The transport of vorticity and heat through fluids in turbulent motion

Abstract 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 that 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, that is when the mean flow is, say, horizontal and constant in direction and magnitude at any given height, the components of stress over a horizontal plane at height z are Fx and Fy where Fx = — ρuw—, Fy = — ρvw—, 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 that mean values have been taken over a large horizontal area and ρ is the density of the fluid. The stress Fx, 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 U0 of the layer from which it originated. At this height it is conceived to mix with its surroundings. If l is small the mean velocity of this layer is U0 + ldU/dz, U being the mean velocity at height z, so that u = — ldU/dz, and hence Fx = ρwl—dU/dz. The quantity ρwl— is 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 at all points in the field.