Transport Phenomena.pdf

§5.5 Turbulent Flow in Ducts 165
(b) For the flow between parallel plates, we can use the expression found in Eq. 5.3-12 for the
time-smoothed velocity profile to get the turbulent momentum flux:
dv x
(5.4-13)
where A = 4C(v*/v) 4 . This is in accord with Eq. 5.4-12.
35.5 TURBULENT FLOW IN DUCTS
We start this section with a short discussion of experimental measurements for turbulent
flow in rectangular ducts, in order to give some impressions about the Reynolds stresses.
In Figs. 5.5^ and 2 are shown some experimental measurements of the time-smoothed
quantities v, v' 2 x
, and v x
v z
for the flow in the z direction in a rectangular duct.
In Fig. 5.5-1 note that quite close to the wall, v v
'i is about 13% of the time-smoothed
centerline velocity v zmax
, whereas V^ is about 5%. This means that, near the wall, the
velocity fluctuations in the flow direction are appreciably greater than those in the transverse
direction. Near the center of the duct, the two fluctuation amplitudes are nearly
equal and we say that the turbulence is nearly isotropic there.
In Fig. 5.5-2 the turbulent shear stress T x
nz = pv' v' is compared with the total shear
x z
stress r V7
= Tyl + r ( i across the duct. It is evident that the turbulent contribution is the
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16
12
V гJp
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Fig. 5.5-1. Measurements of H. Reichardt
[Naturwissenschaften, 404 (1938), Zeits.
f. angew. Math. u. Mech., 13,177-180
(1933), 18,358-361 (1938)] for the turbulent
flow of air in a rectangular duct
with v zmax
= 100 cm/s. Here the quantities
Vv' x
v x
and y/v z
v z
are shown.
\
0 0.2 0.4 0.6 0.8 1.0
Fig. 5.5-2. Measurements of Reichardt
(see Fig. 5.5-1) for the quantity v x
v[ in a
rectangular duct. Note that this quantity
differs from T XZ
/p only near the
duct wall.