Abstracts - KTH Mechanics

Scale energy budget for a viscoelastic wall-bounded flow
E. De Angelis ∗ ,N.Marati,C.M.Casciola,R.Piva.
In isotropic conditions polymers affect the turbulent cascade of kinetic energy and
deplete the inertial transfer below Lumley scale 1 . In the context of drag reduction,
anisotropy and inhomogeneity can be included in the analysis by extending to dragreducing
viscoelastic flows the scale energy budget for a Newtonian channel flow 2 ,
∂〈δq 2 δui〉
+
∂ri
∂〈δq2 ∗ δU〉
dU
+2〈δuδv〉 +
∂rx
dy
∂〈v∗δu 2 〉
= −4〈ɛ
∂Y
∗ 〉 +
2ν ∂2 〈δq 2 〉
−
∂ri∂ri
2 ∂〈δp δv〉
+
ρ ∂Y
ν ∂
2
2 〈δq 2 〉
∂Y 2 +4∂〈T ∗ ijδui〉
+
∂rj
∂〈δTi2δui〉
− 4〈ɛ
∂Y
∗ P 〉 ,
where 〈δq 2 〉 = 〈δuiδui〉 is the scale-energy as a funtion of the separation vector ri
and of the mid-point Xi = xi + ri/2, the asterisk denote a mid point average and
ɛ and ɛP are the Newtonian and polymer dissipations. According to the equation,
the local conservation of scale-energy implies that the local source of scale energy
– the sum of local production and the amount intercepted from the spatial flux –
feeds the inertial cascade and the transfer towards the microstructure, see fig. 1. In
the Newtonian plug of mildly drag-reducing flows the behavior is easily explained
given the budget in the log-layer of Newtonian flows 2 and the effect the polymers
have on the cascade 1 . Closer to the wall the polymers are much more active and
directly interfer with the production process. As will be discussed in detail, in this
elastic layer an entirely different picture emerges consistent with the alteration of the
energy containing scales of the flow which entails dramatic consequences for the drag
experienced at the wall.
Figure 1: Scale-energy budget in the Newtonain plug: Force-velocity correlation —,
inertial and polymer transfer (—, —), viscous diffusion —, polymer and Newtonian
dissipation (—, —).
1 De Angelis E., Casciola C.M., Benzi R., Piva R., J. Fluid Mech 476, 105-114 (2003).
2 Marati N., Casciola C. M., Piva R. (2004), J. Fluid Mech 521, 191-215 (2004).
∗ Dip. di Meccanica e Aereonautica, “La Sapienza”, via Eudossiana 18, 00184, Roma, Italy
65