A numerical method for the multiphase viscous flow equations

A numerical technique is developed for the solution of the equations that govern multiphase viscous flow. We demonstrate that the equations can be written as a coupled system of Partial Differential Equations (PDEs) comprising: (i) first order hyperbolic PDEs for the volume fraction of each phase; (ii) a generalised Stokes flow for the velocity of each phase; and (iii) elliptic PDEs for the concentration of nutrients and messengers. Furthermore, the computational domain may vary with time for some applications. Appropriate numerical methods are identified for each of these subsystems.The numerical technique developed is then demonstrated using two exemplar applications: tissue engineering; and avascular tumour development. This allows verification that the technique is appropriate for many features of multiphase viscous flow modelling.     

A hybrid finite element method for stokes flow: Part I—Formulation and numerical studies

A hybrid finite element scheme, based on assumed deviatoric fluid stresses in the element and continuous velocity fields at the element-boundaries, is presented. The deviatoric stress and hydrostatic pressure field are subject a priori to the constraints of balance of momenta, the advantages of the present scheme are discussed, and its versatility is demonstrated through a few numerical examples. Studies of convergence and stability of the method are included in Part II of the paper.     

A finite element based level-set method for multiphase flow applications

A numerical method for simulating incompressible two-dimensional multiphase flow is presented. The method is based on a level-set formulation discretized by a finite-element technique. The treatment of the specific features of this problem, such as surface tension forces acting at the interfaces separating two immiscible fluids, as well as the density and viscosity jumps that in general occur across such interfaces, have been integrated into the finite-element framework. Using a method based on the weak formulation of the Navier-Stokes equations has its advantages. In this formulation, the singular surface tension forces are included through line integrals along the interfaces, which are easily approximated quantities. In addition, differentiation of the discontinuous viscosity is avoided. The discontinuous density and viscosity are included in the finite element integrals. A strategy for the evaluation of integrals with discontinuous integrands has been developed based on a rigorous analysis of the errors associated with the evaluation of such integrals. Numerical tests have been performed. For the case of a rising buoyant bubble the results are in good agreement with results from a front-tracking method. The run presented here is a run including topology changes, where initially separated areas of one fluid merge in different stages due to buoyancy effects. Received: 1 March 1999 / Accepted: 17 June 1999     

A multiphase compressible model for the simulation of multiphase flows

A compressible model able to manage incompressible two-phase flows as well as compressible motions is proposed. After a presentation of the multiphase compressible concept, the new model and related numerical methods are detailed on fixed structured grids. The presented model is a 1-fluid model with a reformulated mass conservation equation which takes into account the effects of compressibility. The coupling between pressure and flow velocity is ensured by introducing mass conservation terms in the momentum and energy equations. The numerical model is then validated with four test cases involving the compression of an air bubble by water, the liquid injection in a closed cavity filled with air, a bubble subjected to an ultrasound field and finally the oscillations of a deformed air bubble in melted steel. The numerical results are compared with analytical results and convergence orders in space are provided.     

A mixed formulation of the Bingham fluid flow problem: Analysis and numerical solution

In this paper we introduce a mixed formulation of the Bingham fluid flow problem. We consider both the original and a regularized version of the problem, where a parameter ε is introduced, forcing the entire domain to be formally a fluid region. In general, common solvers for the regularized problem experience a performance degradation when the parameter ε gets smaller. The method studied here introduces an auxiliary tensor variable and shows enhanced numerical properties for small values of ε. A good performance is also observed for the non-regularized case. The well posedness for the regularized problem and the equivalence – at the continuous level – between the original (primitive variables) and the mixed formulation are demonstrated. We analyze properties of linearized problems that are relevant for the convergence of numerical solvers. A finite element method for the mixed formulation is discussed. Numerical results confirm the predicted better performances of the mixed formulation when compared to the primitive variables formulation. A comparison to a non-regularized solver based on the augmented Duvaut–Lions–Glowinski formulation of the problem is carried out as well.     

Density-driven flow and solute transport problems. A 2-D numerical model based on the network simulation method

The network simulation method, based on the formal equivalence between physical systems and electrical networks, solves numerical problems of relatively mathematical complexity in a versatile, efficient and computationally fast way. In this paper, the method is applied for the first time to the design of a general purpose model for simulating two-dimensional transient density-driven flow and solute transport through porous media, a mathematical model made up by coupled, nonlinear differential equations. Using the Boussinesq approximation and the stream function formulation, the model is used to solve two typical problems related with groundwater flows. Isochlor concentration and stream function curves are presented and successfully compared with those of other authors. Simulation is carried out using the digital computer program Pspice with relatively low computing times.     

Interactively multiphase image segmentation based on variational formulation and graph cuts

In this paper the multiple piecewise constant (MPC) active contour model is extended to deal with multiphase case. This proposed multiphase model can be effectively optimized by solving the minimum cuts problem of a specially devised multilayer graph. Based on the proposed energy functional and its graph cuts optimization, an interactively multiphase partition method for image segmentation is presented. The user places some scribbles with different colors on the image according to the practical application demand and each group of scribbles with the same color corresponds to a potential image region. The distribution of each region can be learned from the input scribbles with some particular color. Then the corresponding multilayer graph can be constructed and its minimum cuts can be computed to determine the segmentation result of the image. Numerical experiments show that the proposed interactively multiphase segmentation method can accurately segment the image into different regions according to the input scribbles with different color.     

Multifluid numerical approximations based on a multipressure formulation

In the present work, we consider the numerical approximations of multifluid compressible flows. We suppose that the flow is composed by nonmixing compressible fluids. We propose a dissipative model insuring, for the mixture, the conservation of the mass, the momentum and the total energy. Entropy balances between phases are described by additional nonconservative equations. At the limit of vanishing viscosity the model is still consistent with the entropy balances. The nonlinear projection scheme is used to preserve, at the discrete level, the main properties of the model. Numerical experiments are conducted to validate the proposed models for the multicomponent shock tube, the bubble-shock interaction and the tank filling problems.     

Self-sensing for electromagnetic actuators. Part I: A coupled reluctance network model approach

A self-sensing arrangement in active magnetic bearings (AMBs) comprises a single electromagnetic transducer to realize the actuation and sensing functions concurrently. Minimizing the number of sensing devices and associated interfacing directly reduces possible failure points, system costs, and system complexity. Currently, self-sensing performance is degraded due to problems such as magnetic cross-coupling, eddy currents, saturation, and high losses. This first paper in a two part series presents an integrated model for self-sensing of an 8-pole heteropolar magnetic bearing. The proposed self-sensing approach addresses mechanisms that contribute to modelling error and uncertainty by using several techniques in an integrated structure. A coupled reluctance network model (RNM) is developed which models the coil impedance at the switching frequency. The accuracy of the model is improved by incorporating terms for air gap fringing, complex permeability, and magnetic material nonlinearity. The RNM is verified and refined through a process of iteration using finite element method (FEM) results and experimental AMB measurements. The results demonstrate that a RNM with only 40 nodes can achieve high levels of accuracy when compared to an 80 000 node FEM analysis.In Part II of the series, the refined RNM is incorporated into a multiple input multiple output (MIMO) parameter estimation self-sensing scheme.     

A neurofuzzy methodology for impedance-based multiphase flow identification

A neurofuzzy methodology for flow identification based on signals obtained from an impedance void meter is presented. The methodology combines the filtering and interpolative capabilities of neural networks with the representational advantages of fuzzy systems for the purpose of mapping idiosyncratic area-averaged impedance measurements to multiphase flow regimes. It has been shown that electrical signals representing the conductance of the intervening medium can be used to infer crucial flow parameters, and that area-averaged signals contain sufficient information about flow regime and the structure of its two-phase constituents. The neurofuzzy approach is a promising means for reconstructing the visual imagery of flow in a process, analogous to tomography, and holds considerable promise for multiphase flow diagnostic and measurement applications in the nuclear as well as in the petroleum, biomedical, and food-processing industries.     

FITOVERT: A dynamic numerical model of subsurface vertical flow constructed wetlands

This paper introduces a mathematical model (FITOVERT) specifically developed to simulate the behaviour of vertical subsurface flow constructed wetlands (VSSF-CWs). One of the main goals of the development of FITOVERT was to keep the complexity of the model to an acceptable level, so as to provide a practical tool for design and operation optimization. The dynamic formulation of the model allows to simulate the typical non stationary feeding-emptying operation of VSSF-CWs. FITOVERT is able to describe the water flow through porous media in unsaturated conditions, combined with evapotranspiration; its biochemical module describes the degradation of both organic matter and nitrogen; the transport in the liquid phase is implemented for both dissolved and particulate components; the oxygen transport in the gaseous phase of the soil and its exchange with the liquid phase are also considered. As a main advantage, compared to the few currently available dedicated numerical models, FITOVERT is able to handle the porosity reduction due to bacteria growth and accumulation of particulate components, so that the clogging process is also simulated as an effect of the pore size reduction on the hydraulic conductivity of the simulated system. The performance of the model was firstly analyzed by comparison with hydrodynamic tests recorded in an experimental VSSF-CW pilot plant: tracer test were carried out in three different saturation conditions (fully saturated, partially saturated, and completely drained). FITOVERT proved to accurately simulate the hydraulic behaviour of VSSF-CWs in both saturated and unsaturated conditions. The needs for model improvements and further calibration are finally discussed.     

A numerical flow simulation based on the rate form of the equation of state

A finite-difference numerical method of solution for the unsteady, incompressible Navier-Stokes equations using primitive variables is presented. The rate form of the equation of state is used for the calculation of pressure. This form of the equation of state is well-suited for use with the unsteady form of the conservation equations (mass, momentum and energy). An implicit algorithm is used for the time integration for greater numerical stability. This method is used to solve a known benchmark problem in steady-state natural convection as a test of steady-state accuracy. The results of the simulation are compared to the benchmark.     

New fast super-dashpot time-dependent techniques for the numerical simulation of steady flows—I: Numerical formulation

This paper describes new fast artificial time-dependent methods leading asymptotically, after a sufficiently long time, to the solution of any steady system of first-order equations. They can, namely, be very useful and efficient for computing steady inviscid transonic mixed flows, as well as for solving the steady hybrid equations of subsonic rotational flows. The time-dependent equations used by the new methods are constructed by adding a purely artificial unsteady operator to the steady physical equations. That operator introduces a strong internal damping of the perturbation waves similar to that due to dashpots on the surface of a vibrating membrane. As a result, a very large rate of convergence of the same order of magnitude as that of the over-relaxation techniques is obtained. The new methods can be applied to any conservative finite differences, finite volumes or finite element discretization of the steady equations. Their level of generality is comparable to that of the classical time-dependent techniques using the unsteady Euler equations, but they are much faster.     

流量装置流量稳定性数值仿真评价方法研究

为了提高装置流量稳定性,提出基于计算流体动力学（ CFD）仿真,采用轴向动量数KU、旋流数KV 和不对称数KA ,分析管路不同截面位置的流场速度特征,利用多个截面位置的特征参数变化,评价装置管路的流量稳定性。对新建水装置流量稳定性进行数值仿真分析与实验测试,得出一致性结论。该方法可以在装置设计阶段,对流量稳定性进行评价,从而优化管路结构,提高装置流量稳定性。     

A model based integration framework for computer numerical control system development

As a typical complex embedded computer control system, Computer Numerical Control (CNC) system development is confronted with a great challenge because of its specific requirements as well as some recent development trends such as ever more complex products while at lower prices and shorter develop cycle. In this paper, a model based integration framework (CNCMIF) for CNC system design and development is presented, which integrates modeling, simulation, verification and implementation in a uniform environment. The CNCML (CNC modeling language) with well defined syntax and unambiguous semantics is developed to describe the CNC system in an accurate and explicit way. Model transformation strategy for formal verification and code automatic generation for implementation in the framework are also presented. The approach is an attempt to create an infrastructure to support the CNC system design in an efficient way, while at the same time guarantees the function and performance requirements with advanced capability of the system such as modularity, flexibility, reusability, etc.     

Laminar flow in rectangular channels. Part I: Entry analysis. Part II: Numerical solution for a square channel?

Laminar incompressible flow in rectangular channels is considered. In Part I, the entry region is evaluated by a boundary layer/potential core analysis. It is shown that the three-dimensional displacement induced potential flow can be described with a pair of two-dimensional potential functions. Second-order boundary layer solutions, with and without surface mass transfer, are determined; an interesting secondary flow reversal is predicted. In Part II, numerical solutions are obtained for the viscous channel equations, which are derived from the asymptotic theory of Part I. A two stream function, velocity, vorticity system, independent of the Reynolds Number, is solved with a combined iterative ADI/point-relaxation numerical procedure. A single calculation applied for all Reynolds numbers, which appears only in the coordinate scaling. The axial flow behavior of Parts I and II are in good agreement in the asymptotic entry region where both analysis apply. Secondary flow reversal is calculated; however, the grid is too crude for quantitative comparisons. Numerical solutions are obtained until fully developed conditions are achieved. Agreement with experimental data is good.     

CLA-DE: a hybrid model based on cellular learning automata for numerical optimization

This paper presents a hybrid model named: CLA-DE for global numerical optimization. This model is based on cellular learning automata (CLA) and differential evolution algorithm. The main idea is to learn the most promising regions of the search space using cellular learning automata. Learning automata in the CLA iteratively partition the search dimensions of a problem and learn the most admissible partitions. In order to facilitate incorporation among the CLA cells and improve their impact on each other, differential evolution algorithm is incorporated, by which communication and information exchange among neighboring cells are speeded up. The proposed model is compared with some evolutionary algorithms to demonstrate its effectiveness. Experiments are conducted on a group of benchmark functions which are commonly used in the literature. The results show that the proposed algorithm can achieve near optimal solutions in all cases which are highly competitive with the ones from the compared algorithms.     

A numerical model for the investigation of the flow and isotope concentration field in an ultracentrifuge

A numerical code is built up with 36000 cards and 343 subroutines to investigate the interconnected fields of velocity, temperature, pressure and isotope concentration in a gas centrigue. The full set of Navier-Stokes equations (i.e. continuity, momentum and energy equations), the gas state law and the diffusion equation, associated with proper boundary conditions, form the basic mathematical model. This system is solved numerically by the use of a finite element method and direct resolution. An important informatic environment gives flexibility to the code. One typical example illustrates the possibilities of computations.