№10 quaint jet
TRANSCRIPT
№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
31