Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

10 1. Newtonian turbulence
1.1.2 Energy balance
The energy balance in absence of external forcing for Navier-Stokes equation follows
from eqs. (1.1)-(1.2). The total kinetic energy of the fluid is
E = 1
�
d
2
3 x|u| 2
(1.9)
and its time derivative is
dE
dt =
�
d 3 �
x[ui∂tui + uiuj∂jui] =
d 3 x[−ui∂iP + νui∂j∂jui]. (1.10)
Assuming periodic boundary conditions on a cubic volume of size L
u(x + nL, y + mL, z + qL) = u(x, y, z) ∀x, y, z ∈ R ∀n, m, q ∈ Z (1.11)
or null boundary conditions on a volume V
u|∂V = 0 (1.12)
the nonlinear and pressure gradient contributions vanish; using the identity
|∇ × u| 2 = (ɛijk∂juk)(ɛilm∂lum) = ∂j(uk∂juk) − ∂juk∂kuj − uk∂j∂juk (1.13)
one finally gets
�
dE
= ν
dt
d 3 �
xu · ∆u = −ν
d 3 x|∇ × u| 2 = −ν
�
d 3 x|ω| 2
(1.14)
where we have introduced the vorticity of the fluid ω = ∇ ×u. Defining the total
enstrophy as
Z = 1
�
d
2
3 x|∇ × u| 2
(1.15)
the energy balance reads:
dE
= −2νZ
dt
(1.16)
and shows that in absence of external forcing and for ν = 0 energy is conserved,
i.e. it is an inviscid invariant. On the other hand, in the limit ν → 0 the energy
dissipation rate does not vanish, but approaches a constant value:
lim 2νZ = ɛ. (1.17)
ν→0
This phenomenon is known as dissipative anomaly and represents one of the fundamental
laws of turbulence. It implies that for ν → 0 the enstrophy must grow as
Z ∼ ν −1 to compensate the decreasing viscosity. The unbounded growth of enstrophy
in three dimensions is the physical origin of the dissipative anomaly, and
it is caused by the vortex stretching which produces diverging velocity gradients
in the limit Re → ∞.
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