Dynamical Similarity Research Articles

Posterior comparison of model dynamics in several hybrid turbulence model forms

Hybrid turbulence models that can accurately reproduce unsteady three-dimensional flow physics across the entire range of grid scales and turbulence dynamics from Reynolds-averaged Navier–Stokes (RANS), through large-eddy simulation (LES), down to direct numerical simulations (DNS) are of increasing interest to the turbulence modeling community. However, despite decades of research and development, the basic tasks of eliminating poor-performing hybrid RANS-LES models and accelerating adoption of superior models through well-designed validation and verification have yet to occur. As a step in this direction, in this work we evaluate thirteen different hybrid RANS-LES models via systematic grid refinement of decaying homogeneous isotropic turbulence. We further derive a novel mathematical framework for assessing the energy partitioning dynamics of each Hybrid RANS-LES model, wherein model-to-model variations in energy partitioning can be interpreted as different feedback mechanisms operating on a low-dimensional nonlinear dynamical system. We found that model forms similar to the flow simulation methodology—also often termed very-large eddy simulation—are dynamically inconsistent with DNS at all resolutions. Additionally, we found a strong dynamical similarity in the feedback mechanisms of all models related to detached eddy simulation and partially averaged Navier–Stokes that is inherent to their general model forms.

Continuation of Bianchi spacetimes through the big bang

In this paper we present a framework in which the relational description of general relativity can be used to smoothly continue cosmological dynamical systems through the big bang without invoking quantum gravity effects. Cosmological spacetimes contain as a key dynamical variable a notion of scale through the volume factor ν. However, no cosmological observer is ever able to separate their measuring apparatus from the system they are measuring, in that sense every measurement is a relative one, and measurable dynamical variables are, in fact, dimensionless ratios. This is manifest in the identification of a scaling symmetry or “dynamical similarity” in the Einstein-Hilbert action associated with the volume factor. By quotienting out this scaling symmetry, we form a relational system defined on a contact manifold whose dynamical variables are decoupled from scale. When the phase space is reduced to shape space, we show that there exists unique solutions to the equations of motion that pass smoothly through the initial cosmological singularity in flat Friedmann-Lemaître-Robertson-Walker, Bianchi I, and Quiescent Bianchi IX cosmologies. Published by the American Physical Society 2024

High-resolution 2D solid-state NMR provides insights into nontuberculous mycobacteria

We present a high-resolution magic-angle spinning (MAS) solid-state NMR (ssNMR) study to characterize nontuberculous mycobacteria (NTM). We studied two different NTM strains, Mycobacterium smegmatis, a model, non-pathogenic strain, and Mycobacterium abscessus, an emerging and important human pathogen. Hydrated NTM samples were studied at natural abundance without isotope-labelling, as whole-cells versus cell envelope isolates, and native versus fixed sample preparations. We utilized 1D13C and 2D 1H-13C ssNMR spectra and peak deconvolution to identify NTM cell-wall chemical sites. More than ∼100 distinct 13C signals were identified in the ssNMR spectra. We provide tentative assignments for ∼30 polysaccharides by using well resolved 1H/13C chemical shifts from the 2D INEPT-based 1H-13C ssNMR spectrum. The signals originating from both the flexible and rigid fractions of the whole-cell bacteria samples were selectively analyzed by utilizing either CP or INEPT based 13C ssNMR spectra. CP buildup curves provide insights into the dynamical similarity of the cell-wall components for NTM strains. Signals from peptidoglycan, arabinogalactan and mycolic acid were identified. The majority of the 13C signals were not affected by fixation of the whole cell samples. The isolated cell envelope NMR spectrum overlap with the whole-cell spectrum to a large extent, where the latter has more signals. As an orthogonal way of characterizing these bacteria, electron microscopy (EM) was used to provide spatial information. ssNMR and EM data suggest that the M. abscessus cell-wall is composed of a smaller peptidoglycan layer which is more flexible compared to M. smegmatis, which may be related to its higher pathogenicity. Here in this work, we used high-resolution 2D ssNMR first time to characterize NTM strains and identify chemical sites. These results will aid the development of structure-based approaches to combat NTM infections.

Jacobian-scaled K-means clustering for physics-informed segmentation of reacting flows

This work introduces Jacobian-scaled K-means (JSK-means) clustering, which is a physics-informed clustering strategy centered on the K-means framework. The method allows for the injection of underlying physical knowledge into the clustering procedure through a distance function modification: instead of leveraging conventional Euclidean distance vectors, the JSK-means procedure operates on distance vectors scaled by matrices obtained from dynamical system Jacobians evaluated at the cluster centroids. The goal of this work is to show how the JSK-means algorithm – without modifying the input dataset – produces clusters that capture regions of dynamical similarity, in that the clusters are redistributed towards high-sensitivity regions in phase space and are described by similarity in the source terms of samples instead of the samples themselves. The algorithm is demonstrated on a complex reacting flow simulation dataset (a channel detonation configuration), where the dynamics in the thermochemical composition space are known through the highly nonlinear and stiff Arrhenius-based chemical source terms. Interpretations of cluster partitions in both physical space and composition space reveal how JSK-means shifts clusters produced by standard K-means towards regions of high chemical sensitivity (e.g., towards regions of peak heat release rate near the detonation reaction zone). The findings presented here illustrate the benefits of utilizing Jacobian-scaled distances in clustering techniques, and the JSK-means method in particular displays promising potential for improving former partition-based modeling strategies in reacting flow (and other multi-physics) applications.

Recovery of dynamical similarity from lossy representations of collective behavior of midge swarms.

Understanding emergent collective phenomena in biological systems is a complex challenge due to the high dimensionality of state variables and the inability to directly probe agent-based interaction rules. Therefore, if one wants to model a system for which the underpinnings of the collective process are unknown, common approaches such as using mathematical models to validate experimental data may be misguided. Even more so, if one lacks the ability to experimentally measure all the salient state variables that drive the collective phenomena, a modeling approach may not correctly capture the behavior. This problem motivates the need for model-free methods to characterize or classify observed behavior to glean biological insights for meaningful models. Furthermore, such methods must be robust to low dimensional or lossy data, which are often the only feasible measurements for large collectives. In this paper, we show that a model-free and unsupervised clustering of high dimensional swarming behavior in midges (Chironomus riparius), based on dynamical similarity, can be performed using only two-dimensional video data where the animals are not individually tracked. Moreover, the results of the classification are physically meaningful. This work demonstrates that low dimensional video data of collective motion experiments can be equivalently characterized, which has the potential for wide applications to data describing animal group motion acquired in both the laboratory and the field.

Low-altitude scaled similarity analysis and flight tests of near space airships with pressure resistance mechanism

In this paper, considering the low cost and high effectiveness-cost ratio of near space airships, as well as the insufficiency of theoretical methods for verifying space-ground equivalents, flight control technology validation of a typical airship was considered, and a method for describing the airship motion characteristics is presented in detail. According to the similarity and π theorems, dimensional analysis of the prototype airship’s motion characteristics was completed, similarity criteria were tabulated, and the similarity index was determined. The design of the scaled parameters under the single-value constraint of “geometric similarity, kinematic similarity, and dynamical similarity” between the prototype airship and its low-altitude scaled model was performed, and a low-altitude scaled flight test platform and flight test were developed. Theoretical methods for the similarity analysis of the prototype airship and its low-altitude scaled model, as well as the basics of flight test engineering applications, were systematically introduced, describing a low-cost, repeatable low-altitude scaled flight test verification method for key technology verification of high-cost and high-risk near space airships. This study provides a foundation for follow-up studies of relevant similar theories and applications.

Data-driven reduction and decomposition with time-axis clustering

A new approach for modal decomposition through re-interpretation of unsteady dynamics, termed time-axis clustering, is developed in this work and is demonstrated on an experimental turbulent reacting flow dataset consisting of simultaneously measured planar OH-PLIF and PIV fields in a model combustor. The method executes a K-Means clustering algorithm on an alternative representation of the input snapshot dataset: the dataset is interpreted here as a collection of one-dimensional time series, where each time series represents the time evolution of some flow quantity of interest (QoI) at a fixed point in physical space (i.e. pixel locations). The clustering algorithm in the modal decomposition context produces (a) spatial modes (called time-axis modes) that identify localized regions of dynamical similarity in physical space and (b) temporal coefficients that represent average trajectories of the flow QoI conditioned on the regions in physical space identified by the corresponding spatial mode. Due to the non-overlapping nature of K-Means clusters, visualization of the modes provides a unique pathway for flow feature extraction based on dynamical similarity. Ultimately, this work shows how time-axis clustering provides a promising avenue for domain-localized data-based modelling of complex fluid flows.

The linear-time-invariance notion of the Koopman analysis. Part 2. Dynamic Koopman modes, physics interpretations and phenomenological analysis of the prism wake

This serial work presents a linear-time-invariance (LTI) notion to the Koopman analysis, finding consistent and physically meaningful Koopman modes and addressing a long-standing problem of fluid mechanics: deterministically relating the fluid excitations and corresponding structure reactions. Part 1 (Li et al., Phys. Fluids, vol. 34, no. 12, p. 125136) developed the Koopman-LTI architecture and applied it to a pedagogical prism wake. By a systematic analytical procedure, the Koopman-LTI generated sampling-independent linear models that captured all the recurring dynamics embedded in the input data, finding six corresponding, orthogonal, and in-synch fluid–structure mechanisms. This Part 2 analyses the six modal duplets to underpin their physical implications, providing a phenomenological analysis of the subcritical prism wake. Visualizing the newly proposed dynamic Koopman modes, results show that two mechanisms at St1 = 0.1242 and St5 = 0.0497 describe shear layer dynamics, the associated Bérnard–Kármán shedding and turbulence production, which together overwhelm the upstream and crosswind walls by instigating a reattachment-type of reaction. The on-wind walls’ dynamical similarity renders them a spectrally unified fluid–structure interface. Another four harmonic counterparts, namely the subharmonic at St7 = 0.0683, the second harmonic at St3 = 0.2422, and two ultra-harmonics at St7 = 0.1739 and St13 = 0.1935, govern the downstream wall. Finally, this work discovered the vortex breathing phenomenon, describing the constant energy exchange in the wake's circulation-entrainment-deposition processes. With the Koopman-LTI, one may pinpoint the exact excitations responsible for a specific structure reaction, benefiting future investigations into fluid–structure interactions and nonlinear, stochastic systems.

Options for Dynamic Similarity of Interacting Fluids, Elastic Solids and Rigid Bodies Undergoing Large Motions

Motivated by a design of a vertical axis wind turbine, we present a theory of dynamical similarity for mechanical systems consisting of interacting elastic solids, rigid bodies and incompressible fluids. Throughout, we focus on the geometrically nonlinear case. We approach the analysis by analyzing the equations of motion: we ask that a change of variables take these equations and mutual boundary conditions to themselves, while allowing a rescaling of space and time. While the disparity between the Eulerian and Lagrangian descriptions might seem to limit the possibilities, we find numerous cases that apparently have not been identified, especially for stiff nonlinear elastic materials (defined below). The results appear to be particularly adapted to structures made with origami design methods, where the tiles are allowed to deform isometrically. We collect the results in tables and discuss some particular numerical examples.

B -hadron spectroscopy study based on the similarity of double bottom baryon and bottom meson

The dynamical similarity which exists between the $\lambda$-mode excited $bbq$ baryons ($q$ refers to the $u$, $d$, and $s$ quarks) and the $\bar{b}q$ mesons inspires us to carry out a combined study of their spectroscopy. In this work, the masses and strong decays of these low-lying $b\bar{q}$ and $bbq$ states are studied by the same theoretical methods, and the dynamical similarity which is implied in their mass spectra and strong decays are also discussed. The recent discovered $\bar{b}q$ states, including the $B_J(5840)$, $B_J(5970)$, $B_{sJ}(6064)$, and $B_{sJ}(6114)$, are analyzed. According to our result, the $B_J(5840)$ could be assigned as a 2$S$ state, while the $B_J(5970)$ could be regarded as a member of the $1D(2^-,~3^-)_{j_q=5/2}$ doublet. The $B_{sJ}(6064)$ and $B_{sJ}(6114)$ are probably the $D$-wave states. Especially, they could be explained as the members of the $1D(1^-,~2^-)_{j_q=3/2}$ and $1D(2^-,~3^-)_{j_q=5/2}$ doublets, respectively. The predicted masses and decay properties of other unknown $\bar{b}q/bbq$ states may provide useful clues to the future experiment.

Dynamical Similarity of EEG State Transitions for Scoring Performance of a Mental Arithmetic Task

Dynamical Similarity of EEG State Transitions for Scoring Performance of a Mental Arithmetic Task

When scale is surplus

We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton’s principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology.

Smallest scale of wrinkles of a Huygens front in extremely strong turbulence.

By analyzing the statistically stationary stage of propagation of a Huygens front in homogeneous, isotropic, constant-density turbulence, a length scale l_{0} is introduced to characterize the smallest wrinkles on the front surface in the case of a low constant speed u_{0} of the front when compared to the Kolmogorov velocity u_{K}. The length scale is derived following a hypothesis of dynamical similarity that highlights a balance between (i) creation of a front area due to advection and (ii) destruction of the front area due to propagation. Consequently, the front speed is compared with the magnitude of the fluid velocity difference in two points separated by a distance smaller than the Kolmogorov length scale. Appropriateness of the smallest wrinkle scale is demonstrated by applying a fractal approach to evaluating the mean area of the instantaneous front surface. Since the scales of the smallest and larger wrinkles belong to different subranges (dissipation and inertial, respectively) of the Kolmogorov turbulence spectrum, the front is hypothesized to be a bifractal characterized by two different fractal dimensions in the two subranges. Both fractal dimensions are evaluated adapting the aforementioned hypothesis of dynamical similarity. Such a bifractal model yields a linear relation between the mean fluid consumption velocity, which is equal to the front speed u_{0} multiplied with a ratio of the mean area of the instantaneous front surface to the transverse projected area, and the rms turbulent velocity u^{'} even if a ratio of u_{0}/u^{'} tends to zero.

Dynamical ergodicity DDA reveals causal structure in time series.

Determining synchronization, causality, and dynamical similarity in highly complex nonlinear systems like brains is challenging. Although distinct, these measures are related by the unknown deterministic structure of the underlying dynamical system. For two systems that are not independent on each other, either because they result from a common process or they are already synchronized, causality measures typically fail. Here, we introduce dynamical ergodicity to assess dynamical similarity between time series and then combine this new measure with cross-dynamical delay differential analysis to estimate causal interactions between time series. We first tested this approach on simulated data from coupled Rössler systems where ground truth was known. We then applied it to intracranial electroencephalographic data from patients with epilepsy and found distinct dynamical states that were highly predictive of epileptic seizures.

Stability Analysis of the World Energy Trade Structure by Multiscale Embedding

This study investigates the dynamic trading network structure of the international crude oil and gas market from year 2012 to 2017. We employed the dynamical similarity analysis at different time scales by inducing a multiscale embedding for dimensionality reduction. This analysis quantifies the effect of a global event on the dependencies and correlation stability at both the country and world level, which covers the top 53 countries. The response of China’s trading structure toward events after the unexpected 2014 price drop is compared with other major traders. China, as the world’s largest importing country, lacks strong stability under global events and could be greatly affected by a supply shortage, especially in the gas market. The trend of multi-polarization on the market share gives a chance for China to construct closer relationships with more stable exporters and join in the trade loop of major countries to improve its position in the energy trading networks. The hidden features of trade correlation may provide a deeper understanding of the robustness of relationship and risk resistance.

A General Metric for the Similarity of Both Stochastic and Deterministic System Dynamics.

Many problems in the study of dynamical systems—including identification of effective order, detection of nonlinearity or chaos, and change detection—can be reframed in terms of assessing the similarity between dynamical systems or between a given dynamical system and a reference. We introduce a general metric of dynamical similarity that is well posed for both stochastic and deterministic systems and is informative of the aforementioned dynamical features even when only partial information about the system is available. We describe methods for estimating this metric in a range of scenarios that differ in respect to contol over the systems under study, the deterministic or stochastic nature of the underlying dynamics, and whether or not a fully informative set of variables is available. Through numerical simulation, we demonstrate the sensitivity of the proposed metric to a range of dynamical properties, its utility in mapping the dynamical properties of parameter space for a given model, and its power for detecting structural changes through time series data.

Dynamical similarity of multi-block catenary arches and rocking blocks subjected to horizontal base excitation

In this paper, the dynamical similarity of multi-block catenary arches and columns is discussed, which can be used for the simplified design of rocking arches. The basic hypothesis is that the dynamic response of multi-block arches and columns is similar when the shape of the arch follows the thrust line, i.e. it is a catenary arch. It is validated numerically that the responses of slender catenary arches are safe and reliable approximations of those of not slender arches and then that the overturning acceleration (response) spectra of rigid, slender monolithic blocks can be directly applied for catenary arches. The similarity is based on two parameters, on the limit peak ground acceleration (under which the structure will not move at all) and on the frequency parameter (defined by Housner for rigid blocks).

A new blind source separation approach based on dynamical similarity and its application on epileptic seizure prediction

Blind source separation is an important field of study in signal processing, in which the goal is to estimate source signals by having mixed observations. There are some conventional methods in this field that aim to estimate source signals by considering certain assumptions on sources. One of the most popular assumptions is the non-Gaussianity of sources which is the basis of many popular blind source separation methods. These methods may fail to estimate sources when the distribution of two or more sources is Gaussian. Hence, this study aims to introduce a new approach in blind source separation for nonlinear and chaotic signals by using a dynamical similarity measure and relaxing non-Gaussianity assumption. The proposed approach assumes there are dynamical stability in source signals and dynamical independence between them. The efficiency of the proposed approach is evaluated by synthetic simulation. Also, to evaluate the ability of the method in real-world applications and featuring its flexibility, the proposed approach is employed in epileptic seizure prediction by using EEG signals. The results show the potential and ability of the proposed method in nonlinear and chaotic signal processing.

Modeling and Fuzzy Decoupling Control of an Underwater Vehicle-Manipulator System

This paper presents the detailed modeling and simulation of the dynamic coupling between an autonomous underwater vehicle (AUV) and a manipulator. The modeling processes are described with the incorporation of the most dominating hydrodynamic effects such as added mass, lift and drag forces. The hydrodynamic coefficients are derived using strip theory and are adjusted according to dynamical similarity. A fuzzy decoupling controller (FDC) is proposed for an autonomous underwater vehicle-manipulator system (UVMS) which consists of two subsystems, an underwater vehicle and a manipulator. The proposed controller uses a fuzzy algorithm (FA) to adaptively tune the gain matrix of the error function (EF). The EF is described by the integral sliding surface function. This technique allows the off-diagonal elements developed for decoupling the system to be incorporated in the gain matrix. Tracing the FA and EF back to the principle of feedback linearization, one further obtains evidence about the decoupling and stability of the system. Moreover, a desired trajectory with the consideration of the dynamic coupling of the AUV is designed to reduce the thruster forces and manipulator's torques. This technique provides high performance in terms of tracking error norms and expended energy norms. A major contribution of this study is that it adopts the off-diagonal elements to exploit the dynamic coupling between the degrees of freedom of the subsystem and the dynamic coupling between the two subsystems. Simulation results demonstrate the effectiveness and robustness of the proposed technique in the presence of parameter uncertainties and external disturbances.

Robust Impedance-Matching of Manipulators Interacting With Uncertain Environments: Application to Task Verification of the Space Station's Dexterous Manipulator

A systematic design method is developed for the identification and control of a simulating robot in order to adjust its contact force and impact dynamics to match those of a reference robot when both robots couple to a similar physical environment with unknown impedance. The control architecture, which is based on closed-loop impedance matching between two robots, is utilized by a hydraulic robot testbed facility for high-fidelity task verification of the space station's dexterous manipulator without requiring contact dynamics modeling. First, the uncertain environment is modeled in terms of parameter uncertainty bounds for a class of environment impedances. Then, the control goal is specifically defined as to minimize the contact-force error between the two robots, where the contact force is caused either by time-varying operator commands or impacts due to a nonzero preimpact velocity. Next, a hierarchical control architecture is formulated in the general framework of linear fractional transformation system by making use of the frequency-domain specifications of the robots together with the environment's parameter uncertainty bounds. It is shown that the overall system uncertainty can be represented by a perturbation block in the feedback form, which can be specified by a block structure array by making use of a unitary transformation. Subsequently, a $\mu$-synthesis-based controller is developed and implemented to achieve the dynamical similarity while maintaining contact stability. Experimental results demonstrate that the simulating robot can generate high-fidelity contact-force profile and impulsive force for different environments without contact dynamics modeling.