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0
for an incompressible turbulence. For
an axisymmetrical turbulence whose axis of symmetry is
x
,
the invariants
I
2
and
I
3
are reduced to
The invariant
I
1
=
1
⎡
2
2
2
2
⎤
I
=−
b
+
b
+
b
+
2
b
⎣
⎦
2
rr
xx
θ
rx
2
[2.82]
⎡
2
⎤
I
=
bbbb
−
⎣
⎦
3
rr
xx
rx
θθ
All the “realizable”
20
states of the turbulence are found in
the region delimited by the Lumley triangle (boundaries
included). The eigenvalues [2.80] outside of the triangle are
either negative or complex, and are not “realizable”. The
point
I
1
=
0
clearly corresponds to isotropic turbulence.
Supposing that the structural state of the turbulence from
the origin is axisymmetrical with
b
rx
I
2
=
0
and
b
rr
, two
=
=
b
θθ
situations will arise. Equation [2.82] is thus reduced to
(
)
3/2
I
=−
2
I
3
3
2
[2.83]
(
)
3/2
I
=−
2
−
I
3
3
2
3/2
The curve
I
3
(
)
0
extends until the components
=
2
−
I
2
3
>
become small and disappear, eventually reaching a
configuration with a single component. Conversely, the
components
b
rr
b
rr
=
b
θθ
3/2
0
,
leading to an axisymmetrical state with two Reynolds stress
components. The upper part of the “triangle” connects the
stress state to a component by way of the relation
I
3
=−
dominate on the line
I
3
(
)
=
b
θθ
=−
2
−
I
2
3
<
(
)
, as shown in Figure 2.26.
1/9
+
I
2
/3
20 See [LUM 78, pp. 131-133] for a detailed discussion.
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