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Turbulence-induced Vibration
W. Schnell, 13 Nov 2000
Inevitably, the flow of cooling water induces vibrations.
Alignment tolerances are tightest in the quadrupole magnets but
there, water-cooled coils can be (and probably have to be)
mechanically isolated from the yokes. In the accelerating structures
this is impossible, so that a fundamental problem may arise.
Water or ethylene-glycol (C2H6O2) may be considered. The
following properties are for 250 C:
Water
Mass density 
103
C2H6O2
1.11 103
kg m3
Viscosity 
0.89  103
16.1 103
Ns m2
Cinematic viscosity /
0.89 106
14.5 106
m2 s1
Heat capacity c
4.19 106
Heat conductivity
0.6
2.4 106
Ws m3K1
?
Wm1K1
Water is assumed unless stated otherwise – and:
Total dissipation per unit length
5 kWm1
Number of cooling channels n per structure
4
Temperature rise along channel
100Cm1
Equivalent channel diameter d
8 mm
2
Volume flow rate per channel:
V/t = 3.0  10 5 m3/s = 1.8 l/min
Average flow Velocity:
u = 0.593 m/s
Reynold’s number:
Re = udr / = 5330
Moderately above turbulence limit of about 2000
Laminar flow by subdivision in small subchannels
E.g. 64 subchannels of 1mm diameter in a cooling bar
formed from extruded Cu strips
7 strips + 2 half-strips form 64 channels of 1mm diam.
Re = 666 with water or 71 with C2H6O2 (at 5 bar/m)
0.2 m2 surface per m  t = 20C for heat conduction of
1.25 kW across 0.2mm of water
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Turbulent flow with Re  5000
Pressure drop along circular pipe of length l :
p 
u 2
2

l
d
The coefficient  depends weakly on Re. For smooth pipe
and moderate turbulence take Blasius’ formula:
 = 0.316 Re1/4
or published diagrams. Result:  = 0.04
Pump power = exit rate of turbulent kinetic energy:
V
V  2
p 
t
t 2
( = turbulent velocity seen from frame moving with u).
Conversion of hydrostatic energy density p to turbulent
kinetic energy density. Thus:
2  2
p
l
  u2

d
4
Isotropy (marginally valid):
 x2   y2   z2
Local vertical (say) momentum density:
 yrms  u 
l
3d
Rough order-of-magnitude approximation:
Nearly all energy is in coherence-cells of length d/2 thus
of volume Ad/2 (with A = d2/4). Momentum per cell:
rms
Pycell

Ad rms Ad
l
y 
u 
2
2
3d
Factor (2nl/d)1/2 for 2l/d such coherence-cells in length l
and n cooling pipes, net momentum, therefore:
rms
Pytot
 Alu
n
n
 mu
6
6
(m = mass of water in one pipe).
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Assume equivalent rigid mass M for struct.-cum-support,
find rms. velocity transmitted to structure:
rms
 ytot
u
m
M
n
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Dominant frequency():
crit  2
u
d
Rms. vibration amplitude:
s rms

y
1
2
n m
d
6 M
(Note that 1/2  (Re)1/8). With d = 8mm thus m = 50 g/m,
M = 11 kg/m (i.e. a copper rod of 40 mm diam.) and n = 4
s rms
 0.94 m
y
 crit 2  74 Hz
Order of magnitude estimate! Pessimistic? 
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 Order-of-magnitude considerations of turbulence
spectrum by L.D. Landau, E.M. Lifschitz vol IV § 3.3:
For Re/Recrit »1 the range of frequencies observed in a
hardware-frame is
(u/d) «  « (u/d)(Re/Recrit)3/4
Within that range power spectrum  5/3
But here (Re/Recrit)3/4  2 only.
So taking  = 2(u/d) is not so unreasonable
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srms   (dm)/M   d3/M
  (Re)1/8
Increase M as much as structural resonances permit.
(Lowest bending mode of copper rod 0.5 m long with
40 mm diameter is at 540 Hz. The nodes are at 0.276 l )
Decrease d (i.e. increase Re) ?
Not much; since crit  d3 acoustic resonances will
inevitably be excited for small d.
Or else, very high flow velocity and crit “ above”
structural resonances??
Note: l and m refer to the length of pipe of diameter d – not
directly to the length of the structure. Arrange for laminar
flow in input/output pipes (with tapered transitions).
Conclusion: The problem is not fatal but needs experiment
and careful design