Efficient computation of two‐dimensional steady free‐surface flows

We consider a family of steady free‐surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally used for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian‐free Newton‐Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface‐piercing object and bottom topographies.     

Axisymmetric boundary integral formulation for a two‐fluid system

A 3D axisymmetric Galerkin boundary integral formulation for potential flow is employed to model two fluids of different densities, one fluid enclosed inside the other. The interface variables are the velocity potential and the normal velocity, and they can be solved for separately, the second linear system being symmetric. The algorithm is validated by comparing with the analytic solutions for a static interior spherical drop over a range of values for the relative densities of exterior and interior fluids and various boundary conditions. For time‐dependent simulations utilizing a level set method for the interface tracking, the accuracy has been checked by comparing against the known oscillation frequency of the sphere. Pinch‐off profiles corresponding to an initial two‐lobe geometry drop and D = 6 are also presented. Published in 2011 by John Wiley & Sons, Ltd.     

A promising boundary element formulation for three‐dimensional viscous flow

In this paper, a new set of boundary‐domain integral equations is derived from the continuity and momentum equations for three‐dimensional viscous flows. The primary variables involved in these integral equations are velocity, traction, and pressure. The final system of equations entering the iteration procedure only involves velocities and tractions as unknowns. In the use of the continuity equation, a complex‐variable technique is used to compute the divergence of velocity for internal points, while the traction‐recovery method is adopted for boundary points. Although the derived equations are valid for steady, unsteady, compressible, and incompressible problems, the numerical implementation is only focused on steady incompressible flows. Two commonly cited numerical examples and one practical pipe flow problem are presented to validate the derived equations. Copyright © 2004 John Wiley & Sons, Ltd.     

Simulation of three‐dimensional bubbles using desingularized boundary integral method

A very simple model based on the three‐dimensional desingularized boundary integral method is applied to study the evolution of bubble(s) with or without the presence of solid structures. The choice of the desingularization parameters, which is crucial to the success of the method, is studied in the context of bubble dynamics. With the proper choice of parameters, the new model is far more efficient than previous models with virtually the same level of accuracy being achieved. This is largely attributed to the simplicity of the desingularization method. Furthermore, the new model offers a simple and attractive way for mesh refinement. Although it has limitations in the sense that, with this model the time stepping tends to slow down as two surfaces approach each other, this can be easily rectified by switching over to a direct method so that the two surfaces can be drawn closer as required in the context of jet impact. After this the new model can be reinstated to treat the complicated doubly connected geometry involving toroidal bubbles that would otherwise be very difficult to deal with. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical study of hysteresis in annular swirling jets with a stepped‐conical nozzle

This study investigates the experimentally observed hysteresis in the mean flow field of an annular swirling jet with a stepped‐conical nozzle. The flow is simulated using the Reynolds‐averaged Navier–Stokes (RANS) approach for incompressible flow with a k–ε and a Reynolds stress transport (RSTM) turbulence model. Four different flow structures are observed depending on the swirl number: ‘closed jet flow’, ‘open jet flow low swirl’, ‘open jet flow high swirl’ and ‘coanda jet flow’. These flow patterns change with varying swirl number and hysteresis at low and intermediate swirl numbers is revealed when increasing and subsequently decreasing the swirl. The influence of the inlet velocity profile on the transitional swirl numbers is investigated. When comparing computational fluid dynamics with experiments, the results show that both turbulence models predict the four different flow structures and the associated hysteresis and multiple solutions at low and intermediate swirl numbers. Therefore, a good agreement exists between experiments and numerics. Copyright © 2006 John Wiley & Sons, Ltd.     

Riemann solvers and boundary conditions for two‐dimensional shallow water simulations

Most existing algorithms for two‐dimensional shallow water simulations treat multi‐dimensional waves using wave splitting or time splitting. This often results in anisotropy of the computed flow. Both wave splitting and time splitting are based on a local decomposition of the multi‐dimensional problem into one‐dimensional, orthogonal problems. Therefore, these algorithms handle boundary conditions in a very similar way to classical one‐dimensional algorithms. This should be expected to trigger a dependence of the number of boundary conditions on the direction of the flow at the boundaries. However, most computational codes based on alternate directions do not exhibit such sensitivity, which seems to contradict the theory of existence and uniqueness of the solution. The present paper addresses these issues. A Riemann solver is presented that aims to convert two‐dimensional Riemann problems into a one‐dimensional equivalent Riemann problem (ERP) at the interfaces between the computational cells. The ERP is derived by applying the theory of bicharacteristics at each end of the interface and by performing a linear averaging along the interface. The proposed approach is tested against the traditional one‐dimensional approach on the classical circular dambreak problem. The results show that the proposed solver allows the isotropy of the solution to be better preserved. Use of the two‐dimensional solver with a first‐order scheme may give better results than use of a second‐order scheme with a one‐dimensional solver. The theory of bicharacteristics is also used to discuss the issue of boundary conditions. It is shown that, when the flow is subcritical, the number of boundary conditions affects the accuracy of the solution, but not its existence and uniqueness. When only one boundary condition is to be prescribed, it should not be the velocity in the direction parallel to the boundary. When two boundary conditions are to be prescribed, at least one of them should involve the component of the velocity in the direction parallel to the boundary. Copyright © 2003 John Wiley & Sons, Ltd.     

A new mesh relaxation approach and automatic time‐step control method for boundary integral simulations of a viscous drop

We present a numerical methodology for the simulation of a viscous drop under simple shear flows by using the boundary integral method. The present work treats only a single drop in an unbounded fluid‐flow, but the results can be directly applied to studies on the rheology of dilute emulsions, in which the hydrodynamic interactions between two or more drops can be neglected. Singular and non‐singular integral representations of the velocity field are considered. Several aspects of the method are presented, including a new mesh relaxation approach and an automatic time‐step control method. The relaxation strategy is used in order to contain the distortion of the mesh and is performed by using relaxation iterations in a virtual temporal march between each physical time step of the simulation and monitoring the standard deviation of the areas of the elements. The automatic time‐step control method uses a global quantity related to the drop deformation in order to automatically set the temporal integration time step. It is carried out in a way to keep the local integration error less than a given tolerance. This strategy reduces the computational cost of the simulation by dramatically reducing the number of time steps in the temporal integration process. Copyright © 2016 John Wiley & Sons, Ltd.     

A Lagrangian boundary element approach to transient three‐dimensional free surface flow in thin cavities

The lubrication theory is extended for transient free‐surface flow of a viscous fluid inside a three‐dimensional thin cavity. The problem is closely related to the filling stage during the injection molding process. The pressure, which in this case is governed by the Laplace's equation, is determined using the boundary element method. A fully Lagrangian approach is implemented for the tracking of the evolving free surface. The domain of computation is the projection of the physical domain onto the (x, y) plane. This approach is valid for simple and complex cavities as illustrated for the cases of a flat plate and a curved plate. It is found that the flow behavior is strongly influenced by the shape of the initial fluid domain, the shape of the cavity, and inlet flow pressure. Copyright © 2001 John Wiley & Sons, Ltd.     

A weakly compressible MPS method for modeling of open‐boundary free‐surface flow

A mesh‐free particle method, based on the moving particle semi‐implicit (MPS) interaction model, has been developed for the simulation of two‐dimensional open‐boundary free‐surface flows. The incompressibility model in the original MPS has been replaced with a weakly incompressible model. The effect of this replacement on the efficiency and accuracy of the model has been investigated. The new inflow–outflow boundary conditions along with the particle recycling strategy proposed in this study extend the application of the model to open‐boundary problems. The final model is able to simulate open‐boundary free surface flow in cases of large deformation and fragmentation of free surface. The models and proposed algorithms have been validated and applied to sample problems. The results confirm the model's efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

An approach of dynamic mesh adaptation for simulating 3‐dimensional unsteady moving‐immersed‐boundary flows

In this paper, we present an approach of dynamic mesh adaptation for simulating complex 3‐dimensional incompressible moving‐boundary flows by immersed boundary methods. Tetrahedral meshes are adapted by a hierarchical refining/coarsening algorithm. Regular refinement is accomplished by dividing 1 tetrahedron into 8 subcells, and irregular refinement is only for eliminating the hanging points. Merging the 8 subcells obtained by regular refinement, the mesh is coarsened. With hierarchical refining/coarsening, mesh adaptivity can be achieved by adjusting the mesh only 1 time for each adaptation period. The level difference between 2 neighboring cells never exceeds 1, and the geometrical quality of mesh does not degrade as the level of adaptive mesh increases. A predictor‐corrector scheme is introduced to eliminate the phase lag between adapted mesh and unsteady solution. The error caused by each solution transferring from the old mesh to the new adapted one is small because most of the nodes on the 2 meshes are coincident. An immersed boundary method named local domain‐free discretization is employed to solve the flow equations. Several numerical experiments have been conducted for 3‐dimensional incompressible moving‐boundary flows. By using the present approach, the number of mesh nodes is reduced greatly while the accuracy of solution can be preserved.     

A low‐dimensional spectral approach for transient axisymmetric free‐surface flow inside thin cavities of arbitrary shape

A low‐dimensional spectral method is used to solve the transient axisymmetric free surface flow inside thin cavities of arbitrary shape. The flow field is obtained on the basis of the lubrication equations, which are expanded in terms of orthonormal functions over the cavity gap. The formulation accounts for nonlinearities stemming from inertia and front location. The work is of close relevance to the filling stage during die casting, and injection molding, or the flow inside annular (extrusion) dies. Both flows under an imposed flow rate, and an imposed pressure at the cavity entrance are examined. The influence of inertia, aspect ratio, gravity, and wall geometry on the evolution of the front, flow rate, and pressure is assessed particularly in the early stage of flow, when a temporal behavior of the ‘boundary‐layer’ type develops. The multiple‐scale method is applied to obtain an approximate solution at small Reynolds number, Re. Comparison with the exact (numerical) solution indicates a wide range of validity for the multiple‐scale approach, including the moderately small Re range. Copyright © 2002 John Wiley & Sons, Ltd.     

Three‐dimensional immersed boundary conditions for moving solids in the lattice‐Boltzmann method

This paper establishes the range of validity for a previously published three‐dimensional moving solid boundary condition for the lattice‐Boltzmann method. This method was reasonably formulated from a mass and momentum balance perspective, but was only verified for a small range of (primarily two‐dimensional) problems. One of the advantages of this boundary condition is that it offers resolution at the sub‐grid scale, allowing for accurate and stable calculation of the force and torque for solids which are moving through a lattice, even for small solid sizes relative to the computational grid size. We verify the boundary condition for creeping flows by comparison to analytical solutions that include both the force and the torque on fixed and moving spheres, and then follow this with comparisons to experimental and empirical results for both fixed as well moving spheres in inertial flows. Finally, we compare simulation results to numerical results of other investigators for the settling of an offset sphere and the drafting–kissing–tumbling of two sedimenting spheres. We found that an accurate calculation of the collision‐operator weighting used to obtain sub‐grid‐scale resolution was necessary in order to prevent spikes in the velocities, forces, and moments when solid objects cross‐computational cells. The wide range of comparisons collected and presented in this paper can be used to establish the validity of other numerical models, in addition to the one examined here. Published in 2007 by John Wiley & Sons, Ltd.     

The improved space–time conservation element and solution element scheme for two‐dimensional dam‐break flow simulation

A new numerical scheme, namely space–time conservation element and solution element (CE/SE) method, has been used for the solution of the two‐dimensional (2D) dam‐break problem. Distinguishing from the well‐established traditional numerical methods (such as characteristics, finite difference, finite element, and finite‐volume methods), the CE/SE scheme has many non‐traditional features in both concept and methodology: space and time are treated in a unified way, which is the most important characteristic for the CE/SE method; the CEs and SEs are introduced, both local and global flux conservations in space and time rather than space only are enforced; an explicit scheme with a stagger grid is adopted. Furthermore, this scheme is robust and easy to implement. In this paper, an improved CE/SE scheme is extended to solve the 2D shallow water equations with the source terms, which usually plays a critical role in dam‐break flows. To demonstrate the accuracy, robustness and efficiency of the improved CE/SE method, both 1D and 2D dam‐break problems are simulated numerically, and the results are consistent with either the analytical solutions or experimental results. Copyright © 2011 John Wiley & Sons, Ltd.     

Shape reconstruction of an inverse boundary value problem of two‐dimensional Navier–Stokes equations

This paper is concerned with the problem of the shape reconstruction of two‐dimensional flows governed by the Navier–Stokes equations. Our objective is to derive a regularized Gauss–Newton method using the corresponding operator equation in which the unknown is the geometric domain. The theoretical foundation for the Gauss–Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the boundary curve in the sense of a domain derivative. The numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. Copyright © 2009 John Wiley & Sons, Ltd.     

A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom

An accurate three‐dimensional numerical model, applicable to strongly non‐linear waves, is proposed. The model solves fully non‐linear potential flow equations with a free surface using a higher‐order three‐dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second‐order explicit Taylor series expansions with adaptive time steps. The model is applicable to non‐linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16‐node cubic ‘sliding’ quadrilateral elements, providing local inter‐element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well‐posed in all cases of mixed boundary conditions. Higher‐order tangential derivatives, required for the time updating, are calculated in a local curvilinear co‐ordinate system, using 25‐node ‘sliding’ fourth‐order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio‐temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two‐dimensional solution proposed earlier. Finally, three‐dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright © 2001 John Wiley & Sons, Ltd.     

Fitted finite element discretization of two‐phase Stokes flow

We propose a novel fitted finite element method for two‐phase Stokes flow problems that uses piecewise linear finite elements to approximate the moving interface. The method can be shown to be unconditionally stable. Moreover, spherical stationary solutions are captured exactly by the numerical approximation. In addition, the meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations, a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments for our numerical method, which demonstrate the accuracy and robustness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Finite‐element/level‐set/operator‐splitting (FELSOS) approach for computing two‐fluid unsteady flows with free moving interfaces

The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two‐fluid interfacial flows, having in mind possible interface topology changes (like merger or break‐up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator‐splitting for temporal discretization and the level‐set method for interface representation. We show that the finite element implementation of the level‐set approach brings some additional benefits as compared to the standard, finite difference level‐set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second‐order accuracy of the interface normal, curvature and mass conservation. The operator‐splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal‐order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh–Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of liquid crystalline polymer flow into two‐dimensional thin cavity moulds

In this paper, flows of liquid crystalline polymers into two‐dimensional thin cavity moulds are simulated. The flows are modelled by Ericksen–Leslie equations of motion in the high viscosity limit. An elliptic pressure equation is derived under Hele–Shaw approximations, and the non‐isothermal natures of the flow are modelled. The equations are solved using the finite‐difference technique. A new boundary‐mapping technique is developed in this study to solve the difficulty in the finite‐difference treatment of arbitrarily shaped boundaries, which possess no natural coordinate system. This new method avoids the difficult mesh control in the body‐fitted mapping process and makes the mapping process easy to implement. It can also solve the problems caused by the uneven distribution of grid nodes in the traditional body‐fitted mapping technique. Copyright © 2008 John Wiley & Sons, Ltd.     

An adaptive Lagrangian boundary element approach for three‐dimensional transient free‐surface Stokes flow as applied to extrusion,thermoforming, and rheometry

An adaptive (Lagrangian) boundary element approach is proposed for the general three‐dimensional simulation of confined free‐surface Stokes flow. The method is stable as it includes remeshing capabilities of the deforming free surface and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free‐surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several flow problems are presented to illustrate the utility of the approach, particularly as encountered in polymer processing and rheology. These problems illustrate the transient nature of the flow during the processes of extrusion and thermoforming, the elongation of a fluid sample in an extensional rheometer, and the coating of a sphere. Surface tension effects are also explored. Copyright © 2001 John Wiley & Sons, Ltd.