36. Suppressed contraction. Submerged jet

Suppressed contraction occurs when efflux through an orifice is affected by the proximity of the walls of a reservoir. When the ori­fice is concentric with the centre line of the reservoir (Fig. 85) the walls tend to guide the liquid approaching the orifice, thus prevent­ing full contraction beyond the orifice. The contraction is less than in efflux from a reservoir of infinite size; the coefficient of contrac­tion increases, and as a consequence, the coefficient of discharge increases as well.

A theoretical investigation of the efflux of an ideal fluid from a flat reservoir of finite height and infinite width through a slot in the end wall was carried out by N. E. Joukowski as far back as 1890.

For a low-viscous liquid flowing through a round orifice in the centre of the end wall of a circular cylindrical reservoir the coeffi­cient of contraction can be found in terms of the contraction coef­ficient for complete contraction from the following empirical for­mula:

(9.9)

where is the ratio of the orifice area to the cross-sectional area of the tank.

When contraction. of the jet is suppressed the loss coefficient ζand the velocity coefficient φ can be regarded as being independent of n (if, of course, n is not too close to unity); for low-viscous fluids the approximate values are

and

The discharge coefficient is easily found from the relationship

and the rate of discharge is given by the formula

In applying this equation to the case of suppressed contraction it should be remembered that the rated head is the total head

This means that besides the piezometric head the velocity head in the reservoir must also be taken into consideration. As the velocity head is usually unknown, it is desirable to have a formula expressing the rate of discharge for suppressed contraction not in terms of the total head H but in terms of the piezometric head.

Such a formula is easily obtained from Bernoulli's equation and the continuity equation written between sections 1-1 and 2-2 (see Fig. 85):

From this

and

(9.10)

where

(9.11)

If an orifice is close to a side wall of a tank the contraction is sup­pressed over a part of the perimeter of the jet as shown in Fig. 86. In this case, too, the contraction and discharge coefficients ε₂ and ε₂ are greater than for unsuppressed contraction.

The coefficient μ₂ can be found from the empirical formula

(9.12)

where ζ = form coefficient, equal to 0.128 for circular and 0.152 for square orifices;

П = perimeter of orifice;

ΔП = fraction of the perimeter adjoining the wall.

Many engineering problems are concerned with discharge of a liquid below the surface of the same liquid in another vessel (Fig. 87). This is known as a submerged jet.

The total kinetic energy of the jet is dissipated in eddy formation, as in the case of an abrupt expansion. The Bernoulli equation be­tween sections 7-7 and 3-3 (where the velocities can be assumed zero) takes the form

or

where H = rated head;

v = velocity at the vena contracta;

ξ = loss coefficient of the orifice, which is approximately the same as in the case of a free jet.

Hence,

and

Thus, the equations are the same as for a free jot, only in the pres­ent case the head H is the difference between the piezoraetric heads on both sides of the wall.

The coefficients of contraction and discharge of a submerged jet can be assumed the same as of a free jet.