CfCA User Meeting NAOJ, 29-30 November 2016 A simple subgrid-scale model for astrophysical turbulence Nobumitsu Yokoi Institute of Industrial Science (IIS), Univ. of Tokyo Collaborators Axel Brandenburg Nordic Institute for Theoretical Physics (NORDITA), KTH and Univ. of Stockholm JILA and Department of Astrophysics, LASP, Univ. of Colorado Annick Pouquet, Duane Rosenberg, Raffaele Marino National Center for Atmospheric Research (NCAR)
Contents Subgrid-scale (SGS) model Turbulence simulations Theoretical formulation Direct numerical simulations Physical origin Application: angular-momentum transport
http://www.inflowimages.com/ Lake Michigan (Hess et al. 1988) Vortical structure at the solar surface (Wedemeyer-Böhm et al. 2012) on the Mars, imaged by rover (Greeley et al. 2007)
Subgrid-scale modelling SGS viscosity in the Smagorinsky model Fluctuating vorticity (Robinson, Kline & Spalart 1988) Smagorinsky Dissipative constant CS nature Isotropic flow more Mixing-layer flow Channel flow Vortical structures weak intermediate less strong Coherent vortical structures may be related to the less dissipative nature
Helicity SGS model E(k,t) k -5/3 SGS helicity GS velocity equation O(l -1 ) GS O( -1 ) O(η -1 ) SGS k SGS stress : GS velocity strain Evaluation of SGS helicity by DNS Production ~ Dissipation Local equilibrium of the SGS helicity SGS helicity GS quantities
Inverse cascade Dual, bidirectional, constant-flux cascade Simulation parameters Froude # Rossby # Reynolds # Forcing at intermediate scale k F /k 1 10 not k F /k 1 3 4 Re~4000 for points 1024 3 ; Re~8000 for points 1536 3 Energy spectra Energy flux Vertical velocity (horizontal cut) Conditions for the inverse cascade Relative frequency
Emergence of helicity Re ~ 4000 for points 256 3 ; Re ~ 10000 for points 512 3 Rossby wave frequency Brunt-Väisälä frequency Temporal evolution of total helicity More turbulent eddies at higher Froude number E(k,t) Coarse-grained DNS k -5/3 O(l -1 ) GS O( -1 ) SGS O(η -1 ) k
Nonlinear terms appear under the divergence operator Integrated over the volume of the system Fourier representations No net contribution only transfer Nonlinear term The dynamics of k mode is governed by its interaction with all other modes
Scales in turbulence Integral scale Kolmogorov microscale Required grid points Re NG Re NG Walking Cars Airplanes Earth s outer core Solar convection zone Galaxies
Direct Numerical Simulation (DNS) E(k,t) Large Eddy Simulation (LES) E(k,t) Reynolds-averaged Navier Stokes (RANS) E(k,t) k -5/3 k -5/3 k -5/3 O(l -1 ) O(η -1 ) k O(l -1 ) O( -1 ) O(η -1 ) k O(l -1 ) O(η -1 ) k Reality of DNS GS SGS E(k,t) / Kl E(k,t) / ηv 2 1 1 kl Effective viscosity kinematic viscosity due to molecular processes model of the unresolved-scales numerical integration scheme 1 1 kη : Resolved : Unresolved
Vortex generation Vorticity Mean vorticity cf., Biermann battery cf., Mean magnetic field Vortexmotive (Ponderomotive) force V M = hu 0! 0 i Reynolds stress K = hu 02 i/2
Transport suppression in turbulent swirling flow z U z U z θ U θ r Laminar Turbulent Turbulent swirl U θ θ r Reynolds stress Experimental studies (Kitoh, 1991; Steenbergen, 1995) circumferential 0.5 z/d = 4.3 (Re = 50,000) 0.5 z/d = 7.7 (Re = 50,000) 0.5 z/d = 11.5 (Re = 50,000) 0.5 z/d = 21.3 (Re = 50,000) 0.5 z/d = 28.1 (Re = 50,000) 0.5 z/d = 34.9 (Re = 50,000) U θ /U M 0.0 U θ /U M 0.0 U θ /U M 0.0 U θ /U M 0.0 U θ /U M 0.0 U θ /U M 0.0-0.5-1.0-0.5 0.0 0.5 1.0-0.5-1.0-0.5 0.0 0.5 1.0-0.5-1.0-0.5 0.0 0.5 1.0-0.5-1.0-0.5 0.0 0.5 1.0-0.5-1.0-0.5 0.0 0.5 1.0-0.5-1.0-0.5 0.0 0.5 1.0 r /a r /a r /a r /a r /a r /a 1.5 z/d = 4.3 (Re = 50,000) 1.5 z/d = 7.7 (Re = 50,000) 1.5 z/d = 11.5 (Re = 50,000) 1.5 z/d = 21.3 (Re = 50,000) 1.5 z/d = 28.1 (Re = 50,000) 1.5 z/d = 34.9 (Re = 50,000) 1.0 1.0 1.0 1.0 1.0 1.0 U z /U M U z /U M 0.5 0.0-1.0-0.5 0.0 0.5 1.0 r /a axial U z /U M 0.5 0.0-1.0-0.5 0.0 0.5 1.0 r /a U z /U M 0.5 0.0-1.0-0.5 0.0 0.5 1.0 r /a U z /U M 0.5 0.0-1.0-0.5 0.0 0.5 1.0 r /a U z /U M 0.5 0.0-1.0-0.5 0.0 0.5 1.0 r /a 0.5 0.0-1.0-0.5 0.0 0.5 1.0 r /a (Yokoi & Yoshizawa, PoF 1993)
Theoretical formulation (Yokoi & Yoshizawa, Phys. Fluids 1993) Basic field: homogeneous isotropic but non-mirrosymmetric Calculation of the Reynolds stress where mixing length Eddy viscosity Helicity-related coefficient helicity inhomogeneity is essential
Numerical set-up (DNS) z y z z +z 0 +z 0 0 x ω F 0 H dh dz z 0 z 0 Set-up of the turbulence and rotation ωf (left), the schematic spatial profile of the turbulent helicity H (= uʹ ωʹ ) (center) and its derivative dh/dz (right). Rotation Inhomogeneous turbulent helicity Summary of DNS results Run k f /k 1 Re Co /( T 2 ) A 15 60 0.74 0.22 B1 5 150 2.6 0.27 B2 5 460 1.7 0.27 B3 5 980 1.6 0.51 C1 30 18 0.63 0.50 C2 30 80 0.55 0.03 C3 30 100 0.46 0.08
Global flow generation Axial flow component U y on the periphery of the domain Turbulent helicity uʹ ωʹ (top) and mean-flow helicity U 2ωF (bottom)
Reynolds stress Early stage Developed stage Reynolds stress uʹyuʹz (top), helicityeffect term ( H) z 2ωF y (middle), and their correlation (bottom). Mean axial velocity U y (top), turbulent helicity multiplied by rotation 2ωFH (middle), and their correlation (bottom).
Spectra Run k f /k 1 Re Co /( T 2 ) A 15 60 0.74 0.22 B1 5 150 2.6 0.27 B2 5 460 1.7 0.27 B3 5 980 1.6 0.51 C1 30 18 0.63 0.50 C2 30 80 0.55 0.03 C3 30 100 0.46 0.08 Run A Run B2 Run C2
Physical origin V M (r 2 H) Ω * δu = τ δu δω δu + = τ δu δω + Local angular momentum conservation (Coriolis force) δh δω u δω + δh + Inhomogeneous helicity δω = δu δu = τu Ω *
Angular-momentum transport in the solar convection zone Angular momentum around the rotation axis Vector flux of angular momentum Miesch (2005) Liv. Rev. Sol. Phys. 2005-1 Helicity effect
Helicity effect in the stellar convection zone Duarte, et al, (2016) MNRAS 456, 1708 Meridional circulation in the poleward direction at the surface Helicity more negative and positive in the shallow region Angular momentum transport in the outward direction
Summary In turbulent momentum transport in hydrodynamics Mean velocity strain (symmetric part of velocity shear) + Energy Transport enhancement (structure destruction) Mean absolute vorticity (antisymmetric part of velocity shear) + Inhomogeneous helicity Transport suppression (structure formation) N. Yokoi & A. Brandenburg, Phys. Rev. E 34, 033125 (2016) N. Yokoi, Geophys. Astrophys. Fluid Dyn. 107, 114 (2013) N. Yokoi & A. Yoshizawa, Phys. Fluids A 5, 464 (1993)
Helicity Thinkshop 2017 12-15 June 2017 Institute of Industrial Science (IIS), Univ. of Tokyo or National Astronomical Observatory of Japan (NAOJ) S.O.C. Nobumitsu Yokoi, Kirill Kuzanyan Takashi Sakurai, Dmitry Sokoloff Axel Brandenburg, Hongqi Zhan Manolis Georgoulis
Appendices A. Introduction to turbulence B. Theory for inhomogeneous turbulence C. Subgrid-scale (SGS) modelling with helicity effect
A. Introduction to turbulence
Turbulence Research Uriel Frisch, Turbulence: The Legacy of A. N. Kolmogorov
Equation of fluctuating velocity turbulence mean velocity interaction turbulence turbulence interaction Instability approach Linear in and, each (Fourier) mode evolves independently Closure approach Homogeneous turbulence, no dependence on large-scale inhomogeneity
Nonlinearity (Closure theory) Inhomogeneity (Instability theory) Absence of large-scale field Dynamics of nonlinear interaction Homogeneous and isotropic Fully nonlinear Presence of large-scale field Response to large-scale field Inhomogeneous and anisotropic Linear (or weakly nonlinear) Fully developed turbulence Simple boundary conditions Transition to turbulence Nonlinear boundary conditions
Phenomenology of turbulence Nonlinearity Scales of turbulence Scaling of turbulence Dissipation Turbulent transport
Reynolds number High Reynolds number (Re 1) (nonlinear term) (viscous term) Reynolds number (diffusion time) (advection time) (advection length) (diffusion length)
Nonlinear terms appear under the divergence operator Integrated over the volume of the system Fourier representations No net contribution only transfer Nonlinear term The dynamics of k mode is governed by its interaction with all other modes
Closure Problem
Eddy-viscosity representation : eddy viscosity (turbulent viscosity) enhanced transport spatial dependence How to express νt?
Inhomogeneous turbulence Flow Large-scale structure - Homogeneous turbulence Turbulence energy externally injected Transport suppression Transport coefficients Eddy viscosity Turbulent magnetic diffusivity Velocity shear Electric current Magnetic shear Vorticity Production rates - Large-scale inhomogeneities Turbulent energy Intensity of turbulence: energy Turbulence Structure of turbulence: helicities Turbulent cross helicity Turbulent residual helicity
Turbulence effects in mean-fields Mean-field equations Turbulent correlations Mass Momentum Magnetic field Energy Reynolds (+ turb. Maxwell) stress Turbulent electromotive force enhancement enhancement suppression suppression Transport coefficients
Transport coefficients An example: Turbulent magnetic diffusivity - Parameters - Mixing length - Turbulence energy - Propagators - Transport equations
B. Theory for inhomogeneous turbulence
Crazy
Two-Scale Direct-Interaction Approximation (TSDIA) Yoshizawa, 1984: mirrorsymmetric case Yokoi & Yoshizawa, 1993: non-mirrosymmetric case DIA Multiple-scale analysis A closure theory (propagator renormalization) for homogeneous isotropic turbulence Fast and slowly varying fields Introduction of two scales Fourier transform of the fast variables Scale-parameter expansion Introduction of the Green s function Statistical assumptions on the basic fields Calculation of the statistical quantities using the DIA
(i) Introduction of two scales Fast and slow variables Slow variables X and T change only when x and t change much. Velocity-fluctuation equation where
(ii) Fourier transform of the fast variables The fluctuation fields are homogeneous with respect to the fast variables: Velocity-fluctuation equation in the wave-number space: where
(iii) Scale-parameter expansion Eliminating the pressure term, we have (iv) Introduction of the Green s function
(v) Statistical assumptions on the basic fields Basic field: homogeneous isotropic non-mirrorsymmetry
1st-order field Green s function
Formal solution in terms of (vi) Calculation of the statistical quantities using the DIA
Calculation of the Reynolds stress where Eddy viscosity simplest case mixing length Helicity-related coefficient helicity inhomogeneity is essential
C. Subgrid-scale (SGS) modelling with helicity effect
Drawbacks of Smagorinsky model Lack of self-adaptavity SGS viscosity should vanish in regions where the flow is fully resolved; decay correctly in the near-wall region; Needs for the wall damping function Problem of constant adjustment Smagorinsky model with CS = 0.18 is too much dissipative for channel-flow turbulence too less dissipative for isotropic turbulence
Problem of Constant Adjustment Smagorinsky constant Isotropic flow Mixing-layer flow Channel flow need to be adjusted such as To alleviate Dynamic procedure to determine the coefficient CS Alternatives to the generic form Evolution equations of the SGS quantities
Fluctuating vorticity (Robinson, Kline & Spalart 1988) SGS viscosity in the Smagorinsky model Smagorinsky Dissipative constant CS nature Isotropic flow more Mixing-layer flow Channel flow Vortical structures weak intermediate less strong Coherent vortical structures may be related to the less dissipative nature