Aspects of hydrodynamic turbulence in which intermittency plays no major role 1.Decay of homogeneous turbulence 2.Energy spectral density 3.Vortex reconnection 4.Numerical simulations 5.Particle-vortex interactions Focus on doubly turbulent systems Parts to be discussed overlap with several excellent reviews, e.g., W.F. Vinen & J.J. Niemela, JLTP 128, 167 (2002) L. Skrbek, in Vortices and Turbulence at Very Low Temperatures, eds. C.F. Barenghi & R.A. Sergeev, pp (2008) W.F. Vinen, JLTP 161, 419 (2010)

“new” standard grid Niemela, KRS & Donnelly JLTP 138, 537 (2005) Stalp, Skrbek & Donnelly, PRL 82, 4831 (1999) Stalp, Niemela, Vinen & Donnelly Phys. Fluids 14, 1377 (2002) original grid ? Donnelly Stalp

 L, s -1

The energy decay in the hydrodynamic case 1.Energy decay (definition) d/dt =  (the correction due to non-stationarity is demonstrably small) 2. Dissipative anomaly, or the so-called “zeroth law” (Taylor 1915, 1935)  = C 3/2 /  O(1) The growth of  with time is unknown power law, and so (1) and (2) cannot be solved.  grows with time and, after some time, could be limited by D, the width of the apparatus. It is reasonable to assume then that  = constant = D. Consequences  u   (D 2 /C 2 ) (t-t 0 ) -2  = (D 2 /C 2 ) (t-t 0 ) -3 3.Using the exact relation    we obtain      D/C    (t-t 0 ) -3/2      (T)  L Simulations: Roche, Barenghi & Leveque EPL 87, (2009)

By writing  ’  L  2, we have ’/ =  (T); by writing  (T), we have ’  ), say. and L = [D/(C  1/2  3/2 )]  t-t 0   ,  =  1/2 L , v =  ) 1/2  L 1/2, v    0.1, insensitive to T. A B

Walmsley, Golov, Hall, Levchenko & Vinen

some parameters AB  /D u   /D1.3× ×10 -3 vortex Re1010  /L -1/ D/L -1/ For geometric mean scale during decay 5-10 (e.g., 10 vortices at B → D/8)

Classical turbulence behind pull-through grid U Nearly isotropic turbulence is generated. grid turbulence in air: Corke & Nagib square grid of bars tank of water

slope   1.2 slope   1.5 substantially different from -2 L  D/10 Van Doorn, White & KRS, Phys. Fluids 11, 2387 (1999)

Recommendations on Topic 1 Perform a detailed experiment of hydrodynamic turbulence behind a pull- through grid for conditions applicable to the quantum case. Develop alternative theoretical possibilities to explain observations.

Maurer & Tabeling, Europhys. Lett. 43, 29 (1998) a.2.3 K b.2.08 K c.1.4 K

Assumptions 1. Write p m (t)  p(t) +  Uu(t), while ignoring the fluctuation term u 2 ; this is not reasonable because u/U  in this flow. 2. Assume that p <<  Uu(t). Not true for any flow with shear. Pressure follows the Poisson equation  p = − 2  U i /  x j )(  u j /  x i ) −    x i  x j [  ( u i u j − )], The first terms on RHS represents the shear contributions. Pressure fluctuation = O(  u 2 ) for the shear-free case, but O(  U 2 ) for flows with shear, and the pressure spectrum has a slope of -5/3. 3. The situation in the superfluid case is probably different from what the authors assumed.

Shear turbulence in outer boundary layer p’/  u’ 2 non-dimensional frequency (or wave number) Tsuji et al. JFM 585, 1-40 (2007 ) pressure spectrum non-dimensional frequency (or wave number) outer scaling slope = -1.6 mean-square pressure

Roche et al. Europhys. Lett. 77, (2007)

Recommendation on Topic 2 Repeat the Maurer-Tabeling experiment in homogeneous grid flow in a helium wind tunnel.

What is being measured and how to interpret it? The signal (attenuation of the second sound) is averaged along the line of sight. Think of this process as integrating the original function f(x,t) over x (the coordinate in the direction of line of sight). The Fourier transform of the integral of a function is (i  ) -1 times the FT of the function. For spectrum one should multiply by complex conjugate and get  -2 times the spectrum of the function. There may be some technical problems with this thinking, but the basic idea must hold.

From Van Dyke’s Album

At the point of contact between the two tubes, adjacent vortex lines belonging to the two tubes cancel each other by viscous cross-diffusion connecting the remainder of the lines on either end of the region. F. Hussain & K. Duraisamy, Phys. Fluids 23, (2011)

Koplik & Levine, PRL 71, 1375 (1993) R.M. Kerr PRL (2011)

Superfluid case Details presumably depend on initial conditions.  b =  a   1/2 |t-t 0 | , where the most probable value of  = 0.5 [Paoletti, Fisher, KRS & Lathrop, PRL (2008)]; see Tebbe. Youd & Barenghi, JLTP 162, 314 (2011), who numerically studied the approach (but apparently not the repulsion). Would be nice to have a theory. Reconnection complete [Koplik & Levine, PRL 71, 1375 (1993)] Duration of interaction not known; what role in dissipation? No singularities (at least within GP, which is has smooth solutions) Hydrodynamic case Presumably generic.  b = a{Re(t 0 -t)} 3/4 ;  a = b{Re(t-t 0 )} 2 Reconnection is not complete Duration of interaction ~ Re  3/4 Claimed to be a source of aerodynamic noise. Singularities may exist for = 0 (not necessarily ruled out even for  ≠ 0)

Numerical search for 3D Euler singularities (compiled by J.D. Gibbon, 2007) 1. Morf, Orszag & Frisch (1980): Padé-approximation, complex time singularity of 3D Euler {see also Bardos et al (1976)}: Singularity: yes. 2. Chorin (1982): Vortex-method. Singularity: yes. 3. Brachet, Meiron, Nickel, Orszag & Frisch (1983): Taylor-Green calculation. Saw vortex sheets and the suppression of singularity. Singularity: no. 4. Siggia (1984): Vortex-filament method; became anti-parallel. Singularity: yes. 5. Ashurst & Meiron (1987): Singularity: yes. 6. Kerr & Pumir (1987): Singularity: no. 7. Pumir & Siggia (1990): Adaptive grid. Singularity: no. 8. Brachet, Meneguzzi, Vincent, Politano & P-L Sulem (1992): pseudospectral code, Taylor-Green vortex. Singularity: no. 9. Kerr (1993, 2005): Chebyshev polynomials with anti-parallel initial conditions; resolution 5122 × 256. Observed |ω| ~ (T − t) −1. Singularity: yes. 10. Grauer & Sideris (1991): 3D axisymmetric swirling flow. Singularity: yes. 11. Boratav & Pelz (1994, 1995): Kida’s high symmetry. Singularity: yes. 12. Pelz & Gulak (1997): Kida’s high symmetry. Singularity: yes. 13. Grauer, Marliani & Germaschewski (1998): Singularity: yes. 14. Pelz (2001, 2003): Singularity: yes. 15. Kida has edited a memorial issue for Pelz in Fluid Dyn. Res. 36 (2005): –Cichowlas & Brachet: Singularity: no. –Pelz & Ohkitani: Singularity: no.Yes: 11 –Gulak & Pelz: Singularity: yes. No: Hou & Li (2006): Singularity: no.

How confident are you, on a 0-10 scale, that solutions to the Euler and Navier- Stokes equations can develop finite-time singularities? (Survey at the 2007 meeting on: Euler equations: 250 Years On) Scale # of votes (Euler: 43) # of votes (N-S: 43)

Kaneda and Ishihara (2006) Intense-vorticity isosurfaces showing the region where ω > ω + 4σω. Rλ = 732. (a) The size of the display domain is (59842 × 1496) η3, periodic in the vertical and horizontal directions. (b) Close-up view of the central region of (a) bounded by the white rectangular line; the size of display domain is (29922 × 1496) η3. (c) Close-up view of the central region of (b); 14963η3 (d) Close-up view of the central region of (c); (7482 × 1496) η3.

Pr(v) dv = Pr(t) dt v =   (t  t 0 )  0.5 Pr(v) ~ |v|  3 No instances (away from solid boundary) where power-law tails exist for velocity distributions in classical turbulence. But what if we looked only at intense vortices in classical turbulence? Nearly homogeneous turbulence following a counterflow Paoletti, Fisher, KRS & Lathrop, Phys. Rev. Lett. (2008)

large simulations of D.A. Donzis and P.K. Yeung velocity PDFs conditioned on intense vorticity Even by conditioning velocity PDFs on intense vorticity in classical turbulence, one finds no sign of anything other than a Gaussian. velocity normalized by conditional rms normalized PDF

Recommendations on Topic 3 What role does vortex reconnection play in hydrodynamic turbulence? If it does occur, it seems to leave no telltale signs unlike in the quantum case. Produce in the superfluid case some new and better pictures than those of Koplik & Levine nearly 20 years ago. Get data on the symmetry or otherwise of the reconnection events. Study the importance of initial conditions on dynamical and topological features of reconnection of quantized vortices Understand the extent to which a reconnection event in quantum turbulence is dissipative.

kinematic nonlinearity Almost certainly hold for hydrodynamic turbulence; seem to hold also for some extreme conditions such as near the critical point [see Halperin & Hohenberg RMP 49, 435 (1977)]. Hydrodynamic turbulence Quantum turbulence Local induction approximation (let’s move past it!) Bio-Savart; dynamically inconsistent procedures (super fluid not affecting the normal fluid), etc GPE (used for T = 0); how best to account for finite T? RANS-type averaged Euler-like equation arising from GPE What are we doing exactly in solving GPE? computers

R

Simulations (some, not all): C. Nore, M. Abid & M.E. Brachet, Phys. Rev. Lett. 78, 3896 (1997): GP in TG configuration, T = 0 T. Araki, M. Tsubota & S.K. Nemirovskii, Phys. Rev. Lett. 89, (2002): Bio-Savart in TG, T = 0 M. Kobayashi & M. Tsubota, Phys. Rev. Lett. 94, (2005): slightly modified GP in periodic box. J. Yepez et al. PRL 103, (2009): GP in periodic box For a comment, on Yepez et al., see: Krstulovic & Brachet, PRL 105, (2010)

Recommendation on Simulations Make a more convincing and large simulation of quantum turbulence using GPE, especially better resolving scales smaller than the inter- vortex spacing.

Requirements on particle properties Must be small enough to follow the flow with fidelity (i.e., must respond to the smallest scales of the flow with fast response); in particular, must have the same density as the fluid Must be large enough to be imaged with ‘usable’ illumination and detection equipment Must not cluster In liquid helium Because of small apparatus and large Reynolds numbers, small scales are smaller than in water; in particular, the density of helium I is 1/8 that of water; quantized vortices on the size of Angstroms Very small particles cannot be imaged Mutual attraction of particles and clustering cannot be suppressed by using surfactants as in water.

Particles are not always passive tracers!

sphere is trapped by vortex simulations of C. Barenghi and colleagues sphere escapes vortex

helium/hydrogen, by volume clumps dia, nm A suggestive picture of the particle size Many hydrogen atoms even in the smallest particle! We still don’t have the ideal particle.

Thank you for your attention