General Points - Transport and Coherent Structures in Wall Turbulence - page 13

approximations, which we are about to develop, are based on
an important peculiarity concerning the order of magnitude
of u , v and w . The continuity equation for the fluctuating
components is written thus:

∂∂∂
∂∂ ∂

u v w
x

++ = 0

y z

However, a structure responsible for turbulent
fluctuations has no prevailing direction (in a rough sense),
and its characteristic scales in the three directions are,
locally, of the same order of magnitude

l l ≅≅= l l

x y z

needs to be interpreted as being of the order
of rather than approximately equal to . The continuity
equation thus implies

The symbol

≅

′ ′ ′

uvw ′ ≅≅ = u

(Figure 1.3). The order of magnitude of the fluctuations is,
therefore, estimated by

2 2

′′ ′′

≅ <

∂ uu ∂ uv

′ ′

u u

≅

∂ x L ∂ y δ

because

L . It follows that

δ <<

2 2

∂ UU

x y
PP PPx
y

∂

<<

∂

2

∂

2

∂ ∂

()

<< ⇒ =

∞

∂ ∂ x

1 dP

dU

−

∞

= U

∞

∞

ρ dx dx