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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