Page 47 - Fluid-Structure Interactions Slender Structure and Axial Flow (Volume 1)

30 SLENDER STRUCTURES AND AXIAL FLOW

where t is a time delay in the measurement of u; and u,. For homogeneous turbulence,
Rjj depends only on the separation between the two points r = Ilrll. For a uniform flow
field in a given direction, e.g. for fully developed turbulent flow in a pipe, R;j depends
on the separation I, but also on the direction, hence on r. In this latter case,

Rij(r, t) = u;(O, t)u,(r, t + r). (2.86b)

Keeping with this latter form, one distinguishes sparial correlations,
(2.86~)

in which u; and uj are associated with different points in space, but the same time; and
temporal correlations, involving the same point in space and a time delay 5,
t
Ri,(r, t) = ui(r, t)u,(r, + t), (2.86d)
autocorrelations for i = j, and cross-correlations for i # j.
The spatial correlation, when plotted versus a particular component of r, indicates the
distance over which motion at one point significantly affects that at another. It may be used
to assign a length scale to the turbulence, defined as Lk = ( l/v2) R;j(rk, 0) drk, where
-
v2 is a normalizing factor, e.g. v2 = u!, and rk is a particular component of r = (rl, r2, r3)T
in the (XI, x2, x3}-frame used here.' For flow in the x-direction, e.g. for fully developed
pipe flow, the integral (or macro-) scale, associated mainly with the largest, most energetic
eddies, is defined by

Lrn Rll(r1,O)drl
LI = - (2.87)
u:
For points r;! apart, in the cross-stream direction, L2 may be defined in a similar way,
with r2 taking the place of rl; in terms of the normalized form of the correlation function
(the coherence), R11, L2 is given by

(2.88)

The correlation in the streamwise (longitudinal) direction generally decays from 1 at
r = 0 to zero at sufficiently large r, smoothly and without change in sign (Figure 2.6);
whereas the cross-stream (lateral) correlation generally has a negative part for intermediate
r, before it too decays to zero for large enough r (Tritton 1988).
The temporal correlations are functions of the time delay t for measurements at the
same point; they give a measure of the rime scale of turbulence. For small times, or over
small enough distances, turbulence may be considered to be advected past the point of
observation without change in structure. This is Taylor's hypothesis, as a result of which a
temporal correlation is equal to the corresponding spatial correlation for t = rl/Ul; thus,
according to this hypothesis, the eddies of the turbulence are convected without change
over a sufficiently short distance, r, as further discussed in Chapter 9.

+Alternative definitions, for experimental convenience, are sometimes utilized; e.g. by defining the scale as
the distance to where Ri, plotted versus rk becomes negative, or to where it is reduced to l/e.