Lagrangian Transformation Research Articles

Detailed analysis on primary and secondary break up characteristics of liquid fuel jet in crossflow using homogeneous mixture model and large eddy simulation

The detailed breakup characteristics of the liquid jet fuel, Jet-A, in crossflow to various ranges of momentum flux ratios and Weber numbers have been investigated using a high-fidelity compressible multi-phase numerical technique. Multi-phase large eddy simulation with adaptive mesh refinement and Eulerian to Lagrangian transformation are applied to the homogeneous mixture model. The liquid surface instabilities and their frequencies inside the injector orifice are observed and analyzed. Interactions between liquid jet and crossflow result in phenomena such as horse-shoe vortex formation, boundary separation, liquid trailing, and counter-rotating vortex pair. Liquid column breakup characteristics and wave structures are analyzed from both temporal and spatial viewpoints. The Sauter mean diameter distribution and cumulative distributions of droplets resulting from secondary breakup are presented, along with the droplet size distribution derived from the momentum flux ratio and Weber number. Several engineering models, including the instability frequencies, the breakup length, and the penetration depth, are proposed in terms of the momentum flux ratio and Weber number, providing valuable insights for injector and combustor design.

Conformal Yang-Mills field in (A)dS space

Ordinary-derivative (second-derivative) Lagrangian formulation of classical conformal Yang-Mills field in the (A)dS space of six, eight, and ten dimensions is developed. For such conformal field, we develop two gauge invariant Lagrangian formulations which we refer to as generic formulation and decoupled formulation. In both formulations, the usual Yang-Mills field is accompanied by additional vector and scalar fields where the scalar fields are realized as Stueckelberg fields. In the generic formulation, the usual Yang-Mills field is realized as a primary field, while the additional vector fields are realized as auxiliary fields. In the decoupled formulation, the usual Yang-Mills field is realized as massless field, while the additional vector fields together with the Stueckelberg are realized as massive fields. Some massless/massive fields appear with the wrong sign of kinetic terms, hence demonstrating explicitly that the considered models are not unitary. The use of embedding space method allows us to treat the isometry symmetries of (A)dS space manifestly and obtain conformal transformations of fields in a relatively straightforward way. By accompanying each vector field by the respective gauge parameter, we introduce an extended gauge algebra. Levy-Maltsev decomposition of such algebra is noted. Use of the extended gauge algebra setup allows us to present concise form for the Lagrangian and gauge transformations of the conformal Yang-Mills field. Higher-derivative representation of the Lagrangian is also obtained.

Subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder

This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli’s quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. An invertible modified Lagrangian transformation is introduced to overcome this difficulty and straighten the contact discontinuity. The key elements in our analysis are to utilize the deformation-curl decomposition introduced in [S. Weng and Z. Xin, A deformation-curl decomposition for three dimensional steady Euler equations, Sci. Sin. Math. 49 (2019) 307–320 (in Chinese): doi:10.1360/N012018-00125] to effectively decouple the hyperbolic and elliptic modes in the steady axisymmetric Euler system and to use the implicit function theorem to locate the contact discontinuity.

A New Framework of RIS-Aided User-Centric Cell-Free Massive MIMO System for IoT Networks

With its merit of inter-cell interference elimination and enhanced throughput, cell-free (CF) massive multiple-input multiple-output (MIMO) system, has attracted considerable interests in evolution of 6G Internet-of-Things (IoT) networks. Meanwhile, reconfigurable intelligent surface (RIS) has shown its potential benefit in enhancing both capacity and energy efficiency (EE). In this paper, a new framework of RIS-aided CF massive MIMO system is proposed for IoT networks, in which extra active APs are deployed closely to the passive RISs, in that way, the RIS-user channels can be approximately obtained by parameters estimated at those extra APs with acceptable loss. To avoid extra power consumption, user-centric AP selection strategy in CF system is suggested, on the basis of which, an optimization problem related to power control, precoding and RIS phase shift is formulated to maximize the sum rate. To deal with this tough three-variable optimization, a Lagrangian dual transformation and fractional programming based algorithm is proposed. In particular, the proposed algorithm can extend to optimize EE and well adapt to other RIS-aided CF systems. Simulation results reveal that the proposed algorithm can achieve superior performance in terms of both sum rate (SR) and EE.

Gauged D = 4 N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supergravity

We present the full Lagrangian and supersymmetry transformation rules for the gauged D = 4, N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 (half-maximal) supergravity coupled to an arbitrary number of vector multiplets. Using the embedding tensor formulation, the final results are universal and valid in arbitrary symplectic frames. We also analyze the conditions for the critical points of the scalar potential and specify the full spectrum of the quadratic fluctuations about Minkowski vacua. This allows us also to exclude the appearance of quadratic divergences in the 1-loop corrections to the scalar potential for any Minkowski vacuum fully breaking supersymmetry. We also provide some interesting byproducts of our analysis, like the field equations and the quadratic constraints for the fermion shifts characterizing the gauging (also known as T-tensor identities).

Backscatter-enabled CR-NOMA based cooperative V2X communication with imperfect CSI

Non orthogonal multiple access (NOMA) and backscatter communications are considered to be promising technology due to their applications in large scale communications and spectral efficiency. Cognitive radio (CR) also has distinct advantage in terms of allowing secondary users to share spectrum with the primary users. Thus CR and NOMA provides a new way to enable vehicle to everything (V2X) communication through cooperation, in terms of the achievable rate (throughput), which is further enhanced with use of backscatter devices. Moreover, in V2X it is critical to inspect imperfect channel state information (CSI) because, mobility of vehicles leads to more channel estimation uncertainties. Therefore, CR based approach which is supported by NOMA with imperfect CSI, called as B-CR-NOMA is proposed in which several vehicles connect with the base stations (BSs) through different road side units (RSUs) and backscatter devices. To optimize the performance of the proposed approach, a problem of optimal (maximization) of throughput has been formulated. Moreover, the non-convexity associated with the signal to interference plus noise ratio (SINR) is removed by the use of Lagrangian dual transformation by introducing auxiliary variables, which is further solved under Karush–Kuhn–Tucker (KKT) conditions along with using sub-gradient techniques. The exceptional attainment of the throughput for the proposed approach in comparison to the contemporary algorithms in terms of: power as well as number of RSU, number of vehicle users and channel estimation error, accentuate the capability of the proposed approach. The fortitude of B-CR-NOMA is further substantiated in terms of fairness (fairness index), which is measured and compared based on the channel CSI: perfect and imperfect. Also, the effect of the auxiliary variables and iterations on the throughput for the proposed approach is also analyzed. Along with this, two key metrics: average outage probability and average fairness index, which measures the cooperative V2X communication, are also considered for in depth analysis of the proposed approach.

Sum Rate Optimization Scheme of UAV-Assisted NOMA under Hardware Impairments

In the unmanned aerial vehicle (UAV) assisted non-orthogonal multiple access (NOMA) networks, the practical hardware impairments (HIs) and resource allocation is still a challenging problem. Most existing research on resource allocation algorithms for UAV communication is considered with the ideal hardware condition. However, the impact of HIs on system performance cannot be ignored, especially in the case of high bit rates. Considering the HIs, most studies are from the perspective of performance analysis. The resource allocation of UAV relay-assisted NOMA systems is investigated in this paper with HIs. We aim to maximize the sum rate by jointly optimizing the deployment of UAV and transmit power. To address this problem, we first transformed the mixed integer programming problem (MIPP) into a standard convex optimization problem based on successive convex approximation (SCA) technology. Then, we introduced the Lagrangian dual transformation and quadratic transform methods to solve the power allocation problem. Finally, we propose an effective iterative algorithm to achieve an approximate optimal solution. Numerical results demonstrate that the proposed algorithm achieved better performance in terms of the sum rate compared with other benchmark schemes.

Global Existence to Cauchy Problem for 1D Magnetohydrodynamics Equations

Magnetohydrodynamics are widely used in medicine and biotechnology, such as drug targeting, molecular biology, cell isolation and purification. In this paper, we prove the existence of a global strong solution to the one-dimensional compressible magnetohydrodynamics system with temperature-dependent heat conductivity in unbounded domains and a large initial value by the Lagrangian symmetry transformation, when the viscosity μ is constant and the heat conductivity κ, which depends on the temperature, satisfies κ=κ¯θb(b>1).

IRS Empowered UAV Wireless Communication With Resource Allocation, Reflecting Design and Trajectory Optimization

As revolutionary technologies that can actively change the communication link signal, intelligent reflecting surface (IRS) and unmanned aerial vehicle (UAV) have emerged as reliable, economical and convenient wireless communication solutions for a variety of practical scenarios. Therefore, this paper focuses on an IRS empowered UAV downlink communication network, where the dynamic UAV establishes a cascade link via IRS to provide signal enhancement services for multiple users. Considering constraints of transmit power, flight speed and area at the UAV and the reflecting constraints at the IRS, the block coordinate descent (BCD) method based on resource allocation, reflecting design and trajectory optimization is adopted to maximize the sum-rate of all users. The proposed problem is converted by using quadratic transformation and Lagrangian dual transformation. Then applying for the approximate linear method and Iterative Rank Minimization (IRM) to optimize the transmit power of UAV and phase shift of IRS respectively. Since additional reflection propagation paths by IRS, the complexity of the channel model makes the trajectory design difficult. To tackle this problem, this paper proposes a UAV trajectory optimization method based on enhanced reinforcement learning with the fixed initial location and destination. In the end, the convergence of the proposed scheme is effectively verified by simulations. Moreover, abundant simulation comparisons between the proposed scheme and other benchmark schemes demonstrate the validity and high performance gains of the proposed algorithm.

The Lp-Lq decay estimate for the multidimensional compressible flow with free surface in the exterior domain

The aim of this paper is to develop the general Lp theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in RN (N≥3). By the spectral analysis, we obtain the classical Lp-Lq decay estimate for the linearized model problem (with variable coefficients) in view of the partial Lagrangian transformation.

Local Well-Posedness of a Two-Component Novikov System in Critical Besov Spaces

In this paper, we establish the local well-posedness for a two-component Novikov system in the sense of Hadamard in critical Besov spaces Bp,11+1p(R)×Bp,11+1p(R),1≤p<∞. We first provide a uniform bound for the approximate solutions constructed by iterative scheme, then we show the convergence and regularity; afterwards, based on the Lagrangian coordinate transformation techniques, we prove the uniqueness result; finally, we show that the the solution map is continuous.

Thermal pollution level reduction by sweeping jet-based enhanced heat dissipation: A numerical study with calibrated Generalized k-ω (GEKO) model

Thermal pollution is commonly the damage to natural ecosystems caused by industrial water discharge into the environment above ambient temperature. The highly efficiently dissipative sweeping jet actuator is thus considered as the solution to the reduced thermal pollution level. The thermal dissipation capacities of free steady, pulsed, and sweeping jets with a Reynolds number of 49,761 and a Strouhal number of 0.0152 are compared by numerical simulation. The Reynolds stress model (RSM) is used as the turbulence model, with the Generalized k-ω (GEKO) model innovatively used to calibrate jet dissipation rates. The temperature fields are first compared to confirm the overwhelming temperature dilution effect of the sweeping jet in both streamwise and normalwise directions. Then the reasons for the enhanced heat dissipation by sweeping jet are explained by numerous time-averaged and time-resolved contours of the distributions of flow-related physical quantities. The time-averaged flow fields directly show that the sweeping jet has the fastest streamwise velocity decay and the most uniform spanwise momentum distribution due to the continuous transverse oscillation, and hence has the moderate turbulence kinetic energy (TKE) over a wide spanwise range due to the destruction of the potential core commonly seen in a steady jet. The relative turbulence kinetic energy fields, excluding the contribution to turbulence generation from velocity magnitudes, show that the alternating formation and dissipation of shear layer-induced vortices most contribute to the inherent unsteady patterns. In the planes perpendicular to the streamwise direction, the spacings between adjacent velocity contour lines of the sweeping jet increase at the fastest rate with the streamwise distance, indicating the excellent entrainment capacity. The difference among the three primary Reynolds stresses of the sweeping jet is smaller than that of the other two jets, indicating that the sweeping jet converts more streamwise momentum into spanwise and normalwise velocity and turbulence and thus has higher spatial homogeneity. Pathlines of massless particles dispersed in the near-field show that the sweeping jet has the most substantial entrainment capability due to the longest motion distance of the traces originating outside the jet shear layer, although it is difficult for the ambient fluids to penetrate the central jet impact region. Finally, three mixing potential metrics, i.e., jet decay rate, jet spreading rate, and entrainment factor, are used to evaluate the three jets' more general mixing characteristics qualitatively. To extract the sweeping jet's centerline velocity and spreading width, the Lagrangian transformation, which invariably aligns the jet ejection direction with x-axis, is performed on the phase-averaged velocity flow fields. The near-field results show that compared with the steady jet, the sweeping jet boasts a 176% increase in the jet decay rate, a 241% increase in the jet spreading rate, and a 369% increase in the entrainment factor. In the far-field, the entrainment capacity of the sweeping jet is not as good as that of the pulsed jet but is still 68% higher than that of the steady jet. In all, this paper establishes a theoretical basis for the application of fluid oscillators to heat pollution level reduction and more general scenarios involving requirements for enhanced heat and mass transfer by proposing the standard mixing indicators of sweeping jets similar to those of conventional jets, and also serves as a pioneering example for the future possible extension of the GEKO model to a broader range of industrial CFD simulations.

A flamelet LES of turbulent dense spray flame using a detailed high-resolution VOF simulation of liquid fuel atomization

A numerical framework used to model dense spray flames is proposed. In this framework, the liquid fuel (acetone) atomization is solved by a detailed high-resolution VOF simulation, and the Eulerian components of liquid droplets are transformed into Lagrangian droplets, which are stored in a database at a certain downstream cross-section. Then, the combustion process is solved by a LES/FPV (flamelet progress variable) adopting the pre-stored database of Lagrangian droplets (i.e., the position, size, and velocity of each droplet) as the inlet boundary conditions. This framework is a one-way coupling between a VOF simulation and a combustion simulation. The validity of this approach is investigated by comparing the computations with the experiments of the Sydney Piloted Needle Spray Burner. The VOF simulation shows that the volume flux of the droplets at the nozzle exit fluctuates both temporally and spatially and the larger droplets tend to be located away from the center axis compared to the small droplets. The computed breakup length is in good agreement with the empirical correlation. In the database of the Lagrangian droplets for the LES/FPV of spray flames, the location of the sampling cross-section, the sampling time, and the threshold value for Eulerian–Lagrangian (E-L) transformation strongly affect the properties of the Lagrangian droplets, and are critical for the successful use of the LES/FPV. Two spray flames with different recess distances are computed using their optimal pre-stored droplets databases and both show generally good agreement with the experiments in terms of the gas temperature and droplet size distributions. The spray flame with a longer recess distance, which is more representative of a dilute spray, is considered to have a longer and wider premixed core than that with a shorter recess distance representing a dense spray. The discrepancy in the prediction of denser spray flames becomes more evident leading to over-predictions of gas temperature further downstream. Reasons for this behavior are discussed in the text.

Nakanishi–Kugo–Ojima quantization of general relativity in Heisenberg picture

The Chern–Weil topological theory is applied to a classical formulation of general relativity in four-dimensional spacetime. Einstein–Hilbert gravitational action is shown to be invariant with respect to a novel translation (co-translation) operator up to the total derivative; thus, a topological invariant of a second Chern class exists owing to Chern–Weil theory. Using topological insight, fundamental forms can be introduced as a principal bundle of the spacetime manifold. Canonical quantization of general relativity is performed in a Heisenberg picture using the Nakanishi–Kugo–Ojima formalism in which a complete set of quantum Lagrangian and BRST transformations including auxiliary and ghost fields is provided in a self-consistent manner. An appropriate Hilbert space and physical states are introduced into the theory, and the positivity of these physical states and the unitarity of the transition matrix are ensured according to the Kugo–Ojima theorem. The nonrenormalizability of quantum gravity is reconsidered under the formulation proposed herein.

On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform Wq2−1/q domain in RN (N≥2). We prove the local in the time unique existence theorem for our problem in the Lp in time and Lq in space framework with 2<p<∞ and N<q<∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal Lp-Lq regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal Lp-Lq regularity theorem.

Age of Processing: Age-Driven Status Sampling and Processing Offloading for Edge-Computing-Enabled Real-Time IoT Applications

The freshness of status information is of great importance for time-critical Internet-of-Things (IoT) applications. A metric measuring status freshness is the Age of Information (AoI), which captures the time elapsed from the status being generated at the source node (e.g., a sensor) to the latest status update. However, in intelligent IoT applications such as video surveillance, the status information is revealed after some computation-intensive and time-consuming data processing operations, which would affect the status freshness. In this article, we propose a novel metric, Age of Processing (AoP), to quantify such status freshness, which captures the time elapsed of the newest received processed status data since it is generated. Compared with AoI, AoP further takes the data processing time into account. Since an IoT device has limited computation and energy resources, the IoT device can choose to offload the data processing to the nearby edge server under constrained status sampling frequency. We aim to minimize the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">average</i> AoP in a long-term process by jointly optimizing the status sampling frequency and processing offloading policy. We first formulate this online problem as an infinite-horizon constrained Markov decision process (CMDP) with an average reward criterion. We then transform the CMDP problem into an unconstrained Markov decision process (MDP) by leveraging a Lagrangian method, and accordingly propose a Lagrangian transformation framework for the original CMDP problem. Furthermore, we integrate the framework with a perturbation-based refinement mechanism for achieving the optimal policy of the CMDP problem. Our investigation shows that to minimize the average AoP: 1) for processing offloading: the policy exploits good channel state to offload processing to the edge server and 2) for status sampling: the waiting time presents a threshold structure. Extensive numerical evaluations show that the proposed algorithm outperforms the benchmarks, with an average AoP reduction up to 30%.

Structural Stability of the Transonic Shock Problem in a Divergent Three-Dimensional Axisymmetric Perturbed Nozzle

In this paper, we prove the structural stability of the transonic shocks for three-dimensional axisymmetric Euler system with swirl velocity under the perturbations for the incoming supersonic flow, the nozzle boundary, and the exit pressure. Compared with the known results on the stability of transonic shocks, one of the major difficulties for the axisymmetric flows with swirls is that corner singularities near the intersection point of the shock surface and nozzle boundary and the artificial singularity near the axis appear simultaneously. One of the key points in the analysis for this paper is the introduction of an invertible modified Lagrangian transformation which can straighten the streamlines in the whole nozzle and help to represent the solutions of transport equations explicitly. Furthermore, the simple but useful modified Lagrangian transformation makes the treatment for the singularities near the axis easy and clean. Such a technique may be helpful in studies of other problems for axisymmetric flows.

Dynamics of a diffusive competitive model on a periodically evolving domain

This article concerns a two-species competitive model with diffusive terms in a periodically evolving domain and study the impact of the spatial periodic evolution on the dynamics of the model. The Lagrangian transformation approach is adopted to convert the model from a changing domain to a fixed domain with the assumption that the evolution of habitat is uniform and isotropic. The ecological reproduction indexes of the linearized model are given as thresholds to reveal the dynamic behavior of the competitive model. Our theoretical results show that a lager evolving rate benefits the persistence of competitive populations for both sides in the long run. Numerical experiments illustrate that two competitive species, one of which survive and the other vanish in a fixed domain, both survive in a domain with a large evolving rate, and both vanish in a domain with a small evolving rate. For more information see https://ejde.math.txstate.edu/Volumes/2020/86/abstr.html

Perfectly matched layer for computing the dynamics of nonlinear Schrödinger equations by pseudospectral methods. Application to rotating Bose-Einstein condensates

In this paper, we first propose a general strategy to implement the Perfectly Matched Layer (PML) approach in the most standard numerical schemes used for simulating the dynamics of nonlinear Schrödinger equations. The methods are based on the time-splitting Bao et al. (2003)[15] or relaxation Besse (2004)[24] schemes in time, and FFT-based pseudospectral discretization method in space. A thorough numerical study is developed for linear and nonlinear problems to understand how the PML approach behaves (absorbing function and tuning parameters) for a given scheme. The extension to the rotating Gross-Pitaevskii equation is then proposed by using the rotating Lagrangian coordinates transformation method Antonelli et al. (2012), Bao et al. (2013), García-Ripoll et al. (2001)[13,16,38], some numerical simulations illustrating the strength of the proposed approach.

Theoretical and experimental analysis of an unbalanced and cracked cardan shaft in the vicinity of the critical speed

This paper presents a theoretical and experimental analysis of a coupled lateral and torsional vibrations of two identical rotors interconnected by a flexible Hooke’s joint and modelled as a multibody system with a small misalignment angle. Using energy principle and a Lagrangian transformation, the governing equation of the propeller shaft system is established by considering a nonlinear elastic shaft time-dependent perturbation. To study the sensitivity of the crack for a rotating shaft, the model is enriched by considering the periodical feature of the time-varying stiffness deriving from the crack breathing model. The nonstationary response of a cracked rotor system in the presence of unbalance has been evaluated using orbit patterns and Fast Fourier Transform. The highly oscillated feature of the rotors system is theoretically obtained and experimentally analysed. The analysis demonstrated that the crack parameters in the input shaft tend to inhibit the occurrence of unstable oscillations in lateral deflection, orbit and frequency spectrum of the secondary response. It is also found that the passage of the cracked primary shaft near to an integral multiple of the critical speed leads to the phenomenon of sup-harmonic resonance. Subsequently, the experimental analysis conducted equally indicated that the quantitative relation between the faults and the performance of the transmission is impacted by the time-varying stiffness and is the main cause of the frequency-modulated feature in the Cardan shaft system. Finally, the experimental results were informative for the transient response exploration and comparable to the theoretical findings for validating the proposed twin-rotor model.