Learning the Latent dynamics of Fluid flows from High-Fidelity Numerical Simulations using Parsimonious Diffusion Maps

Abstract

We use parsimonious diffusion maps (PDMs) to discover the latent dynamics of high-fidelity Navier-Stokes simulations with a focus on the 2D fluidic pinball problem. By varying the Reynolds number, different flow regimes emerge, ranging from steady symmetric flows to quasi-periodic asymmetric and turbulence. We show, that the proposed non-linear manifold learning scheme, identifies in a crisp manner the expected intrinsic dimension of the underlying emerging dynamics over the parameter space. In particular, PDMs, estimate that the emergent dynamics in the oscillatory regime can be captured by just two variables, while in the chaotic regime, the dominant modes are three as anticipated by the normal form theory. On the other hand, proper orthogonal decomposition (POD)/PCA, most commonly used for dimensionality reduction in fluid mechanics, does not provide such a crisp separation between the dominant modes. To validate the performance of PDMs, we also computed the reconstruction error, by constructing a decoder using Geometric Harmonics. We show that the proposed scheme outperforms the POD/PCA over the whole Reynolds number range. Thus, we believe that the proposed scheme will allow for the development of more accurate reduced order models for high-fidelity fluid dynamics simulators, thus relaxing the curse of dimensionality in numerical analysis tasks such as bifurcation analysis, optimization and control.

Figures (18)

• Figure 1: Schematic representation of the fluidic pinball configuration.
• Figure 2: Initial condition (left panels) and instantaneous snapshot after the transient (right panels) of the vorticity field $\omega(x,y)$ by varying the Reynolds number: $Re = 10$ ((a)-(b)); $30$ ((c)-(d)); $80$ ((e)-(f)); $105$ ((g)-(h)); $130$ ((i)-(j)).
• Figure 3: Left panels: temporal evolution of vorticity $\omega(x = 10, y, t)$ at the vertical locations $y=-1$ (red curves) and $y=1$ (blue curves). Right panels: $\omega(x = 10, y=1, t)$ plotted as a function of $\omega(x = 10, y=-1, t)$ including (black dots) and excluding (red circles) the transient, reporting also the -1 slope line (black dashed curve). From top to bottom: $Re = 10$ ((a)-(b)); $30$ ((c)-(d)); $80$ ((e)-(f)); $105$ ((g)-(h)); $130$ ((i)-(j)).
• Figure 4: Mean absolute value (blue curve) and standard deviation (red curve) of the quantity $\Delta \omega$ (see (\ref{['eq:Deltaom']})) by varying the Reynolds number $Re$. The cases $Re=10$, $30$, $80$, $105$ and $130$ discussed so far are highlighted by vertical blue dashed lines, while the vertical black continuos lines denote the regime transition thresholds $Re \approx 18$ (from steady symmetric to periodic symmetric), $Re \approx 68$ (from periodic symmetric to periodic asymmetric), $Re \approx 104$ (from periodic to quasi-periodic asymmetric) and $Re \approx 115$ (from quasi-periodic asymmetric to chaotic statistically symmetric) reported by Deng_Noack_2020.
• Figure 5: Schematic representation of the encoding (blue dots and arrows) via the Nyström method to the DMs and the decoding operations via Geometric Harmonics and "double" DMs (red dots and arrows). The black dots denote observations in the ambient space $\mathbb{R}^{N}$ and their representations on the DMs and "double" DMs spaces $\mathbb{R}^{D}$ and $\mathbb{R}^{K}$, respectively. The Nyström method is performed for obtaining the image $\mathcal{R}\left(\boldsymbol{\omega}_{l}^{n}\right)$ of the new point $\boldsymbol{\omega}_{l}^{n}$. The reconstruction with GHs is performed for obtaining the pre-image $\mathcal{L}\left(\boldsymbol{\psi}_{l}^{n}\right)$ of a new point $\boldsymbol{\psi}_{l}^{n}=\mathbf{u}_l^n \boldsymbol{\Lambda}$. As described in § \ref{['subsec:lifting']} for the GHs extension, projection to the "double" DMs space is first performed for defining a new basis in $\mathbb{R}^{K}$ through which one obtains the reconstructed state in $\mathbb{R}^{N}$.