.8 1

-2

-1

0

1

2

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

1,1.0,1 0 rneu

Non-dimensional frequency

Maximum amplitude of sheet

Weir flow Forced vibrations

5 Hz 8 Hz 10 Hz 15 Hz

h0=2.3 cm, u0=15 cm/s (Q=0.13 m3/min)

3m

1.5m

Theory 3. Shear Wave Instability Theory 3. Shear Wave Instability

020

20

10

00

gy

u

y

uv

x

uu

t

uawww

010

y

v

x

u

txvytxvytxyvtxvtyxv

txuytxyutxutyxu

,,,,,,

,,,,,

133

122

11101

022

01000

Governing Equations of Water FlowGoverning Equations of Water Flow

Thin sheet approximation (Lowest order)

Water sheet ― Long wave + Thin sheet approximations Air flow ― Boundary layer approximation for steady flow

21

21

11

01

y

v

y

vv

x

vu

t

v

y

pww

w

02 0200

0000

ugx

uu

t

uEqMx www

03

1

2

1 3002

2001000

010

ppp

p uuuxt

vEqKp

03

1

2

1 3002

2001000

010

nnn

n uuuxt

vEqKn

0222,, 02002001000

x

Suuux

txPSnEqSn ppwpwwpwpp

02 00201 nwwnn uuStEqSt

02 00201 pwwpp uuStEqSt

Equations for Water Sheet MovementsEquations for Water Sheet Movements

0222,, 02002001000

x

Suuux

txPSnEqSn nnwnwwnwnn

･ 7 unknowns 　 for 7 equationswmnp pvuuu ,,,,,, 1002010000

x

uv

x

vu

x

u

x

uu

x

u

xt

uy

x

uv

x

vu

x

u

t

vytxPtyxP

w

ww

w

wwwmw

0110

1001

02200

2

00

2

000022

0010

1000

0110

22

,,,

Momentum eq.

Kinematic eq.

Normal stress eq.

Tangential stress eq.

Governing Equations of Air FlowGoverning Equations of Air Flow

0

0

Vy

P

y

VV

x

VU

t

V

Ux

P

y

UV

x

UU

t

U

aa

aa

0

y

V

x

U

0,,

0,,2

002001000

2002001000

nnnnn

ppppp

uuuyxUEqu

uuuyxUEqu

03

1

2

1,,

03

1

2

1,,

0230

0120

000100

0230

0120

000100

x

u

x

u

x

uvyxVEqv

x

u

x

u

x

uvyxVEqv

nnnnnn

pppppp

Navier-Stokes equation

No slip boundary conditions at the sheet surfaces

･ 4 constitutive relations for 4 equations

nnppnnnnppnn

nnppppnnpppp

UUVUStStUUVUSnSn

UUVUStStUUVUSnSn

,,,,,,,

,,,,,,,,

Steady Flow ―― Boundary layer approximation Steady Flow ―― Boundary layer approximation

02

2

y

U

y

UV

x

UU S

aS

SS

Sa

0

y

V

x

U SS

―→　 Falkner-Skan similarity solution ―→　 Falkner-Skan similarity solution

0;1;0;03

2

2;4

3;

3

4

00

2

2

2

3

3

00000

0000

d

df

d

dff

d

df

d

fdf

d

fd

xgux

uYf

u

xu

a

a

a

aS

xV

yU S

FSS

FS

,

y=5 cm

0 20 40 60 80 100Distance from the weir crestHcmL

0

1

2

3

4

5

y

Velocity vectors

Suction flow

w

a

aa

www

a

a

a

a

S

xxuWb

x

xd

xu

xx

xxu

,/

/,,

3

4,Re

200

00

00

VVVUUU FSFSˆ,ˆ

Non-dimensionalization

xuxxxu 0000 /variableTemporal,variablesSpatial,Velocity

Governing Parameters

Gradually Varying Flow

tdxxkiyxAVU exp,ˆ,ˆ

Linear Stability Analysis for Gradually Varying FlowLinear Stability Analysis for Gradually Varying Flow

Fundamental Solutions of the Orr-Sommerfelds Equation

yByAy

yByAy

nnnnn

ppppp

21

21

Eigen-value Relation for the Sinuous Mode

yyy

yyy

np

np

222

111

0''''''''''''''''''''''''

''''''''''''

211222112221122

211222112221122

FED

CBA

23222

334222

1/1211232

312112

cwccRkik

kciRkcRckA

be

ee

321121 222 ikRkcRckcB ee

22221/212 22 ikccwRkRckC bee

22 112 kciD

ciE 122

22 12 ciF

Neutral curve of a water sheet with uniform thickness

10000 15000 20000Reynolds number: kRe

0.86

0.88

0.9

0.92

cNeutral curve: Wb=10000, l =0.5

10000 15000 20000Reynolds number: kRe

0.1

0.15

0.2

0.25

0.3

0.35

k

Neutral curve: Wb=10000, l =0.5

Falkner-Skan flow

Falkner-Skan flow

Water sheet

Water sheet

Frequency and wavelength of the most unstable mode at x=100 cm and 150 cm.

0 20 40 60 80 100Distance from the weir crestcm20

30

40

50

60

fzH

Frequency of unstable wave

0 20 40 60 80 100Distance from the weir crestcm2

3

4

5

6

htgneL

Wave length

cmHz

<Free-fall experiments> 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.

<Forced-vibration experiments> 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.

<Experiments with the back-wall> The confined air between the walls and the water sheet

causes a modulation of the sheet amplitude.

CONCLUSIONS

<Potential flow theory> 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.

<Viscous flow theory> Incipient oscillations of the sheet observed in the free-fall experime

nts were characterized by using a locally uniform-flow model subjected to a linear stability anal

ysis of the corresponding Navier–Stokes equations.