5.4: Pancakes and noodles- the geometry of turbulence

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In incompressible flow, the sum of the principal strains is zero:

\[\lambda^{(1)}+\lambda^{(2)}+\lambda^{(3)}=\operatorname{Tr}(e)=\vec{\nabla} \cdot \vec{u}=0. \nonumber \]

This is easily seen in the principal frame, and can be shown to be true in any frame because both the eigenvalues and the trace are scalars. (Exercise: prove this.)

If we order the eigenvalues \(\left{\{\lambda(1)\ ,\lambda(2)\ ,\lambda(3) \right\}\) from smallest to largest, then \(\lambda(1) < 0\) and \(\lambda(3) > 0\), whereas \(\lambda(2)\) can have either sign. In any region of space where \(\lambda(2) > 0\) there are two extensional strains and one compressive strain. As a result, a spherical fluid parcel would be distended into an oblate spheroid or, less technically, a “pancake”. In regions where \(\lambda(2) < 0\), there are two compressive strains and one extensional strain resulting in a prolate spheroid, or “noodle”.

In the 1990s, computational hardware advances allowed the simulation of turbulent flows to determine which flow geometry is dominant. The procedure was to compute the eigenvalues of the strain rate tensor at every point in space, then see whether positive or negative values of \(\lambda(2) < 0\) were more common. Invariably, it was found that \(\lambda(2)\) has a tendency to be positive, i.e., a turbulent strain field is more likely to produce pancakes than noodles. An example from a flow similar to that shown in Figures 5.3.4 and 5.3.5 is shown in Figure 5.3.12 (middle frame).

The rotation field is dominated by one-dimensional regions of rapid rotation as sketched in figure 5.3.1, i.e., vortices. These tend to wrap the pancakes around themselves to form what we might think of as crepes.