Concentration from an Instantaneous Source in Stationary, Homogeneous Turbulence

Mean Concentration from an Instantaneous Source in Stationary, Homogeneous Turbulence 

The key problem in using Eq. (3.1) is the specification of p. We ask whether we can derive an expression for p. The velocity components u, v, and w, although random, are related through conservation of mass and momentum for the flow, that is, they are governed by the stochastic Navier-Stokes and continuity equations. In general, as we have noted, an exact solution for u, v, and w is unobtainable. We can, however, consider an idealized situation in which the statistical properties of u, v, and w are specified a priori. Then, in so doing, we wish to see if we can obatin an exact solution of Eq. (2.4) from which p can be obtained through Eq. (2.6). 

We seek therefore to evaluate p through its relationship as the expected value of the stochastic Green s function G. To do so requires that we make appropriate assumptions that allows us to solve Eq. (2.2). Our basic assumption is that the turbulence is stationary and homogeneous, and 

The mean velocity components are expressed as //, v, and ii . We assume that the velocity components are stationary Gaussian random processes, so that, based on the preceding discussion, the autocovariances of u, V, and w can be written as (Papoulis, 1965, p. 397) 

The mean square concentration is given by Eq. (2.10), which can be rewritten as 

---