Flow velocity

“ The relation between gradient and velocity is evident; that between volume and velocity is illustrated by every stream in time of flood, when its rate of flow is greatly increased. The relation between velocity and load is less obvious, but none the less definite. Every particle of sediment carried by a stream makes a draught on its energy, and energy expended in this way reduces the velocity. The draught on a stream’s energy of a particle carried in suspension is measured by its mass into the distance it would fall in a unit of time in still water. ”

“ Again, the energy of a clear stream is entirely consumed in the friction of flow; and the friction bears a direct relation to its velocity. But if detritus be added to the water, then a portion of its energy is diverted to the transportation of the load; and this is done at the expense of the friction of flow, and hence at the expense of velocity. As the energy expended in transportation increases, the velocity diminishes. If the detritus be composed of uniform particles, then we may also say that as the load increases the velocity diminishes. ”

“ Velocity of Whirl and Velocity of Flow.—Let acb (fig. 193) be the path of the particles of water in a turbine wheel. That path will be in a plane normal to the axis of rotation in radial flow turbines, and on a cylindrical surface in axial flow turbines. At any point c of the path the water will have some velocity v, in the direction of a tangent to the path. ”

“ The work done, however, by a stream of air or other fluid in a given time, will vary as the cube of the velocity; for if the velocity of a stream of air be doubled, there will not only be four times the pressure exerted per square foot, but twice the quantity of air will be employed; and in windmills, accordingly, it is found, that the work done varies nearly as the cube of the velocity of the wind. If, however, the work done by a given quantity of air moving at different speeds be considered, it will vary as the squares of the speeds. ”

“ The validity of the above reasoning is unquestionable if we are dealing with a fluid whose properties are those of the Newtonian medium; the energy of turbulence being represented by a variation in the individual velocities of the particles such as will not affect their mean velocity. ”

“ The fluid which is falling from a higher to a lower level, carries with it the velocity due to its revolution with the earth at a greater distance from its centre. It will therefore accelerate, although to an almost infinitesimal extent, the earth's daily rotation. The sum of all these increments of velocity, arising from the descent of all the falling waters on the earth's surface, would in time become perceptible, did not nature, by the process of evaporation, convey the waters back to their sources ”

“ The speed with which the surface waves move outwards in all directions, varies not only with the force of the wave, but also with the kind of material through which they pass. This velocity may be in the neighborhood of twenty miles per second, while in others the velocity is as great as 140 miles per second. ”

“ II. The spaces run thro’ by falling bodies are proportional to one another as the squares of their last or acquired velocities.
These two principles are demonstrated by the writers on mechanics.
III. Water forced out of a larger chanel thro’ one or more smaller passages, will have the streams thro’ those passages contracted in the ratio of 25 to 21.
This is shewn in the 36th prop. of the 2d book of Newton’s Principia.
IV. In any stream of water, the velocity is such, as would be acquired by the fall of a body from a height above the surface of that stream. ”

“ No one has yet examined what happens when a rough sphere enters a liquid with a very high velocity. That the motion set up must differ from that at a low velocity is apparent to any one who has thrown stones from a low bridge into deep water below. ”

“ But as it can be shown theoretically that a horizontal stream of air of constant cross section and uniform velocity at each section, can not greatly fluctuate in velocity from point to point, without more pronounced changes of density than the barometer records, it naturally follows that the stream must broaden where the air speed lags, and narrow where it accelerates ”

“ The velocity with which a denser fluid flows into a rarer one is equal to the velocity a heavy body acquires in falling through a height equal to the difference of altitude of two columns of the heavier fluid of such heights as will produce the respective pressures; and, therefore, when the difference of pressure or amount of rarefaction in the chimney is known, it is easy to tell the velocity of motion which ought to be produced by it. ”

“ The velocities vary in different parts of the same transverse section of a stream, for the air upon the surface of the water, as well also as the solid bottom of the stream, has a certain effect in retarding the current. The velocity is found to be greatest in the middle, where the water is deepest, Fig. ”

“ A.--All bodies fall with the same velocity, when there is no resistance from the atmosphere, as is shown by the experiment of letting fall, from the top of a tall exhausted receiver, a feather and a guinea, which reach the bottom at the same time. The velocity of falling bodies is one that is accelerated uniformly, according to a known law. When the height from which a body falls is given, the velocity acquired at the end of the descent can be easily computed. It has been found by experiment that the square root of the height in feet multiplied by 8.021 will give the velocity. ”

“ Fig. 105 shows the vertical and horizontal velocity curves and the contours of equal velocity in a rectangular channel, from one of Bazin’s gaugings.
§ 103. Experimental Observations on the Vertical Velocity Curve.—A preliminary difficulty arises in observing the velocity at a given point in a stream because the velocity rapidly varies, the motion not being strictly steady. If an average of several velocities at the same point is taken, or the average velocity for a sensible period of time, this average is found to be constant. ”

“ Irrotationality of a fluid is secured, as stated above, when the velocity of each element in any direction is the rate of variation per unit distance in that direction of a certain function of the coordinates, the distances, taken parallel to three lines perpendicular to one another and drawn from a point, which specify the position of the particle. In fact, what is called a velocity-potential exists, similar to the potential described in Chapter IV above, for an electric field. This condition, together with the specified motion of the surface, suffices to determine the motion of the fluid. ”

“ The water was banked up by its own mass so that the surface fall was less than at the point of previous measurements. "In obedience to known hydraulic law, this decreased surface-inclination was indicated by decreased velocity of surface-current. But the power of the whole moving mass was greater in proportion to its depth. And thus, while to the superficial gaze the velocity was less, the mean velocity was greater; and the river swept with more resistless energy over its bed. ”

“ Turbulence.—The steady state of viscous motion depicted in Figs. 26 and 27, on which the laws of viscosity and skin-friction have been based, is found in practice to obtain over a moderate range of velocity only. When a certain critical velocity is exceeded the continuity becomes broken and the phenomenon of turbulence manifests itself. ”

“ But there are certain local conditions that are continually modifying the phenomena of air movement. The velocity of winds as they occur from day to day varies very greatly with the height above the surface of the earth; ordinarily the velocity at 1000 feet above the earth will be more than three times greater than it is at 50 or 60 feet above, and even at 60 feet the velocity is much greater than at the surface of the earth. ”

“ Now, the velocity of wave propagation can be seen, without the aid of any mathematical analysis, to depend on the elasticity of the medium and its density; for we can see that if a medium is highly elastic the disturbance would be propagated at a great speed. ”

“ The uncorrected velocities, however, are in all cases too large, and by greater and greater amounts the higher the velocity. At 60 miles per hour the observed velocities are about 12 miles per hour too high, and for an indicated velocity of 90 miles the experiments show that the actual velocity is but a fraction over 69 miles per hour. ”

“ We see at once that, where u=0, the velocity has its “normal” value, while where u is positive the velocity is in excess, and where u is negative the velocity is in defect of the normal value. If, then, a (fig. 10) represents the displacement curve of a train of waves, b will represent the pressure excess and particle velocity, and from (11) we see that while the nodal conditions of b, with ῶ=0 and u=0, travel with velocity √ (E/ρ) , the crests exceed that velocity by 1/4 (γ + 1) u. ”

“ We shall investigate the velocity of such plane waves by a method which is only a slight modification of a method given by W. J. M. Rankine (Phil. Trans., 1870, p. 277) .
Whatever the form of a wave, we could always force it to travel on with that form unchanged, and with any velocity we chose, if we could apply any “external” force we liked to each particle, in addition to the “internal” force called into play by the compressions or extensions. For instance, if we have a wave with displacement curve of form ABC (fig. 8) , and we require it to travel
Fig. 8. ”

“ In the case of winds, the inertia of the moving mass becomes what is commonly termed “wind pressure” against any object not moving with it at an equal speed. The following table gives the measurements of wind pressure, as recorded at the station on the Eiffel Tower, for differing velocities of wind:
In applying this table, the velocity to be considered is the net velocity of the movements of the airship 32 and of the wind. ”

“ In a moment, Leonard; then I want you and Jim to ask me any questions you like. All this that I have said, is prefatory. The question is one solely of velocity. The rest is automatic. The natural laws governing the attributes of matter in relation to its velocity are these:
“First: A substance whose velocity is increased loses density proportionately. For instance, our vehicle, I consider it as a whole, you understand. ”

“ Thus, if the curve be plotted for a one foot diameter circular plane and one foot per second velocity is represented by one inch, then for a two foot circular plane a one foot per second velocity will be represented by two inches, that is to say, a given pressure will be developed at one-half the velocity. ”

“ The slowness with which powders subside, depends partly on the specific gravity of the substance, and partly on the magnitude of the particles themselves. Bodies, in falling through a resisting medium, after a certain time acquire a uniform velocity, which is called their terminal velocity, with which they continue to descend: when the particles are very small, and the medium dense, as water, this terminal velocity is soon arrived at. ”