Lagrangian Approximation Research Articles

Closure Approximations for Passive Scalar Turbulence: A Comparative Study on an Exactly Solvable Model with Complex Features

Some standard closure approximations used in turbulence theory are analyzed by examining systematically the predictions these approximations produce for a passive scalar advection model consisting of a shear flow with a fluctuating cross sweep. This model has a general geometric structure of a jet flow with transverse disturbances, which occur in a number of contexts, and it encompasses a wide variety of possible spatio-temporal statistical structures for the velocity field, including strong long-range correlations. Even though the Eulerian and Lagrangian velocity statistics are not equal and the passive scalar statistics exhibit broader-than-Gaussian intermittency, this model is nevertheless simple enough so that many passive scalar statistics can be computed exactly and compared systematically with the predictions of the closure approximations. Our comparative study illustrates the strength and weaknesses of the closure approximations and points out the physical phenomena that these approximations are able or not able to describe properly. In particular it is shown that the direct interaction approximation (DIA), one of the most sophisticated closure approximations available, fails to reproduce adequately the statistical features of the scalar and may even lead to absdurd predictions, even though the equations it produces are rather complicated and difficult to analyze. Two alternative closure approximations, the Modified DIA (MDIA) and the Renormalized Lagrangian Approximation (RLA), with different levels of sophistication, both are simpler to use than the DIA and perform better. In particular, it is shown that both closure approximations always reproduce exactly the second order statistics for the scalar and that the MDIA is even able to capture intermittency effects.

Relative diffusion of a pair of fluid particles in the inertial subrange of turbulence

Turbulent diffusion of a pair of fluid particles in 3-dimensional homogeneous and isotropic turbulence was studied using a high-resolution direct numerical simulation (DNS) with 10243 grid points. The DNS showed that the mean square of the distance δx between the two fluid particles grows with time t as 〈|δx|2〉∼Cεt3 in the inertial subrange, which is in agreement with Richardson (1926) and Obukhov (1941), where C≈0.7 and ε is the mean dissipation rate per unit mass. A simple Lagrangian closure approximation for 〈|δx|2〉 is shown to be in good agreement with the DNS.

Turbulent thermal diffusion and barodiffusion of passive scalar and dispersed phase of particles in turbulent flows.

It is shown that direct interaction approximation closure solutions for passive scalar and dispersed inertial particles in compressible turbulent flows result in the phenomena, proposed by Elperin et al. [Phys. Rev. E 58, 3113 (1998)], of turbulent thermal diffusion and turbulent barodiffusion. A more rigorous analysis of Lagrangian history direct interaction approximation for the dispersed phase in the kinetic approach framework is used to accurately quantify the phenomena.

Extending Lagrangian perturbation theory to a fluid with velocity dispersion

We formulate a perturbative approximation to gravitational instability, based on Lagrangian hydrodynamics in Newtonian cosmology. We take account of `pressure' effect of fluid, which is kinematically caused by velocity dispersion, to aim hydrodynamical description beyond shell crossing. Master equations in the Lagrangian description are derived and solved perturbatively up to second order. Then, as an illustration, power spectra of density fluctuations are computed in a one-dimensional model from the Lagrangian approximations and Eulerian linear perturbation theory for comparison. We find that the results by the Lagrangian approximations are different from those by the Eulerian one in weakly non-linear regime at the scales smaller than the Jeans length. We also show the validity of the perturbative Lagrangian approximations by consulting difference between the first-order and the second-order approximations.

Dispersion fitted finite difference method with applications to molecular quantum mechanics

An approach to finite difference approximation is presented based on the idea of fitting the dispersion relation up to a limiting accuracy. The resulting approximations to the second derivative can be more accurate than the standard, Lagrangian finite difference approximations by an order of magnitude or more. The locality of the methods makes them well suited to parallel computation, in contrast with pseudospectral methods. The approach is illustrated with application to a simple bound state problem and to a more challenging three dimensional reactive scattering problem.

A Lagrangian approximation to the water-wave problem

We describe the derivation by a variational approach of the Camassa-Holm model for periodic shallow water waves.

On Local Approximations to the Nonlinear Evolution of Large‐Scale Structure

We present a comparative analysis of several methods, known as local Lagrangian approximations, which are aimed to the description of the nonlinear evolution of large-scale structure. We have investigated various aspects of these approximations, such as the evolution of a homogeneous ellipsoid, collapse time as a function of initial conditions, and asymptotic behavior. As one of the common features of the local approximations, we found that the calculated collapse time decreases asymptotically with the inverse of the initial shear. Using these approximations, we have computed the cosmological mass function, finding reasonable agreement with N-body simulations and the Press-Schechter formula.

Passive scalar spectrum in the viscous-convective range in two-dimensional steady turbulence

The spectrum of a passive scalar field in the viscous-convective range convected by two-dimensional steady turbulence was studied using the Lagrangian Renormalized Approximation (LRA) and high resolution DNS with a resolution up to 40962. It was found that the scalar spectrum F(k) using the LRA scales as k−1 in the range, and decays Gaussianly in the far diffusive range, and these wavenumber dependencies in both ranges agree with the DNS data. A new Lagrangian closure approximation that is invariant under random Galilean transformation and random uniform rotation is proposed. The approximation makes better quantitative predictions than the LRA and yields F(k)=CVCχ̄(ν/ε̄)1/2k−1 with CVC=5.25 in the viscous-convective range, which is in good agreement with the DNS value CVC=5.11. The importance of molecular diffusion in the far diffusive range when the viscous-convective range exists is also discussed.

A New QP-Free, Globally Convergent, Locally Superlinearly Convergent Algorithm For Inequality Constrained Optimization

In this paper, we propose a new QP-free method, which ensures the feasibility of all iterates, for inequality constrained optimization. The method is based on a nonsmooth equation reformulation of the KKT optimality condition, by using the Fischer--Burmeister nonlinear complementarity problem function. The study is strongly motivated by recent successful applications of this function to the complementarity problem and the variational inequality problem. The method we propose here enjoys some advantages over similar methods based on the equality part of the KKT optimality condition. For example, without assuming isolatedness of the accumulation point or boundedness of the Lagrangian multiplier approximation sequence, we show that every accumulation point of the iterative sequence generated by this method is a KKT point if the linear independence condition holds. And if the second-order sufficient condition and the strict complementarity condition hold, the method is superlinearly convergent. Some preliminary numerical results indicate that this new QP-free method is quite promising.

Design of raft-pile foundation using combined optimization and finite element approach

This paper describes the application of a structural optimization approach combined with the finite element method for the optimal design of a raft–pile foundation system. The analysis takes into account the non-linear behaviour of the soil medium and the piles. For the optimization process, the sensitivity analysis is carried out using the approximate semi-analytical method while the constraint approximation is obtained from the combination of extended Bi-point and Lagrangian polynomial approximation methods. The objective function of the problem is the cost of the foundation. The design variables are the raft thickness, cross-section, length and number of piles. The maximum displacement and differential displacement are selected as the constraints. The proposed method is shown to achieve an optimum design of raft–pile foundation efficiently and accurately. Copyright © 1999 John Wiley & Sons, Ltd.

Passive scalar spectrum in isotropic turbulence: Prediction by the Lagrangian direct-interaction approximation

The Lagrangian direct-interaction approximation developed previously by the present authors [S. Kida and S. Goto, J. Fluid Mech. 345, 307 (1997)] is applied to a passive scalar field in isotropic turbulence. We examine the behavior of solutions to the resultant closure equations for the correlation function of the scalar field for arbitrary values of the Schmidt number, and show systematically that the solutions are completely consistent with the phenomenological theories on the scalar spectral function by Obukhov (1949), Corrsin (1951), Batchelor et al. (1959), and Batchelor (1959). The universal forms of the function in the statistically stationary state are obtained by solving the closure equations numerically in the whole wave number range for each case of moderate, extremely large, and small values of the Schmidt number.

Optimization approach applied to nonlinear analysis of raft-pile foundations

Optimal design of raft-pile foundations is examined by combining finite element technique and the optimization approach. The piles and soil medium are modeled by three dimensional solid elements while the raft is modelled by shell elements. Drucker-Prager criterion is adopted for the soil medium while the raft and the piles are assumed to be linear elastic. For the optimization process, the approximate semi-analytical method is used for calculating constraint sensitivities and a constraint approximation method which is a combination of the extended Bi-point approximation and Lagrangian polynomial approximation is used for predicting the behaviour of the constraints. The objective function of the problem is the volume of materials of the foundation while the design variables are raft thickness, pile length and pile spacing. The generalized reduced gradient algorithm is chosen for solving the optimization process. It is demonstrated that the method proposed in this study is promising for obtaining optimal design of raft-pile foundations without carrying out a large number of analyses. The results are also compared with those obtained from the previous study in which linear analysis was carried out.

Collisions of domain walls in a supersymmetric model

The process of collision of two parallel domain walls in a supersymmetric model is studied both in effective Lagrangian approximation and by numerical solving of the exact classical field problem. For small initial velocities we find that the walls interaction looks like elastic reflection with some delay. It is also shown that in such approximation internal parameter of the wall may be considered as a time-dependent dynamical variable.

On the Eulerian Approach to the Evolution of Large‐Scale Structures

Within the classical Eulerian theory of gravitational instability, and in an Ω = 1 universe, we introduce a method for obtaining iterative solutions for the density contrast and peculiar velocity that can be extended to any order of approximation in the quasilinear regime. We compute the solution up to the fourth order (E4) and establish a standard criterion on the validity of Eulerian perturbative studies. The accuracy of E1-E4 is analyzed by using a power-law spherical initial profile for the average density contrast. Within the standard validity range, predictions of E4 generally have an accuracy of 95% or better. Concordance between perturbative predictions and the exact values depends on the index of the initial profile, the nature of the structure (cluster or void), and the magnitude under study (density contrast, δ, or relative deviation from the Hubble flow, β). From a global point of view, the E2 approximation describes the final status of clusters quite well (with a relative error of under 10%), while it is poorer in studying the inner parts of voids. So E2 may be considered as a sufficiently good approximation for the study of the evolution of the halo of overdense regions. We also compare the Eulerian approximation with the two main Lagrangian approximations (L1 = Zeldovich and L2) and show that, in a quasilinear regime (| δ | 1) and within the standard validity range, E2 is better than L2. There is one exception to this general behavior, for instance, the contrast density for clusters with a steep initial density profile or voids with a smooth one. The Zeldovich approximation is particularly inefficient at tracing the evolution of peculiar motions (E1 is clearly better). This L1 approximation leads to an artificial behavior of the inner regions of voids, in disagreement with previous work based on a homogeneous void (top-hat spherical underdensity). Our results warn of the risks of a systematic and indiscriminate use of Lagrangian approximations in the study of the large-scale structures evolution.

The permeable element method for modelling of deformation processes in porous and powder materials: theoretical basis and checking by experiments

Abstract The permeable element method (PEM), as the development of an arbitrary Eulerian–Lagrangian finite element approximation, intended for the description of densification of porous bodies, is elaborated. The Eulerian frame of reference, for which the material movement is independent of the movement of a discretizing network, is the basis for the consideration of the deformation process. The shape of the elements and network movement are deliberately determined from the point of view of convenience for the analysis of the calculation results. Using the PEM, the pressing of hollow cylinders in a rigid die is considered. The computed spatial density distributions are compared with the experimental data obtained, using quantitative metallography analysis. The problem of pressing an article of a stepped shape is solved. The peculiarities of material flow and density distributions are discussed.

Testing tree-level perturbation theory for large-scale structure with the local Lagrangian approximation

We test tree-level perturbation theory for Gaussian initial conditions with power spectra P(k)∞ kn by comparing the probability distribution function (PDF) for the density predicted by the local Lagrangian approximation (LLA) with the results of numerical gravitational clustering simulations. Our results indicate that our approximation correctly reproduces the evolved density PDF for n = − 1 and − 2 power spectra up to the weakly non-linear regime, while it shows marginal agreement for power indices n = 0 and +1 in the linear regime and poor agreement beyond this point. This suggests that tree-level perturbation theory (as realized in the LLA) can accurately predict the density distribution function for n ≤ − 1, but fails for n ≥ 0.

Local Lagrangian approximations for the evolution of the density distribution function in large-scale structure

We examine local Lagrangian approximations for the gravitational evolution of the density distribution function. In these approximations, the final density at a Lagrangian point q at a time t is taken to be a function only of t and of the initial density at the same Lagrangian point. A general expression is given for the evolved density distribution function for such approximations, and we show that the vertex generating function for a local Lagrangian mapping applied to an initially Gaussian density field bears a simple relation to the mapping itself. Using this result, we design a local Lagrangian mapping which reproduces nearly exactly the hierarchical amplitudes given by perturbation theory for gravitational evolution. When extended to smoothed density fields and applied to Gaussian initial conditions, this mapping produces a final density distribution function in excellent agreement with full numerical simulations of gravitational clustering. We also examine the application of these local Lagrangian approximations to non-Gaussian initial conditions.

Accuracy of Lagrangian approximations in voids

We study the behaviour of spherical Voids in Lagrangian perturbation theories L(n), of which the Zel'dovich approximation is the lowest order solution L(1). We find that at early times higher order L(n) give an increasingly accurate picture of Void expansion. However at late times particle trajectories in L(2) begin to turnaround and converge leading to the {\em contraction} of a Void, a sign of pathological behaviour. By contrast particle trajectories in L(3) are well behaved and this approximation gives results in excellent agreement with the exact top-hat solution as long as the Void is not too underdense. For very underdense Voids, L(3) evacuates the Void much too rapidly leading us to conclude that the Zel'dovich approximation L(1), remains the best approximation to apply to the late time study of Voids. The behavior of high order approximations in spherical voids is typical for asymptotic series and may be generic for Lagrangian perturbation theory.

A new mixed finite element method for viscoelastic flows governed by differential constitutive equations

A new mixed finite element method is presented that has improved numerical stability and has similar numerical accuracy compared to the EVSS/FEM developed by Rajagopalan et al. (J. Non-Newtonian Fluid Mech. 36 (1990) 159–192). The new method, denoted EVSS-G/FEM, is based on separate bilinear interpolation of all relevant components of the velocity gradient tensor. Lagrangian bilinear approximations of this variable are formed by least-squares interpolation of the derivatives of a Lagrangian biquadratic approximation to the velocity field that is generated by solution of the momentum and continuity equations using a standard mixed formulation for velocity and pressure. Elastic components of the stress are computed by solving the constitutive equation using either the streamline upwinding (SU) or streamline upwind Petrov-Galerkin (SUPG) methods with bilinear interpolation. The motivation for the interpolation of the velocity gradients and for the stress interpolation is to force the compatibility between these variables that is needed to satisfy differential viscoelastic constitutive equations in the limit where the velocity field vanishes, as it does near solid boundaries. The enhanced numerical stability of the EVSS-G/FEM over the EVSS/FEM formulation is demonstrated for the calculation of the linear stability of planar Couette flow of a UCM fluid, where closed form results exist for the linear stability of the steady-state flow; hence, any instability detected in the numerical solution is a result of either the finite element approximation or the time integration method. The second possibility is eliminated by using a fully implicit time integration method. Calculations with the EVSS-G/FEM are stable for De>100, whereas calculations with the EVSS/FEM become numerically unstable for De>5. The EVSS-G formulation is combined with finite element discretizations that incorporate local adaptive mesh refinement for increasing the accuracy of the calculations in regions where the stress and velocity gradients change rapidly. The accuracy of the EVSS-G/FEM is demonstrated by steady-state calculations for the flow between eccentric rotating cylinders, flow through a wavy-walled tube, and for flow through a square array of cylinders; each has been used before as a test problem for the accuracy of viscoelastic flow calculations. As in previous calculations, discretization of the constitutive equation by SUPG gives superior accuracy at high values of De compared to the application of SU.

Time‐dependent liquid metal flows with free convection and a deformable free surface

AbstractThe finite element method is employed to investigate time‐dependent liquid metal flows with free convection, free surfaces and Marangoni effects. The liquid circulates in a two‐dimensional shallow trough with differentially heated vertical walls. The spatial formulation incorporates mixed Lagrangian approximations to the velocity, pressure, temperature and free surface position. The time integration is performed with the backward Euler and trapezoid rule methods with step size control. The Galerkin method is used to reduce the problem to a set of non‐linear equations which are solved with the Newton–Raphson method. Calculations are performed for conditions relevant to the electron beam vaporization of refractory metals. The Prandtl number is 0·015 and Grashof number are in the transition range between laminar and turbulent flow. The results reveal the effects of flow intensity, surface tension gradients, mesh refinement and time integration strategy.