Causally coherent structures in turbulent dynamical systems.

The characteristics of turbulent boundary layers (TBLs) subjected to adverse pressure gradients are analysed through well-resolved large-eddy simulations. The geometries under study are the NACA0012 and NACA4412 wing sections, at and angle of attack, respectively, both of them at a Reynolds number based on inflow velocity and chord length of . The turbulence statistics show that adverse pressure gradients (APGs) have a significant effect on the mean velocity, velocity fluctuations and turbulent kinetic energy budget, and this effect is more prominent on the outer region of the boundary layer. Furthermore, the effect of flow history is assessed by means of an integrated Clauser pressure-gradient parameter (Vinuesa et al., Flow Turbul. Combust., vol. 99, 2017, pp. 565–587), through the study of cases with matching local values of and the friction Reynolds number to isolate this effect. Our results show a noticeable effect of the flow history on the outer region, however the differences in the near-wall peak of the tangential velocity fluctuations appear to be mostly produced by the local APG magnitude. The one-dimensional power-spectral density shows energetic small scales in the outer region of APG TBLs, whereas these energetic scales do not appear in zero-pressure-gradient (ZPG) TBLs, suggesting that small scales near the wall are advected towards the outer layer by the APG. Moreover, the linear coherence spectra show that the spectral outer peak of high-Reynolds-number ZPG TBLs is highly correlated with the near-wall region (Baars et al., J. Fluid Mech., vol. 823, 2017, R2), unlike APG TBLs which do not show such a correlation. This result, together with the different two-dimensional spectra of APG and high-Reynolds-number ZPG TBLs, suggests different energisation mechanisms due to APG and increase in Reynolds number. To the authors’ knowledge, this is the first in-depth analysis of the TBL characteristics over wings, including detailed single-point statistics, spectra and coherence.

The near-wall, subgrid-scale (SGS) model [Chung and Pullin, Large-eddy simulation and wall-modeling of turbulent channel flow, J. Fluid Mech. 631, 281--309 (2009)] is used to perform large-eddy simulations (LES) of the incompressible developing, smooth-wall, flat-plate turbulent boundary layer. In this model, the stretched-vortex, SGS closure is utilized in conjunction with a tailored, near-wall model designed to incorporate anisotropic vorticity scales in the presence of the wall. The composite SGS-wall model is presently incorporated into a computer code suitable for the LES of developing flat-plate boundary layers. This is then used to study several aspects of zero- and adverse-pressure gradient turbulent boundary layers. First, LES of the zero-pressure gradient turbulent boundary layer are performed at Reynolds numbers Reθ based on the free-stream velocity and the momentum thickness in the range Reθ = 103 - 1012. Results include the inverse skin friction coefficient, √2/Cf, velocity profiles, the shape factor H, the Karman constant, and the Coles wake factor as functions of Reθ. Comparisons with some direct numerical simulation (DNS) and experiment are made, including turbulent intensity data from atmospheric-layer measurements at Reθ = O(106. At extremely large Reθ, the empirical Coles-Fernholz relation for skin-friction coefficient provides a reasonable representation of the LES predictions. While the present LES methodology cannot of itself probe the structure of the near-wall region, the present results show turbulence intensities that scale on the wall-friction velocity and on the Clauser length scale over almost all of the outer boundary layer. It is argued that the LES is suggestive of the asymptotic, infinite Reynolds-number limit for the smooth-wall turbulent boundary layer and different ways in which this limit can be approached are discussed. The maximum Reθ of the present simulations appears to be limited by machine precision and it is speculated, but not demonstrated, that even larger Reθ could be achieved with quad- or higher-precision arithmetic. Second, the time series velocity signals obtained from LES within the logarithmic region of the zero-pressure gradient turbulent boundary layer are used in combination with an empirical, predictive inner--outer wall model [Marusic et al., Predictive model for wall-bounded turbulent flow, Science 329, 193 (2010)] to calculate the statistics of the fluctuating streamwise velocity in the inner region of the zero-pressure gradient turbulent boundary layer. Results, including spectra and moments up to fourth order, are compared with equivalent predictions using experimental time series, as well as with direct experimental measurements at Reynolds numbers Reτ based on the friction velocity and the boundary layer thickness, Reτ =7,300, 13,600 and 19,000. LES combined with the wall model are then used to extend the inner-layer predictions to Reynolds numbers Reτ =62,000, 100,000 and 200,000 that lie within a gap in log(Reτ) space between laboratory measurements and surface-layer, atmospheric experiments. The present results support a log-like increase in the near-wall peak of the streamwise turbulence intensities with Reτ and also provide a means of extending LES results at large Reynolds numbers to the near-wall region of wall-bounded turbulent flows. Finally, we apply the wall model to LES of a turbulent boundary layer subject to an adverse pressure gradient. Computed statistics are found to be consistent with recent experiments and some Reynolds number similarity is observed over a range of two orders of magnitude.

: This is an interim annual report for an ongoing research program studying coherent structure in turbulent boundary layers. The design and development of a water channel facility for use in flow visualization studies of coherent turbulent boundary structure is described. The particularly unique aspects of this facility include the incorporation of (1) a moving reference (viewing) platform for study of relative motion effects and (2) a closed circuit television monitoring and recording system. Preliminary flow visualization results are presented for both a fixed and moving reference frame, and a tentative vortex-loop model of coherent turbulent boundary layer structure is proposed. Computer simulations of a transverse structure in a turbulent boundary layer are presented showing a remarkable similarity with experimental flow visualization results. Research directions for the following year are outlined. (Author)

Modeling of Microblogging Social Networks: Dynamical System vs. Random Dynamical System

The effects of surface roughness on a spatially developing turbulent boundary layer (TBL) are investigated by performing direct numerical simulations of TBLs over rough and smooth walls. The Reynolds number based on the momentum thickness was varied in the range Reθ = 300 ∼ 1400. The roughness elements were periodically arranged two-dimensional spanwise rods, and the roughness height was k = 1.5θin, where θin is the momentum thickness at the inlet, which corresponds to k/δ = 0.045 ∼ 0.125, δ being the boundary layer thickness. To avoid generating a rough-wall inflow, which is prohibitively difficult, a step change from smooth to rough was placed 80θin downstream from the inlet. The spatially developing characteristics of the rough-wall TBL were examined. Along the streamwise direction, the friction velocity approached a constant value, and self-preserving forms of the turbulent Reynolds stress tensors were obtained. Introduction of the roughness elements affected the turbulent stress not only in the roughness sublayer but also in the outer layer. Despite the roughness-induced increase of the turbulent Reynolds stress tensors in the outer layer, the roughness had only a relatively small effect on the anisotropic Reynolds stress tensor in the outer layer. Inspection of the triple products of the velocity fluctuations revealed that introducing the roughness elements onto the smooth wall had a marked effect on vertical turbulent transport across the whole TBL. By contrast, good surface similarity in the outer layer was obtained for the third-order moments of the velocity fluctuations.

Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot.

The excellent drag reduction effect of the bubble drag reduction technique has been proved through many experiments since it was proposed. In this paper, the authors investigate the bubble-turbulence interaction and the corresponding drag reduction effect with a two-way coupled Euler–Lagrange code. The liquid phase is simulated by using a large eddy simulation method with the immersed bubbles treated using a nonlinear collision model to accurately simulate the bubble–wall interaction. A Gaussian distributed method is adopted to obtain the void fraction and interphase forces in the two-way coupled algorithm. Two typical wall-bounded turbulent flow problems (turbulent channel flow and boundary layer flow) are simulated to validate the accuracy and stability in bubbly flows and investigate the drag reduction mechanism. First, the effect of bubbles on the turbulent flow is studied in the channel flow cases in which the bubbles are observed attaching to the upper plate and swaying in the spanwise direction. In this case, Reynolds stress near the wall is decreased, which contributes to the drag reduction. Moreover, drag reduction of a turbulent boundary layer flow with bubble injection is studied in which the drag reduction under different air flow rates is in good agreement with experimental results. The contribution of turbulence and different liquid forces to the migration of bubbles away from the wall is investigated. The bubble trajectory in the turbulent boundary layer is divided into three distinct stages and discussed in detail finally.

A theory for natural convection turbulent boundary layers next to heated vertical surfaces

This paper considers the influence of higher harmonics in dynamic action systems due to their complex movement in the process of interaction with the technological load. The object of research is the process of propagation of oscillations in complex dynamic systems. One of the problems in the application of oscillatory processes is the consideration of higher harmonics in the overall movement of systems. To solve the problem, the idea of using a hybrid model that takes into account both discrete and distributed parameters was proposed. The resulting mathematical discrete model in the analytical equations of motion of the dynamic system preserves continuous properties in the form of wave coefficients. These coefficients in their analytical form take into account the contribution of higher harmonics of both the reactive (elastic-inertial) and active (dissipative) components of the resistance force. The studies were carried out on a model of a plant with a multimode spectrum of oscillations and a nonlinear dynamic system, which is a system with piecewise linear characteristics.A series of experimental studies with a wide variation of the change in the frequency of oscillations was carried out on the installation with a multimode spectrum of oscillations. Zones of manifestation of higher harmonics along the vertical axis of force action were revealed. The given spectrum at the exciter frequency of 35 Hz showed the manifestation of the spectrum component (around 70 Hz) along the X axis, which is an important result for practical application. For a system with piecewise linear characteristics, the manifestation of multimode, which manifests itself in the form of subharmonic and superharmonic oscillations, was determined. The contribution of each harmonic is determined by applying the obtained dependences. The results were used in the development of algorithms and calculation methods of a new class of dynamic action systems taking into account the contribution of higher harmonics

Dynamical systems ideas have recently gained increased momentum in the study of turbulent flows. The reason is that dynamical systems could be used to describe low dimensional structures, such as the coherent structures in the turbulent boundary layer, which play an important role in the dynamics of the flow. In this study, we perform threedimensional simulations of a turbulent flow past a circular cylinder at Re=1000. We record 1000 snapshots of the velocity field data for more than a dozen vortex shedding cycles. We compute the POD modes and project the Navier-Stokes equations onto these modes. In order to model higher frequency modes, we propose an LES-type approach and include an additional term in the reduced-order model for closure.

In the present study, we performed direct numerical simulations for a hypersonic turbulent boundary layer over the windward side of a lifting body, the HyTRV model, at Mach number $6$ and attack angle 2 $^{\circ }$ to investigate the global and local turbulent features, and evaluate its difference from canonical turbulent boundary layers. By scrutinizing the instantaneous and averaged flow fields, we found that the transverse curvature on the windward side of the HyTRV model induces the transverse opposing pressure gradients that push the flow on both sides towards the windward symmetry plane, yielding significant effects of the azimuthal inhomogeneity and large-scale cross-stream circulations, moderate and azimuthal independent influences of adverse pressure gradient, and negligible impact of the mean flow three-dimensionality. Further inspecting the local turbulent statistics, we identified that the mean and fluctuating velocity become increasingly similar to the highly decelerated turbulent boundary layers over flat plates in that the mean velocity deficit is enhanced, and the outer layer Reynolds stresses are amplified as it approaches the windward symmetry plane, and prove to be self-similar under the scaling of Wei & Knopp (J. Fluid Mech., vol. 958, 2023, A9) for adverse-pressure-gradient turbulent boundary layers. Conditionally averaged Reynolds stresses based on strong sweeping and ejection events demonstrated that the Kelvin–Helmholtz instability of the strong embedded shear layer induced by the large-scale cross-stream circulations is responsible for the turbulence amplification in the outer layer. The strong Reynolds analogy that relates the mean velocity and temperature was refined to incorporate the non-canonical effects, showing considerable improvements in the accuracy of such a formula. On the other hand, the temperature fluctuations are still transported passively, as indicated by their resemblance to the velocity. The conclusions obtained in the present study provide potentially profitable information for turbulent modelling modification for the accurate predictions of skin friction and wall heat transfer.

In this paper, we consider an observer design by using a formal linearization based on Fourier expansion for nonlinear dynamic and measurement systems. A non-linear dynamic system is given by a nonlinear ordinary differential equation, and a measurement sysetm is done by a nonlinear equation. Defining a linearization function which consists of the trigonometric functions considered up to the higher-order, a nonlinear dynamic system is transformed into an augmented linear one with respect to this linearization function by using Fourier expansion. Introducing an augmented measurement vector which consists of polynomials of measurement data, a measurement equation is transformed into an augmented linear one with respect to the linearization function in the same way. To these augmented linearized systems, a linear estimation theory is applied to design a new non-linear observer.

Detection of coherent structures in a turbulent boundary layer using a scanning LDV system

Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering, including climate, material, and neural science. The existence of a strange attractor in the turbulent systems containing a large number of positive Lyapunov exponents results in the rapid growth of small uncertainties from imperfect modeling equations or perturbations in initial values, naturally requiring a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification (UQ) in turbulent dynamical systems is a grand challenge whose goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in their nonlinear responses to changes in external forcing parameters or uncertain initial data. In the development of a proper UQ scheme for systems of high or infinite dimensionality with instabilities, significant model errors compared with the true natural signal are always unavoidable due to both the imperfect understanding of the underlying physical processes and the limited computational resources available through direct Monte Carlo integration. One central issue in contemporary research is the development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium (model fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity). \indent A general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems is discussed here. There are generally three stages in the modeling strategy: imperfect model selection, calibration of the imperfect model in a training phase, and prediction of the responses with UQ to a wide class of forcing and perturbation scenarios. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. Several mathematical ideas will be introduced to improve the prediction skill of the imperfect reduced-order models. Most importantly, empirical information theory and statistical linear response theory are applied in the training phase for calibrating model errors to achieve optimal imperfect model parameters, and total statistical energy dynamics are introduced to improve the model sensitivity of the prediction phase, especially when strong external perturbations are exerted. The validity of the general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed. The UQ and statistical response for these complex models are accurately captured by the reduced-order models with only $2\times10^{2}$ modes in a highly turbulent system with $1\times10^{5}$ degrees of freedom. Less than 0.15% of the total spectral modes are needed in the reduced-order models.