Getting to know turbulence - Turbulence in the Atmosphere - page 49

2.9 Reynolds-number similarity
The statistics of the energy-containing eddies in a turbulent flow of a given type,
when nondimensionalized with the scales of the energy-containing range, u and
, are often observed not to depend significantly on Reynolds number when
it is larger than a threshold value. A physical interpretation is that since the
turbulent Reynolds number R t is large in turbulent flow, the energy-containing
eddies do not experience significant viscous forces and their statistics eventu-
ally become independent of Reynolds number. This is called Reynolds-number
similarity .
Reynolds-number similarity, while only approximate, is useful because it allows
us to make geometrically identical but smaller-scale laboratory models of geo-
physical flows with some confidence that their energy-containing structure will be
representative of the much-larger-Reynolds-number geophysical flow. Thus, it pro-
vides the basis for the physical modeling we discussed in Chapter 1 . Laboratory
flows can provide less expensive measurements and have much smaller required
averaging times (Problem 2.6) than atmospheric flows. Much of our knowledge
of turbulent diffusion from sources in the convective atmospheric boundary layer
was actually gained through measurements in laboratory convection tanks at vastly
smaller Reynolds number ( Willis and Deardorff , 1974 ).
Reynolds-number similarity also provides some justification for carrying out
direct numerical simulation (DNS) of geophysical turbulent flows. Coleman et al .
( 1990 ) used DNS to simulate an atmospheric boundary layer at Reynolds numbers
far lower than in nature. Their simulation corresponded to a layer with wind speed
on the order of 1 cm s − 1 .
2.10 Coherent structures
Among the largest eddies in a turbulent flow are apt to be “coherent structures.”
These are quasi-steady, significant-amplitude circulations that exist in essentially
the same form, intensity, and location in each realization of the flow. With their
Reynolds stresses they extract kinetic energy from the mean flow, and they lose it
through the Reynolds stresses exerted on them by smaller eddies. Their form and
intensity depend on the structure of the parent flow.
According to Holmes et al . ( 1996 ), Liu ( 1988 ) identified the first appearance
of the idea of coherent structures in turbulent flows as being in the late 1930s.
Townsend's (1956) book on turbulent shear flow presents analyses of the velocity
and length scales of the coherent structures in various shear flows. The Holmes et al .
monograph has extensive discussions of analytical methods for isolating them and
predicting their form.