3.4.1 Double Correlations and Typical Scales

Isotropy implies that the two-point second-order correlation tensor

(time is omitted for the sake of brevity) can be expressed as R _ij = A( r) ? _ij + B( r) r _i r _j, or

introducing the scaling factor and using the longitudinal correlation function

and its tranverse counterpart

in which n is a unit vector normal to r (see Fig. 3.7).

Figure 3.7: Schematic view of multipoint correlations. Top: general sketch of the correlation between two velocity components taken at two differents points A and B. Bottom: illustration of the physical meaning of the longitudinal correlation function f(r) and its transverse counterpart g(r).

The scalar correlation functions f and g are linked by the incompressibility constraint. Using one obtains

Finally, reintroducing the time dependency, the evolution equation for the two-point second-order tensor amounts to the single scalar equation, e.g., for f, as follows:

which is referred to as the Karman Howarth equation. The term R _{LL ,L} represents the longitudinal two-point third-order correlation function, which is involved by means of the quadratic nonlinearity. It is defined as

A slightly different form can be found in Mathieu and Scott (2000).

Typical spatial length scales of turbulence can be defined by functions f( r) and g( r). The longitudinal and transverse integral length scales, denoted...