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 Ecully R.J. Perkins 20091VII Séminaire Transalpin de Physique - Atmospheric 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… What is turbulence? Clouds over Madeira NASA R.J. Perkins 20092VII Séminaire Transalpin de Physique - Atmospheric Turbulence

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence3 Reynolds experiment What is turbulence?

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence4 Reynolds’ analysis of his pipe flow experiment What is turbulence? Critical Reynolds number for transition Re  2000Re  2000 Flow is stable (laminar)Flow becomes unstable (turbulent)

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence5 The role of Reynolds number What is turbulence? The wake behind a cylinder

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence6 A wide range of Length and Time Scales What is turbulence?

Conservation of mass ― for an incompressible fluid R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence7 The Governing Equations

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence8 Conservation of momentum – the Navier-Stokes equations 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: R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence9 The Governing Equations

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence10 In Dimensionless Form: The Governing Equations 4 variables ( u 1, u 2, u 3, 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

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence11 Flow between parallel plates Laminar flow

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence12 Flow between parallel plates Laminar flow

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence13 What happens at higher Reynolds numbers? Turbulent flow 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 LaminarTurbulent Fransson, Talamelli, Brandt & Cossu (PRL, 2006). Could we do the same analysis, using just the average velocities?

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence14 Reynolds Decomposition Turbulent flow For a steady flow we can take a time average of the velocity: For unsteady flow we need to take an ensemble average

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence15 Reynolds Decomposition applied to the Continuity Equation Turbulent flow Conclusions The average velocities satisfy the continuity equation The fluctuating velocities satisfy the continuity equation, at every instant

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence16 Reynolds Decomposition applied to the Navier-Stokes Equations Turbulent flow Conclusions The average velocities do not satisfy the Navier-Stokes equations! Correlations between the fluctuating velocities contribute to the mean transport of momentum.

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence17 The Reynolds stress term Turbulent flow 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

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence18 The closure problem Turbulent flow Need a model for the Reynolds stress terms to close the system of equations.

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence19 Closure models Turbulent flow

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence20 Numerical solutions of the Navier-Stokes equations Turbulent flows Direct Numerical Simulation – DNS All the terms are computed explicitly  Spatial resolution Δx, Δy ~ k η – Kolmogorov length scale Large Eddy Simulation – LES The large scales are calculated explicitly ( Δx, Δy  k η ) The effect of the small scales is modelled using a sub-grid scale model Express the derivatives as Finite Differences: e.g.

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence21 Vertical Structure of the Atmospheric Boundary layer Turbulence in the Atmospheric Boundary Layer

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence22 Length and Time Scales Turbulence in the Atmospheric Boundary Layer

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence23 Synoptic Scales – Radioactive plume from Chernobyl Turbulence in the Atmospheric Boundary layer

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence24 Diurnal variations Turbulence in the Atmospheric Boundary Layer

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence25 Effect of density gradient of air Thermal Effects in the ABL Hydrostatic pressure: Ideal gas : Adiabatic movement: Potential temperature :

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence26 Thermal Stability Thermal Effects in the ABL NeutralStableUnstable

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence27 Effects on Dispersion Thermal Effects in the ABL

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence28 Inversion layers Thermal Effects in the ABL Beirut, April 2000

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence29 The dispersion of hot smoke in a tunnel The effect of stratification on turbulence

Mechanical Production of Turbulence R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence30 The effect of stratification on turbulence

Buoyant production/destruction of turbulence R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence31 The effect of stratification on turbulence

Vertical Heat flux R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence32 The effect of stratification on turbulence 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: ρ and T vary very little in the first 10m-50m Þ At low altitudes, stability is determined principally by mechanical production Þ At higher altitudes, stability is determined principally by buoyant production

The Richardson number R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence33 The effect of stratification on turbulence

R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence34 The Richardson number The Effect of Stratification on Turbulence

The Monin-Obukhov Length R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence35 The Effect of Stratification on Turbulence

Lagrangian dispersion Consider the trajectories of particles passing through the source: R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence36 Turbulent dispersion coefficient In the absence of molecular diffusion, the concentration transported by a particle remains constant.

Trajectory of a single particle R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence37 Turbulent dispersion coefficient

Lagrangian analysis R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence38 Turbulent dispersion coefficient

Diffusion by continuous movements (Taylor, 1921) R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence39 Turbulent dispersion coefficient

Time dependence of the dispersion coefficient R.J. Perkins 2009VII Séminaire Transalpin de Physique - Atmospheric Turbulence40 Turbulent dispersion coefficient K T varies with distance from the source