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Aviation History
1917
1917 - 0815.PDF
AUGUST 9, 1917. :i,.; ii?. THEORY OF PRESSURE ON A PLANE SURFACE DUE TO RELATIVE WIND.-.¥.•-r By A. E. IN dealing from the theoretical standpoint with what, in a general way, may be termed wind-pressure on a surface, we are concerned only with relative movement of the surface and gaseous medium under consideration. It is convenient to consider the medium, such as air, to be at rest, and the relative wind to be due to movement of the surface through the medium. In establishing an expression for the pressure, the Newtonian method of determining the change of momentum at the surface in unit time is usually adopted, the reasoning being somewhat as follows. Conceive a plane- of unit area to be moving through still air of density p at a velocity v, all points of the surface moving parallel to a line making an angle 0 with the surface. In unit time the surface sweeps out a column of air of cross-sectional area sin 0 and length v, its volume being v sin 9 and its mass p v sin B. To this air is imparted a velocity v sin 0, the component of the velocity v normal to the surface. The momentum imparted is therefore p v2 sin2 0, and is a measure of the force acting on unit surface (pressure) throughout the motion. When 0 is a right-angle the relative wind is normal to the surface, and sin2 0 is unity. The above reasoning leads to an expression of the form pressure — k p v* when the relative wind impinges normally on the surface, the coefficient k depending on such considerations as the form and area of the surface, and being introduced to make theory fit practice. The underlying principle of determining the force acting frdm the change of momentum is, of course, physically sound, but the particular application given above appears to the writer to be unsound, for reasons which we endeavour to make clear in what follows. The aim of this article is, how- ever, not to present a cut-and-dried theory with an air of finality, but to suggest a line of thought which is at least interesting, and which, it is to be hoped, will prove fruitful. According to the expression given above' the pressure varies as the square of the velocity, but there is reason to believe that this is not strictly accurate. At low velocities the pressure appears to vary as the velocity, and at very high velocities as the cube or higher power. A possible explana- tion of this will be found in the development of theory given herein. When v is zero the above expression gives zero pressure. The condition v = o would correspond to the surface being at rest in still air. A theory which says that the pressure is zero under conditions in which, as a matter of fact, if we take the atmospheric pressure as 14-7 lbs. on the square inch, the pressure on a square foot is nearly one ton, is obviously unsatisfactory. To this it may be objected that the expression is only intended to give the increase of pressure due to relative wind, and that the total pressure would be (Obtained by adding the atmospheric pressure, so that the complete expression would be of the form pressure — kpv* + c where c is the atmospheric pressure. Such procedure is, however, arbitrary and unscientific, and moreover is not justified by the physical considerations on which the expres- sion is established. Further, such procedure does not remove a second objection, that the theory given does not enable us to calculate or account for the decrease of pressure on the back of the moving plane. / It will be seen that the " square " law set out above rests on an assumption that each and every particle or molecule of the gas has the same relative velocity with respect to the plane, that this relative velocity is determined entirely by the movement of the plane through the medium, and that when such movement ceases relative movement of necessity ceases also. In short, the gaseous medium is regarded as though its constituent molecules had no movement of their own. It should be obvious that no consistent thinker can assent to this view of the gaseous medium and at the same time assent to the kinetic theory of gases. According to the kinetic theory the molecules of a gas (which, in an ordinary way and considered as a whole, is at rest) are individually possessed of rapid translatory move- ment, and the so-called static pressure exerted by the gas on any surface with which it is in contact is explained by the impacts of the gas particles against the surface. This move- ment decreases as the temperature of the gas falls, and is accompanied by a corresponding decrease in the pressure exerted. The movement ceases and the pressure simul- taneously becomes zero when the temperature reached is that known as absolute zero. In this condition the gas would conform to the conception of the medium used above in establishing the expression for pressure due to relative WATSON. wind, and the " square " law would hold good. At ordinary temperatures, however, the " molecular velocity," as the velocity of the constituent particles according to kinetic theory may be termed, is, in the case of air at atmospheric pressure, of the order of a thousand miles or more an hour, which is by no means negligible. If, then, the molecular velocity of kinetic theory has any counterpart in fact, such molecular velocity must be taken into account in any theory of pressure due to relative wind which has any pretence to completeness. In short, such theory must be based on the relative movement of individual particles (or of streams ofN such particles as have the same relative movement) with respect to the plane, as determined by compounding the velocity v of the plane with the molecular velocity V of such particles or streams. At first thought it might be concluded hastily that such treatment of the problem belongs to the realm of impossibility. As a matter of fact, on the assump- tions made in developing accepted kinetic theory, the treat- ment is comparatively simple. In establishing an expression for the static pressure exerted by a gas on a surface in contact therewith, assumptions are made which, in effect, amount to assuming that the surface is bombarded equally from all possible directions by streams of gas particles, the streams having the same density (d, say) and velocity V. Thus, if AB (Fig. 1) represents a plane, and we confine our attention • to a small element of area approximating in magnitude to a point C, the streams may be represented, as regards velocity and direction, by the radii of a sphere DEF having its centre at C, the radii being regarded as directed towards the centre. As we are here concerned with a relative movement deter- mined entirely by the molecular velocity V, the Newtonian method given above is applicable, and the pressure on the element of area at C due to any stream represented, for instance, by GC, is given by the expression dV2 sin2 0, where M 9 is the angle GCF. The total pressure oh the element of area is the sum of the pressures due to the individual streams. If we distinguish the inclinations of these streams by sub- scripts i, 2, 3, &c, the total pressure p is given by p = dV* (sin2 flj + sin2 02 + sin21?, &c.) (i) If there are n streams, there will be n terms within the brackets, and we may write the above expression in the form .... ton terms\ ,2% The expression within brackets is the mean value of sin2 8 for all the streams, and nd is the total density of all the streams, and is therefore equal to p, the density of the gas. To find the mean value of sin2 e for all possible radii uni- formly distributed throughout the hemisphere, it is con- venient to associate mentally the distribution of the radii with the distribution of their outer extremities over the hemispherical surface. These extremities will be uniformly distributed, and the mean value of sin2 0 is then obtained from the expression ' N «-•.• : (3) where A is the area of the surface over which the integration is performed, and d\ is an element of that surface. In the present case A is the area of the hemispherical surface, given by 2xVs (since V is the radius of the sphere of reference). 815
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