The effect of homogeneous turbulence on material lines and surfaces

Abstract

A material line consisting always of the same particles of fluid is lengthened by the convective action of turbulent motion, at a rate which is shown to be exponential at a sufficient time after the initial choice of the line. The area of material surface elements also increases exponentially, although with a different exponential coefficient. It is then a consequence of conservation of mass of the fluid that the normal distance between two neighbouring parallel material surfaces decreases at an exponential rate which is simply related to the rate at which material lines lengthen. These kinematical results are relevant to the convective action of the turbulence on the distribution of two kinds of local property of the fluid, viz. quantities, represented by $\theta $, of which the total amount in a material volume of the fluid remains constant (e.g. mass density of a foreign substance), and quantities, represented by F, of which the total flux across a material surface remains constant (e.g. vorticity). F is proportional, at all times, to the vector representing a material line element and G, = $\nabla \theta $, is proportional to the vector representing a material surface element, and hence they both increase exponentially in time. The last two sections of the paper are concerned with the combined effect on F and G of convection and molecular diffusion, and a condition for the effect of the convection to be dominant is obtained. In the case of F, convection is found to be dominant if the appropriate molecular diffusivity is small compared with the kinematic viscosity of the fluid, and ${\bf F}^{2}$ will then increase indefinitely, unless some additional effect comes into play (as it does in the case of magnetic field strength). In the case of G, convection will be dominant if the initial distribution of G has a large enough length scale; the effect of the convection is to decrease this length scale so that ultimately the two effects reach an equilibrium. The length scale of the ultimate distribution of G, as obtained herein, is different from that put forward recently by Obukhoff.