Manifold-Adapted Sparse RBF-SINDy: Unbiased Library Construction and Unsupervised Discovery of Dynamical States in Turbulent Wall Flows

Abstract

The turbulent attractor of wall bounded flows is not a structureless strange set but contains a skeleton of dynamically distinct states connected by rare directed transitions whose geometry is reflected in the invariant measure of the phase space trajectory. We show that this skeleton can be recovered from wall measurements alone, namely wall pressure and wall shear stress, without physical labels or prior knowledge, provided that the data driven function library used to identify the dynamics respects the intrinsic geometry of the attractor rather than the variance hierarchy of the POD representation. Standard sparse identification approaches introduce two structural biases during library construction. First, the steep decay of POD spectra causes Euclidean distances in k means clustering to be dominated by leading modes, collapsing basis function centres into a low dimensional subspace and leaving transitional dynamics poorly represented. Second, turbulent trajectories slow near quasi invariant states, so uniform time sampling over represents these regions and under samples rapid transitions. Both biases are corrected by resampling the trajectory uniformly in arc length and replacing the Euclidean metric with a Mahalanobis metric derived from the local cluster covariance. A single sparse regression on this corrected library yields a reduced model. Applied to a minimal turbulent channel at low Reynolds number, unsupervised clustering reveals two phases of the near wall cycle: stable streak states and burst initiating instabilities corresponding to the coherent structure skeleton of the flow. The model reproduces the invariant measure, reaches the Lyapunov predictability horizon and provides a differentiable vector field on which invariant solutions can be located by Newton iteration.

Figures (11)

• Figure 1: Schematic of the modelling pipeline. Wall measurements $\{P_\mathrm{wall},\tau_x,\tau_z\}$ at time $T_i$ are encoded onto POD latent coefficients $\{\alpha_j\}$, advanced by the learned RBF-SINDy ODE, and decoded back to wall fields at time $T_i + \Delta t$.
• Figure 2: Comparison of isotropic Gaussian RBF (left) and Mahalanobis Gaussian RBF (right) for the same cluster centroid. The isotropic function assigns equal bandwidth in all directions regardless of the local data geometry; the Mahalanobis function stretches along low-variance directions and compresses along high-variance ones, matching the ellipsoidal shape of the cluster.
• Figure 3: Three representative clusters (1, 10, 20) visualised in the space of the first three standardised POD coefficients. The ellipsoids represent the one-sigma Mahalanobis surface $(\bm{a} - \bm{c}_k)^\top\bm{\Sigma}_k^{-1}(\bm{a}-\bm{c}_k) = 1$. The varying size and orientation reflect the strongly anisotropic and heterogeneous geometry of the attractor.
• Figure 4: Left: cumulative energy as a function of POD mode index, with the 95% threshold at mode 83 indicated. Right: wall-pressure $P_\mathrm{wall}$, streamwise shear $\tau_x$, and spanwise shear $\tau_z$ contours for modes 1, 3 and 5 (columns), illustrating the physical content of the leading modes.
• Figure 5: Average residence time versus velocity coherence for the 32 clusters returned by G-means on the arc-length resampled, Mahalanobis-corrected trajectory. Two populations are clearly separated: high-residence / low-coherence clusters (upper left) correspond to quasi-steady streak states; low-residence / high-coherence clusters (lower right) correspond to burst-initiating instabilities.