Chapter V flow through pipes. Hydrodynamic similarity

19. Flow through pipes

Two types of fluid flow through pipes are known from experience: laminar and turbulent.

Laminar flow occurs when a fluid moves in stratified layers, without mixing or velocity fluctuations. The shape of the streamlines is de­termined by the form of the channel through which the fluid is flowing. In laminar flow through a straight pipe of uniform cross-section all the streamlines are straight lines parallel to the pipe axis. There is no transverse displacement of fluid particles and the fluid does not mix. A piezometer tapped to a pipe in which laminar flow is taking place shows a constant pressure (and velocity) which does not change with time. Thus, laminar flow is uniform and, if the-head is constant, steady.

At the same time, however, laminar flow cannot be regarded as being completely eddiless. For, although there are no marked eddies in it, each fluid particle, besides being in translatory motion, is in. uniform rotational motion about its instantaneous centres, the angu­lar velocity being a real quantity.

In turbulent flow the fluid mixes and.velocity and pressure fluctuate. The stream lines are only approximately determined by the channel form. The motion of fluid particles is irregular and the path lines-are erratic curves. The reason for this is that in turbulent flow, besides mixing along the pipe, there occur mixing perpendicular to-the flow and rotational motions of small volumes of the fluid.

Laminar and turbulent flow can be observed with the help of a setup like the one shown schematically in Fig. 39. It consists of a tank A filled with water from which emerges a glass pipe В with a faucet C, and a smaller vessel D containing dye which can be injected into pipe В in a thin stream.

When the faticet is opened slightly the flow is slow; a filament of. the dye injected into the stream by opening valve E will produce a coloured streak which does not mix with the stream of water and is apparent along the whole length of the pipe. This indicates the strat­ified nature of the flow with no mixing: laminar flow.

When the velocity of the stream is gradually increased by further opening the faucet, the flow pattern first remains unchanged. Then, at a certain moment, a rapid change takes place. The streak of dye begins to oscillate and then diffuses into the water, eddies and rota­tional motion of the liquid being quite apparent. This means that turbulent flow has set in (inset of Fig. 39). When the velocity is re­duced the flow again becomes laminar.

The change of flow in any given pipe takes place at a certain critical velocity v_cr. Experiments show that the critical velocity is directly proportional to the kinematic viscosity v of the fluid and inversely proportional to the pipe diameter d:

The dimensionless coefficient of proportionality к was found to be the same for all liquids and gases and any pipe diameters.

This means that the change in the flow takes place at a certain ratio between velocity, pipe diameter and viscosity

This dimensionless number is called the critical Reynolds number, in honour of the English scientist Osborne Reynolds who presented it, and it is denoted:

(5.1)

Experimental data set the critical Reynolds number at about 2,300.

In addition to the critical Reynolds number corresponding to the change in flow regime, there is also introduced the actual Reynolds number for a specific stream, which is expressed in terms of its ac­tual velocity:

(5.2)

Thus, we have a criterion according to which we can determine whether the flow in a pipe will be laminar or turbulent. As laminar flow takes place at low velocities, it follows that at any value of Re < Re_cr the flow is laminar. At Re > Re_rr the flow is usually turbulent.

Knowing the velocity of flow, the pipe diameter and the viscos­ity of the liquid, it is possible to predict the type of motion. This is very important for hydraulic calculations.

In practice laminar flow occurs when very viscous liquids, such as lubricants and glycerine, flow through pipes.

Turbulent flow usually occurs in water mains and in pipes in which gasoline, kerosene, alcohols and acids are transported. Thus, in aircraft systems both laminar and turbulent flow is encountered: the former in oil and hydraulic transmission systems, the latter in fuel systems.

The change in the type of flow on reaching the critical Reynolds number is explained by the fact that the one type of flow becomes unsteady and the other'steady. At Re < Re_cr laminar flow is steady and any artificially induced turbulence (by shaking the pipe, sub­merging a vibrating body in the stream, etc.) is dampened by the viscosity of the fluid and laminar flow is restored as the turbulent regime in this case is unsteady. At Re > Re_cr, on the other hand, turbulent flow is steady and laminar flow is unsteady.

Actually, the critical Reynolds number corresponding to the changeover from laminar to turbulent flow is usually somewhat greater than Re_cr for the reverse change. In laboratory conditions, when all factors facilitating turbulence are removed, it is possible to sustain laminar flow at Reynolds numbers much higher than the critical. Laminar flow in such cases, however, is so unstable that the slightest jolt is sufficient to change it immediately into turbulent flow. In practice, notably in aircraft pipelines, conditions usually contribute to turbulence (vibrations of the pipes, local disturb­ances, irregular discharge, etc.), and, in hydraulics, such sustained laminar flow is rather of theoretical than practical interest.*

The theory of steady laminar flow and turbulence has not yet bееn developed. Experiments show, however, that in a pipe of given cross-section turbulence is facilitated by such factors as distance from walls, velocity and the velocity gradient dvldy across a section. The distance from the walls is greatest arid the velocity is highest along the centre of the stteam, the velocity gradient being zero. At the walls, on the other hand, the velocity gradient is largest, while the velocity and distance у vanish. Consequently, initial tur­bulence in laminar flow in a straight pipe of uniform cross-section begins somewhere between the centre line and the wall of the pipe, closer to the latter.