Fundamentals of Large Eddy Simulation Basic Equations

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Fundamentals of Large Eddy SimulationBasic
Equations

• Heiko Jansen
• University of Hannover
• LES / PALM - Seminar
• Zingst, July 2004

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Contents

• Motivation
• The role of turbulence
• The three classes of turbulence models
• Direct numerical simulation
• Reynolds-average simulation
• Large eddy simulation
• The concept of Large Eddy Simulation
• Filtering
• Parameterization
• Basic equations

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The role of turbulence 1/2

• Most flows in nature technical applications are
  turbulent
• Significance of turbulence
• Meteorology Transport processes of momentum,
  heat, water as well as substances and pollutants
• Health care Pollution
• Aviation, engineering Wind
• Characteristics of turbulence
• non-periodical, 3D stochastic movements
• mixes air and its properties on scales between
  large-scale advection and molecular diffusion
• non-linear ? energy is distributed smoothly with
  wavelength
• wide range of spatial and temporal scales

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The role of turbulence 2/2

• Large eddies 103 m, 1 hSmall eddies 10-3 m,
  0.1 s
• Energy production and dissipation
• on different scales
• Large scales shear and buoyant production
• Small scales viscous dissipation
• Energy-containing range
• Large eddies contain most energy.
• Energy-cascade
• In the inertial subrange large eddies are broken
  up by instabilities and handled down to smaller
  scales.

Stull (1988) Garratt (1992)
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The Reynolds number (Re)
Number of gridpoints for 3D simulation
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Classes of turbulence models 1/3

• Direct numerical simulation (DNS)
• Most straight-forward approach
• Resolve all scales of turbulent flow explicitly
• Advantage
• (In principle) a very accurate turbulence
  representation
• Problem
• Limited computer resources (1996 108, today
  109 gridpoints)
• Consequences
• DNS is restricted to moderately turbulent flows.
• Highly turbulent atmospheric turbulent flows
  cannot be simulated.

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Classes of turbulence models 2/3

• Reynolds average simulation (RAS)
• Opposite strategy
• For applications that only require average
  statistics of the flow.
• Integrate merely the ensemble-averaged equations.
• Parameterize turbulence over the whole eddy
  spectrum.
• Advantage
• Computationally inexpensive, fast.
• Problems
• Turbulent fluctuations not explicitly captured.
• Parameterizations are very sensitive to
  large-eddy structure that depends on
  environmental conditions such as geometry and
  stratification. ?Parameterizations are not valid
  for a wide range of different flows.
• Consequence
• Not suitable for detailed turbulence studies.

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Classes of turbulence models 3/3

• Large eddy simulation (LES)
• Seeks to combine advantages and avoid
  disadvantages of DNS and RAS by treating large
  scales and small scales separately, based on
  Kolmogorov's (1941) similarity theory of
  turbulence.
• Large eddies are explicitly resolved.
• The impact of small eddies on the large-scale
  flow is parameterized.
• Advantages
• Highly turbulent flows can be simulated.
• Computationally expensive, but parameter studies
  are still feasible.
• Local homogeneity and isotropy at large Re
  (Kolmogorov's 1st hypothesis) leaves
  parameterizations uniformly valid for a wide
  range of different flows.

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Concept of Large Eddy Simulation 1/2

• Filtering
• Spectral cut at wavelength ?x
• Structures larger than ?x areexplicitly
  calculated (resolvedscales).
• Structures smaller than ?x mustbe filtered out
  (subgrid scales),formally known as low-pass
  filtering.
• Reynolds averaging
• split variables in mean part and fluctuation,
  e.g.where
• spatially average the model equations
• ? lecture by M. Schröter, Thur 9am

Stull (1988)
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Concept of Large Eddy Simulation 2/2

• Parameterization
• The filter procedure removes the small scales
  from the model equations, but it produces new
  unknowns, mainly averages of fluctuation
  products.
• e.g.,
• These unknowns describe the effect of the
  unresolved, small scales on the resolved, large
  scales therefore it is important to include them
  in the model.
• But we do not have information about the
  variables (e.g., vertical wind component and
  potential temperature) on these small scales of
  their fluctuations.
• Therefore these unknowns have to be parameterized
  using information from the resolved scales.
• A typical example is the flux-gradient
  relationship, e.g.,
• ? lectures tomorrow 9am

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Basic equations, unfiltered

• Navier-stokes equations

• First principle of thermodynamics and equation
  for any passive scalar ?

• continuity equation

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Basic equations, unfiltered(in flux-form for
incompressible flows)

• Navier-stokes equations

• First principle of thermodynamics and equation
  for any passive scalar ?

• continuity equation

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Symbols
pressure density geopotential height Coriolis
parameter alternating symbol molecular
diffusivity sources or sinks
velocity components spatial coordinates potentia
l temperature passive scalar actual temperature
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Summary

• Motivation
• The importance of turbulence
• Three classes of turbulence models
• DNS, RAS and LES
• Key points of LES
• Filtering
• Parameterization
• Basic equations