General Points - Transport and Coherent Structures in Wall Turbulence - page 3

1.2.1.1. Continuity equation
The continuity equation expresses the conservation of
mass throughout an elementary volume of fluid. It can be
written in various forms, including

∂ρ
∂∂

∂ρ U

+

i

= 0 [1.2]

t x

i

where is the density. This equation can also be written as ρ

D
Dt

ρ ∂ U

[1.3]

+ ρ

i

= 0

∂ x

i

using the definition of the material derivative. In the context
of the applications envisaged in this topic, the density is

considered to be constant in this equation; so the continuity
equation is reduced to

ρ

∂
∂

U
x

i

= 0 [1.4]

i

1.2.1.2. Momentum balance equations
We obtain the momentum balance equations by applying
Newton's first law to an elementary volume. We obtain

DU ρ

∂

[1.5]

i

= σ + f

ji ext i ,

Dt ∂ x

j

In this relation, f represents the external forces and

ext i ,

is the shear stress tensor defined by

σ

ji

U
P D x

⎛ ⎞ ∂

2
3

μ j

[1.6]

σ =− δ + μ − δ

⎜ ⎜ ⎝ ⎠

ji ji ji ji

∂

j