Inertial Range Dynamics and Mixing

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Inertial Range Dynamics and Mixing
Kinetic Theory Representation for Turbulence
Modeling and Computation

• Hudong Chen

Collaborators S. Orszag, S. Succi, I.
Staroselsky, V. Yakhot
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Contents

• Brief introduction and motivation
• Some basics in kinetic theory
• Brief review connection to Navier-Stokes
• New observations and insights in extended
  regimes
• Attempt on turbulence modeling
• Phenomenological argument
• Expanded analogy
• Discussions

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Brief introduction
Navier-Stokes (incompressible)
Averaged (or filtered) fluid equation
Reynolds stress
fluctuating velocity field
averaged velocity field
The averaged fluid equation is not closed
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Brief introduction
The central task has been to form a closure for
stress tensor
expressed in mean flow quantities
Boussinesq assumption
Eddy viscosity
rate of strain tensor of averaged flow

• Assume fluctuating eddies interacting like
  molecules
• Similar models also in wave-number
  representations
• Commonly used in turbulence RANS and LES
• By redefining viscosity, the Navier-Stokes form
  retained for mean velocity
• Questions arise due to lack of scale separation
  in turbulence
• Higher order extensions based in Navier-Stokes
  have other issues

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Basics of Kinetic Theory
Boltzmann Equation
single particle pdf at velocity value at

• The description is not only applicable to
  rarified gases
• W links to higher pdfs
• Collisions drives the system towards equilibrium
• Collisions conserve mass, momentum and energy
• The Naver-Stokes is one of its limiting
  situations

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Basics of Kinetic Theory
Boltzmann Equation

• Collisions obey conservation laws (vanishing 1st
  three moments)

mass, momentum, energy
hydrodynamic variables correspond to continuity
relations
Mass density
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Basics of Kinetic Theory
mass continuity
Fluid velocity
momentum continuity
Stress

• P is fully determined as soon as f (or P) is
  solved
• In macroscopic description, closure model is
  required for P

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Basics of Kinetic Theory
Collisions drives the system to an equilibrium
distribution
with a characteristic relexation time

• t is determined by micro-properties, can be
  different for hydro-moments
• Equilibrium distribution is Maxwell-Boltzmann
  (a Gaussian form)

• Closeness to equilibrium is an interplay
  between (large scale) fluid shear
• and (small scale) interactions
• It is estimated by a Weissenberg number Wi
  t S, ratio between
• hydrodynamic and collision times (related to
  Kn, ratio between spatial scales)

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Basics of Kinetic Theory
If separation of scales exists
Hierarchical relations via Chapman-Enskog
Distribution of n-th order is approximated
by a lower (n-1)-th order distribution
, so that all expressible via
Higher order non-equilibrium distributions and
moments are expressible as time and spatial
derivatives of the equilibrium (Gaussian)
distribution
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Basics of Kinetic Theory
Stress tensor calculable via a series of
Gausssian moments
Euler
Navier-Stokes
n-th order
Leading orders (for conventional fluids)
Euler
Navier-Stokes
Stress tensor and local strain tensor have same
structure except for a scalar proportionality
factor (i.e., Newtonian fluids)
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New observations
Higher order physics (Chen et al, 04)
2nd order

• Memory effects Important for very fast time
  fluid modes
• Nonlinear constitutive relations Secondary flow
  phenomenon,
• also imply complex tensorial fluid diffusion

• Smaller scales have greater non-equilibrium
  effects
• Extended fluid equations are closures based on
  higher truncations
• Boltzmann representation is effective
  re-summation of all orders

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New observations
Properties for a fixed collision time scale t

• An exact relationship between full pdf and
  equilibrium pdf in
• boundary free situation and long after initial
  transient
• Both memory and non-local spatial effects
  present when
• scales are comparable to t
• Flows at smaller scales are further away from
  equilibrium
• (and less Gaussian)
• Plugging this pdf into stress tensor
  definition, resulting in a
• Navier-Stokes like equation but has a
  differential-integral form

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New observations
Exact equation for unidirectional flows for
arbitrary Wi (Kn) (Chen et al, 07)

• Finite t (and l t sqrt(q) ) leads to
  non-local dependence in
• both space and time
• A diffusion equation recovered at large scale
  limit (t 0, l 0)

• Dispersion relation exhibits oscillations at
  large Wi (Yakhot et al, 07)

Eddy generation by a fast oscillating object of
extremely small Re is observed (impossible from
Navier-Stokes)
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New observations
Elementary example
A particle motion in a random fluctuating media
Longevin
Mean square displacement
A diffusion process
A ballistic process
In PDF, the system is approximated by
Fokker-Planck
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New observations
Effective viscosity vs Wi (Yakhot, et al, 07)

• Effective viscosity is significantly reduced
  for small scales
• To be careful This only applicable to linear
  transverse shear modes
• (other modes may have opposite behavior
  enhancement)

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New observations
Channel flows (theoretical result) (Chen et al,
07)
Navier-Stokes
Higher order physics

• Slip exhibited at wall due to finite t (or
  mean-free path)
• Boundary layer exists when mean-free path is
  confined by distance to wall

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Summary Remarks

• Boltzmann equation describes a wider scale
  range of physics
• When separation of scales exists, closed fluid
  equations are
• in principle constructed via expansions
• The Navier-Stokes equation is a 1st-order
  trancation taking into
• account of only linear departure from
  equilibrium
• Higher order non-equilibrium effects are
  important when spatial and
• time scales become comparable to the
  microscopic ones
• It is well accepted that the Navier-Stokes is
  fully adequate for describing flows of common
  macroscopic scales, then why we bother worrying
  about Boltzmann-like representation??
• The turbulent mean field part alone is not
  described by a closed N-S,
• and its small scales bear resemblance to
  micro-fluids, and may need
• to modeled as such
• Boussinesq approximation is perhaps fine for
  very large eddies

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Attempt for turbulence
Conjecture Averaged (or coarse grained)
velocity field at scales comparable to that
of fluctuating eddies behaves like micro-fluids
Longevin-DIA form (Kraichanan)

• It is non-Markovian (at lease at short times),
  and higher order diffusion
• terms are known to have tensorial forms
• At large scale, its behavior is dominated by
  eddy viscous diffusion.
• But nonlocal and memory effect s are important
  at short scales
• N-S moment closure beyond the Boussinesq is
  very difficult
• A finite form pdf representation is called for,
  containing non-equilibrium
• effects to all orders
• Fluctuating eddies defines relaxation time
  rather than eddy viscosity

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Attempt for turbulence

• An expanded analogy (Chen, 04)
• A single valued collision time for all scales,
• A single valued 2nd variant for all scales,

e.g.
Results
At very long wavelength
At shorter wavelength, it exhibits a leading
order correction
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Attempt for turbulence
Comparison to representative non-linear
turbulence models
Matching the linear term result in coefficients
for nonlinear terms
Kinetic approach
Rubinstein-Barton
Yoshizawa
Speziale
Kinetic model also includes effects of all higher
orders
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Discussion

• Conventional approaches in RANS (and LES) is to
  apply closure models
• for Reynolds stress (or alike) terms, in a
  modified Navier-Stokes
• It has issues going beyond Boussinesq
  Realizability, well posed-ness,
• BC, and limitation only for a finite moments
  orders
• (similar to modeling micro-fluids via extending
  the Navier-Stokes)
• Effects deep into non-equilibrium (non-Gaussian)
  regime are important for
• comparable scales, hence a Boltzmann-like pdf
  formulation is desirable
• Besides mean velocity field, it also gives
  information about turbulent
• kinetic energy equation, etc.
• However, though extended analogy is plausible,
  formulation from first
• principle is lacking
• Hopf formulation has so far not seen helpful,
  certainly not the
• Hamiltonian-Liouville based ones

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Discussion

• Assuming a Boltzmann-kinetic theory based
  formulation is
• meaningful, then
• What is a truly appropriate relaxation time? A
  single valued or broad band?
• Indeed, eddies have all sizes (or even
  self-similar scales) ,
• as opposed to molecules in a fluid (to say the
  least!)
• Insights may be gained from a particle systems
  having a broad
• range of molecular sizes or types
• Furthermore, what would be the appropriate
  (Gaussian) variant? Anything
• to do with those common turbulence
  nomenclatures?
• e.g., structure functions? dissipations?
• The effective collision process must also ensure
  incompressibility,
• without introducing thermal relevant
  dimensional quantities
• Earlier attempt Start on a (un-averaged) N-S
  equivalent Boltzmann eq.,

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Computation
Implication to LES Common LES models rely on
similar Boussinesq approximation, i.e., stress
and local large-eddy strain tensors are
proportional , and an LES viscosity It is
robust computationally, and commonly
used Besides many conventionally known issues,
the afore-mentioned issues also exist concerning
resolved velocity scales near grid (filter)
scale Non-Boussinesq LES models are less robust
and may have other difficulties, as mentioned
earlier An alternative computational approach is
perhaps Lattice Boltzmann based It gives the
same LES viscosity model at very large scales
than grid (filter) size, but it has different
effects at both time and space close to grid
scale But existing LBM schemes are not fully
addressing the above needs