Geoscience Reference
In-Depth Information
Today the principal purpose of averaging the turbulent-flow equations is to
enable their numerical solution, so we shall denote a spatially filtered variable
through a superscript “r”, this part of the variable being computationally
resolvable
.
We write
f
filt
f
filt
)
f
( f
=
f
r
+
f
s
,
=
+
−
(3.32)
so that spatial filtering decomposes a turbulent variable into
resolvable
(r) and
subfilter-scale
(s) parts. In general subsequent applications of a filter also have
effects;
†
that is,
=
f
r
.
(3.33)
From
Eq. (3.32)
it follows that in general the spatially filtered subfilter-scale field
does not vanish:
( f
r
)
r
( f
s
)
r
0
.
(3.34)
Unlike ensemble averaging, spatial filtering can preserve the turbulent character
of the flow variable if the spatial scale of the filter function
G
is much less than
the spatial scale of the turbulence.
‡
It can profoundly reduce the computational
requirements for numerical solution of the equations, as we shall see.
=
3.4.2 Spatially filtered governing equations
We can now apply the local average (interpreted as Leonard's more general spatial
filter,
Eq. (3.31)
) to the governing equations. With the decomposition of
Eq. (3.32)
the continuity
equation (3.1)
for the full velocity implies
∂ u
i
∂ u
i
∂
u
i
∂x
i
=
˜
∂
∂x
i
(
u
i
+˜
u
i
)
˜
=
∂x
i
+
∂x
i
=
0
.
(3.35)
Assuming that spatial filtering commutes with differentiation,
§
applying it to the
full continuity
equation (3.1)
yields
∂
r
∂ u
i
u
i
∂x
i
˜
=
∂x
i
=
0
.
(3.36)
Subtracting
Eq. (3.36)
from
Eq. (3.35)
then gives
u
i
∂
˜
∂x
i
=
0
.
(3.37)
u
i
and
u
i
are divergence free if
u
i
is.
Thus, both
†
The exception is the “top-hat” filter,
Chapter 6
.
‡
Equation (3.31)
shows that when
G
is vanishingly narrow - a “delta function” - it leaves the function unchanged.
§
Leonard
(
1974
) shows this is true if the function vanishes on the boundaries.
