A procedure based on energy stability arguments is presented as a
method for extracting large-scale, coherent structures from fully
turbulent shear flows. By means of two distinct averaging operators,
the instantaneous flow field is decomposed into three components: a
spatial mean, coherent field and random background fluctuations. The
evolution equations for the coherent velocity, derived from the
Navier–Stokes equations, are examined to determine the mode that
maximizes the growth rate of volume-averaged coherent kinetic
energy. Using a simple closure scheme to model the effects of the
background turbulence, we find that the spatial form of the maximum
energy growth modes compares well with the shape of the empirical
eigenfunctions given by the proper orthogonal decomposition. The
discrepancy between the eigenspectrum of the stability problem and
the empirical eigenspectrum is explained by examining the role of
the mean velocity field. A simple dynamic model which captures the
energy exchange mechanisms between the different scales of motion is
proposed. Analysis of this model shows that the modes which attain
the maximum amplitude of coherent energy density in the model
correspond to the empirical modes which possess the largest
percentage of turbulent kinetic energy. The proposed method provides
a means for extracting coherent structures which are similar to
those produced by the proper orthogonal decomposition but which
requires only modest statistical input.