Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 117

Two of the eigenvalues are equal in case of
axisymmetrical turbulence - for example, = λ 3 . Because, λ 2

at the same time, 0 , equation [2.82] enables λ 1 + λ 2 + λ 3 = b ii =

us to write

3
4

1
4

λ 2 , I 3 λ 3 I 2 =− =

Figure 2.27. Lumley anisotropy “triangle”
with the structural states. This figure is
adapted from [SIM 08]

The invariant I 3 is negative when

0 , in which case λ 1 <

the principal stress σ 1 is lesser than = σ 3 , the turbulence σ 2

is squashed in the direction of σ 1 , and structurally begins to

assume the shape of a disk (an oblate spheroid). On the
opposite I 3 0 side, the structural form of the turbulence

(axisymmetrical) is similar to a cigar or an American football
(a prolate spheroid). We show the structural states in
Figure 2.27.

>