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

Classical (viscous) turbulence • In a 3D classical turbulent ∂ t + ( v · r ) v = � 1 ∂ v flow, large scale eddies ρ r p + ν r 2 v break up into smaller eddies, these into smaller ones and so on...(Richardson Cascade) v = sin( x ) ⇒ v ∂ v ∂ x ∼ sin(2 x ) • 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 / 3 k − 5 / 3

Classical Vorticity ω = r ⇥ u

Quantum Fluids

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

Kuchemann : “ vortices are the sinews and muscles of fluid motions ”

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

<latexit sha1_base64="jyBL1KM+gbkPaO/UeKaqXwkHDk=">ACGXicbVDLSgMxFM34tr6qLt0Ei+BC6ozvTaHoRnCjYG2hU0omvbXBTGZM7ohl6G+48VfcuFDEpa78G9NpF1o9EHI4596bmxPEUh03S9nbHxicmp6ZjY3N7+wuJRfXrkyUaI5VHgkI10LmAEpFRQoIRarIGFgYRqcHPS96t3oI2I1CV2Y2iE7FqJtuAMrdTMuz7CPWZz0kAm0EsvS15x/+Bsi/q3CWtRX3eiptnOLlU6DXzBbfoZqB/iTckBTLEeTP/4bcinoSgkEtmTN1zY2ykTKPgEno5PzEQM37DrqFuqWIhmEabdSjG1Zp0Xak7VFIM/VnR8pCY7phYCtDh0z6vXF/7x6gu2jRipUnCAoPnionUiKEe3HRFtCA0fZtYRxLeyulHeYZhxtmDkbgjf65b/kaqfo7RZ3LvYK5eNhHDNkjayTeKRQ1Imp+ScVAgnD+SJvJBX59F5dt6c90HpmDPsWSW/4Hx+A6WloAs=</latexit> ������� ���������� ����� ���������������� <latexit sha1_base64="ERTrTxM/p8YBZYG6rgJVkexURhQ=">ACF3icbVDLSgMxFM34tr6qLt0Ei+BCxpkq6KYguhHcVGhV6JSydzaYCYzJnfEMvQv3Pgrblwo4lZ3/o3p2IWvAyGHc+69uTlhKoVBz/twxsYnJqemZ2ZLc/MLi0vl5ZUzk2SaQ5MnMtEXITMghYImCpRwkWpgcSjhPLw6GvrnN6CNSFQD+ym0Y3apRFdwhlbqlN0A4RaLObmGaJA3alW3erJFg+uMRTQvaRjtotL1bxBp1zxXK8A/Uv8EamQEeqd8nsQJTyLQSGXzJiW76XYzplGwSUMSkFmIGX8il1Cy1LFYjDtvNhnQDesEtFuou1RSAv1e0fOYmP6cWgrY4Y989sbiv95rQy7+1cqDRDUPzroW4mKSZ0GBKNhAaOsm8J41rYXSnvMc042ihLNgT/95f/krOq6+41dPdysHhKI4ZskbWySbxyR45IMekTpqEkzvyQJ7Is3PvPDovzutX6Zgz6lklP+C8fQI2LZ9M</latexit> Yet we still see ‘classical’ behaviour φ ( k ) Vortex shedding 10 − 4 -5/3 10 − 5 p ( v ) 10 ≠ 3 10 − 6 10 ≠ 4 v ≠È v Í 10 ≠ 5 σ kL 0 / (2 π ) − 4 − 2 0 2 4 10 − 7 10 − 1 10 0 10 1 Probe cut-o ff T = 2 . 2 K, ρ s / ρ n = 0 10 0 T = 1 . 56 K, ρ s / ρ n = 6 / ( ‘ r ) È δ v 3 Í 0 3 / 2 È δ v 2 Í 10 − 1 ” v 3 , ≠ 0 . 1 + − 5 4 10 − 2 ≠ 0 . 2 r/L 0 10 − 1 10 0 r/L 0 Salort et al., 2011 10 − 1 5 · 10 − 1

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. D ω Dt = ( ω · r ) v + ν r 2 ω

Mathematical approach 3 distinct scales/numerical approaches Barenghi et al. (2014) Gross-Pitaevskii Point Vortex/VFM Course-Grained HVBK

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

Andronikashvili, 1946 Mutual friction Counterflow Turbulence 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: v ext n ( s , t ) = ( c, 0 , 0)

Generation of bundles at finite temperatures Vortex Locking - Morris, Koplik & Rouson, PRL, 2008 Gaussian normal fluid vortex – Samuels, PRB, 1993

Reconnections: Bundles remain intact Numerical simulations using both GPE and vortex filament method. Alamri et al. 2008

Decomposition of a tangle 1 0.8 0.6 0.4 0.2 0 AWB, Laurie & Barenghi, 2012

Motivation Roussel, Schneider & Farge, 2005

Numerical results 1 0.8 0.6 0.4 0.2 0 − 1 10 f − 5 / 3 a − 2 − 3 10 10 k − 1 − 3 10 E ( k ) b PSD − 4 10 − 4 10 − 5 10 k − 5 / 3 − 5 10 − 6 10 − 7 10 0 1 2 2 3 10 10 10 10 10 f k Left, frequency spectra (red polarised ; black total), right energy spectrum, upper random component, lower polarised component.

Experimental detection Rusaouen et al., 2017 Transmission shaft Pumped He bath P [arbitr. units and offset] Heat exchanger Pressurized HeI / He II (Ø 780 mm cell) Top propeller 702 mm Mixing layer Parietal pressure probes ( Ø 1 mm tap holes ρ s / ρ = 0 % , Re = 6.6e7 [ θ =0.12] 34 mm below equator) ρ s / ρ = 19 %, Re = 8.6e7 [ θ =0.12] ρ s / ρ = 83 %, Re = 8.9e7 [ θ =0.11] Bottom propeller 0 20 40 60 80 100 Time [number of turns] ρ s/ ρ = 0 %, Re=5.5e7 [ θ =0.20] ρ s/ ρ = 0 %, Re=5.5e7 [ θ =0.12] ρ s/ ρ = 0 %, Re=6.6e7 [ θ =0.12] Presence of 10 -1 ρ s/ ρ = 19 % Re=5.9e7 [ θ =0.20] ρ s/ ρ = 19 % Re=8.6e7 [ θ =0.12] ρ s/ ρ = 19 % Re=1.1e8 [ θ =0.12] coherent structures Probability density ρ s/ ρ = 79 % Re=1.3e8 [ θ =0.20] ρ s/ ρ = 79 % Re=1.3e8 [ θ =0.11] 10 -2 ρ s/ ρ = 83 % Re=8.9e7 [ θ =0.11] gaussian (standard deviation=1) inferred from 10 -3 intermittent pressure drops 10 -4 -10 -5 0 P [standard deviation unit]

<latexit sha1_base64="KlG78PnM1cDk81IXtyIu4X3+Bxg=">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</latexit> Hall-Vinen-Bekarevich-Khalatnikov Equations Course-grained, macroscopic model ∂ t + ( v n · r ) v n = � 1 ∂ v n ρ r P + µ r 2 v n + ρ s ρ F , r · v n = 0 , ∂ v s ∂ t + ( v s · r ) v s = � 1 ρ r P � ρ n ρ F , r · v s = 0 . F ' αρ s h | ω s | i ( v s � v n ) , r 2 P ⇠ ρ s 2 ( ω 2 s � σ 2 ρ s � ρ n : s ) ρ s/ ρ = 0 %, Re=5.5e7 [ θ =0.20] ρ s/ ρ = 0 %, Re=5.5e7 [ θ =0.12] ρ s/ ρ = 0 %, Re=6.6e7 [ θ =0.12] 10 -1 ρ s/ ρ = 19 % Re=5.9e7 [ θ =0.20] ρ s/ ρ = 19 % Re=8.6e7 [ θ =0.12] ρ s/ ρ = 19 % Re=1.1e8 [ θ =0.12] Probability density ρ s/ ρ = 79 % Re=1.3e8 [ θ =0.20] ρ s/ ρ = 79 % Re=1.3e8 [ θ =0.11] 10 -2 ρ s/ ρ = 83 % Re=8.9e7 [ θ =0.11] gaussian (standard deviation=1) 10 -3 10 -4 -10 -5 0 P [standard deviation unit]

A single bundle in isolation ∂ v s ∂ t + ( v s · r ) v s = � 1 ρ r P � ρ n ρ F , ✓ ◆ 0 , N Γ v s = ( v r , v θ , v z ) = 2 ⇡ r, 0 ! � | k | 2 ˆ F ( | k | ) = exp 2 k 2 P = P 0 � ⇢ s N 2 Γ 2 f 8 ⇡ 2 r 2 , P ( N ) ⇠ � N 2 . min V 10 7 1 0 10 8 -1 -2 10 7 -3 -4 -5 10 6 -6 -7 -8 10 5 -9 10 1 10 2 -0.5 0 0.5 AWB & Laurie arxiv:1910.00276

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Random ‘Vinen’ Tangle Quasiclassical Ultraquantum Walmsley et al. 2013 10 0 F ' αρ s h | ω s | i ( v s � v n ) , 6 6 5 10 -1 5 4 3 2 4 1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 3 10 -2 2 1 0 10 -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5