High‐accuracy upwind method using improved characteristics speeds for incompressible flows

In this study, the Nervier–Stokes equations for incompressible flows, modified by the artificial compressibility method, are investigated numerically. To calculate the convective fluxes, a new high‐accuracy characteristics‐based (HACB) scheme is presented in this paper. Comparing the HACB scheme with the original characteristic‐based method, it is found that the new proposed scheme is more accurate and has faster convergence rate than the older one. The second order averaging scheme is used for estimating the viscose fluxes, and spatially discretized equations are integrated in time by an explicit fourth‐order Runge–Kutta scheme. The lid driven cavity flow and flow in channel with a backward facing step have been used as benchmark problems. It is shown that the obtained results using HACB scheme are in good agreement with the standard solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparison of Symmetric and Asymmetric Schemes with Arithmetic and Harmonic Averaging for Fracture Flow on Cartesian Grids

Performance of four finite-difference schemes for fluid flow in rough-walled fractures on regular Cartesian grids is evaluated numerically. The four schemes are an asymmetric scheme with arithmetic averaging, an asymmetric scheme with harmonic averaging, a symmetric scheme with arithmetic averaging, and a symmetric scheme with harmonic averaging. The schemes are compared with respect to their simulated hydraulic aperture and the mass balance error. 1320 flow simulations with different grid sizes, mean fracture aperture and root mean square (RMS)/mean aperture ratio are completed. The asymmetric scheme with arithmetic averaging arises naturally, without any extra assumptions about the correct transmissivity averaging procedure, when one uses second-order finite differences to approximate the generalized Laplace operator expanded as a derivative of a product. Hydraulic apertures obtained with harmonic averaging are found to usually be smaller than those obtained with arithmetic averaging, especially when the ratio of aperture RMS to the mean aperture is larger. The traditionally used asymmetric schemes are found to be superior to symmetric schemes in terms of mass balance accuracy.

     

Effect of crack orientation statistics on effective stiffness of mircocracked solid

This paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics. To do so, the representative unit cell approach has been used. The microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks; a sufficient number is taken to account for the microstructure statistics. The developed method combines the superposition principle, the technique of complex potentials and certain new results in the theory of special functions. A proper choice of potentials provides reducing the boundary-value problem to an ordinary, well-posed set of linear algebraic equations. The exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields. The convergence study has been performed; the statistically meaningful results obtained show dependence of the effective elastic stiffness on angular scattering of cracks. Comparison has been made with the selected simple micromechanical models, namely, non-interaction approximation, differential scheme and modified differential scheme. It is found that, among these models, the differential scheme provides the best fit of the numerical data.     

Inertial Manifolds for the 3D Modified-Leray-$$\alpha $$ Model with Periodic Boundary Conditions

The existence of an inertial manifold for the modified Leray-\(\alpha \) model with periodic boundary conditions in three-dimensional space is proved by using the so-called spatial averaging principle. Moreover, an adaptation of the Perron method for constructing inertial manifolds in the particular case of zero spatial averaging is suggested.     

A new characteristic approach for incompressible thermo‐flow in Cartesian and non‐Cartesian grids

A virtual‐characteristic approach is developed for thermo‐flow with finite‐volume methodology in which a multidimensional characteristic (MC) scheme is applied along with artificial compressibility. To obtain compatibility equations and pseudo‐characteristics, energy equation is taken into account in the MC scheme. With this inherent upwinding of convective fluxes, no artificial viscosity is required even at high Reynolds numbers. Another remarkable advantage of the MC scheme lies in its faster convergence rate with respect to the averaging scheme that is found to exhibit substantial delays in convergence. As benchmarks, forced and mixed convections in a cavity and in flow over cylinder and between parallel plates are examined for a wide range of Reynolds, Grashof, and Prandtl numbers. The MC and averaging schemes are applied for simulation purposes. Results show the better performance of the MC scheme in forced and mixed convections. Results confirm the robustness of the MC scheme in terms of accuracy and convergence. Copyright © 2015 John Wiley & Sons, Ltd.     

A stabilized extremum-preserving scheme for nonlinear parabolic equation on polygonal meshes

In this paper, a stabilized extremum-preserving scheme is introduced for the nonlinear parabolic equation on polygonal meshes. The so-called harmonic averaging points located at the interface of heterogeneity are employed to define the auxiliary unknowns and can be interpolated by the cell-centered unknowns. This scheme has only cell-centered unknowns and possesses a small stencil. A stabilized term is constructed to improve the stability of this scheme. The stability analysis of this scheme is obtained under standard assumptions. Numerical results illustrate that the scheme satisfies the extremum principle with anisotropic full tensor coefficient problems and has optimal convergence rate in space on distorted meshes.     

Ultrasonic attenuation in polycrystals using a self-consistent approach

The self-consistent method of averaging elastic moduli to define the effective medium of a polycrystal is used to investigate the dynamic problem of wave propagation. An alternative covariance tensor describing the elastic moduli fluctuations of the polycrystal containing self-consistent elastic properties is derived and found to be significantly smaller than the covariance tensor formed through traditional Voigt averaging. Attenuation curves are generated using the self-consistent elastic moduli and covariance tensors and these results are compared with previous Voigt-averaged estimates. The second-order polycrystalline dispersion relation for the self-consistent scheme is found for cases of low and high crystallite anisotropy. The attenuation coefficients and dispersion relations derived through the self-consistent scheme are considerably different than previous estimates. Experimentally measured longitudinal attenuation coefficients support the use of the self-consistent scheme for estimation of attenuation.     

Evolution of perturbed rotations of an asymmetric Gyro in a gravitational field and a resisting medium

We study the fast rotational motion of a dynamically asymmetric satellite with a spherical cavity filled with a highly viscous liquid about the center of mass under the action of gravitational torque and medium drag torques. The system obtained by averaging over the Euler–Poinsotmotion and by using a modified averaging method is analyzed. An analytic study and numerical analysis are carried out.     

An improved approach for random parametric response of dynamic systems with non-linear inertia

A non-Gaussian closure scheme is developed for determining the stationary response of dynamic systems including non-linear inertia and stochastic coefficients. Numerical solutions are obtained and examined for their validity based on the preservation of moments properties. The method predicts the jump phenomenon, for all response statistics at an excitation level very close to the threshold level of the condition of almost sure stability. In view of the increased degree of non-linearity, resulting from the non-Gaussian closure scheme, the mean square of the response displacement is found to be less than those values predicted by other methods such as the Gaussian closure or the first order stochastic averaging.     

On Simulating Flows with Multiple Time Scales Using a Method of Averages

We present a new method, based on averaging, to simulate certain systems with multiple time scales efficiently and demonstrate its utility in the context of the shallow-water equations. We first develop the method in a simple linear setting and analytically prove its stability. This is followed by an extension to the full equations and a presentation of a computational model for it. In this preliminary study, we find that the new method produces results that are very close to a fully explicit (spatially and temporally) second-order accurate scheme and much better than a fully explicit (spatially and temporally) first-order accurate scheme, while costing less than the first-order accurate scheme. Received 25 September 1996 and accepted 11 April 1997     

An arbitrary Lagrangian Eulerian (ALE) formulation for free surface flows using the characteristic‐based split (CBS) scheme

An arbitrary Lagrangian Eulerian (ALE) method for non‐breaking free surface flow problems is presented. The characteristic‐based split (CBS) scheme has been employed to solve the ALE equations. A simple mesh smoothing procedure based on coordinate averaging (Laplacian smoothing) is employed in the calculations. The mesh velocity is calculated at each time step and incorporated as part of the scheme. Results presented show an excellent agreement with the available experimental data. Copyright © 2005 John Wiley & Sons, Ltd.     

A modified method of averaging for solving a class of nonlinear equations

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A scale-dependent equation for solute transport in porous media

The governing equation describing solute transport in porous media is reformulated using standard volume averaging techniques. The alternative formulation is based on a modified definition of the deviation, which allows for variation of macroscopic velocity across the REV. The new equation contains additional scale-dependent terms which are functions of the size of the averaging volume (REV). This result indicates that the scale-dependent nature of the dispersion phenomenon is inherent even at the scale of the REV.     

Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media

Abstract. The Picard and modified Picard iteration schemes are often used to numerically solve the nonlinear Richards equation governing water flow in variably saturated porous media. While these methods are easy to implement, they are only linearly convergent. Another approach to solve the Richards equation is to use Newton's iterative method. This method, also known as Newton–Raphson iteration, is quadratically convergent and requires the computation of first derivatives. We implemented Newton's scheme into the mixed form of the Richards equation. As compared to the modified Picard scheme, Newton's scheme requires two additional matrices when the mixed form of the Richards equation is used and requires three additional matrices, when the pressure head-based form is used. The modified Picard scheme may actually be viewed as a simplified Newton scheme.Two examples are used to investigate the numerical performance of different forms of the 1D vertical Richards equation and the different iterative solution schemes. In the first example, we simulate infiltration in a homogeneous dry porous medium by solving both, the h based and mixed forms of Richards equation using the modified Picard and Newton schemes. Results shows that, very small time steps are required to obtain an accurate mass balance. These small times steps make the Newton method less attractive.In a second test problem, we simulate variable inflows and outflows in a heterogeneous dry porous medium by solving the mixed form of the Richards equation, using the modified Picard and Newton schemes. Analytical computation of the Jacobian required less CPU time than its computation by perturbation. A combination of the modified Picard and Newton scheme was found to be more efficient than the modified Picard or Newton scheme.     

Periodic solutions of strongly non-linear Mathieu oscillators

The purpose of this paper is to continue our investigation into periodic solutions of strongly non-linear Mathieu oscillators. The modified version of the generalized averaging method which we developed recently is applied to derive highly accurate analytical expressions for these periodic solutions. These analytical results are used, together with the perturbation methods of multiple time scaling, to obtain second order expressions for the stability regions of these periodic solutions. The analytical research results are verified with numerical computations. Very good agreement is found, which shows the applicability of the modified version of the generalized averaging method to this kind of strongly non-linear oscillators. These oscillators may be used to model the beam-beam interaction in particle accelerators.     

Responses of a non-linearly coupled pitch-roll ship model under harmonic excitation

Stationary responses of nonlinearly coupled pitch and roll ship modes are studied using a modified averaging method, along with two second order multiple time scale (MTS) methods for comparison. Stability of the solutions is also studied. In the case of harmonic excitation all the three methods give fairly accurate results to the original problem but the averaging method is the most efficient. Analytic solutions are obtained from the averaged equations, which can be used to predict stationary responses both for small and for large excitations. From the averaging method several qualitatively different phenomena which cannot be addressed by the first order theory have been obtained: (i) the saturation phenomenon is lost, (ii) the bifurcation points are altered and (iii) a drift term is present which, although small, appears to have a significant effect on the accuracy of the solutions.     

A LOCAL MESH REFINEMENT ALGORITHM APPLIED TO TURBULENT FLOW

This paper presents a local mesh refinement procedure based on a discretization over internal interfaces where the averaging is performed on the coarse side. It is implemented in a multigrid environment but can optionally be used without it. The discretization for the convective terms in the velocity and the temperature equation is the QUICK scheme, while the HYBRID-UPWIND scheme is used in the turbulence equations. The turbulence model used is a two-layer k–ϵ model. We have applied this formulation on a backward-facing step at Re=800 and on a three-dimensional turbulent ventilated enclosure, where we have resolved a geometrically complex inlet consisting of 84 nozzles. In both cases the concept of local mesh refinements was found to be an efficient and accurate solution strategy. © 1997 by John Wiley & Sons, Ltd.     

具有多个极限环非线性动力系统的解析近似

应用一种新的解析方法------同伦分析法，研究了一种具有多个 极限环的Rayleigh振子问题. 与所有其他传统方法不同，该方法不依赖于小参数， 且提供了一个简便的途径以确保级数解的收敛, 因此，特别适用于强非线性问题. 将同伦分析法与平均法以及四阶的龙格库塔方法(数值解)做了比较. 结果 表明，平均法在强非线性情况失效, 四阶的龙格库塔法不能找到非稳定的极限环，而同伦分析法不仅适用于强非线性情 况，而且给出了非稳定的极限环.     

Averaging of particle data from phase Doppler anemometry in unsteady two-phase flow: Validation by numerical simulation

A novel post-processing algorithm is proposed to correct statistical bias observed in the treatment of time series obtained by a phase Doppler anemometer (PDA) at flow locations with variable particle velocity and concentration. Extensive properties of each validated particle are weighted with their inverse measuring (validation) volume to account for the procedure of particle sampling and fluctuations in the particle concentration. To compensate for the short characteristic length of the validation volume, the properties of particles are expressed by properties of fields of particle groups, using a local averaging time. A window shift and a decorrelation scheme are applied on the fields to increase their frequency resolution. This algorithm has been tested on numerical time series, provided by an Eulerian/Lagrangian code representing a gas/solids flow past a bluff body. Moments and spectral estimates of concentration and velocity of particle groups were successfully validated by the numerical simulation using the PDA data algorithm and control volume averaging. The control volume was much larger than the PDA validation volume, but the centre positions of the two volumes were identical.