Renormalization group in stochastic theory of developed turbulence 4

Similar documents
Renormalization group in stochastic theory of developed turbulence 3

Effects of forcing in three-dimensional turbulent flows

arxiv: v1 [cond-mat.stat-mech] 19 Nov 2018

K e sub x e sub n s i sub o o f K.. w ich i sub s.. u ra to the power of m i sub fi ed.. a sub t to the power of a

Dimensionality influence on energy, enstrophy and passive scalar transport.

Quantum Fields in Curved Spacetime

arxiv: v1 [cond-mat.stat-mech] 16 Dec 2017

Diffusive Transport Enhanced by Thermal Velocity Fluctuations

Characteristics of Linearly-Forced Scalar Mixing in Homogeneous, Isotropic Turbulence

Calculation of the dynamical critical exponent in the model A of critical dynamics to order ε 4

Simulations for Enhancing Aerodynamic Designs

A new approach to the study of the 3D-Navier-Stokes System

Energy Spectrum of Quasi-Geostrophic Turbulence Peter Constantin

Tutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace

1 Running and matching of the QED coupling constant

Non-equilibrium statistical mechanics of turbulence

Lifshitz Hydrodynamics

Asymptotically safe Quantum Gravity. Nonperturbative renormalizability and fractal space-times

Basic hydrodynamics. David Gurarie. 1 Newtonian fluids: Euler and Navier-Stokes equations

Renormalization Group: non perturbative aspects and applications in statistical and solid state physics.

Turbulence: Basic Physics and Engineering Modeling

Dissipation Scales & Small Scale Structure

PHYS 611, SPR 12: HOMEWORK NO. 4 Solution Guide

Homogeneous Rayleigh-Bénard convection

arxiv: v1 [hep-ph] 30 Dec 2015

Energy spectrum of isotropic magnetohydrodynamic turbulence in the Lagrangian renormalized approximation

TURBULENCE IN FLUIDS AND SPACE PLASMAS. Amitava Bhattacharjee Princeton Plasma Physics Laboratory, Princeton University

Before we consider two canonical turbulent flows we need a general description of turbulence.

Topological Derivatives in Shape Optimization

Development of high vorticity structures in incompressible 3D Euler equations.

The Truth about diffusion (in liquids)

Validation of a Pseudo-Sound Theory for the Pressure- Dilatation in DNS of Compressible Turbulence

Statistical studies of turbulent flows: self-similarity, intermittency, and structure visualization

Quantum Gravity and the Renormalization Group

Turbulence and Bose-Einstein condensation Far from Equilibrium

Hydrodynamic Modes of Incoherent Black Holes

Does the flatness of the velocity derivative blow up at a finite Reynolds number?

Critical Region of the QCD Phase Transition

MHD turbulence in the solar corona and solar wind

Energy spectrum in the dissipation range of fluid turbulence

The effect of asymmetric large-scale dissipation on energy and potential enstrophy injection in two-layer quasi-geostrophic turbulence

Bäcklund Transformations: a Link Between Diffusion Models and Hydrodynamic Equations

arxiv: v1 [physics.flu-dyn] 17 Jul 2013

Turbulence and Bose-Einstein condensation Far from Equilibrium

Hydrodynamics, Thermodynamics, and Mathematics

Locality of Energy Transfer

MCRG Flow for the Nonlinear Sigma Model

Max Planck Institut für Plasmaphysik

Dynamics of the Coarse-Grained Vorticity

Kaluza-Klein Masses and Couplings: Radiative Corrections to Tree-Level Relations

Analysis of Turbulent Free Convection in a Rectangular Rayleigh-Bénard Cell

Spatiotemporal correlation functions of fully developed turbulence. Léonie Canet

Implementation of the Quasi-Normal Scale Elimination (QNSE) Model of Stably Stratified Turbulence in WRF

Computational Fluid Dynamics 2

Intermittency, Fractals, and β-model

Unitarity, Dispersion Relations, Cutkosky s Cutting Rules

Spacetime emergence via holographic RG flow from incompressible Navier-Stokes at the horizon. based on

J. Szantyr Lecture No. 4 Principles of the Turbulent Flow Theory The phenomenon of two markedly different types of flow, namely laminar and

Logarithmically modified scaling of temperature structure functions in thermal convection

Amplification of magnetic fields from turbulence in the early Universe

UV-IR Mixing on the Noncommutative Minkowski Space in the Yang-Feldman Formalism

UV completion of the Standard Model

Energy Cascade in Turbulent Flows: Quantifying Effects of Reynolds Number and Local and Nonlocal Interactions

MACROSCOPIC VARIABLES, THERMAL EQUILIBRIUM. Contents AND BOLTZMANN ENTROPY. 1 Macroscopic Variables 3. 2 Local quantities and Hydrodynamics fields 4

Energy and potential enstrophy flux constraints in quasi-geostrophic models

NONLINEAR FEATURES IN EXPLICIT ALGEBRAIC MODELS FOR TURBULENT FLOWS WITH ACTIVE SCALARS

Turbulent Vortex Dynamics

Dimensional reduction near the deconfinement transition

Turbulent spectra generated by singularities

An Overview of Computational Method for Fluid-Structure Interaction

FIELD THEORY MODEL FOR TWO-DIMENSIONAL TURBULENCE: VORTICITY-BASED APPROACH

ρ Du i Dt = p x i together with the continuity equation = 0, x i

Resonances, Divergent series and R.G.: (G. Gentile, G.G.) α = ε α f(α) α 2. 0 ν Z r

arxiv:chao-dyn/ v1 17 Jan 1996

Evolution of the pdf of a high Schmidt number passive scalar in a plane wake

Geostrophic turbulence and the formation of large scale structure

Multiscale Hydrodynamic Phenomena

Turbulentlike Quantitative Analysis on Energy Dissipation in Vibrated Granular Media

Analytic continuation of functional renormalization group equations

Turbulence models of gravitational clustering

Passive Scalars in Stratified Turbulence

Small-Scale Statistics and Structure of Turbulence in the Light of High Resolution Direct Numerical Simulation

Lagrangian intermittency in drift-wave turbulence. Wouter Bos

The zeta function, L-functions, and irreducible polynomials

An Introduction to Theories of Turbulence. James Glimm Stony Brook University

Formation of Inhomogeneous Magnetic Structures in MHD Turbulence and Turbulent Convection

A simple subgrid-scale model for astrophysical turbulence

WALL RESOLUTION STUDY FOR DIRECT NUMERICAL SIMULATION OF TURBULENT CHANNEL FLOW USING A MULTIDOMAIN CHEBYSHEV GRID

Introduction to Magnetohydrodynamics (MHD)

Influence of high temperature gradient on turbulence spectra

Project Topic. Simulation of turbulent flow laden with finite-size particles using LBM. Leila Jahanshaloo

c k i εi, (9.1) g k k i=1 k=1

3D Large-Scale DNS of Weakly-Compressible Homogeneous Isotropic Turbulence with Lagrangian Tracer Particles

Local flow structure and Reynolds number dependence of Lagrangian statistics in DNS of homogeneous turbulence. P. K. Yeung

Causal RG equation for Quantum Einstein Gravity

On the Perturbative Stability of des QFT s

Buoyancy Fluxes in a Stratified Fluid

July 2, SISSA Entrance Examination. PhD in Theoretical Particle Physics Academic Year 2018/2019. olve two among the three problems presented.

Incompressible MHD simulations

NUMERICAL SIMULATION OF THE TAYLOR-GREEN VORTEX

Transcription:

Renormalization group in stochastic theory of developed turbulence 4 p. 1/14 Renormalization group in stochastic theory of developed turbulence 4 Low-dimensional fluctuations and improved ε expansion Juha Honkonen National Defence University, Helsinki, Finland

Renormalization group in stochastic theory of developed turbulence 4 p. 2/14 Outline Two-parameter expansion Improved ε expansion Two-loop results Kolmogorov constant Prandtl number Conclusion

Renormalization group in stochastic theory of developed turbulence 4 p. 3/14 Skewness factor in the inertial range Large-scale pumping: ε 2, m = 1 L 0 d f(k) δ(k).

Renormalization group in stochastic theory of developed turbulence 4 p. 3/14 Skewness factor in the inertial range Large-scale pumping: ε 2, m = 1 L 0 d f(k) δ(k). Connect to experimental data through (m = 0) E = (d 1) 2(2π) d dkd f (k) D 10 = 4(2 ε) Λ2ε 4 E S d (d 1), Λ = (E/ν 3 0) 1/4.

Renormalization group in stochastic theory of developed turbulence 4 p. 3/14 Skewness factor in the inertial range Large-scale pumping: ε 2, m = 1 L 0 d f(k) δ(k). Connect to experimental data through (m = 0) E = (d 1) 2(2π) d dkd f (k) D 10 = 4(2 ε) Λ2ε 4 E S d (d 1), Λ = (E/ν 3 0) 1/4. Experimental C K at ε 2, but C K (ε) is ambiguous via order-of-magnitude estimate of Λ.

Renormalization group in stochastic theory of developed turbulence 4 p. 3/14 Skewness factor in the inertial range Large-scale pumping: ε 2, m = 1 L 0 d f(k) δ(k). Connect to experimental data through (m = 0) E = (d 1) 2(2π) d dkd f (k) D 10 = 4(2 ε) Λ2ε 4 E S d (d 1), Λ = (E/ν 3 0) 1/4. Experimental C K at ε 2, but C K (ε) is ambiguous via order-of-magnitude estimate of Λ. Attempts to relate D 10 and E in the momentum-shell approach not flawless.

Renormalization group in stochastic theory of developed turbulence 4 p. 3/14 Skewness factor in the inertial range Large-scale pumping: ε 2, m = 1 L 0 d f(k) δ(k). Connect to experimental data through (m = 0) E = (d 1) 2(2π) d dkd f (k) D 10 = 4(2 ε) Λ2ε 4 E S d (d 1), Λ = (E/ν 3 0) 1/4. Experimental C K at ε 2, but C K (ε) is ambiguous via order-of-magnitude estimate of Λ. Attempts to relate D 10 and E in the momentum-shell approach not flawless. Use independent of D 10 quantity - the skewness factor [Adzhemyan, Antonov, Kompaniets & Vasil ev (2003)]: S = S 3 /S 3/2 2.

Renormalization group in stochastic theory of developed turbulence 4 p. 4/14 Unambiguous Kolmogorov constant For ε 3 2 the structure function S 2(r) const, replace in S by the function with powerlike asymptotics r r S 2 (r) and define: Q(ε) r rs 2 (r) S 3 (r) 2/3 = r rs 2 (r) [ S 3 (r)] 2/3.

Renormalization group in stochastic theory of developed turbulence 4 p. 4/14 Unambiguous Kolmogorov constant For ε 3 2 the structure function S 2(r) const, replace in S by the function with powerlike asymptotics r r S 2 (r) and define: Q(ε) r rs 2 (r) S 3 (r) 2/3 = r rs 2 (r) [ S 3 (r)] 2/3. Calculate Kolmogorov constant and skewness factor unambiguously as C K = [ ][ ] 2/3 3Q(2) 12, S = 2 d(d + 2) [ ] 3/2 3Q(2). 2

Renormalization group in stochastic theory of developed turbulence 4 p. 5/14 Effect of low-dimensional fluctuations Two-loop corrections to C K and S large: 100 % change for d = 3 but rapidly decreasing with growing d. Drastic growth in the limit d 2 due to singular graphs.

Renormalization group in stochastic theory of developed turbulence 4 p. 5/14 Effect of low-dimensional fluctuations Two-loop corrections to C K and S large: 100 % change for d = 3 but rapidly decreasing with growing d. Drastic growth in the limit d 2 due to singular graphs. Summing up singularities calls for additional renormalization near d = 2: to make it multiplicative, introduce (m = 0) d f (k) = D 10 k 4 d 2ε + D 20 k 2 (JH & Nalimov, 1996) with D 20 to be renormalized.

Renormalization group in stochastic theory of developed turbulence 4 p. 5/14 Effect of low-dimensional fluctuations Two-loop corrections to C K and S large: 100 % change for d = 3 but rapidly decreasing with growing d. Drastic growth in the limit d 2 due to singular graphs. Summing up singularities calls for additional renormalization near d = 2: to make it multiplicative, introduce (m = 0) d f (k) = D 10 k 4 d 2ε + D 20 k 2 (JH & Nalimov, 1996) with D 20 to be renormalized. Coarse-graining of finite band-width forcing always generates the local term (Forster, Nelson & Stephen, 1977).

Renormalization group in stochastic theory of developed turbulence 4 p. 6/14 2d vs. 3d turbulence Why 2d fluctuations of importance for 3d? Fluctuations present in all d s, sum in low dimensions! Then extrapolate.

Renormalization group in stochastic theory of developed turbulence 4 p. 6/14 2d vs. 3d turbulence Why 2d fluctuations of importance for 3d? Fluctuations present in all d s, sum in low dimensions! Then extrapolate. Different physics in 2d and 3d: is it legal to extrapolate?

Renormalization group in stochastic theory of developed turbulence 4 p. 6/14 2d vs. 3d turbulence Why 2d fluctuations of importance for 3d? Fluctuations present in all d s, sum in low dimensions! Then extrapolate. Different physics in 2d and 3d: is it legal to extrapolate? Borderline between direct and inverse cascades near the point (2,2) in the d, ε plane (Fournier & Frisch, 1977):

Renormalization group in stochastic theory of developed turbulence 4 p. 6/14 2d vs. 3d turbulence Why 2d fluctuations of importance for 3d? Fluctuations present in all d s, sum in low dimensions! Then extrapolate. Different physics in 2d and 3d: is it legal to extrapolate? Borderline between direct and inverse cascades near the point (2,2) in the d, ε plane (Fournier & Frisch, 1977): ε

Renormalization group in stochastic theory of developed turbulence 4 p. 6/14 2d vs. 3d turbulence Why 2d fluctuations of importance for 3d? Fluctuations present in all d s, sum in low dimensions! Then extrapolate. Different physics in 2d and 3d: is it legal to extrapolate? Borderline between direct and inverse cascades near the point (2,2) in the d, ε plane (Fournier & Frisch, 1977): ε Yes, inverse energy cascade far from the linear extrapolation path.

Renormalization group in stochastic theory of developed turbulence 4 p. 7/14 Two-parameter expansion Additional U V -renormalization near d = 2 required S R = 1 2 v ( D 1 k 4 d 2ε + D 2 Z D2 k 2) v v [ t v + (v )v νz ν 2 v ] with ν 0 = νz ν and g 01 = D 10 ν0 3 = g 1 µ 2ε Zν 3, g 20 = D 20 ν 3 0 = g 2 µ 2 d Z D2 Z 3 ν.

Renormalization group in stochastic theory of developed turbulence 4 p. 7/14 Two-parameter expansion Additional U V -renormalization near d = 2 required S R = 1 2 v ( D 1 k 4 d 2ε + D 2 Z D2 k 2) v v [ t v + (v )v νz ν 2 v ] with ν 0 = νz ν and g 01 = D 10 ν0 3 = g 1 µ 2ε Zν 3, g 20 = D 20 ν 3 0 = g 2 µ 2 d Z D2 Z 3 ν. The RG solution [m = 0, UV cutoff Λ imposed] G(k,g 10,g 20,ν 0, Λ) = (D 10 /ḡ 1 ) 2/3 k 2 d 4ε/3 R Λ (1,ḡ 1,ḡ 2, Λ/k).

Renormalization group in stochastic theory of developed turbulence 4 p. 7/14 Two-parameter expansion Additional U V -renormalization near d = 2 required S R = 1 2 v ( D 1 k 4 d 2ε + D 2 Z D2 k 2) v v [ t v + (v )v νz ν 2 v ] with ν 0 = νz ν and g 01 = D 10 ν0 3 = g 1 µ 2ε Zν 3, g 20 = D 20 ν 3 0 = g 2 µ 2 d Z D2 Z 3 ν. The RG solution [m = 0, UV cutoff Λ imposed] G(k,g 10,g 20,ν 0, Λ) = (D 10 /ḡ 1 ) 2/3 k 2 d 4ε/3 R Λ (1,ḡ 1,ḡ 2, Λ/k). Near d = 2 IR-stable fixed point giving rise to double expansion in ε and 2 = d 2.

Renormalization group in stochastic theory of developed turbulence 4 p. 8/14 Minimal subtractions on rays Two-parameter renormalization not entirely trivial; problems analytic renormalization: no MS scheme,

Renormalization group in stochastic theory of developed turbulence 4 p. 8/14 Minimal subtractions on rays Two-parameter renormalization not entirely trivial; problems analytic renormalization: no MS scheme, stable nontrivial fixed point at d > 2: dimensional regularization insufficient for thermal fluctuations.

Renormalization group in stochastic theory of developed turbulence 4 p. 8/14 Minimal subtractions on rays Two-parameter renormalization not entirely trivial; problems analytic renormalization: no MS scheme, stable nontrivial fixed point at d > 2: dimensional regularization insufficient for thermal fluctuations. Renormalize on a ray with intermediate Λ renormalization (Adzhemyan, JH, Kompaniets, Vasil ev 2005):

Renormalization group in stochastic theory of developed turbulence 4 p. 8/14 Minimal subtractions on rays Two-parameter renormalization not entirely trivial; problems analytic renormalization: no MS scheme, stable nontrivial fixed point at d > 2: dimensional regularization insufficient for thermal fluctuations. Renormalize on a ray with intermediate Λ renormalization (Adzhemyan, JH, Kompaniets, Vasil ev 2005): fix the ratio (d 2)/ε = 2ζ to restore MS scheme,

Renormalization group in stochastic theory of developed turbulence 4 p. 8/14 Minimal subtractions on rays Two-parameter renormalization not entirely trivial; problems analytic renormalization: no MS scheme, stable nontrivial fixed point at d > 2: dimensional regularization insufficient for thermal fluctuations. Renormalize on a ray with intermediate Λ renormalization (Adzhemyan, JH, Kompaniets, Vasil ev 2005): fix the ratio (d 2)/ε = 2ζ to restore MS scheme, introduce explicit cutoff Λ, renormalize out large Λ terms [replace primary (physical) bare parameters by secondary ones],

Renormalization group in stochastic theory of developed turbulence 4 p. 8/14 Minimal subtractions on rays Two-parameter renormalization not entirely trivial; problems analytic renormalization: no MS scheme, stable nontrivial fixed point at d > 2: dimensional regularization insufficient for thermal fluctuations. Renormalize on a ray with intermediate Λ renormalization (Adzhemyan, JH, Kompaniets, Vasil ev 2005): fix the ratio (d 2)/ε = 2ζ to restore MS scheme, introduce explicit cutoff Λ, renormalize out large Λ terms [replace primary (physical) bare parameters by secondary ones], the remainder is analytic continuation from d < 2.

Renormalization group in stochastic theory of developed turbulence 4 p. 9/14 Two expansions for the skewness factor Two complementary ways to calculate the universal ratio Q:

Renormalization group in stochastic theory of developed turbulence 4 p. 9/14 Two expansions for the skewness factor Two complementary ways to calculate the universal ratio Q: In ε, expansion on the ray ζ = (d 2)/2ε = const: Q(ε) = r rs 2 (r) = ε1/3 ( S 3 (r)) 2/3 k=0 Ψ k (ζ)ε k.

Renormalization group in stochastic theory of developed turbulence 4 p. 9/14 Two expansions for the skewness factor Two complementary ways to calculate the universal ratio Q: In ε, expansion on the ray ζ = (d 2)/2ε = const: Q(ε) = r rs 2 (r) = ε1/3 ( S 3 (r)) 2/3 Ψ k (ζ)ε k. k=0 In ε expansion with coefficients singular, when d 2: Q(ε) = ε 1/3 k=0 Q k (d)ε k.

Renormalization group in stochastic theory of developed turbulence 4 p. 9/14 Two expansions for the skewness factor Two complementary ways to calculate the universal ratio Q: In ε, expansion on the ray ζ = (d 2)/2ε = const: Q(ε) = r rs 2 (r) = ε1/3 ( S 3 (r)) 2/3 Ψ k (ζ)ε k. k=0 In ε expansion with coefficients singular, when d 2: Q(ε) = ε 1/3 Q k (d)ε k. These are two different subsequences of the double series k=0 Q(ε,d) = ε 1/3 k=0 l=0 [2ε/(d 2)] k q kl [(d 2)/2] l.

Renormalization group in stochastic theory of developed turbulence 4 p. 10/14 Improved ε expansion Combine the information from both expansions [ n 1 Q (n) eff = ε1/3 n 1 + k,l=0 n 1 k=0 ( 2ε d 2 k=0 Ψ k ( d 2 2ε ) k q kl Q k (d)ε k ) ε k ( ) ] l d 2 2. Subtraction term to account for double counting in the overlap region.

Renormalization group in stochastic theory of developed turbulence 4 p. 10/14 Improved ε expansion Combine the information from both expansions [ n 1 Q (n) eff = ε1/3 n 1 + k,l=0 n 1 k=0 ( 2ε d 2 k=0 Ψ k ( d 2 2ε ) k q kl Q k (d)ε k ) ε k ( ) ] l d 2 Subtraction term to account for double counting in the overlap region. 2. } } x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x xx x x x x x xx x x x x x x x x x x x x x x x d x x x x x x x e,d xx x x x xx x x x xx x x x x x x x x x x x x x x x xx x x x x x x x x x x xx x xxx xxx xxx xx x xx x xxx x xxx x x x x x xx xxx xxx xxx xxx x xxx xx x x xx xxx x xxx xxx xx xx xxx xxxx xxx x x x x xx x x xxx xxx xxx xxx xx x x x x x x x x x xx xx xxx x x xxx x xx xxx xxx xxx x x x e } }

Renormalization group in stochastic theory of developed turbulence 4 p. 11/14 Improved two-loop Kolmogorov constant Comparison of one-loop and two-loop results for C K : C ε ε expansion C ε, double expansion C δ overlap correction n C ε C ε, C δ C eff 1 1.47 1.68 1.37 1.79 2 3.02 3.57 4.22 2.37 C eff improved ε expansion

Renormalization group in stochastic theory of developed turbulence 4 p. 11/14 Improved two-loop Kolmogorov constant Comparison of one-loop and two-loop results for C K : C ε ε expansion C ε, double expansion C δ overlap correction n C ε C ε, C δ C eff 1 1.47 1.68 1.37 1.79 2 3.02 3.57 4.22 2.37 C eff improved ε expansion Recommended experimental value: C K = 2.0 (Sreenivasan, 1995).

Renormalization group in stochastic theory of developed turbulence 4 p. 12/14 Turbulent Prandtl number Prandtl number for thermal conduction: Pr = ν 0 /κ 0 = 1/u.

Renormalization group in stochastic theory of developed turbulence 4 p. 12/14 Turbulent Prandtl number Prandtl number for thermal conduction: Pr = ν 0 /κ 0 = 1/u. Define turbulent (effective) inverse Prandtl number: u eff Γ θθ (k,ω = 0)/Γ vv (k,ω = 0).

Renormalization group in stochastic theory of developed turbulence 4 p. 12/14 Turbulent Prandtl number Prandtl number for thermal conduction: Pr = ν 0 /κ 0 = 1/u. Define turbulent (effective) inverse Prandtl number: u eff Γ θθ (k,ω = 0)/Γ vv (k,ω = 0). Singular in d 2 contributions cancel: two-loop correction small [Adzhemyan, JH, Kim & Sladkoff (2005)]: u eff = u (0) (1 0.0358ε) + O(ε 2 ), u (0) = 43/3 1 2, d = 3.

Renormalization group in stochastic theory of developed turbulence 4 p. 12/14 Turbulent Prandtl number Prandtl number for thermal conduction: Pr = ν 0 /κ 0 = 1/u. Define turbulent (effective) inverse Prandtl number: u eff Γ θθ (k,ω = 0)/Γ vv (k,ω = 0). Singular in d 2 contributions cancel: two-loop correction small [Adzhemyan, JH, Kim & Sladkoff (2005)]: u eff = u (0) (1 0.0358ε) + O(ε 2 ), u (0) 43/3 1 = 2, d = 3. At ε = 2 the turbulent Prandtl number Pr t close to accepted experimental value Pr t 0.81: Pr (0) t 0.72, Pr t 0.77.

Renormalization group in stochastic theory of developed turbulence 4 p. 13/14 Ramifications of the Navier-Stokes problem advection of passive scalar hydrodynamic fluctuations, momentum-shell RG: Forster, Nelson & Stephen (1976), LR correlated injection, field-theoretic RG: Adzhemyan, Vasil ev & Pis mak (1983), decaying scalar, hydrodynamic fluctuations, LR correlated injection, field-theoretic RG: Hnatich (1990, reflecting boundary), Hnatich, JH (2000, absorbing boundary);

Renormalization group in stochastic theory of developed turbulence 4 p. 13/14 Ramifications of the Navier-Stokes problem advection of passive scalar hydrodynamic fluctuations, momentum-shell RG: Forster, Nelson & Stephen (1976), LR correlated injection, field-theoretic RG: Adzhemyan, Vasil ev & Pis mak (1983), decaying scalar, hydrodynamic fluctuations, LR correlated injection, field-theoretic RG: Hnatich (1990, reflecting boundary), Hnatich, JH (2000, absorbing boundary); compressible fluid LR correlated injection, momentum-shell RG: Staroselsky, Yakhot, Kida & Orszag (1990), LR correlated injection, expansion in Mach number, FTRG, OPE: Adzhemyan, Nalimov & Stepanova (1995);

Renormalization group in stochastic theory of developed turbulence 4 p. 13/14 Ramifications of the Navier-Stokes problem advection of passive scalar hydrodynamic fluctuations, momentum-shell RG: Forster, Nelson & Stephen (1976), LR correlated injection, field-theoretic RG: Adzhemyan, Vasil ev & Pis mak (1983), decaying scalar, hydrodynamic fluctuations, LR correlated injection, field-theoretic RG: Hnatich (1990, reflecting boundary), Hnatich, JH (2000, absorbing boundary); compressible fluid LR correlated injection, momentum-shell RG: Staroselsky, Yakhot, Kida & Orszag (1990), LR correlated injection, expansion in Mach number, FTRG, OPE: Adzhemyan, Nalimov & Stepanova (1995); anisotropic random forcing LR, momentum-shell RG, weak anisotropy: Rubinstein & Barton (1987), LR, FTRG, weak anisotropy: Adzhemyan, Hnatich, Horvath & Stehlik (1995); Kim & Serdukov (1995); LR, FTRG, strong anisotropy: Buša, Hnatich, JH & Horvath (1997).

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback