41. Unsteady flow through pipes

The general case of unsteady flow is fairly complicated, so we shall restrict ourselves to the main special cases which are found in aeronautical engineering: unsteady flow in a pipe of uniform cross-section and in compound pipes in series.

Take a pipe of length / and diameter d arbitrarily located in space (Fig. 106) and denote by z_x and z₂ the respective elevation heads of the initial (1-1) and terminal (2-2) cross-sections. Let a liquid be flowing through the pipe with an acceleration

which, in the general case, may vary with time.

The velocityv and the accelera­tion /, evidently, are the same at all cross-sections of the pipe at a given instant.

We neglect friction losses at first and assume the velocity distribu­tion across the sections to be uni­form.

Take a differential cylindrical control volume of length dl and cross-section dS (Fig. 106) and de­velop its equation of motion.

Projection of the pressure force and the gravity force on the tangent to the centre line of the pipe yields

or

As

then

(remembering also that p is a function not only of I but of t as well). Integrating along the pipe for a specific moment of time, we have

or

where

The equation resembles Bernoulli's equation for relative motion. The term h_in is also called the inertia head, not to be confused with ΔH_n, however, as their physical meanings are quite different. The quantity h_in, as is evident from Eq. ( 10.4), represents the difference between the specific energies of the liquid at sections 1-1 and 2-2 at a given time due to the acceleration (or retardation) of the flow in the pipe. The difference is positive when there is acceleration, i.e., the specific energy decreases along the stream; in the case of retardation it is negative, which means that the specific energy is increasing from the first section to the second.

By analogy with the Bernoulli equation, any energy losses (minor or friction) must be written down in the right-hand side of Eq. (10.4):

Remember that this equation is valid only for pipes of uniform cross-section. If a pipeline consists of a series of pipes of different cross-sectional areas, S_v S₂, etc., then, obviously, the inertia head along the whole of the pipeline must be found as the sum of the iner­tia heads of each portion. The respective accelerations in this case are found from the following equations, which are obtained by dif­ferentiating the continuity equations with respect to time:

Furthermore, it follows from the energy considerations cited be­fore that in this case the velocity heads must be considered at the initial and end sections of the pipe.

The equation of unsteady flow between sections 1-1 and n-n takes the form

This equation is used in computing the starting and transient re­gimes of aircraft hydraulic systems, notably fuel supply systems for liquid-propellant rocket motors.

The following exampleixvill illustrate this equation. Let the pis­ton in Fig. 107 be moving to the left with a positive acceleration j'. Applying Eq. (10.6) between sections 0-0 and 2-7, and then between sections 2-2 and 3-3, and constructing the hydraulic gradient for the moment of time under consideration, we have

and

Thus, in the first case the inertia head, which is added to the head loss, causes a still greater pressure drop at the piston than in the case of uniform motion. A vacuum forms at section 1-1 and the liq­uid may even separate from the piston. In the second case, addition of the heads h_f and h_tn causes a pressure increase at the piston.

When the acceleration / is negative, i. e., the flow is retarded, the inertia head is negative in both cases and, consequently, it compen­sates, to a greater or lesser extent, for the loss of head, reducing the vacuum on the one side and increasing the pressure on the other.

Example. Determine the absolute pressure at the intake of the pump of the aircraft lubricating system in the example to Chapter VI (see Fig. 51), if the aircraft dives with a negative £-load of n_g= - 1 (inertia force is directed upwards).

Solution. Tho unit inertia force is (see Sec. 11)

and the inertia head

From Bernoulli's equation for relative motion,

or

The pressure at the pump intake is too low, which means that steps must be taken to improve the performance of the lubricating system at high altitudes.