Parametric reduced-order modeling and mode sensitivity of actuated cylinder flow from a matrix manifold perspective

TL;DR

This work develops a geometry-driven framework for parametric POD-Galerkin ROMs by exploiting matrix-manifold structures. By representing POD subspaces on the Grassmann manifold $\mathrm{Gr}(N,r)$ and POD modes on the Stiefel manifold $\mathrm{St}(N,r)$, the authors define subspace and mode sensitivities via exponential/logarithmic maps and geodesic distances, enabling tangent-space interpolation to construct parametric ROMs. Key findings show that the inverse subspace-sensitivity scales with the Roshko number $Ro=Re\cdot St$, approaching zero near critical regimes such as Hopf bifurcation, and that direct POD-mode interpolation can fail, whereas subspace interpolation yields orthonormal pseudo-POD modes and improved flow-field reconstructions. The results guide subspace sampling strategies, suggesting denser sampling in high-sensitivity regions to achieve accurate ROMs with limited data, and demonstrate the approach on flow around a rotating cylinder. Overall, the paper provides a rigorous, geometry-based pathway to efficient and reliable parametric ROMs for complex fluid dynamics problems.

Abstract

We present a framework for parametric proper orthogonal decomposition (POD)-Galerkin reduced-order modeling (ROM) of fluid flows that accommodates variations in flow parameters and control inputs. As an initial step, to explore how the locally optimal POD modes vary with parameter changes, we demonstrate a sensitivity analysis of POD modes and their spanned subspace, respectively rooted in Stiefel and Grassmann manifolds. The sensitivity analysis, by defining distance between POD modes for different parameters, is applied to the flow around a rotating cylinder with varying Reynolds numbers and rotation rates. The sensitivity of the subspace spanned by POD modes to parameter changes is represented by a tangent vector on the Grassmann manifold. For the cylinder case, the inverse of the subspace sensitivity on the Grassmann manifold is proportional to the Roshko number, highlighting the connection between geometric properties and flow physics. Furthermore, the Reynolds number at which the subspace sensitivity approaches infinity corresponds to the lower bound at which the characteristic frequency of the Kármán vortex street exists (Noack & Eckelmann, JFM, 1994). From the Stiefel manifold perspective, sensitivity modes are derived to represent the flow field sensitivity, comprising the sensitivities of the POD modes and expansion coefficients. The temporal evolution of the flow field sensitivity is represented by superposing the sensitivity modes. Lastly, we devise a parametric POD-Galerkin ROM based on subspace interpolation on the Grassmann manifold. The reconstruction error of the ROM is intimately linked to the subspace-estimation error, which is in turn closely related to subspace sensitivity.

Figures (24)

• Figure 1: Schematic of the representation of sets of POD modes extracted from the flow field dataset in a wide range of flow parameters in terms of a matrix manifold $\mathcal{M}$. The variation of the characteristic structures of the fluid flow around a rotating cylinder with the Reynolds number $Re$ and rotation rate $\alpha$ are described as curves on $\mathcal{M}$. The relationship between the matrix manifold $\mathcal{M}$ and a tangent vector space at $p$, represented as $T_{p}\mathcal{M}$, is also described. Tangent vectors in the tangent-vector space in the Reynolds-number and rotation-rate directions are represented as $\Delta_{Re}$ and $\Delta_{\alpha}$, respectively.
• Figure 2: Schematic of the flow field to be described on matrix manifolds in this study: (a) sketch of the flow field around a rotating cylinder; (b,c) instantaneous spatial distributions of $x$-component velocity at $Re=100$ and $160$ without rotation; (d,e) with the rotation rate of $\alpha=1.0$.
• Figure 3: Comparison of the Strouhal--Reynolds number between the results obtained by the numerical simulation (circle symbol) and an empirical theory (solid line).
• Figure 4: Spatial distributions of the first POD modes at $Re=100$, $120$, $140$, and $160$: (a,b,c,d) at the rotation rate of $\alpha=0.0$; (e,f,g,h) first POD modes at $\alpha=0.8$; (i,j,k,l) first POD modes at $\alpha=1.6$.
• Figure 5: Sensitivity of the subspace with respect to the Reynolds-number variation as a function of the Reynolds number when the dimension of the subspace is 12.