Source-linked AI summary
Cluster-based reduced-order modelling of a mixing layer
Eurika Kaiser, Bernd R. Noack, Laurent Cordier, Andreas Spohn, Marc Segond, Markus Abel, Guillaume Daviller, Jan Östh, Siniša Krajnović, Robert K. Niven
TL;DR
The paper addresses reduced-order modelling of complex, high-dimensional unsteady flows and the identification of their physical mechanisms. It proposes CROM, which clusters snapshots into representative states and models state transitions probabilistically, and shows that the framework identifies quasi-attractors and transition processes in several flow examples. Its conclusions are bounded by cluster-size diffusion and by the dynamics represented in the data and Markov assumption.
Problem
High-dimensional turbulent-flow models make physical mechanisms difficult to understand and complicate realistic optimisation and real-time control.
Method
CROM clusters flow snapshots into representative centroid states, estimates their probabilities, and models transitions among them with a Markov process.
Results
CROM identifies quasi-attractors and transition processes in an unsupervised manner across the Lorenz attractor, mixing layer, and turbulent wake examples.
Takeaways & Limitations
CROM provides tools for identifying transition regions, singular events, metastable states, bimodality, hysteresis, and multi-attractor behaviour from observational data.
Takeaways & Limitations
CROM has uncertainty from cluster size and can reveal only dynamics covered by the data ensemble; its Markov assumption can fail when flow solutions depend on history.
Abstract
from arXiv · showhide
We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Secondly, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.
1. Introduction
The paper introduces CROM, a cluster-based reduced-order modelling strategy that combines snapshot clustering with Markov modelling of transitions between flow states. It applies this framework to canonical and turbulent flows to identify quasi-attractors and transition processes without supervision.
- Model reduction addresses the challenge of understanding and controlling high-dimensional turbulent flows by retaining degrees of freedom relevant to the phenomenon and modelling purpose.
- CROM combines cluster analysis of flow observations with a Markov model for transitions between the resulting flow states.
- Clustering organises snapshots into geometrically close groups and represents each group by a centroid, using k-means to maximise within-cluster similarity and minimise between-cluster similarity.
- CROM generalises the Ulam-Galerkin method for finite-rank approximation of the Perron-Frobenius operator, while using cluster analysis to encode information through the distance metric.
- The framework is applied to the Lorenz attractor, a two-dimensional spatially developing mixing layer, and a three-dimensional turbulent Ahmed-body wake.
- Across these examples, CROM identifies quasi-attractors and transition processes in an unsupervised manner.
2. Cluster-based reduced-order modelling
CROM combines a kinematic snapshot-based description of flow states with a dynamical model for transitions among cluster states, then characterises the resulting attractor.
- CROM combines cluster analysis of snapshot ensembles for a low-order kinematic description with a dynamical model for the cluster states.
- Criteria defined in the methodology characterise the attractor and are later applied to the Lorenz attractor and mixing layer.
2.1. Kinematic analysis
The kinematic component of CROM represents snapshot ensembles through a similarity metric, k-means clusters, and centroid-based reduced states. The procedure can operate directly on snapshots or through a POD coefficient representation for computational convenience.
- A chosen metric measures snapshot similarity, and k-means partitions the state space into clusters whose means provide a reduced description.
- CROM begins with an ensemble of snapshots describing velocity fields at discrete times, which may come from experiments or numerical simulations.
- CROM is designed primarily for attractor data, although snapshots may also originate from transient processes.
- POD coefficient vectors can represent snapshots for clustering and greatly reduce the cost of subsequent distance evaluations, but POD preprocessing has a comparable correlation-integral cost.
- The clustering objective represents the snapshot ensemble by K clusters and centroids while minimising total cluster variance.
- For periodic flows, sufficiently many clusters produce centroids that represent phase averages of snapshots in each cluster.
2.2. Dynamical model
The dynamical component augments cluster states with probabilities and a cluster transition matrix, yielding a discrete-time Markov model for probability evolution. Its validity depends on a memoryless approximation whose applicability is constrained by flow history effects and boundary conditions.
- CROM estimates cluster probabilities from snapshot counts and uses a cluster transition matrix as the propagator for the probability distribution.
- The transition matrix is left stochastic: its entries are non-negative and each column sums to one, preserving probability normalisation.
- CROM orders clusters using state and transition probabilities to extract a most probable path and identify dynamically important clusters through iteration and eigenvectors.
- The Markov assumption makes the next state depend only on the current state, but Navier-Stokes dynamics can retain history under discontinuous boundary conditions or history-dependent constitutive laws.
- The probability vector evolves by the discrete-time iteration p_l+1 = P p_l, with P encoding one-step transitions between clusters.
- The transition matrix approximates the Perron-Frobenius operator for discrete states and discrete times.
- For smoothly varying boundary conditions, the model may need to be time-dependent rather than time-homogeneous.
2.3. Attractor characterisation
The attractor is characterised geometrically through snapshot and cluster statistics, and dynamically through propagator mixing, transition-matrix comparison, and entropy-based measures.
- The attractor diameter provides an upper bound and reference level for the considered distances.
- Cluster diameters and standard deviations quantify the size and homogeneity of each cluster.Similar cluster diameters indicate a relatively homogeneous state-space partition.
- The cluster distance matrix measures distances between centroids and estimates trajectory lengths for direct cluster transitions.
- The finite-time Lyapunov exponent characterises the mixing property of the propagator through the divergence of initially close probability distributions.Its dependence on the initial condition decreases as the propagation time increases, and it vanishes when trajectories converge to the fixed point.
- The dynamically asymptotic propagator is compared with the cluster-analysis transition matrix, which is constructed from weighted-average realisations.The two matrices should approach a common value for an ergodic system with very large sample numbers.
- Kullback-Leibler entropy measures the uncertainty of finite-time transitions relative to the transition matrix and provides an ergodicity measure through convergence toward H = 0.The associated divergence is zero when P = Q and positive when P ≠ Q.
3. Lorenz attractor as illustrating example
CROM clusters the Lorenz attractor into nearly homogeneous regions and uses the cluster transition matrix to recover its two oscillatory ears, branching region, transition pathways, and convergence behaviour without prior attractor knowledge.
- Setup: CROM is applied to the Lorenz equations as an introductory example, using snapshot data to represent the attractor’s dynamics.The trajectory oscillates around the two non-zero fixed points before switching between the ears.
- Cluster structure: Using Kc = 10 clusters, CROM partitions the attractor into colour-coded sectors that resolve three groups: transition clusters T and ear groups E+ and E−.The distance matrix confirms that centroids within each of these regions are comparably close.
- Transition dynamics: The cluster transition matrix identifies cyclic motion within each ear and state k = 1 as a branching point connecting trajectories to the two ears.Transitions proceed through clusters 3 →4 →5 →6 around E+, while the reverse-numbered sequence represents E−; the final state returns to k = 1 to complete an orbit.
- Transition dynamics: After l = 10 iterations, transitions from cluster k = 3 cannot yet reach the intermediate state k = 1, whereas a full circulation around one fixed point requires more than 10 time steps.From k = 5, transitions can reach k = 1, restart the same ear orbit at k = 3, or enter the opposite orbit at k = 10.
- Convergence: Iterating the transition matrix smooths probabilities toward an asymptotic distribution, with convergence of multiple dynamical and statistical properties at approximately l ≈102.The converged distribution agrees with the dominant eigenvector, while entropy also indicates the validity regime of the continuous-time Markov approximation.
- Conclusions: The Lorenz analysis recovers attractor structure without prior knowledge, and its main conclusions persist under changes in cluster number and snapshot time step.Increasing K adds state resolution and branching detail, while transition probabilities depend approximately linearly on the snapshot time step.
4. Mixing layer model
CROM models the mixing layer as clustered flow states and probabilistic transitions, resolving Kelvin–Helmholtz shedding, vortex pairing, and the transition between them. Its centroids capture phase-aligned flow evolution, while the asymptotic state and prediction horizon characterize forecast behavior.
- Flow model: The DNS mixing layer contains successive Kelvin–Helmholtz roll-up and vortex-pairing events, including mergers that broaden the shear layer.The simulation uses velocity ratio r = 3 and Reynolds number Re = 500.
- Cluster representation: CROM uses ten clusters of POD coefficient vectors to separate vortex-pairing-dominated flows from Kelvin–Helmholtz-dominated flows.The temporal POD coefficients show that decreasing first-mode-pair amplitudes and increasing second-mode-pair amplitudes indicate the regime change.
- Cluster representation: The centroids are ordered along dynamical evolution: vortex-pairing centroids advance by about one-sixth wavelength, Kelvin–Helmholtz centroids by about one-third wavelength, and cluster k = 1 is intermediate.Unlike POD modes, CROM resolves structures by phase and explicitly captures transient states.
- Transition dynamics: The transition structure contains two cyclic groups for Kelvin–Helmholtz shedding and vortex pairing, connected through an intermediate bifurcation or flipper cluster.The distance matrix and transition matrix expose the geometric and temporal relations among the VP, KH, and transition clusters.
- Prediction and convergence: The model converges toward an asymptotic state that defines its prediction horizon and whose invariant distribution corresponds to the trajectory’s fraction of time in each cluster.Probability distributions diffuse after trajectories diverge and converge after the prediction horizon; the snapshot time step has little effect when small relative to the characteristic period.
5. Interpretation of CROM as a generalised Ulam-Galerkin method
CROM is interpreted as a data-driven finite-rank approximation of the Perron-Frobenius operator, extending Ulam-Galerkin through cluster-based state-space partitioning.
- The probabilistic formulation describes evolution of ensembles through probability densities rather than individual trajectories.
- The Ulam-Galerkin method approximates the Perron-Frobenius operator by projecting the Liouville equation onto characteristic functions of Voronoi cells.
- Its transition entries represent the fraction of a cell mapped into another cell after one application of the dynamical map.
- CROM’s cluster transition matrix provides another finite-rank approximation of the Perron-Frobenius operator.
- CROM extends Ulam-Galerkin by replacing geometrical tessellation with k-means-based clustering, allowing the distance metric to be modified.
6. Discussion
The discussion contrasts CROM’s probabilistic, cluster-based representation with POD’s deterministic projection models and highlights complementary strengths, uncertainty, and control potential.
- CROM yields a probabilistic dynamical model for state-variable evolution, while POD models provide deterministic reduced descriptions.
- POD modes span a linear subspace and may not correspond to physical flow realisations, whereas CROM centroids approximate snapshots within clusters.
- For periodic flows, POD produces harmonic mode pairs, while CROM discretises the cycle into phase bins and covers snapshots geometrically.
- CROM represents states with bounded cluster probabilities, unlike POD coefficients that can assume arbitrary values by construction.
- A CROM-based control strategy is outlined to move the probability distribution toward a predefined distribution.
7. Conclusions
The conclusions present CROM as an unsupervised framework for exposing attractor structure and transitions, while noting uncertainty from clustering and dependence on the data ensemble.
- CROM identifies physical mechanisms and approximates the Perron-Frobenius operator through cluster analysis and ensemble probability dynamics.
- Cluster size introduces probability diffusion, and the model can reveal only dynamics represented in the data ensemble.
- The framework connects cluster transitions with symbolic-dynamics entropy through correspondences among clusters, distances, sampling time, and exit times.
- When attractors are separated, CROM’s cluster analysis can coincide with ergodic partitioning.
- CROM exposes attractor features and can detect quasi-attractors, transitions, singular events, metastable states, bimodality, hysteresis, and multi-attractor behaviour.
Appendix A. Example with broadband turbulence and simple
Applied to the turbulent Ahmed-body wake, CROM resolves two asymmetric quasi-attractors, their transition region, force signatures, and a dominant oscillatory frequency.
- The Ahmed-body wake is a highly turbulent, broadband flow with a documented bimodal switch between two asymmetric states.
- Ten clusters form periodic groups T, B1, and B2, with B1 and B2 weakly connected and T acting as their transition region.
- B1 and B2 centroids contain strong vortical structures in opposite directions, whereas T is weaker and more symmetric about z = 0.
- Clusters k = 5 and k = 8 are branching clusters connecting the asymmetric groups to the transition region.
- The side-force coefficient changes sign between asymmetric states, while drag and lift show no significant drift.
- St = 0.17 is the dominant frequency of the least-damped oscillatory mode in the transition-matrix spectrum.
Appendix B. Towards CROM-based flow control
Appendix B develops a CROM-based control design that steers the Markov-model probability distribution toward a desired target distribution. The strategy uses volume-force actuation, linear full-state feedback, and an optimization horizon that tunes control aggressiveness.
- The control objective is to bring the probability distribution p close to a desired distribution ptarget represented by Ptarget.
- The controlled Galerkin model combines unforced dynamics f with an actuation term g in the Liouville equation for p(a, t).
- The actuation model assumes a single volume force and searches for a linear full-state feedback control law.
- For each cluster, the actuation is estimated from its centroid, yielding a corresponding actuation transition matrix for the Markov model.
- The control design determines K from an optimality condition, with short horizons producing aggressive laws and long horizons producing conservative laws.
- The linear control law incorporates nonlinear actuation dynamics resolved by the clustered state space and may exploit effects such as frequency cross-talk.
Appendix C. Perron-Frobenius and Koopman operators
Appendix C relates the Liouville, Perron-Frobenius, and Koopman operators for nonlinear dynamical systems. It explains how probability densities and observables evolve through linear operators and connects this framework to CROM and DMD.
- The Liouville, Perron-Frobenius, and Koopman operators are linear despite describing nonlinear system dynamics.
- The Perron-Frobenius operator advances probability density functions forward in time and is formally expressed as exp(tL̂).
- The Perron-Frobenius operator also admits an interpretation through the flow map, whose kernel represents a conditional probability density for future trajectory locations.
- The Koopman operator governs observable evolution and is adjoint to the Perron-Frobenius operator.
- CROM is strongly connected to Koopman analysis and offers an approach for complex dynamical systems where analytical mode construction is impractical.
Appendix D. Visualisation of the cluster topology
Appendix D describes low-dimensional visualization of CROM cluster topology using projections of centroids. The projection is designed to approximately preserve centroid distances, supporting interpretation of attractor dynamics.
- High-dimensional trajectory visualization can be challenging for velocity snapshots or POD coefficients while preserving important data properties.
- CROM visualizes cluster arrangements in state space to support understanding of dynamics on the attractor, using centroids instead of velocity snapshots.
- The centroids are projected onto leading covariance eigenvectors, producing low-dimensional points whose distances approximate those between the original centroids.
- For visualization, the method uses r = 2, with the projected cluster-centroid coordinates represented by POD mode amplitudes αk.
- Multidimensional scaling provides a more general distance-preserving low-dimensional representation, including non-Euclidean distance metrics.