The further investigation of the problem is difficult. It has been carried forward to some extent by Orr, and more recently by Rayleigh ‡, in whose papers other references will be found. 369. Reynolds, in a remarkable paper§, has attacked the question from a different point of view. Taking the turbulent motion as already existing, he seeks to establish a criterion which shall decide whether the turbulent character will increase or diminish or be stationary. For this purpose the velocity (u, v, w) is resolved into two constituents. We may, for instance, write 1 ö= so that u, v, w are the mean values of u, v, w at the point (x, y, z), taken over an interval of time extending from t Ttot+T. Again, we might consider the mean values at the instant t over a space S (e.g. a sphere) surrounding the point (x, y, z); thus 1 S dx го 1 } ///udzdydz, = } ///vdzdydz, = }}]][wdzdydz. S fwdx dz......(2) Or, again, we might take a double mean, for times ranging over an interval τ, and points ranging over a space S. The actual velocities are in each case denoted by u = π + u', v = v + v′, w = w + w', ..(3) where u', v', w' may be called the components of the turbulent motion. This implies that where the bar placed over a symbol denotes the mean value, taken according to the particular convention adopted. * Cf. Stokes, Camb. Trans. t. x. p. 106 (1857) [Papers, t. iv. p. 77]. For the relation between (15) and Riccati's and Bessel's equations see Forsyth, Differential Equations, Art. 111. ‡ “Stability of Viscous Fluid Motion," Phil. Mag. (6), t. xxviii. (1914); “On the Stability of the Simple Shearing Motion of a Viscous Incompressible Fluid," Phil. Mag. (6), t. xxx. p. 329 (1915). § "On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion," Phil. Trans. A, t. clxxxvi. p. 123 (1894) [Papers, t. ii. p. 535]. For simplicity we will adopt the definition of mean value which is embodied in the formulae (1). Reynolds starts from the dynamical equations in the forms which are seen to be equivalent to Art. 328 (1) in virtue of the equation of continuity ди av ди + + дх dy az = 0. .(6) These forms are not essential to the argument, but are interesting as an application of the method employed by Maxwell* in the kinetic theory of gases. They express the rate of variation of the momentum contained in a fixed rectangular space dx dysz, as a consequence partly of the forces acting on the substance which at the moment occupies this space, and partly of the flux of matter across the boundary, carrying its momentum with it. Thus the fluxes of x-momentum across unit areas perpendicular to Ox, Oy, Oz are pu. u, pv. u, and pw. u, respectively; and taking the difference of the fluxes across opposite faces of the elementary space dx dydz, we obtain a gain of x-momentum equal to a dy per unit time. a дх (pu dydz. u) 8x a (pvdzdx. u) dy (pw dx dy. u) dz az We now take the mean value of each member of the equations (5), using the substitutions (3). It is assumed that we may, without sensible error, take the mean values of ū, uu', uv', uw', to be u, 0, 0, 0, ., respectively. This is not exact, but is permissible provided the fluctuations of u, v, w about their mean values are sufficiently numerous within the time-interval T. It follows that ... In this way we obtain ии = uu + u'u', uv = ñv + u ́v', uw = ñw + u'w', .... ..(7) These are the equations of mean motiont. It is to be noticed that the dynamical equations have the same form as the exact equations (5), provided we introduce additional stress-components This recalls the explanation of gaseous viscosity by Maxwell (l.c.). The equations (8) may be written, in virtue of (9), .(10) † Or rather 'mean-mean-motion,' in the phraseology of Reynolds. He applies the term 'mean-motion' to the system of velocities (u, v, w), to distinguish it from 'molecular motion.' The turbulent motion (u', v', w') is called by him 'relative-mean-motion.' Let us first suppose that there are no extraneous forces X, Y, Z; and let us apply (12) to the case of a region bounded by fixed walls at which u, v, w, and therefore also u, v, . all vanish. If we write The formula (14) gives the rate of variation of the energy of the mean motion (ū, v, w). The first term on the right-hand represents the dissipation due to the mean motion alone, and is essentially negative. The second term represents the rate at which work is being done by the fictitious stresses (10). Now if T be the true kinetic energy, we may write, in virtue of assumptions already made, where T .(17) (18) i.e. T' is the kinetic energy of the eddying motion. By the method of Art. 344 it may be shewn that on the present supposition of fixed boundaries at which there is no slipping, the total dissipation is, on the average, equal to the sum of the dissipations due to the mean motion and the eddying motion respectively. Thus *It should be noticed that we are here virtually taking the differential time-element ôt to be of the order of magnitude of the interval 7 employed in the definitions (1). The procedure in The sign of the expression on the right-hand determines whether the mean energy T' of the eddying motion (u', v', w') will increase or diminish. The first part, which alone involves the viscosity μ, is essentially negative; the second part depends on the inertia of the fluid, and may be positive or negative according to circumstances. When there are extraneous forces X, Y, Z to be taken into account, and when the velocities u, v, w do not necessarily vanish at the boundary of the region considered, the equation (14) requires to be amended by the addition of terms which represent partly the convection of kinetic energy of mean motion into the region, partly the work done by the forces X, Y, Z, and partly the work done at the boundary by the mean stresses Pxx, Pyx, Pzx, and by the fictitious stresses Pax, Pyx, Pzx, .... The equation (21), on the other hand, requires only the addition of a term representing the convection of the energy of turbulent motion across the boundary. The derivation of the remarkable formulae (14) and (21), and of the modifications just referred to, appears to be free from objection, on the convections adopted. But, in applying these formulae to actual conditions, the restrictions and assumptions which have been introduced as to the character of the turbulent motions must be borne in mind. One or two consequences of the formula (21) may be noted*. In the first place, the relative magnitude of the two terms on the right-hand is unaffected if we reverse the signs of u', v', w', or if we multiply them by any constant factor. The stability of a given state of mean motion should not therefore depend on the scale of the disturbance. On the other hand, certain combinations of u', v', w' appear to be more favourable to stability than others. Thus, in the case of disturbed laminar motion parallel to Ox, between two rigid planes y = b, the formula (16) reduces to y=pu'v' ди .(22) so that the types of disturbance which tend to increase are those in which (for y>0) combinations of u', v' with the same sign preponderate. This indicates a tendency to equalization of the velocity in the different strata. Again, the relative importance of the second term in (21), which alone can contribute to the increase of T', is greater the greater the rates of strain du/dx, ... in the mean motion. This suggests a reason why a given type of mean motion does not begin to break down until a certain critical velocity is ⚫ reached. If we apply the (modified) formulae to the case of flow in a uniform cylindrical pipe, on the supposition that the pressure gradient (– dp/dx) is zero, we find * Cf. Lorentz, “Ueber die Entstehung turbulenter Flüssigkeitsbewegungen und über den Einfluss dieser Bewegungen bei der Strömung durch Röhren," Abhandlungen über theoretische Physik, Leipzig, 1907, t. i. p. 43. The paper is a revised form of one published in 1897. The region here considered is that contained between two cross-sections (of area „a2) at unit distance apart; the axis of x coincides with that of the pipe; and q denotes the velocity at right angles to this axis. It is assumed of course that q O and du/dx = 0; also that the mean state of things is in all respects the same at each section. The conditions of steady motion are obtained by equating the right-hand members of (23) and (24) to zero. = Reynolds discusses in detail the two-dimensional form of the problem, where there is a flow parallel to x between two fixed plane walls y = ±b. Assuming that ū varies as b2 - y2, in conformity with Art. 330, he seeks to determine a minimum value of the flux consistent with the condition dT'/dt = 0; but for this we must refer to the original paper. The result obtained is that the critical ratio ub/v, where u。 is the mean value of u between the limits y=b, must exceed 258*. Resistance of Fluids. 370. This subject is important in relation to many practical questions, e.g. the propulsion of ships, the flight of projectiles, and the effect of wind on structures. Although it has recently been studied with renewed energy, owing to its bearing on the problems of artificial flight, our knowledge of it is still mainly empirical. It has been seen that in the case of an isolated body moving in frictionless liquid, at a distance from the boundaries (if any), there is no abstraction of energy; in particular, if the motion of the fluid has been. started from rest, and is therefore irrotational and acyclic, its influence can be completely allowed for by a modification of the inertia of the solid† (Arts. 92, 117). The first attempt to obtain, on exact theoretical lines, a result less opposed to ordinary experience is contained in the investigations of Kirchhoff and Rayleigh relating to the two-dimensional form of the problem of the motion of a plane lamina (Arts. 76, 77). It is to be noticed that the motion of the fluid in such problems is no longer strictly irrotational, a surface of discontinuity being equivalent to a vortex-sheet (Art. 151). This theory is open to the objection that the unlimited mass of 'dead water' following the lamina implies an infinite kinetic energy, and it has been somewhat severely criticised, on this and other grounds, by Kelvin‡, who maintained that the only legitimate application of the methods of Helmholtz and Kirchhoff is to the case of free surfaces, as of a jet. It gives us, however, a scheme which we can to some extent complete in imagination, * A different result is obtained by Sharpe, "On the Stability of the Motion of a Viscous Liquid," Trans. Amer. Math. Soc. t. vi. p. 496 (1905), where also the case of flow through a cylindrical pipe is investigated. These problems, together with that of uniform shearing motion between parallel planes, have been treated more fully by Orr (l.c.). The differences in the numerical results obtained appear to arise from differences in the types of disturbance considered. The last-mentioned problem has also been treated by Lorentz (l.c.). † The absence of resistance, properly so called, in such cases is often referred to by continental writers as the 'paradox of d'Alembert.' Nature, t. 1. pp. 524, 549, 573, 597 (1894) [Papers, t. iv. p. 215]. |