Numerical Simulations of Fractional Systems: An Overview of Existing Methods and Improvements

An overview of the main simulation methods of fractional systems is presented. Based on Oustaloups recursive poles and zeros approximation of a fractional integrator in a frequency band, some improvements are proposed. They take into account boundary effects around outer frequency limits and simplify the synthesis of a rational approximation by eliminating arbitrarily chosen parameters.     

Bulk Flow Regimes and Fractional Flow in 2D Porous Media by Numerical Simulations

We investigate a two-dimensional network simulator that models the dynamics of two-phase immiscible bulk flow where film flow can be neglected. We present a method for simulating the detailed dynamical process where the two phases are allowed to break up into bubbles, and bubbles are allowed to merge together. The notions of drainage and imbibition are not adequate to describe this process since there is no clear front between the fluids. In fact, the simulator is constructed so that one can study the behaviour of the system far from inlets and outlets, where the two fluids have been mixed together so much that all initial fronts have broken up. The simulator gives the fractional flow as a function of the saturation of each of the fluids. For the case of two fluids with equal viscosity, we classify flow regimes that are parametrized by the capillary number.     

Large Eddy Simulations of Swirling Non-premixed Flames With Flamelet Models: A Comparison of Numerical Methods

This work investigates the application of large eddy simulation (LES) to selected cases of the turbulent non-premixed Sydney swirl flames. Two research groups (Loughborough University, LU and Imperial College, IC) have simulated these cases for different parameter sets, using two different and independent LES methods. The simulations of the non-reactive turbulent flow predicted the experimental results with good agreement and both simulations captured the recirculation structures and the vortex breakdown without major difficulties. For the reactive cases, the LES predictions were less satisfactory, and using two independent simulations has helped to understand the shortcomings of each. Furthermore one of the flames (SMH2) was found to be exceptionally hard to predict, which was supported by the lower amount of turbulent kinetic energy that was resolved in this case. However, the LES has identified modes of flame instability that were similar to those observed in some of the experiments.     

Impulse Responses of Fractional Damped Systems

Considered are systems of single-mass oscillators with different fractional damping behaviour. Similar to the classical model, where the damping terms are represented by first derivatives, the eigensystem can be used to decompose the fractional system in frequency domain, if mass, stiffness and damping matrices are linearly dependent. The solution appears as a linear combination of single-mass oscillators. This is true even in the general case such that stability and causality are insured by the same argumentation as used in the linear dependent case.     

分数因子与分数阶完整力学系统的运动方程和循环积分

引入分数因子和分数增量,给出了分数阶微积分的定义和性质;基于分数阶导数的定义,证明了含有分数因子的等时变分与分数阶算子的交换关系;提出了分数阶完整保守和非保守系统的Hamilton原理;建立了分数阶完整保守系统和非保守系统的运动微分方程;得到了分数阶完整保守系统的循环积分;并利用分数阶循环积分导出分数阶罗兹方程．最后给出了两个例子．研究表明利用分数因子给出的分数阶微分方程是一个含有分数因子的通常的微分方程,那么分数阶系统运动微分方程的求解都可以采用通常微分方程的求解方法．     

Impulse Responses of Fractional Damped Systems

Considered are systems of single-mass oscillators with different fractional damping behaviour. Similar to the classical model, where the damping terms are represented by first derivatives, the eigensystem can be used to decompose the fractional system in frequency domain, if mass, stiffness and damping matrices are linearly dependent. The solution appears as a linear combination of single-mass oscillators. This is true even in the general case such that stability and causality are insured by the same argumentation as used in the linear dependent case.     

Correction to: Time and Space Fractional Diffusion in Finite Systems

Additive manufacturing technology, or 3D printing, with silica sand has enabled the manufacture of porous rock analogues for the use in experimental studies of geomechanical properties of reservoir rocks. The accurate modelling of the fluid flow phenomena within a reservoir and improving the performance of hydrocarbon recovery require an understanding of physical and chemical interactions of the reservoir fluids and the rock matrix. Therefore, for the 3D printed samples to serve as rock analogues, flow properties have to be equivalent to the petrophysical properties of their natural counterparts, such as Berea sandstone. In this study, sandstones that were 3D printed with silica sand and Poly-Furfuryl alcohol (PFA) binder, were used to investigate interactions between porous media with different fluids. Wettability preference of 3D printed samples was characterized through contact angle measurements, as well as co-current and counter-current spontaneous imbibition experiments. Results indicated that 3D printed sandstones had mixed-wet characteristics due to the high preference of silica grains for polar fluids and the affinity PFA binder to the oleic phase. Printing configurations including binder saturation were found to greatly influence the wettability preference of the 3D printed analogue rocks as higher PFA concentrations resulted in more strongly oil-wet preferences. Efforts to optimize the printing process and challenges to control the wettability preferences of the 3D printed samples are also highlighted.

     

Numerical Simulations of High-Speed Turbulent Cavity Flows

Detached-Eddy Simulations (DES) of flows over clean and controlled cavities with and without doors are presented in this paper. Mach and Reynolds numbers (based on cavity length) were 0.85 and one million respectively. Spectral analyses showed that the DES computations were able to correctly predict the frequencies of the Rossiter modes for both uncontrolled and controlled cases. Flow visualisations revealed that the impact of the shear layer formed along the cavity on a slanted aft wall no longer creates a large source of acoustic noise. Therefore little acoustic propagation was seen up the cavity. This was confirmed by the analysis of the cavity wall forces, which showed that the oscillations of the shear layer were reduced when the wall was slanted. This aided in reducing the overall Sound Pressure Levels throughout the cavity and far-field. Comparisons of the flow-fields suggested that the addition of the doors also aided in stabilising the shear layer, which was also shown in the analysis of the wall forces. As a result, the addition of the doors was found to affect the clean cavity configuration significantly more than the controlled one.     

Time and Space Fractional Diffusion in Finite Systems

Spatiotemporal nonlocal diffusion in a bounded system is addressed by considering fractional diffusion in a linear, composite system. By considering limiting conditions, solutions for combinations of Neumann and Dirichlet boundary conditions (either zero or nonzero) at the ends of a finite system are derived in terms of Mittag–Leffler functions by the Laplace transformation. Computational viability is demonstrated by inverting the solutions numerically and comparing resulting calculations with asymptotic solutions. Time and space fractional derivatives, defined by variables \(\alpha \) and \(\beta \), respectively, are employed in the Caputo sense; a single-sided, asymmetric space derivative is used. Inspection of the asymptotic solutions leads to insights on the structure of the solutions that may not be available otherwise; the resulting deductions are verified through the numerical inversions. For pure superdiffusion, characteristics of some of the solutions presented here are very similar to those of classical diffusion but combined effects for the corresponding situation result in power-law behaviors. Incidentally, to our knowledge, the pressure distribution for space fractional diffusion at long enough times in a finite system is derived based on first principles for the first time.     

船舶碰撞动力学过程的数值仿真研究

在分析显式非线性有限元基本理论和关键技术的基础上 ,探讨了船舶动力学过程的数值仿真方法 ,并对碰撞过程中船体的力学性能进行了分析 ,获得并讨论了碰撞力、能量吸收和损伤变形的时序结果 ,所得结论具有一般性。碰撞仿真分析中 ,被撞船的舷侧受撞区域和撞击船的首部区域作为可变形结构处理 ,而其余区域则视为刚体。被撞船周围的水划分成Euler有限体积网格 ,船体结构则划分成Lagrange有限元网格 ,并采用耦合技术将它们联系在一起。撞击船周围的水的影响采用附连水质量进行处理。     

The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview

Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New developments have been proposed in order to examine nonlinear systems, among which the theory based on nonlinear normal modes is indubitably the most appealing. In this paper, a different approach is adopted, and proper orthogonal decomposition is considered. The modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data. The utility of the method for dynamic characterization and order reduction of linear and nonlinear mechanical systems is demonstrated in this study.     

A Pore-Scale Model for Permeable Biofilm: Numerical Simulations and Laboratory Experiments

Transport in Porous Media - In this paper, we derive a pore-scale model for permeable biofilm formation in a two-dimensional pore. The pore is divided into two phases: water and biofilm. The...     

Collision between high and low viscosity droplets: Direct Numerical Simulations and experiments

Binary droplet collisions are of importance in a variety of practical applications comprising dispersed two-phase flows. In the present work we focus on the collision of miscible droplets, where one droplet is composed of a high viscous liquid and the other one is of lower viscosity. This kind of collisions take place in, for instance, spray drying processes when droplets with different solid content collide in recirculation zones. The aim of this paper is to investigate the details of the flow inside the colliding droplets. For this purpose, two prototype cases are considered, namely the collision of equal sized droplets and the collision between a small and highly viscous droplet and a bigger low viscous droplet. A new experimental method has been developed in order to visualize the penetration and mixing process of two colliding droplets, where a fluorescence marker is added to one liquid and the droplets are excited by a laser. The results show a delay in the coalescence which takes place during the initial stage of a collision of droplets with different viscosities. Direct Numerical Simulations based on the Volume-of-Fluid method are used to study these collisions and to allow for a more detailed inspection of the mixing process. The method is extended to consider a second liquid with a different viscosity. In order to reproduce the delay of coalescence, an algorithm for the temporal suppression of the coalescence is applied. A predictive simulation of the delay is not possible, because the extremely thin air gap separating the droplets cannot be resolved by the numerics. This approach is validated by comparison with experimental data. The results provide local field data of the flow inside the collision complex, showing in particular a pressure jump at the liquid–liquid interface although no surface tension is present. The detailed analysis of the terms in the momentum balance show that the pressure jump results from the viscosity jump at the liquid–liquid interface.     

惰性颗粒抑爆过程的数值模拟

对高温火球诱导的爆炸及惰性颗粒的抑爆过程进行了数值研究。由于扰动特征时间 ,两相弛豫特征时间和化学平衡特征时间的差异 ,这种带激波的两相化学反应流守恒方程具有刚性。利用处理包括组元守恒方程的全耦合TVD格式和可处理方程源项的Lax Wendroff Rubin格式分别求解气相和颗粒相 ,并根据各特征时间的比值用分步法处理方程的刚性。计算结果反映了均温燃烧火球在可燃环境中形成加速火焰和激波 ,进而成长为爆轰波的过程 ,以及惰性颗粒对爆轰波的抑制过程 ,反映了激波、化学反应和惰性颗粒之间的相互作用和耦合。计算结果表明 ,仅当颗粒浓度大于某值时才能有效抑制爆轰 ,否则无论多大的颗粒散布区域 ,皆不能使爆轰波完全抑制 ,爆轰波或是在其中发展为两相爆轰波 ,或是穿越该区域后重新成长为气相爆轰。     

Identification of Fractional Systems Using an Output-Error Technique

An original method for modeling, simulation and identification of fractional systems in the time domain is presented in this article. The basic idea is to model the fractional system by a state-space representation, where conventional integration is replaced by a fractional one with the help of a non-integer integrator. This operator is itself approximated by a N-dimensional system composed of an integrator and of a phase-lead filter. An output-error technique is used in order to estimate the parameters of the model, including the fractional order N. Simulations exhibit the properties of the identification algorithm. Finally, this methodology is applied to the modeling of the dynamics of a real heat transfer system.     

Identification of Fractional Systems Using an Output-Error Technique

An original method for modeling, simulation and identification of fractional systems in the time domain is presented in this article. The basic idea is to model the fractional system by a state-space representation, where conventional integration is replaced by a fractional one with the help of a non-integer integrator. This operator is itself approximated by a N-dimensional system composed of an integrator and of a phase-lead filter. An output-error technique is used in order to estimate the parameters of the model, including the fractional order N. Simulations exhibit the properties of the identification algorithm. Finally, this methodology is applied to the modeling of the dynamics of a real heat transfer system.     

Numerical Simulations of Flows in a Heterogeneous Porous Medium

Numerical simulations are employed to investigate the fluid flow and pressure loss in a heterogeneous block within a composite porous medium. The mean permeability of the heterogeneous block is seen to affect the overall effective flux significantly. The heterogeneous parameters of the permeability field, such as the correlation length and variance, affect it quite differently. Because of the channelling effects, the effective flux depends strongly on the realization of the permeability for larger correlation length. Under a specific permeability field, higher effective flux results from smaller variances. The influences of the inertial factor are found to be insignificant within the range of practical interests.