Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.

This paper introduces a new method for analyzing solution-gas production to determine the ultimate recovery of a well or a field. The procedure developed and outlined in this paper requires very little input data and is easily implemented. By modifiying the equations for dimensionless rate and dimensionless cumulative production derived for the single phase model, a new set of equations is developed to approximate the ultimate recovery of a solution-gas well. Using the approximate material balance equation based on numerical results by Vogel1 and the equation for the production rate derived by Fetkovich et al.2,3,4, this new set of dimensionless rate and dimensionless cumulative production equations are derived. Using the relationship between these equations and an iterative calculation procedure, the ultimate recovery for the solution-gas well can be easily determined. All that is needed as input is the producing bottom-hole pressure, the initial pressure, and the oil production data. The method has been validated with twelve simulator cases, six under constant bottom hole pressure production constraint and six under variable bottom hole pressure production constraint. Furthermore, several field cases have been analyzed. The synthetic and field cases validate the procedure. Using the early pseudo-steady state production data in the analysis the results generated by the method are consistent with the actual ultimate recoveries.

We present a new complete set of Lagrangian relativistic hydrodynamical equations describing the transfer of energy and momentum between a standard fluid and a radiation fluid in a general non-stationary spherical flow. The new set of equations has been derived for a particular application in the study of the cosmological quark--hadron transition, but can also be used in other contexts.

The article is devoted to the research of the effects of wave processes on above-water constructions using a mathematical model of wave processes based on the system of Navier–Stokes equations, which includes three equations of motion in areas with dynamically changing geometry of the computational domain. The pressure correction method is used for the approximation of a hydrodynamic model. The difference schemes describing the mathematical model of the wave exit to the shore are constructed on the basis of the integro-interpolation method using the scheme with weights. An adaptive alternating-triangular iterative method is used to solve the system of grid equations. The practical significance of numerical algorithms and the complex of programs implementing them consists in the possibility of their application in the study of hydrophysical processes in coastal water systems, as well as for the construction of the velocity field and pressure of the aquatic environment, as well as the assessment of the hydrodynamic effect on coastal protection structures and riparian constructions in the presence of surface waves.

The basic problem of order reduction of linear and nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via Time Periodic Center Manifold Theory. A ‘reducibility condition’ is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show applications to real problems. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘combination’ resonances are discussed.

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler-Poincare (EP) variational framework of fluid dynamics, for an averaged Lagrangian. This is the Lagrangian averaged Euler-Poincare (LAEP) theorem. Next, we derive a set of approximate small amplitude GLM equations (glm equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the glm equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The glm derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction. Next, the new glm EP motion equations for incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or glm) fluid theory with a Taylor hypothesis closure. Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha equations. Thus, by using the LAEP theorem, we bridge between the GLM equations and the Euler-alpha closure equations, through the small-amplitude glm approximation in the EP variational framework. We conclude by highlighting a new application of the GLM, glm, and alpha-model results for Lagrangian averaged ideal magnetohydrodynamics. (c) 2002 American Institute of Physics.

The physics involving the emission of coherent undulator radiation from a bunch of correlated electrons wiggling inside an undulator has been very well established in the frequency domain. With the aim of describing the process of emission of coherent undulator radiation in time domain while incorporating the effect of hyperbolic field profile of the undulator and various electron beam parameters, we present a time-domain analysis which can serve as a complimentary formulation to the frequency-domain analysis of the coherent undulator radiation. The formulation requires revisiting the equation of motion of an electron moving inside a planar undulator. An electron beam with substantial spread along the direction of the magnetic field will undergo betatron oscillations due to the hyperbolic profile of the undulator fields. We show that the betatron motion can be conveniently expressed in terms of Mathieu functions. The new set of equations of motion have been used to study the effect of betatron motion on the emitted undulator radiation. The single-particle time-domain analysis is then easily extended to multiparticle systems by correctly accounting for the phase-difference between the individual electromagnetic waves emitted by different electrons occupying different phase-space in the bunch.

A new dynamic theory of die swell for entangled polymeric liquids in a steady simple shear flow is proposed which can be used to predict the correlation of the time-dependent and time-independent extrudate swelling behaviors to the molecular parameters of polymers and the operational variables. The theory is based on the O-W-F constitutive equation and the free recovery from Poioeuille flow with different ratios. Experiments show that the magnitudes of the simple shear in the steady simple shear flow may be resolved into the free recoil resulting from the recoverable elastic strains and the viscous heating resulting from the unrecoverable viscoelastic strains. For distinguishing the recoil from the viscous heating, a partition function and two exponential fractions of conformation for the recoil and the viscous heating were defined. Thus the instantaneous, delayed and ultimate recoverable strain, and recoil in the free recovery were correlated to the partition function, the fraction of recoverable conformation, the molecular parameters, and the operational variables. Also the dynamics of the growth equation on the delayed viscoelastic strain and the delayed recoil in free state were deduced. After introducing the condition of uniform two dimensional extensions, the definition of swell ratio and the operational variables into the above correlation expressions and growth equations, then the correlations of the delayed extrudate swelling effect and the ultimate extrudate swelling effects to the molecular parameters and the operational variables were derived. Finally, two new sets of equations on the growth variables and ultimate extrudate swelling ratios under the dynamic and equilibrium states were also deduced from this dynamic theory of die swell. The first set of equations on the ultimate extrudate swelling ratio under the free and equilibrium states was verified by HDPE experimental data at two temperatures and different operational variables. The second set of equations on the growth extrudate swelling ratios under free and dynamic states was verified by PBD experimental data with different molecular weights and different operational variables. An excellent agreement is obtained, which shows that the two sets of equations for the growth and ultimate extrudate swelling ratios can be used directly to predict the correlation of extrudate swelling ratios to the molecular parameters and the operational variables.

The motion of two planets around a Sun-like star under the combined effects of mutual interaction and tidal dissipation is investigated. The secular behaviour of the system is analyzed using two different approaches. First, we solve the exact equations of motion through the numerical simulation of the system evolution. In addition to the orbital decay and circularization, we show that the final configuration of the system is affected by the shrink of the inner orbit. Our second approach consist in the analysis of the stationary solutions of mean equations of motion based on a Hamiltonian formalism. We consider the case of a hot super-Earth planet with a more massive outer companion. As a real example, the CoRoT-7 system is analyzed solving the exact and mean equations of motion. The star-planet tidal interaction produces orbital decay and circularization of the orbit of CoRoT-7b. In addition, the long-term tidal evolution is such that the eccentricity of CoRoT-7c is also circularized and a pair of final circular orbits is obtained. A curve in the space of eccentricities can be constructed through the computation of stationary solutions of mean equations including dissipation. The application to CoRoT-7 system shows that the stationary curve agrees with the result of numerical simulations of exact equations. A similar investigation performed in a super-Earth-Jupiter two-planet system shows that the doubly circular state is accelerated when there is a significant orbital migration of the inner planet, in comparison with previous results were migration is neglected.

Edge noise is generated if turbulence interacts with solid edges. Reduction of trailing edge noise of airfoils can be achieved by replacing the solid material at the trailing edge by inlays of porous permeable material. The acoustic benefit of approximately 6 dB of such treatment is known from experiments. Enroute to numerically optimized porous properties, this paper presents a first principle based Computational Aeroacoustics (CAA) method for predicting the acoustic effect of a porous NACA0012 trailing edge. In a hybrid two-step CFD/CAA procedure the turbulence statistics from a solution of the Volume Averaged Navier-Stokes (VANS) equations is used as a basis for the prediction of turbulent-boundary-layer trailing-edge noise (TBL-TEN). For the acoustic part of the calculation, the Acoustic Perturbation Equations (APE) are solved in the flow field. Inside the porous regions, a different set of governing equations, referred to as Linear Perturbation Equations (LPE) will be solved. The LPE represent a modified form of the Linearized Euler Equations (LEE) with the APE vorticity source term shifted to the right-hand side. The new set of equations is derived by volume averaging the Navier-Stokes equations and decomposing the flow variables into a time-averaged mean part and a fluctuating part and isolating the vorticity source term to the right-hand side of the momentum equation. The LPE are verified by an analytical solution. The simulation results of a NACA0012 airfoil geometry with and without porous trailing edge treatment are compared to wind tunnel measurements. The noise reduction effect of such a trailing edge treatment is successfully demonstrated.

A general tool for solving MHD and hydrodynamical problems typical of astrophysical applications is designed and implemented. The code allows the user to solve a hyperbolic system of partial differential equations with a variety of modern high resolution numerical schemes on 1, 2 or 3D non-uniform Cartesian grids with slab or axial symmetry. The equations may contain fluxes and source terms. The boundary conditions can be chosen from a number of implemented types, — fixed, periodic, reflective or ‘free’ — but the code is structured in a manner that allows the user to write his/her own subroutines for a new type of boundary condition, or even to implement a new set of equations. The choice of algorithms allows the user to find the optimal one for the application at hand and/or to compare results of different schemes thus gaining insight into the numerical effects. An emphasis is put on independence of computer platforms. Efficiency of the code is also a high priority, the options available for the user should not compromise the performance.

On the numerical modeling of an oldroyd-type constitutive equation

The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.

Global and localised temporal structures in driven ring quantum cascade lasers

Equations of motion for optimal maneuvering with global aerodynamic model