Biomedical Engineering Reference
In-Depth Information
Fig. 5.13
Laminar flow over a cylinder showing the wake region that oscillates periodically over
time
The purpose of turbulence modelling is to account for the averaged flow equa-
tions with the inclusion of a fluctuating component. The continuity and momentum
equations are known as the Navier-Stokes equations (named after famous math-
ematicians) and its averaged form is called the Reynolds Averaged Navier-Stokes
(RANS). This produces a set of time averaged equations with the turbulent features
encapsulated by a new term called the Reynolds stress. It is this term that research
effort and development in turbulence modelling aims to describe and resolve.
The RANS form of the continuity and momentum equations in 2D are
∂
∂
u
x
+
∂
∂
v
y
=
0
(5.20)
(
)
(
)
∂
uu
∂
vu
∂
u
1
∂∂∂ ∂∂ ∂∂
p
u
u
u
+
+
=−
+
ν
+
ν
+
ν
∂ ∂
t
x
∂
y
ρ
∂ ∂ ∂ ∂∂ ∂ ∂
xx x y y x x
(
)
(
)
∂
uu
′′
∂
uv
′′
(5.21)
∂∂
v
+
ν
−
+
∂∂ ∂ ∂
y
x
x
y
(
)
(
)
∂
uv
∂
vv
∂
v
1
∂∂∂ ∂∂ ∂∂
p
v
v
u
+
+
=−
+
ν
+
ν
+
ν
∂ ∂
t
x
∂
y
ρ
∂ ∂ ∂ ∂∂ ∂ ∂
yx x y y x y
(
)
(
)
(5.22)
∂
uv
′′
∂
vv
′′
∂∂
v
+
ν
−
+
∂∂ ∂ ∂
y
y
x
y
where
u
,
v
,
p
are mean values and
uʹ
,
vʹ
,
pʹ
are turbulent fluctuations. The equations
ab
ove
ar
e s
imilar
to
those formulated for laminar flows, except for the presence
of
′′
uu
,
′′
uv
, and
′′
vv
terms. As a result, we have three additional unknowns (in
three dimensions, we will have nine additional unknowns), known as the Reynolds
stresses, in the time-averaged momentum equations. How the Reynolds stresses is
resolved is the key feature of all RANS-based turbulence models.
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