Classical (viscous) turbulence In a 3D classical turbulent t + ( - - PowerPoint PPT Presentation

Classical (viscous) turbulence

• In a 3D classical turbulent

flow, large scale eddies break up into smaller eddies, these into smaller ones and so on...(Richardson Cascade)

• If there is a large inertial

range between the forcing and dissipation scale (i.e. high Re) then the flow of energy through scales is characterized by a constant energy flux .

• Dimensional analysis leads

to a power-law scaling for the energy spectrum,

E(k) = C✏2/3k−5/3

∂v ∂t + (v · r)v = 1 ρrp + νr2v

v = sin(x) ⇒ v ∂v ∂x ∼ sin(2x)

Classical Vorticity

ω = r ⇥ u

Quantum Fluids

Γ = ∮$𝐰 ⋅ 𝑒𝐦 ∈ ℝ Γ = ∮$𝐰 ⋅ 𝑒𝐦 = 2𝜌ℏ 𝑛 𝑜

Kuchemann:

“vortices are the sinews and muscles of fluid motions”

If this is true then Quantum Turbulence represents the ‘skeleton’

Yet we still see ‘classical’ behaviour

• Salort et al., 2011

10−1 100 101 10−7 10−6 10−5 10−4

• 5/3

kL0/(2π) φ(k)

Probe cut-off Vortex shedding

−4 −2 2 4 10≠5 10≠4 10≠3

p(v) 10−1 5 · 10−1 10−2 10−1 100 r/L0 − 5

4

+ ”v3, /(‘r)

10−1 100 ≠0.2 ≠0.1 r/L0 Èδv3Í Èδv2Í

T = 2.2K, ρs/ρn = 0

T = 1.56K, ρs/ρn = 6

Coherent structures

• In classical turbulence vorticity

is concentrated in vortical ‘worms’ (She & al, Nature, 1990 ; Goto, JFM, 2008)

• Are there vortex bundles in

quantum turbulence ?

• Would allow a mechanism for

vortex stretching, i.e. stretch the bundle.

Dω Dt = (ω · r)v + νr2ω

Mathematical approach

3 distinct scales/numerical approaches

Gross-Pitaevskii Point Vortex/VFM Course-Grained HVBK

Barenghi et al. (2014)

Vortex filament method

Biot-Savart Integral Model reconnections algorithmically ‘cut and paste’

Mutual friction

Normal viscous fluid coupled to inviscid superfluid via mutual friction. Superfluid component extracts energy from normal fluid component via Donelly- Glaberson instability, amplification of Kelvin waves. Kelvin wave grows with amplitude:

Counterflow Turbulence

vext

n (s, t) = (c, 0, 0)

Andronikashvili, 1946

Generation of bundles at finite temperatures

Vortex Locking - Morris, Koplik & Rouson, PRL, 2008 Gaussian normal fluid vortex – Samuels, PRB, 1993

Reconnections: Bundles remain intact

Alamri et al. 2008

Numerical simulations using both GPE and vortex filament method.

Decomposition of a tangle

0.2 0.4 0.6 0.8 1 AWB, Laurie & Barenghi, 2012

Motivation

Roussel, Schneider & Farge, 2005

Numerical results

10 10

1

10

2

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

PSD

f f−5/3

10

2

10

3

10

−5

10

−4

10

−3

k E(k) k−5/3 k−1

a b

Left, frequency spectra (red polarised ; black total), right energy spectrum, upper random component, lower polarised component.

Experimental detection

Presence of coherent structures inferred from intermittent pressure drops

Pumped He bath Pressurized HeI / He II (Ø 780 mm cell) Bottom propeller Heat exchanger 702 mm Parietal pressure probes ( Ø 1 mm tap holes 34 mm below equator) Transmission shaft Top propeller Mixing layer

20 40 60 80 100

Time [number of turns] P [arbitr. units and offset]

ρs/ρ = 0 % , Re = 6.6e7 [θ=0.12] ρs/ρ= 19 %, Re = 8.6e7 [θ=0.12] ρs/ρ= 83 %, Re = 8.9e7 [θ=0.11]

Rusaouen et al., 2017

• 10
• 5

10-4 10-3 10-2 10-1

P [standard deviation unit] Probability density

∂vn ∂t + (vn · r) vn = 1 ρrP + µr2vn + ρs ρ F, r · vn = 0, ∂vs ∂t + (vs · r) vs = 1 ρrP ρn ρ F, r · vs = 0.

F ' αρsh|ωs|i(vs vn),

ρs ρn : r2P ⇠ ρs 2 (ω2

s σ2 s)

• 10
• 5

10-4 10-3 10-2 10-1

P [standard deviation unit] Probability density

Hall-Vinen-Bekarevich-Khalatnikov Equations

Course-grained, macroscopic model

A single bundle in isolation

∂vs ∂t + (vs · r) vs = 1 ρrP ρn ρ F,

vs = (vr, vθ, vz) = ✓ 0, NΓ 2⇡r, 0 ◆

P = P0 ⇢sN 2Γ2 8⇡2r2 ,

min

V

P(N) ⇠ N 2.

• 0.5

• 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

ˆ F(|k|) = exp |k|2 2k2

f

!

AWB & Laurie arxiv:1910.00276

Turbulent Tangle

• 5
• 4
• 3
• 2
• 1

ˆ F(|k|) = exp |k|2 2k2

f

!

r kf = 2⇡/lf tering process

vn

• 5
• 4
• 3
• 2
• 1

ωs

P

Random ‘Vinen’ Tangle

Walmsley et al. 2013

• 5
• 4
• 3
• 2
• 1

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 10 -3 10 -2 10 -1 10 0

F ' αρsh|ωs|i(vs vn),

• 5
• 4
• 3
• 2
• 1

Summary

• Coherent vortical structures are present in the quasi-

classical regime of Quantum Turbulence.

• Important (essential?) for K41 like statistical

properties of QT.

• Good agreement between macroscopic HVBK model

and mesoscale vortex approach.

• Interesting high pressure signal found in the Vinen

regime.

The End