Jet Space Research Articles

Discrete Moving Frames on Lattice Varieties and Lattice-Based Multispaces

In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalisation of the jet bundle that also generalises Olver’s one-dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame.

Numerical study on the effect of jet spacing on the Swirl flow and heat transfer in the turbine airfoil leading edge region

ABSTRACTIn this paper, flow and heat transfer of a swirl chamber that models an internal cooling passage for a gas turbine airfoil leading edge is studied with numerical simulation. The geometry consists of a circular pipe, and rectangular section inlets that lead inlet flow to impinge tangentially on the circular pipe. The effects of the ratio of jet spacing to swirl chamber radius and Reynolds numbers on swirl cooling performance are investigated. The results indicate how the pressure loss and globally averaged Nusselt number on the swirl chamber wall increase with increases of Reynolds number and the ratio of jet spacing to swirl chamber radius. A Nusselt number correlation on these parameters is suggested. Also shown is how Nusselt numbers on the swirl chamber surface increase with the ratio of jet spacing to swirl chamber radius.

Geometry of fanning curves in divisible Grassmannians

We study the geometry of fanning curves in the Grassmann manifold of n-dimensional subspaces of Rkn; we construct a complete system of invariants which solve the congruence problem. The geometry of the invariants themselves in terms of equivariant models of the relevant spaces of jets is studied; in particular we provide a geometrical characterization of the flat curves.

Lagrangian and Hamiltonian Duality

We propose a setting for De Donder–Hamilton field theory in jet bundles, generalizing the usual multisymplectic formalism. Using a reformulation of Hamilton theory for the family of local Lagrangians related to a global Euler–Lagrange form, we construct a dual Hamiltonian bundle and corresponding Legendre maps, linking a Lagrangian system on a jet bundle with a canonical Hamiltonian system on the affine dual. Our approach significantly extends the family of regular variational problems that can be treated directly within a dual Hamiltonian formalism, thus avoiding the necessity to use the Dirac constraint formalism.

Geometric Interpretation of Fractional-Order Derivative

Abstract A new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

Strong topologies for spaces of smooth maps with infinite-dimensional target

In this article we study two “strong” topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modelled on a locally convex space. Namely, we construct Whitney type topologies for these spaces and a certain refinement corresponding to Michor’s FD-topology. Then we establish the continuity of certain mappings between spaces of smooth mappings, e.g. the continuity of the joint composition map. As a first application we prove that the bisection group of an arbitrary Lie groupoid (with finite-dimensional base) is a topological group (with respect to these topologies).For the reader’s convenience the article includes also a proof of the folklore fact that the Whitney topologies defined via jet bundles coincide with the ones defined via local charts.

Shape maps for second order partial differential equations

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of shape maps. The trace of such maps represents a congruence volume whose collapse we study through an appropriate evolution equation, akin to Raychaudhuri's equation. We develop the necessary geometric framework on a suitable jet space in which the shape maps appear naturally associated with certain linear connections. Explicit computations are given, along with a nontrivial example.

Foliation, Jet Bundle and Quantization of Einstein Gravity

In \cite{Park:2014tia} we proposed a way of quantizing gravity with the Hamiltonian and Lagrangian analyses in the ADM setup. One of the key observations was that the physical configuration space of the 4D Einstein-Hilbert action admits a three-dimensional description, {thereby making gravity renormalization possible through a metric field redefinition.} Subsequently, a more mathematical and complementary picture of the reduction based on foliation theory was presented in \cite{Park:2014qoa}. With the setup of foliation the physical degrees of freedom have been identified with a certain leaf. Here we expand the work of \cite{Park:2014qoa} by adding another mathematical ingredient - an element of jet bundle theory. With the introduction of the jet bundle, the procedure of identifying the true degrees of freedom outlined therein is made precise and the whole picture of the reduction is put on firm mathematical ground.

Multiobjective Fractional Variational Problem on Higher-Order Jet Bundles

The main goal of this paper is to introduce necessary efficiency conditions for a class of multi-time vector fractional variational problems with nonlinear equality and inequality constraints involving higher-order partial derivatives. We consider the multi-time multiobjective variational problem (MFP) of minimizing a vector of path-independent curvilinear integral functionals quotients subject to PDE and/or PDI constraints, developing an optimization theory on the higher-order jet bundles.

Performance Analysis of a Multiple Micro-Jet Impingements Cooling Model

The present study investigates the thermal performance of a multiple micro-jet impingements model for electronics cooling. The fluid flow and heat transport characteristics were investigated for steady incompressible laminar flow by solving three-dimensional (3D) Navier-Stokes equations. Several parallel and staggered micro-jet configurations (ie. inline 2 Å~ 2, 3 Å~ 3 and 4 Å~ 4 jets, and staggered five-jet and 13-jet arrays with the jet diameter to the channel height ratios from 0.25–0.5) were analyzed at various flow rates for the maximum temperature rise, pressure drop, heat-transfer coefficient, thermal resistance, and pumping power characteristics. The parametric investigation was carried out based on the number of jets and the jet diameters at various mass flow rates and jet Reynolds numbers. Temperature uniformity and coefficient of performance were evaluated to find out the trade-off among the various designs investigated in the present study. The maximum temperature rise and the pressure drop decreased with an increase in the number of jets except in the case of staggered five-jet array. A higher temperature uniformity was observed at higher flow rates with a decrease in the coefficient of performance. The performance parameters, such as thermal resistance and pumping power, showed a conflicting nature with respect to design variables (viz. jet diameter to stand-off ratio and interjet spacing or number of jets) at various Reynolds numbers within the laminar regime.

Forced convective heat transfer on a horizontal circular cylinder due to multiple impinging circular jets

A computational study is undertaken to investigate convective heat transfer on a horizontal cylinder due to a bank of vertically oriented circular jets. The importance of this study stems from the real-world application of convective cooling of rotary cement kilns by the use of large axial fans. A computational model is developed which considers one spatially-periodic section of the domain, and solutions of the conservation equations combined with an appropriate two-equation turbulence model are obtained using the commercial software Fluent™. The computational model is validated by a comparison to previous studies of a single circular jet impinging on a cylinder. A parametric study is presented which considers the impact on average heat transfer due to: jet-to-cylinder spacing (y/d), axial jet spacing (z/d), jet-to-cylinder diameter ratio (d/D), jet offset from the axial centerline (x/d), and jet Reynolds number. Results of the parametric study show that for jet-to-cylinder ratios of d/D⩾0.23, that movement away from the cylinder axis, increased axial spacing and lateral offset all lead to degradation of heat transfer from the cylinder. For d/D=0.15, similar degradation occurs, but in part of the parametric space studied, increased distance from the cylinder and lateral offset leads to enhancements of the heat transfer. The complete set of results is presented as a correlation wherein the influence of all parameters studied are included as power-law corrections to the average Nusselt number.

On the Influence of Array Size and Jet Spacing on Jet Interactions and Confluence in Round Jet Arrays

This work uses computational models to study the effects of confluence and jet-to-jet interactions for inline matrices of confluent round jets. In total, 12 different confluent jet arrangements, having various jet array sizes and dimensionless jet spacing, S/d0, have been investigated. The array size varies from 6 × 6 to 10 × 10 jets, while S/d0 varies between 1.75≤S/d0≤4.0. The Reynolds number, based on the nozzle exit diameter, is between 2200 and 6600. The results show that both jet spacing and jet array size largely influence the jet-to-jet interactions and flow field development in confluent jet arrays. The jet interactions in the investigated setups result in regions of negative static pressure between jets, jet deformation, high spanwise velocity, and jet displacement. Generally, smaller jet spacing and larger array size result in stronger influence of jet interactions. After the jets have combined, the confluent jets form a zone with constant maximum streamwise velocity and decay of turbulence intensity, called a confluent core zone (CCZ). During the CCZ, the combined jet will have asymmetric spreading rates leading to axis-switching. The entrainment rate of the CCZ is constant, but the volumetric flow rate of the combined jet is substantially affected by the degree of entrainment before the jets have combined.

Irreducible components of quiver Grassmannians

We prove a decomposition theorem for irreducible components of Grassmannians of submodules, as well as for other schemes arising from representation theory, thus generalising the result of Crawley-Boevey and Schroer for module varieties. The method is based on jet space calculations, using that the formation of direct sums induces a separable morphism of schemes.

The turbulent behavior of novel free triple-impinging jets with large jet spacing by means of particle image velocimetry

A novel reactor that achieves rapid liquid–liquid mixing via free triple-impinging jets (FTIJs) is developed to improve mixing efficiency at unequal flow rates for liquid–liquid reactions. The flow characteristics of FTIJs were investigated using particle image velocimetry (PIV). The instantaneous and mean velocities data at different Reynolds numbers (Re) were analyzed to provide insights into the velocity distributions in FTIJs. The effect of jet spacing on the stagnation points, instantaneous velocity, mean velocity, profiles of the x- and y-components of mean velocity, and turbulent kinetic energy (TKE) distributions of FTIJs were investigated at Re=4100 with a volumetric flow rate ratio of 0.5. The characteristics of the turbulent flows are similar for all jet spacings tested. Two stagnation points are observed, which are independent of jet spacing and are not located in the center of the flow field. However, velocity and TKE distributions are strongly dependent on the jet spacing. Decreasing jet spacing increases the expansion angle and the values of TKE, leading to strong turbulence, improving momentum transfer and mixing efficiency in FTIJs. The present study shows that optimization of the operating parameters is helpful for designing FTIJs.

Overconnections and the energy-tensors of gauge and gravitational fields

A geometric construction for obtaining a prolongation of a connection to a connection of a bundle of connections is presented. This determines a natural extension of the notion of canonical energy-tensor which suits gauge and gravitational fields, and shares the main properties of the energy-tensor of a matter field in the jet space formulation of Lagrangian field theory, in particular with regards to symmetries of the Poincaré–Cartan form. Accordingly, the joint energy-tensor for interacting matter and gauge fields turns out to be a natural geometric object, whose definition needs no auxiliary structures. Various topics related to energy-tensors, symmetries and the Einstein equations in a theory with interacting matter, gauge and gravitational fields can be viewed under a clarifying light. Finally, the symmetry determined by the “Komar superpotential” is expressed as a symmetry of the gravitational Poincaré–Cartan form.

Comparison of the single/multi transverse jets under the influence of shock wave in supersonic crossflow

In this study, the effects of the shock wave on sonic transverse hydrogen through single and multi-jets for supersonic combustion were investigated numerically. This study presents the fundamental flow physics of the interaction between fuel jets (single or multi array) and incident shock waves into a Mach 4.0 crossflow. Parametric studies were conducted on the performance of the shock wave by using the RANS equations with Menter's Shear Stress Transport turbulence model. In a parametric study, both the streamwise spacing and jet-to-freestream total pressure ratio are varied. For all downstream mixing, the associated flow behavior was found to be a direct result of both the type of injection (single/Multi jet) and interactions between shock waves and injectors. According to the results, shock wave reduces the maximum concentration of the hydrogen jet more than 20% in both single and multi jet. Furthermore, a significant increase (approximately 40%) occurs in the mixing of the hydrogen jet at downstream when shock generator is presented in the multi jet with PR=0.27. Moreover, hydrogen-air mixing rate extends in streamwise direction as the jet space increases. Thus, an enhanced mixing zone occurs in the in far downstream of the jet and the shock wave.

Bicrossed products induced by Poisson vector fields and their integrability

First we show that, associated to any Poisson vector field [Formula: see text] on a Poisson manifold [Formula: see text], there is a canonical Lie algebroid structure on the first jet bundle [Formula: see text] which, depends only on the cohomology class of [Formula: see text]. We then introduce the notion of a cosymplectic groupoid and we discuss the integrability of the first jet bundle into a cosymplectic groupoid. Finally, we give applications to Atiyah classes and [Formula: see text]-algebras.

Experimental Investigation of Jet Impingement Cooling On Ribbed Target Surface

Experimental investigation of jets impacting at right angleon a flat surfaceis introduced in the present study. Round jet holes of inline arrays arrangements with diameter (5mm) and jet to jet spacing of 4 jet hole diameter are considered. Jet Reynolds numbers of 4000 to 16000 and jet height to diameter ratio (space between the jet plate and target plate) of 2.0 is maintained. The heat transfer model consists of amultiple jet holes, ribbed target plate having back side resistive film. Threetarget models have been considered; model (1) is a clean surface target acting as the baseline case, the model (2) the target surface consists of lateral rib rows and model (3) the target surface consists grids rib arrangement. The ribs height for models (2 and 3) is 0.86 of jet diameter of the triangle shape. The average wall cooling effectiveness and heat transfer coefficients and consequently the Nusselt numbers for each model are estimated. Numerical computations of jets flow field are made.Theresults show that model (3) enhanced both heat transfer coefficients in the target plat inner surface and wall cooling effectiveness in the target plat outer surface greatly,the maximum increments in heat transfer coefficient and wall coolingeffectiveness are 14.5% and 4.5% respectively.

Numerical and Experimental Investigation of Near-Field Mixing in Parallel Dual Round Jets

Parallel underexpanded round jets system has been widely used in engineering applications, and the flow field structures are very complex because of the jets interaction. In this paper, we studied the near-field mixing phenomenon in parallel dual underexpanded jets numerically by solving the Reynolds-Averaged Navier-Stokes Equations. The numerical results agree well with experimental data acquired by particle image velocimetry. Similar to plane jets, to some degree, two round jets are deflected towards the dual nozzle symmetry plane; the flow field can also be divided into three regions. Meanwhile, attempts have been made to predict merge point and combine point locations on certain cross profile of computational domain by correlating them with jet spacing and jet pressure ratio. The jet spacing plays an important role in jets interaction, and jet interaction decreases with the increase in jet spacing. The jets interaction in terms of merge (combine) point and pressure varies significantly while the jet spacing differs. Additionally, as pressure ratio increases, the effect of jet interaction decreases, and the merge (combine) point location moves downstream.

Jet Bundles on Projective Space II

In previous papers the structure of the jet bundle as P-module has been studied using different techniques. In this paper we use techniques from algebraic groups, sheaf theory, generliazed Verma modules, canonical filtrations of irreducible SL(V)-modules and annihilator ideals of highest weight vectors to study the canonical filtration Ul (g)Ld of the irreducible SL(V)-module H0 (X, iX(d))* where X = i‡(m, m + n). We study Ul (g)Ld using results from previous papers on the subject and recover a well known classification of the structure of the jet bundle il (i(d)) on projective space i(V*) as P-module. As a consequence we prove formulas on the splitting type of the jet bundle on projective space as abstract locally free sheaf. We also classify the P-module of the first order jet bundle iX1 (iX (d)) for any d ≥ 1. We study the incidence complex for the line bundle i(d) on the projective line and show it is a resolution of the ideal sheaf of I l (i(d)) - the incidence scheme of i(d). The aim of the study is to apply it to the study of syzygies of discriminants of linear systems on projective space and grassmannians.