Nan Deng Luc R. Pastur School of Mechanical Engineering and Automation Unité de Mécanique, IMSIA Harbin Institute of Technology, Shenzhen ENSTA-Paris, Institut Polytechnique de Paris 518055, People's Republic of China F-91120 Palaiseau, France dengnan@hit.edu.cn luc.pastur@ensta-paris.fr Marek Morzyński Department of Virtual Engineering Poznań University of Technology PL 60-965 Poznań, Poland morzynski@virtual.edu.pl Bernd R. Noack School of Mechanical Engineering and Automation Harbin Institute of Technology, Shenzhen 518055, People's Republic of China bernd.noack@hit.edu.cn
ABSTRACT
We propose three kinds of mean-field modeling strategies for the complex dynamics generally found in fluid mechan- ics. A key enabler is a mean-field assumption, where slowly- varying mean-field deformations are due to the fluctuating field through the Reynolds stress, resulting in a Reynolds- like decomposition. We have developed projection-based and cluster-based reduced-order models, i.e., a least-order mean- field model for the successive bifurcations (Deng et al. , 2020), an aerodynamic force model associated with a Galerkin model (Deng et al. , 2021), and a hierarchical network model to au- tomate the identification of multi-attractor dynamics (Deng et al. , 2022).  These mean-field models are exemplified for a challenging test case of the fluidic pinball at Re = 80, charac- terized by six invariant sets (three steady solutions and three limit cycles) induced by the first two successive bifurcations of pitchfork and Hopf types. This work shows a paradigm for automatable reduced-order modeling of complex flows using first principles and machine learning techniques, balancing be- tween data-driven and physics-driven approaches and improv- ing model interpretability and generalizability.
MODELING COMPLEX FLOW DYNAMICS
The complexity of the fluid flow comes from its high di- mensionality, nonlinearity, and multiscale spatial and tempo- ral behavior. A massive amount of flow detail information will be generated for increasing resolution when dealing with fluid flow problems. It is hard to understand all these details of dif- ferent scales, and the control optimization is impossible to be applied.  Reduced-order modeling aims to obtain simplified, interpretable models of complex systems. Discussing and an- alyzing the dynamics in this low-dimensional approximated system makes it possible to understand the underlying mech- anisms and design the control laws.  Such low-dimensional models are essential for flow dynamics prediction, effective control, and optimization (Rowley & Dawson, 2017). In the era of big data, extracting physical laws of complex Flow configuration of the fluidic pinball.  A vor- Figure  l. ticity field is color-coded in the range [− 1.5, 1.5] from blue to red. Drag and lift coefficients are denoted with CD and CL . nonlinear phenomena and constructing parsimonious models from data is one of the frontier hotspots.  Artificial Intelli- gence (AI) methods and Machine Learning (ML) techniques are revolutionizing our traditional paradigm of modeling and control in fluid mechanics (Brunton et al. , 2020).  However, data-driven approaches cannot guarantee the models' stability, generalization, and physical interpretability.  Combining first principles and a priori physical information into the modeling process is the key to promising solutions to this problem. In this presentation, we will start with the weakly non- linear analysis under the mean-field consideration and develop three kinds of modeling strategies for the complex wake dy- namics of the fluidic pinball.  The resulting models are used to describe the transient and post-transient, multi-attractor dy- namics of the successive bifurcations, the instantaneous aero- dynamic force, and a multiscale topography of global and local dynamics .
Fluidic pinball
As shown in Fig. 1, the fluidic pinball consists of three fixed cylinders, whose axes are placed on the vertices of an equilateral triangle in the ( x, y ) plane and perpendicularly to this plane.  The gap distance is one radius between every two 1 cylinders. The upstream flow U ∞ is uniform in the streamwise direction e x at the inlet. The transient and post-transient dynamics at different Reynolds numbers are numerically investigated by Direct Nu- merical Simulation (DNS). At Re = 80, three steady solutions and three periodic solutions (limit cycles) can be found due to two successive instabilities, as shown in Fig. 2, henceforth providing a challenging test case for the derivation of a pre- dictive force model. We consider multiple trajectories starting from the vicinity of the three steady solutions.  More details for quasi-periodic and chaotic dynamics at higher Reynolds numbers can be found in Deng et al. (2020).
MEAN-FIELD ANSATZ
Under the mean-field consideration, the triple decompo- sition of the flow field reads
where the dominant angular frequency ω c is defined as the dominant peak in the Fourier spectrum of the velocity field. The velocity field is decomposed into a slowly-varying mean- flow field {u}r, a coherent component on timescales of order 2π/ως , involving coherent structures й, and a remaining (sup- posedly) non-coherent small scale fluctuations u ′ . This kind of decomposition follows the weakly nonlinear analysis in build- ing the low-order Galerkin models of Tadmor et al. (2011). From the mean-field theory, the slowly-varying mean- flow field evolves out of the steady solution under the action of the Reynolds stress associated with the most unstable eigen- mode(s). The mean-flow field deformation u A is used to de- scribe the difference between the slowly-varying mean flow and the invariant base flow u s (x), which reads
The linear dynamics only involve a minimal neighbor- hood of the base flow u 5 , as the perturbation is tiny at the onset of the instability.  As the perturbation develops, the nonlinear term cannot be ignored and will start to modify the base flow. Hence, the mean flow deformation comes from the effect of the Reynolds stresses of the coherent component ( ũ · V ) ũ and of the non-coherent small scale fluctuations (u' · V)u'. During the mean-field distortion from the base flow u s to the mean flow, the original linear dynamics is no longer valid, and the nonlinear interactions will drive the transient dynamics until saturation.  For the transient and post-transient dynamics, the slaving relation between the active modes (fluctuations) and the shift modes (mean-field distortion) and their timescale dif- ference can be used to improve model interpretability in a data- driven modeling process.
MEAN-FIELD MODELS
In this section, we briefly introduce the three kinds of mean-field modeling methodologies and discuss their results.
Least-order mean-field model
For the Galerkin framework, the velocity field u(x,t) is decomposed in a basic mode u 0 (x) with amplitude a 0 ≡ 1 and a fluctuating contribution
where the fluctuation is represented by a Galerkin approxi- mation of N orthonormal space-dependent modes u i (x), i = 1 , . . . , N , with time-dependent amplitudes a i ( t ) . The Galerkin expansion in Eq.(3) satisfies the incom- pressibility condition and the boundary conditions by con- struction.  The evolution equation for the mode amplitudes a j is derived by a Galerkin projection of the Navier-Stokes equa- tions onto the modes u i : with ( u , ∇ · u , ∇ u , ) cous, convective and pressure terms in the Navier-Stokes equations, respectively. The Galerkin system, as in Eq.(4), directly derived from the Galerkin projection of the leading Proper Orthogonal De- composition (POD) modes, cannot capture the transient and post-transient dynamics correctly.  It introduces extremely long transients due to the underestimated growth rate.  In a least-order mean-field model, we represent the flow dynam- ics with the fewest number of degrees of freedom, as shown in Fig. 3(left). The basic modes are chosen with the knowl- edge of the instabilities that the system has undergone while. increasing the Reynolds number.  Based on an optimal low- dimensional basis, a general Galerkin framework leads to the linear-quadratic Ordinary Differential Equations (ODE) with normal forms of the corresponding bifurcations, as shown in Fig. 3(right).  We allow the cross-terms to exist for the cou- pling relationship between the degrees of freedom of two bi- furcations. Our strategy is to correct some of the system coef- ficients by the knowledge provided by the linear stability anal- ysis and the asymptotic dynamics. The remaining coefficients of the nonlinear interaction terms are identified by a sparse re- gression algorithm with physics-based constraints, namely the constrained sparse Galerkin regression (Brunton et al. , 2016; Loiseau & Brunton, 2018). The least-order model is only five- dimensional but can reproduce the key features of the full dy- namics (Deng et al. , 2020).
Galerkin force model
Force computations on immersed bodies are of particular concern in engineering fluid mechanics. Despite the wide use of POD-Galerkin for reduced-order modeling in industry, sys- tematic investigations and interpretations of the aerodynamic force in the Galerkin framework are mostly missing.  Previ- ous studies of Noca et al. (1999) revealed that the force on an immersed body can be expressed simply in terms of the velocity field and its derivatives.  Based on this idea, Liang & Dong (2014) derived a force expression with the forces of each velocity-based POD mode and the forces from the inter- action between the POD modes.  The Galerkin force model proposed in our study reveals that the force on the body is a constant-linear-quadratic function of the mode amplitudes from a Galerkin expansion.  In addition, physical constraints and sparse calibration can be further used to improve the hu- man interpretability of the resulting models. Based on the Galerkin expansion of Eq.(3), the instanta- neous force on the body can be derived as a function of the mode amplitudes from first principles.  Let Γ be the bound- ary of the body in the flow domain Ω and n the unit normal pointing outward from the surface element dS.  The viscous force component in direction eα of the velocity mode u j is expressed as
where the strain rate tensor Sα,β = ( ∂α ι β + ∂β ια ) /2 with in- dices α,β = x,y.  The pressure force involves two velocity modes from the pressure Poisson equation
and the force component in α -direction is expressed as
Hence, the total force can be expressed as a constant- linear-quadratic expression in terms of the mode coefficients As a 0 ≡ 1, the drag (in direction e x ) and lift (in direction e x ) coefficients can be derived as a constant-linear-quadratic ex- pression in terms of the mode coefficients,
The drag and lift formulae are simplified for the fluidic pinball exploiting the symmetry of the modes. Approximately half of the terms can be discarded on the grounds of symme- try. A second simplification is performed with a sparse calibra- tion of the remaining coefficients. We employ the constrained SINDy (sparse identification of nonlinear dynamics) algorithm (Brunton et al. , 2016; Loiseau & Brunton, 2018), and the spar- sity parameter λ penalizes any non-vanishing term and yields sparse human-interpretable expressions. The resulting drag and lift formulae show long-term agreement with the force coefficients in the full-flow dynam- ics from DNS, based on a mean-field Galerkin model with only seven degrees of freedom, as shown in Fig. 4. The sparse force model successfully describes multi-attractor behaviour even for the complex dynamics with six exact solutions.  We provide a detailed discussion on the challenges of applying the purely projection-based approach and using standard POD modes in Deng et al. (2021). In summary, we proposed an aerodynamic force formulae complementing mean-field POD-Galerkin models for the un- forced fluidic pinball. We envision successful applications of sparse regression for aerodynamic forces for turbulent flows. The force formula may be particularly instructive for drag re- duction with active control. Given a Galerkin model, the force formula indicates beneficial regions of the state space.  Thus, an upfront kinematical insight is gained in which direction control needs to 'push' the attractor.
Hierachical cluster-based network model
Automated reduced-order modeling is one of the most challenging and exciting directions for complex nonlinear dy- namics.  The traditional modeling techniques, like Galerkin modeling, always project the original system onto a low- dimensional subspace and model an approximate dynamical system with an optimal basis (Taira et al. , 2017). The projec- tion basis decides the closeness of the approximate dynamics to the full dynamics.  Data-driven modeling can liberate us from the issue of choosing the projection basis and provide us with novel and promising modeling strategies from math- ematics, data science, and statistical physics. In this subsec- tion, we briefly overview the hierarchical modeling strategy, which presents great consistency with the Reynolds decompo- sition and shows its great potential for multi-scale and multi- frequency modeling. Modal analysis-based modeling relies on the designer's empirical and theoretical knowledge to select modes and con- struct subspace.  In contrast, Cluster-based ROM (CROM) or Network Model (CNM) can effectively avoid these challenges (Kaiser et al. , 2014; Fernex et al. , 2021).  The clustering al- gorithm achieves a kinematic compression of the input data u m , m = 1, . . . , M , by grouping the similar states into the clus- ters & k , k = 1, . . . , K . The snapshots are featured by statistical averages of the grouping data (centroids) in the original data space by k -means algorithm and are labeled with a cluster in- dex k m = k, k = 1,...,K.  As in the state space without ap- proximation, the identified manifold can preserve the original structure depending on the cluster distribution. The CROMs are expected to replace the traditional modeling methods as a more generic means of modeling flow dynamics. According to the minimization strategy of k -means algo- rithm, the clustering is optimal for the spatial distributions of the centroids but not for the dynamics.  The classical CROM may face difficulties with multi-scale problems in transient and post-transient dynamics because the clustering results are highly dependent on the spatial distances in the state space. The critical enabler for applying CNM to the transient and post-transient dynamics is the triple decomposition under the mean-field consideration, as in Eq.(1).  By decomposing the flow into a hierarchy of components, we can systematically model the dynamics at different scales, as shown in Fig. 5. A self-supervised hierarchical clustering is performed from top to bottom, which prioritizes the modeling of global trends for the mean flow distortion, then refines the local dy- namics with sub-clusters. Both the global trends and the local structure during the transition can be well preserved by a fewer number of clusters in the hierarchical structure, which leads to a better understanding of the physical mechanisms involved in the flow dynamics. In summary, the Hierarchical Cluster-based Network Model (HiCNM) introduces hierarchical structure into the CNM from the mean-field considerations, which enables sys- tematically identifying the transient and post-transient dynam- ics between multiple invariant sets in a self-supervised man- ner and steps towards automated ROM of complex dynam- ics.  HiCROM inherits the excellent recognition performance of CROM and provides a generalized modeling strategy for complex dynamics with multiple scales and frequencies.
CONCLUSIONS
We propose combining the mean-field assumption in fluid flow and data-driven ML methods to optimize the ROM meth- ods for complex flow dynamics. In the classic POD-Galerkin framework, the mean-field assumption leads to a least-order constrained Galerkin system, where the degrees of freedom are optimal with respect to the underlying instabilities in the sys- tem. Based on the Galerkin expansion, the instantaneous force on the body is derived as a constant-linear-quadratic function of the mode amplitudes from first principles. The force model can be further simplified using symmetry properties and sparse The HiCROM can systematically identify the calibration. multi-scale dynamics, including the transitions between dif- ferent Navier-Stokes solutions, the bifurcating dynamics into different attractors, and the local structures from destabiliza- tion to saturation stages.  All the above-mentioned modeling strategies are exemplified for the transient and post-transient dynamics of the unforced fluidic pinball at Re = 80 and re- sult in low-dimensional, sparse, human-interpretable mean- field models. This will be a fruitful area for advancing automated ROM with first-principles and AI methods for complex dynamical systems in flow mechanics and industrial applications (like vehicles, trains, airplanes, and other nonlinear dynamics) to achieve a deeper understanding of physical mechanisms, dy- namics estimation, state control, and optimization.