Real-space Manifestations of Bottlenecks in Turbulence Spectra Rahul Pandit Centre for Condensed Matter Theory, Department of Physics Indian Institute of Science, Bangalore, India. 28 May 2012 Wolfgang Pauli Institute, Vienna, Austria.
Work done with: ◮ Uriel Frisch, Laboratoire Lagrange, OCA, UNS, CNRS, BP 4229, 06304 Nice Cedex 4, France; ◮ Samriddhi Sankar Ray, Laboratoire Lagrange, OCA, UNS, CNRS, BP 4229, 06304 Nice Cedex 4, France; ◮ Ganapati Sahoo, Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 G¨ ottingen, Germany; ◮ Debarghya Banerjee, Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, India. ◮ Support: OTARIE (France), CSIR, DST, UGC (India) and SERC (IISc)
Outline 1. Energy-spectra bottlenecks: Direct numerical simulations (DNS), experiments, and earlier studies. 2. Hyperviscous hydrodynamical equations: ◮ 1D deterministic hyperviscous Burgers (DHB) equation; ◮ 1D stochastic hyperviscous Burgers (SHB) equation; ◮ 3D deterministic hyperviscous Navier-Stokes (3DHNS) equation. 3. Real-space manifestations of bottlenecks: ◮ DHB: boundary-layer theory and (DNS); ◮ SHB: DNS; ◮ 3DHNS: DNS. 4. Conclusions
Bottlenecks : DNS !"#$ %$ &'()*+,-'. '/ 012 *32+*425 '.265,(2.-,'.*7 7'.4,085,6 Bottleneck effect in three-dimensional turbulence simulations , W. Dobler, N. Erland, L. Haugen,T. A. Yousef, and A. Brandenburg, Phys. Rev. E , 68 , 026304 (2003).
Bottlenecks : DNS 3.5 a k 5/3 E ( k )/ ε 2/3 ∏ ( k )/ ε 3.0 R λ = 167 R λ = 167 257 257 471 471 732 732 2.5 1131 1131 2.0 1.5 1.0 0.5 0 0.001 0.01 0.1 1 k η Study of HighReynolds Number Isotropic Turbulence by Direct Numerical Simulation , Takashi Ishihara, Toshiyuki Gotoh, and Yukio Kaneda, Annu. Rev. Fluid Mech. , 41 , 165 (2009).
Bottlenecks : DNS ( a ) 2.5 2.0 1000 10 0 Ψ ( k η ) 90 1.5 400 650 140 38 240 11/3 1.0 10 –3 10 –2 10 –1 Ψ ( k η ) 2.2 ( b ) 0 2.0 10 –1 − 1 1.8 d[log E ( k )] d[log k ] R λ 1.6 − 2 10 –2 10 –1 − 3 10 –3 10 –2 10 –1 10 0 10 –3 10 –2 10 –1 k η k η The bottleneck effect and the Kolmogorov constant in isotropic turbulence , D. A. Donzis and K. R. Sreenivasan, J. Fluid Mech. , 657 , 171 (2010).
Bottlenecks : Experiments On the universal form of energy spectra in fully developed turbulence , Z-S. She and E. Jackson, Phys. Fluids A , 5 , 1526 (1993).
Bottlenecks : Experiments Local isotropy in turbulent boundary layers at high Reynolds number , S. G. Saddoughi and S. V. Veeravalli, J. Fluid Mech. , 268 , 333 (1994).
Hyperviscous Hydrodynamical Models 1. Deterministic hyperviscous Burgers equation (DHB). 2. Stochastic hyperviscous Burgers equation (SHB). 3. 3D hyperviscous Navier–Stokes equation (3DHNS).
DHB equation ∂ t u + u ∂ x u = − ν α k − 2 α (− ∂ 2 x ) α u + f ( x , t ) r ◮ u - velocity field ◮ ν α - coefficient of hyperviscosity ◮ k r - a reference wavevector ◮ α - dissipativity ◮ We use f ( x , t ) = sin ( x )
SHB equation ∂ t u + u ∂ x u = − ν α k − 2 α (− ∂ 2 x ) α u + f ( x , t ); r f ( x , t ) is a zero-mean, space-periodic Gaussian random force with � ^ f ( k 1 , t 1 )^ f ( k 2 , t 2 ) � = 2 D 0 | k | − 1 δ ( t 1 − t 2 ) δ ( k 1 + k 2 ) .
3DHNS equation ∂ u − ∇ 2 � α u ( x , t ) + f ( x , t ); � ∂ t + u · ∇ u ( x , t ) = − ∇ p − ν α ∇ · u 0 . = p - pressure ; ρ = 1. The force f ( x , t ) is specified in terms of its spatial Fourier transform, as f ( k , t ) = P Θ ( k f − k ) ^ 2 E ( k f , t ) ^ u ( k , t ) , where Θ ( k f − k ) is 1 if k ≤ k f and 0 otherwise, P is the power input, and E ( k f , t ) = � k ≤ k f E ( k , t ) ; we choose k f = 1.
Principal Results ◮ We develop a quantitative, analytical understanding of bottlenecks in the hyperviscous Burgers equation. ◮ For this it is crucial to examine the solution in real space, where we can use boundary-layer-type analysis, in the vicinities of shocks, to uncover oscillations in the velocity profile. ◮ We validate our DHB solutions with a pseudospectral DNS.
Principal Results ◮ The key feature of real-space oscillations carries over to oscillations in velocity correlation functions in hyperviscous hydrodynamical equations that display genuine turbulence. ◮ We show this in the second part of our study by using DNS. ◮ This association of bottlenecks and oscillations in velocity correlation functions, similar the association of peaks in the static structure factor S ( k ) , of a liquid in equilibrium, with damped oscillations in the pair correlation function, has not been made so far.
S ( k ) and g ( r ) : liquid-state of Ar at 85 K Structure Factor and Radial Distribution Function for Liquid Argon at 85K , J. L. Yarnell et al., Phys. Rev. A, 7 , 2130 (1973).
Earlier Theoretical Models ◮ Inhibited cascade model (G. Falkovich, 1994). ◮ Local scaling exponents (D. Lohse and A. M¨ uller-Groeling, 1995). ◮ Aborted Thermalisation (Frisch et al. , 2008).
Inhibited cascade model (Falkovich) ◮ Evolution equation of pair correlation functions. ◮ Lowest non-linearity expressed via a triple correlation function. ◮ Pair-correlator expressed as a power function. ◮ The non-linear interaction in the inertial range and in the dissipation range gives a term, which, if greater than the viscous dissipation, yields a bottleneck. Bottleneck phenomenon in developed turbulence , G. Falkovich, Phys. Fluids , 6 , 1411 (1994).
Local scaling exponents (Lohse and M¨ uller-Groeling) ◮ Analysis of experimental data to obtain the energy spectrum and the second-order structure function. ◮ The local scaling exponent of the second-order structure function decreases monotonically. ◮ The local scaling exponent of the spectrum has a minimum and a maximum. ◮ Energy pile-up using Falkovich’s argument. Bottleneck effects in turbulence: Scaling phenomenon in r versus p space , D. Lohse and A. M¨ uller-Groeling, Phys. Rev. Lett. , 74 , 1747 (1995).
Hyperviscous Burgers and Navier-Stokes Equations ∂ t v + v ∂ x v = − µ K − 2 α (− ∂ 2 x ) α v G ∂ t v + v · ∇ v = − ∇ p − µ K − 2 α (− ∇ 2 ) α v , ∇ · v = 0 G ◮ µ > 0, K G > 0, and α is the dissipativity. ◮ Dissipation rate µ ( k / K G ) 2 α tends to zero for all k < K G and to infinity for k > K G , when α → ∞ . ◮ For fixed µ and K G , the solution of the hyperviscous equations tend to the Galerkin-truncated equations as α tends to ∞ . ◮ True for Navier-Stokes, Burgers, MHD, DIA and EDQNM. ◮ False for MRCM and resonant wave interaction theory. ◮ Galerkin-truncation leads to thermalization ( Lee, 1952; Hopf, 1952; Kraichnan, 1958 ).
EDQNM: Hyperviscosity # !" * ! 1! ".!0' ! 1# ! 1) ! 10 ! !" ".!/0 ! 1#2 ! 1/! ! 12#0 & '() *+,&- & #3'() " !" ".!/# # ) $ ' !" !" !" !" ! ! !" ! 1! & 4 1!" ' ! # !" * " # $ % !" !" !" !" & Hyperviscosity, Galerkin truncation and bottlenecks in turbulence , U. Frisch, S. Kurien, R. Pandit, W. Pauls, S. S. Ray, A. Wirth, and J-Z Zhu, Phys. Rev. Lett. , 101 , 144501 (2008).
Hyperviscous Burgers Equation : Large α 0 t = 1 −1 t = 2 t = 4 t = 10 −2 t = 20 t = 30 −3 k −2 scaling log(E(k)) −4 −5 α = 1000 −6 −7 0 0.5 1 1.5 2 2.5 3 log(k) Hyperviscosity, Galerkin truncation and bottlenecks in turbulence , U. Frisch, S. Kurien, R. Pandit, W. Pauls, S. S. Ray, A. Wirth, and J-Z Zhu, Phys. Rev. Lett. , 101 , 144501 (2008).
Thermalization and Bottlenecks ◮ Large α produces a big bottleneck. ◮ Thermalization is accompanied by Gaussianization . ◮ Spurious effects expected: depletion of intermittency.
Methods ◮ DHB : Boundary–layer theory and DNS ◮ SHB : DNS ◮ 3DHNS : DNS
DHB : Boundary–layer Theory ◮ The velocity eventually goes to a steady state, which is a solution of the ordinary differential equation (ODE) that is obtained by dropping the time-derivative term. ◮ When α � = 1, this nonlinear ODE is not integrable, but its limit as ν α → 0 is the same as for ordinary dissipation, namely, it has a shock at x = π , where the solution jumps from u − = + 2 to u + = − 2.
DHB: Boundary–layer Theory ◮ For small but finite ν α , the shock is broadened and its structure can be analyzed by a boundary-layer technique using the stretched spatial variable X ≡ ( x − π ) /ν β , with β = 1 2 α − 1 , and expanding the boundary-layer velocity in powers of ν α . ◮ To leading order � u 2 = (− 1 ) α + 1 d 2 α � d 0 dX 2 α u 0 , u 0 ( ± ∞ ) = ∓ 2 . dX 2
DHB : Boundary–layer Theory ◮ For large X the equation can be linearized because u 0 is close to its asymptotic constant value. ◮ For example, for large negative X , we set u 0 = 2 + w , discard the quadratic term in w , and obtain, after integrating once, (− 1 ) α + 1 d 2 α − 1 w / dX 2 α − 1 = 2 w .
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