MOTION OF FLUIDS The motion of a fluid may be of two kinds, viz., stream line and turbulent. In stream line motion the filaments move in definite paths and the resistance to flow is due purely to the shear of adjacent layers and is directly proportional to the viscosity and to the velocity. In turbulent motion the water moves in an eddying mass and the motion at a given point varies in an ir regular manner from instant to instant. The resistance is only to a slight degree dependent on the viscosity and is proportional to th, nth power of the velocity where n is approximately equal to 2.
At very low velocities, the motion is usually streamline, but as the velocity is increased the motion breaks down and becomes turbulent. For any particular case there is some particular ve locity at which the change over from one type of motion to the other takes place, and this is known as the "critical velocity." Several conditions combine to determine whether the motion of a fluid shall be streamline or turbulent. Osborne Reynolds, who first investigated the two manners of motion, came to the conclusion that the conditions tending to the maintenance of streamline motion are: (I) an increase in the viscosity of the fluid ; (2) converging solid boundaries; (3) free (exposed to air) surfaces; (4) curvature of the path with the greatest velocity at the outside of the curve; (5) a reduced density of the fluid. The reverse of these conditions tends to produce turbulence. The effect of solid boundaries in causing turbulence appears to be due rather to their tangential than to their lateral stiff ness. One remarkable instance of this is shown by the effect of a film of oil on the surface of water exposed to the wind. The oil film exerts a very small but appreciable tangential constraint, with the result that the motion of the water below the film tends to become unstable. This results in the formation of eddies below the surface, and the energy, which is otherwise imparted by the action of the wind to form and maintain stable wave motion, is now absorbed in the institution of eddy motion, with the well known effect as to the stilling of the waves.
Where two streams of fluid are moving with different velocities the common surface of separation is in a very unstable condi tion. Generally speaking, wherever the velocity of flow is in creasing and the pressure diminishing, as where lines of flow are converging, there is an overwhelming tendency to stability of flow. In a tube with converging boundaries this effect is suffi ciently great to overcome the tendency to turbulent motion to which all solid boundaries, of whatever form, give rise, and the motion in such tubes is stable for very high velocities. On the other hand, the tendency to eddy formation is very great wher ever the lines of flow are diverging and the velocity is diminish ing in the direction of flow.
At low velocities this fluid is drawn out into a single colour band extending through the length of the tube. This appears to be motionless unless a slight movement of oscillation is given to the water in the supply tank when the colour band sways from side to side, but without losing its definition. As the velocity of flow is gradually increased, by opening the outlet valve, the colour band becomes more attenuated, still retaining its defini tion, until at a certain velocity eddies begin to be formed, at first intermittently, near the out let end of the tube (fig. 2) . As the velocity is still further in creased the point of eddy initia tion approaches the mouthpiece, and finally the motion becomes sinuous throughout. The appar ent lesser tendency to eddy for mation near the inlet end of the tube is due to the stabilizing in fluence of the convergent mouth piece. The velocity at which eddy formation is first noted in such experiments is termed the "higher critical velocity." There is also a "lower critical velocity," at which the eddies in originally turbulent flow die out, and this is, strictly speaking, the true critical velocity. It has a much more definite value than the higher critical velocity, which is extremely sensitive to any disturbance, either of the fluid before entering the tube, or at the entrance. Over the range of velocities between the two critical values, the fluid, if mov ing with streamline flow, is in an essentially unstable state, and the slightest disturbance may cause it to break down into turbulent motion.
The determination of the lower critical velocity is not possible by the colour band method, and Reynolds took advantage of the fact that the law of resistance changes at the critical velocity, to determine the values by measuring the loss of head accom panying different velocities of flow in pipes of different diameters. On plotting a curve showing velocities and losses of head, it is found that up to a certain velocity the points lie on a straight line passing through the origin of co-ordinates. Following this there is a range of velocities over which the plotted points are very irregular, indi cating general instability, while for still greater velocities the points lie on a smooth curve, indicating that the loss of head is possibly proportional to v..
To test this, and if so to determine the value of n, the logarithms of the loss of head h and of the velocity were plotted (fig. 3). Then if the equation to a straight line inclined at an angle of n to the axis of logy, and cutting off an intercept logk on the axis of logh. On doing this it is found that with motion initially unsteady the plotted points lie on a straight line up to a certain point A, the value of n for this portion of the range being unity. At A, which marks the lower critical velocity, the law suddenly changes and h increases rapidly. There is, however, no definite relationship between h and v until the point B is reached. Above this point the relationship again becomes definite, and within the limits of experimental error, over a fairly large range of velocities, the plotted points lie on a straight line whose inclination varies with the roughness of the pipe walls. The values of n determined in this way by Reynolds are: those for cast iron being deduced from experiments by Darcy.
Between A and B the value of n is greater than between B and C, and the increased resistance accompanying a given change in velocity is greater even than when the motion is entirely turbulent. This is due to the fact that within this range of velocities eddies are being initiated in the tube, and the loss of head is due not only to the maintenance of a more or less uniform eddy regime, but also to the energy absorbed in the initiation of eddy motion.
As a result of his experiments, Reynolds concluded that the critical velocity vk is inversely proportional to the diameter d of the pipe, and is given by the formula where b is a numerical constant, and where P is proportional to the viscosity divided by the density, or µ/p. If the unit of length is the foot, b equals 25.8 for the lower critical velocity, and 4.06 for the higher critical velocity; while if t= temperature in degrees Centigrade, More recent experiments show that by taking the greatest care to eliminate all disturbance at entry to the tube, values of the higher critical velocity considerably greater than (up to 3.66 times as great as) those given by the above formula may be obtained. The probability is, in fact, that there is no definite higher critical velocity, but that this always increases with decreasing dis turbances.
A general expression for the lower critical velocity in a parallel pipe, applicable to any fluid and any system of units, is 2000/2 _ vkdP Thus for water at µ/p= 1.92 X in foot-pound second units, so that °3g4 Vk = f t./sec., where d is in feet.
d Bernoulli's Theorem.—Water in motion possesses energy in virtue of its velocity, its pressure and its elevation. Thus water in motion with velocity v f.s. has kinetic energy ft.lb. per pound. Its pressure energy is p±w ft.lb. per pound where p is its pressure in pounds per square foot, and w its weight per cubic foot, and its potential energy is z ft.lb. per pound where z is its height in feet above datum level. Each of these expressions is equivalent to a height or head in feet. Thus is the height through which a body falling freely would attain a velocity v, while p--w is height of a column of water which would produce the pressure p at its base. p- w is therefore called the pressure head.
The total energy per lb. is equal to - + +z ft.lb.
The Venturi meter (fig. 4) consists essentially of such a con verging pipe, which is extended beyond the throat to its original diameter. The meter is usually constructed with an upstream cone having an angle of convergence of about 20°, connected to a downstream cone whose angle of divergence is about 5° 30', by easy curves. One annular chamber surrounds the entrance to the meter, and a second surrounds the throat, the mean pressures in the pipe at these sections being transmitted to the chambers through a series of small holes in the wall of the pipe. The cham bers are connected to the two limbs of a differential pressure gauge which records their difference of pressure h in feet of water. For this purpose a U-tube containing mercury may be used as in fig. 4. In this case if the connecting pipes are full of water it may readily be shown that the difference of pressure in feet of water is equal to 12.59 times the difference of level of the tops of the mercury columns. By using an inverted U-tube, with compressed air sup plied to the highest portion of the tube, the difference of pressure may be directly recorded in feet of water. Actually, owing to fric tional losses the discharge is slightly less than is indicated by for mula (I), and is given by where C varies from about .96 to •99, increasing slightly with the size of meter. The ratio A :a is usually between 4:i and 9:1, depending on the range of discharges to be measured.
Change of Pressure Along a Radius in Curvilinear Mo tion.—If water be moving in a curved path the pressure along the radius of curvature varies.
This change in pressure may be determined by considering the equilibrium of an elementary column of fluid of sectional area 8 a, having its axis radial, and its two ends in regions where the pressures are p and (p-}-S p) re spectively. The centrifugal force on the column is balanced by the difference of pressure on the two ends and we have, for equilib rium, Vortex Motion—Pressure in a notating Liquid.—If a mass of liquid has a rotary motion about some axis, it is termed a Vortex. Such a vortex may be either of two types—forced or free.
An example of the first is seen when a vessel containing water is rotated for a sufficient time for the water to adopt the motion of the vessel. The second is seen when water flows freely through a hole in the bottom of a vessel. Here, the water moves spirally towards the centre with streamline motion, so that, neglecting viscosity, its energy per unit mass is everywhere the same. This is termed a "free spiral vortex." Forced Vortex Motion with Uniform Angular Velocity. —Since the angular velocity co is constant, we have at any radius r, v = co r. The increase in pressure radially is given by which is the equation to a parabola.
Since the pressure at any point in the fluid is that equivalent to the column of water supported at the point, it follows that But p1=po=o both being atmospheric (datum) pressures, and if the area of the tank is large compared with that of the orifice, .•. \/2g = V2gh where h is the difference of level between vena contracta and the free surface in the tank.
Allowance for the small loss of energy up to the point of dis charge is made by writing 2gh where the coefficient of velocity, usually has a value of about .975.
If a and are the areas of the orifice and of the vena contracta, the ratio is termed the "coefficient of contraction," and the discharge is given by V 2gh = Ca V 2gh, where C is termed the "coefficient of discharge." The values of and of C vary with the type of orifice, with its situation, and to a small degree with its size. For a small sharp edged circular orifice remote from the sides of the tank, C is approximately 0.62. Its extreme values are about 0.52 and 0•99. The former is obtained with a sharp-edged re-entrant mouth piece and the latter with a bell-mouthed orifice.