BASIC FLUID MECHANICS BEHIND THE FORMATION OF HIGH SPEED WATER JETS

In the previous section the formation of the jet has been introduced. The jet is formed as the high pressure fluid exits through a small opening called an annulus into a region of less pressure. For the purpose here this region will be considered at atmospheric pressure. If a streamline is followed the writing of Bernoulli’s Equation is justified. Bernoulli’s Equation is basically a restatement of the work energy theorem. Ideally, the work done on a system goes into changing the energy of the system or

Work = Change in Energy .......(1)

In fluid mechanics this becomes

P₂ – P₁ = 1/2r ^. (v₂² – v₁²) + r g ^. (h₂ - h₁).....(2)

Where the left hand side, the change in pressure, is the work done on the system and the first term on the left is the change in kinetic energy and the second is the change in potential energy all normalized by a unit volume. More traditionally this is written as

.....(3)

where the left side is the individual terms outside the annulus of the jet nozzle and the right side is inside the annulus. Since the pressure inside the pipe is much greater than atmospheric and the values of h₂ and h₁ are the same and the velocity outside is much greater than the pipe velocity then from Eq 3

.....(4)

Where the subscript th represents the theoretical value of the jet exit velocity.

In practice the jet velocity is somewhat less that the theoretical value because wall friction, fluid flow disturbances, etc. Therefore the actual jet speed is some fraction m of the theoretical or

V₀ = m ^. V_0th.....(5)

Typically m is between 0.80 and 0.98 [1,2]

The jet velocity profile changes as the one gets further from the jet nozzle. The change would be thought to be a gradual decrease in jet velocity but this is not quite so because the jet consists of several different regions. Fig. 1 below gives a profile of the jet.

Fig.1 Development of the Jet after Exiting from the Nozzle.

The region immediately in front of the nozzle is known as the core zone which is conical in shape. At the nozzle it has the same diameter as the nozzle opening but as the distance along the centerline increases, the radius of the core zone decreases. In this zone the velocity profile, stagnation pressure and density are uniform. However this core zone lasts for only a distance x_c after which a region of nonuniform velocity profile is evident. This next region actually starts at the nozzle bordering the core zone and develops as air interacts with the jet. This next region is called the transition zone and the velocity profile develops a bell shape. The boundary of this region is the line separating the water jet from the air and the velocity is zero. However this boundary is not well defined. One can imagine an almost mist quality to this boundary so that it is difficult to tell where the fluid stops and the air begins. In the transition zone air is being sucked into the water flow, such that the density of the jet is continually decreasing. Despite this uncertainty in the boundary it is possible to talk about the velocity profile in the transition region. As the distance from the nozzle increase, the jet spreads. Consequently the width of the velocity bell shaped profile expands and its height decreases. Because the velocity profile is decreasing in height, the stagnation pressure also decreases. Thus, the further away from the jet nozzle, the smaller is the maximum magnitude of both jet velocity and the stagnation pressure. . The length of this transition zone is approximately 90 to 600 nozzle diameters [4]. The region after the transition zone is the furthest from the nozzle and is a mixture of air and liquid drops.

In the core of the jet as mentioned the velocity profile is uniform and the other properties of the jet are uniform. Because the velocity is uniform within the core zone, the kinetic energy with the core zone will also be a constant. The stagnation pressure is uniform and has the value of

P_stag= r ^. V₀² /2 .....(6)

Where r is the density of the abrasive water jet and V₀ is the exit velocity. The length of the jet core has been found by several researchers [5] to vary from 20 to 150 nozzle diameters and depends on the Reynolds-number of the fluid as well as the orifice geometry and quality.

In the transition zone, the centerline velocity, the stagnation pressure, and the density are decreasing functions of the distance from the nozzle. Since the velocity profile is bell shaped, the stagnation pressure has been modeled by several different functions, three of which are exponential are tabulated below in Table.1

Table 1Radial stagnation-pressure distribution functions

Function	Reference
	Yanaida [6] Davies and Jackson [7]
	Shavlovsky [4]
	Leach and Walker [8]
	Yahiro and Yoshida [9]
	Rehbinder [10]

a. Air Content [7]

b. Jet Diameter [6]

Fig.2 Air content and structure of a high speed water jet in the radial direction.

Fig.2 part a shows that for an axial distance of about 100 jet nozzle orifice diameters the jet is 100% water. After that the percentage of water decreases and the percentage of air in the jet increases. Part b shows that the jet diameter continually increases as a function along the jet axis.

In the transition zone the fluid density has the empirical form of

.....(7)

and can be seen to depend on the distance from the nozzle as well as the diameter of the nozzle [ 4 ]. From a simple momentum balance of

Momentum at the nozzle = Momentum at some value of x in the transition zone,

The value of the maximum jet velocity which would occur at the center line at a given value of x from the nozzle could be obtained for a unit flow length from

d₀² V_{0 r w}/4 = r _jet (x) V(x,r) rdr .....(8)

where V(x,r) would have a form similar to the pressure profile given in Table 3.1 of the form

V(x,r) = V_max(x) f(r) ......(9)

and the integration on the right hand side would have limits of zero to d_jet. The reader will also recognize Eq 8 to be a statement of continuity with no mass storage._. The jet diameter in the transition zone has been proposed [11] as a relation with respect to the centerline distance from the nozzle to be

....(10)

And by Nienhaus [ 12] to have the form

......(11)

As can be seen V_max(x) will decrease as the distance from the nozzle, x, increases. Therefore the kinetic energy of the jet will decrease as the distance from the nozzle increases.