34. Local features in aircraft hydraulic systems

Local features in aircraft hydraulic systems and transmissions include filters, taps, valves, bends and other fittings and units of diverse geometrical shapes. Flow through such features may be either laminar or turbulent, depending on the velocity and temperature (viscosity) of the fluid. The Reynolds number varies within a fairly wide range and includes Re_cr. Accordingly, the loss coefficient ζ for these features must be regarded as functions of Re.

The logarithmic scale in Fig. 81 presents plots of ζ as a function of Re for some typical local features in aircraft hydraulic systems obtained experimentally by the Soviet engineer N. V, Levkoyeva, The values of Re and ζ were computed according to the velocity of the stream in the pipe and the diameter of the latter.

Note that the dependence of £ on Re in a felt filter is a linear func­tion up to Re = 5,000. The reason is that in laminar flow through the filter pores friction is very great and practically no turbulence takes place. In the plots for a tap, valve and bend the linear sections are much shorter; they are followed by considerable tran­sient sections and, finally, a quadratic resistance law (ζ = const).

The steeper slope of the nonreturn valve curve is ac­counted for by the fact that with Re increasing the open­ing of the valve increases (owing to faster flow through it), i. e., the geometric charac­teristics change.

The Reynolds' number is usually much higher in air­craft fuel systems than in hy­draulic systems and it can be assumed, without much error, that the local loss coefficients of fuel pipes do not depend on Re.

Table 4 presents loss coeffi­cients for some common units and local features of fuel systems. The values of ζ are referred to the velocity head at the intake pipe of the unit or feature.

Table 4

Local feature	ζ
Flexible pipe connection Standard 90° elbow (bored body) Tee Fuel tap Nonreturn valve Gauze filter Flowmeter sensor with impeller rotating with impeller braked Pipe entrance (tank outlet) Pipe outlet (tank inlet)	0.3 1.2-1.3 3.5 1-2.5 2.0 1.5-2.5 7.0 11-12 0.5-1.0 1.0

Chapter IX flow through orifices, tubes and nozzles

35. Sharp-edged orifice in thin wall

In this chapter we shall examine various cases of fluid efflux from reservoirs, tanks or boilers through orifices, tubes and nozzles of dif­ferent kind into the atmosphere or a vessel filled with a gas or the same fluid. Characteristic of this case of fluid motion is that in the course of efflux the overall potential energy of a liquid (or at least most of it) is converted, with greater or lesser losses, into the kinetic energy of a free jet or a falling liquid.

In aircraft engineering flow through orifices and mouthpieces must be investigated in connection with problems of fuel supply to combustion chambers of gas turbine and liquid-propellant rocket engines, shock-absorber operation and flow through jets in fuel and other aircraft systems. The question of primary interest to us is that of determining the velocity and rate of flow of liquids through orifices and tubes of different shape.

Consider a large tank filled with a liquid under pressure p₀, with a small orifice in the wall at a fairly large depth H₉ from the free surface (Fig. 82). The liquid flows out of the orifice into the atmos­phere, the pressure at section 1-1 being p_x.

Let the orifice be shaped as in Fig. 83a (a hole with a square shoul­der in a thin wall) or a^in Fig. 836 (a hole with a sharp edge in a thick wall). The conditions of flow through both orifices are identical: the liquid particles approach the aper­ture from all sides of the adjacent volume, moving with acceleration along converging streamlines as shown in Fig. 83a; the jet separates from the wall at the edge of the orifice and converges Into what is called the vena contracta. At about one orifice diameter away from the edge the streamlines become parallel. The vena contracta is caused by the necessarily smooth change in the direction of motion of the particles approaching the orifice (including those approaching it at right angles along the tank wall).

In the case of a small orifice as compared with the size of the reser­voir and the head H_o, when the free surface and walls do not affect the approach of the liquid to the aperture, the jet contraction is complete (suppressed jet contraction will be examined later on).

The degree of contraction is characterised by the coefficient of contraction e, which is the ratio of the area of the vena contracta to the area of the orifice:

(9.1)

Let Bernoulli's equation be written between the free surface of the reservoir (section 0-0 in Fig. 82), where the pressure is p₀ and the velocity can be assumed zero, and section 1-1 where the jet is cylin­drical and the pressure is p_x. Assuming the velocity distribution in the jet to be uniform,

where ζ = loss coefficient of the orifice. Introducing the notation

for the rated head, we obtain

whence the velocity of efflux is

(9.2)

where

(9.3)

is the so-called coefficient of velocity.

In the case of an ideal liquid ζ = 0, and consequently φ = 1, whence the theoretical, or ideal, velocity of efflux is

(9.4)

An investigation of Eq. (9.2) reveals that the coefficient of velocity φ is the ratio of the actual to the theoretical velocity:

(9.5)

Because of friction, the actual velocity of efflux is always less than the theoretical velocity, and the coefficient of velocity is less than unity.

Velocity distribution, it should be noted, is uniform only in the central portion of a jet; the external layer is retarded by friction (see Fig. 83fc). Experiments show that the velocity in the central portion of a jet is practically equal to the theoretical velocity , therefore the velocity coefficient φ refers actually to the mean veloc­ity. When efflux is into the atmosphere, the pressure across the diam­eter of the jet is atmospheric, which is confirmed by experiments.

Let us compute the rate of discharge through an orifice as the prod­uct of the actual velocity and the actual area of the jet. Application of Eqs (9.1) and (9.2) yields:

(9.6)

The product of the two coefficients

μ =eφ

is called the coefficient of discharge.

Thus, Eq. (9.6) can be rewritten

(9.6')

where p = rated pressure under which efflux is taking place.

The expression (9.6') is fundamental as it solves the basic problem of determining the rate of discharge through an orifice and it is valid for all types of efflux. The only difficulty of using it is the accurate determination of the discharge coefficient μ.

It follows from Eq. (9.6') that

i. e. the coefficient of discharge is the ratio of the actual rate of dis­charge Q to the theoretical rate of discharge Q_t (discharge without friction or contraction of the jet). The term "theoretical discharge", incidentally, should not be understood absolutely, as is not the rate of discharge of an ideal liquid: an ideal liquid would also display a vena contracta.

Due to contraction of the jet and friction, the actual rate of flow is always less than the theoretical discharge and the coefficient of discharge is always less than unity. Whether contraction or friction is the governing factor will depend on conditions of efflux.

The coefficients of contraction ε, loss ζ, velocity φ and discharge μ depend, first of all, on the type of orifice or tube and, like all dimen-sionless hydraulic coefficients, on the Reynolds number, the basic criterion of hydrodynamic similarity.

Fig. 84 presents a diagram in which the coefficients φ, ε and μ for a circular orifice are plotted as functions of the Reynolds number computed for the theoretical velocity:

The graph shows that, with increasing (i. e., when viscosityceases to be of importance) φ increases thanks to ζ decreasing, while e decreases as a result of less velocity retardation at the edge of the orifice and greater permissible radii of curvature at the vena con-tracta. Both (φ and ε asymptotically approach their values correspond­ing to an ideal liquid, i. e., when and

The coefficient of discharge, which is determined by the product of ε and φ, first increases with increasing due toφ increasing. On reaching a maximum fa_max'= 0.69 at ) it falls due to ε rapidly decreasing, and at high values of Re, when μ = 0.59-0.60, it becomes practically constant.

When is very low ( < 25) the effects of viscosity come into play and the retardation of the velocity of efflux at the orifice edge becomes so great that there is no contraction (ε = 1) and φ = μ. The rate of discharge Q in this regime of flow is proportional to the first power of the head, and the coefficient of discharge is proportional to .For this regime the following theoretical formula, con­firmed by experiment, can be used:

(9.7)

to which corresponds

(9.8)

The efflux coefficients for low-viscous liquids (water, gasoline, kerosene) through circular orifices in thin walls vary insignificantly and in engineering problems the following mean values are commonly taken:

ε = 0.63; φ = 0.97; μ = 0.61; ζ = 0.065.

In the case of flow of low-viscous liquids through a round orifice in a thin wall the jet contraction is considerable and the resistance is small; the coefficient of discharge is much less than unity, mainly due to jet contraction.