Numerical simulations of non-stationary flows of non-Newtonian fluids between concentric and eccentric cylinders by stream-tube method and domain decomposition

This study involves a theoretical formulation of the stream-tube method in non-stationary flows. Initially, this approach allowed flow computations by determining an unknown transformation between the physical domain and a mapped domain where the streamlines are rectilinear and parallel. To take into account vortex zones, we define local transformations of subregions of the physical domain that are mapped into rectangular domains where the transformed streamlines are still parallel and straight. The local functions must be determined numerically from the governing equations and boundary conditions put together with compatibility equations. The method enables to compute streamlines and flow data at every time, using distinguishing properties, as verification of mass conservation and definition of rectangular meshes allowing to adopt finite-difference schemes. The numerical simulations concern different non-Newtonian fluids under various geometrical and kinematic specifications related to flows between concentric and eccentric cylinders, leading to comparisons with literature data. The results also highlight the influence of the rheological properties on the flow characteristics in unsteady conditions.     

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of three-dimensional duct flows of incompressible fluids by using the stream-tube method

Numerical simulation of three-dimensional flows generally involves solving large-scale problems. In this paper we consider the stream-tube method in three-dimensional duct flows. The analysis uses the concept of stream tubes in a mapped computational domain of the physical domain where the streamlines are parallel and straight. The incompressibility equation is automatically verified and the primary unknowns of the problem are, in addition to the pressure, the transformation functions between the two domains. It is also shown that the flow may be computed by considering successive subdomains (the stream tubes). This results in a reduction of computing time and storage area. Incompressible viscous and elastic liquids involving memory-integral equations may be considered in the flow simulations. This part of the paper examines three-dimensional flows of Newtonian fluids. The method is applied to the flow in a duct involving a threefold rotational symmetry, where the discretized relevant equations are solved by using the Levenberg-Marquardt algorithm.     

The computation of the eddy along the upper wall in the three‐dimensional flow over a backward‐facing step

A three‐dimensional laminar flow over a backward‐facing step is studied as a numerical experiment by solving the steady‐state, isothermal and incompressible Navier–Stokes equations using two different finite element codes. The Reynolds number ranges from 100 to 1050. The expansion ratio is 1:1.94, and the aspect ratio is 1:36.7. The numerical experiment reveals both eddies along the lower and upper walls downstream of the step. Results of computations regarding positions of detachment of the eddy along the upper wall and positions of reattachments of the eddies along both the lower and upper walls are tabulated along with positions and magnitudes of global extrema of shear rate within the eddies. The wall effects are shown by calculating streamlines along planes parallel/normal to the lateral walls of the domain and depicting how the streamlines are distorted close to the walls and how they assume a two‐dimensional configuration in the plane of symmetry. Comparisons are made with available numerical results and laboratory measurements. Copyright © 2011 John Wiley & Sons, Ltd.     

A parallel monolithic algorithm for the numerical simulation of large‐scale fluid structure interaction problems

A novel parallel monolithic algorithm has been developed for the numerical simulation of large‐scale fluid structure interaction problems. The governing incompressible Navier–Stokes equations for the fluid domain are discretized using the arbitrary Lagrangian–Eulerian formulation‐based side‐centered unstructured finite volume method. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant–Kirchhoff material, and the classical Galerkin finite element method is used to discretize the governing equations in a Lagrangian frame. A special attention is given to construct an algorithm with exact total fluid volume conservation while obeying both the global and the local discrete geometric conservation law. The resulting large‐scale algebraic nonlinear equations are multiplied with an upper triangular right preconditioner that results in a scaled discrete Laplacian instead of a zero block in the original system. Then, a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned sub‐domains is utilized for the modified system. The accuracy and performance of the proposed algorithm are verified for the several benchmark problems including a pressure pulse in a flexible circular tube, a flag interacting with an incompressible viscous flow, and so on. John Wiley & Sons, Ltd.     

A numerical investigation of wall effects in three‐dimensional,laminar flow over a backward facing step with a constant aspect and expansion ratio

The wall effects are investigated in the three‐dimensional laminar flow over a backward‐facing step. For this purpose, a numerical experiment is designed under actual laboratory conditions. The aspect ratio of the computational domain is 1:40 and the expansion ratio is 1:2. The Reynolds number ranges from 100 to 950. The governing equations are the steady state, isothermal and incompressible Navier–Stokes equations for Newtonian fluids. They are solved with a homemade Galerkin finite element code. The computations are validated with data from available laboratory and numerical experiments. The results focus on the variation of both velocity profiles and lengths of eddies along the lower and upper wall in the spanwise direction. Calculated streamlines in the streamwise and transverse direction show how the flow is distorted near the lateral wall and how it develops up to the plane of symmetry. The study of skin friction lines along the top and bottom wall of the domain reveals a flow that takes place in the spanwise direction. This spanwise component of the flow becomes more dominant with increasing Reynolds number and is impossible to be sustained at steady state for Reynolds numbers higher than 950 for this particular geometry. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite volume solution of the Navier–Stokes equations in velocity–vorticity formulation

For the incompressible Navier–Stokes equations, vorticity‐based formulations have many attractive features over primitive‐variable velocity–pressure formulations. However, some features interfere with the use of the numerical methods based on the vorticity formulations, one of them being the lack of a boundary conditions on vorticity. In this paper, a novel approach is presented to solve the velocity–vorticity integro‐differential formulations. The general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so‐called generalized Biot–Savart formula combined with a fast summation algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well‐known fractional step approaches are used to solve the vorticity transport equation. The paper describes in detail how we accurately impose no normal‐flow and no tangential‐flow boundary conditions. We impose a no‐flux boundary condition on solid objects by the introduction of a proper amount of vorticity at wall. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential‐flow boundary condition. As application examples, the impulsively started flows through a flat plate and a circular cylinder are computed using the method. The present results are compared with the analytical solution and other numerical results and show good agreement. Copyright © 2005 John Wiley & Sons, Ltd.     

计算离心泵叶轮流场的扫描流束有限元方法

本文提出一种求解离心式叶轮流场的数值方法，将流动求解区域离散为有限个由流线构成其边界的单元，采用伽辽金法建立的单元方程在一条流束上集合为方程组，流线上的节点坐标亦作为未知量包含在有限元方程中，通过扫描计算，逐步解得流线位置及流动参数。本文应用叶轮的通流理论流动模型，采用扫描流速有限元方法对离心泵叶轮流场进行了计算，并与有关文献作了比较。     

Explicit formulations for evaluation of velocity gradients using boundary‐domain integral equations in 2D and 3D viscous flows

In this paper, explicit boundary‐domain integral equations for evaluating velocity gradients are derived from the basic velocity integral equations. A free term is produced in the new strongly singular integral equation, which is not included in recent formulations using the complex variable differentiation method (CVDM) to compute velocity gradients (Int. J. Numer. Meth. Fluids 2004; 45 :463–484; Int. J. Numer. Meth. Fluids 2005; 47 :19–43). The strongly singular domain integrals involved in the new integral equations are accurately evaluated using the radial integration method (RIM). Considerable computational time for evaluating integrals of velocity gradients can be saved by using present formulation than using CVDM. The formulation derived in this paper together with those presented in reference (Int. J. Numer. Meth. Fluids 2004; 45 :463–484) for 2D and in (Int. J. Numer. Meth. Fluids 2005; 47 :19–43) for 3D problems constitutes a complete boundary‐domain integral equation system for solving full Navier–Stokes equations using primitive variables. Three numerical examples for steady incompressible viscous flow are given to validate the derived formulations. Copyright © 2007 John Wiley & Sons, Ltd.     

Modified method of characteristics for solving population balance equations

This paper presents a new numerical method for solving the population balance equation using the modified method of characteristics. Aggregation and break‐up are neglected but the density function variations in the three‐dimensional space and its dependence on the external fields are accounted for. The method is an interpretation of the Lagrangian approach. Based on a pre‐specified grid, it follows the particles backward in time as opposed to forward in the case of traditional method of characteristics. Unlike the direct marching method, the inverse marching method uses a fixed grid thus, making it compatible with other numerical schemes (e.g. finite‐volume, finite elements) that may be used to solve other coupled equations such as the mass, momentum, and energy conservation equations. The numerical solutions are compared with the exact analytical solutions for simple one‐dimensional flow cases. Very good agreement between the numerical and the theoretical solutions has been obtained confirming the validity of the numerical procedure and the associated computer program. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of complex flows of non-Newtonian fluids using the stream tube method and memory integral constitutive equations

In this paper a memory integral viscoelastic equation is considered for simulating complex flows of non-Newtonian fluids by stream tube analysis. A formalism is developed to take into account co-deformational memory equations in a mapped computational domain where the transformed streamlines are parallel and straight. The particle-tracking problem is avoided. Evolution in time and related kinematic quantities involved with a K-BKZ integral constitutive model are easily taken into account in evaluating the stresses. Successive subdomains, the stream tubes, may be considered for computing the main flow in abrupt axisymmetric contractions from the wall to the central flow region. The ‘peripheral stream tube’ close to the duct wall is determined by developing a non-conventional modified Hermite element. A mixed formulation is adopted and the relevant non-linear equations are solved numerically by the Levenberg-Marquardt algorithm. Although the singularity at the section of contraction is not involved explicitly, the results obtained for the peripheral stream tube clearly show the singularity effects and the extent of the recirculating zone near the salient corner. The algorithm is stable even at high flow rates and provides satisfactory solutions when compared with similar calculations in the literature.     

A least‐square weighted residual method for level set formulation

In this paper, a least‐square weighted residual method (LSWRM) for level set (LS) formulation is introduced to achieve interface capturing in two‐dimensional (2D) and three‐dimensional (3D) problems. An LSWRM was adopted for two semi‐discretized advection and reinitialization equations of the LS formulation. The present LSWRM provided good mathematical properties such as natural numerical diffusion and the symmetry of the resulting algebraic systems for the advection and reinitialization equations. The proposed method was validated by solving some 2D and 3D benchmark problems such as those involving a rotating slotted disk, the rotation of a slotted sphere, and a time‐reversed single‐vortex flow and a deformation problem of a spherical fluid. The numerical results were compared with those obtained from essentially non‐oscillatory type formulations and particle LS methods. Further, the proposed LSWRM for the LS formulation was coupled with a splitting finite element method code to solve the incompressible Navier–Stokes equations, and then, the collapse of a 3D broken dam flow was well simulated; in the simulation, the entrapping of air and the splashing of the surge front of water were reproduced. The mass conservation of the present method was found to be satisfactory during the entire simulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Comparison of wide and compact fourth‐order formulations of the Navier–Stokes equations

In this study the numerical performances of wide and compact fourth‐order formulation of the steady 2D incompressible Navier–Stokes equations will be investigated and compared with each other. The benchmark driven cavity flow problem will be solved using both wide and compact fourth‐order formulations and the numerical performances of both formulations will be presented and also the advantages and disadvantages of both formulations will be discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

A finite element method for liquid with memory

Steady, two-dimensional flows of viscoelastic liquid with memory are analyzed by means of a nonlinear integral constitutive equation, novel streamlined finite elements, Galerkin's method of weighted residuals, and Newton iteration. By their relation to streamlines the new elements allow, for the first time, full Newton iteration of the algebraic equation set to which the governing integrodifferential system reduces. Streamlines are computed simultaneously with velocity components and pressure, the primary unknowns; to track the history of liquid particles the deformation equations are solved analytically in a Protean system of coordinates that conforms to the streamlines. In the cases studied the Newton iteration converges quadratically to a creeping flow state of Weissenberg number (product of upstream wall shear rate and relaxation time of the liquid) up to about 20 in channel flow, 10 in film flow, and 2 in die-swell flow. Shear-thinning shrinks the domain of convergence; it also reduces die-swell. A slip boundary condition in the vicinity of the contact line extends the domain of convergence.     

Fictitious domain approach for numerical modelling of Navier–Stokes equations

This study investigates a fictitious domain model for the numerical solution of various incompressible viscous flows. It is based on the so‐called Navier–Stokes/Brinkman and energy equations with discontinuous coefficients all over an auxiliary embedding domain. The solid obstacles or walls are taken into account by a penalty technique. Some volumic control terms are directly introduced in the governing equations in order to prescribe immersed boundary conditions. The implicit numerical scheme, which uses an upwind finite volume method on staggered Cartesian grids, is of second‐order accuracy in time and space. A multigrid local mesh refinement is also implemented, using the multi‐level Zoom Flux Interface Correction (FIC) method, in order to increase the precision where it is needed in the domain. At each time step, some iterations of the augmented Lagrangian method combined with a preconditioned Krylov algorithm allow the divergence‐free velocity and pressure fields be solved for. The tested cases concern external steady or unsteady flows around a circular cylinder, heated or not, and the channel flow behind a backward‐facing step. The numerical results are shown in good agreement with other published numerical or experimental data. Copyright © 2000 John Wiley & Sons, Ltd.     

A parallel adaptive numerical method with generalized curvilinear coordinate transformation for compressible Navier–Stokes equations

A fourth‐order finite‐volume method for solving the Navier–Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth‐order quadrature rules for evaluating face‐averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge–Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth‐order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non‐rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of plane and axisymmetric flows of incompressible fluids with the stream tube method: Numerical simulation by trust-region optimization algorithm

New concepts for the study of incompressible plane or axisymmetric flows are analysed by the stream tube method. Flows without eddies and pure vortex flows are considered in a transformed domain where the mapped streamlines are rectilinear or circular. The transformation between the physical domain and the computational domain is an unknown of the problem. In order to solve the non-linear set of relevant equations, we present a new algorithm based on a trust region technique which is effective for non-convex optimization problems. Experimental results show that the new algorithm is more robust compared to the Newton-Raphson method.     

A CIP/multi‐moment finite volume method for shallow water equations with source terms

A novel finite volume method has been presented to solve the shallow water equations. In addition to the volume‐integrated average (VIA) for each mesh cell, the surface‐integrated average (SIA) is also treated as the model variable and is independently predicted. The numerical reconstruction is conducted based on both the VIA and the SIA. Different approaches are used to update VIA and SIA separately. The SIA is updated by a semi‐Lagrangian scheme in terms of the Riemann invariants of the shallow water equations, while the VIA is computed by a flux‐based finite volume formulation and is thus exactly conserved. Numerical oscillation can be effectively avoided through the use of a non‐oscillatory interpolation function. The numerical formulations for both SIA and VIA moments maintain exactly the balance between the fluxes and the source terms. 1D and 2D numerical formulations are validated with numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.