Influence of incombustible vapours on inflammability of gases 539 This work was carried out in the Department of Chemical Technology, Imperial College, London as part of the research programme of the Joint Fire Research Organization of the Department of Scientific and Industrial Research and Fire Offices’ Committee. The authors are indebted to the Director for permission to publish this account.
540 G. K. Batchelor and A. A. Townsend 1. Introduction In a previous paper (Batchelor & Townsend 1947—referred to hereafter as I) the authors have analyzed measurements of the rate of change of mean square vorticity in isotropic turbulence. It has been shown theoretically that two separate physical processes contribute to the change in the vorticity. The first process is an average extension of the vortex lines due to the random diffusive motion, giving a positive contribution to do)'2jdt, where a/2 is the mean square of any vorticity component, which is directly related to the skewness of the probability distribution of the rate of extension of fluid elements aligned in any direction. The second process is a dis sipation of the vorticity due to the effect of viscosity, and the contribution to da)'2ldt depends on the fourth derivative at the origin of the double velocity correla tion function. Both of these contributions and the value of itself were measured at different stages in the decay of the turbulence. The agreement made it clear that it is permissible to apply the theory of isotropic turbulence to the motion behind a grid in a uniform stream and that the measurements described were essentially correct.
Decay of isotropic turbulence in the initial period 541 direct measurements obtained by a suitable electrical analysis of the wire output have been preferred. In this way direct measurements of u', A, k% A3 and/ovA4 as described in I, and of | fdr, g(r) and k(r) have been made. Details of the methods Jo are described fully in a separate paper by one of us (Townsend 1947). The wind tunnel used in the experiments is situated in the Cavendish Laboratory, Cambridge, and has been described in I. As is customary, the time of decay in the idealized problem of homogeneous isotropic turbulence will be identified with the time x/U which has elapsed since any point distance x from the turbulence-producing grid and moving with the mean stream was at the grid.
542 G. K. Batchelor and A. A. Townsend The measurements reported in I were made behind several grids ( = 1-27, 2*54 and 5-08 cm.) over ranges which lay within the limits x/M = 20 to x/M = 130 and the turbulence was in every case evidently within the initial period of decay. The range x/M < 20 is of course excluded from the system of classification since the turbulence takes a little time to become uniform (in the lateral direction) and isotropic. It remains to be seen whether it is possible to define an initial period of decay by the laws (3-1) for all values of the grid Reynolds number. The experiments of I covered the range RM— 5-5 x 103 to 4-4 x 104 and experiments to be described herein confirm the validity of the laws (3-1) in the initial period of decay for values of Rm at least down to 600.
Decay of isotropic turbulence in the initial period 543 The experiments of I established for the initial period of decay and 0*5 x 104 < Rm < 4-5 x 104 that the skewness factor s is approximately an absolute constant with a value close to 0-39. Hence the factor which governs the relative importance of the viscous and inertia effects on the rate of change of mean square vorticity is 22 1+iV (3*7) The number Rx clearly plays a fundamental role in the theory of isotropic tur bulence and the authors propose to call it the Reynolds number of the turbulence. Since Rx is constant during the initial period of the decay, the value of RA is a con venient and significant index to the type of turbulence occurring throughout that period. Three roughly separate regimes depending on the magnitude of Rx can be distinguished. The regime of large Reynolds numbers of turbulence is such that 22/Rx<^l and the contributions to d(i)'2jdt from the viscous and inertia terms are approximately equal and opposite. At intermediate Reynolds numbers the factor 22/Rx is of order unity and the ratio of the two contributions to do/2/dt is significantly different from unity. Finally, at small Reynolds numbers of turbulence, 22/i?A> 1, and the effect of the triple correlations on d(o'2jdt is negligible compared with the viscous dissipation.
544 G. K. Batchelor and A. A. Townsend at t increases. Since there is no means of predicting the rate of approach to (3*1), measurement of the energy decay law is not sufficient to choose between the two solutions. However they make different predictions about the correlation functions and the evidence presented below is sufficient to show that one solution is correct. When Rx is small, it has been found possible (see II) to predict the form of the function/(r) for as large a range of r as f(r) is self-preserving during decay.
546 G. K. Batchelor and A. A. Townsend for an initial range of values of x/M, where a and x0 are constants. The energy equa tion (5*1) then requires lOv 10 M2 / x £0\ A2 = (x-x0) (5-3) Rm \M ~m )' the rather curious but well-known result that A2 depends only on the viscosity and time of decay—depending not at all on Mr o directly—is a consequence of any decay law which makes u'2 vary as some power of x or t. In the plot of A in figure 4, lines are drawn with slope 10 U/vnad they represent well the experimental x (cm.) Figure 4. A2 vs. x Curves I and II, slope = 2-26 cm. = -— ^'2 /£ Figure 3. —U*x —M ( U = 643 cm.sec.-1).
Decay of isotropic turbulence in the initial period 547 the turbulence is effectively created with infinite energy. Values of xQ given by the straight lines best fitting observations of U and A2 (in the case of A2 the slope of the lines was taken as 10 U/vnad only the position was varied to improve the fit) are set out in table 1 and are approximately in accord.