  We propose a customized convolutional neural network based autoencoder called
a hierarchical autoencoder, which allows us to extract nonlinear autoencoder
modes of flow fields while preserving the contribution order of the latent
vectors. As preliminary tests, the proposed method is first applied to a
cylinder wake at $Re_D$ = 100 and its transient process. It is found that the
proposed method can extract the features of these laminar flow fields as the
latent vectors while keeping the order of their energy content. The present
hierarchical autoencoder is further assessed with a two-dimensional $y-z$
cross-sectional velocity field of turbulent channel flow at $Re_{\tau}$ = 180
in order to examine its applicability to turbulent flows. It is demonstrated
that the turbulent flow field can be efficiently mapped into the latent space
by utilizing the hierarchical model with a concept of ordered autoencoder mode
family. The present results suggest that the proposed concept can be extended
to meet various demands in fluid dynamics including reduced order modeling and
its combination with linear theory-based methods by using its ability to
arrange the order of the extracted nonlinear modes.
