Modeling and simulation of the viscoelastic fluid mold filling process by level set method

A model for unifying a viscoelastic fluid and a Newtonian fluid is established, in which the governing equations for the viscoelastic fluid and the Newtonian fluid are successfully united into a system of generalized Navier–Stokes equations. A level set method is set up to solve the model for capturing the moving interface in the mold filling process. The physical governing equations are solved by the finite volume method on a non-staggered grid and the interpolation technique on the collocated grid is used for the pressure-velocity and the stress-velocity decoupling problems. The level set and its reinitialization equation are solved by the finite difference method, in which the spatial derivatives are discretized by the 5th-order Weighted Essentially Non-Oscillatory (WENO) scheme, and the temporal derivatives are discretized by the 3rd-order Total Variation Diminishing Runge–Kutta (TVD-R–K) scheme. The validity of the method is verified by some benchmark problems. Then a simulation of viscoelastic fluid mold filling process is pursued with the method. The moving interface and all the information of the physical quantities during the injection process are captured. The die swelling phenomenon is found in the simulation. The influences of elasticity and viscosity on the physical quantities such as stresses etc. in the mold filling process are analyzed. Numerical results show that elastic characteristics such as the stretch and die swelling etc. reinforce accordingly as Weissenberg number increases. Pressures increase continuously in the mold filling process and the pressure maintains the maximum value at the inlet. Injection velocity is proportional to injection pressure. A higher viscosity leads to a higher pressure distribution, that is, the pressure decreases as Reynolds number increases.     

An algorithm that accelerates convergence rates for incompressible Navier-Stokes problems

An algorithm, called the Algebraic Continuity Equations Solver (ACES), is developed based on the concept that two algebraic equations (three for 3D problems) can be generated from rearranging the discretized continuity equations. These rearranged equations are used to re-compute the two velocity components (three for 3D problems), whose values are already obtained from solving the momentum equations. When written in a Navier-Stokes computer code, this algorithm is equivalent to a fairly concise set of statements and can be implemented immediately after the computation of the continuity equation. In our analysis, ACES is used in conjunction with a grid having nodal velocity components at the vertices and the nodal pressure at the centre of each computational cell. With the aid of ACES, correction of velocity components during the iteration can be inexpensively made, leading to faster convergence rates or rendering otherwise divergent computations convergent. Test problems include benchmark problems such as lid-driven cavity flows and buoyancy-driven cavity flows of various parametric values and grid sizes. A 3D time-dependent flow in an irregular geometry is also investigated. Discussions are presented to clarify some relevant issues. A possible reason why we think ACES is capable of improving the convergence rates is also given.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

ADAPTIVE SOLUTIONS FOR UNSTEADY LAMINAR FLOWS ON UNSTRUCTURED GRIDS

An adaptive finite volume method for the simulation of time-dependent, viscous flow is presented. The Navier–Stokes equations are discretized by central schemes on unstructured grids and solved by an explicit Runge–Kutta method. The essential topics of the present study are a new concept for a local Runge–Kutta time-stepping scheme, called multisequence Runge–Kutta, which reduces the severe stability restriction in unsteady problems, a common grid generation and adaptation procedure and the application of dynamic grids for capturing moving flow structures. Results are presented for laminar, separated flow around an aerofoil with a flap.     

Vorticity–velocity formulation of the 3D Navier–Stokes equations in cylindrical co‐ordinates

A finite difference method is presented for solving the 3D Navier–Stokes equations in vorticity–velocity form. The method involves solving the vorticity transport equations in ‘curl‐form’ along with a set of Cauchy–Riemann type equations for the velocity. The equations are formulated in cylindrical co‐ordinates and discretized using a staggered grid arrangement. The discretized Cauchy–Riemann type equations are overdetermined and their solution is accomplished by employing a conjugate gradient method on the normal equations. The vorticity transport equations are solved in time using a semi‐implicit Crank–Nicolson/Adams–Bashforth scheme combined with a second‐order accurate spatial discretization scheme. Special emphasis is put on the treatment of the polar singularity. Numerical results of axisymmetric as well as non‐axisymmetric flows in a pipe and in a closed cylinder are presented. Comparison with measurements are carried out for the axisymmetric flow cases. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical solution of the incompressible Navier–Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered‐mesh grid system. The convection terms are discretized with a fifth‐order‐accurate upwind compact difference approximation, the viscous terms are discretized with a sixth‐order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth‐order difference approximation on a cell‐centered mesh. Time advancement uses a three‐stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical solution of three‐dimensional velocity–vorticity Navier–Stokes equations by finite difference method

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false‐transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence‐free velocity field. The vorticity transport equations in conservative form are solved using the second‐order accurate Adams–Bashforth central difference scheme in order to assure divergence‐free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri‐diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri‐diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three‐dimensional problems, which otherwise requires enormous computational effort. For three‐dimensional‐driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100?Re?2000. Copyright © 2004 John Wiley & Sons, Ltd.     

A multigrid method for steady incompressible navier-stokes equations based on flux difference splitting

The steady Navier–Stokes equations in primitive variables are discretized in conservative form by a vertex-centred finite volume method Flux difference splitting is applied to the convective part to obtain an upwind discretization. The diffusive part is discretized in the central way. In its first-order formulation, flux difference splitting leads to a discretization of so-called vector positive type. This allows the use of classical relaxation methods in collective form. An alternating line Gauss–Seidel relaxation method is chosen here. This relaxation method is used as a smoother in a multigrid method. The components of this multigrid method are: full approximation scheme with F-cycles, bilinear prolongation, full weighting for residual restriction and injection of grid functions. Higher-order accuracy is achieved by the flux extrapolation method. In this approach the first-order convective fluxes are modified by adding second-order corrections involving flux limiting. Here the simple MinMod limiter is chosen. In the multigrid formulation the second-order discrete system is solved by defect correction. Computational results are shown for the well known GAMM backward-facing step problem and for a channel with a half-circular obstruction.     

Linear and non-linear iterative methods for the incompressible Navier-Stokes equations

In this study, the discretized finite volume form of the two-dimensional, incompressible Navier-Stokes equations is solved using both a frozen coefficient and a full Newton non-linear iteration. The optimal method is a combination of these two techniques. The linearized equations are solved using a conjugate-gradient-like method (CGSTAB). Various types of preconditioning are developed. Completely general sparse matrix methods are used. Investigations are carried out to determine the effect of finite volume cell anisotropy on the preconditioner. Numerical results are given for several test problems.     

A CELL VERTEX ALGORITHM FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS ON NON-ORTHOGONAL GRIDS

The steady, incompressible Navier–Stokes (N–S) equations are discretized using a cell vertex, finite volume method. Quadrilateral and hexahedral meshes are used to represent two- and three-dimensional geometries respectively. The dependent variables include the Cartesian components of velocity and pressure. Advective fluxes are calculated using bounded, high-resolution schemes with a deferred correction procedure to maintain a compact stencil. This treatment insures bounded, non-oscillatory solutions while maintaining low numerical diffusion. The mass and momentum equations are solved with the projection method on a non-staggered grid. The coupling of the pressure and velocity fields is achieved using the Rhie and Chow interpolation scheme modified to provide solutions independent of time steps or relaxation factors. An algebraic multigrid solver is used for the solution of the implicit, linearized equations. A number of test cases are anlaysed and presented. The standard benchmark cases include a lid-driven cavity, flow through a gradual expansion and laminar flow in a three-dimensional curved duct. Predictions are compared with data, results of other workers and with predictions from a structured, cell-centred, control volume algorithm whenever applicable. Sensitivity of results to the advection differencing scheme is investigated by applying a number of higher-order flux limiters: the MINMOD, MUSCL, OSHER, CLAM and SMART schemes. As expected, studies indicate that higher-order schemes largely mitigate the diffusion effects of first-order schemes but also shown no clear preference among the higher-order schemes themselves with respect to accuracy. The effect of the deferred correction procedure on global convergence is discussed.     

Calculations of stratified wavy two-phase flow in pipes

Calculations of fully developed, stratified wavy gas–liquid pipe flow is presented. The wavy interface is represented by an equivalent interfacial roughness obtained from experimental data, which is made non-dimensional following the Charnock formulation. The two-dimensional, steady-state axial momentum equation is solved together with a two-layer turbulence model, which is modified to account for the roughness introduced at the interface. The governing equations are discretized using a finite difference method on a composite, overlapping grid with local grid refinement near the interface and the wall. The immersed interface method is used to make the numerical scheme well-defined across the interface, and a level set function is used to represent the interface. Numerical calculations are found to compare satisfactorily with experimental data.     

Three-dimensional numerical predictions of internally heated free convective flows

Steady laminar free convection in cylindrical tanks containing high Prandtl number fluids, heated with localized point or line heat sources, is simulated in three dimensions. The governing system of equations in primitive variables, simplified with the Boussinesq approximation is solved using a segregated numerical formulation with skewed time-like marching procedure. The discretized pressure correction equation, which links the continuity and momentum equations is solved using a multigrid method. Flow and temperature fields are predicted for a variety of heat source strengths, lengths and locations and heat transfer coefficients at the convective boundaries. The effects of these variables on the thermal and hydrodynamic conditions in the tank are presented and analysed.     

水下爆炸非均熵二维定常流的三族特征线解法

针对二维定常可压缩超声速非等熵柱状流,提出一种特征线差分解法,通过在沿马赫线的相容方程中添加沿流线的熵变项以描述非等熵效应,得到等熵流和非等熵流均适用的三族特征线方程组。根据水下爆炸近场特点,建立无限长柱状装药的定常模型,将三族特征线方程组用有限差分法离散求解,通过构造合适的网格保证计算格式可以数值上收敛,由此编制程序并计算几种柱状炸药的水下爆炸近场冲击波。对比有限元模拟结果和实验结果发现,特征线差分法可以比较准确地捕捉冲击波形状并计算冲击波后流场,从而验证了所提出的三族特征线差分法的准确性。     

模拟相变问题的精细积分边界元法

将精细积分边界元法和界面追踪法相结合求解相变问题。因为边界元法只需要将待求解空间域的边界离散,方便连续追踪移动界面位置和重构网格,所以边界元法适合应用于移动边界问题的模拟。首先,利用精细积分边界元法在固相区域和液相区域分别求解相应的瞬态热传导控制方程,从而求得温度场和边界热流密度。然后,根据固-液相变界面上的能量平衡方程,利用热流密度求得相变界面的移动速度,再采用界面追踪法预测移动相变界面的位置变化。最后,给出了几个数值算例,并通过与参考解的对比验证本文方法的准确性。     

Depth‐averaged two‐dimensional curvilinear explicit finite analytic model for open‐channel flows

A depth‐averaged two‐dimensional model has been developed in the curvilinear co‐ordinate system for free‐surface flow problems. The non‐linear convective terms of the momentum equations are discretized based on the explicit–finite–analytic method with second‐order accuracy in space and first‐order accuracy in time. The other terms of the momentum equations, as well as the mass conservation equation, are discretized by the finite difference method. The discretized governing equations are solved in turn, and iteration in each time step is adopted to guarantee the numerical convergence. The new model has been applied to various flow situations, even for the cases with the presence of sub‐critical and supercritical flows simultaneously or sequentially. Comparisons between the numerical results and the experimental data show that the proposed model is robust with satisfactory accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips

In this paper, the Generalized Differential Quadrature (GDQ) method is used to obtain bending solution of moderately thick rectangular plates. The plate is resting on two-parameter elastic (Pasternak) foundation or strips with a finite width. Various combinations of clamped, simply supported and free boundary conditions are considered. According to the first-order shear deformation theory, the governing equations of the problem consist of three second-order partial differential equations (PDEs) in terms of displacement and rotations of the plate. The governing equations and solution domain is discretized based on the GDQ method. It is demonstrated that the method converges rapidly while providing accurate results with relatively small number of grid points. Accuracy of the results is examined using available data in the literature for Pasternak foundation. Furthermore, due to lack of data for Pasternak strips, all predictions are verified by finite element analysis which can be used as benchmark in future studies.     

Numerical scheme for riser motion calculation during 3-D VIV simulation

This paper presents a numerical scheme for riser motion calculation and its application to riser VIV simulations. The discretisation of the governing differential equation is studied first. The top tensioned risers are simplified as tensioned beams. A centered space and forward time finite difference scheme is derived from the governing equations of motion. Then an implicit method is adopted for better numerical stability. The method meets von Neumann criteria and is shown to be unconditionally stable. The discretized linear algebraic equations are solved using a LU decomposition method. This approach is then applied to a series of benchmark cases with known solutions. The comparisons show good agreement. Finally the method is applied to practical riser VIV simulations. The studied cases cover a wide range of riser VIV problems, i.e. different riser outer diameter, length, tensioning conditions, and current profiles. Reasonable agreement is obtained between the numerical simulations and experimental data on riser motions and cross-flow VIV a/D. These validations and comparisons confirm that the present numerical scheme for riser motion calculation is valid and effective for long riser VIV simulation.     

Oscillations in high‐order finite difference solutions of stiff problems on non‐uniform grids

This work investigates the mitigation and elimination of scheme‐related oscillations generated in compact and classical fourth‐order finite difference solutions of stiff problems, represented here by the Burgers and Reynolds equations. The regions where severe gradients are anticipated are refined by the use of subdomains where the grid is distributed according to a geometric progression. It is observed that, for multi‐domain solutions, both the classical and compact fourth‐order finite difference schemes can exhibit spurious oscillations. When present, the oscillations are initially generated around the interface between the uniform and non‐uniform grid subdomains. Based on a thorough study of the grid distribution effects, it is shown that the numerical oscillations are caused by inadequate geometric progression ratios within the non‐uniformly discretized subdomains. Indeed, accurate solutions are obtainable if and only if the grid ratios in the non‐uniform subdomains are greater than a critical threshold ratio. It is concluded that high‐order classical and compact schemes can be used with confidence to efficiently solve one‐ or two‐dimensional problems whose solutions exhibit sharp gradients in very thin regions, provided that the numerically generated oscillations are eliminated by an appropriate choice of grid distribution within the non‐uniformly discretized subdomains. Copyright © 1999 John Wiley & Sons, Ltd.     

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

In large‐scale shallow flow simulations, local high‐resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non‐uniform Cartesian grid system as an alternative to the existing approaches to provide local high‐resolution mesh. On generating a structured but non‐uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second‐order finite volume Godunov‐type scheme in a similar way as that on a uniform grid. Copyright © 2010 John Wiley & Sons, Ltd.     

基于边界元法的非均质薄板弯曲问题的解

本文用边界元法研究非均质无限域弹性薄板弯曲问题.在数值实施过程中,对于夹杂和基体分别形成边界积分方程.通过离散边界积分方程,得到相应的方程组,然后结合界面条件,最终获得问题的求解方程组.在界面的相关量求得之后,可以根据需要来求解基体和夹杂中的有关位置的弯矩.数值结果与已有的解做了对比.