Fig 32.5 Velocity Correlation
- A statistical correlation can be applied to fluctuating velocity terms in turbulence. Turbulent motion is by definition eddying motion. Not withstanding the circulation strength of the individual eddies, a high degree of correlation exists between the velocities at two points in space, if the distance between the points is smaller than the diameter of the eddy. Conversely, if the points are so far apart that the space, in between, corresponds to many eddy diameters (Figure 32.5), little correlation can be expected.
- Consider a statistical property of a random variable (velocity) at two points separated by a distance r. An Eulerian correlation tensor (nine terms) at the two points can be defined by
In other words, the dependence between the two velocities at two points is measured by the correlations, i.e. the time averages of the products of the quantities measured at two points. The correlation of the
components of the turbulent velocity of these two points is defined as
It is conventional to work with the non-dimensional form of the correlation, such as
A value of R(r) of unity signifies a perfect correlation of the two quantities involved and their motion is in phase. Negative value of the correlation function implies that the time averages of the velocities in the two correlated points have different signs. Figure 32.6 shows typical variations of the correlation R with increasing separation r .
The positive correlation indicates that the fluid can be modelled as travelling in lumps. Since swirling motion is an essential feature of turbulent motion, these lumps are viewed as eddies of various sizes. The correlation R(r) is a measure of the strength of the eddies of size larger than r. Essentially the velocities at two points are correlated if they are located on the same eddy
- To describe the evolution of a fluctuating function u'(t), we need to know the manner in which the value of u' at different times are related. For this purpose the correlation function
- The correlation studies reveal that the turbulent motion is composed of eddies which are convected by the mean motion . The eddies have a wide range variation in their size. The size of the large eddies is comparable with the dimensions of the neighbouring objects or the dimensions of the flow passage.
The size of the smallest eddies can be of the order of 1 mm or less. However, the smallest eddies are much larger than the molecular mean free paths and the turbulent motion does obey the principles of continuum mechanics.
Fig 32.6 Variation of R with the distance of separation, r
