Page 171 - Bird R.B. Transport phenomena

§5.1 Comparisons of Laminar and Turbulent Flows 155

Over the same range of Reynolds numbers the mass rate of flow and the pressure
drop are no longer proportional but are related approximately by
9 \7/4/ u l/47
4
5
- ® L « 0.198 w 7/4 (10 < Re < 10 ) (5.1-6)
The stronger dependence of pressure drop on mass flow rate for turbulent flow results
from the fact that more energy has to be supplied to maintain the violent eddy motion in
the fluid.
The laminar-turbulent transition in circular pipes normally occurs at a critical
Reynolds number of roughly 2100, although this number may be higher if extreme care is
2
taken to eliminate vibrations in the system. The transition from laminar flow to turbu-
lent flow can be demonstrated by the simple experiment originally performed by
Reynolds. One sets up a long transparent tube equipped with a device for injecting a
small amount of dye into the stream along the tube axis. When the flow is laminar, the
dye moves downstream as a straight, coherent filament. For turbulent flow, on the other
hand, the dye spreads quickly over the entire cross section, similarly to the motion of
particles in Fig. 2.0-1, because of the eddying motion (turbulent diffusion).

Noncircular Tubes
For developed laminar flow in the triangular duct shown in Fig. 3B.2(b), the fluid parti-
cles move rectilinearly in the z direction, parallel to the walls of the duct. By contrast, in
turbulent flow there is superposed on the time-smoothed flow in the z direction (the pri-
mary flow) a time-smoothed motion in the xy-p\ane (the secondary flow). The secondary
flow is much weaker than the primary flow and manifests itself as a set of six vortices
arranged in a symmetric pattern around the duct axis (see Fig. 5.1-2). Other noncircular
tubes also exhibit secondary flows.

Flat Plate
In §4.4 we found that for the laminar flow around a flat plate, wetted on both sides, the
solution of the boundary layer equations gave the drag force expression
2
F = 1328VpiJLLW vi (laminar) 0 < Re L < 5 X 10 5 (5.1-7)
in which Re L = Lv^p/fi is the Reynolds number for a plate of length L; the plate width is
W, and the approach velocity of the fluid is v K.

Fig. 5.1-2. Sketch showing the secondary flow patterns
for turbulent flow in a tube of triangular cross section
[H. Schlichting, Boundary-Layer Theory, McGraw-Hill,
New York, 7th edition (1979), p. 613].

2
O. Reynolds, Phil. Trans. Roy. Soc, \1\, Part III, 935-982 (1883). See also A. A. Draad and F. M. T.
Nieuwstadt, /. Fluid Mech., 361, 297-308 (1998).