№10Quaint jet

Reporter: Anastasiia Chervinskaia

Team: Russia, Voronezh

1

The problem

When water is forced through a thin slit, the flowsometimes takes the shape of a helix. Describe thephenomenon and explain the dependence of theaspect ratio(s) of the helix on the fluid parameters,parameters of the flow and the shape of the nozzle.

2

The phenomena

3

Plan of the report

4

Qualitative description of the phenomena

Linear model of the process

Nonlinear effects

Description of helix

5

𝛼

45

°vi

ew

fro

nt

view

𝛼 – the angle between the planes

𝛼 = 90o+ 𝛼0

𝛼0 −“twisting angle"

sid

e vi

ew

chain-like structure

Experimental setup

6

Work of the setupScheme and photo

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.00 0.01 0.02 0.03

A

B

C

ℎ0

ℎ,m

𝑣 𝑦,m

/s

A B C

ℎ0

Velocity profile of the jet

7

𝑣𝑦 almost doesn’t depend on

x-coordinate on the height ℎ0.The motion of water layer can be

considered as a whole.

𝑥

𝑦

drops of ink

Forces acting on the stretched water layer

8

Ԧ𝐹σ Ԧ𝐹σ

Ԧ𝑣 Ԧ𝑣

Ԧ𝑣

Ԧ𝑣

Ԧ𝐹σ

Ԧ𝐹σ

Surface energy converts into the kinetic energy and vice versa

Qualitative explanation. “Links of the chain”

«Top view» Inside the jet

9The oscillations of water layer create helix-like shape of jet

Number of mode

10

The higher modes can appear for other shapes of the orifice

3n 4n 5n2n

Experiment with 3-d mode

11

Exp

erim

ent

Nu

mer

ic m

od

elin

g

«Top view»

Similar oscillation can take place on the third mode.

Nozzle

Weber number

Weber number — dimensionless number, describing a measure of the relative importance of the fluid's inertia compared to its surface tension.

𝑊𝑒 =𝜌𝐿𝑣2

𝜎

𝐿 is the characteristic length

𝜌 is the density of the fluid

12

𝑣 is the velocity of the fluid

𝜎 is the coefficient of the surface tension

Twisting

13

1st link 2nd link

Front view

Side view

Qualitative explanation. Twisting

14

1. Initial vorticity.

а. in the funnel b. because of slit’s asymmetry

Г = Г0 Г = න Ԧ𝑣𝑑𝑙 ≈ 𝜔0𝑆

Г is the circulation S is the initial area of jet cross-section𝜔0 is the initial angular velocity

Jet of inkAngle between directions of

velocity

𝛽

The appearance of twisting

Circulation is constant

The less is the area, the bigger is angular velocity

The vertical velocity increases, and the cross

area decreases

Angular velocity increases

Г = const

𝜔0 =Г

𝑆

𝑣𝑦 ↑,

𝜔0 ↑

𝑆 ↓

Experimental setup – the hose

Tap without aerator

Clamp for the hose

16

A

B

C

A B C A B C

0

Varying the shape of hose

Hose

The frequency of oscillations of liquid cylinder

𝑟 = 𝑟0 1 + 𝛼𝑛 cos 𝑛𝜃 cos𝑡

𝐿 =𝜌

2න𝑣2𝑑𝑉 − 𝜎𝑆 ሷ𝛼𝑛 + 2𝛼𝑛 = 0

2 =6𝜎

𝜌𝑟03

𝜎 is the surface tension

𝑟0(𝑧) – hydraulic radius of the jet

𝜌 – density of the liquid

17

linearization

n = 2

Kinetic energy

Potential energy

Eqution of oscilltion

Frequency for the second mode

n = 2 𝑟

𝑟0

𝜃

The hydraulic radius vs height

𝑣𝑆 = 𝑐𝑜𝑛𝑠𝑡

𝑣 = 𝑣02 + 2𝑔𝑧

𝑟0𝑟0 = 𝑅 1 +

2𝑔𝑧

𝑣02

−1/4

𝑅

𝑧

𝑅 – hydraulic radius of the orifice

𝑣0 - initial velocity of the jet

18

𝑆 – is the cross-section area of the jet

Continuity equation

𝑆

𝑣

Hydraulic radius decreases with heightThe frequency of oscillations increases

Velocity of the layer

The aspect ratio of the helix

𝑅≈ 0,95 𝑊𝑒

𝑊𝑒 – Weber number𝑅 – hydraulic radius of the orifice - pitch of a helix

19

cos𝜑 (𝑧) = 0

cos𝜑(𝑧) = 1

𝑧

𝑡 𝑧 =𝑣0𝑔

1 +2𝑔z

𝑣02 − 1

=6𝜎

𝜌𝑅31 +

2𝑔z

𝑣02

3/8

Frequency

Time

𝜑 𝜆 = න0

𝜆

𝜔(𝑧)𝜕𝑡

𝜕𝑧𝑑𝑧 =

𝜋

2

Phase

Helix pitch vs number of the turn

20

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑢𝑟𝑛

𝜆,𝑐𝑚 theory

experiment

2𝜆 2𝜆 2𝜆 2𝜆𝜆

1st 2nd 3rd 4th 5th

The length of the link doesn’t depend on the number of turn

The experiment vs numeric model

5 cm

exp

erim

ent

mo

del

21ex

per

imen

t

mo

del

5 cm

mo

del

Role of nonlinearity

22

linear case nonlinear case

𝑎

𝑏

𝜀 =𝑎 − 𝑏

𝑎 + 𝑏

Relative elongation

𝑎

𝑏

𝜀 ≃ 0.5 𝜀 ≃ 0.95

The technique of the experiment

Using Tracker (software for video analyses) we determine the height of

water in the funnel

The flow rate was calculated for the geometry of funnel

Using Tracker measured the length of the first link

First link length vs velocity

24

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.5 1 1.5 2 2.5

𝑣, m/s

alcohol

water

𝜆, m

𝑣 = 1,5 𝑚/𝑠

𝜆 𝜆

water

𝑣 = 0,7 𝑚/𝑠

First link length vs surface tension

25

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

0.02 0.025 0.03 0.035 0.04 0.045 0.05

water𝜎 = 0,073 𝑁/𝑚

alcohol𝜎 = 0,022 𝑁/𝑚

𝜆 𝜆

𝜎, N/m

λ,cm

The dependence on We

26

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

𝑊𝑒

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.5 1 1.5 2 2.5

v, m/s

alcohol

water

linear model

experiment

𝜆, m𝜆

𝑟0

The first link elongation for the contraction of the slit

27

The increase of amplitude leads to the increase of the period, and, correspondingly, the length of

the links.

b

𝜆0– length of the first link (linear model).0

0.5

1

1.5

2

2.5

0 10 20 30 40

𝑎

𝑏

𝑎/𝑏

𝜆

𝜆0

𝜔 = 𝜔0 1 − 𝛾𝜀2 + 𝑜(𝜀4)

𝜆

𝜆0≃

1

1 − 𝛾𝜀2

linear nonlinear

𝜆 – length of the first link (experiment)

modification of the frequency

𝛾 = 0,57

Damping oscillations

28

The length of links decreases with length (in nonlinear case)

x1

2 x2

2 x2

ሷ𝛼 + 𝛾 ሶ𝛼 + 2𝛼 = 0

𝛾 ~ 𝜇

𝜇 – dynamic viscosity

𝛼 ~ exp −𝛾𝑡

𝜆 = 𝑔 𝛼

𝛼 – amplitude of oscillations

𝛼 ↓ 𝜆 ↓

Damped oscillator

Only for nonlinear case

𝜔 = 𝑓 𝛼

Length of the link vs number of the turn (nonlinear case)

29

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

Len

gth

of

the

link,

cm

Number of the turn

numeric modelling

experiment

55% water-glycerol solution

Conclusions

• Qualitative explanation:• The cause of chain-links – oscillations of water layer

• The cause of twisting – increase of initial vorticity.

• Aspect ratio of the helix is determined by Weber number and by the aspect ratio of the slit.

• The numeric model of the helix was created

• The dependence on viscosity, surface tension, geometry of jet and flow rate was investigated

30

References

1. N. Bohr, Determination of the surface-tension of water by the method of jet vibration

2. Lord Rayleigh, On the capillary phenomena of jets

3. J. Bush, On the collision of laminar jets: fluid chains and fishbones

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