The fluctuations of density in isotropic turbulence

Abstract

In this paper the fluctuations of density in a compressible fluid under conditions of homogeneous isotropic turbulence are considered. It is shown how from the equation of continuity alone, an invariant can be derived. Thus, if {w}(r, t) = {δ ρ δ ρ'} denotes the correlation between the instantaneous fluctuations of the density from the mean, at two points separated by a distance r, then int₀^∞ r²{w}(r, t) dr = constant. The meaning of this invariant is that the largest scales of the fluctuations of density are determined by the initial conditions of the problem and represent permanent features of the system. An equation of motion for {w}(r, t) is also derived which relates the fluctuations in density with the fluctuations in velocity; if, as an approximation, we substitute in this equation of motion the expression for the fundamental correlation tensor {u_iu_j'} which is valid for an incompressible fluid, we obtain a simple equation connecting {w} and the defining scalar, Q, of {u_iu_j'}. When {u²}< < c² (where c denotes the velocity of sound) the equation for {w} is of the same form as that governing the propagation of spherical sound waves except that the velocity of propagation is not c but surd 2c. More generally, it is found that when the term in Q is included, the equation for {w} still admits periodic solutions of the form of spherical waves; but they are distorted for small values of r and are propagated with a velocity (2c²+⅔{u²})½. Also, under the same conditions we can picture {w}(r, t) as a superposition of the fundamental periodic solutions.