Superfluid helium is an intimate mixture of a viscous normal fluid, with continuous vorticity, and an inviscid superfluid, where vorticity is constrained to thin, stable topological defects. One mechanism to generate turbulence in this system is through the application of a heat flux, so-called thermal counterflow. Of particular interest is how turbulence in the superfluid responds to both a laminar and turbulent normal fluid in the presence of walls. We model superfluid vortex lines as reconnecting space curves with fixed circulation, and consider both laminar (Poiseuille) and turbulent normal fluid flows in a channel configuration. Using high resolution numerical simulations we show that turbulence in the normal fluid sustains a notably higher vortex line density than a laminar flow with the same mean flow rate. We examine Vinen's relation,

$\sqrt{L}=\gamma v_{ns}$
L=γvns⁠, between the steady state vortex line density L and the counterflow velocity vns. Our results support the hypothesis that transition to turbulence in the normal fluid is responsible for the TI to TII transition. We also consider the spectral properties of fluctuations of the superfluid vortices, which show a good agreement with previous experimental results.

1.
L.
Skrbek
and
K. R.
Sreenivasan
, “
Developed quantum turbulence and its decay
,”
Phys. Fluids
24
,
011301
(
2012
).
2.
W. F.
Vinen
and
J. J.
Niemela
, “
Quantum turbulence
,”
J. Low Temp. Phys.
128
,
167
(
2002
).
3.
J.
Maurer
and
P.
Tabeling
, “
Local investigation of superfluid turbulence
,”
EPL
43
,
29
(
1998
).
4.
J.
Salort
,
C.
Baudet
,
B.
Castaing
,
B.
Chabaud
,
F.
Daviaud
,
T.
Didelot
,
P.
Diribarne
,
B.
Dubrulle
,
Y.
Gagne
,
F.
Gauthier
 et al, “
Turbulent velocity spectra in superfluid flows
,”
Phys. Fluids
22
,
125102
(
2010
).
5.
C.
Nore
,
M.
Abid
, and
M. E.
Brachet
, “
Kolmogorov turbulence in low-temperature superflows
,”
Phys. Rev. Lett.
78
,
3896
(
1997
).
6.
T.
Araki
,
M.
Tsubota
, and
S. K.
Nemirovskii
, “
Energy spectrum of superfluid turbulence with no normal-fluid component
,”
Phys. Rev. Lett.
89
,
145301
(
2002
).
7.
A. W.
Baggaley
,
C. F.
Barenghi
,
A.
Shukurov
, and
Y. A.
Sergeev
, “
Coherent vortex structures in quantum turbulence
,”
EPL
98
,
26002
(
2012
).
8.
J.
Salort
,
B.
Chabaud
,
E.
Lvque
, and
P.-E.
Roche
, “
Energy cascade and the four-fifths law in superfluid turbulence
,”
EPL
97
,
34006
(
2012
).
9.
L.
Skrbek
,
A. V.
Gordeev
, and
F.
Soukup
, “
Decay of counterflow He II turbulence in a finite channel: Possibility of missing links between classical and quantum turbulence
,”
Phys. Rev. E
67
,
047302
(
2003
).
10.
P.
Walmsley
and
A.
Golov
, “
Quantum and quasiclassical types of superfluid turbulence
,”
Phys. Rev. Lett.
100
,
245301
(
2008
).
11.
R.
Donnelly
,
Quantized Vortices in Helium II
,
Cambridge Studies in American Literature and Culture
Vol.
2
(
Cambridge University Press
,
1991
).
12.
J.
Tough
, in
Progress of Low Temperature Physics
, edited by
D.
Brewer
(
North-Holland Publications
,
Amsterdam
,
1982
), Vol.
VIII
.
13.
W. F.
Vinen
, “
Mutual friction in a heat current in liquid helium II. I. Experiments on steady heat currents
,”
Proc. R. Soc. London, Ser. A
240
,
114
(
1957
).
14.
W. F.
Vinen
, “
Mutual friction in a heat current in liquid helium II. II. Experiments on transient effects
,”
Proc. R. Soc. London, Ser. A
240
,
128
(
1957
).
15.
W. F.
Vinen
, “
Mutual friction in a heat current in liquid helium II. III. Theory of the mutual friction
,”
Proc. R. Soc. London, Ser. A
242
,
493
(
1957
).
16.
W. F.
Vinen
, “
Mutual friction in a heat current in liquid helium II. IV. Critical heat currents in wide channels
,”
Proc. R. Soc. London, Ser. A
243
,
400
(
1958
).
17.
C. F.
Barenghi
,
A. V.
Gordeev
, and
L.
Skrbek
, “
Depolarization of decaying counterflow turbulence in He II
,”
Phys. Rev. E
74
,
026309
(
2006
).
18.
H.
Adachi
,
S.
Fujiyama
, and
M.
Tsubota
, “
Steady-state counterflow quantum turbulence: Simulation of vortex filaments using the full Biot-Savart law
,”
Phys. Rev. B
81
,
104511
(
2010
).
19.
L.
Galantucci
,
C.
Barenghi
,
M.
Sciacca
,
M.
Quadrio
, and
P.
Luchini
, “
Turbulent superfluid profiles in a counterflow channel
,”
J. Low Temp. Phys.
162
,
354
(
2011
).
20.
W.
Guo
,
S. B.
Cahn
,
J. A.
Nikkel
,
W. F.
Vinen
, and
D. N.
McKinsey
, “
Visualization study of counterflow in superfluid 4He using metastable helium molecules
,”
Phys. Rev. Lett.
105
,
045301
(
2010
).
21.
D.
Melotte
and
C.
Barenghi
, “
Normal fluid velocity profile and transition from T-1 to T-2 state of superfluid turbulence
,”
J. Low Temp. Phys.
113
,
573
(
1998
).
22.
K. W.
Schwarz
, “
Three-dimensional vortex dynamics in superfluid 4He: Line-line and line-boundary interactions
,”
Phys. Rev. B
31
,
5782
(
1985
).
23.
R. J.
Donnelly
and
C. F.
Barenghi
, “
The observed properties of liquid helium at the saturated vapor pressure
,”
J. Phys. Chem. Ref. Data
27
,
1217
(
1998
).
24.
K. W.
Schwarz
, “
Three-dimensional vortex dynamics in superfluid 4He: Homogeneous superfluid turbulence
,”
Phys. Rev. B
38
,
2398
(
1988
).
25.
A. W.
Baggaley
,
L. K.
Sherwin
,
C. F.
Barenghi
, and
Y. A.
Sergeev
, “
Thermally and mechanically driven quantum turbulence in helium II
,”
Phys. Rev. B
86
,
104501
(
2012
).
26.
K.
Morris
,
J.
Koplik
, and
D. W. I.
Rouson
, “
Vortex locking in direct numerical simulations of quantum turbulence
,”
Phys. Rev. Lett.
101
,
015301
(
2008
).
27.
D.
Kivotides
,
J. C.
Vassilicos
,
C. F.
Barenghi
,
M. A. I.
Khan
, and
D. C.
Samuels
, “
Quantum signature of superfluid turbulence
,”
Phys. Rev. Lett.
87
,
275302
(
2001
).
28.
D.
Kivotides
, “
Coherent structure formation in turbulent thermal superfluids
,”
Phys. Rev. Lett.
96
,
175301
(
2006
).
29.
M. S.
Paoletti
,
M. E.
Fisher
,
K. R.
Sreenivasan
, and
D. P.
Lathrop
, “
Velocity statistics distinguish quantum turbulence from classical turbulence
,”
Phys. Rev. Lett.
101
,
154501
(
2008
).
30.
M.
Leadbeater
,
T.
Winiecki
,
D. C.
Samuels
,
C. F.
Barenghi
, and
C. S.
Adams
, “
Sound emission due to superfluid vortex reconnections
,”
Phys. Rev. Lett.
86
,
1410
(
2001
).
31.
L.
Kondaurova
and
S. K.
Nemirovskii
, “
Full Biot-Savart numerical simulation of vortices in He II
,”
J. Low Temp. Phys.
138
,
555
(
2005
).
32.
A.
Baggaley
, “
The sensitivity of the vortex filament method to different reconnection models
,”
J. Low Temp. Phys.
168
,
18
(
2012
).
33.
See http://code.google.com/p/incompact3d/ for open source code.
34.
S. K.
Lele
, “
Compact finite difference schemes with spectral-like resolution
,”
J. Comput. Phys.
103
,
16
(
1992
).
35.
S.
Laizet
and
E.
Lamballais
, “
High-order compact schemes for incompressible flows: A simple and efficient method with the quasi-spectral accuracy
,”
J. Comput. Phys.
228
(
16
),
5989
(
2009
).
36.
R. D.
Moser
,
J.
Kim
, and
N. N.
Mansour
, “
Direct numerical simulation of turbulent channel flow up to Reτ = 590
,”
Phys. Fluids
11
,
943
(
1999
).
37.
M.
Tsubota
,
C. F.
Barenghi
,
T.
Araki
, and
A.
Mitani
, “
Instability of vortex array and transitions to turbulence in rotating helium II
,”
Phys. Rev. B
69
,
134515
(
2004
).
38.
S.
Babuin
,
M.
Stammeier
,
E.
Varga
,
M.
Rotter
, and
L.
Skrbek
, “
Quantum turbulence of bellows-driven 4He superflow: Steady state
,”
Phys. Rev. B
86
,
134515
(
2012
).
39.
A.
Baggaley
and
C.
Barenghi
, “
Tree method for quantum vortex dynamics
,”
J. Low Temp. Phys.
166
,
3
(
2012
).
40.
C. F.
Barenghi
and
D. C.
Samuels
, “
Scaling laws of vortex reconnections
,”
J. Low Temp. Phys.
136
,
281
(
2004
).
41.
S. K.
Robinson
, “
Coherent motions in the turbulent boundary layer
,”
Annu. Rev. Fluid Mech.
23
,
601
(
1991
).
42.
V. S.
L'vov
,
S. V.
Nazarenko
, and
O.
Rudenko
, “
Bottleneck crossover between classical and quantum superfluid turbulence
,”
Phys. Rev. B
76
,
024520
(
2007
).
43.
A. W.
Baggaley
, “
The importance of vortex bundles in quantum turbulence at absolute zero
,”
Phys. Fluids
24
,
055109
(
2012
).
44.
H.
Hoch
,
L.
Busse
, and
F.
Moss
, “
Noise from vortex-line turbulence in He II
,”
Phys. Rev. Lett.
34
,
384
(
1975
).
45.
R.
Ostermeier
,
M.
Cromar
,
P.
Kittel
, and
R.
Donnelly
, “
Fluctuations in turbulent He II counterflow
,”
Phys. Lett. A
77
,
321
(
1980
).
46.
C. F.
Barenghi
,
C. E.
Swanson
, and
R. J.
Donnelly
, “
Induced vorticity fluctuations in counterflowing He II
,”
Phys. Rev. Lett.
48
,
1187
(
1982
).
47.
P.-E.
Roche
,
P.
Diribarne
,
T.
Didelot
,
O.
Franais
,
L.
Rousseau
, and
H.
Willaime
, “
Vortex density spectrum of quantum turbulence
,”
EPL
77
,
66002
(
2007
).
48.
A. W.
Baggaley
,
J.
Laurie
, and
C. F.
Barenghi
, “
Vortex-density fluctuations, energy spectra, and vortical regions in superfluid turbulence
,”
Phys. Rev. Lett.
109
,
205304
(
2012
).
You do not currently have access to this content.