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wheremolecularviscosityplaysamajorroleinnaturalturbulentflows.Thesimplest
versionof(3.3),with
P
b
=
0(anapproximatebalancenotuncommoninnatural
flows)resultsinabalancebetweenproductionofTKEbyshearanddissipation,and
provides a framework for discussing the scales involved in transferring kinetic en-
ergyfromlargescaleinstabilities(“energy-containingeddies”)whereitisextracted
fromthemeanflow,toscaleswheremolecularinteractionisimportant.Specificdis-
cussionoftheenergy-containingscalesinIOBLturbulenceisdeferredtoChapter5,
butherewemayanticipatethattheshearproductiontermcanbeexpressedinterms
D
=
ofthefrictionvelocityattheboundary,
u
∗
=
√
τ
isthekinematicReynolds
stressmagnitude,andalengthscalecharacterizingtheenergy-containingcontaining
eddies,
where
τ
λ
:
3
P
S
=
τ
·
∂
U
/
∂
z
=
u
∗
/
λ
(3.5)
01ms
−
1
and
TypicalvelocityandlengthscalesintheIOBLare
u
∗
=
0
.
λ
=
2m,so
10
−
7
m
2
s
−
3
(theunitsareequivalentto Wkg
−
1
).
A fundamentalconceptin turbulencetheoryis thatin flowswith largeReynolds
number,energyis“cascaded”fromlarge(production)scalestoscalessmallenough
thatinertialandviscousforcesarecomparable,
3
i.e.,wherethe
local
Reynoldsnum-
ber of the flow is close to unity,
×
areasonableestimate of
ε
is5
being the small scale veloc-
ity and length scales, respectively.Kolmogorovhypothesized (see Batchelor 1967;
Hinze 1975)thatat these small scales, turbulenceis statistically in equilibriumand
uniquely determined by
υη
/
ν
∼
1
,
υ
and
η
. The velocity and length scales then follow from
basicdimensionalanalysis(e.g.,Barenblatt1996)
ε
and
ν
ν
1
/
4
3
η
=
ε
1
/
4
υ
=(
νε
)
10
−
6
m
2
s
−
1
,sotheKolmogorovscales
Theviscosityofcoldseawaterisabout1
.
8
×
1mms
−
1
and
for our example are about
2mm. The scale velocities of
eddiesintheproductionandviscoussubrangesdifferbyafactorofabout10,while
the length scales differ by threeordersof magnitude.Thus there is a large rangeof
length scales across which the “smaller whirls” feed. For comparison, if the fluid
inourexamplewereglycerineinsteadofwater,with akinematicviscosityofabout
1
υ
=
η
=
10
−
3
m
2
s
−
1
, thelengthscalewouldbe
.
2
×
η
=
24cm,muchcloserto
λ
.
3.4 Scalar Variance Conservation
Conservation equations for scalar (
T
S
, or other contaminants) variance may be
derived by analog with the TKE equation. The temperature variance equation
for steady, horizontally homogeneous turbulent flow comprises a balance among
,
3
The most concise description is still L. F. Richardson's famous verse: “Big whirls have smaller
whirlsthatfeed on theirvelocity, andlittlewhirlshave lesser whirls, andso ontoviscosity.”
