General Points - Transport and Coherent Structures in Wall Turbulence - page 4

(

G

)

for a Newtonian fluid. 2 In this equation, Pxt , represents

the field of local instantaneous pressure, is the dynamic

μ

viscosity and is the Kronecker delta ( if j = i , and 0

δ δ = 1

ji ji

if not). The value

i D is the strain tensor, which is expressed

by

∂ U

∂ U j

[1.7]

D = +

i

ji

∂ x ∂ x

i j

The momentum balance equation [1.4] can be reduced to
the NS equation for an incompressible Newtonian fluid with
constant viscosity. The NS equation is written, in tensor
form, as

2

DU ∂ U ∂ U 1 ∂ P ∂ U 1

[1.8]

i

=

i

+ U

i

= − + ν

i

+ f

j ext i ,

Dt ∂ t ∂ x ρ ∂ x ∂ x ∂ x ρ

j i j j

where is the kinematic viscosity. A slightly different

form of the NS equation is

νμρ =

∂ UU

∂ U 1 ∂ P Uf 1

[1.9]

i j 2

i

+ = − + ν ∇ +

i ext i ,

∂ t ∂ x ρ ∂ x ρ

j i

As a general rule, the Einstein summation over repeated
indices is adopted in this topic, except in certain very specific
cases, which will be clearly indicated. The viscous terms are
contained in the Laplacian operator in

2 2

∇= ∂∂ ∂ x x

j j

equation [1.9].

2 The discussion in this topic is strictly limited to incompressible flows of
Newtonian fluids.