Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 92

2.9. Pressure

As suggested by the acoustic analogy put forward by
Lighthill [LIG 52, LIG 54], the turbulent fluctuations are
considered sources of noise for a turbulent flow with a low
Mach number. The acoustic radiation coming from small
local zones of turbulence in an infinite non-turbulent
medium with pressure p 0 and density ρ 0 , at great distances,

engenders turbulent fluctuations in density ρ = ρ − ρ 0 , which

behave in the same way as sound waves, and satisfy the
propagation equation

′

2

c 2 ∂

1 ρ ′

2

[2.51] −∇ ρ = ′ 0

t 2 ∂

By combining the continuity and momentum equations, we
are able to write the Lighthill equation, which is reduced to

( )

2

= ρ 0 ∂

u i u j − u i u j

2 p

c 2 ∂

1

[2.52]

2 p

−∇

t 2

∂ x i ∂ x j

∂

for low Mach numbers and high Reynolds numbers [BLA 70,
BLA 86]. However, in these conditions, the linear acoustic
approximation is expressed by

( ) = c 2

[2.53]

c 2

p − p 0 = ρ − ρ 0 ρ ′

Consequently, the fluctuations in density which cause
noise are directly linked to the pressure transport
equation [2.52] applicable in these turbulent parcels. The
study of the pressure fluctuations in turbulent wall flows has
multiple applications in aerodynamics, land transport, and
in submarine acoustics. From a fundamental point of view,
the terms of pressure/velocity correlations play a very
important role in redistributing the turbulence between the
different velocity components, as we showed in the foregoing
sections. A multitude of studies have been published on the
dynamics of pressure fluctuations and the scales which
govern such fluctuations. In this section, we summarize
some of these studies.