CHAPTER 3
FLUID-FLOW THEORY (cont'd)

STREAMLINE AND TURBULENT FLOW

When a liquid flowing in a pipe is observed carefully, it will be seen that the pattern of flow becomes more disturbed as the velocity of flow increases. Perhaps this phenomenon is more commonly seen in a river or stream. When the flow is slow the pattern is smooth, but when the flow is more rapid, eddies develop and swirl in all directions and at all angles to the general line of flow.

At the low velocities, flow is calm. In a series of experiments, Reynolds showed this by injecting a thin stream of dye into the fluid and finding that it ran in a smooth stream in the direction of the flow. As the velocity of flow increased, he found that the smooth line of dye was broken up until finally, at high velocities, the dye was rapidly mixed into the disturbed flow of the surrounding fluid.
From analysis, which was based on these observations, Reynolds concluded that this instability of flow could be predicted in terms of the relative magnitudes of the velocity and the viscous forces that act on the fluid. In fact the instability which leads to disturbed, or what is called "turbulent" flow, is governed by the ratio of the kinetic and the viscous forces in the fluid stream. The kinetic (inertial) forces tend to maintain the flow in its general direction but as they increase so does instability, whereas the viscous forces tend to retard this motion and to preserve order and reduce eddies..

rv²D/mv = Dvr/m

This ratio is very important in the study of fluid flow. As it is a ratio, it is dimensionless and so it is numerically independent of the units of measurement so long as these are consistent. It is called the Reynolds number and is denoted by the symbol (Re).

From a host of experimental measurements on fluid flow in pipes, it has been found that the flow remains calm or "streamline" for values of the Reynolds number up to about 2100. For values above 4000 the flow has been found to be turbulent. Between above 2100 and about 4000 the flow pattern is unstable; any slight disturbance tends to upset the pattern but if there is no disturbance, streamline flow can be maintained in this region.

To summarise for flow in pipes:

For (Re)  < 2100 streamline flow,
For 2100 < (Re) < 4000 transition,
For (Re)  > 4000 turbulent flow.

EXAMPLE 3.9. Flow of milk in a pipe
Milk is flowing at 0.12 m³ min^-1 in a 2.5-cm diameter pipe. If the temperature of the milk is 21°C, is the flow turbulent or streamline?

Viscosity of milk at 21°C     = 2.1 cP = 2.10 x 10^-3 N s m^-2
Density of milk at 21°C        = 1029 kg m^-3.
Diameter of pipe                  = 0.025 m.
Cross-sectional area of pipe = (p/4)D²
                                         = p/4 x (0.025)²
                                         = 4.9 x 10^-4 m²
Rate of flow                        = 0.12 m³ min^{-1  = (0.12/60)} m³ s^{-1 = 2 x 10} m³ s^-1

So velocity of flow               = (2 x 10^-3)/(4.9 x 10^-4)
                                         = 4.1 m s^-1,
and so                        (Re) = (Dvr/m)
                                         = 0.025 x 4.1 x 1029/(2.1 x 10^-3)
                                         = 50,230
and this is greater than 4000 so that the flow is turbulent.

As (Re) is a dimensionless ratio, its numerical value will be the same whatever consistent units are used.
However, it is important that consistent units be used throughout, for example the SI system of units as are used in this book. If; for example, cm were used instead of m just in the diameter (or length) term only, then the value of (Re) so calculated would be greater by a factor of 100. This would make nonsense of any deductions from a particular numerical value of (Re).
On the other hand, if all of the length terms in (Re), and this includes not only D but also v (m s^-1), r
(kg m^-3) and m (N s m^-2), are in cm then the correct value of (Re) will be obtained. It is convenient, but not necessary to have one system of units such as SI. It is necessary, however, to be consistent throughout.


Fluid-flow theory > ENERGY LOSSES IN FLOW

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