By P. Bradshaw
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Extra info for An Introduction to Turbulence and its Measurement
When we speak of the length scale of the energy00 containing eddies we mean a length of order j Rar (actually called the "integral scale"). 4. Time Correlations (Autocorrelations or Autocovariances) The correlation between the same (Greek autos = self or same) fluctuating quantity measured at two different times (at the same point 30 TURBULENCE AND ITS MEASUREMENT in space) is not itself very relevant to the behaviour of turbulence and its measurement requires a time delay mechanism (usually a taperecorder with movable heads or a digital sample-and-delay system): the usual reason for interest is that if the turbulent velocity fluctuations are small compared with the mean velocity, the eddies or vortex lines do not change appreciably in shape as they pass a given point and therefore the autocorrelation of the v component (say) with time delay T, written as JR 2 2( T ) = KO v(* + r)lv2> will t>e the same as the space correlation with separation — JJx in the x direction (which we suppose for simplicity to be the direction of the mean velocity).
As in solids, whether a given arrangement of stresses is "normal" or "shear" or both depends solely on the choice of axes and, again, as in solids, we can always find a set of "principal" axes with respect to which only normal stresses remain. e. a scalar) is u2 + v2 + w2, written as q2 for brevity and to show that it is the mean square of the resultant velocity fluctuation q. The sum of the principal stresses is — qq2, and \qq2 is called the turbulent (kinetic) energy per unit volume. The justification for the name is that we can deduce, from the Navier-Stokes equations, a conservation equation for the value of \qq2 in the elementary control volume of Fig.
The equation can be regarded as an equation for the rate of change of \$q2 along a mean streamline in terms of local quantities: the actual value of q2 would have to be obtained by integrating the equation along the whole length of the mean streamline, starting at the point where it entered the region of turbulent flow. In non-mathematical language, q2 depends on the whole history of the flow, in particular the history of the mean rate of strain which appears in the production terms [(3) above].