Some Measurements on Fluctuating Components

A principal problem with the Reynolds-Average NS Equations is that we introduce 6 additional stress components.

In general, engineers are interested only in the mean motion,--- the motion described by Eqn. (7.11). However, Eqn (7.11) can not be calculated without knowing the turbulent stress terms. This is the well known closure problem of turbulent flows. We shall now turn to some classic measurements. We start with a typical signal from a hot wire anemometer (Fig 7.4). It is easy to see that the fluctuations do not follow any clear pattern.

Figure 7.4

If the records shown in Figure are taken for a sufficiently long time (the order of few seconds), the fluctuations average out to zero. A measure of the level of fluctuations is taken to be the root mean square of the fluctuating quantity, Thus may be called the intensity of turbulence as well. Such quantities have been measured for the flow of air in a circular pipe by Laufer in 1954. Let us look at his results.

Figure 7.5

Figure 7.5 shows turbulent intensities as a function of non-dimensional distance from the pipe wall. The intensities are normalized with respect to the maximum time average velocity in the pipe. The maximum intensities occur in the main flow direction (along the axis) while the minimum intensities occur in a direction normal to the pipe wall. This is explained by the fact that the wall damps out the turbulent intensity in the normal direction in a much stronger manner than in the other directions.

Figure 7.6

Figure 7.6 shows measurements close to the wall, again by Laufer. This is one of the early measurements that were carried out so close to the wall. Note that the first point used for the streamwise intensity is at 0.0003 of the pipe radius. For other two intensities (sensors have to be larger), he could go upto 0.002 t 0.003 of the pipe radius. The figure also shows another form of non-dimensionalization using u* or, also known as friction velocity.

(7.12)

Where is the wall shear stress. The distance can also be non-dimensionalized using u*

(7.13)