Transcript Boundary Layer Meteorology Lecture 3

Boundary Layer Meteorology Lecture 3
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Review summation (Einstein) notation
Introduce some Non-Dimensional Numbers
Reynolds averaging and Reynolds Stresses
Review chapter 2 of Garratt
Summation (Einstein) notation
ui ui  u1v1  u2 v2  u3 v3
For example :
 ij   pij   (eij  (2 / 3)uk / xk
  11   p  [u1 / x1  u1 / x1  (2 / 3) 
(u1 / x1  u2 / x2  u3 / x 3 )]
  12  [u1 / x2  u2 / x1  (2 / 3) 
(u1 / x1  u2 / x2  u3 / x 3 )]
Some Non-Dimensional Numbers
• Reynolds number: Re = VL/
– Reynolds number is ratio of acceleration (or “inertial force”) to friction
force. It governs transition to turbulence (at high Reynolds numbers , e.g.
about 2300 for pipes; highly variable, depending on shape of the flow!).
– For more detail, see:
http://physics.mercer.edu/hpage/friction/ajp/reynolds.html
• Richardson numbers: ratio of buoyant production (or destruction) to
shear production of turbulence.
– Flux: Rf = (g/v)w’’/(u’w’ du/dz + v’w’ dv/dz)
– Gradient: Ri = (g/ddz)/(du/dz)2
– Bulk: RiB = (g/v)z(v-)/(u2+v2)
Reynolds averaging and Reynolds Stresses
define : u = u + u
1
u =
t1
t 0 t1
 u dt
t0
u 0
t1 should be enough larger than t2 so that the average is
independent of time.
Reynolds averaging and Reynolds Stresses
uv  (u  u)(v  v)
 u v  u v uv  uv
 u v  uv

uv  u v  uv
Understanding Reynolds Stress
Random fluctuations
will always tend to remove local

maxima or minima, since, for a maximum they carry with
them momentum from elsewhere, which must be smaller
than the momentum at the maximum. Similarly, they will
tend to remove curvature….