Frederick William Lanchester

“ The mingling of the wake and counterwake may be regarded as a phenomenon quite apart from the initial disturbance, and the turbulence or otherwise of the wake does not materially add to or detract from the pressure on the front face of the body, but concerns merely the ultimate disposal of the energy left behind in the fluid. ”

“ All fluids are viscous to a greater or lesser degree; the viscosity of water is small, that of air still less, whilst lubricating oil and treacle are highly viscous substances. Now just as in the study of ordinary mechanics it is found expedient initially to neglect the effects of friction, so, in connection with the present subject, we can afford to ignore the fluid friction to which the viscosity of the fluid gives rise, and in the first instance deal with resistance as a function of density alone. ”

“ If, on the other hand, as in the Newtonian medium, we assume discontinuity, then it is evident by our definition that streamline form can have no existence, which, again, is what we know to be the case. ”

“ The Doctrine of Kinetic Discontinuity.—The theory of kinetic discontinuity is of modern origin, having been introduced and developed by Kirchhoff, Helmboltz, and others, to account for the phenomenon of resistance in fluid motion. The analytical theory, based on the hypothesis of continuity, does not in general lead to results in harmony with experience. All bodies, according to the Eulerian theory, are of streamline form, provided that the hydrostatic pressure of the fluid is sufficient to prevent cavitation ”

“ When the flow of momentum is equal and opposite, as in the case of a solid in stress, there is no displacement of matter, and it is only when there is a displacement of matter that the Newtonian method can be applied. The case of a gas under pressure is, according to the kinetic theory, an example of the actual communication of momentum, and its pressure and the mean velocity can be correlated on the Newtonian principle ”

“ The basis of the Newtonian method is found in the principle of the conservation of momentum, which maybe taken as corollary to the third law of motion as written: When force acts on a body the momentum generated in unit time is proportional to the force. This method is best studied in connection with a hypothetical medium suggested by Newton, on which he based several of the problems in the "Principia." This medium is defined as consisting of a large number of material particles, equally distributed in space, having no sensible magnitude, but possessing mass ”

“ In the practical shaping of a streamline body it is found essential to avoid corners or sharp curves in the line of flow. Bodies in which due precaution is not taken in this respect offer considerable resistance to motion, and the regions of abrupt curvature give rise to a discontinuity in the motion of the fluid. Thus Fig. 7 represents a double cone moving axially, and it will be noticed that the flow has not time to close in round the run, as it would do in a properly formed streamline body, but shoots past the sharp edge, as indicated in the figure. ”

“ It is generally believed that, in a fluid whose viscosity is negligible, the latter cause would be inoperative, the whole resistance being then due to the excess pressure region in front of the body, the dead-water or wake being at approximately the hydrostatic pressure of the fluid.
The surface separating the live stream and the dead-water constitutes a discontinuity, since the velocity of the fluid, considered as a function of its position in space, is discontinuous. ”

“ If we take away the condition that the enclosure is fixed and suppose the force applied from without, then the problem is not essentially altered, for though the external force F will now impart momentum to the system en bloc, its action in this respect has no relation to phenomena in the interior, and does not provide any data for the determination of the pressure-velocity relation.
The supposition that the body is a plane evades any question relative to the density of the body itself, and thus simplifies the argument. ”

“ The commencement and end of the orbit, where the motion should be in the same direction as the body, is most difficult to observe, though even this detail is visible if the orbit selected be sufficiently near to the axis of flight. ”

“ If, in the case of a body propelled at a constant velocity, the entire run be removed, as in § 23, the consequence is a surface of discontinuity emanating from the periphery of section. Under these circumstances, if we neglect the influence of viscosity and the consequent loss of wake pressure, the work done appears wholly in the counterwake current, on the production of which energy is being continuously expended. ”

“ Now so long as the bends in the pipe are sufficiently gradual, we know that they cause no sensible resistance to the motion of the fluid. We have excluded viscous resistance by hypothesis, and if the areas at the points A and E are equal there is no change in the kinetic energy. Moreover, the sectional area of the pipe between the points A and E may vary so long as the variations are gradual; change of pressure will accompany change of area on well-known hydrodynamic principles, but no net resistance is introduced ”

“ The sternward velocity imparted to the fluid by the propeller is, under proper conditions, small in comparison with the velocity of travel, so that the lines of flow are not radically altered, and instead of a circulation such as arises in the case of a normal plane, there is merely a slight contraction of the stream at the region in which the propeller operates, and a trifling readjustment of the surrounding lines of flow to suit. ”

“ The criterion of compressibility in fluid dynamics involves the relative density of the fluid, and on this basis air is only about eighteen times as compressible as water, the ratio of the velocity of sound in water and air being approximately in the proportion of It is shown later that the influence of compressibility only becomes manifest as the velocity of motion approaches the velocity of sound in the fluid in question, or if the pressures developed involve a serious change in density. ”

“ Let us suppose that the mass acted on per second be divided into parts, and that each part be acted on by the force for th of a second. Then the momentum communicated to each part and the total momentum per second which holds good when the number of parts becomes infinite and the communication of momentum continuous. ”

“ When a body, propelled through an incompressible fluid, contained within a fixed, enclosure, experiences resistance to its motion, the force exerted by the body on the fluid does not impart momentum to the fluid, but is transmitted instantly to the confines of the fluid however remote, and is wholly borne by its boundary surfaces.
Let us suppose (Fig. 1) a body which we will take to be a normal plane C, acted upon by a force F in an enclosure A,
Fig. ”

“ This question could also be eliminated by supposing the body to possess the same density as the surrounding fluid; in any case a force applied to the body to overcome its inertia is a matter external to, and without influence on, the conditions. ”

“ Then this roughness will add to the skin friction and will give rise to some local turbulence, but the general character of the flow system remains as before. We may go further and suppose the bricks so large as to form steps capable of giving rise to surfaces of discontinuity (Fig. 19) . Then the resistance will be increased, and the layer of fluid next the body will be violently stirred up; but if we examine the fluid some distance away we shall still find it comparatively unaffected. ”

“ When, however, the principle is applied to bodies aerodynamically supported in the free atmosphere the matter is not so self-evident; here, for example, we find that the weight of a parachutist is borne by the earth's surface almost from the moment he leaves the car, and his presence overhead, or the presence of a passing flight of birds, could be detected barometrically if we possessed an instrument of sufficient delicacy. ”

“ The fact that the propelling force only comes into existence when the initial spin is sufficient to eject the dead water from the leading side of the rotor blade fully explains the observed fact that a very considerable initial spin is necessary. ”

“ Froude, that it is "blunt tails rather than blunt noses that cause eddies" (and therefore involve a loss of power) , is applicable only to bodies having already some approximation to streamline form. It is
Fig. 17. obviously useless to provide a nice sharp tail if previous attention has not been given to the shoulder and buttock lines. Mr. Froude probably meant that in a well-designed streamline form the tail should be finer in form than the head, a matter that up to his time had presumably been neglected. ”

“ The relation of fineness to speed is not so easy of demonstration, owing to the absence of accurate data. It would, however, seem to be sufficiently obvious as a matter of general experience that our conclusions hold good. ”

“ If this apparatus be held in a current of air with the plane face fronting the wind, as, for instance, by holding it outside the window of a railway carriage in motion, the rotor evinces no tendency to go round in the one direction or the other. If, however, a considerable initial spin be imparted in either direction, the wind will suddenly get a bite, so to speak, and the rotor will gather speed
Fig. ”

“ This may be recognised at once as a result often given in text books as the pressure-velocity equation for air, and is tacitly put forward as if founded on experiment. It is approximately 50 per cent. higher than the true value. ”