Biomedical Engineering Reference
In-Depth Information
ν
∂
ν
∂
ν
∂
∂
v
∂t
+
¯
∂
(
v
)
∂x
u
¯
¯
∂
(
v
)
∂y
¯
v
¯
1
ρ
∂
p
∂y
+
¯
∂
∂x
v
∂x
¯
∂
∂y
v
∂y
¯
∂
∂x
u
∂y
¯
+
=−
+
+
∂
(
u
v
)
∂x
ν
∂
∂
∂y
v
∂y
¯
∂
(
v
v
)
∂y
+
−
+
(5.26)
k
ρC
p
k
ρC
p
∂u
T
∂x
∂v
T
∂y
∂T
∂t
+
∂
(
uT
)
∂x
¯
∂
(
vT
)
∂y
¯
∂
∂x
∂T
∂x
∂
∂y
∂T
∂y
+
=
+
−
+
(5.27)
p
,
T
are mean values and
u
,
v
,
p
,
T
are turbulence fluctuations. The
term
k
/
ρ C
p
is the thermal diffusivity of the fluid. The equations listed above are
similar to those f
or lamina
r flo
ws,
except for the presence of additional unknown
terms of the form
u
u
,
u
v
, and
v
v
. That is, we have three additional unknowns (or
six additional unknowns in three dimensions), known as the Reynolds stresses, in
the time-averaged momentum e
quati
ons.
Simi
larly, the time-averaged temperature
equa
tion shows the extra terms
u
T
and
v
T
(in three dimensions, we also have
w
T
). The earliest and simplest turbulence model was introduced by Boussinesq
(known as the
Boussinesq assumption
) who suggested that the Reynolds stresses can
be related to the mean rates of deformation similar to the way viscous shear stress is
related to the deformation rate. That is,
where
¯
u
,
v
,
¯
¯
Laminar shear stress:
Turbulent shear stress:
μ
t
du
dy
(5.28)
μ
d
dy
·
τ
=
τ
turb
=−
ρu
v
≈
Equation (5.28) assumes that the turbulent momentum transport is proportional to the
mean gradients of velocity. Here
μ
t
is the turbulence or eddy viscosity because the
turbulence mixing is hypothesised to be the consequence of the motion of turbulence
eddies that transport momentum.
Similarly the turbulent transport of temperature is taken to be proportional to the
gradient of the mean temperature (the transported quantity). In another words,
−
ρv
T
=
T
∂T
∂y
(5.29)
where
T
is the turbulence diffusivity. Since the turbulent transport of momentum
and heat is due to the same mechanisms of “eddy mixing”, the value of the turbulence
heat diffusivity can be taken to be close to that of turbulence (eddy) viscosity
μ
T
.
The corresponding turbulence Prandtl number,
Pr
T
, is defined as
μ
T
T
Pr
T
=
(5.30)
Experimental data suggested that this ratio is often nearly constant and around unity.
Most CFD models assume this to be the case and use values of
Pr
T
around unity.
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