Geology Reference
In-Depth Information
Now take the power 3
/
2 on both sides, and rearrange a little:
8
κ
6
μ
gρ
0
aT
.
d
3
=
Now divide by
D
3
and tidy up:
d
D
3
48
κμ
gρ
0
αT D
3
.
The left-hand side is a ratio of lengths, and therefore dimensionless. This means
that the right-hand side must be dimensionless. The Rayleigh number is defined
as the inverse of this collection of quantities, leaving out the numerical factor. In
other words
=
gρ
0
aT D
3
κμ
Ra
≡
.
(5.19)
The Rayleigh number is generally symbolised as Ra, and the 'a' is not a subscript.
We'll encounter other numbers with two-letter symbols shortly. The triple-bar
equality means that this is a definition: the left-hand side equals the right-hand
side by definition. The Rayleigh number is a dimensionless quantity, as already
implied. Its full usefulness will become apparent after we define a couple of other
numbers. First, however, note that the previous expression can now be written in
the compact form
Ra
48
−
1
/
3
d
D
=
.
(5.20)
Now let's look at velocity again. The thermal diffusivity
κ
has dimensions of
length
2
/time. Therefore
κ/D
will have dimensions of velocity (length/time). We
will use this as a velocity scale:
V
=
κ/D
. Now you can use Eq. (5.18) to show,
with a little manipulation, that
Ra
6
2
/
3
v
V
=
,
(5.21)
so we can also write the convective velocity in very compact form.
Finally let's look at heat flow. If the thermal boundary layer has a thickness of
d
, then, from Fourier's 'law' (Eq. (5.10) in Section 5.4), the heat flux through the
boundary layer will be
q
=
KT/d.
This is also the total heat flux transported through the fluid, as it is the heat emerging
at the surface. If the heat had to be conducted through the fluid layer, in other words
