Incompressible flows with combustion simulated by a preconditioning method using multigrid acceleration and MUSCL reconstruction

A numerical algorithm for the steady state solution of three‐dimensional incompressible flows is presented. A preconditioned time marching scheme is applied to the conservative form of the governing equations. The preconditioning matrix multiplies the time derivatives of the system and circumvents the eigenvalue‐caused stiffness at low speed. The formulation is suitable for constant density flows and for flows where the density depends on non‐passive scalars, such as in low‐speed combustion applications. The k–ε model accounts for turbulent transport effects. A cell‐centred finite volume formulation with a Runge–Kutta time stepping scheme for the primitive variables is used. Second‐order spatial accuracy is achieved by developing for the preconditioned system an approximate Riemann solver with MUSCL reconstruction. A multi‐grid technique coupled with local time stepping and implicit residual smoothing is used to accelerate the convergence to the steady state solution. The convergence behaviour and the validation of the predicted solutions are examined for laminar and turbulent constant density flows and for a turbulent non‐premixed flame simulated by a presumed probability density function (PDF) model. Copyright © 2001 John Wiley & Sons, Ltd.     

On the constitutive modeling of coupled thermomechanical phase-change problems

This paper deals with a thermodynamically consistent numerical formulation for coupled thermoplastic problems including phase-change phenomena and frictional contact. The final goal is to get an accurate, efficient and robust numerical model, able for the numerical simulation of industrial solidification processes. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing differential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain fixed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. All the material properties have been considered to be temperature dependent. The constitutive behavior has been defined by a thermoviscous/elastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. The constitutive model proposed accounts for a continuous transition between the initial liquid state, the intermediate mushy state and the final solid state taking place in a solidification process. In particular, a pure viscous deviatoric model has been used at the initial fluid-like state. A thermomecanical contact model, including a frictional hardening and temperature dependent coupled potential, is derived within a fully consistent thermodinamical theory. The numerical model has been implemented into the computational finite element code COMET developed by the authors. Numerical simulations of solidification processes show the good performance of the computational model developed.     

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.     

Algebraic multigrid and incompressible fluid flow

This paper is concerned with the development of algebraic multigrid (AMG) solution methods for the coupled vector–scalar fields of incompressible fluid flow. It addresses in particular the problems of unstable smoothing and of maintaining good vector–scalar coupling in the AMG coarse‐grid approximations. Two different approaches have been adopted. The first is a direct approach based on a second‐order discrete‐difference formulation in primitive variables. Here smoothing is stabilized using a minimum residual control harness and velocity–pressure coupling is maintained by employing a special interpolation during the construction of the inter‐grid transfer operators. The second is an indirect approach that avoids the coupling problem altogether by using a fourth‐order discrete‐difference formulation in a single scalar‐field variable, primitive variables being recovered in post‐processing steps. In both approaches the discrete‐difference equations are for the steady‐state limit (infinite time step) with a fully implicit treatment of advection based on central differencing using uniform and non‐uniform unstructured meshes. They are solved by Picard iteration, the AMG solvers being used repeatedly for each linear approximation. Both classical AMG (C‐AMG) and smoothed‐aggregation AMG (SA‐AMG) are used. In the direct approach, the SA‐AMG solver (with inter‐grid transfer operators based on mixed‐order interpolation) provides an almost mesh‐independent convergence. In the indirect approach for uniform meshes, the C‐AMG solver (based on a Jacobi‐relaxed interpolation) provides solutions with an optimum scaling of the convergence rates. For non‐uniform meshes this convergence becomes mesh dependent but the overall solution cost increases relatively slowly with increasing mesh bandwidth. Copyright © 2006 John Wiley & Sons, Ltd.     

Implicit weighted essentially non‐oscillatory schemes for the incompressible Navier–Stokes equations

A class of lower–upper/approximate factorization (LUAF) implicit weighted essentially non‐oscillatory (ENO; WENO) schemes for solving the two‐dimensional incompressible Navier–Stokes equations in a generalized co‐ordinate system is presented. The algorithm is based on the artificial compressibility formulation, and symmetric Gauss–Seidel relaxation is used for computing steady state solutions while symmetric successive overrelaxation is used for treating time‐dependent flows. WENO spatial operators are employed for inviscid fluxes and central differencing for viscous fluxes. Internal and external viscous flow test problems are presented to verify the numerical schemes. The use of a WENO spatial operator not only enhances the accuracy of solutions but also improves the convergence rate for the steady state computation as compared with using the ENO counterpart. It is found that the present solutions compare well with exact solutions, experimental data and other numerical results. Copyright © 1999 John Wiley & Sons, Ltd.     

On the formulation of coupled thermoplastic problems with phase-change

This paper deals with a numerical formulation for coupled thermoplastic problems including phase-change phenomena. The final goal is to get an accurate, efficient and robust numerical model, allowing the numerical simulation of solidification processes in the metal casting industry. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing differential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain fixed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. The constitutive behavior has been defined by a thermoelastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. A brief summary of the thermomechanical frictional contact model is included. The numerical model has been implemented into the computational Finite Element code COMET developed by the authors. A numerical assessment of the isentropic and isothermal operator splits, regarding the nonlinear stability behavior, has been performed for weakly and strongly coupled thermomechanical problems. Numerical simulations of solidification processes show the performance of the computational model developed.     

注塑成型三维流动的数值模拟

建立了非等温、粘性、不可压缩、非牛顿流体流动的控制方程。为了避免同时求解耦合的压力场、速度场,本文通过修改Galerkin方法的变分方程,导出了关于压力场的拟Poisson方程,用迭代法独立地求解连续性方程、动量方程,并进行速度一粘度迭代求出最终的压力场、速度场。由于直接使用Galerkin方法求解能量方程容易引起温度场的振荡,本文采用隐式格式及“上风”法离散能量方程,用超松驰迭代法求解温度场的代数方程组。比较了模拟结果与等温管道流动的解析解及法兰的实际注射结果,算例表明本文方法可以预测注射成型流动过程中的一些重要特征。与传统Galerkin方法相比,本文方法可以减少内存,提高数值方法的稳定性。     

A reference solution for phase change with convection

A reference solutions for phase change involving convection in the melt is currently missing. In the present study, we focus on the problem of melting of pure tin in a square cavity heated from the side, which is used as a benchmark test problem. The mathematical model used for the simulations is based on the enthalpy formulation. Extensive numerical computations are performed with grids as fine as 800 × 800. The convergence of the numerical solution is demonstrated and its level assessed. Data values and plots are provided for use as a reference solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation

Based on a new global variational formulation, a spectral element approximation of the incompressible Navier–Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems. Iterative solutions of such systems are feasible and attractive for large problems. It is shown that, provided an appropriate pre‐conditioner is chosen for the pressure system, the nested conjugate gradient methods can be applied to obtain rapid convergence rates. Detailed numerical examples are given to prove the quality of the pre‐conditioner. Thanks to the rapid iterative convergence, the global Uzawa algorithm takes advantage of this as compared with the classical iteration by sub‐domain procedures. Furthermore, a generalization of the pre‐conditioned iterative algorithm to flow simulation is carried out. Comparisons of computational complexity between the Navier–Stokes/Euler coupled solution and the full Navier–Stokes solution are made. It is shown that the gain obtained by using the Navier–Stokes/Euler coupled solution is generally considerable. Copyright © 2000 John Wiley & Sons, Ltd.     

An exact non‐linear Navier–Stokes compressible‐flow solution for CFD code verification

An exact similarity solution of the compressible‐flow Navier–Stokes equations is presented, which embeds supersonic, transonic, and subsonic regions. Describing the viscous and heat‐conducting high‐gradient flow in a shock wave, the solution accommodates non‐linear temperature‐dependent viscosity as well as heat‐conduction coefficients and provides the variation of all the flow variables and their derivatives. Also presented are methods to obtain time‐dependent and/or multi‐dimensional solutions as well as verification benchmarks of increasing severity. Comparisons between the developed analytical solution and CFD solutions of the Navier–Stokes equations, with determination of convergence rates and orders of accuracy of these solutions, illustrate the utility of the developed exact solution for verification purposes. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical investigation of the flow front behaviour in the co‐injection moulding process

This paper presents a three‐dimensional (3D) solution algorithm for solving the sequential co‐injection moulding process. The flow of skin and core materials inside a rectangular cavity is investigated both numerically and experimentally. A 3D finite element flow analysis code is used to solve the governing equations of the non‐isothermal sequential co‐injection moulding. The predicted flow front behaviour is compared to the experimental observations for various skin/core volume ratio, injection speed, injection temperature, and core injection delay. Simulation results are in good agreement with experimental data and indicate correctly the trends in solution change when processing parameters are changing. Solutions are also shown for the filling of a spiral‐flow mould. The numerical approach is shown to predict the core expansion phase during which the flow front of core and skin materials advance together without breakthrough. Breakthrough phenomena is also predicted and the numerical solution is in good agreement with the experiment. Copyright © 2005 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Time‐domain BEM solution of convection–diffusion‐type MHD equations

The two‐dimensional convection–diffusion‐type equations are solved by using the boundary element method (BEM) based on the time‐dependent fundamental solution. The emphasis is given on the solution of magnetohydrodynamic (MHD) duct flow problems with arbitrary wall conductivity. The boundary and time integrals in the BEM formulation are computed numerically assuming constant variations of the unknowns on both the boundary elements and the time intervals. Then, the solution is advanced to the steady‐state iteratively. Thus, it is possible to use quite large time increments and stability problems are not encountered. The time‐domain BEM solution procedure is tested on some convection–diffusion problems and the MHD duct flow problem with insulated walls to establish the validity of the approach. The numerical results for these sample problems compare very well to analytical results. Then, the BEM formulation of the MHD duct flow problem with arbitrary wall conductivity is obtained for the first time in such a way that the equations are solved together with the coupled boundary conditions. The use of time‐dependent fundamental solution enables us to obtain numerical solutions for this problem for the Hartmann number values up to 300 and for several values of conductivity parameter. Copyright © 2007 John Wiley & Sons, Ltd.     

Coupled thermo-mechanical analysis of shape memory alloy circular bars in pure torsion

Pure torsion of shape memory alloy (SMA) bars with circular cross section is studied by considering the effect of temperature gradient in the cross sections as a result of latent heat generation and absorption during forward and reverse phase transformations. The local form of energy balance for SMAs by taking into account the heat flux effect is coupled to a closed-form solution of SMA bars subjected to pure torsion. The resulting coupled thermo-mechanical equations are solved for SMA bars with circular cross sections. Several numerical case studies are presented and the necessity of considering the coupled thermo-mechanical formulation is demonstrated by comparing the results of the proposed model with those obtained by assuming an isothermal process during loading–unloading. Pure torsion of SMA bars in various ambient conditions (free and forced convection of air, and forced convection of water flow) subjected to different loading–unloading rates are studied and it is shown that the isothermal solution is valid only for specific combinations of ambient conditions and loading rates.     

2D thermal/isothermal incompressible viscous flows

2D thermal and isothermal time‐dependent incompressible viscous flows are presented in rectangular domains governed by the Boussinesq approximation and Navier–Stokes equations in the stream function–vorticity formulation. The results are obtained with a simple numerical scheme based on a fixed point iterative process applied to the non‐linear elliptic systems that result after a second‐order time discretization. The iterative process leads to the solution of uncoupled, well‐conditioned, symmetric linear elliptic problems. Thermal and isothermal examples are associated with the unregularized, driven cavity problem and correspond to several aspect ratios of the cavity. Some results are presented as validation examples and others, to the best of our knowledge, are reported for the first time. The parameters involved in the numerical experiments are the Reynolds number Re, the Grashof number Gr and the aspect ratio. All the results shown correspond to steady state flows obtained from the unsteady problem. Copyright © 2005 John Wiley & Sons, Ltd.     

p-version least squares finite element formulation for two-dimensional,incompressible, non-Newtonian isothermal and non-isothermal fluid flow

This paper presents a p- version least squares finite element formulation (LSFEF) for two-dimensional, incompressible, non-Newtonian fluid flow under isothermal and non-isothermal conditions. The dimensionless forms of the diffential equations describing the fluid motion and heat transfer are cast into a set of first-order differential equations using non-Newtonian stresses and heat fluxes as auxiliary variables. The velocities, pressure and temperature as well as the stresses and heat fluxes are interpolated using equal-order, C⁰-continuous, p-version hierarchical approximation functions. The application of least squares minimization to the set of coupled first-order non-linear partial differential equations results in finding a solution vector {δ} which makes the partial derivatives of the error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. The paper presents the implementation of a power-law model for the non-Newtonian Viscosity. For the non-isothermal case the fluid properties are considered to be a function of temperature. Three numerical examples (fully developed flow between parallel plates, symmetric sudden expansion and lid-driven cavity) are presented for isothermal power-law fluid flow. The Couette shear flow problem and the 4:1 symmetric sudden expansion are used to present numerical results for non-isothermal power-law fluid flow. The numerical examples demonstrate the convergence characteristics and accuracy of the formulation.     

Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage: Formulation and computational aspects

It is well established that the use of inelastic constitutive equations accounting for induced softening, leads to pathological space (mesh) and time discretization dependency of the numerical solution of the associated Initial and Boundary Value Problem (IBVP). To avoid this drawback, many less or more approximate solutions have been proposed in the literature in order to regularize the IBVP and to obtain numerical solutions which are, at convergence, much less sensitive to the space and the time discretization. The basic idea behind these regularization techniques is the formulation of nonlocal constitutive equations by introducing some effects of characteristic lengths representing the materials microstructure. In this work, using the framework of generalized nonlocal continua, a thermodynamically-consistent micromorphic formulation using appropriate micromorphic state variables and their first gradients, is proposed in order to extend the classical local constitutive equations by incorporating appropriate characteristic internal lengths. The isotropic damage, the isotropic and the kinematic hardenings are supposed to carry the targeted micromorphic effects. First the theoretical aspects of this fully coupled micromorphic formulation is presented in details and the proposed generalized balance equations as well as the fully coupled micromorphic constitutive equations deduced. The associated numerical aspects in the framework of the classical Galerkin-based FE formulation are briefly discussed in the special case of micromorphic damage. Specifically, the formulation of 2D finite elements with additional degrees of freedom (d.o.f.), the dynamic explicit global resolution scheme as well as the local integration scheme, to compute the stress tensor and the state variables at each integration point of each element, are presented. Application is made to the typical uniaxial tension specimen under plane strain conditions in order to chow the predictive capabilities of the proposed micromorphic model, particularly against its ability to give (at convergence) a mesh independent solution even for high values of the ductile damage (i.e., the macroscopic cracks).     

Parallel three‐dimensional simulation of the injection molding process

This paper describes the development of a parallel three‐dimensional unstructured non‐isothermal flow solver for the simulation of the injection molding process. The numerical model accounts for multiphase flow in which the melt and air regions are considered to be a continuous incompressible fluid with distinct physical properties. This aspect avoids the complex reconstruction of the interface. A collocated finite volume method is employed, which can switch between first‐ and second‐order accuracy in both space and time. The pressure implicit with splitting of operators algorithm is used to compute the transient flow variables and couple velocity and pressure. The temperature equation is solved using a transport equation with convection and diffusion terms. An upwind differencing scheme is used for the discretization of the convection term to enforce a bounded solution. In order to capture the sharp interface, a bounded compressive high‐resolution scheme is employed. Parallelization of the code is achieved using the PETSc framework and a single program multiple data message passing model. Predicted numerical solutions for several example problems are considered. The first case validates the solution algorithm for moderate Reynolds number flows using a structured mesh. The second case employs an unstructured hybrid mesh showing the capability of the solver to describe highly viscous flows closer to realistic injection molding conditions. The final case presents the non‐isothermal filling of a thick cavity using three mesh sizes and up to 80 processors to assess parallel performance. The proposed algorithm is shown to have good accuracy and scalability. Copyright © 2008 John Wiley & Sons, Ltd.     

Analytical series solution for the fully developed forced convection duct flow with frictional heating and variable viscosity

Laminar forced convection flow of a liquid in the fully developed region of a circular duct with isothermal wall is analyzed. The effects of viscous dissipation as well as of temperature dependent viscosity are taken into account. The coupled momentum and energy equations are solved analytically by means of a power series method. Then, reference is made to the Poiseuille model for the temperature change of viscosity. For a fixed value of the axial pressure gradient along the duct, dual solutions are found for the velocity and temperature fields. Although dual solutions correspond to the same value of the axial pressure gradient, they lead in general to different values of the average fluid velocity, of the average fluid temperature and of the wall heat flux. It is shown that, for a given fluid and for a fixed duct radius, the absolute value of the axial pressure gradient has an upper bound above which no steady laminar solution can exist.     

Natural convection in composite systems with phase change materials

In this study, we carried out a numerical simulation of transient heat transfer in a composite passive system consisting of air–phase change material–air, arranged as a rectangular enclosure. The vertical boundaries of the enclosure are isothermal and the horizontal ones adiabatic. The enthalpy formulation with a fixed grid is used to study the process of phase change with liquid–solid interface zone controlled by natural convection. The flow in this zone is simulated by a model based on the Darcy porous medium. The numerical solution of the mathematical model is done using finite difference–control volume algorithm. The influence of the geometrical and thermal parameters is studied. It is found that subcooling coefficient is the most important parameter influencing heat transfer, and for a given subcooling, there is an optimum phase change partition thickness.