Turbulence. 2. Reynolds number is an indicator for turbulence in a fluid stream

Transcription

1 Turbulence injection of a water jet into a water tank Reynolds number EF$ 1. There is no clear definition and range of turbulence (multi-scale phenomena) 2. Reynolds number is an indicator for turbulence in a fluid stream Re = u : " : l : µ : # = µ /" : inertial force density viscous force density = "u2 /l # Re = l u µ u /l 2 $ velocity of stream (e.g., 0.1 m/s) density of fluid (surface air: 1.2 kg/m 3 ) diameter of stream (e.g., 1 m) viscosity (e.g., 1.8e-5 kg/sm, varies with temperature) kinematic viscosity (1.5e-5 m 2 /s) limit for turbulence Re =1* 0.1/1.5e " 5 = 6667 > 6000 Atmospheric flow is expected to be turbulent! EG$

2 Navier-Stokes Equation: leftover of the continuity equation "# D# + u.$# = 0 % "t Dt = 0 for incompressible fluid, density is constant for observer moving with the stream nonlinear term ~ u 2 /l makes the Reynolds number large, responsible for turbulence viscous force density equation of motion EH$ Navier-Stokes is transfered into an energy equation: local change of kinetic energy advection of total energy energy dissipation rate! (conversion of mechanical energy into heat by vortices) work by viscous stresses EI$

3 Direct Numerical Simulation of Turbulence G. Levina: Turbulent vortex dynamo theory blue vortex tube with a white streamline vorticity field " # u of turbulent fluid (Imperial College London)! Vorticity in the atmosphere: (y,z) cross-sections across a stream in x-direction. Interaction of gravity wave and turbulence. Intermittency of turbulence EJ$ Back to the picture of a cascade of kinetic energy: P. Sagaut et al., Multiscale and multiresolution approaches in turbulence, ICP book, 2006 EE$

4 Power law energy spectrum and Structure function E(k) ~ k "#, 1 < # < 3 if the energy spectrum has a power law, then the spatial structure function also has a power law: = u(r') " u(r) 2 ~ r'"r # "1 velocity increment ~ r'"r 2 / 3, # = 5 /3 EK$ Kolmogorov s 1941 theory: "u(r,l) # u(r + l) $ u(r) l T: time scale : spatial scale, small dimensional analysis (for homogenous, isotropic smallscale turbulence): ("u(l)) 2 # [ l] 2 / [ T] 2 mean energy dissipation rate per unit mass: " # [ l] 2 /[ T] 3 [ l] 2 / T ("u(l)) 2 = C# 2 / 3 l 2 / 3 ( ) 2 / 3 l [ ] 2 = [ l] 2 / [ T] 3 [ ] 2 / 3 C: universal, dimensionless constant EL$

5 Kolmogorov s 5/3 power law energy spectrum: (l) = ("u(l)) 2 ~ # 2 / 3 l 2 / 3 $ # 2 / 3 k %2 / 3 " # = 5/3 = u(r') " u(r) 2 ~ r'"r # "1 = l # "1 "# +1 $k E(k) ~ k "# E(k) = C K " 2 / 3 k #5 / 3 l ~ k "1 C K : Kolmogorov constant (=1.6 ± 0.2, Ozawa et al.) EM$ From theory to application: Analyzing structures with the Structure Function S structure functions or statistical moments: S n (R) = "u(r) Aragon et al., ( ) n S 1 Luminance fluctuations "u in the painting of Van Gogh scale like Kolmogorov s isotropic turbulence: ~ r'"r 2 / 3 = R 2 / 3 This painting has indeed a turbulent structure! structure functions S n of luminance differences "u as function of spatial scale R EN$

6 Measurement of the Energy Dissipation Rate: Radar signals are back-scattered by small-scale irregularities of atmospheric refractivity (fluctuations of water vapour, air density, electron density). Echo intensity and/or spectral width provide: Structure function of refractive index differences at a spatial scale of about the half of the radar wavelength (e.g., 10 cm for a 1300 MHz UHF radar) 2dsin" = n# $ d ~ # /2, for " = 90, n =1 Energy dissipation rate! can be related to the refractivity structure function with assumption of Kolmogorov s isotropic turbulence (e.g., Hocking and Mu, 1997) KF$ Diurnal variation of energy dissipation rate (Kalapureddy et al., 2007) UHF radar India, two days in July planetary boundary layer (strong turbulence and dissipation)! 0:00 12:00 0:00 12:00 0:00 local time KG$

7 Seasonal variation of energy dissipation rate: (Kalapureddy et al., 2007, UHF radar, India) Enhanced energy injection (Sun) Claire Max and energy dissipation during daytime! Turbulent planetary boundary layer (PBL) grows up to 1.5 or 2 km altitude free troposphere PBL NOAA/ESRL KH$ Problem of the smooth model world: Where is the mesoscale eddy energy? horizontal scales of km Berner et al. (JAS,2009) investigate the dissipation rate of kinetic energy in the numerical weather prediction model of ECMWF. A more realistic energy spectrum was obtained and weather forecast was improved. 1) loss of fluctuations by numerical interpolation,... could be a reason 2) Underestimation of eddy generation by surface-atmosphere interactions could be a reason KI$

8 Impact of atmospheric noise on weather forecast: scheme from J. Berner et al. (ppt presentation) Potential Weak noise Strong noise mesoscale eddies ( noise ) can induce a change from a stable atmospheric state to a nearby stable state PDF Unimodal Multi-modal Realistic consideration of atmospheric noise improves the probability density function of the ensemble forecast of the atmospheric state ensemble forecast Richardson number Ri = static stability shear flow energy = N 2 2 #"u& # % ( + "v & % ( $ "z ' $ "z ' 2 = g") )"z 2 #"u& # % ( + "v & % ( $ "z ' $ "z ' 2 shear flow induced turbulence: Kelvin-Helmholtz instability u,v: eastward and northward component of horizontal wind N: Brunt-Vaisala frequency ": potential temperature g: gravity acceleration Potential temperature of an air parcel (with T,p) is the temperature that the parcel would acquire if adiabatically brought to sea level pressure P 0 atmospheric turbulence is expected for: Ri < 0.25 Richardson number is a good indicator for shear flow-induced turbulence in the atmosphere KE$