Spatiotemporal correlation functions of fully developed turbulence - PowerPoint PPT Presentation

Spatiotemporal correlation functions of fully developed turbulence L´ eonie Canet ERG Trieste 19/09/2016

In collaboration with ... Guillaume Nicol´ as Vincent Bertrand Wschebor Rossetto Balarac Delamotte LEGI Univ. Rep´ ublica LPMMC LPTMC Montevideo Univ. Grenoble Alpes Grenoble INP Univ. Paris 6 LC, B. Delamotte, N. Wschebor, Phys. Rev. E 91 (2015) LC, B. Delamotte, N. Wschebor, Phys. Rev. E 93 (2016) LC, V. Rossetto, N. Wschebor, G. Balarac, arXiv :1607.03098 (2016)

Presentation outline 1 NPRG approach to Navier-Stokes equation Fully developed turbulence Navier-Stokes equation NPRG formalism for NS Leading Order approximation 2 Exact correlation function in the limit of large wave-numbers Exact flow equations in the limit of large wave-numbers Solution in the inertial range Solution in the dissipative range 3 Perspectives

Navier-Stokes turbulence stationary regime of fully developed isotropic and homogeneous turbulence integral scale (energy injection) : ℓ 0 Kolmogorov scale (energy dissipation) : η ℓ 0 injection ǫ energy cascade flux ǫ ℓ 0 η ∼ R 3 / 4 dissipation ǫ η constant energy flux in the inertial range η < r < ℓ 0 Frisch, Turbulence, the legacy of AN Kolmogorov Cambridge Univ. Press (1995)

Scale invariance in the inertial range velocity structure functions velocity increments x + � x )] · � δ v ℓ � = [ � u ( � ℓ ) − � u ( � ℓ structure function S p ( ℓ ) ≡ � ( δ v ℓ � ) p � ∼ ℓ ξ p ξ 2 = 2 / 3 energy spectrum inertial range E ( k ) = 4 π k 2 TF ( � � v ( � x ) · � v (0) � ) ∼ k − 5 / 3 ONERA wind tunnel Anselmet, Gagne, Hopfinger, Antonia, J. Fluid Mech. 140 (1984).

Scale invariance in the inertial range velocity structure functions velocity increments x + � x )] · � δ v ℓ � = [ � u ( � ℓ ) − � u ( � ℓ structure function S p ( ℓ ) ≡ � ( δ v ℓ � ) p � ∼ ℓ ξ p ξ 2 = 2 / 3 energy spectrum dissipative range E ( k ) = 4 π k 2 TF ( � � v ( � x ) · � v (0) � ) ∼ k − 5 / 3 ONERA wind tunnel Anselmet, Gagne, Hopfinger, Antonia, J. Fluid Mech. 140 (1984).

Scale invariance in the inertial range velocity structure functions velocity increments x + � x )] · � δ v ℓ � = [ � u ( � ℓ ) − � u ( � ℓ structure function S p ( ℓ ) ≡ � ( δ v ℓ � ) p � ∼ ℓ ξ p ξ 2 = 2 / 3 energy spectrum E ( k ) = 4 π k 2 TF ( � � v ( � x ) · � v (0) � ) ∼ k − 5 / 3 ONERA wind tunnel Anselmet, Gagne, Hopfinger, Antonia, J. Fluid Mech. 140 (1984).

Kolmogorov K41 theory for isotropic 3D turbulence Kolmogorov original work A.N. Kolmogorov, Dokl. Akad. Nauk. SSSR 30, 31, 32 (1941) Assumptions symmetries restored in a statistical sense : homogeneity, isotropy finite dissipation rate per unit mass ǫ in the limit ν → 0 = ⇒ derivation of energy flux constancy relation exact result “four-fifth law” S 3 ( ℓ ) = − 4 5 ǫ ℓ Assuming universality in the inertial range r , λ� r , � ℓ ) = λ h δ� self-similarity δ� v � ( � v � ( � ℓ ) dimensional analysis = ⇒ scaling predictions S p ( ℓ ) = C p ǫ p / 3 ℓ p / 3 E ( k ) = C K ǫ 2 / 3 k − 5 / 3

Intermittency, multi-scaling deviations from K41 illustration : in experiments and numerical von K´ arman swirling flow simulations S p ( ℓ ) ≡ � ( δ v ℓ � ) p � ∼ ℓ ξ p ξ p � = p / 3 violation of simple scale- invariance = ⇒ multi-scaling • exp. , * num. � non-Gaussian statistics of - - - K41 velocity differences Mordant, L´ evˆ eque, Pinton, = ⇒ intermittency New J. Phys. 6 (2004)

Intermittency, multi-scaling theoretical challenge : understand K41 and intermittency from first principles (microscropic description) Various perturbative RG approaches p ) ∝ p 4 − d − 2 ǫ formal expansion parameter through the forcing profile N αβ ( � early works de Dominicis, Martin, PRA 19 (1979) , Fournier, Frisch, PRA 28 (1983) Yakhot, Orszag, PRL 57 (1986) reviews Zhou, Phys. Rep. 488 (2010) Adzhemyan et al. , The Field Theoretic RG in Fully Developed Turbulence , Gordon Breach, 1999 Non-Perturbative (functional) RG approaches Tomassini, Phys. Lett. B 411 (1997), Mej´ ıa-Monasterio, Muratore-Ginnaneschi, PRE 86 (2012)

Intermittency, multi-scaling theoretical challenge : understand K41 and intermittency from first principles (microscropic description) Various perturbative RG approaches p ) ∝ p 4 − d − 2 ǫ formal expansion parameter through the forcing profile N αβ ( � early works de Dominicis, Martin, PRA 19 (1979) , Fournier, Frisch, PRA 28 (1983) Yakhot, Orszag, PRL 57 (1986) reviews Zhou, Phys. Rep. 488 (2010) Adzhemyan et al. , The Field Theoretic RG in Fully Developed Turbulence , Gordon Breach, 1999 Non-Perturbative (functional) RG approaches Tomassini, Phys. Lett. B 411 (1997), Mej´ ıa-Monasterio, Muratore-Ginnaneschi, PRE 86 (2012)

Intermittency, multi-scaling theoretical challenge : understand K41 and intermittency from first principles (microscropic description) Various perturbative RG approaches p ) ∝ p 4 − d − 2 ǫ formal expansion parameter through the forcing profile N αβ ( � early works de Dominicis, Martin, PRA 19 (1979) , Fournier, Frisch, PRA 28 (1983) Yakhot, Orszag, PRL 57 (1986) reviews Zhou, Phys. Rep. 488 (2010) Adzhemyan et al. , The Field Theoretic RG in Fully Developed Turbulence , Gordon Breach, 1999 NPRG without truncations : exact closure based on symmetries ! LC, Delamotte, Wschebor, PRE 93 (2016), LC, Rossetto, Wschebor, Balarac, arXiv :1607.03098

Microscopic theory Navier Stokes equation with forcing for incompressible fluids ∂� v = − 1 v v · � ∇ p + ν � � ∇ 2 � v + � ∂ t + � ∇ � f ρ � ∇ · � v ( t , � x ) = 0 v ( � � x , t ) velocity field and p ( � x , t ) pressure field ρ density and ν kinematic viscosity � f ( � x , t ) gaussian stochastic stirring force with variance x ) f β ( t ′ , � x ′ ) = 2 δ αβ δ ( t − t ′ ) N ℓ 0 ( | � x ′ | ) . � � f α ( t , � x − � with N ℓ 0 peaked at the integral scale (energy injection)

Non-Perturbative Renormalisation Group for NS MSR Janssen de Dominicis formalism : NS field theory Martin, Siggia, Rose, PRA 8 (1973), Janssen, Z. Phys. B 23 (1976), de Dominicis, J. Phys. Paris 37 (1976) � � ∂ t v α + v β ∂ β v α + 1 � � � ρ∂ α p − ν ∇ 2 v α S 0 = v α ¯ + ¯ p ∂ α v α t ,� x � � � x ′ | ) − x ′ ¯ v α N ℓ 0 ( | � x − � ¯ v α t ,� x ,� Non-Perturbative Renormalization Group approach Wetterich’s equation for scale-dependent effective actions Γ κ ∂ κ Γ κ = 1 � � − 1 = 1 � � Γ (2) ∂ κ R κ κ + R κ ∂ κ R κ · G κ 2 Tr 2 Tr q � � q C. Wetterich, Phys. Lett. B 301 (1993)

Non-Perturbative Renormalisation Group for NS Aim : compute correlation function and response function � � � � v α ( t , � x ) v β (0 , 0) and v α ( t , � x ) f β (0 , 0) Wetterich’s equation for the 2-point functions � � − 1 ∂ κ Γ (2) 2 Γ (4) κ, ij ( p ) = ∂ κ R κ ( q ) · G κ ( q ) · κ, ij ( p , − p , q ) Tr q � +Γ (3) κ, i ( p , q ) · G κ ( p + q ) · Γ (3) κ, j ( − p , p + q ) · G κ ( q ) infinite hierarchy of flow equations approximation scheme : truncation of higher-order vertices Tomassini, Phys. Lett. B 411 (1997), Mej´ ıa-Monasterio, Muratore-Ginnaneschi, PRE 86 (2012) LC, Delamotte, Wschebor, PRE 93 (2016) exact closure exploiting (time-gauged) symmetries of NS action LC, Delamotte, Wschebor, PRE 93 (2016)

Non-Perturbative Renormalisation Group for NS Aim : compute correlation function and response function � � � � v α ( t , � x ) v β (0 , 0) and v α ( t , � x ) f β (0 , 0) Wetterich’s equation for the 2-point functions � � − 1 ∂ κ Γ (2) 2 Γ (4) κ, ij ( p ) = ∂ κ R κ ( q ) · G κ ( q ) · κ, ij ( p , − p , q ) Tr q � +Γ (3) κ, i ( p , q ) · G κ ( p + q ) · Γ (3) κ, j ( − p , p + q ) · G κ ( q ) infinite hierarchy of flow equations approximation scheme : truncation of higher-order vertices Tomassini, Phys. Lett. B 411 (1997), Mej´ ıa-Monasterio, Muratore-Ginnaneschi, PRE 86 (2012) LC, Delamotte, Wschebor, PRE 93 (2016) exact closure exploiting (time-gauged) symmetries of NS action LC, Delamotte, Wschebor, PRE 93 (2016)

Non-Perturbative Renormalisation Group for NS Aim : compute correlation function and response function � � � � v α ( t , � x ) v β (0 , 0) and v α ( t , � x ) f β (0 , 0) Wetterich’s equation for the 2-point functions � � − 1 ∂ κ Γ (2) 2 Γ (4) κ, ij ( p ) = ∂ κ R κ ( q ) · G κ ( q ) · κ, ij ( p , − p , q ) Tr q � +Γ (3) κ, i ( p , q ) · G κ ( p + q ) · Γ (3) κ, j ( − p , p + q ) · G κ ( q ) infinite hierarchy of flow equations approximation scheme : truncation of higher-order vertices Tomassini, Phys. Lett. B 411 (1997), Mej´ ıa-Monasterio, Muratore-Ginnaneschi, PRE 86 (2012) LC, Delamotte, Wschebor, PRE 93 (2016) exact closure exploiting (time-gauged) symmetries of NS action LC, Delamotte, Wschebor, PRE 93 (2016)