Page:EB1911 - Volume 14.djvu/57

The velocity would be infinite at radius 0, if the current could be conceived to extend to the axis. Now, if the motion is steady,

H = p₁/G + v₁²/2g = p₂/G + v₂²/2g;

= p₂/G + r₁² + v₁²/r₂²2g;

(p₂ − p₁)/G = v₁² (1 − r₁²/r₂²)/2g;

(5)

p₂/G = H − r₁²v₁²/r₂²2g.

(6)

Hence the pressure increases from the interior outwards, in a way indicated by the pressure columns in fig. 36, the curve through the free surfaces of the pressure columns being, in a radial section, the quasi-hyperbola of the form xy² = c³. This curve is asymptotic to a horizontal line, H ft. above the line from which the pressures are measured, and to the axis of the current.

Fig. 36.

Free Circular Vortex.—A free circular vortex is a revolving mass of water, in which the stream lines are concentric circles, and in which the total head for each stream line is the same. Hence, if by any slow radial motion portions of the water strayed from one stream line to another, they would take freely the velocities proper to their new positions under the action of the existing fluid pressures only.

For such a current, the motion being horizontal, we have for all the circular elementary streams

H = p/G + v²/2g = constant;

∴ dH = dp/G + vdv/g = 0.

(7)

Consider two stream lines at radii r and r + dr (fig. 36). Then in (2), § 33, ρ = r and ds = dr,

v² dr/gr + vdv/g = 0,

dv/v = −dr/r,

v ∞ 1/r,

(8)

precisely as in a radiating current; and hence the distribution of pressure is the same, and formulae 5 and 6 are applicable to this case.

Free Spiral Vortex.—As in a radiating and circular current the equations of motion are the same, they will also apply to a vortex in which the motion is compounded of these motions in any proportions, provided the radial component of the motion varies inversely as the radius as in a radial current, and the tangential component varies inversely as the radius as in a free vortex. Then the whole velocity at any point will be inversely proportional to the radius of the point, and the fluid will describe stream lines having a constant inclination to the radius drawn to the axis of the current. That is, the stream lines will be logarithmic spirals. When water is delivered from the circumference of a centrifugal pump or turbine into a chamber, it forms a free vortex of this kind. The water flows spirally outwards, its velocity diminishing and its pressure increasing according to the law stated above, and the head along each spiral stream line is constant.

§ 35. Forced Vortex.—If the law of motion in a rotating current is different from that in a free vortex, some force must be applied to cause the variation of velocity. The simplest case is that of a rotating current in which all the particles have equal angular velocity, as for instance when they are driven round by radiating paddles revolving uniformly. Then in equation (2), § 33, considering two circular stream lines of radii r and r + dr (fig. 37), we have ρ = r, ds = dr. If the angular velocity is α, then v = αr and dv = αdr. Hence

dH = α²rdr/g + α²rdr/g = 2α²rdr/g.

Comparing this with (1), § 33, and putting dz = 0, because the motion is horizontal,

dp/G + α²rdr/g = 2α²rdr/g,

dp/G = α²rdr/g,⁠

p/G = α²/2g + constant.

(9)

Let p₁, r₁, v₁ be the pressure, radius and velocity of one cylindrical section, p₂, r₂, v₂ those of another; then

p₁/G − α²r₁²/2g = p₂/G − α²r₂²/2g;

(p₂ − p₁)/G = α² (r₂² − r₁²)/2g = (v₂² − v₁²)/2g.

(10)

That is, the pressure increases from within outwards in a curve which in radial sections is a parabola, and surfaces of equal pressure are paraboloids of revolution (fig. 37).

Fig. 37.

Dissipation of Head in Shock

§ 36. Relation of Pressure and Velocity in a Stream in Steady Motion when the Changes of Section of the Stream are Abrupt.—When a stream changes section abruptly, rotating eddies are formed which dissipate energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the work thus expended internally in the fluid is too important to be neglected, and the energy thus lost is commonly termed energy lost in shock. Suppose fig. 38 to represent a stream having such an abrupt change of section. Let AB, CD be normal sections at points where ordinary stream line motion has not been disturbed and where it has been re-established. Let ω, p, v be the area of section, pressure and velocity at AB, and ω₁, p₁, v₁ corresponding quantities at CD. Then if no work were expended internally, and assuming the stream horizontal, we should have

p/G + v²/2g = p₁/G + v₁²/2g.

(1)