Many of the statistical theories just mentioned do not require the turbulent fluctuations to satisfy the equations of motion nor do they require the fluid motion to be continuous. A statistical theory of turbulence which is applicable to continuous movements and which satisfies the equations of motion was inaugurated in 1935 by Taylor8 and further developed by himself and by von Karman.9 It is the object of this paper to give a connected account of the present state of this particular statistical theory of turbulence. 2. Turbulent fluctuations and the mean motion. As in other theories of turbulent flow, the flow is regarded as a mean motion with velocity components, U, V, and W, on which are superposed fluctuations of the velocity with components of magnitude u, v, and w at any instant. The mean values of u, v, and w are zero. In most cases U, V, and W are the average values at a fixed point over a definite period of time, although in certain problems it is more convenient to take averages over a selected area or within a selected volume at a given instant. The rules for forming mean values were stated by Rey-nolds10 and some further critical discussion by Burgers and others has been recorded in connection with a lecture by Oseen.11 When the turbulent motion is produced in a pipe by the action of a constant pressure gradient or near the surface of an object in a wind tunnel in which the fan is operated at a constant speed, there is considerable freedom 2 K&rman, Th. von, Uber die Stabilitat der Laminarstromung und die Theorie der Turbulenz,
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.