Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 182

3.9. Transport of invariants

Direct approaches, such as a dynamical system, are
relatively rare in the field of wall turbulence. The equations
governing the transport of the invariants and are, in
that sense, highly attractive [CHA 00]. Already, we see again
the equation governing the transport of the terms in the
velocity gradient tensor from the previous chapter:

Q

R

Da δ

ij ij

+ aa − a a = H [3.53]

ik kj km mk ij

Dt 3

The Hessian source term on the right-hand side of this

H ij

equation is

⎛ ⎞

2

δ ∂ a

2 2

1 ∂ P ∂ P

ij ij

[3.54]

H =− − + ν

⎜ ⎟

⎜ ⎟

ij

ρ ∂∂ x x ∂∂ x x 3 ∂∂ x x

⎝ ⎠

i j k k k k

The invariant is according to equation
[3.28]. Its transport equation is obtained by multiplying
equation [3.53] by and plotting the corresponding
tensorial equation. We obtain

Q =− a ik a ki 2

Q

a jk

DQ

+=− 3 R aH [3.55]

ik ki

Dt

Similarly, the dynamic equation of the invariant
is obtained by multiplying equation [3.53] by
, which gives us

R a ik a kn a ni /3

a jk a kp

=−

DR 2

3

2

[3.56]

− QaaH = −

ik kn ni

Dt

The terms on the left-hand side of these equations are
clearly defined. However, the Hessian i H is totally
unknown. In isotropic homogenous turbulence, the tensor