Turbulence in Superfluid Helium-4 atmK Temperatures

Otaniemi, 16. Nov. 2001

W. Schoepe

Institut für Experimentelle und Angewandte Physik

Universität Regensburg

Contents

1. Why is turbulence in superfluids at T = 0 interesting?

2. Basics of classical turbulence.

3. Basics of superfluid turbulence.

4. Classical character of superfluid turbulence at high vortex densities.

5. Quantum turbulence: our experiment.

6. Conclusion.

Literature:

R.J. Donnelly „Quantized Vortices in Helium II“Cambridge University Press, Cambridge 1991.

Lecture Notes in Physics „Quantized Vortex Dynamics andSuperfluid Turbulence“, Vol. 571, Springer 2001.

Classical turbulence

Navier-Stokes-equation:

( ) Vpgrad1VgradVtV rrrr

∆ν+ρ

−=⋅+∂∂

ρην = kinematic viscosity

air 1,4 · 10-5 m2/swater 10-6

HeI (2,2 K) 1,7 · 10-8

HeII (0K) 0

( ) L/V~VgradV 2rr⋅

VL

LVRe

22

ν⋅≡⇒

2L/V~V ν∆⋅νr

Reynoldszahlν

= VLRe

Example

Sphere in water

RV

L = 2 R = 10-2 m⇒ Re = 6

22

101010−

−− ⋅ = 102

V = 10-2 m/s

Re < 1: laminar flowdrag force 6πηRV (Stokes)

Re > 103: turbulent flowdrag force 22

D VRc21 ρπ

definition 22

DVR

21

forcedragcρπ

=

sphere: cD ≈ 0,4

Kolmogorov’s Law

1V l 2V

3/2221 l|VV| ∝−rr

|21 VV|rr

− can depend only on

=ε 3

2

sm

kgWatt and l (m)

⇒ | 3/121 )l(|VV ε∝−rr

3/23/23/23/2221 kl|VV|E −ε∝ε∝−∝rr

3/2

kkdk)k(FE −∫

∞∝= ⇒

3/5k)k(F −∝

Kolmogorov 1941v. Weizsäcker 1945

Decay of turbulence

>ω<ν=−=ε 2

dtdE , V

rrrot=ω

⇒∝ε⇒∝⇒∝ε −− 322/3 ttEE

2/3t−∝ω

Superfluid turbulence (T = 0)

(4He, BEC)

Equation of motion : nonlinearSchrödinger equation

ψψ+ψ∇−=∂ψ∂ 22

2||V

m2ti hh

Madelung transformation: φ=ψ iae

φ∇=m

v hr

2ma=ρ

0)v(t

=ρ⋅∇+∂ρ∂ r

continuity equation

mVav)v(

tv 2

∇−=∇⋅+∂∂ rrr

Euler equation

no viscosity, no Reynolds number

pure potential flow: 0vrot =r

quantized circulation: ∫ κ=⋅ Nrdv rr , vortices with N = 1

superfluid turbulence = tangle of singly quantized vortices

High vortex densities:

classical character

Examples:

1. rotating superfluid:

imitation of classical solid body rotation

rv rrr ×Ω= ( )Ω=rr 2vrot

by n vortices/area:

Ω==κ 2|vrot|n r

2. Tabeling’s “washing machine”:

Kolmogorov’s law

3. Decay of grid turbulence

Quantum turbulence

L line length density

= 23 m

1mm

κ circulation quantum

−

sm10

27

inter vortex spacing )m(L

1

energy/mass E ~ κ 2L

2

2

sm

dissipated power/mass

κε 3

223

smL~

vorticity

κω

s1L~

quantum turbulence when characteristic length scalel <

L1

41

3l

εκ<

Experimental Setup

• A magnetic microsphere is levitating between two ho-

rizontally arranged superconducting electrodes made of

niobium

• no mechanical suspension elements, horizontal stabi-

lity is provided by trapped flux lines

• the sphere is electrostatically charged by applying a high

dc voltage to the capacitor before cooling below Tc

• space between the electrodes is filled with pure 4He

• excitation of vertical oscillations of the sphere around its

equilibrium levitation position by applying a resonant ac

electric field

• measurement of: velocity amplitude of the oscillatons

• parameters: driving force amplitude / temperature

different regimes

0 200 400 600 8000

10

20

30

40

T = 300 mK

v (m

m/s

)

F (pN)

low driving forces: laminar flow, dissipation is givenby ballistic scattering of phonons

large driving forces: turbulent flow, large dissipati-on similiar to classical flow

intermediate range: state of system is unstable, sy-stem switches intermittently between laminar flowand turbulent flow

turbulent regime

0 2 4 6 8 10 120

20

40

60

80

100

T = 100 mK

v (m

m/s

)

F (nN)

In the turbulent regime a driving force similiar toturbulent flow in classical fluids is observed:

Ft = γ(v2−v20)

using the classic turbulent drag coefficient

γ = CDρπR2/2

with CD ≈ 0.4 for spheres, which describes thesolid line through the data.

Intermittent Switching

18

20

22

24

26

28

30

time (100 s / division)

18

20

22

24

26

28

30

v (m

m/s

)

18

20

22

24

26

28

30

Turbulent Statistics

20 30 40 501

10

100

1000

ba

32 mK 100 mK 200 mK 300 mK 403 mK

µ (s

)F-λvt (pN)

0 10 20 30 40 500.001

0.01

0.1

1

T = 300 mKF = 59 pN

t (s)

P(t

)

Laminar Statistics

0 10 20 30 40 500

1

2

3

4

5

6

7

b

+10%

-10%4.79

28mK 100mK 200mK 300mK 403mK

v w (

mm

/s)

F-λvt (pN)0 2 4 6 8 10 12

0.01

1

a

T = 300 mKF = 55 pN

P( ∆

v)

∆v (mm/s)

P (∆v) = exp(−(∆v/vw)2)

Lifetime of Laminar Phases

Cumulative distribution function:

P (t) = P (∆v(t)) = exp

(−

(∆v(t)

vw

)2)

where

∆v(t) = ∆vmax(1 − exp(−t/τ)).

Failure rate:

Λ(t) = −d lnP (t)dt

= v−2w

d(∆v2)dt

for long times t τ :

∆v = ∆vmax = const and therefore P = const

and hence Λ = 0:

stable laminar phases although v > v

Stability of Laminar Phases

0 100 200 300 400 5001E-3

0,01

0,1

1

45,0 pN

47,1 pN

49,1 pN

51,2 pN

300mK

rel.

Anz

ahl l

amin

arer

Pha

sen

t in s

0 200 400 600 800 1000 1200 14000.01

0.1

1

without 60

Co sourceτ = 25 min

T = 300 mKF = 47 pN

with 60

Co sourceτ = 3 min

norm

aliz

ed n

umbe

r of

lam

inar

pha

ses

t (s)

Summary

Statistics of turbulent phase laminar phase vortex nucleation

CDF exponential(t) Gauss(∆v) double exponential(∆v)

PDF exponential Weibull exp · double exponential

Failure rate const linear exponential

dependence Fturb vw vc(T ), ∆vc

Summary

• A levitating magnetic microsphere is oscillating in liquid

helium in the mK range, creating turbulence.

• Between turbulent and laminar regimes there is a unstable

regime where the system switches between both states

intermittently.

• The velocities reached during laminar phases are Weibull

distributed, while the lifetimes of turbulent phases are

exponentially distributed.

• The distributions are independent of temperature.

• Metastable laminar phases exist, their lifetime is only

limited by natural background radiation

Future

• using different sized spheres

• perform the experiment in superfluid 3He