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for these are the only combinations of and ν that produce a length scale and
a velocity scale. The Reynolds number of the dissipative eddies is therefore
υη/ν
=
1, which confirms that they are strongly influenced by viscosity.
u 3 / we can write, using Eq. (1.34)
Using
u
ν
3 / 4
1 / 4
ν 3 / 4
u 3 / 4
1 / 4 ν 3 / 4
η =
= R 3 / 4
=
,
t
(1.35)
u
υ =
u
(ν) 1 / 4
u
(u 3 ν/) 1 / 4
= R 1 / 4
.
t
Equation (1.35) implies that in large- R t turbulent flows the dissipative eddies are
quite weak and quite small compared to the energy-containing eddies. If, for exam-
ple, u
1ms 1 and
10 3 m, as is typical in the atmospheric boundary layer,
10 3 m.
The ratio of vorticities typical of the dissipative and energy-containing eddies is
10 8 so that υ
10 2 ms 1 and η
then R t
vorticity of dissipative eddies
vorticity of energy-containing eddies
υ/η
u/ =
υ
u
η
R 1 / 2
;
(1.36)
t
at large R t the dissipative eddies contain essentially all the turbulent vorticity.
The smallness of η/ in large- R t turbulence puts severe limits on the R t that can
be reached in direct numerical simulations of turbulence (Problem 1.9) . Although
few turbulent flows of practical importance have R t values small enough to be
calculated in this way, the concept of “Reynolds number similarity” (Chapter 2)
does make them useful.
The Kolmogorov microscale η , which from Eq. (1.35) can be written as
ν 3 / 4 1 / 4
u 3 / 4
η
,
(1.37)
is seldom smaller than 10 4 m in engineering flows and about 10 3 mintheatmo-
sphere. This is almost always large enough to ensure the applicability of continuum
fluid mechanics.
1.6.5 Mathematical intractability
Generally speaking only linear differential equations can be directly and straight-
forwardly solved by analytical means. The advective acceleration term in the
Navier-Stokes equation (1.26) involves a product of velocity and velocity gra-
dient, making the equation nonlinear and mathematically intractable. This has a
simple physical intepretation. This nonlinear term produces the vortex stretching
term in Eq. (1.28) ; vortex stretching is believed to be a principal mechanism in the
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