Fuel injection model for Euler–Euler and Euler–Lagrange large-eddy simulations of an evaporating spray inside an aeronautical combustor

Large-Eddy Simulations (LES) of an evaporating two-phase flow in an experimental burner are investigated. Two different numerical approaches for the simulation of the dispersed phase are coupled to the same gaseous solver: a mesoscopic Eulerian method and a Lagrangian particle tracking technique. The spray is represented by a single droplet size owing to the locally monodisperse formulation of the employed mesoscopic Eulerian approach. Both approaches use the same drag and evaporation models. They do not take into account the atomization process and a simplified injection model is applied instead. The presented methodology, referred as FIM-UR (Fuel Injection Method by Upstream Reconstruction) defines injection profiles for the monodisperse spray produced by a pressure-swirl atomizer. It is designed so as to ensure similar spray characteristics for both approaches and allows for a direct comparison between them. After a validation of the purely gaseous flow in the burner, liquid-phase dynamics and droplet dispersion are qualitatively and quantitatively evaluated for the Eulerian and Lagrangian simulations. Results obtained for both approaches are in very good agreement and compare reasonably with experiments, indicating that simplified injection methods are appropriate for the simulation of realistic combustor geometries.     

Euler–Lagrange simulation of high pressure shock waves

This paper presents the numerical simulation of overdriven detonation (or O.D.D.) that occurs when a high velocity object impacts an explosive. The pressure and the velocity at this state are higher than those of the Chapman–Jouguet (C–J) state. First, before the simulation of this event, a study of PBX air blast by using multi-material Eulerian method is presented. Pressure peaks are computed for several distances from the explosive. Second, the O.D.D. phenomenon is modeled by the Euler–Lagrange penalty coupling, which permits to couple a Lagrangian mesh of the flyer plate to multi-material Eulerian mesh of explosives and air. This coupling gives us the high detonation velocities in the acceptor explosive and demonstrates that it is able to handle shock–structure interaction problems.     

Fractional Euler–Lagrange equations revisited

This paper presents the necessary and sufficient optimality conditions for the Euler?CLagrange fractional equations of fractional variational problems with determining in which spaces the functional must exist where the functional contains right and left fractional derivatives in the Riemann?CLiouville sense and the upper bound of integration less than the upper bound of the interval of the fractional derivative. In order to illustrate our results, one example is presented.     

Artificial boundary conditions for Euler–Bernoulli beam equation

In a semi-discretized Euler-Bernoulli beam equa- tion, the non-nearest neighboring interaction and large span of temporal scales for wave propagations pose challenges to the effectiveness and stability for artificial boundary treat- ments. With the discrete equation regarded as an atomic lattice with a three-atom potential, two accurate artificial boundary conditions are first derived here. Reflection co- efficient and numerical tests illustrate the capability of the proposed methods. In particular, the time history treatment gives an exact boundary condition, yet with sensitivity to nu- merical implementations. The ALEX （almost EXact） bound- ary condition is numerically more effective.     

Formation–containment control for networked Euler–Lagrange systems with input saturation

Nonlinear Dynamics - This paper studies the formation–containment control problem for networked nonlinear Euler–Lagrange systems with input saturation. To realize the concurrency of...     

On exact solutions of a class of fractional Euler–Lagrange equations

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where _a ^c D _t ^α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange

Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations

This paper presents the fractional order Euler–Lagrange equations and the transversality conditions for fractional variational problems with fractional integral and fractional derivatives defined in the sense of Caputo and Riemann–Liouville. A fractional Hamiltonian formulation was developed and some illustrative examples were treated in detail.

     

Distributed finite-time coordinated tracking control for multiple Euler–Lagrange systems with input nonlinearity

Nonlinear Dynamics - In this paper, distributed finite-time coordinated tracking control for multiple Euler–Lagrange systems with input nonlinearity is investigated by using backstepping...     

Using the Euler–Lagrange variational principle to obtain flow relations for generalized Newtonian fluids

The Euler–Lagrange variational principle is used to obtain analytical and numerical flow relations in cylindrical tubes. The method is based on minimizing the total stress in theflow duct using the fluid constitutive relation between stress and rate of strain. Newtonian and non-Newtonian fluid models, which include power law, Bingham, Herschel–Bulkley, Carreau, and Cross, are used for demonstration.     

Triangular form for Euler–Lagrange systems with application to the global output tracking control

This paper presents a practical method to solve the problem of global output feedback tracking trajectories for a class of Euler–Lagrange systems when the variables of velocity are the unmeasured part of the state. We exhibited a new output feedback control scheme, which globally exponentially stabilize trajectories. It relies on the determination of a change of coordinates which gives to the systems a triangular form. Results are illustrated on the academic example of the two-link direct drive robot manipulator.     

Task-space time-varying formation tracking for networked heterogeneous Euler–Lagrange systems via hierarchical predefined-time control approach

This paper studies the task-space predefined-time time-varying formation tracking (TVFT) problem in networked heterogeneous Euler–Lagrange systems (NHELSs) subject to parametric uncertainties and external disturbances. A novel hierarchical predefined-time control algorithm based on the use of non-singular terminal sliding mode control method is proposed, which can achieve the TVFT with digraphs in a predefined time. In this way, the settling time can be predetermined arbitrarily in advance as a control parameter. By virtue of the Lyapunov stability and the nearest neighbor-interactions rules, some sufficient criteria for realizing the task-space predefined-time TVFT of the NHELSs are obtained. Eventually, numerical simulation results are provided to demonstrate the effectiveness of the provided results.

     

Stability of Transonic Shock Solutions for One-Dimensional Euler–Poisson Equations

In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler–Poisson systems are investigated. First, a steady transonic shock solution with a supersonic background charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with a supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbations of the initial data, provided the electric field is not too negative at the shock location. The proof of the first stability result relies on a monotonicity argument for the shock position and the downstream density, and on a stability analysis for subsonic and supersonic solutions. The dynamical stability of the steady transonic shock for the Euler–Poisson equations can be transformed to the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. The analysis for the associated linearized problem plays an essential role.     

Euler–Euler large eddy simulations for dispersed turbulent bubbly flows

In this paper we present detailed Euler–Euler Large Eddy Simulations (LES) of dispersed bubbly flow in a rectangular bubble column. The motivation of this study is to investigate the potential of this approach for the prediction of bubbly flows, in terms of mean quantities. The physical models describing the momentum exchange between the phases including drag, lift and wall force were chosen according to previous experiences of the authors. Experimental data, Euler–Lagrange LES and unsteady Euler–Euler Reynolds-Averaged Navier–Stokes results are used for comparison. It is found that the present model combination provides good agreement with experimental data for the mean flow and liquid velocity fluctuations. The energy spectrum obtained from the resolved velocity of the Euler–Euler LES is presented as well.     

Bounded states for breathers–soliton and breathers of sine–Gordon equation

Nonlinear Dynamics - The Wronskian solutions to the sine–Gordon (sG) equation that can provide interaction of different kinds of solutions are revisited. And a novel expression of N-soliton...