Bundle Geometry Research Articles

Sheaves and symplectic geometry of cotangent bundles

Sheaves and symplectic geometry of cotangent bundles

Quantization of symplectic fibrations and canonical metrics

We relate Berezin–Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including the spectral gap of the Berezin transform and the convergence rate of Donaldson’s iterations toward balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson’s iterations toward balanced metrics on Kähler manifolds with constant scalar curvature.

Flow field characteristics of a 127-pin rod bundle with hexagonal spacer grids

A determination of nominal flow phenomena in liquid metal fast reactor (LMFR) fuel assemblies is critical toward generation-IV reactor development. Axially positioned spacer grids are used to maintain the geometry of hexagonal rod bundles and simultaneously introduce perturbations in the flow. Three-dimensional (3D) printed asymmetric honeycomb spacer grids were installed in a prototypical 127-pin LMFR fuel assembly model to study complex fluid dynamics interactions induced by the spacer grid and rods. To characterize flow dynamics in this intricate geometry, time-resolved particle image velocimetry (TR-PIV) using the matched-index-of-refraction method was employed to obtain non-intrusive velocity measurements for three axial planes (one near-wall and two interior planes) at a Reynolds number of 6000. The statistical TR-PIV results compared sub-channel-dependent normalized time-averaged velocity, velocity fluctuations, Reynolds stress, vorticity, and turbulence kinetic energy distributions. TR-PIV line profiles characterized downstream spacer grid flow dynamics. Two-point spatial and spatial–temporal cross-correlation fields revealed local coherent structures and quantified convection velocities of traveling vortices. Spatial–temporal decomposition using dynamic mode decomposition (DMD) applied to the near-wall vorticity fields extracted turbulent structures and flow instabilities in the wake region of the spacer grid, along with their decay and frequency rates. Reduced-order velocity fields from DMD reconstructions identified the most energy-containing coherent structures persistent in the near-wall region. This research provides experimental data sets and analyses of flow behavior in rod bundles with hexagonal spacer grids. The results are critical toward LMFR design and geometry optimization, crucial for the validation of computational fluid dynamics and reduced-order flow models.

BIO13: Effect of Artificial Lung Fiber Bundle Geometric Design on Micro- and Macro-scale Clot Formation

Purpose of Study: The hollow fiber membrane bundle is the functional component of artificial lungs, transferring oxygen and carbon dioxide to and from the blood. It is also the primary location of blood clot formation and propagation in these devices. The geometric design of fiber bundles is defined by a narrow range of parameters that determine gas exchange efficiency and blood flow resistance, such as fiber packing density, path length, and frontal area. However, these parameters also affect thrombosis. Methodology: This study investigated the effect of these parameters on clot formation using 3-D printed flow chambers that mimic the geometry and blood flow patterns of fiber bundles. Hollow fibers were represented by an array of vertical micro-rods (380 micron diameter) arranged with varying packing densities (40, 50, and 60%) and path lengths (2 and 4 cm), all of which are fiber bundle parameters in commercially available artificial lung devices. Blood was pumped through the device corresponding to three mean blood flow velocities (16, 20, and 25 cm/min), which were scaled from 4 L/min flowrates in an Adult Quadrox oxygenator. Results: Results showed that (1) clot formation decreases dramatically with decreasing packing density and increasing blood flow velocity, (2) clot formation at the outlet of fiber bundle enhances deposition upstream, and consequently (3) greater path length provides more clot-free fiber surface area for gas exchange than a shorter path length. Summary: These results can be used to create less thrombogenic, more efficient artificial lung designs.Figure 1. a) The experimental flow chamber was designed to be 2 or 4 cm and with a packing density of 60%, 50% or 40% rods to simulate the flow path of blood through an artificial lung hollow fiber bundle. (b) The flow circuit involved one-way flow from a syringe pump, connected to the flow chamber and then to a waste container. A manometer measured the pressure drop across each experimental device. (c) The averaged clot volumes from the microCT scans show more clot (red) at the outlets of all devices. There was also less clot in the chambers with a longer path length device, faster flow rate, and less density.

Consistent truncations from the geometry of sphere bundles

In this paper, we present a unified perspective on sphere consistent truncations based on the classical geometric properties of sphere bundles. The backbone of our approach is the global angular form for the sphere. A universal formula for the Kaluza-Klein ansatz of the flux threading the n-sphere captures the full nonabelian isometry group SO(n + 1) and scalar deformations associated to the coset SL(n + 1, ℝ)/SO(n + 1). In all cases, the scalars enter the ansatz in a shift by an exact form. We find that the latter can be completely fixed by imposing mild conditions, motivated by supersymmetry, on the scalar potential arising from dimensional reduction of the higher dimensional theory. We comment on the role of the global angular form in the derivation of the topological couplings of the lower-dimensional theory, and on how this perspective could provide inroads into the study of consistent truncations with less supersymmetry.

On the Geometry of Tangent Bundle and Unit Tangent Bundle with Deformed-Sasaki Metric

Let $(M^{m}, g)$ be a Riemannian manifold and $TM$ its tangent bundle equipped with a deformed Sasaki metric. In this paper, firstly we investigate all forms of Riemannian curvature tensors of $TM$ (Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature). Secondly, we study the geometry of unit tangent bundle equipped with a deformed Sasaki metric, where we presented the formulas of the Levi-Civita connection and also all formulas of the Riemannian curvature tensors of this metric.

Topological insulators and geometry of vector bundles

For a long time, band theory of solids has focused on the energy spectrum, or Hamiltonian eigenvalues. Recently, it was realized that the collection of eigenvectors also contains important physical information. The local geometry of eigenspaces determines the electric polarization, while their global twisting gives rise to the metallic surface states in topological insulators. These phenomena are central topics of the present notes. The shape of eigenspaces is also responsible for many intriguing physical analogies, which have their roots in the theory of vector bundles. We give an informal introduction to the geometry and topology of vector bundles and describe various physical models from this mathematical perspective.

Geometric tool kit for higher-spin gravity (Part I): An introduction to the geometry of differential operators

This paper provides an introduction to the algebra and geometry of differential operators and jet bundles. Its point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalizations of Lie derivatives and diffeomorphisms. Nevertheless, the material covered here may be of general interest to anyone working on topics where geometrical (coordinate-free, global, generic) and mathematically rigorous definitions of differential operators are required.

О группе автоморфизмов алгебры плюральных чисел

The algebra of dual numbers was first introduced by V. K. Clifford in 1873. The algebras of plural and dual numbers are analogous to the algebra of complex numbers. Dual numbers form an algebra, but not a field, because only dual numbers with a real part not equal to zero have an inverse element. In this work, automorphisms of algebras of plural numbers, which are a generalization of the algebra of dual numbers, are studied. Algebras of plural numbers were in the center of attention of the professor of Kazan University A. P. Shirokov. Studying the geometry of higher-order tangent bundles, he established that higher-order tangent bundles over smooth manifolds have the structure of a smooth manifold over algebras of plural numbers. This allowed him in the 70s of the twentieth century to construct a theory of lifts of tensor fields and linear connections from a smooth manifold to its tangent bundles of arbitrary order. In this paper, we study automorphisms of the algebra of plural numbers. It is proved that the set of all automorphisms of the algebra of plural numbers forms a group. The structure of this group is described. The groups of automorphisms of the algebra of plural numbers with small dimension are indicated as examples.

Origin of Internal Symmetries of the Fundamental Interactions,the Family Problem, Fractional Quark Charges, and Unification in the Tangent Bundle Geometry

Origin of Internal Symmetries of the Fundamental Interactions,the Family Problem, Fractional Quark Charges, and Unification in the Tangent Bundle Geometry

A novel friction factor model for wire-wrapped rod bundles

Wire-wrapped rod bundles are often considered for fast spectrum nuclear reactors, especially liquid metal cooled fast reactors (LMFRs). Accurate calculation of the pressure drops of these bundles is key for proper thermal–hydraulic design of the reactor. Existing models, while able to accurately predict bundle friction factor, are complex and can be difficult to use without utilizing a numerical implementation. In this paper, a new model for bundle friction factor is proposed based on exact laminar solutions of the Navier-Stokes equations for a simplified rod bundle geometry. These exact solutions can then be adjusted via geometric and empirical parameters to the prototypic geometry. The proposed model is able to calculate the bundle friction factor with an average absolute percent difference of 10.8 % compared to the available experimental data. This compares favourably to existing correlations, offering similar performance while minimizing the number of empirical coefficients and the overall effort required. The model is also valuable for determining a theoretical lower bound for bundle friction factor for a given geometry, with some data found to be lower (>10 %) than this bound. The overall methodology can also in principal be applied to other geometries where similar simplifications could be made.

ALLEGRO Gas cooled Fast Reactor Rod Bundle investigations with CFD and PIV method

The development of a new fuel bundle usually relies heavily on research methods based on computer codes and numerical simulations. The reliability of these computer codes should be compared with measurement results for validation purposes.The main objective of this paper is to present a new measuring system developed at Budapest University of Technology and Economics (BME) Institute of Nuclear Techniques (NTI). The system is designed to investigate the emerging flow structures in GFR relevant 7 pin rod bundle geometry. The created rod bundle is a simplified and reduced version of the ALLEGRO GFR fuel assembly (P/D = 1.208). Our test facility named PIROUETTE (PIv ROd bUndlE Test faciliTy at bmE) is capable of capturing 2D velocity flow fields inside the rod bundle. The fuel rods made of special polymer (FEP), which are practically transparent in water, therefore provide with the opportunity to investigate the effect of different spacer grids even inside the rod bundle. With high resolution 3D printing method different mixing vane spacer grid types (e.g. spacer grid without vane, and spacers with SPLIT and TWISTED type mixing vanes) were manufactured and tested. An uncertainty estimation related to the PIV methodology was also performed.CFD models have been developed for the flow domain, which includes the test section of the PIROUETTE facility and the rod bundle inside it. Separate CFD models were made to describe the different mixing vanes. Mesh independence study was created to provide suitable discretisation of the models. Steady-state RANS calculations were performed with different turbulence models. The measurement results were compared with the results of the CFD modelling. Transient calculations were created to achieve more accurate results. All the calculations were carried out using ANSYS CFX 19.2 code.

Iterative computed tomography reconstruction of void fraction in a fuel bundle on unstructured mesh

The experimental investigation of internal two-phase flow structures in high pressure/high temperature systems is technologically challenging and limited to a few instrumentation techniques. For complex fuel bundle geometries, the use of Computed Tomography (CT) allows for relatively high spatial resolution (0.2 to 1 mm) and non-intrusive measurement of time-averaged void fraction. One of the main issues using standard structured reconstruction mesh is due to the loss of accuracy near the test section boundaries, for instance within the pixels overlapping both the two-phase coolant and the heated rods. In this work, a void fraction reconstruction methodology based on unstructured mesh is developed to accurately bound relevant attenuation regions within the considered test section. This method eliminates any overlapping reconstruction pixels and allows straightforward applications of knowledge-based, low noise, algebraic iterative reconstruction techniques (ART, SIRT, etc.). Within the same consistent framework, CT scan forward projections can be simulated for various selected cases (single-phase, arbitrary-two-phase void fraction distributions, etc.) and reconstructed to assess the performance of both the CT scan setup (e.g., projection data sampling and number of projections) and the reconstruction technique. The method is applied to the void fraction reconstruction of gamma CT data in a 5 × 5 fuel bundle geometry measured at the Westinghouse FRIGG thermal–hydraulic test facility operating under prototypical conditions of Boiling Water Reactor. The unstructured mesh is designed to match the features of the test section (isolation, pressure vessel, water gap, fuel channel, heated rods and coolant) with high-resolution in the domain of interest (coolant) and very low-resolution in all other domains with known attenuation coefficients. As compared to previous void fraction reconstruction attempts for this data set, the new reconstruction methodology allows significant improvements in void fraction resolution and noise reduction.

On Complete, Horizontal and Vertical Lifts From a Manifold With fλ(6,4) Structure to Its Cotangent Bundle

Manifolds with fλ(6,4) structure was defined and studied in the past. Later the geometry of tangent and cotangent bundles in a differentiable manifold with fλ(6,4) structure was studied. The aim of the present paper is to study complete, horizontal and vertical lifts from a manifold with fλ(6,4)- structure to its cotangent bundle.

Experimental study of bottom reflooding for 5×5 non-uniformly heated rod bundle with spacer grid

Water heat transfer effectiveness during reflooding process in rod bundle geometry is very important for reactor safety under loss of coolant accident. A series of reflooding experiment is carried out at the REFB_NUSOL test facility for a 5 × 5 rod bundle to study the effect of several key parameters on overall reflooding process, including subcooling of coolant (20 °C to 80 °C), coolant flow velocity (2.36 cm/s to 11.19 cm/s), initial peak surface temperature (500 °C–900 °C) and steam temperature on quench front progress. The temperature, heat flux and heat transfer coefficient on the heated rod surfaces have been achieved via a self-developed Reflooding Experimental Data Analysis (REDA) code. Studies on surface temperature variation, quench front progress, heat transfer enhancement caused by spacer grid and effect rules of minimum film boiling temperature are discussed to shed a light on the thermal-hydraulic behaviors during reflooding process in different conditions for rod bundles. Additionally, the prediction accuracy of three published empirical correlations on minimum film boiling temperature are evaluated using our experiment data. The meaning of this work is to explain the reflooding phenomenon of 5 × 5 rod bundle meanwhile provide validation data and suggestions for empirical equation development for nuclear thermal hydraulic analysis code to better predict reflooding process.

Metal fuel relocation experiments with pressure injection

A pool-type sodium-cooled fast reactor (SFR) using metal fuels has a number of inherent safety features that can support benign consequences for design basis accidents (DBAs). Even in postulated severe accident conditions, the core is designed to remain under sub-critical condition in a passive coolable geometry. In case of the postulated severe accidents in SFRs, fuel relocation in the core region along the coolant channel is an important negative reactivity feedback factor that lowers the reactor power level and consequently eliminates the possibility of recriticality. Therefore, understanding the relocation behavior of fuels and the coolability of the relocated fuels in the postulated severe accident is one of the most important factors in the safety assessment of SFRs. In the present study, the relocation behavior of the metal fuel in a pin bundle geometry was investigated by injecting the metal fuels into the coolant channels with pressure. The first metal fuel relocation with pressure injection (RPI-1) experiment showed that many of the metal fuels levitated to upper plenum. In case of RPI-2 experiment, the Real Time X-ray Video System (RTXVS) was constructed and used to obtain a real time X-ray video of the experiment. This real time X-ray video clearly showed the relocation behavior of the pressure injected metallic uranium in the sodium coolant channel. Through this video, it was confirmed that part of the fuel was dispersed downward and part of the fuel was dispersed upward, and the relocation behavior of the injected fuel proceeded simultaneously in the downward and upward directions. It can be concluded that in case of pressure injection of the metallic fuel, there is a possibility that the metallic fuel can be quickly removed from the core region, resulting in a negative reactivity feedback effect.

Fundamental forces and their dynamics

In this essay, we wish to propose a general principle: the equation of motion or dynamics of a fundamental force should not be prescribed but instead be entirely driven by geometry of the appropriate spacetime manifold, and the equation is then obtained by employing only the geometric property without appeal to an action. The motivation for this pronouncement comes from the fact that the equation of motion of general relativity follows from the geometry of Riemannian spacetime manifold without appeal to anything else from outside. The driving differential geometric property is the Bianchi identity satisfied by the Riemann curvature tensor. Similarly it is geometry of the principal tangent bundle of fibre spacetime manifold that may account for dynamics of the gauge vector fields. It is the classical electric force for the Abelian gauge symmetry group while the non-Abelian symmetry leads to the non-Abelian forces, the weak and the strong. We shall also reflect on a unified picture of the basic forces, and the duality correspondences it may inspire.

Friction factor and Reynolds number correlation for finned tube bundle of the air cooler of Monju reactor

Applicability of the analysis model of the cooling ability under a high temperature and low flowrate condition should be improved to evaluate the plant safety in case of the severe accident due to long-term station blackout.The present study reviews experiments on pressure drop behavior for complicated tube bundle geometry to the flow path and then develops a new correlation equation based on a computational fluid dynamics analysis of the Monju air cooler reflecting the actual geometry and plant data.After a basic simulation model being developed for a typical pressure drop experiment, the simulation is applied to the Monju air cooler that has a finned tube bundle. The obtained relationship between friction factors and Reynolds numbers ranging from 10 to 10,000 are fitted to a power function to derive a correlation equation of the fin tube bundle friction factor. The derived correlation equation can estimate pressure loss in the finned tube bundle more precisely than that used in the Monju design. It is applicable to the future reactor design of the air cooler, especially when the cooling ability in low Reynolds number is requested.

Research on the thermal performance of a heat exchanger with meso-scale twisted helical tube bundles

Heat exchangers with helical tubes are applied widely in various industrial fields and cryogenic applications. Currently, the most common ones are the spiral wound heat exchangers, which consists of many layers of spiral tubes with different coil diameters in a closed shell. Most of the fluid in the shell-side passes directly through the gap between two adjacent layers, which weakens the turbulence and the associated heat transfer coefficient. A novel twisted helical bundle geometry is proposed in this paper of which the thermal performance is explored using CFD simulation. The heat exchanger uses helium gas as the working fluid and operates between 300 K and 30 K. The curve of each tube in the bundle is generated by different 3D sinusoidal equations, and the tubes at adjacent layers are twisted together thereby producing the same end-to-end length for each tube. By simulating a spiral wound heat exchanger using different turbulent equations and comparing the results with experimental data from literature, the k-omega SST model is found to be the most suitable one to simulate this kind of geometry. Nusselt numbers and friction factors are obtained under different Re values and compared with those of standard spiral wound heat exchangers. The creative twisted geometry can improve the Nu number although also increasing the friction factor. Geometry parameters such as pitch length are being further explored to obtain an optimal thermal performance.

Simultaneous measurement of the structures of the velocity and thermal boundary layers in the rod bundle channel

The complex geometry of the rod bundle fuel assemblies makes the measurement of velocity and temperature difficult, which hinders the design and optimization of the thermohydraulic performance of the advanced fuel assemblies. In the present study, Particle Image Velocimetry (PIV) and Laser-Induced Fluorescence (LIF) techniques are used to simultaneously measure and reconstruct the structures of the velocity and thermal boundary layers. The velocity and temperature distributions near the rod surface under different flow and heat transfer conditions are quantitatively analyzed and compared. The experimental results indicate that increasing the heat flux would increase the near-wall velocity gradient and decrease the temperature gradient. The buoyancy increases with the increase of the heat flux, reducing the turbulent fluctuation in the channel, thereby weakening the generation and diffusion of the turbulent vortex. The dimensionless velocity distribution is obtained by fitting the experimental data to the Spalding formula. Based on the structure of the boundary layer, the increase of buoyancy significantly reduces the proportion of the logarithmic law layer in comparison to the inner layer of the boundary layer. This weakens the convective heat transfer in the channel. The increase of the Reynolds number increases the turbulent fluctuation in the channel and weakens the influence of the buoyancy. Thus, under constant heat flux, the increase of the Reynolds number increases the proportion of the logarithmic law layer in comparison to the inner layer, which enhances the convective heat transfer in the channel.