Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 74

and an antisymmetrical part

( )

2 A ≡ A ij = ∂ U i ∂ x j − ∂ U j ∂ x i

The terms on the left-hand side of equation [2.29] can
easily be decomposed into

⎡ ⎣ ⎤

∂ a ij

∂

U k ∂

a ij
∂

DS ij
Dt

+

x k +

a ik a kj = + A ik A kj + S ik S kj

+ ⎢ ⎥

t

⎦

[2.34]

⎡ ⎣ ⎤

DA ij
Dt

+ A ik S kj + S ik A kj

⎢ ⎥

⎦

The first term in square brackets on the right of the above
equality is symmetrical, and the second term below it is
antisymmetrical. However, the Hessian

x j is
symmetrical. By identifying the symmetrical and
antisymmetrical parts of the transport equation [2.30] after
decomposition, we successively come to

2 P

−∂ ∂ x i ∂

2 S ij

DS ij
Dt

2 P

∂

1
ρ

∂

[2.35] + A ik A kj + S ik S kj =−

x j + ν ∂ x i ∂ ∂ x k ∂ x k

and

2 A ij

DA ij
Dt

∂

[2.36]

+ A ik S kj + S ik A kj = ν

∂ x k ∂ x k

We can easily show that the antisymmetrical tensor A ij is
linked to the vorticity component

k by

Ω = Ω + ω

k k

1
2 ε ijk Ω k

[2.37]

A ij =−

The antisymmetrical tensor

ε ijk is defined such that:

-

0 if two of the indices are arbitrarily identical ε =

ijk

( 0 ); ε 112 = ε 122 = ε 313 =