Extracting self-similarity from data

TL;DR

This work tackles the challenge of extracting self-similarity from data without relying on known governing equations or boundary conditions. It introduces a two-stage framework in which Step 1 searches for similarity variables through a dilation/translation-based transformation to collapse profiles across times, and Step 2 uses symbolic regression to obtain analytic forms of the similarity transformations under dimensional-homogeneity constraints. The approach is demonstrated on five influential fluid-mechanics problems, successfully recovering classical similarity relations for four and revealing an empirical Taylor-microscale-like length scale in decaying turbulence. The framework provides a general, data-driven pathway to uncover scaling laws and similarity transformations in complex systems, with publicly available code to facilitate application and extension.

Abstract

Identifying self-similarity is key to understanding and modelling a plethora of phenomena in fluid mechanics. Unfortunately, this is not always possible to perform formally in highly complex flows. We propose a methodology to extract the similarity variables of a self-similar physical process directly from data, without prior knowledge of the governing equations or boundary conditions, based on an optimization problem and symbolic regression. We analyze the accuracy and robustness of our method in five problems which have been influential in fluid mechanics research: a laminar boundary layer, Burger's equation, a turbulent wake, a collapsing cavity, and decaying turbulence. Our analysis considers datasets acquired via both numerical and wind tunnel experiments. The algorithm recovers the known self-similarity expressions in the first four problems and generates new insights on single length scale theories of homogeneous turbulence.

Figures (8)

• Figure 1: Data-driven identification of self-similarity in the Blasius boundary layer. (a) Streamwise velocity field (Blasius solution). The dashed vertical lines denote the nine velocity profile sampling stations. (b) Sampled velocity profiles. (c) Algorithmically collapsed velocity profiles. (d) Algorithmically identified scaling $\alpha$ of the wall-normal coordinate $y$. (e) Algorithmically identified scaling $\beta$ of the streamwise velocity $u$.
• Figure 2: Data-driven identification of self-similarity in Burgers' equation. (a) Spatiotemporal evolution of the velocity. (b) Extracted profiles at different time instants. Markers show the data given to the algorithm. (c) Algorithmically collapsed profiles.
• Figure 3: Data-driven identification of self-similarity in the wake of a porous plate. (a) Schematic of the experimental apparatus showing the flume, porous plate and PIV configuration. (b) Mean streamwise velocity profiles at different locations downstream of the plate. (c) Algorithmically collapsed velocity profiles.
• Figure 4: Data-driven identification of self-similarity in a collapsing cavity. Three-dimensional visualization of the liquid-gas interface at (a) $t/\tau=0$, (b) $t/\tau=0.25$, and (c) $t/\tau=0.5$, where $\tau=\sqrt{\rho R_0^3 / \gamma}$ is the inertio-capillary timescale. (d) Liquid-gas interface time evolution, $t/\tau = \left(0, 0.05, \dots, 0.5 \right)$. (e) Interface profiles near cavity collapse. (f) Algorithmically collapsed interface profiles near cavity collapse. (g-i) Identified transformations $\alpha$, $\beta$, and $\gamma$. Comparison with theoretical scaling laws.
• Figure 5: Data-driven identification of self-similarity in decaying turbulence. (a) Schematic of the experimental set-up. (b) Experimentally measured power spectral densities. The dashed vertical line delineates the range of the spectrum that is used. (c) Measured spectrum normalized by the inertial scales. (d) Measured spectrum normalized by the Kolmogorov scales. (e) Measured spectrum normalized by the algorithmically identified expression (Eq. \ref{['eq:compositeturb']}).