The Transport of Vorticity and Heat through Fluids in Turbulent Motion

G. I. Taylor
1932 Proceedings of the Royal Society A  
In Reynolds' well-known theory of turbulent flow the effect of turbulence on the mean flow of a fluid is conceived as the same as th a t of a system of stresses which, like those due to viscosity, may have tangential as well as normal components across any plane element. Taking the case of laminar mean flow, th a t is when the mean flow is, say, horizontal and constant in direction and magnitude a t any given height, the components of stress over a horizontal plane a t height z are F a and Fv
more » ... z are F a and Fv where F^ = -p and u, v, w are the components of turbulent velocity parallel to two horizontal axes x and y and the vertical axis z. The bar denotes th a t mean values have been taken over a large horizontal area and p is the density of the fluid. The stress ¥ x is therefore due to the existence of a correlation between u and w. In the extension of Reynolds' theory due to Prandtl this correlation depends on the rate of change in mean velocity. In its most simplified form the theory may be expressed as follows. A portion of fluid possessing the mean velocity of a level z0 may be conceived to move upwards to a layer of height z0 -\-l preserving the mean velocity U 0 of the layer from which it originated. At this height it is conceived to mix with its surroundings. If l is small the mean dXJ velocity of this layer is U 0 + l, U being the mean velocity a t heigh 11/Z cZTT th a t u --l -r -, and hence dz The quantity p w li s therefore of the same dimensions as viscosity and in Prandtl's theory it is treated as though it were in fact a coefficient of viscosity, though not necessarily as one which has the same value a t all points in the field. In deriving the expression (1) it is assumed th a t the pressure gradients on the fluid which accompany the eddying motion have no effect on the final result, so th a t each particle continues moving with the horizontal momentum of the layer from which it originated till at some stage it mixes with the fluid on July 20, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from
doi:10.1098/rspa.1932.0061 fatcat:wuq2gbbc2zfbddfbk5hcuogu3y