Jökull - 01.12.1968, Side 27

where z0 is the equivalent sand roughness of
the bottom and x is von Kármán’s constant
(x ~ 0.4). vf is the friction velocity defined by
Vf = V—= V gOI (6)
v e
where x0 is the shear stress at the bottom, o
the specific mass (o = Y/g). g the acceleration
of gravity, D the depth and I is the slope of
the energy line. The equation (5) is not ac-
curate close to the surface and it does not hold
in the laminar layer at the bottom.
The diffusion coefficient KTz is of the same
order of magnitude as the corresponding coeffi-
cient for momentum transfer, KJf (the eddy
viscosity), which is defined by:
v r —________________K
’ Y •' V. ~~
3vx
3z“
the transfer of heat being similar to transfer
of momentum (Reynolds analogy). An expres-
sion for Km in our case is obtained as follows:
Km =
VX T/e
dv/dz dv/dz
x is the shear stress at height z:
z
T = T„ 1 —
and from (5)
With this we get
D
dv
: Vf2 6 1
dz xz
D
Km = «vfz f
D
(7)
The ratio KM/KT is the dimensionless turbu-
lent Prandtl number,
(8)
In turbulent flow the value of Pt is close to
unity at a boundary wall but decreases to ap-
proximately 0.5 far away from it (Schlichting
1960, p. 499).
A comparison between k/yc and KTz can
now be made for a practical case, say a river
with D = 2.0 m and I = 0.002. It is assumed
that Pt = 0.8. For this case the following values
of KTz are found:
z/D 0.95 0.75 0.50
KTz [m2 s-1] 0.0094 0.0372 0.0496
k/yc is a function of the water temperature:
T [°C] 0 10
k/yc [m2 s-1] 1.3 - 10-7 1.4 • 10-v
It is seen that in this case the heat exchange
by turbulent diffusion is about 100 000 times
greater than by conduction, except very close
to the surface and the bottom where KTz^> 0.
KTx can not be calculated as KTz. It must be
a function of z and probably ol the same order
of magnitude as KTz. For sufficiently high flow
velocities we have then
3
3x
3T 3T
---«v----
3x 3x
so that in many cases diffusion in the flow
direction can be neglected. Doing this and in-
serting the above expressions for v and KTz we
get a differential equation for the water temp-
erature in turbulent flow in a wide rectangular
channel:
3T
"aT
■+ Vf
3
3z
— ln — + 8.48) —
x z„ / 3x
k
--+
■ xv,z 1
3T
3z
YC »t
The boundary condition at the surface is
3T
(9)
'dz
— — S;
(10)
where S is the heat loss froin surface per unit
time and unit area. The boundary condition
at the bottom is similar and in most practical
cases we can neglect the heat exchange with
the bottom.
To find the dimensionless groups on which
the solution to (9) must depend we introduce
dimensionless quantities, denoted by star:
JÖKULL 18. ÁR 361