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Plasma Instabilities and Turbulence-I: basic ideas of fluid turbulence

A ThyagarajaUKAEA/EURATOM Fusion Association

Culham Science Centre, Abingdon, OX14 3DB, UK

QUAMP Lecture, Durham, September 18, 2006

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Suggested Reading

1. P.A. Davidson: “Turbulence”- Oxford University Press, 20042. L.D. Landau and E.M. Lifshitz: “Fluid Mechanics”-Pergamon, (2nd

Edition)19893. P. Drazin and W. Reid: “Hydrodynamic stability”-Cambridge U.P. 19814. A.S. Monin and A.M. Yaglom: “Statistical fluid mechanics”-MIT Press,

19735. S. Chandrasekhar: “Hydrodynamic and hydromagnetic stability”-Oxford

U.P. 19616. R.D. Hazeltine and J.D. Meiss:”Plasma confinement”-Dover, 20037. J.A. Wesson:”Tokamaks”-Oxford U.P, (3rd Edition), 20048. A. Thyagaraja and F.A. Haas: “Conceptual and experimental bases of

theories of anomalous transport in tokamaks”-Physics Reports, 143, 19869. A. R. Choudhuri: “The physics of fluids and plasmas”, Camb. U.P. 1999

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Fluid turbulence-1

• Flow in a pipe: Reynolds’ experiment, what is “laminar flow”; when do we get “turbulence”?

• When the applied pressure is “small”, the flow is “smooth” and leads to Poiseuille’s formula

• Viscosity balances pressure; inertia is negligible. The flow profile is parabolic (velocity vanishes on wall).

• Increasing pressure gradient leads to instability. Small disturbances grow rapidly-very complicated to calculate, with confusing results!

• Turbulent state is rapidly time and space-dependent. Average flow profile nearly flat in core with strong boundary layers.

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Fluid turbulence-2

• Instability excites many “modes”; some are linearly unstable, most are stable. The number of excited modes depends upon the degree of “forcing”.

• The external drive (ie imposed pressure gradient) creates instability which redistributes energy among many stable modes. Viscosity degrades the energy at the shortest wave lengths (highest wave numbers).

• This “dissipation rate” is determined by the energy input at long wave lengths. It is actually independent of the value of viscosity (provided it is non-zero!)

• Dynamics of systems with one or two d.f. trivial. Any higher, the motion can be “chaotic” and effectively unpredictable! Lyapunov instability (“sensitivity to initial conditions/perturbations”).

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Fluid turbulence-3

• Turbulence is chaos in a driven dissipative system with very large numbers of dynamically active degrees of freedom. Distinct from chaos in conservative systems.

• Sources of instability: “Nature dislikes gradients!”

• In pipe flow, the velocity gradient “causes” instability if too large (Kelvin-Helmholtz shear flow instability).

• A heavy fluid resting on a lighter one can be in equilibrium but will be unstable to the smallest corrugation of the interface (Rayleigh-Taylor interchange instability)

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Navier-Stokes Equations of Motion of Hydrodynamics

0( ). v

m[ . ] p

t

vv v v

vW

m VL/μR

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Fluid turbulence-4

• The system is completed by specifying the domain of flow, boundary conditions and initial conditions.

• Does a solution exist? Is it unique? How sensitive is it to “perturbations”?

• No general proofs known! • Incompressibility a good assumption when flow speed is small

compared with speed of sound; density is treated constant and pressure responds to flow.

• Reynolds number measures ratio of “inertial accelerations” to viscous forces. “Inviscid” hydrodynamics leaves out viscosity-surprisingly good for some questions but solutions may “blow up”!

• When viscosity is very high, life becomes easy! At low Reynolds numbers, inertia is negligible and the system evolves, like glycerine or treacle to simple steady states.

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Fluid turbulence-5

• Turbulence is a high Reynolds number phenomenon: at low enough values of Re “steady” solutions can be calculated (sometimes analytically).

• This flow is smooth and “laminar”. When it is “perturbed” one obtains evolution equations for the perturbations. Assuming they are “small” one linearizes the equations. How sensitive is it to “perturbations”?

• If the linearized equations have exponentially decaying (in time) solutions, the system is linearly stable, and the flow remains laminar.

• If the linearized equations have exponentially growing solutions (often these “eigen solutions” are called unstable modes) the linearization breaks down. One may get a new, laminar flow (this is called a

bifurcation).

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Real turbulence (from Van Dyke, An Album of fluid motion, 1982)

a) Wake behind an inclined flat plate Re=4300 and b) the wake of a grounded tank ship: Re 10,000,000!

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Fluid turbulence-6

• The new “bifurcated” state may or may not be time-dependent! However, in general, the starting solution will usually have a high degree of symmetry which will be “spontaneously broken” at the transition.

• As Re increases further, the system finally makes a transition to a highly chaotic, time-dependent state: the motion is bounded but unstable to tiny perturbations.

• All flow properties acquire statistical distributions and the number of independent “modes” required to describe the motion becomes large like some power of Re.The system is “driven” at long wave-lengths. In steady state, the energy is dissipated at the very short “viscous” scales.

• Vorticity is a key property of flow. The amplification of it is the basic process by which energy “cascades” from the “inertial” scales to the viscous scales of motion.

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A paradigmatic model for turbulence

• All turbulent systems can be qualitatively understood in terms of a model (cf. A Thyagaraja, Proc. Camb. Phil. Soc. 80, (1976), p. 153.) (includes in principle, neutral fluids, plasmas: fluid and gyrokinetic models-generalizing Landau-Hopf)

.. ijkij

iii i j j kj jk

d

dt

aa a a aVU S

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A paradigmatic model for turbulence

• The a(t): “generalized Fourier coefficients”. S - “sources”. i-in a potentially infinite “index set” determined by the system. Represent collective degrees of freedom.

• U,V couplings “conservative”. Lambda’s > 0.

..i ijk

i j kjki

d

dt

bb b bW

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A paradigmatic model for turbulence-II

• Equilibrium: solve for a’s given S’ in steady state.

• Stability: “perturb”, find linear growth rate for “modes”, “b’s”.

• Modes will nonlinearly “saturate” and ultimately dissipate.

• Stable ones will have negative gamma’s; unstable ones, positive. The W’s describe turbulence spreading.

• Separation of equilibrium and turbulence: convention, symmetry and time-scale.

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A paradigmatic model for turbulence-III

• The b’s are linear combinations of a’s; indexed by the address “i” which encodes spatial information.

• “Profiles” form a special sub-class; may vary more slowly and respond to sources imposed on system. Gradients drive

turbulence (but can also suppress/quench!)

• Stable and unstable branches are both crucial! Two rapidly growing “unstable” modes can “beat” together to drive a stable one at the sum of the two growth rates. This could be relevant to zonal flows and “streamer” genesis.

• The “turbulent state” is fundamentally nonlinear.

• Linear growth rates have little relevance to dynamics.

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A paradigmatic model for turbulence-IV

• The free energy flux into the unstable modes is redistributed by the nonlinearity (saturation).

• All turbulent transport occurs with “diffusivity” (turbulent Prandtl number O(1)) of order:

2

)( turb turbturbLD

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The nature of homogeneous isotropic turbulence

• Simple dimensional considerations imply “turbulence energy spectrum”, wave number k and the dissipation rate ε: Kolmogorov’s famous Law:

• This type of turbulence tells us general things about universal features of the spectrum.

• It is known that turbulence behaves differently in different dimensions.

• Fully developed “grid-turbulence” experimentally realizes homogeneous, isotropic turbulence models.

2/3

5/3S(k) = C εk

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The nature of inhomogeneous turbulence

• Driven, dissipative systems with very large numbers of degrees of freedom exhibit “turbulent evolution”.

• Energy from the driving sources is redistributed among the unstable and stable degrees of freedom and dissipated/transported out of the system.

• Free energy redistribution occurs in both real space (“turbulence spreading”) and wave number space (“cascading”). Enstrophy (vorticity) cascades to high-k!

• Typically excitation occurs at macro/meso scales.

• Dissipation usually occurs at very short length scales.

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Examples: pipe flows, rotating cylinders, thermal convection

• Flow in a pipe is surprisingly complicated; linear instability neither necessary nor sufficient!

• Inviscid instability, Rayleigh criterion is inadequate.

• Sources of instability, “free energy”; Rayleigh-Taylor and Kelvin-Helmholtz paradigms;

• Convective instability of atmospheres: “interchange idea”.

• Turbulence is not totally chaotic! Coherent structures and “relaminarization” possible. Turbulence can also be beneficial (turbulent boundary layers are crucial to cricket, golf, foot ball..!)

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Convection in a rotating cylinder

(from Van Dyke, An Album of fluid motion, 1982) The flow pattern in a rotating dishpan heated at rim and cooled in centre resembles mid-lattitude atmosphere.

Water 4 cm deep in a cylinder 19.5 cm is rotating counterclockwise at 30 rpm and heated 150 W at the wall.

Streak photo of aluminium particles show narrow jets and irregular vortices suggest hemispheric weather map for upper troposphere.

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Problems in fluid turbulence theory

1. If the viscosity is zero and the flow takes place in a bounded domain with normal component of velocity equal to zero, show that the kinetic energy of an incompressible fluid is conserved if no external forces act on it.

2. Show that the momentum equations of an incompressible fluid moving under a constant viscosity take the form, where is the so-called “kinematic viscosity”.

3. If the flow is steady and inviscid prove “Bernoulli’s theorem” that

m

2p 1( )

2t

v

W v Wv

m

2p 1( ) 0

2

v v

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4. If R is a bounded region, and the velocity is prescribed to be zero on the boundary of R, show that

2

R

d 1dV( ) 0mdt 2

v

Evaluate the RHS. What can you say about the long-term behaviour of the system from this integral inequality?

5. Derive an equation for the time evolution of the vorticity W and discuss the time evolution of the enstrophy, defined below, in 2 and 3 dimensions, in both inviscid and viscous flows.

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R

dVW