Jökull - 01.12.1968, Side 26

(3)
3 (•
— yc T dR
3t J K
and this must equal the heat added through
conduction and advection:
k div grad T — yc v • grad 1’ — yc div (T'v')
3T
= YC -
3t
/,
k grad T • dA
v • dA
3
3t
YC T dR
The term yc div (T'v') is the turbulent dif-
fusion and it must be connected to the time-
averages of T and v in order that we can
handle it. In technical fluid dynamics it is
assumed that
The instantaneous values of T and v can be
written down as sums of time-averages and
fluctuations:
T = T + T'; v = v + v'.
The time-averages of the fluctuations are equal
to zero:
TV = -ktx-
3x
and corresponding for the other components.
It must be emphasized that the diffusion coeffi-
cients, Kt, are not constant like the conduc-
tivity. With this and writing out in coordinates
and rearranging we get:
T' =0; v' = 0.
Inserting this and taking time-averages we get:
f k grad T • dA — f yc T v • dA
J A *Z A
_ J yc TV' • dA = f yc -J- dR (1)
/,
3t
The surface integrals can be transformed to
volume integrals:
| k div grad T dR — f yc div (T v) dR
•z r R
- f yc div (TÝ) dR = f yc dR. (2)
J E J R 3t
By making use of the continuity equation for
an incompressible fluid,
div v = 0,
we have
div (T v) = T div v + v • grad T = v • grad T;
'3T 3T 3T 3T
'iT + Vx "3^ + Vy + Vz
3 / k \ 3T
3x Uc + Tx/ 3x
+
3y \ yc
k '*-)?
3T
3z
3 / k
+ , ' ( + KTz
dz \ yc
(4)
Here the bars have been omitted as all values
are time-averages.
This equation will now be studied closer for
the simplest case: A broad rectangular channel
where the temperature is uniform transversely
across the channel. With x in the flow direct-
ion and z as the ordinate (z = 0 at the bottom)
we have 3T/3y = 0, vy = vz = 0 and vx = v(z).
It is further assumed that the velocity field is
independent of the temperature field; this is
bclieved to hold in swift rivers.
For fully developed turbulence the velocity
distribution in a broad channel is given by the
well-known equation (Prandtl-Nikuradse (Brett-
ing 1960)):
and since equation (2) must hold for any
volume the integrands must vanish:
360 JÖKULL 18. ÁR