Non-Newtonian turbulence: viscoelastic fluids and binary mixtures.

26 1. Newtonian turbulence
The linear dissipative term on the right-hand side of (1.74) accounts for friction
between the thin layer of fluid which is considered and the rest of the three dimensional
environment. The term fω represents an external forcing that counteracts
dissipation by viscosity ν and friction α and allows to obtain a statistically steady
state.
To solve eq. (1.74) it is necessary to specify boundary conditions, which are
required to solve the Poisson equation (1.73) for the stream function. In most
studies on two-dimensional turbulence these are assumed to be of periodic type
in both the directions. The presence of realistic no-slip boundaries gives rise to a
source of vorticity fluctuations.
1.3.2 Inviscid invariants
The main difference between the two and three-dimensional problem is the conservation
of vorticity along fluid trajectories in 2D, when viscosity, friction and
forcing are ignored.
The origin of this phenomenon is due to the absence of the vortex-stretching
term (ω ·∇)u in the two-dimensional vorticity equation. In three dimensions this
term is responsible for the unbounded growth of enstrophy in the limit of infinite
Reynolds number.
In the inviscid limit ν = 0, with no forcing and friction, eq. (1.74) simply
states that the material derivative of ω vanishes:
Dω
Dt
= ∂ω
∂t
+ u · ∇ω = 0 (1.75)
thus implying conservation of vorticity of fluid particles. Moreover, all the integrals
of the form � d 2 rf(ω) are inviscid invariants of the flow. In particular, this
yields conservation of the of the enstrophy
Z = 1
�
2
d 2 r|ω| 2
If f = α = 0 in eq. (1.74), the enstrophy balance equation
(1.76)
dZ
dt = −ν〈|∇ω|2 〉 = −2νP (1.77)
states that Z is bounded since, by definition, the palinstrophy P is a positivedefinite
quantity:
P = 1
2 〈|∇ω|2 〉 = 1
�
d
2
2 r|∇ω 2 | (1.78)
26