Spectral element modelling and analysis of a pipeline conveying internal unsteady fluid

In this paper, a spectral element model is developed for the uniform straight pipelines conveying internal unsteady fluid. Four coupled pipe-dynamics equations are derived first by using the Hamilton's principle and the principles of fluid mechanics. The transverse displacement, the axial displacement, the fluid pressure and the fluid velocity are all considered as the dependent variables. The coupled pipe-dynamics equations are then linearized about the steady-state values of the fluid pressure and velocity. As the final step, the spectral element model represented by the exact dynamic stiffness matrix, which is often called spectral element matrix, is formulated by using the frequency-domain solutions of the linearized pipe-dynamics equations. The fast Fourier transform (FFT)-based spectral dynamic analyses are conducted to evaluate the accuracy of the present spectral element model and also to investigate the structural dynamic characteristics and the internal fluid transients of an example pipeline system.     

Numerical computation of fluid pressure transients in pumping installations with air entrainment

In pumping installations such as sewage pumping stations, where gas content and air entrainment exist, the computation of fluid pressure transients in the pipelines becomes grossly inaccurate when constant wave speed and constant friction are assumed. A numerical model and computational procedure have been developed here to better compute the fluid pressure transient in a pipeline by including the effects of air entrainment and gas evolution characteristics of the transported fluid. Free and dissolved gases in the fluid and cavitation at the fluid vapour pressure are modelled. Numerical experiments show that entrained, entrapped or released gases amplify the pressure peak, increase surge damping and produce asymmetric pressure surges. The transient pressure shows a longer period for down-surge and a shorter period for up-surge. The up-surge is considerably amplified and the down-surge marginally reduced when compared with the gas-free case. These observations are consistent with the experimental observations of other investigators. Numerical experiments also show that the use of a variable loss factor in the pressure transient analysis produces marginally higher maximum and lower minimum pressure transients when compared with the constant-loss-factor model for pipelines where the pressures are above the fluid vapour pressure.     

Numerical modelling and computation of fluid pressure transients with air entrainment in pumping installations

In pumping installations such as sewage pumping stations, where gas content and air entrainment exist, the computation of fluid pressure transients in pipelines becomes grossly inaccurate when a constant wave speed is assumed. An accurate numerical model with gas release and absorption has been developed in this paper and used to compute the fluid pressure transients in the pumping mains of selected pumping installations. Free and dissolved gases in the transported fluid and cavitation at vapour pressure are also modelled. When compared with the gas-free case, computations show that entrained, entrapped or released gases amplify the positive pressure peak, increase surge damping and produce asymmetric pressure surges. While the upsurge with air entrainment in the pipelines was considerably amplified, the downsurge was only marginally reduced. The computed results show good agreement with the data available.     

Transient gas flow simulation using an Adaptive Method of LinesSimulation d'écoulements dans un gazoduc par la méthode adaptative de lignes

Designing natural gas pipelines to safely and efficiently handle unsteady flows, requires knowledge of pressure drop, flowrate and temperature distribution throughout the system. The accurate prediction of these parameters is essential in order to achieve optimum cumulative deliverability, and safe and reliable operation. An Adaptive Method of Lines algorithm is formulated for the solution of Euler system of equations, which fully simulates slow and fast transients. Two test cases present the improvement of the numerical solution from grid adaptation. Good results are obtained both for slow and fast transients simulations proving that the suggested numerical procedure is appropriate for such predictions. To cite this article: E. Tentis et al., C. R. Mecanique 331 (2003).     

Numerical solution of the eXtended Pom-Pom model for viscoelastic free surface flows

In this paper we present a finite difference method for solving two-dimensional viscoelastic unsteady free surface flows governed by the single equation version of the eXtended Pom-Pom (XPP) model. The momentum equations are solved by a projection method which uncouples the velocity and pressure fields. We are interested in low Reynolds number flows and, to enhance the stability of the numerical method, an implicit technique for computing the pressure condition on the free surface is employed. This strategy is invoked to solve the governing equations within a Marker-and-Cell type approach while simultaneously calculating the correct normal stress condition on the free surface. The numerical code is validated by performing mesh refinement on a two-dimensional channel flow. Numerical results include an investigation of the influence of the parameters of the XPP equation on the extrudate swelling ratio and the simulation of the Barus effect for XPP fluids.     

Development of projection and artificial compressibility methodologies using the approximate factorization technique

Predictions for two-dimensional, steady, incompressible flows under both laminar and turbulent conditions are presented. The standard k-? turbulence model is used for the turbulent flows. The computational method is based on the approximate factorization technique. The coupled approach is used to link the equations of motion and the turbulence model equations. Mass conservation is enforced by either the pseudocompressibility method or the pressure correction method. Comparison of the two methods shows a superiority of the pressure correction method. Second- and fourth-order artifical dissipation terms are used in order to achieve good convergence and to handle the turbulence model equations efficiently. Several internal and external test cases are investigated, including attached and separated flows.     

Numerical solution of the generalized Serre equations with the MacCormack finite-difference scheme

This paper describes a two-dimensional numerical model to solve the generalized Serre equations. In order to solve the system equations, written in the conservative form, we use an explicit finite-difference method based on the MacCormack time-splitting scheme. The numerical method and the computational model are validated by comparing one- and two-dimensional numerical solutions with theoretical and experimental results. Finally, the two-dimensional model (in a horizontal plane) is tested in a domain with complicated boundary conditions.     

A PRESSURE-BASED ALGORITHM FOR HIGH-SPEED TURBOMACHINERY FLOWS

The steady state Navier–Stokes equations are solved in transonic flows using an elliptic formulation. A segregated solution algorithm is established in which the pressure correction equation is utilized to enforce the divergence-free mass flux constraint. The momentum equations are solved in terms of the primitive variables, while the pressure correction field is used to update both the convecting mass flux components and the pressure itself. The velocity components are deduced from the corrected mass fluxes on the basis of an upwind-biased density, which is a mechanism capable of overcoming the ellipticity of the system of equations, in the transonic flow regime. An incomplete LU decomposition is used for the solution of the transport-type equations and a globally minimized residual method resolves the pressure correction equation. Turbulence is resolved through the k–ε model. Dealing with turbomachinery applications, results are presented in two-dimensional compressor and turbine cascades under design and off-design conditions. © 1997 John Wiley & Sons, Ltd.     

Two-equation (k, ϵ) turbulence computations by the use of a finite element model

A finite element technique is presented and applied to some one- and two-dimensional turbulent flow problems. The basic equations are the Reynolds averaged momentum equations in conjunction with a two-equation (k, ?) turbulence model. The equations are written in time-dependent form and stationary problems are solved by a time iteration procedure. The advection parts of the equations are treated by the use of a method of characteristics, while the continuity requirement is satisfied by a penalty function approach. The general numerical formulation is based on Galerkin's method. Computational results are presented for one-dimensional steady-state and oscillatory channel flow problems and for steady-state flow over a two-dimensional backward-facing step.     

Numerical solution of 2D and 3D turbulent internal flow problems

The paper describes a method for solving numerically two-dimensional or axisymmetric, and three-dimensional turbulent internal flow problems. The method is based on an implicit upwinding relaxation scheme with an arbitrarily shaped conservative control volume. The compressible Reynolds-averaged Navier-Stokes equations are solved with a two-equation turbulence model. All these equations are expressed by using a non-orthogonal curvilinear co-ordinate system. The method is applied to study the compressible internal flow in modern power installations. It has been observed that predictions for two-dimensional and three-dimensional channels show very good agreement with experimental results.     

基于遗传算法的低压液压管路模型参数识别

为了合理预测伴随气泡和气穴的低压液压管路压力瞬态脉动,提出了用改进遗传算法对低压液压管路压力瞬态脉动模型进行参数辨识的新方法.给出了用来描述管路流动特性的瞬态脉动数学模型,建立了用来计算伴随气泡和气穴的液压管路瞬态下气泡体积和气穴体积的数学模型.构造了基于最小二乘法的适应度模型,探讨了遗传操作方式及算法终止准则,采用了算术交叉同线性逼近相结合的改进算术交叉算子进行交叉操作,给出了模型参数寻优的算法流程.实现了对低压液压管路压力瞬态脉动数学模型的参数识别,得到了参数优化后的低压液压管路压力瞬态脉动模型.仿真结果与实验数据的比较表明在低压液压管路瞬态模型中,用改进遗传算法来识别模型中的未知参数的方法是可行的、有效的.     

Finite element modelling of local scour below a pipeline under steady currents

Local scour has been identified as the main factor that causes failures of structures in offshore engineering. Numerous research efforts have been devoted to local scour around offshore pipelines in the past. In this paper, a finite element numerical model is established for simulating local scour below offshore pipelines in steady currents. The flow is simulated by solving the unsteady Reynolds-averaged Navier–Stokes equations with a standard k ? ? turbulent model closure. A sand slide scheme is proposed for the scour calculation, and bed load is considered in the proposed scour model. To account for changes in bed level, the moving mesh method is adopted to capture the water–sediment interface (bed), and the change of bed level is calculated by solving Exner–Polya equation. All the equations are discretised within the two-step Taylor–Galerkin algorithm in this paper. It is found that the sand slide model works well for the simulation of the scour, and the numerical results are shown to be in good agreement with the available experimental data.     

An adaptive ALE method for underwater explosion simulations including cavitation

The aim of this paper is to develop a numerical procedure for simulating a simplified mathematical model of underwater explosion phenomena. The Euler set of equations is selected as the governing equations and the ideal gas and Tammann equations of state (EOS) are used to obtain pressure in the gas bubble and the surrounding water zone, respectively. The modified Schmidt EOS is used to simulate the cavitation regions. An arbitrary Lagrangian–Eulerian method is used to integrate the governing equations over an unstructured moving grid. A mesh adapting technique is applied to increase the accuracy as well as for better capturing of flow physics. Moreover, a least-square smoother is employed to moderate the undesirable effects of gas–water interface irregularities. The numerical results verify that the proposed method is capable of predicting complex physics involved in a spherical underwater explosion. The method also shows a very good performance in smoothing the interface while minimizing the loss of mass and momentum in two-dimensional problems.     

Direct Numerical Simulation of Self-Similar Turbulent Boundary Layers in Adverse Pressure Gradients

Direct numerical simulations of the Navier–Stokes equations have been carried out with the objective of studying turbulent boundary layers in adverse pressure gradients. The boundary layer flows concerned are of the equilibrium type which makes the analysis simpler and the results can be compared with earlier experiments and simulations. This type of turbulent boundary layers also permits an analysis of the equation of motion to predict separation. The linear analysis based on the assumption of asymptotically high Reynolds number gives results that are not applicable to finite Reynolds number flows. A different non-linear approach is presented to obtain a useful relation between the freestream variation and other mean flow parameters. Comparison of turbulent statistics from the zero pressure gradient case and two adverse pressure gradient cases shows the development of an outer peak in the turbulent energy in agreement with experiment. The turbulent flows have also been investigated using a differential Reynolds stress model. Profiles for velocity and turbulence quantities obtained from the direct numerical simulations were used as initial data. The initial transients in the model predictions vanished rapidly. The model predictions are compared with the direct simulations and low Reynolds number effects are investigated.     

孔板消减气流脉动的数值模拟及实验研究

添加孔板是一种消减压缩机管道系统内气流脉动有效而简便的方法,尽管在工业生产中已被广泛应用,但是其设计和制作所需的各个参数尚处于靠经验取值的阶段.针对这种情况,首先阐述了孔板消减管道内气流脉动的机理;然后使用流体仿真计算软件Fluent建立了管道内气体的二维非稳定流动模型,计算了孔板对管道内气流的压力脉动的影响;并在数值模拟的基础上,搭建了往复式压缩机管道系统实验平台,在进气管线研究了孔板对气流脉动的消减作用.通过数值模拟和实验研究分析了孔板孔径比对气流脉动的影响,并指出选用恰当孔径比的孔板不仅能有效降低主管线和缓冲器至孔板段管线的压力脉动幅度而且对压缩机进口段管线内压力脉动同样具有良好的消减效果.     

NUMERICAL PREDICTION OF TWO-PHASE FLOW IN BUBBLE COLUMNS

A numerical model is described for the prediction of turbulent continuum equations for two-phase gas–liquid flows in bubble columns. The mathematical formulation is based on the solution of each phase. The two-phase model incorporates interfacial models of momentum transfer to account for the effects of virtual mass, lift, drag and pressure discontinuities at the gas–liquid interface. Turbulence is represented by means of a two-equation k–ϵ model modified to account for bubble-induced turbulence production. The numerical discretization is based on a staggered finite-volume approach, and the coupled equations are solved in a segregated manner using the IPSA method. The model is implemented generally in the multipurpose PHOENICS computer code, although the present appllications are restricted to two-dimensional flows. The model is applied to simulate two bubble column geometries and the predictions are compared with the measured circulation patterns and void fraction distributions.     

OPTIMAL CONTROL OF GAS PIPELINES VIA INFINITE-DIMENSIONAL ANALYSIS

A general optimal control approach employing the principles of calculus of variations has been developed to determine the best operating strategies for keeping the outlet pressure of gas transmission pipelines around a predetermined value while achieving reasonable energy consumption. The method exploits analytical tools of optimal control theory. A set of partial differential equations characterizing the dynamics of gas flow through a pipeline is directly used. The necessary conditions to minimize the specific performance index come from the infinite-dimensional model. The optimization scheme has been tested on a pipeline subject to stepwise change in demand.     

Inverse laminar boundary-layer problems with assigned shear. The mechul function revisited

An inverse method is presented which accurately determines the pressure distribution for assigned wall shear in a two-dimensional, laminar, incompressible boundary layer. The method reformulates the mechul function scheme of Cebeci and Keller to produce a stable solution in the marching direction and to increase accuracy in the normal direction. In the reformulation a modified pressure gradient parameter variation in the normal direction is used in conjunction with three-point backward differences for streamwise derivatives and fourth-order accurate splines for normal derivatives. The resulting spline-finite difference equations are solved by Newton-Raphson iteration together with partial pivoting. Numerical solutions are presented for self-similar and non self-similar flows and compared with published results.     

The numerical solution of the transient two‐phase flow in rigid pipelines

Consideration is given in this paper to the numerical solution of the transient two‐phase flow in rigid pipelines. The governing equations for such flows are two coupled, non‐linear, hyperbolic, partial differential equations with pressure dependent coefficients. The fluid pressure and velocity are considered as two principle dependent variables. The fluid is a homogeneous gas–liquid mixture for which the density is defined by an expression averaging the two‐component densities where a polytropic process of the gaseous phase is admitted. Instead of the void fraction, which varies with the pressure, the gas–fluid mass ratio (or the quality) is assumed to be constant, and is used in the mathematical formulation. The problem has been solved by the method of non‐linear characteristics and the finite difference conservative scheme. To verify their validity, the computed results of the two numerical techniques are compared for different values of the quality, in the case where the liquid compressibility and the pipe wall elasticity are neglected. Copyright © 1999 John Wiley & Sons, Ltd.