The outcome of the foregoing discussion is that dynamical similarity will exist between all cases for which vDp/µ is a constant; the characteristic constant will be different for each characteristic feature of the flow that may be examined.
Further elucidation is obtained if we examine the various experimental results which have been obtained for various velocities, diameters and fluids. These include fluids as diverse as air and water. If vDp/,u be plotted horizontally on a single diagram against vertically, for all cases a single curve is obtained, the ordinates decreasing at first, then rapidly increas ing for a short range of values of the abscissae and then grad ually falling continuously. The diagram can be compressed into less space if logvDphz is plotted horizontally instead of the quantity itself (fig. 1). Any single point on the curve repre sents a' whole group of cases which are dynamically similar. The most characteristic case is the position of the almost dis continuous middle region where a sudden change in the law of flow is indicated. This region is, in fact, that in which the flow changes from linear to turbulent. This change occurs in each case for a certain velocity V but this critical velocity depends upon the other data for the same point on the diagram; it occurs, in fact, for a definite value of the same for the whole group. Doubling the viscosity alone, doubles the critical velocity-, using the same fluid (i.e., µ/p = const.) and doubling the diam eter of the tube halves the critical velocity and so on. The value of this critical constant is called the Reynolds number after Osborne Reynolds who first examined the problem both theoretically and experimentally. It is, from what has been said, a pure number independent of the system of units employed; its value is a little over 2,000. Any lack of perfect definiteness can be attributed to fortuitous circumstances such as different roughnesses of the tube, accidental tremors, etc. It is evident from the experimental curve that these accidents play only a very subordinate part. In passing along the curve the change of (which is also a numeric), in the critical region is from -0038 to .0054 or thereabouts (Stanton and Pannell Phil. Trans. Roy. Soc. A. vol. 214, p. 199, [1914. Professor C. H. Lees from an examination of this diagram has concluded that the resist ance per unit area, R, can be represented by the following formulae in the region of turbulent flow; for water at is° C, R = o.o191v and for air at 15°C and 76 cm.
pressure, 0.00000110'0 where C.G.S. units are employed. These formulae show that the unknown function can be represented sufficiently well by two terms, one of which is practically independent of the diameter of the tube. The velocity referred to is the mean velocity for the whole cross section of the tube. The formulae are considered to be valid for at least a range of diameters from 0.3 to 12 cm. (Roy. Soc. Proc. A. Vol. 91, p. 6o, 1914). If a single term be used instead of two the best index of v is not far from i•75.
It was by no means certain that a single curve would repre sent the facts for tubes of various diameters because the prin ciple of dynamical similarity requires that the eddies when present should have diameters proportional to those of the tubes and this is not known to be the case. Our knowledge concern ing the curve must, in this repect, rest on experimental results.
Flow Past Obstacles.—From the dimensional point of view the flow past fixed obstacles is dealt with in the same way as that through pipes. The diameter, D, of the obstacle takes the place of the diameter of the pipe and further we naturally deal with the total force, F, which the obstacle experiences from the fluid instead of a force per unit area. The final result obtained as before is velocity of the fluid at a great distance from the obstacle (i.e., what may be called the undisturbed velocity) to that of this obstacle (which we have supposed fixed). The same equation also represents the case of a body moving with velocity v through a fluid which is stationary at a great distance and in which the body is completely immersed. No factor dependent on the shape is introduced; therefore for similarity between the several members of a group they must all have the same shape. The formula shows that for this family will have a constant value provided vDp/ ,u remains constant. The corresponding values of these terms can be determined by experiments on small-scale models and once determined should be applicable to all cases having the same geometrical shape. All the results for different media and different sizes and the different forces that correspond can be plotted as a curve with ordinates and abscissae either vDp/µ or its logarithm (or any other convenient function of it). Any one point on such a diagram gives a group of cases which are dynamically similar to one another.