arXiv subjects

Peter Schmid

Publications and source records attributed to Peter Schmid.

8 recordsLinked to original sources

arXiv2026

Data-driven modeling of hypersonic flows in chemical non-equilibrium with catalytic surfaces

Hypersonic flows involve extreme thermochemical non-equilibrium, where strong energy dissipation leads to tightly coupled chemical reactions, radiation, and energy exchange. In this regime, surface chemistry, particularly catalytic wall reactions, can significantly affect boundary-layer composition and surface heat transfer. Accurate simulations of such flows may require repeated evaluations of detailed thermochemical libraries, which represent a major computational bottleneck in high-fidelity reactive-flow simulations. To mitigate this cost, we employ the data-driven reduced-order framework introduced by Scherding et al. (2023), which combines nonlinear dimensionality reduction, community clustering, and local surrogate models to efficiently approximate high-dimensional thermochemical mappings. In this work, this framework is extended for the first time to hypersonic reactive flows with localized catalytic surface discontinuities, introducing sharp variations in wall chemistry and heat transfer. To address the increased complexity of the thermochemical state space, the dimensionality reduction method is enhanced with a Sammon-type stress penalty that mitigates topological folding of the latent manifold and improves the robustness of the clustering and surrogate stages. The resulting model accurately captures the effects of discontinuous catalytic properties, including sharp gradients in wall species mass fractions, diffusion fluxes, and surface heat transfer, while reducing the overall simulation cost by 50% without compromising accuracy.

physics.flu-dyn↗

arXiv2025

HFNO: an interpretable data-driven decomposition strategy for turbulent flows

Fourier Neural Operators (FNOs) have demonstrated exceptional accuracy in mapping functional spaces by leveraging Fourier transforms to establish a connection with underlying physical principles. However, their opaque inner workings often constitute an obstacle to physical interpretability. This work introduces Hierarchical Fourier Neural Operators (HFNOs), a novel FNO-based architecture tailored for reduced-order modeling of turbulent fluid flows, designed to enhance interpretability by explicitly separating fluid behavior across scales. The proposed architecture processes wavenumber bins in parallel, enabling the approximation of dispersion relations and non-linear interactions. Inputs are lifted to a higher-dimensional space, Fourier-transformed, and partitioned into wavenumber bins. Each bin is processed by a Fully Connected Neural Network (FCNN), with outputs subsequently padded, summed, and inverse-transformed back into physical space. A final transformation refines the output in physical space as a correction model, by means of one of the following architectures: Convolutional Neural Network (CNN) and Echo State Network (ESN). We evaluate the proposed model on a series of increasingly complex dynamical systems: first on the one-dimensional Kuramoto-Sivashinsky equation, then on the two-dimensional Kolmogorov flow, and finally on the prediction of wall shear stress in turbulent channel flow, given the near-wall velocity field. In all test cases, the model demonstrates its ability to decompose turbulent flows across various scales, opening up the possibility of increased interpretability and multiscale modeling of such flows.

physics.flu-dyn↗

arXiv2025

Learning dissipation and instability fields from chaotic dynamics

To make predictions or design control, information on local sensitivity of initial conditions and state-space contraction is both central, and often instrumental. However, it is not always simple to reliably determine instability fields or local dissipation rates, due to computational challenges or ignorance of the governing equations. Here, we construct an alternative route towards that goal, by estimating the Jacobian of a discrete-time dynamical system locally from the entries of the transition matrix that approximates the Perron-Frobenius operator for a given state-space partition. Numerical tests on one- and two-dimensional chaotic maps show promising results.

nlin.CD↗

arXiv2024

A Hybrid Discrete Exterior Calculus Discretization and Fourier Transform of the Incompressible Navier-Stokes Equations in 3D

The simulation of fluid flow problems, specifically incompressible flows governed by the Navier-Stokes equations (NSE), holds fundamental significance in a range of scientific and engineering applications. Traditional numerical methods employed for solving these equations on three-dimensional (3D) meshes are commonly known for their moderate conservation properties, high computational intensity and substantial resource demands. Relying on its ability to capture the intrinsic geometric and topological properties of simplicial meshes, discrete exterior calculus (DEC) provides a discrete analog to differential forms and enables the discretization of partial differential equations (PDEs) on meshes.We present a hybrid discretization approach for the 3D incompressible Navier-Stokes equations based on DEC and Fourier transform (FT). An existing conservative primitive variable DEC discretization of incompressible Navier-Stokes equations over surface simplicial meshes developed by Jagad et al. [1] is considered in the planar dimension while the Fourier expansion is applied in the third dimension. The test cases of three-dimensional lid-driven cavity and viscous Taylor-Green three-dimensional vortex (TGV) flows show that the simulation results using this hybrid approach are comparable to literature.

physics.comp-ph↗

arXiv2024

Higher DNS-resolution requirements for expanded overlap region and confirmation of a convergence criterion

Direct numerical simulations (DNS) stand out as formidable tools in studying turbulent flows. Despite the fact that the achievable Reynolds number remains lower than those available through experimental methods, DNS offers a distinct advantage: the complete knowledge of the velocity field, facilitating the evaluation of any desired quantity. This capability should extend to compute derivatives. Among the classic functions requiring derivatives is the indicator function, $\Xi(y^+) = y^+\frac{{\rm d}\overline{U}_x^+}{{\rm d}y^+}$. This function encapsulates the wall-normal derivative of the streamwise velocity, with its value possibly influenced by mesh size and its spatial distribution. The indicator function serves as a fundamental element in unraveling the interplay between inner and outer layers in wall-bounded flows, including the overlap region, and forms a critical underpinning in turbulence modeling. Our investigation reveals a sensitivity of the indicator function on the utilized mesh distributions, prompting inquiries into the conventional mesh-sizing paradigms for DNS applications.

physics.flu-dyn↗

arXiv2023

Sensitivity study of resolution and convergence requirements for extended overlap region in wall-bounded turbulence

Direct Numerical Simulations (DNSs) are one of the most powerful tools for studying turbulent flows. Even if achievable Reynolds numbers are lower than those obtained with experimental means, there is a clear advantage since the entire velocity field is known, and any desired quantity can be evaluated. This also includes the computation of derivatives of all relevant terms. One such derivative provides the indicator function, which is the product of wall distance by wall-normal derivative of the mean streamwise velocity. This derivative may depend on mesh spacing and distribution. However, it is extremely affected by the convergence of the simulation. The indicator function is a cornerstone to understanding inner and outer interactions in wall-bounded flows and describing the overlap region between them. We find a clear dependence of this indicator function on mesh distributions we examined, raising questions about classical mesh and convergence requirements for DNS and achievable accuracy. Within the framework of the logarithmic plus linear overlap region, coupled with a parametric study of channel flows and some pipe flows, sensitivities of extracted overlap parameters are examined, and a path is revealed to establishing their high-$Re_\tau$ or near-asymptotic values at modest Reynolds numbers accessible by high-quality DNS with reasonable ``cost''.

physics.flu-dyn↗

arXiv2021

Symmetry-Aware Autoencoders: s-PCA and s-nlPCA

Nonlinear principal component analysis (NLPCA) via autoencoders has attracted attention in the dynamical systems community due to its larger compression rate when compared to linear principal component analysis (PCA). These model reduction methods experience an increase in the dimensionality of the latent space when applied to datasets that exhibit invariant samples due to the presence of symmetries. In this study, we introduce a novel machine learning embedding for autoencoders, which uses Siamese networks and spatial transformer networks to account for discrete and continuous symmetries, respectively. The Siamese branches autonomously find a fundamental domain to which all samples are transformed, without introducing human bias. The spatial transformer network discovers the optimal slicing template for continuous translations so that invariant samples are aligned in the homogeneous direction. Thus, the proposed symmetry-aware autoencoder is invariant to predetermined input transformations. This embedding can be employed with both linear and nonlinear reduction methods, which we term symmetry-aware PCA (s-PCA) and symmetry-aware NLPCA (s-NLPCA). We apply the proposed framework to the Kolmogorov flow to showcase the capabilities for a system exhibiting both a continuous symmetry as well as discrete symmetries.

physics.flu-dyn↗

arXiv2014

Thermoacoustic instability - a dynamical system and time domain analysis

This study focuses on the Rijke tube problem, which includes features relevant to the modeling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source, and nonlinearities. This thermo-acoustic system features a complex dynamical behavior. In order to synthesize accurate time-series, we tackle this problem from a numerical point-of-view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness, in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by means of the projection method proposed by Jarlebring (2008), which alleviates the linearization with respect to the retarded time. The results are then compared to the analytical solution of the undamped system, and to Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We illustrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case, predicted by using Galerkin expansion, show spurious oscillations resulting from the Gibbs phenomenon. By comparing the modes of the linear to that of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching. Finally, time-series in the fully nonlinear regime, where a limit cycle is established, are analyzed and dominant modes are extracted. The analysis of the saturated limit cycles shows the presence of higher frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not captured when the coupling between different frequencies is not accounted for.

physics.comp-ph↗