Biomedical Engineering Reference
In-Depth Information
The turbulence stress is also referred to as the
Reynolds stress
and is an unknown
that needs to be modelled. While the turbulence stress is a physical stress, its consti-
tutive modelling is different from the viscous stress that has a constant coefficient of
viscosity. This is because the Reynolds stress is the consequence of the flow veloc-
ity fluctuations unlike the viscous stress which is due to molecular fluctuations. The
time and length scales of the Reynolds stresses are comparable with those of the flow
itself; therefore, there is strong statistical coupling of the turbulence stresses with the
mean fluid motion. Reynolds was the first to provide the decomposition of the flow
variables into the statistical mean and fluctuating components and to perform aver-
aging of the governing equations. The statistical averaging forms the basis of many
semi-empirical equations which have been developed to model the Reynolds stress
and to provide the so-called mathematical
closure
to the averaged governing equa-
tions of turbulence motion. These semi-empirical equations are called
turbulence
models
.
5.3.2
Introduction to Turbulence Modelling
When the governing equations for classical conservation laws are averaged, they
lead to the generation of new unknown stresses and fluxes that are a consequence
of turbulence fluctuations and the nonlinearity of the convective transport terms.
Turbulence modelling is referred to the inclusion of additional algebraic or transport
equations to augment the governing averaged equations (continuity, momentum, and
energy) to account for the Reynolds stresses as given by Eq. (5.28) and turbulence
fluxes. Turbulence modelling has a long history and many models have been derived
and improved over time by many researchers, based on experimental measurements
and physical and mathematical reasoning. There are a number of turbulence models
that range in complexity, which are summarised in this section.
The Reynolds Averaged Navier Stokes (RANS) based turbulence models are most
commonly used. They get their name from the Reynolds decomposition of the flow
variables into average and fluctuating components and from averaging the governing
equations. The RANS equations for flow and heat transfer in a two-dimensional
domain are given as
∂
u
∂x
+
¯
v
∂y
=
∂
¯
0
(5.24)
ν
∂
ν
∂
ν
∂
∂
u
∂t
+
¯
∂
(
u
)
∂x
u
¯
¯
∂
(
u
)
∂y
v
¯
¯
1
ρ
∂
p
∂x
+
¯
∂
∂x
u
∂x
¯
∂
∂y
u
∂y
¯
∂
∂x
u
∂x
¯
+
=−
+
+
∂
(
u
u
)
∂x
ν
∂
∂
∂y
v
∂x
¯
∂
(
u
v
)
∂y
+
−
+
(5.25)
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