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Atmospheric turbulence

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  1. Atmospheric turbulence Richard Perkins Laboratoire de Mécanique des Fluides et d’Acoustique Université de Lyon CNRS – EC Lyon – INSA Lyon – UCBL 36, avenue Guy de Collongue 69134 Ecully Richard.Perkins@ec-lyon.fr VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  2. What is turbulence? • One of the great unsolved problems From a theoretical point of view: • Einstein/Heisenberg, Cray prize From a practical point of view: • Most ‘engineering’ and geophysical flows are turbulent • Impossible to define satisfactorily But usually easy to recognise Is it random? Is it unpredictable? • Often described in terms of how it occurs… Clouds over Madeira NASA VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  3. What is turbulence? • Reynolds experiment VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  4. What is turbulence? • Reynolds’ analysis of his pipe flow experiment VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  5. What is turbulence? • The role of Reynolds number The wake behind a cylinder VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  6. What is turbulence? A wide range of Length and Time Scales VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  7. The Governing Equations • Conservation of mass for an incompressible fluid VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  8. The Governing Equations • Conservation of momentum – the Navier-Stokes equations VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  9. The Governing Equations • Dimensional Analysis • The physical problem can be characterised by: • the fluid density, ρ • a characteristic length scale, L • a characteristic velocity scale, U • The dimensionless variables then become: VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  10. The Governing Equations • In Dimensionless Form: 4 variables (u1, u2, u3, p) and 4 equations 1 independent parameter – the Reynolds number Re(=UL/ν)  Family of solutions, as a function of Re Very few analytical solutions available  Need to solve the equations numerically VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  11. Laminar flow • Flow between parallel plates VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  12. Laminar flow • Flow between parallel plates VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  13. Turbulent flow • What happens at higher Reynolds numbers? If Re  1000 the flow will start to become turbulent, and the velocities will fluctuate in space and in time. Poiseuille flow close to the boundary, visualised with smoke Laminar Turbulent Fransson, Talamelli, Brandt & Cossu (PRL, 2006). Could we do the same analysis, using just the average velocities? VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  14. Turbulent flow • Reynolds Decomposition For a steady flow wecantake a time average of the velocity: For unsteady flow weneed to take an ensemble average VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  15. Turbulent flow • Reynolds Decomposition applied to the Continuity Equation • Conclusions • The average velocities satisfy the continuity equation • The fluctuating velocities satisfy the continuity equation, at every instant VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  16. Turbulent flow • Reynolds Decomposition applied to the Navier-Stokes Equations • Conclusions • The average velocities do not satisfy the Navier-Stokes equations! • Correlations between the fluctuating velocities contribute to the mean transport of momentum. VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  17. Turbulent flow • The Reynolds stress term Reynolds stresses in the boundary layer • Fluctuating velocities towards the wall transport faster fluid towards the wall • Fluctuating velocities away from the wall transport slower fluid away from the wall • Reynolds stresses transport momentum down the momentum gradient • The action of the Reynolds stresses is similar to the action of viscosity. • But, the Reynolds stresses are much more effective than viscosity •  They cannot be neglected VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  18. Turbulent flow • The closure problem Need a model for the Reynolds stress terms to close the system of equations. VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  19. Turbulent flow • Closuremodels VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  20. e.g. • Direct Numerical Simulation – DNS • All the terms are computed explicitly • Spatial resolution Δx, Δy ~ kη– Kolmogorov length scale Turbulent flows • Numerical solutions of the Navier-Stokes equations Express the derivatives as Finite Differences: • Large Eddy Simulation – LES • The large scales are calculated explicitly (Δx, Δy kη) • The effect of the small scales is modelled using a sub-grid scale model VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  21. Turbulence in the Atmospheric Boundary Layer • Vertical Structure of the AtmosphericBoundary layer VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  22. Turbulence in the Atmospheric Boundary Layer Length and Time Scales VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  23. Turbulence in the Atmospheric Boundary layer Synoptic Scales – Radioactive plume from Chernobyl VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  24. Turbulence in the Atmospheric Boundary Layer Diurnal variations VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  25. Idealgas : Adiabaticmovement: Hydrostatic pressure: Potentialtemperature : Thermal Effects in the ABL Effect of density gradient of air VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  26. Neutral Stable Unstable Thermal Effects in the ABL Thermal Stability VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  27. Thermal Effects in the ABL • Effects on Dispersion VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  28. Thermal Effects in the ABL Inversion layers Beirut, April 2000 VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  29. The effect of stratification on turbulence • The dispersion of hot smoke in a tunnel VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  30. The effect of stratification on turbulence • Mechanical Production of Turbulence VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  31. The effect of stratification on turbulence • Buoyant production/destruction of turbulence VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  32. The effect of stratification on turbulence • Vertical Heat flux • For an unstable (convective) boundary layer H>0: • upward heat flux adds to the turbulence • For a stable boundary layer H<0: • downward heat flux suppresses turbulence • Buoyant production is almost independent of height: • ρand T vary very little in the first 10m-50m • At low altitudes, stability is determined principally by mechanical production • At higher altitudes, stability is determined principally by buoyant production VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  33. The effect of stratification on turbulence • The Richardson number VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  34. The Effect of Stratification on Turbulence • The Richardson number VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  35. The Effect of Stratification on Turbulence • The Monin-ObukhovLength VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  36. Turbulent dispersion coefficient • Lagrangian dispersion • Consider the trajectories of particles passing through the source: In the absence of molecular diffusion, the concentration transported by a particle remains constant. VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  37. Turbulent dispersion coefficient • Trajectory of a single particle VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  38. Turbulent dispersion coefficient • Lagrangian analysis VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  39. Turbulent dispersion coefficient • Diffusion by continuous movements (Taylor, 1921) VII Séminaire Transalpin de Physique - Atmospheric Turbulence

  40. Turbulent dispersion coefficient • Time dependence of the dispersion coefficient KTvaries with distance from the source VII Séminaire Transalpin de Physique - Atmospheric Turbulence