On energy flux and group velocity

Inhomogeneous plane wave solutions to the wave equations for a linear isotropic elastic solid and a linear isotropic dielectric are shown to possess energy flux velocity vectors which are non-coincident with corresponding group velocity vectors.In contrast to free surface waves, these examples imply a driving constraint and have an associated non-zero Lagrangian energy density.     

3D two phase flows numerical simulations by SL‐VOF method

This paper describes the development of a semi‐Lagrangian computational method for simulating complex 3D two phase flows. The Navier–Stokes equations are solved separately in both fluids using a robust pseudo‐compressibility method able to deal with high density ratio. The interface tracking is achieved by the segment Lagrangian volume of fluid (SL‐VOF) method. The 2D SL‐VOF method using the concepts of VOF, piecewise linear interface calculation (PLIC) and Lagrangian advection of the interface is herein extended to 3D flows. Three different test cases of SL‐VOF 3D are presented for validation and comparison either with 2D flows or with other numerical methods. A good agreement is observed in each case. Copyright © 2004 John Wiley & Sons, Ltd.     

Rate theory of nonlocal gradient damage-gradient viscoinelasticity

A general concept for the analysis of damage evolution in heterogeneous media is proposed. Since macroscopic failure is governed by physical mechanisms on two different length-scale levels, the macro- and mesolevel, we introduce a 6-dimensional kinematical model with manifold structure accounting for discontinuous fields of microcracks, microvoids and microshear bands. As point of departure a variational functional is introduced with a Lagrangian density depending on macro- and microdeformation gradients and of a damage variable representing scalar-, vector- and/or tensor-type quantities. To derive the equations of motion for viscoinelastic damage evolution on macro- and mesolevel, we introduce into the Lagrangian the macro- and microdeformation gradients, damage variable and also their gradients and time rates. The equations of motion on macro- and mesolevel are derived for non-equilibrium states. We assume that the Lagrangian can be split into two contributions, a time-independent and a time-dependent one which can be identified with the Helmholtz free energy and a dissipation potential. This split of the Lagrangian can be used to decompose the stresses and forces into reversible and irreversible ones. The latter can be considered as dissipative driving stresses and driving forces, respectively, on defects. The model presented in this paper can be considered as a framework, which enables to derive various nonlocal and gradient, respectively, damage theories by introducing simplifying assumptions. As special cases a scalar damage and a solid-void model are considered.     

Conservation of charge and second-order gauge-tensor field theories

In a space of four-dimensions I determine all possible second-order gauge-tensor field equations which are derivable from a variational principle, compatible with the notion of conservation of charge, and in agreement with the Yang-Mills field equations in flat space. The Lagrangian which yields these fields equations differs from the Lagrangian of the Einstein-Yang-Mills field theory (with cosmological term included) by one term.     

Complexification-averaging method via the averaged Lagrangian

In this paper, the complexification-averaging (CX-A) method for multi-DOF nonlinear vibratory systems is rederived in a new way based upon the averaged Lagrangian. The complex variables are introduced to represent the original displacements and velocities, and then the fast–slow decomposition of the complex variables is made. The time averaging of the Lagrangian over the fast variables is performed. Two different expressions for the kinetic energy are presented, and this results in two schemes for deriving the governing equations of the slow variables. For the scheme I, through the order analysis of the derivatives of the slow variables, it is shown that the second-order terms appeared in the averaged Lagrangian can be omitted, and thus a reduced averaged Lagrangian is obtained. Via the reduced averaged Lagrangian, the corresponding Lagrangian equations are derived. For the scheme II, through time averaging, the averaged Lagrangian is obtained, and then the corresponding equations for the slow variables can be obtained. Finally, two nonlinear vibratory systems with two-DOF and four-DOF, respectively, are given as examples to illustrate the new procedure for the CX-A method. The loci of nonlinear normal modes on the potential surface are studied in the first example, and the frequency-energy plot is investigated in the second example.     

A hybrid Lagrangian–Eulerian particle‐level set method for numerical simulations of two‐fluid turbulent flows

A coupled Lagrangian interface‐tracking and Eulerian level set (LS) method is developed and implemented for numerical simulations of two‐fluid flows. In this method, the interface is identified based on the locations of notional particles and the geometrical information concerning the interface and fluid properties, such as density and viscosity, are obtained from the LS function. The LS function maintains a signed distance function without an auxiliary equation via the particle‐based Lagrangian re‐initialization technique. To assess the new hybrid method, numerical simulations of several ‘standard interface‐moving’ problems and two‐fluid laminar and turbulent flows are conducted. The numerical results are evaluated by monitoring the mass conservation, the turbulence energy spectral density function and the consistency between Eulerian and Lagrangian components. The results of our analysis indicate that the hybrid particle‐level set method can handle interfaces with complex shape change, and can accurately predict the interface values without any significant (unphysical) mass loss or gain, even in a turbulent flow. The results obtained for isotropic turbulence by the new particle‐level set method are validated by comparison with those obtained by the ‘zero Mach number’, variable‐density method. For the cases with small thermal/mass diffusivity, both methods are found to generate similar results. Analysis of the vorticity and energy equations indicates that the destabilization effect of turbulence and the stability effect of surface tension on the interface motion are strongly dependent on the density and viscosity ratios of the fluids. Copyright © 2007 John Wiley & Sons, Ltd.     

On the Passage of a Shock Wave through a Layer of Charged Gas

A class of exact solutions of the ideal electrohydrodynamics equations is presented. These solutions describe the propagation of a plane shock wave along a static background with decreasing density in the presence of gravity and longitudinal electric fields. This class of solutions contains an arbitrary function of the Lagrangian variable which makes it possible to consider many physically different cases.     

Analysis of multifluid flows with large time steps using the particle finite element method

Multifluids are those fluids in which their physical properties (viscosity or density) vary internally and abruptly forming internal interfaces that introduce a large nonlinearity in the Navier–Stokes equations. For this reason, standard numerical methods require very small time steps in order to solve accurately the internal interface position. In a previous paper, the authors developed a particle‐based method (named particle finite element method (PFEM)) based on a Lagrangian formulation and FEM for solving the fluid mechanics equations for multifluids. PFEM was capable of achieving accurate results, but the limitation of small time steps was still present. In this work, a new strategy concerning the time integration for the analysis of multifluids is developed allowing time steps one order of magnitude larger than the previous method. The advantage of using a Lagrangian solution with PFEM is shown in several examples. All kind of heterogeneous fluids (with different densities or viscosities), multiphase flows with internal interfaces, breaking waves, and fluid separation may be easily solved with this methodology without the need of small time steps. Copyright © 2014 John Wiley & Sons, Ltd.     

Sediment flow in inclined vessels calculated using a multiphase particle-in-cell model for dense particle flows

Sedimentation of particles in an inclined vessel is predicted using a two-dimensional, incompressible, multiphase particle-in-cell (MP-PIC) method. The numerical technique solves the governing equations of the fluid phase using a continuum model and those of the particle phase using a Lagrangian model. Mapping particle properties to an Eulerian grid and then mapping back computed stress tensors to particle positions allows a complete solution of sedimentation from a dilute mixture to close-pack. The solution scheme allows for distributions of types, sizes and density of particles, with no numerical diffusion from the Lagrangian particle calculations. The MP-PIC solution method captures the physics of inclined sedimentation which includes the clarified fluid layer under the upper wall, a dense mixture layer above the bottom wall, and instabilities which produce waves at the clarified fluid and suspension interface. Measured and calculated sedimentation rates are in good agreement.     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems

The automatic derivation of motion equations is an important problem of multibody system dynamics. Firstly, an overview of the matrix calculus related to Kronecker product of two matrices is presented. A new matrix form of Lagrangian equations with multipliers for constrained multibody systems is then developed to demonstrate the usefulness of Kronecker product of two matrices in the study of dynamics of multibody systems. Finally, the equations of motion of mechanisms are derived using the proposed matrix form of Lagrangian equations as application examples.     

Sediment flow in inclined vessels calculated using a multiphase particle-in-cell model for dense particle flows

Sedimentation of particles in an inclined vessel is predicted using a two-dimensional, incompressible, multiphase particle-in-cell (MP-PIC) method. The numerical technique solves the governing equations of the fluid phase using a continuum model and those of the particle phase using a Lagrangian model. Mapping particle properties to an Eulerian grid and then mapping back computed stress tensors to particle positions allows a complete solution of sedimentation from a dilute mixture to close-pack. The solution scheme allows for distributions of types, sizes and density of particles, with no numerical diffusion from the Lagrangian particle calculations. The MP-PIC solution method captures the physics of inclined sedimentation which includes the clarified fluid layer under the upper wall, a dense mixture layer above the bottom wall, and instabilities which produce waves at the clarified fluid and suspension interface. Measured and calculated sedimentation rates are in good agreement.     

Conditional Fluid-Particle Accelerations in Turbulence

Simple closures for average fluid-particle accelerations, conditional on fixed local fluid velocity, are considered in isotropic, homogeneous and stationary turbulence using exact probability density transport equations and are compared with direct numerical simulations (DNS). Such accelerations are common ingredients in Lagrangian stochastic models for fluid-particle trajectories in turbulence. One-particle accelerations are essentially trivial, so the focus is on two-particle relative accelerations, which are important in the relative dispersion process. The closure is simply a quadratic form in the velocity variable and this special form also defines the Eulerian velocity probability density function (pdf), and comparisons with DNS (for grids up to 512³) of both the acceleration closure and velocity pdf's are encouraging. Received 2 June 1997 and accepted 29 December 1997     

On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative

Fractional mechanics describe both conservative and nonconservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics, the equivalent Lagrangians play an important role because they admit the same Euler–Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant, the Lagrangian we obtain are the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann–Liouville fractional derivatives. As a consequence of applying this procedure, the classical results are reobtained as a special case. The fractional generalization of Faà di Bruno formula is used in order to obtain the concrete expression of the fractional Lagrangians which differs from a given fractional Lagrangian by adding a fractional derivative. The fractional Euler–Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated, and two examples are analyzed in detail.     

Equations of Motion of an Asymmetric Timoshenko Shaft

The equations of motion of an asymmetric Timoshenko shaft, that is having unequal principal moments of inertia, are derived within the framework of the Lagrangian formulation for continuous systems and fields. The Lagrangian density of the system is calculated in a moving frame, that is a rotating frame attached to the deformed shaft, and proves to depend on the four Lagrangian variables (fields) of the system and their first derivatives w.r.t. space and time.On account of general results of the theory of continuous systems and fields, the four Lagrange's equations of motion are derived from the Lagrangian density and are successively reduced to the two usual equations in the displacements.The procedure described in this work is compared with both a different Lagrangian formulation, based on the use of a floating frame, that is a rotating frame attached to the undeformed shaft, and the well-known Newtonian approach adopted by Dimentberg. Sommario. Si applica la formulazione lagrangiana per i sistemi continui e i campi per ricavare le equazioni del moto di un albero di Timoshenko asimmetrico, la cui sezione presenta cioé momenti principali dinerzia diversi. La densità di lagrangiana è calcolata in un sistema di riferimento rotante solidale allalbero deformato e risulta essere funzione delle quattro variabili lagrangiane (campi) del sistema e delle loro derivate prime rispetto allo spazio e al tempo.In accordo con i risultati generali della teoria dei sistemi continui e dei campi, si ricavano, a partire dalla densità di lagrangiana, le quattro equazioni di Lagrange successivamente ridotte alle due classiche equazioni rispetto ai soli spostamenti.Il procedimento proposto viene messo a confronto con una diversa formulazione lagrangiana, basata sulluso di un sistema di riferimento rotante solidale allalbero indeformato, e con la ben nota formulazione newtoniana adottata da Dimentberg.     

Incompressible Flows with Piecewise Constant Density

We investigate the incompressible Navier–Stokes equations with variable density. The aim is to prove existence and uniqueness results in the case of discontinuous initial density. In dimension n = 2,3, assuming only that the initial density is bounded and bounded away from zero, and that the initial velocity is smooth enough, we get the local-in-time existence of unique solutions. Uniqueness holds in any dimension and for a wider class of velocity fields. In particular, all those results are true for piecewise constant densities with arbitrarily large jumps. Global results are established in dimension two if the density is close enough to a positive constant, and in n dimensions if, in addition, the initial velocity is small. The Lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper.     

Noether-Type Symmetries and Conservation Laws Via Partial Lagrangians

We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived. Examples are given including those that admit a standard Lagrangian such as the Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations in more than two independent variables.