Turbulence in Astrophysics (Theory) Wolfram Schmidt Institut für theoretische Physik und...

Turbulence in Astrophysics Turbulence in Astrophysics (Theory)(Theory)

Wolfram SchmidtWolfram SchmidtInstitut für theoretische Physik und AstrophysikInstitut für theoretische Physik und Astrophysik

Universität WürzburgUniversität Würzburg

21 September 2004 Astrophysical Turbulence 2

Stirring of FluidStirring of Fluid

• Mechanical force stirring fluid into rotational motion

• Turn-over time T, wavelength L• What happens in the limit t → ∞?

T

LLV 21Re

It depends on the Reynolds number!


Laminar vs. Turbulent FlowLaminar vs. Turbulent Flow

• If Re is relatively small, only eddies of size L are produced

• For Re ~ 1000, the motion of adjacent fluid layers becomes unstable

Reynolds, 1883


Fluid motion forces vortices to Fluid motion forces vortices to stretch, and a stretching vortex stretch, and a stretching vortex

must fold to accomodate an must fold to accomodate an increasing length in a fixed volume. increasing length in a fixed volume.

To the extent that the flow is To the extent that the flow is scaling, I conjecture the vortex scaling, I conjecture the vortex

tends toward a fractal.tends toward a fractal.

Mandelbrot, Mandelbrot, The Fractal Geometry of The Fractal Geometry of NatureNature


VorticesVortices

Turbulent fluid motion is inherently rotational


Strain and VorticityStrain and Vorticity

ikkiik vvW 21

v ikikWWω 2

v iiSd

ikkiik vvS 21

ikik SSS 2

Symmetric derivative Antisymmetric derivative

Rate of strain

Dilatation

Vorticity


Vortex FormationVortex Formation

Vortices are streched and folded in three dimensions

Port

er

et

al.

AS

CI, 1

99

7


The Turbulence CascadeThe Turbulence Cascade

• Breaking up of laminar flow structure due to large |S| produces high vorticity ω

• Force of wavelength L produces structure on scales much smaller than L for high Re

• Small vortices are quasi random

Turbulence is a non-linear multi-scale phenomenon

Richardson, 1922; Onsager, 1945


Isotropic TurbulenceIsotropic Turbulence

• Statistically, there is no prefered direction (random orientation of vortices)

• In nature, turbulence is never exactly isotropic on large scales (forcing, BCs)

• However, turbulent flows tend to become asymptotically isotropic towards small scales (randomisation by non-linear energy transfer)

Taylor, 1935


The Kolmogorov TheoryThe Kolmogorov Theory• Hypothesis of local isotropy: At sufficiently high

Re, the dynamics on small scales tends to become statistically isotropic

• First similarity hypothesis: The statistics of isotropic velocity fluctuations on sufficiently small scales are universal und uniquely determined by the viscosity and the rate of kintetic energy dissipation

• Second similarity hypothesis: There is a subrange of scales for which the statistics of turbulent fluid motions are independent of the mechanism and the length scale of dissipation


The 5/3 Power Law (K41)The 5/3 Power Law (K41)

3/53/2)( kεCkE

4/14/3K

ενη

4/3

K

Reη

L

Rate of dissipation ε Wave number k = 2π/l

log E

log k

k -5/3

L-1 ηK-1

Length scale of viscous dissipation

transfer


But the hope that „homogeneous But the hope that „homogeneous turbulence“ would be a sensible model turbulence“ would be a sensible model was dashed by Landau & Lifschitz 1953-was dashed by Landau & Lifschitz 1953-1959, which notes that some regions are 1959, which notes that some regions are marked by very high dissipation, while marked by very high dissipation, while other regions seem by contrast nearly other regions seem by contrast nearly

free of dissipationfree of dissipation..

Mandelbrot, Mandelbrot, The Fractal Geometry of The Fractal Geometry of NatureNature


Realistic TurbulenceRealistic Turbulence

• Convective boundary layers: Anisotropy in stratified medium (convection zones, atmospheres)

• Turbulent combustion: Anisotropy across flame surface, transient flow (thermonuclear supernovae)

• Gravoturbulence: Inhomogeneous and supersonic turbulence in self-gravitating fluids (star formation)

• MHD turbulence: Instability of fluid motion due to interaction with magnetic field, multi-scale anisotropy (ionized gas in ISM, jets, accretion disks)


The Navier-Stokes EquationThe Navier-Stokes Equation

σfPv ρt

ρD

D

vttD

D

Conservation of momentum

Lagrangian time derivative

ikikikik δSρνρνSσ 3122

Viscous dissipation tensor

Mechanical, magnetic, gravitational forces

Non-linear advection


Further EquationsFurther Equations

0D

D vρρ

t

vfvσv ρTχρcPet

ρ PD

D

Gρ 4

Mass conservation

Conservation of energy

Poisson equation

Maxwell equationsin the case of MHD


Statistical TheoriesStatistical Theories

• Mixing length theory: one characteristic length scale lM =αHP (Kolmogorv spectrum → δ-peak)

• ODT models: one-dimensional stochastic process for eddy size (reproduces Kolmogorv spectrum)

• PDF models: determine probability distributions for velocity fluctuations etc.

• Reynolds stress models: dynamical equations for moments of fluctuating fields


Stellar ConvectionStellar Convection• Full Reynolds stress model for compressible

turbulence (Canuto, 1997): multitude of coupled, non-linear PDEs → hopeless

• Feasible model: reduced set of eqns. for mean K , Fc , ε and average squared fluctuations of temperature and horizontal velocity (Kupka, 1999)

Closure relations for higher order momentsNon-local & anisotropic


Stellar Convection: Stellar Convection: Convective FluxConvective Flux

Kup

ka

MPA

, 2

004


Stellar Convection: Vertical Stellar Convection: Vertical RMS Velocity RMS Velocity

Kup

ka

MPA

, 2

004


Numerical SimulationsNumerical Simulations

• Direct numerical simulation (DNS):Direct numerical simulation (DNS): Static grid, NSE or numerical viscosityStatic grid, NSE or numerical viscosity

• Large Eddy Simulation (LES):Large Eddy Simulation (LES): Subgrid Subgrid scale model for unresolved turbulencescale model for unresolved turbulence

• Smooth particle hydrodynamics (SPH):Smooth particle hydrodynamics (SPH): Particle ensemble represents the flowParticle ensemble represents the flow

• Adaptive mesh refinement (AMR):Adaptive mesh refinement (AMR): Hierarchy of dynamically generated Hierarchy of dynamically generated grid patchesgrid patches


Thermonuclear SupernovaeThermonuclear Supernovae

• Runaway turbulent deflagration of C+O in a Chandrasekhar-mass white dwarf

• PPM for hydrodynamics (Fryxell et al., 1989)

• Subgrid scale model for turbulent flame speed (Niemeyer & Hillebrandt, 1995)

• Level set method for flame surface tracking (Reinecke et al., 1999 )

• Homologous grid expansion to follow the explosion (Röpke, 2004)


History of a SN Ia History of a SN Ia ExplosionExplosion

Röpke et al. MPA, 2004

t = 0 s

t = 0.3 s

t = 0.6 s

t = 2 s


Turbulence in the ISMTurbulence in the ISM

• Supersonic turbulence in self-gravitating gas• Thermal processes, magnetic fields• Paradigm of turbulent star formation:

Turbulence can induce local gravitational collapse, albeit it provides global support

• SPH treatment: e.g. Klessen, 2001• AMR treatment with PPM/ZEUS: e.g. Kritsuk

& Norman, 2002; Abel et al., 2002


RésuméRésumé• Significant developments in the treatment of

turbulent convection via statistical models• Three-dimensional simulations with

sophisticated codes running on extremely powerful computers offer exciting insights

• However, most simulations are ignorant of small-scale turbulence (SGS models!)

• AMR is excellent for inhomogeneous and transient and astrophysical flows, but is it appropriate for turbulence?

Turbulence in Astrophysics (Theory) Wolfram Schmidt Institut für theoretische Physik und...

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