Geoscience Reference
In-Depth Information
This says that following the fluid motion and neglecting molecular diffusion,
c
does
not change. We call such a scalar a
conserved
scalar.
†
In flows where heating due to radiation, phase change, chemical reactions, and
viscous effects is negligible the thermal energy equation reduces to the same form
as
Eq. (1.31)
,
∂
2
T
∂x
i
∂x
i
,
DT
Dt
=
∂T
∂t
+
u
i
∂T
∂x
i
=
α
(1.32)
where
T
is temperature and
α
is the thermal diffusivity of the fluid.
Equation (1.32)
says that under these conditions temperature is a conserved variable, changing only
through conduction heat transfer.
1.6 Key properties of turbulence
Equations (1.17)
,
(1.26)
,
(1.28)
,
(1.31)
,and
(1.32)
govern the evolution of the fluid
mass, velocity, vorticity, conserved scalar constituent, and temperature fields in a
constant-density, Newtonian fluid. Their turbulent solutions have properties that
distinguish them from other three-dimensional, time-dependent flow fields.
‡
1.6.1 Vortex stretching and tilting: viscous dissipation
Vortex stretching
is one of the mechanisms contained in the first term on the far right
of the vorticity
equation (1.28)
. To illustrate, let's consider a vortex with its axis in
the
x
1
direction, say, so the initial vorticity is
ω
i
=
(ω
1
,
0
,
0
)
.
Equation (1.28)
says
that ignoring viscous effects the vorticity initially evolves as
Dω
1
ω
1
∂u
1
Dω
2
ω
1
∂u
2
Dω
3
ω
1
∂u
3
Dt
=
∂x
1
,
Dt
=
∂x
1
,
Dt
=
∂x
1
.
(1.33)
If
∂u
1
/∂x
1
is positive
(1.33)
says the vortex is stretched in the
x
1
direction,
increasing the magnitude of
ω
1
.
∂u
2
/∂x
1
and
∂u
3
/∂x
1
can generate
ω
2
and
ω
3
from
ω
1
; this is sometimes called
vortex tilting
.
In two-dimensional turbulence the velocity field is
u
i
=[
u
1
(x, y), u
2
(x, y),
0
]
,
say. Then
ω
i
=
(
0
,
0
,ω
3
)
and the vortex-stretching term in
Eq. (1.28)
for
ω
3
is
ω
3
∂u
3
/∂x
3
=
0
.
This demonstrates that three dimensionality is necessary for
vortex stretching.
A cascade of kinetic energy through eddies of diminishing size (
Chapters 6
and
7
)thatterminatesin
viscous dissipation
- the conversion of kinetic energy
into internal energy by viscous forces in the smallest eddies - is a defining feature
†
Unfortunately, if the velocity divergence is nonzero the density of a mass-conserving species is not a conserved
scalar. The ratio of its density and the fluid density is a conserved scalar, however
(Part II)
.
‡
Parts of this discussion are adapted from
Lumley and Panofsky
(
1964
).
