Barycentric Mapping Research Articles

Large eddy simulation of highly underexpanded sonic jets from elliptical nozzles

The injection of high-pressure jets into quiescent air poses significant challenges in fluid dynamics, pertinent to various industrial engineering applications. This study used large eddy simulations on a massively parallel computational framework, employing a grid of over 600 million nodes, to investigate the behavior of highly underexpanded sonic jets from elliptical nozzles at a nozzle pressure ratio of 15. Three elliptical nozzles, with aspect ratios of 1.5, 2.2, and 3.0, each having a sectional area equivalent to that of a circular jet with a diameter of D=1 mm, were analyzed. The aim was to clarify the gasdynamic and mixing characteristics of these jets to guide the design of next-generation injectors. A detailed analysis of the flow provided insights into the mechanisms of turbulence generation and Reynolds stress anisotropy. This was achieved using the componentality contour approach and a modified barycentric color mapping scheme, offering valuable reference data for developing lower-order models. The results indicate a non-axisymmetric radial expansion of the jet boundary in all elliptical injectors, leading to an axis switch phenomenon. The use of elliptical orifices was found to reduce jet penetration, mitigating issues such as fuel impingement in small engine combustion chambers and promoting improved air–fuel mixing quality.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Direct numerical simulation of a pressurized thermal shock scenario at higher reynolds number

Direct Numerical Simulation (DNS) is performed for a pressurized thermal shock scenario in a reactor downcomer geometry. In the present work, a simplified representative geometry is adopted where cold fluid injected from a square duct is injected in a planar downcomer. A secondary hot inlet at the top of the downcomer is employed to enforce a downward flow of the injected emergency core cooling water, to mimic the effect of density driven flow. The Reynolds number of the impinging flow, based on bulk velocity and duct height, equals 12,500, while a unity Prandtl number fluid is used. The simulation is performed for two thermal boundary conditions, namely an isotemperature condition and an adiabatic condition, representing the two extreme scenarios of a conjugate heat transfer problem. The instantaneous and mean fields are first analysed in order to study the flow pattern and formation of vortical structures within the downcomer. The impinging flow is deflected radially outwards in the vicinity of the barrel wall, which then interacts with the surrounding hot fluid forming large vortical structures and bringing the cold fluid back to the vessel wall. The mixing of flow and penetration of colder temperatures in the downcomer is observed to be greatly influenced by interaction of these vortical structures. The descending cold plume in the downcomer is observed to form three branches separated by low-speed regions. The maximum values of turbulent kinetic energy are observed only in the vicinity of flow impingement, while high values of its production and dissipation are also observed within the shear layers of the descending plume. The maximum values of temperature fluctuations are found at the interface of the descending cold plume and surrounding hot fluid, while relatively high values are also seen at the locations of large vortical strucutres. Anisotropy in turbulence is also analysed using the componentality contour approach and a modified barycentric colour mapping scheme. Also reported herein is a comparison of mean and statistical profiles with those at lower Reynolds number reported in the literature. At higher Reynolds number, the cold impinging jet is deflected farther outwards in the downcomer. The locations of peak fluctuations in velocity and temperature are also farther away from the impingement. The magnitudes of fluctuating temperature and turbulent kinetic energy are observed to be higher for the present higher Reynolds number case. The peak Nusselt number at the barrel wall is observed to scale with a factor of Re1/2 in the present configuration.

Mapping the shape and dimension of three-dimensional Lagrangian coherent structures and invariant manifolds

We introduce maps of Cauchy–Green strain tensor eigenvalues to barycentric coordinates to quantify and visualize the full geometry of three-dimensional deformation in stationary and non-stationary fluid flows. As a natural extension of Lagrangian coherent structure diagnostics, which provide separate scalar fields and a one-dimensional quantification of fluid deformation, our barycentric mapping visualizes the role of all three Cauchy–Green eigenvalues (or rates of stretching) in a single plot through a novel stretching coordinate system. The coordinate system is based on the distance from three distinct limiting states of deformation that correspond with the dimension of the underlying invariant manifolds. One-dimensional axisymmetric deformation (sphere to rod deformation) corresponds to one-dimensional unstable manifolds, two-dimensional axisymmetric deformation (sphere to disk deformation) corresponds to two-dimensional unstable manifolds and the rare three-dimensional isometric case (sphere to sphere translation and rotation) corresponds to shear-free elliptic Lagrangian coherent structures (LCSs). We provide methods to visualize the degree to which fluid deformation approximates these limiting states, and tools to quantify differences between flows based on the compositional geometry of invariant manifolds in the flow. We also develop a simple analogue for bilinearly representing and plotting both rates of stretching and rotation as a single vector. As with other LCS techniques, these diagnostics define frame-indifferent material features in the flow. We provide multiple computed examples of LCS and momentum transport barriers, and show advantages over other coherent structure diagnostics.

Sources of anisotropy in the Reynolds stress tensor in the stable boundary layer

AbstractData collected by 12 sonic anemometers over the Plaine Morte Glacier (Swiss Alps) during the Snow Horizontal Arrays Turbulence Study are used to investigate sources of anisotropy in the Reynolds stress tensor for stable boundary layers. A coarse‐graining approach is applied to evaluate transfers of kinetic energy across scales, and internal gravity waves (IGWs) are detected with a suitable criterion. Both approaches are combined with a classification of anisotropy based on the barycentric map of 1 min Reynolds stress tensors. A wind‐speed threshold is found that discriminates between regimes with a different dominant topology of the Reynolds stress tensor. One‐component and isotropic states are frequent for low wind speed and strong stratification, whereas two‐component axisymmetric states dominate the high wind speed regime with strong vertical shear. Results suggest that the presence of IGWs is mostly responsible for one‐component states, and additionally influences the relative amount of kinetic and potential energy in the perturbation field. To provide supporting evidence, a complementary analysis of a clean IGW detected during a field campaign in Dumosa (Australia) is presented. This case study highlights how waves contribute to drive the Reynolds stress tensor towards the one‐component limit. Cases are shown where a linear detrending procedure may be effective in filtering out waves and avoid their spurious contributions in turbulence statistics.

Direct numerical simulation of planar turbulent jets: Effect of a pintle orifice

The effects of a pintle-shaped orifice on a planar turbulent jet flow at Reynolds number 4000, based on the inlet bulk mean velocity and the jet width, are studied using direct numerical simulations. Flapping of the jet along with a low-frequency modulation of the Kelvin–Helmholtz (KH) instability, in the presence of a pintle-shaped orifice, is observed. To compare the pintle-jet behavior, a free-jet is simulated as a reference case. The effects of the near-field region on the far-field flow characteristics have been investigated. In both the cases, the KH instability in the near-field influences the far-field jet, whereas the pintle-jet also exhibits a low-frequency flapping. In addition, oblique vortex pattern has been observed in the case of pintle-jet. The far-field flow statistics of the pintle-jet with a top-hat inlet interestingly agree with those of the free-jet with a hyperbolic tangent inlet. Temporal variation of the jet characteristics has been analyzed using spatiotemporal plots. In addition, the large- and small-scale turbulent motion have been studied using three anisotropic invariant maps (turbulence triangles, eigenvalue, and barycentric maps). Moreover, that the barycentric map gives a better visualization of the anisotropic behavior has been observed in the current study.

Anisotropy characterization of turbulent fluidization

The turbulence anisotropy in moderately dense turbulent fluidization is characterized with the aid of the barycentric anisotropy map (BAM). Therein, the phase-filtered anisotropy Reynolds stress tensor is considered to classify the possible states of turbulence into: 1C (one-dimensional), 2C (two-dimensional isotropic) and 3C (three-dimensional isotropic). The turbulence trajectories on the gas phase, as highlighted in the side Figure, have revealed a prevalent 2-D ellipse-like turbulence in the dilute regions, changing to 1-D turbulence in transition areas, and tend towards 3-D turbulence in elongated pancake-like at the interface and inside the clusters.

Data-driven quantification of model-form uncertainty in Reynolds-averaged simulations of wind farms

Computational fluid dynamics using the Reynolds-averaged Navier–Stokes (RANS) remains the most cost-effective approach to study wake flows and power losses in wind farms. The underlying assumptions associated with turbulence closures are the biggest sources of errors and uncertainties in the model predictions. This work aims to quantify model-form uncertainties in RANS simulations of wind farms at high Reynolds numbers under neutrally stratified conditions by perturbing the Reynolds stress tensor through a data-driven machine-learning technique. To this end, a two-step feature-selection method is applied to determine key features of the model. Then, the extreme gradient boosting algorithm is validated and employed to predict the perturbation amount and direction of the modeled Reynolds stress toward the limiting states of turbulence on the barycentric map. This procedure leads to a more accurate representation of the Reynolds stress anisotropy. The data-driven model is trained on high-fidelity data obtained from large-eddy simulation of a specific wind farm, and it is tested on two other (unseen) wind farms with distinct layouts to analyze its performance in cases with different turbine spacing and partial wake. The results indicate that, unlike the data-free approach in which a uniform and constant perturbation amount is applied to the entire computational domain, the proposed framework yields an optimal estimation of the uncertainty bounds for the RANS-predicted quantities of interest, including the wake velocity, turbulence intensity, and power losses in wind farms.

Predictive Data-Driven Model Based on Generative Adversarial Network for Premixed Turbulence-Combustion Regimes

ABSTRACT Premixed flames exhibit different asymptotic regimes of interaction between heat release and turbulence depending on their respective length scales. At high Karlovitz number, the dilatation caused by heat release does not have any relevant effect on turbulent kinetic energy with respect to non-reacting flow, while at low Karlovitz number, the mean shear is a sink of turbulent kinetic energy, and counter-gradient transport is observed. This latter phenomenon is not well captured by closure models commonly used in Large Eddy Simulations that are based on gradient diffusion. The massive amount of data available from Direct Numerical Simulation (DNS) opens the possibility to develop data-driven models able to represent physical mechanisms and non-linear features present in both these regimes. In this work, the databases are formed by DNSs of two planar hydrogen/air flames at different Karlovitz numbers corresponding to the two asymptotic regimes. In this context, the Generative Adversarial Network (GAN) gives the possibility to successfully recognize and reconstruct both gradient and counter-gradient phenomena if trained with databases where both regimes are included. Two GAN models were first trained each for a specific Karlovitz number and tested using the same dataset in order to verify the capability of the models to learn the features of a single asymptotic regime and assess its accuracy. In both cases, the GAN models were able to reconstruct the Reynolds stress subfilter scales accurately. Later, the GAN was trained with a mixture of both datasets to create a model containing physical knowledge of both combustion regimes. This model was able to reconstruct the subfilter scales for both cases capturing the interaction between heat release and turbulence closely to the DNS as shown from the turbulent kinetic budget and barycentric maps.

On a Linear Gromov-Wasserstein Distance.

Gromov-Wasserstein distances are generalization of Wasserstein distances, which are invariant under distance preserving transformations. Although a simplified version of optimal transport in Wasserstein spaces, called linear optimal transport (LOT), was successfully used in practice, there does not exist a notion of linear Gromov-Wasserstein distances so far. In this paper, we propose a definition of linear Gromov-Wasserstein distances. We motivate our approach by a generalized LOT model, which is based on barycentric projection maps of transport plans. Numerical examples illustrate that the linear Gromov-Wasserstein distances, similarly as LOT, can replace the expensive computation of pairwise Gromov-Wasserstein distances in applications like shape classification.

Scale-dependent anisotropy, energy transfer and intermittency in bubble-laden turbulent flows

Data from direct numerical simulations of disperse bubbly flows in a vertical channel are used to study the effect of the bubbles on the carrier-phase turbulence. We developed a new method, based on an extension of the barycentric map approach, that allows us to quantify and visualize the anisotropy and componentiality of the flow at any scale. Using this we found that the bubbles significantly enhance anisotropy in the flow at all scales compared with the unladen case, and that for some bubble cases, very strong anisotropy persists down to the smallest scales of the flow. The strongest anisotropy observed was for the cases involving small bubbles. Concerning the energy transfer among the scales of the flow, our results indicate that for the bubble-laden cases, the energy transfer is from large to small scales, just as for the unladen case. However, there is evidence of an upscale transfer when considering the transfer of energy associated with particular components of the velocity field. Although the direction of the energy transfer is the same with and without the bubbles, the behaviour of the energy transfer is significantly modified by the bubbles, suggesting that the bubbles play a strong role in altering the activity of the nonlinear term in the flow. The skewness of the velocity increments also reveals a strong effect of the bubbles on the flow, changing both its sign and magnitude compared with the single-phase case. We also consider the normalized forms of the fourth-order structure functions, and the results reveal that the introduction of bubbles into the flow strongly enhances intermittency in the dissipation range, but suppresses it at larger scales. This strong enhancement of the dissipation-scale intermittency has significant implications for understanding how the bubbles might modify the mixing properties of turbulent flows.

Contribution of Reynolds shear stress to near-wall turbulence in Rayleigh–Bénard convection

Direct numerical simulations of Rayleigh–Bénard convection flows are performed for Pr=0.7 with Ra=5×108 and 2×1010, and for Pr=0.021 with Ra=107 and 5×108, where Pr and Ra are the Prandtl number and the Rayleigh number, respectively. The velocity-gradient production based on the short-time averaging shows the near-wall intensive positive peak and negative region. The correlations between the Reynolds shear stresses and the wall-normal and transverse gradients of the horizontal mean velocities locally and temporally result in the positive and negative velocity-gradient production near the wall. The eigenvalues of the anisotropic Reynolds stress tensors on the barycentric map show that the short-lived anisotropy related to the one-directional stretching occurs along the wall-normal distance; this behavior is similar to that of a canonical turbulent channel flow. The results indicate that the local and temporal Reynolds shear stresses near the wall with the shear of the horizontal mean velocities play an important role in the near-wall velocity-gradient production, which leads to the turbulent regime in the Rayleigh–Bénard convection flow with an increase in Rayleigh number.

On the unsteady Reynolds-averaged Navier–Stokes capability of simulating turbulent boundary layers under unsteady adverse pressure gradients

Predictive capabilities of unsteady Reynolds-averaged Navier–Stokes (URANS) techniques using the k−ω shear stress transport and Spalart–Allmaras models are assessed for the simulation of turbulent boundary layers under unsteady adverse pressure gradients by comparing their results with direct numerical simulation (DNS) results. Simulations are conducted for separating and reattaching turbulent boundary layers under periodic adverse pressure gradients. Phase-wise comparisons of the velocity, the Reynolds stress, and the skin friction coefficient obtained by URANS simulations and DNS are carried out. URANS techniques are found to qualitatively well predict the formation of the separation bubble and the phase response of the shear layer height, while they predict earlier separation and a larger recirculation bubble compared with those in DNS. Phase responses of the skin friction predicted by URANS simulations are found not to be an accurate indication of flow separation and reattachment of the turbulent boundary layer. The main causes of discrepancies among DNS and URANS results in the near-wall region are attributed to the different anisotropy of the Reynolds stress, which can be characterized by a barycentric map.

Characterization of anisotropic turbulence behavior in pulsatile blood flow

Turbulent-like hemodynamics with prominent cycle-to-cycle flow variations have received increased attention as a potential stimulus for cardiovascular diseases. These turbulent conditions are typically evaluated in a statistical sense from single scalars extracted from ensemble-averaged tensors (such as the Reynolds stress tensor), limiting the amount of information that can be used for physical interpretations and quality assessments of numerical models. In this study, barycentric anisotropy invariant mapping was used to demonstrate an efficient and comprehensive approach to characterize turbulence-related tensor fields in patient-specific cardiovascular flows, obtained from scale-resolving large eddy simulations. These techniques were also used to analyze some common modeling compromises as well as MRI turbulence measurements through an idealized constriction. The proposed method found explicit sites of elevated turbulence anisotropy, including a broad but time-varying spectrum of characteristics over the flow deceleration phase, which was different for both the steady inflow and Reynolds-averaged Navier–Stokes modeling assumptions. Qualitatively, the MRI results showed overall expected post-stenotic turbulence characteristics, however, also with apparent regions of unrealizable or conceivably physically unrealistic conditions, including the highest turbulence intensity ranges. These findings suggest that more detailed studies of MRI-measured turbulence fields are needed, which hopefully can be assisted by more comprehensive evaluation tools such as the once described herein.

Parametric numerical study of passive scalar mixing in shock turbulence interaction

Turbulent mixing of passive scalars is studied in the canonical shock–turbulence interaction configuration via shock-capturing direct numerical simulations, varying the shock Mach number ( ), turbulence Mach number ( ), Taylor microscale Reynolds number ( ) and Schmidt number ( , 1, 2). The shock-normal evolution of scalar variance and dissipation transport equations, spectra and probability density functions (PDFs) are examined. Scalar dissipation, its production and destruction increase across the shock with higher , lower and lower . Mixing enhancement for different flow topologies across the shock is studied from changes in the PDFs of velocity gradient tensor invariants and conditional distributions of scalar dissipation. The proportion of the stable-focus-stretching flow topology is the highest among all the topologies in the flow both before and after the shock. Unstable-node/saddle/saddle topology is the most dissipative throughout the flow domain, despite variations across the shock. Preshock and postshock distributions of the alignment between the strain-rate tensor eigenvectors and the scalar gradient, vorticity and the mean streamwise vector conditioned on flow topology are studied. A novel barycentric map representation is introduced for a more direct visualization of the alignments and conditioned scalar dissipation distributions. Interaction with the shock increases alignment of the scalar gradient with the most extensive eigenvector, decreasing it with the most compressive, which is still dominant. The barycentric map of the passive scalar gradient also reveals that, across the shock, the most probable alignment between scalar gradient and strain eigendirections converges towards the alignment that provides the most dissipation. This also leads to an enhancement of scalar dissipation immediately downstream of the shock.

Anisotropic mesh quality measures and adaptation for polygonal meshes

Anisotropic mesh quality measures and adaptation are studied for convex polygonal meshes. Three sets of alignment and equidistribution measures are developed, based on least squares fitting, generalized barycentric mappings, and the singular value decomposition, respectively. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given anisotropic metrics. Based on the second set of quality measures and using a moving mesh partial differential equation, an anisotropic adaptive polygonal mesh method is constructed for the numerical solution of second-order elliptic equations. Numerical examples are presented to demonstrate the effectiveness of the method.

Accurate coarse soft tissue modeling using FEM-based fine simulation

Finite element method is a well-known approach in soft tissue modeling. However, it introduces nonconforming deformations for a soft tissue in different resolutions in response to the same applied force. These deformations make the approach inefficient in data-driven enrichment schemes which demand more accurate conforming models of an object in both low and high resolutions at the same time. This paper presents two methods based on (1) Sampling and (2) Barycentric mapping to overcome this problem and to generate geometrically conforming deformations in different resolutions. In proposed methods, first, the soft tissue is modeled in high resolution by using finite element method to achieve the desired accuracy. The coordinates of this accurate model are then used to find the corresponding coordinates of the coarse model. This step is done by using either Sampling or Barycentric mapping. Quantitative evaluation of the simulation results confirms the efficiency of suggested methods in modeling geometrically conforming soft tissues in different resolutions.

Existence and uniqueness of the Karcher mean on unital C⁎-algebras

The Karcher mean on the cone Ω of invertible positive elements of the C⁎-algebra B(E) of bounded operators on a Hilbert space E has recently been extended to a contractive barycentric map on the space of L1-probability measures on Ω. In this paper we first show that the barycenter satisfies the Karcher equation and then establish the uniqueness of the solution. Next we establish that the Karcher mean is real analytic in each of its coordinates, and use this fact to show that both the Karcher mean and Karcher barycenter map exist and are unique on any unital C⁎-algebra. The proof depends crucially on a recent result of the author giving a converse of the inverse function theorem.

Convergence theorems for barycentric maps

We first develop a theory of conditional expectations for random variables with values in a complete metric space [Formula: see text] equipped with a contractive barycentric map [Formula: see text], and then give convergence theorems for martingales of [Formula: see text]-conditional expectations. We give the Birkhoff ergodic theorem for [Formula: see text]-values of ergodic empirical measures and provide a description of the ergodic limit function in terms of the [Formula: see text]-conditional expectation. Moreover, we prove the continuity property of the ergodic limit function by finding a complete metric between contractive barycentric maps on the Wasserstein space of Borel probability measures on [Formula: see text]. Finally, the large deviation property of [Formula: see text]-values of i.i.d. empirical measures is obtained by applying the Sanov large deviation principle.

Improving Initial Flattening of Convex-Shaped Free-Form Mesh Surface Patches Using a Dynamic Virtual Boundary

This study proposes an efficient algorithm for improving flattening result of triangular mesh surface patches having a convex shape. The proposed approach, based on barycentric mapping technique, incorporates a dynamic virtual boundary, which considerably improves initial mapping result. The dynamic virtual boundary approach is utilized to reduce the distortions for the triangles near the boundary caused by the nature of convex combination technique. Mapping results of the proposed algorithm and the base technique are compared by area and shape accuracy metrics measured for several sample surfaces. The results prove the success of the proposed approach with respect to the base method.