Finite element framework for describing dynamic wetting phenomena

The finite element simulation of dynamic wetting phenomena, requiring the computation of flow in a domain confined by intersecting a liquid–fluid free surface and a liquid–solid interface, with the three‐phase contact line moving across the solid, is considered. For this class of flows, different finite element method (FEM) implementations have been used in the literature, and in some cases, these produced apparently contradictory results. In the present paper, a robust framework for the FEM simulation of dynamic wetting flows is developed, which, by consistently adhering to the FEM methodology, leaves no room for ad hoc ‘optional’ variations in the numerical handling of these flows. The developed approach makes it possible to conduct a convergence study, assess the spatial resolution required to achieve a preset accuracy and provide the corresponding benchmark calculations. This analysis allows one to identify numerical artefacts, which had previously been interpreted as physical effects, and demonstrates that suppressing numerical errors using a ‘strong’ implementation of a boundary condition creates bigger and less detectable errors elsewhere in the computational domain. We provide practical recommendations on the spatial resolution required by a numerical scheme for a given set of non‐dimensional similarity parameters and give a user‐friendly step‐by‐step guide specifying the entire implementation, which allows the reader to easily reproduce all presented results including the benchmark calculations. It is also shown how the developed framework accommodates generalizations of the mathematical model accounting for additional physical effects, such as gradients in surface tensions. Copyright © 2011 John Wiley & Sons, Ltd.     

Convected level set method for the numerical simulation of fluid buckling

‘Fluid buckling’ is a phenomenon characterized mainly by the existence of fluid toroidal oscillations during flow. It appears when a high viscosity fluid flows vertically against a flat surface and may occur in industrial applications, as in injection molding of a propergol in complex‐shaped cavities. These coiling or folding oscillations appear during the mold filling stage, leading to air entrapment. To understand and to model this free surface flow problem, a convected level set method is proposed. First, a sinus filter is applied to the distance function to get a smooth truncation far from the interface. Second, the reinitialization is embedded in the transport equation model, avoiding it as a separate step during calculation. In order to validate the method, numerical results are presented on classical interface capturing benchmarks. Finally, results are shown on two‐dimensional and three‐dimensional viscous jet buckling problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Parallel solution of a coupled flow and transport model for shallow water

We present an integrated approach for the concurrent solution of a 3D hydrodynamical model coupled with a 3D transport model. Since both models are quite similar in nature, the same numerical method has been employed. This leads to a code that is more efficient than when two existing codes would have been combined. Discretization of the spatial differential operators, and the boundary conditions, results in a stiff initial value problem. To cope with the stiffness, we select an implicit time‐integration formula, viz. the second‐order, L‐stable BDF method because of its excellent stability properties. To reduce the huge amount of linear algebra involved in solving the implicit relations, an Approximate Factorization technique has been used. Essentially, this technique replaces a ‘multi‐dimensional’ system by a series of ‘one‐dimensional’ systems. Since the output of the hydrodynamical model (i.e., the flow field) serves as input for the transport model, we solve the hydrodynamical model one time step ahead in time. This allows us to solve the models in parallel, using two different groups of processors. By a little tuning of the parameters in the algorithm, a load‐balancing has been obtained that is close to optimal. As a result, both models require roughly the same amount of CPU time, so that one of them, effectively, can be considered as obtained ‘for free’. Copyright © 2002 John Wiley & Sons, Ltd.     

A physically based flux limiter for QUICK calculations of advective scalar transport

Transient, advective transport of a contaminant into a clean domain will exhibit a moving sharp front that separates contaminated and clean regions. Due to ‘numerical diffusion’—the combined effects of ‘cross‐wind diffusion’ and ‘artificial dispersion’—a numerical solution based on a first‐order (upwind) treatment will smear out the sharp front. The use of higher‐order schemes, e.g. QUICK (quadratic upwinding) reduces the smearing but can introduce non‐physical oscillations in the solution. A common approach to reduce numerical diffusion without oscillations is to use a scheme that blends low‐order and high‐order approximations of the advective transport. Typically, the blending is based on a parameter that measures the local monotonicity in the predicted scalar field. In this paper, an alternative approach is proposed for use in scalar transport problems where physical bounds C_Low?C?C_High on the scalar are known a priori. For this class of problems, the proposed scheme switches from a QUICK approximation to an upwind approximation whenever the predicted upwind nodal value falls outside of the physical range [C_Low, C_High]. On two‐dimensional steady‐state and one‐dimensional transient test problems predictions obtained with the proposed scheme are essentially indistinguishable from those obtained with monotonic flux‐limiter schemes. An analysis of the modified equation explains the observed performance of first‐ and second‐order time‐stepping schemes in predicting the advective transport of a step. In application to the transient two‐dimensional problem of contaminate transport into a streambed, predictions obtained with the proposed flux‐limiter scheme agree with those obtained with a scheme from the literature. Copyright © 2007 John Wiley & Sons, Ltd.     

The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models

A new scheme for differentiating complex mesh‐based numerical models (e.g. finite element models), the Independent Set Perturbation Adjoint method (ISP‐Adjoint), is presented. Differentiation of the matrices and source terms making up the discrete forward model is realized by a graph coloring approach (forming independent sets of variables) combined with a perturbation method to obtain gradients in numerical discretizations. This information is then convolved with the ‘mathematical adjoint’, which uses the transpose matrix of the discrete forward model. The adjoint code is simple to implement even with complex governing equations, discretization methods and non‐linear parameterizations. Importantly, the adjoint code is independent of the implementation of the forward code. This greatly reduces the effort required to implement the adjoint model and maintain it as the forward model continues to be developed; as compared with more traditional approaches such as applying automatic differentiation tools. The approach can be readily extended to reduced‐order models. The method is applied to a one‐dimensional Burgers' equation problem, with a highly non‐linear high‐resolution discretization method, and to a two‐dimensional, non‐linear, reduced‐order model of an idealized ocean gyre. Copyright © 2010 John Wiley & Sons, Ltd.     

A level set approach for computing solutions to inviscid compressible flow with moving solid boundary using fixed Cartesian grids

A level set approach for computing solutions to inviscid compressible flow with moving solid surface is presented. The solid surface is considered to be sharp and is described as the zero level set of a smooth explicit function of space and time. The finite volume TVD–MacCormack's two‐step procedure is used. The boundary conditions on the solid surface are easily implemented by defining the smooth level set function. The present treatment of the level set method allows the handling of fluid flows in the presence of irregularly shaped solid boundaries, escaping from the bookkeeping complexity in the so‐called ‘surface‐tracking’ method. Using the proposed numerical techniques, a two‐dimensional numerical simulation is made to investigate the aerodynamic phenomena induced by two high‐speed trains passing by each other in a tunnel. Copyright © 2001 John Wiley & Sons, Ltd.     

A finite element model for the simulation of lost foam casting

In this paper, we present a numerical model to simulate the lost foam casting process. We introduce this particular casting first in order to capture the different physical processes in play during a casting. We briefly comment on the possible physical and numerical models used to envisage the numerical simulation. Next we present a model which aims to solve ‘part of’ the complexities of the casting, together with a simple energy budget that enables us to obtain an equation for the velocity of the metal front advance. Once the physical model is established we develop a finite element method to solve the governing equations. The numerical and physical methodologies are then validated through the solution of a two‐ and a three‐dimensional example. Finally, we discuss briefly some possible improvements of the numerical model in order to capture more physical phenomena. Copyright © 2004 John Wiley & Sons, Ltd.     

Two-state dynamic gain scheduling control applied to an F16 aircraft model

The feasibility and benefits of applying a novel multi-variable dynamic gain scheduling (DGS) approach to a complex ‘industry-scale’ aircraft model are investigated; the latter model being a non-linear representation of the intrinsically unstable F16 aircraft incorporating detailed aerodynamic data. DGS is a novel control approach, which involves scheduling controller gains with one (or more) of the system states whilst accounting for the ‘hidden coupling terms’ ensuring a near-ideal response. It is effective for non-linear systems exhibiting rapid dynamic changes between operating points. Recently, this approach has been extended to a multi-variable and multi-input context. Hence, unlike previous DGS work on realistic aircraft models, relevant feedback gains are to be scheduled with all (i.e. two) state variables in order to demonstrate the ability of the approach to compensate for non-linearity during rapid manoeuvres and consequently achieving better flying qualities over a range of conditions than standard gain scheduling. Time history simulations are used to draw comparisons with the more traditional ‘static’ gain scheduling and input gain scheduling methods.     

Fifth‐order Hermitian schemes for computational linear aeroacoustics

We develop a class of fifth‐order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth‐order accuracy. A one‐dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy–Kovaleskaya (the ‘Δ‐P5 scheme’); the second one relies on a semi‐discrete form and evolves in time the discrete unknowns by using a five‐stage Runge–Kutta method (the ‘RGK‐P5 scheme’). Although the RGK‐P5 scheme shares the same local spatial interpolator with the Δ‐P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi‐dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210 :133–170; J. Comput. Phys. 2006; 217 :530–562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the ‘reconstruction stage’, we define a fifth‐order local spatial interpolator based on an upwind stencil. In the ‘decomposition stage’, we decompose the time derivatives into simple wave contributions. In the ‘evolution stage’, we use these fluctuations to update either by a Cauchy–Kovaleskaya procedure or by a five‐stage Runge–Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the ‘evolution stage’, two fifth‐order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK‐P5 appears as the best compromise between simplicity and accuracy, while the Δ‐P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical modelling of 3D stratified free surface flows: a case study of sediment dumping

A three‐dimensional numerical model has been developed to simulate stratified flows with free surfaces. The model is based on the Reynolds‐averaged Navier–Stokes (RANS) equations with variable fluid density. The equations are solved in a transformed σ‐coordinate system with the use of operator‐splitting method (Int. J. Numer. Meth. Fluids 2002; 38 :1045–1068). The numerical model is validated against the one‐dimensional diffusion problem and the two‐dimensional density‐gradient flow. Excellent agreements are obtained between numerical results and analytical solutions. The model is then used to study transport phenomena of dumped sediments into a water body, which has been modelled as a strongly stratified flow. For the two‐dimensional problem, the numerical results compare well with experimental data in terms of mean particle falling velocity and spreading rate of the sediment cloud for both coarse and medium‐size sediments. The model is also employed to study the dumping of sediments in a three‐dimensional environment with the presence of free surface. It is found that during the descending process an annulus‐like cloud is formed for fine sediments whereas a plate‐like cloud for medium‐size sediments. The model is proven to be a good tool to simulate strongly stratified free surface flows. Copyright © 2005 John Wiley & Sons, Ltd.     

A multidimensional HLL‐Riemann solver for non‐linear hyperbolic systems

This article presents a numerical model that enables to solve on unstructured triangular meshes and with a high order of accuracy, Riemann problems that appear when solving hyperbolic systems. For this purpose, we use a MUSCL‐like procedure in a ‘cell‐vertex’ finite‐volume framework. In the first part of this procedure, we devise a four‐state bi‐dimensional HLL solver (HLL‐2D). This solver is based upon the Riemann problem generated at the barycenter of a triangular cell, from the surrounding cell‐averages. A new three‐wave model makes it possible to solve this problem, approximately. A first‐order version of the bi‐dimensional Riemann solver is then generated for discretizing the full compressible Euler equations. In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third‐order accuracy. The resulting over determined system is solved by using a least‐square methodology. To enforce monotonicity conditions into the polynomial interpolation, we use and adapt the monotonicity‐preserving limiter, initially devised by Barth (AIAA Paper 90‐0013, 1990). Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model. Copyright © 2010 John Wiley & Sons, Ltd.     

A meshless method for numerical simulation of depth‐averaged turbulence flows using a k‐ϵ model

A three‐dimensional numerical model is developed to analyze free surface flows and water impact problems. The flow of an incompressible viscous fluid is solved using the unsteady Navier–Stokes equations. Pseudo‐time derivatives are introduced into the equations to improve computational efficiency. The interface between the two phases is tracked using a volume‐of‐fluid interface tracking algorithm developed in a generalized curvilinear coordinate system. The accuracy of the volume‐of‐fluid method is first evaluated by the multiple numerical benchmark tests, including two‐dimensional and three‐dimensional deformation cases on curvilinear grids. The performance and capability of the numerical model for water impact problems are demonstrated by simulations of water entries of the free‐falling hemisphere and cone, based on comparisons of water impact loadings, velocities, and penetrations of the body with experimental data. For further validation, computations of the dam‐break flows are presented, based on an analysis of the wave front propagation, water level, and the dynamic pressure impact of the waves on the downstream walls, on a specific container, and on a tall structure. Extensive comparisons between the obtained solutions, the experimental data, and the results of other numerical simulations in the literature are presented and show a good agreement. Copyright © 2015 John Wiley & Sons, Ltd.     

Near‐wall profiles of mean flow and turbulence quantities predicted by eddy‐viscosity turbulence models

This paper presents for the simple flow over a flat plate the near‐wall profiles of mean flow and turbulence quantities determined with seven eddy‐viscosity turbulence models: the one‐equation turbulence models of Menter and Spalart & Allmaras; the k‐ω two‐equation model proposed by Wilcox and its TNT, BSL and SST variants and the $k-\sqrt{k}L$ two‐equation model. The results are obtained at several Reynolds numbers ranging from 10⁷ to 2.5 × 10⁹. Sets of nine geometrically similar Cartesian grids are adopted to demonstrate that the numerical uncertainty of the finest grid predictions is negligible. The profiles obtained numerically have relevance for the application of so‐called ‘wall function’ boundary conditions. Such wall functions refer to assumptions about the flow in the viscous sublayer and the ‘log law’ region. It turns out that these assumptions are not always satisfied by our results, which are obtained by computing the flow with full near‐wall resolution. In particular, the solution in the ‘log‐law’ region is dependent on the turbulence model and on the Reynolds number, which is a disconcerting result for those who apply wall functions. Copyright © 2009 John Wiley & Sons, Ltd.     

A necklace model for vesicles simulations in 2D

The aim of this paper is to propose a new numerical model to simulate 2D vesicles interacting with a Newtonian fluid. The inextensible membrane is modeled by a chain of circular rigid particles, which are maintained in cohesion by using two different types of forces. First, a spring force is imposed between neighboring particles in the chain. Second, in order to model the bending of the membrane, each triplet of successive particles is submitted to an angular force. Numerical simulations of vesicles in shear flow have been run using FEM and the FreeFem++ software. Exploring different ratios of inner and outer viscosities, we recover the well‐known ‘tank‐treading’ and ‘tumbling’ motions predicted by a theory and experiments. Moreover, for the first time, 2D simulations of the ‘vacillating‐breathing’ regime without special ingredient such as thermal fluctuations are recovered. Copyright © 2014 John Wiley & Sons, Ltd.     

A projection method for the spectral solution of non‐homogeneous and incompressible Navier–Stokes equations

This paper is devoted to the development of a parallel, spectral and second‐order time‐accurate method for solving the incompressible and variable density Navier–Stokes equations. The method is well suited for finite thickness density layers and is very efficient, especially for three‐dimensional computations. It is based on an exact projection technique. To enforce incompressibility, for a non‐homogeneous fluid, the pressure is computed using an iterative algorithm. A complete study of the convergence properties of this algorithm is done for different density variations. Numerical simulations showing, qualitatively, the capabilities of the developed Navier–Stokes solver for many realistic problems are presented. The numerical procedure is also validated quantitatively by reproducing growth rates from the linear instability theory in a three‐dimensional direct numerical simulation of an unstable, non‐homogeneous, flow configuration. It is also shown that, even in a turbulent flow, the spectral accuracy is recovered. Copyright © 2012 John Wiley & Sons, Ltd.     

Divergence preserving reconstruction of the nodal components of a vector field from its normal components to edges

We have developed a new divergence preserving method for the reconstruction of the Cartesian components of a vector field from the orthogonal projection of a vector field to the normals to edges in two dimensional. In this method, discrete divergences computed from the nodal components and from the normal ones are exactly the same. Our new method consists of two stages. At the first stage, we use an extended version of the local procedure described in [J. Comput. Phys., 139 :406–409, 1998] to obtain a ‘reference’ nodal vector. This local procedure is exact for linear vector fields; however, the discrete divergence is not preserved. Then, we formulate a constrained optimization problem, in which this reference vector plays the role of a target, and the divergence constraints are enforced by using Lagrange multipliers. It leads to the solution of ‘elliptic’ like discrete equations for the cell‐centered Lagrange multipliers. The new global divergence preserving method is exact for linear vector fields. We describe all details of our new method and present numerical results, which confirm our theory. Copyright © 2016 John Wiley & Sons, Ltd.     

A three‐dimensional Cartesian cut cell method for incompressible viscous flow with irregular domains

A three‐dimensional Cartesion cut cell method is presented for the simulations of incompressible viscous flows with irregular domains. A new model (referred to as ‘6+N’ model) is proposed to describe arbitrarily shaped cut cells and treat all the cells as polyhedrons with 6+N faces. The finite volume discretization of the Navier–Stokes equation is then implemented by using the ‘6+N’ model to separate the surface flux integrals into two parts, that is, the fluxes through the basic face of the hexahedron and those through the cutting surfaces. The previously proposed Kitta Cube algorithm and volume computer‐aided design platform (J. Comput. Aided. Des. 2005; 37(4): 1509–1520. Doi:10.1016/j.cad.2005.03.006) are adopted to generate cut cells and provide shape data and physical attributes for the numerical analysis. A modified SIMPLE‐based smoothing pressure correction scheme is applied to suppress checkerboard pressure oscillations caused by the collocated arrangement of velocities and pressure. The calculation accuracy of the numerical method expressed by L₁ and L _∞ norm errors is first demonstrated by the simulation of a pipe flow. Then its feasibility, efficiency, and potential in engineering applications are verified by applying it to solve natural convections between concentric spheres and between eccentric spheres. The heat transfer patterns in eccentric spheres are also obtained by using the numerical method. Copyright © 2011 John Wiley & Sons, Ltd.     

Computation of turbulent free‐surface flows around modern ships

This paper presents the calculated results for three classes of typical modern ships in modelling of ship‐generated waves. Simulations of turbulent free‐surface flows around ships are performed in a numerical water tank, based on the FINFLO‐RANS SHIP solver developed at Helsinki University of Technology. The Reynolds‐averaged Navier–Stokes (RANS) equations with the artificial compressibility and the non‐linear free‐surface boundary conditions are discretized by means of a cell‐centred finite‐volume scheme. The convergence performance is improved with the multigrid method. A free surface is tracked using a moving mesh technology, in which the non‐linear free‐surface boundary conditions are given on the actual location of the free surface. Test cases recommended are a container ship, a US Navy combatant and a tanker. The calculated results are compared with the experimental data available in the literature in terms of the wave profiles, wave pattern, and turbulent flow fields for two turbulence models, Chien's low Reynolds number k–εmodel and Baldwin–Lomax's model. Furthermore, the convergence performance, the grid refinement study and the effect of turbulence models on the waves have been investigated. Additionally, comparison of two types of the dynamic free‐surface boundary conditions is made. Copyright © 2003 John Wiley& Sons, Ltd.     

Computation of a drastic flow pattern change in an annular swirling jet caused by a small decrease in inlet swirl

This paper investigates the flow pattern change in an annular jet caused by a sudden change in the level of inlet swirl. The jet geometry consists of an annular channel followed by a specially designed stepped‐conical nozzle, which allows the existence of four different flow patterns as a function of the inlet swirl number. This paper reports on the transition between two of them, called the ‘open jet flow high swirl’ and the ‘Coanda jet flow.’ It is shown that a small sudden decrease of 4% in inlet swirl results in a drastic and irreversible change in flow pattern. The objective of this paper is to reveal the underlying physical mechanisms in this transition by means of numerical simulations. The flow is simulated using the unsteady Reynolds‐averaged Navier–Stokes (URANS) approach for incompressible flow with a Reynolds stress turbulence model. The analysis of the numerical results is based on a study of different forces on a control volume, which consists of the jet boundaries. The analysis of these forces shows that the flow pattern change consists of three different regimes: an immediate response regime, a quasi‐static regime and a Coanda regime. The simulation reveals that the pressure–tangential velocity coupling during the quasi‐static regime and the Coanda effect at the nozzle outlet during the Coanda regime are the driving forces behind the flow pattern change. These physical mechanisms are validated with time‐resolved stereo‐PIV measurements, which confirm the numerical simulations. Copyright © 2008 John Wiley & Sons, Ltd.