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					   Fundamentals of
Large Eddy Simulation
   Basic Equations

     Heiko Jansen
    Zingst, July 2005
• 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
      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
           The role of turbulence 2/2
• Large eddies: 103 m, 1 h
  Small eddies: 10-3 m, 0.1 s
• Energy production and
  on different scales
   – Large scales: shear and
      buoyant production
   – Small scales: viscous
• Large eddies contain most
• Energy-cascade
  In the inertial subrange large
  eddies are broken up by
  instabilities and handled down to
                                      Stull (1988); Garratt (1992)
  smaller scales.
          The Reynolds number (Re)

         u        u
                                     in e r tiao r c e s    u   3D wind vector

          u                         v is c o u o r c e s
                                               fs               kinematic molecular viscosity
                                                                outer scale of turbulence
         R e4
                  1 0 ( in thaetmo s p h e r e
                                                            L   characteristic velocity scale
                                                                inner scale of turbulence
                                                            U   (Kolmogorov dissipation length)

Number of gridpoints for 3D simulation

     3          9
 L                          18
             R e4                                   )
                          1 0 ( i n t haet mo s p h e r e
   Classes of turbulence models
• 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, but ~ 1018 gridpoints needed,
          see prior slide)
       • 1 h simulation of 4*108 gridpoints on 128 processors of the
          HLRN supercomputer needs 10 h CPU time
   – Consequences:
       • DNS is restricted to moderately turbulent flows.
       • Highly turbulent atmospheric turbulent flows cannot be
     Classes of turbulence models
• Reynolds average simulation (RAS)
   – Opposite strategy:
      • 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.
    Classes of turbulence models
• 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
   – Advantages:
      • Highly turbulent flows can be simulated.
      • Local homogeneity and isotropy at large Re (Kolmogorov's 1st
         hypothesis) leaves parameterizations uniformly valid for a wide
         range of different flows.
              Concept of Large Eddy
                 Simulation 1/2
• Filtering:
• Spectral cut at wavelength Δx
• Structures larger than Δx are
  explicitly calculated (resolved
• Structures smaller than Δx must
  be filtered out (subgrid scales),                 Stull (1988)
  formally known as low-pass

• Reynolds averaging: split
  variables in mean part and              
                                       w w w  
  fluctuation, spatially average the    
                                       w w 
  model equations

→ lecture Wednesday, 9 am
           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., w 
   – 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.
   – We do not have information about the variables (e.g., vertical wind
     component and potential temperature) on these small scales of their
   – Therefore, these unknowns have to be parameterized using
     information from the resolved scales.
       • A typical example is the flux-gradient relationship, e.g.,
                                                          w     Kh
→ lecture Wednesday, 9am                                               z
              Basic equations, unfiltered
•   Navier-stokes equations
     ui                  ui           1       p                     ui   1      uk
               uk                                  ijkf j uk         2
     t                   xk                   xi               xi   xk   3 xi   xk
•   Equation for any passive scalar ψ
                                            
               uk                             2   Q
      t                  xk                   xk

•   First principle of thermodynamics
              uk              h    2      Q
      t             xk            xk

•   Continuity equation
              u k
          t               xk
         Basic equations, unfiltered
     (in “flux-form“ for incompressible
•   Navier-stokes equations
     ui     uku i             1 p                     ui
                                     ijkf ju k         2
      t         xk              xi               xi   xk
•   Equation for any passive scalar ψ
          u k        
                        2  Q
      t     xk         xk
•   First principle of thermodynamics
          uk              2
                      h    2   Q
     t     xk             xk
•   continuity equation
              u k
      t              xk
u i ( i  1,2 , 3)
                                              p            pressure

                     velocity components
u ,v , w          

x i ( i  1,2 , 3)
                                                           geopotential height

                     spatial coordinates
x, y ,z           
                                                          Coriolis parameter

                                               ijk
                      potential temperature                alternating symbol

                      passive scalar
                                               ,         molecular diffusivity

T                     actual temperature
                                              Q, Q         sources or sinks
• Motivation
  – The importance of turbulence
• Three classes of turbulence models
  – DNS, RAS and LES
• Key points of LES
  – Filtering
  – Parameterization
• Basic equations