Orbitally shaken bioreactors are an emerging alternative to stirred-tank bioreactors for large-scale mammalian cell culture, but their fluid dynamics is still not well defined. Among the theoretical and practical issues that remain to be resolved, the characterization of the liquid free surface during orbital shaking remains a major challenge because it is an essential aspect of gas transfer and mixing in these reactors. To simulate the fluid behavior and the free surface shape, we developed a numerical method based on the finite element framework. We found that the large density ratio between the liquid and the gas phases induced unphysical results for the free surface shape. We therefore devised a new pressure correction scheme to deal with large density ratios.

The simulations operated with this new scheme gave values of wave amplitude similar to the ones measured experimentally. These simulations were used to calculate the shear stress and to study the mixing principle in orbitally shaken bioreactors. Copyright © 2012 John Wiley & Sons, Ltd.

Orbitally shaken bioreactors are an emerging alternative to stirred-tank bioreactors for large-scale mammalian cell culture, but their fluid dynamics is still not well-defined. Among the theoretical and practical issues that remain to be resolved, the characterization of the liquid free surface during orbital shaking remains a major challenge because it is an essential aspect of gas transfer and mixing in these reactors.

Particle suspensions play an important role in many engineering applications, yet their behavior in a number of respects remains poorly understood. In conjunction with careful experiments, modeling and simulation of these systems can provide key insight into their complex behavior. However, these two-phase systems pose the challenge of simultaneously, accurately, and efficiently capturing the complex geometric structure, kinematics, and dynamics of the particulate discrete phase and the discontinuities it introduces into the variables (e.g., velocity, pressure, density) of the continuous phase. To this end, a new conformal decomposition finite element method (CDFEM) is introduced for solid particles in a viscous fluid. The method is verified in several simple test problems that are representative of aspects of particle suspension behavior. In all cases, we find the CDFEM to perform accurately and efficiently leading to the conclusion that it forms a prime candidate for application to the full direct numerical simulation of particle suspensions. Copyright © 2012 John Wiley & Sons, Ltd.

Nonlinearities arise in aerodynamic flows as a function of various parameters, such as angle of attack, Mach number and Reynolds number. These nonlinearities can cause the change from steady to unsteady flow or give rise to static hysteresis. Understanding these nonlinearities is important for safety validation and performance enhancement of modern aircraft. A continuation method has been developed to study nonlinear steady state solutions with respect to changes in parameters for two-dimensional compressible turbulent flows at high Reynolds numbers. This is the first time that such flows have been analysed with this approach. Continuation methods allow the stable and unstable solutions to be traced as flow parameters are changed. Continuation has been carried out on two-dimensional aerofoils for several parameters: angle of attack, Mach number, Reynolds number, aerofoil thickness and turbulent inflow as well as levels of dissipation applied to the models. A range of results are presented. Copyright © 2012 John Wiley & Sons, Ltd.

A spectral approach is proposed to determine the flow field of a thin film inside narrow channels of arbitrary shape. Although the method is easily extended to transient flow, only steady flow is considered here. The flow field is represented spectrally in the depthwise direction in terms of orthonormal shape functions, which together with the Galerkin projection lead to a system of ordinary differential equations that can be solved using standard methods. The method is particularly effective for nonlinear flow, including nonlinearities of geometrical or material origins. The validity of the proposed method is demonstrated for a flow with inertia, and, unlike the depth-averaging method, is not limited to a flow at small Reynolds number. The problem is closely related to high-speed lubrication flow. The validity of the spectral representation is assessed by examining the convergence of the method, and comparing it with the fully two-dimensional finite-element solution, and the widely used depth-averaging method from shallow-water theory. It is found that a low number of modes are usually sufficient to secure convergence and accuracy. The influence of inertia is examined on the velocity and pressure fields. The pressure distributions reflect excellent agreement between the low-order spectral method and the finite-element solution, even at moderately high Reynolds number. The depth-averaging solution is unable to predict accurately (qualitatively and quantitatively) the high-inertia flow. Comparison of the velocity field reflects the expected discrepancy in a boundary layer formulation.Copyright © 2012 John Wiley & Sons, Ltd.

The search for the temperature disturbance causing transition between regular and Mach reflections in the dual solution domain is addressed in an optimization statement. The gradient of the discrepancy between the current and target flow fields was calculated using adjoint equations. The control was determined by gradient-based optimization. The flow field simulation is verified via a posteriori error estimates using the solution of an additional adjoint problem. Copyright © 2012 John Wiley & Sons, Ltd.

Efficient preconditioning for Oseen-type problems is an active research topic. We present a novel approach leveraging stabilization for inf-sup stable discretizations. The Grad-Div stabilization shares the algebraic properties with an augmented Lagrangian-type term. Both simplify the approximation of the Schur complement, especially in the convection-dominated case. We exploit this for the construction of the preconditioner. Solving the discretized Oseen problem with an iterative Krylov-type method shows that the outer iteration numbers are retained independent of mesh size, viscosity, and finite element order. Thus, the preconditioner is very competitive. Copyright © 2012 John Wiley & Sons, Ltd.

This paper presents multirate explicit time-stepping schemes for solving partial differential equations with discontinuous Galerkin elements in the framework of Large-scale marine flows. It addresses the variability of the local stable time steps by gathering the mesh elements in appropriate groups. The real challenge is to develop methods exhibiting mass conservation and consistency. Two multirate approaches, based on standard explicit Runge–Kutta methods, are analyzed. They are well suited and optimized for the discontinuous Galerkin framework. The significant speedups observed for the hydrodynamic application of the Great Barrier Reef confirm the theoretical expectations. Copyright © 2012 John Wiley & Sons, Ltd.

An eighth-order filter method for a wide range of compressible flow speeds (H. C. Yee and B. Sjogreen, Proceedings of ICOSAHOM09, June 22–26, 2009, Trondheim, Norway) is employed for large eddy simulations (LES) of temporally evolving mixing layers (TML) for different convective Mach numbers (M_c) and Reynolds numbers. The high-order filter method is designed for accurate and efficient simulations of shock-free compressible turbulence, turbulence with shocklets, and turbulence with strong shocks with minimum tuning of scheme parameters. The value of the M_c considered is for the TML range from the quasi-incompressible regime to the highly compressible supersonic regime. The three main characteristics of compressible TML (the self-similarity property, compressibility effects, and the presence of large-scale structures with shocklets for high M_c) are considered for the LES study. The LES results that used the same scheme parameters for all studied cases agree well with experimental results and published direct numerical simulations (DNS). Copyright © 2012 John Wiley & Sons, Ltd.

In this paper the dynamics of a two-layered liquid, made of two immiscible shallow-layers of different density, has been investigated within the framework of the lattice Boltzmann method (LBM). The LBM developed in this paper for the two-layered, shallow-water flow has been obtained considering two separate sets of LBM equations, one for each layer. The coupling terms between the two sets have been defined as external forces, acted on one layer by the other. Results obtained from the LBM developed in this paper are compared with numerical results obtained solving the two-layered, shallow-water equations, with experimental and other numerical results published in literature. The results are interesting. First, the numerical results obtained by the LBM and by the shallow-water model can be considered as equivalent. Second, the LBM developed in this paper is able to simulate motion conditions on nonflat topography. Third, the agreement between the LBM (and also shallow-water model) numerical results and the experimental results is good when the evolution of the flow does not depend on the viscosity, that is, during the initial phase of the flow, dominated by gravity and inertia forces. When the viscous forces dominate the evolution of the flow the agreement between numerical and experimental results depends strongly on the viscosity; it is good if the numerical LBM viscosity has the same order of magnitude of the liquid's kinematic viscosity. Copyright © 2012 John Wiley & Sons, Ltd.

Discontinuous Galerkin (DG) methods have shown promising results for solving the two-dimensional shallow water equations. In this paper, the classical Runge–Kutta (RK) time discretisation is replaced by the eigenvector-based reconstruction (EVR) that allows the second-order time accuracy to be achieved within a single time-stepping procedure. Moreover, the EVRDG approach yields stable solutions near drying and wetting fronts, whereas the classical RKDG approach yields instabilities. The proposed EVRDG technique is compared with the original RKDG approach on various test cases with analytical solutions. The EVRDG solutions are shown to be as accurate as those obtained with the RKDG scheme. Besides, the EVRDG scheme is 1.6 times faster than the RKDG method. Simulating dambreaks involving dry beds confirms that EVRDG scheme gives correct solutions, whereas the RKDG method yields instabilities. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, the standard Smagorinsky's algorithm is embedded into the multiple relaxation time (MRT) lattice Boltzmann model (LBM) for large eddy simulation (LES) of turbulent shallow water flows (MRT-LABSWE^TM). The model is based on the two-dimensional nonlinear shallow water equations, giving the depth-averaged features. It is verified by applying the model in three typical cases in engineering with turbulence: (i) the flow around a square cylinder, (ii) plane cavity flow, and (iii) flows in a junction of 90°. The results obtained by the MRT-LABSWE^TM are compared with BGK-LABSWE^TM results and experimental data. The objectives of this study are to validate the MRT-LABSWE^TM in a turbulence simulation and perform a comparative analysis between the results of BGK-LABSWE^TM and MRT-LABSWE^TM. Copyright © 2012 John Wiley & Sons, Ltd.

A semi-implicit scheme is presented for large eddy simulation of turbulent reactive flow and combustion in reciprocating piston engines. First, the governing equations in a deforming coordinate system are formulated to accommodate the moving piston. The numerical scheme is made up of a fourth-order central difference for the diffusion terms in the transport equations and a fifth-order weighted essentially nonoscillatory (WENO) scheme for the convective terms. A second- order Adams–Bashforth scheme is used for time integration. For higher density ratios, it is combined with a predictor–corrector scheme. The numerical scheme is explicit for time integration of the transport equations, except for the continuity equation which is used together with the momentum equation to determine the pressure field and velocity field by using a Poisson equation for the pressure correction field. The scheme is aimed at the simulation of low Mach number flows typically found in piston engines. An efficient multigrid method that can handle high grid aspect ratio is presented for solving the pressure correction equation. The numerical scheme is evaluated on two test engines, a laboratory four-stroke engine with rectangular-shaped engine geometry where detailed velocity measurements are available, and a modified truck engine with practical cylinder geometry where lean ethanol/air mixture is combusted under a homogeneous charge compression ignition (HCCI) condition. Copyright © 2012 John Wiley & Sons, Ltd.

The applicability and performance of the lattice-Boltzmann (LB) and meshless point collocation methods as CFD solvers in flow and conjugate heat transfer processes are investigated in this work. Lid-driven cavity flow and flow in a slit with an obstacle including heat transfer are considered as case studies. A comparison of the computational efficiency accuracy of the two methods with that of a finite volume method as implemented in a commercial package (ANSYS CFX, ANSYS Inc., Canonsburg, PA) is made. Utilizing the analogy between heat and mass transfer, an advection–diffusion LB model was adopted to simulate the heat transfer part of the slit flow problem followed by a rigorous mapping of the mass transfer variables to the heat transfer quantities of interest, thus circumventing the need for a thermal LB model. Direct comparison among the results of the three methods revealed excellent agreement over a wide range of Reynolds and Prandtl number values. Furthermore, an integrated computational scheme is proposed, utilizing the rapid convergence of the LB model in the flow part of the conjugate heat transfer problem with that of the meshless collocation method for the heat transfer part. The meshless treatment remains sufficiently rapid even for conduction-controlled processes in contrast to the LB method, which is very rapid in the convection-controlled case only. A single, common computational grid, composed of regularly distributed nodes is used, saving significant computational and coding time and ensuring convergence of the discrete Laplacian operator in the heat transfer part of the computations. Copyright © 2012 John Wiley & Sons, Ltd.

Mesh generation has been frequently the most time consuming step in typical CFD analysis studies. In the past two decades, adaptive Cartesian mesh methods have gained increasing popularity among CFD researches, mainly because of its simplicity and the possibility of automating mesh generation step. In contrast to body-fitted mesh, cells in Cartesian mesh are aligned with coordinate axes. In adaptive Cartesian mesh, cells near the objects’ boundary are recursively refined using quad-tree (two-dimensional) or octree (three-dimensional). Then, cells intersecting the objects’ boundary are clipped by the surfaces, leaving numerous small irregular shaped cells, called cut-cells. Most of the computational efforts required to generate adaptive Cartesian mesh is concentrated on the cut-cell clipping operation. To achieve the computational accuracy in the subsequent numerical solver, the number of cut-cells can be easily over millions, demanding substantial amount of computation time. Reducing mesh generation time matters more especially for unsteady flow simulation involving moving objects, which requires frequent regeneration of meshes for varied postures of the object. In this paper, we report an efficient novel approach to generating adaptive Cartesian mesh by parallelization using the graphics processing unit. The proposed method consists of the following three steps: (1) computing cross-sectional curves of object boundary, (2) octree refinement based on the section curves, and (3) cut-cell clipping. Because each step is designed to be highly parallelizable, we also implemented it on a graphics processing unit, showing orders of magnitude faster performance than the CPU version.Copyright © 2012 John Wiley & Sons, Ltd.

This paper presents a general strategy for designing adaptive space–time finite element discretizations of the nonstationary Navier–Stokes equations. The underlying framework is that of the dual weighted residual method for goal-oriented a posteriori error estimation and automatic mesh adaptation. In this approach, the error in the approximation of certain quantities of physical interest, such as the drag coefficient, is estimated in terms of local residuals of the computed solution multiplied by sensitivity factors, which are obtained by numerically solving an associated dual problem. In the resulting local error indicators, the effects of spatial and temporal discretization are separated, which allows for the simultaneous adjustment of time step and spatial mesh size. The efficiency of the proposed method for the construction of economical meshes and the quantitative assessment of the error is illustrated by several test examples. Copyright © 2012 John Wiley & Sons, Ltd.

We study the adaptive nonlinear filtering in the Leray regularization model for incompressible, viscous Newtonian flow. The filtering radius is locally adjusted so that resolved flow regions and coherent flow structures are not ‘filtered out’, which is a common problem with these types of models. A numerical method is proposed that is unconditionally stable with respect to time step and decouples the problem so that the filtering becomes linear at each time step and is decoupled from the system. Several numerical examples are given that demonstrate the effectiveness of the method. Copyright © 2012 John Wiley & Sons, Ltd.

We demonstrate that radically differing implementations of finite element methods (FEMs) are needed on multi-core (CPU) and many-core (GPU) architectures, if their respective performance potential is to be realised. Our numerical investigations using a finite element advection–diffusion solver show that increased performance on each architecture can only be achieved by committing to specific and diverse algorithmic choices that cut across the high-level structure of the implementation. Making these commitments to achieve high performance for a single architecture leads to a loss of performance portability. Data structures that include redundant data but enable coalesced memory accesses are faster on many-core architectures, whereas redundancy-free data structures that are accessed indirectly are faster on multi-core architectures. The Addto algorithm for global assembly is optimal on multi-core architectures, whereas the Local Matrix Approach is optimal on many-core architectures despite requiring more computation than the Addto algorithm. These results demonstrate the value in making the correct choice of algorithm and data structure when implementing FEMs, spectral element methods and low-order discontinuous Galerkin methods on modern high-performance architectures. Copyright © 2012 John Wiley & Sons, Ltd.

Hessian-based model reduction was previously proposed as an approach in deriving reduced models for the solution of large-scale linear inverse problems by targeting accuracy in observation outputs. A control-theoretic view of Hessian-based model reduction that hinges on the equality between the Hessian and the transient observability gramian of the underlying linear system is presented. The model reduction strategy is applied to a large-scale ( degrees of freedom) three-dimensional contaminant transport problem in an urban environment, an application that requires real-time computation. In addition to the inversion accuracy, the ability of reduced models of varying dimension to make predictions of the contaminant evolution beyond the time horizon of observations is studied. Results indicate that the reduced models have a factor speedup in computing time for the same level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.

This paper describes the development of a mesh deformation method used for aero-thermo-mechanical coupling of turbo-engine components. The method is based on the nonlinear solution of an elastic medium analogy, solved using finite element discretisation and modified to let the boundary nodes be free to slide over the deflected surfaces. This sliding technique relies on a B-spline reconstruction of the moving boundary and increases the robustness of the method in situations where the boundary deflection field presents significant gradients or large relative motion between two distinct boundaries. The performance of the method is illustrated with the application to an interstage cavity of a turbine assembly, subjected to the deformations computed by a coupled thermo-mechanical analysis of the engine component. Copyright © 2011 John Wiley & Sons, Ltd.

A grid adaptation technique for two-dimensional unstructured grids of triangles and quadrilaterals is presented. The error estimation procedure is formulated in terms of a node pair-based data structure that allows for a unified description of the finite element and finite volume schemes. The adaptation algorithm is based on a strategy of hierarchical corrections, where a suitable number of intermediate adapted grids are generated and successively corrected by employing a simple node insertion technique at the midpoint of the element edges. Coarsening of the grid is obtained in an implicit fashion by avoiding the insertion of new nodes during the correction phase. The adaptation history, from the initial to the current grid, including all intermediate grids, is stored and updated through the whole process. No intermediate grid is therefore required to be stored explicitly. The adapted grid is anisotropic, thanks to the adoption of both regular triangular and quadrilateral elements with high aspect ratio that are gathered in, for example, boundary layer or wake regions. Numerical experiments of steady compressible flows, including both inviscid and viscous flows, are presented to support the suitability of the adaptation technique. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, two repair techniques are proposed for diamond schemes of anisotropic diffusion problems to ensure that the repaired solutions satisfy the discrete maximum principle. One of them is an extension of that in [Liska R, Shashkov M. Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems. Communications in Computational Physics 2008; 3(4):852–877.] for linear finite element solutions, which is a local repair technique, and another is a new global repair technique. Both of them keep total energy conservation and are easy to be implemented in existing codes. Numerical examples show that these two repair techniques do not destroy the accuracy of solution for the diamond schemes on distorted meshes. Copyright © 2012 John Wiley & Sons, Ltd.

This paper considers the problem of estimating the strengths of two time-varying heat sources simultaneously, from measurements of the temperature inside the square domain in a porous medium, when prior knowledge of the source functions is not available. This problem is an inverse natural convection problem. In order to circumvent this problem, we define several optimization criteria (objective functionals) that measure discrepancies between model and measured data, where objective functionals depend on two heat sources and use multi-criteria optimization to identify Nash equilibria, which are solutions to the non-cooperative game according to game theory. Two non-cooperative game strategies are considered: competitive (Nash) game and hierarchical (modified Stackelberg) game. The methodology that we employ relies on a combination of mixed finite element space approximations, finite difference time discretizations, adjoint equation and sensitivity equation techniques, and nonlinear conjugate gradient algorithms for the solutions of estimating two heat sources. Applying the Sobolev gradient for the noise removal is investigated. The performance of the present technique of inverse analysis is evaluated, by means of several numerical experiments, and is found to be very accurate as well as efficient. Copyright © 2012 John Wiley & Sons, Ltd.

The Chimera technique for moving grids is used to take into account nonhomogeneous unsteady inflow conditions in the simulation of aerodynamic flows. The method is applied to simulate the transport of a large-scale vortex by a mean velocity field over a large distance, where it finally interacts with an airfoil. The Chimera approach allows one to resolve the vortex on a fine grid, whereas the unstructured background grid covering most of the computational domain can be much coarser. This method shows the same low numerical dissipation as a simulation on a globally fine grid. Several precursor tests are performed with a finite modified analytical Lamb–Oseen type vortex to study the influence of spatial and temporal resolution and the employed numerical scheme. Then, the interaction of an analytical vortex with a NACA0012 airfoil and with an ONERA-A airfoil near stall is studied. Finally, a realistic vortex is generated by a ramping airfoil and is transported on a moving Chimera block and then interacts with a two-element airfoil, which allows one to simulate a typical setup for a gust generator in aerodynamic facilities. Copyright © 2012 John Wiley & Sons, Ltd.

The design of the mold and the choice of the injection parameters for metal injection molding (MIM) is required to maintain homogeneity of the filled mixture. However, powder segregation is unavoidable in MIM because of the significant difference in densities of the metallic powder and the polymer binder. To achieve an effective prediction of segregation effect, a biphasic model based on mixture theory is employed. The viscous behaviors of each phase and the interaction coefficient between the flows of the two phases should be determined. The solution of two coupled Navier–Stokes equations results in a tremendous computation effort. The previous development of an explicit algorithm makes the biphasic simulation much faster than that of the classic methods. However, it is strongly desired to reduce or even eliminate the numerous global solutions for pressure fields at each time step. Hence, a new vectorial algorithm is proposed and developed to perform the simulation only by vectorial operations. It provides the anticipated efficiency in the simulation of biphasic modeling, and the advantage to use the classic elements of equal-order interpolations. Some results produced by the two algorithms are compared with the experimental values to validate the new vectorial algorithm. Copyright © 2012 John Wiley & Sons, Ltd.

This work describes a methodology to simulate free surface incompressible multiphase flows. This novel methodology allows the simulation of multiphase flows with an arbitrary number of phases, each of them having different densities and viscosities. Surface and interfacial tension effects are also included. The numerical technique is based on the GENSMAC front-tracking method. The velocity field is computed using a finite-difference discretization of a modification of the Navier–Stokes equations. These equations together with the continuity equation are solved for the two-dimensional multiphase flows, with different densities and viscosities in the different phases. The governing equations are solved on a regular Eulerian grid, and a Lagrangian mesh is employed to track free surfaces and interfaces. The method is validated by comparing numerical with analytic results for a number of simple problems; it was also employed to simulate complex problems for which no analytic solutions are available. The method presented in this paper has been shown to be robust and computationally efficient. Copyright © 2012 John Wiley & Sons, Ltd.

In this work, a corrected symmetric and periodic density reinitialized SPH (CSPDR-SPH) method is proposed and extended to simulate the viscoelastic free surface flows based on the Phan–Thien–Tanner model. The improvements mainly lie in deriving a corrected symmetric kernel gradient, and combining it with a periodic density reinitialization procedure. In addition, a simple artificial viscosity and a simple artificial stress form are adopted. Thus, the CSPDR-SPH method has higher accuracy and better stability than the SPH method, and conserves both linear and angular momentums. The consistency and convergence of the CSPDR-SPH method are justified by approximating a function in one and two dimensions. The merits of CSPDR-SPH method are demonstrated by several benchmarks. The simple flow in a two-dimensional channel is investigated to show the capability of the CSPDR-SPH method to simulate the viscoelastic free surface flow. Then the CSPDR-SPH method is extended to simulate the impacting drop problem. Numerical results show that the CSPDR-SPH method can precisely capture the viscoelastic free surface. The Reynolds number, Weissenberg number and elongation parameter have remarkable effect on the flows. Copyright © 2012 John Wiley & Sons, Ltd.

This paper presents a local moving least square-one-dimensional integrated radial basis function networks method for solving incompressible viscous flow problems using stream function-vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square and one-dimensional integrated radial basis function networks techniques. The major advantages of the proposed method include the following: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker- δ property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local radial basis function approximations. Several examples including two-dimensional (2D) Poisson problems, lid-driven cavity flow and flow past a circular cylinder are considered, and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, we firstly apply generalized difference methods to solve a fluid mixture model. The model is usually used to describe the tissue deformations and contains a nonlinear hyperbolic equation and an elliptic equation. Most people have used finite difference methods for solving the elliptic equation and other schemes for solving the hyperbolic equation. It is well known that the accuracy of traditional finite difference method is not high. This may be a serious disadvantage in the fluid mixture model, which describes cell movements and tissue deformations. The numerical methods we propose to improve accuracy are based on generalized Galerkin methods and dual decomposition. By choosing suitable trial function space and test function space, our generalized upwind difference schemes exhibit second-order convergence in space for smooth problems and can eliminate numerical oscillations for discontinuous problems. Some numerical results are presented to demonstrate the advantages of our methods. Copyright © 2012 John Wiley & Sons, Ltd.

The lattice-Boltzmann method is being applied to a diversity of fluid flow and heat transfer problems nowadays. Because of its microscale nature, strict attention should be paid when introducing macroscopic inputs to the model. One of the challenging issues dealing with macroscale and microscale treatment is the implementation of boundary conditions. In this regard constant-temperature boundaries are frequently used in energy transfer problems. Such boundaries are simply modeled in Navier–Stokes based solvers, but they are not so harnessed in lattice-Boltzmann models. One of the problems is that the calculated tangential heat flux is not zero along such boundaries in most of the previous models. In the present paper, a model has been developed, which has the capability of controlling tangential heat flux along the constant-temperature boundaries. It aims to set the heat flux nearly zero along the boundary in midplane grid schemes. Copyright © 2012 John Wiley & Sons, Ltd.

In the present analysis, the influence of heat and mass transfer on the peristaltic flow of a hyperbolic tangent fluid in an asymmetric channel has been discussed. The highly nonlinear equations are simplified under lubrication approach. The perturbation and numerical solutions of the problem are not only discussed but the validity of the results is also being checked. The graphical results of the problem under discussion are also being brought under consideration to see the behavior of various physical parameters. Copyright © 2012 John Wiley & Sons, Ltd.

A unified model of an induction-stirred ladle in two and three dimensions is presented. Induction stirring of molten steel is a coupled multiphysics phenomena involving electromagnetic and fluid flow. Models presented in this paper gives a more accurate description of the real stirring conditions and flow pattern by taking into account the multiphysics behavior of the induction-stirring process in an induction-stirred ladle. This paper presents formulation of coupled electromagnetic and fluid flow equations. The coupled electromagnetic and fluid flow equations are solved with the use of the finite element method in two and three dimensions. The model is used to predict values of steel velocities and magnetic flux density. The model is also used to predict the effect of increased current density on flow velocity. Magnetic flux density values obtained from the model are verified against the experimental values. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, a simple and efficient analytical method, Hankel–Padé method, is employed in solving boundary-layer problems. Two important types of boundary-layer problems, the problem of the boundary-layer flow of an incompressible viscous fluid over a nonlinear stretching sheet and the problem of the nano boundary-layer flows with Navier boundary condition, are considered. The analytical solutions of the governing nonlinear boundary-layer problems are developed as rational approximation solutions. The comparison of the obtained results with other available results shows that, in general, the Hankel–Padé is much simpler and more accurate than other approaches that were proposed by other authors recently. Copyright © 2011 John Wiley & Sons, Ltd.

Finite volume methods on structured and unstructured meshes commonly use second-order, upwind-biased linear reconstruction schemes to approximate the convective terms. Limiters are employed to reduce the inherent variable overshoot and undershoot of these schemes in discontinuous regions; however, they also can significantly increase the numerical dissipation of a solution in smooth regions. This paper presents a novel nonlocal, nonmonotonic (NLNM) limiter developed by enforcing cell minima and maxima on dependent variable values projected to cell faces. The minimum and maximum values for a cell are determined primarily through recursive reference to the minimum and maximum values of its upwind neighbors. The new limiter has been implemented into an existing flow solver. Various test cases are presented, which highlight the ability of the NLNM limiter to eliminate nonphysical oscillations while maintaining negligible levels of limiter-induced dissipation into the solution. Results from the new limiter are compared with those from other limited and unlimited second-order upwind and first-order upwind schemes. For the cases examined in the study, the NLNM limiter was found to improve accuracy without significantly decreasing overall simulation efficiency and robustness.Copyright © 2011 John Wiley & Sons, Ltd.

Simulation of unsteady fluid behaviour with arbitrary boundary motion or topological change remains restricted owing to mesh deformation limitations, and usually requires the application of special techniques using overlapping meshes, sliding planes, remeshing or immersed boundaries. This work presents the application of a spacetime interpretation of the fluid conservation laws that unifies meshes in space and time. This effectively replaces the problem of mesh deformation with the problem of mesh generation and, because connectivity is no longer restricted to being constant in time, any motion or topological change may be simulated. Examples are given applying the method to a piston, a pitching NACA0012 aerofoil, an appearing/disappearing object, a two-dimensional store separation and a rotation case to validate and then demonstrate the capabilities of the method. Results for the pitching aerofoil case are compared with a conventional moving mesh unsteady computation and shown to be consistent, whereas the demonstration cases show qualitatively correct behaviour and illustrate the general nature of the technique. Copyright © 2011 John Wiley & Sons, Ltd.

This paper is concerned with the development of a high-order numerical scheme for the modelling of two-phase Newtonian flows. The companion paper, herein referred to as Part 2, extends the scheme to two-phase viscoelastic flows. The particular problem of the collapse of a two-dimensional bubble in the vicinity of a rigid boundary is considered. The governing equations are discretized using the spectral element method, and the two phases are modelled using a marker particle method. The marker particle scheme is validated using the Zalesak slotted disk rotation test problem. A comprehensive set of results is presented for the problem of bubble collapse near a rigid wall, and qualitative agreement is obtained with other numerical studies and experimental observations. Viscous effects are shown to inhibit bubble collapse and prevent jet formation and are therefore likely to have a mitigating effect on cavitation damage.Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we continue to study the entropy dissipation scheme developed in former. We start with a numerical study of the scheme without the entropy dissipation term on the linear advection equation, which shows that the scheme is stable and numerical dissipation and numerical dispersion free for smooth solutions. However, the numerical results for discontinuous solutions show nonlinear instabilities near jump discontinuities. This is because the scheme enforces two related conservation properties in the computation. With this study, we design a so-called ‘minimums-increase-and-maximums-decrease’ slope limiter in the reconstruction step of the scheme and delete the entropy dissipation in the linear fields and reduce the entropy dissipation terms in the nonlinear fields. Numerical experiments show improvements of the designed scheme compared with the results presented in former. However, the minimums-increase-and-maximums-decrease limiter is still not perfect yet, and better slope limiters are still sought.Copyright © 2011 John Wiley & Sons, Ltd.

A class of axisymmetric horizontal flows dominated by strong heat or mass sources are considered here, which systematically arise in multiscale atmospheric and oceanic flows in various spatiotemporal scales such as hurricane embryos. Various linear and nonlinear analytical solutions are proposed, and several examples are given to illustrate them. On the basis of the proposed exact solutions, stability of the flow under various conditions is examined. The proposed exact solutions serve as benchmarks for validation of numerical models. A Chebyshev pseudospectral model is also developed for these flows. The numerical results are compared with the developed analytical solutions as well as a high-resolution numerical solution, which showed an excellent performance of the numerical model in terms of numerical diffusion and oscillation. The numerical method is also efficient because expensive calculations such as LU decomposition are only needed once, at the beginning of calculations. The model could be used in studying small-scale and multiple-scale phenomena such as hot towers and hurricane embryos, which are the subject of future studies. Copyright © 2011 John Wiley & Sons, Ltd.

This paper describes a numerical study of the two-dimensional and three-dimensional unsteady flow over two square cylinders arranged in an in-line configuration for Reynolds numbers from 40 to 1000 and a gap spacing of 4D, where D is the cross-sectional dimension of the cylinders. The effect of the cylinder spacing, in the range G = 0.3D to 12D, was also studied for selected Reynolds numbers, that is, Re = 130, 150 and 500. An incompressible finite volume code with a collocated grid arrangement was employed to carry out the flow simulations. Instantaneous and time-averaged and spanwise-averaged vorticity, pressure, and streamlines are computed and compared for different Reynolds numbers and gap spacings. The time averaged global quantities such as the Strouhal number, the mean and the RMS values of the drag force, the base suction pressure, the lift force and the pressure coefficient are also calculated and compared with the results of a single cylinder. Three major regimes are distinguished according to the normalized gap spacing between cylinders, that is, the single slender-body regime (G < 0.5), the reattach regime (G < 4) and co-shedding or binary vortex regime (G ≥4).

Hysteresis with different vortex patterns is observed in a certain range of the gap spacings and also for the onset of the vortex shedding. Copyright © 2011 John Wiley & Sons, Ltd.

This paper considers the computation of flow sensitivities that arise in the context of design optimization. The scheme is based on the solution of a continuous adjoint problem, for which two complementary, although analytically equivalent, approaches have been routinely used for some time now, yielding expressions for the sensitivities that contain, respectively, boundary and domain integrals. These concepts are clarified in a unified framework and their equivalence at the continuous level is demonstrated through appropriate algebraic manipulations. Equivalence at the discrete level is assessed through numerical testing for various aerodynamic shape-optimization problems. Copyright © 2011 John Wiley & Sons, Ltd.

In this article, we present a finite element variational multiscale (VMS) method for incompressible flows based on the construction of projection basis functions and compare it with common VMS method, which is defined by a low-order finite element space L_h on the same grid as X_h for the velocity deformation tensor and a stabilization parameter α. The best algorithmic feature of our method is to construct the projection basis functions at the element level with minimal additional cost to replace the global projection operator.

Finally, we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method. Copyright © 2011 John Wiley & Sons, Ltd.

A three-dimensional extended finite element method is presented to simulate Stokes flow in complex geometries with internal moving parts. Instead of re-meshing the flow domain, the kinematics of the internal objects are imposed on the conservation equations using a constraint, implemented with a Lagrangian multiplier. To capture discontinuities of field variables, such as pressure and velocity, on the intersected elements, XFEM is used. To validate our method, it is first applied to a relatively simple problem, that is, the flow around a cylinder in a channel. The results are verified by comparing with a boundary-fitted solution. After validation of the model and its implementation, the three-dimensional flow in a twin-screw extruder is simulated and the results are compared with experimental data from literature. XFEM shows very good accuracy for complex geometries with internal moving parts and narrow gap regions where the shear rate is orders of magnitude higher than in other regions. Copyright © 2011 John Wiley & Sons, Ltd.

Four-dimensional variational (4D-Var) data assimilation method is used to find the optimal initial conditions by minimizing a cost function in which background information and observations are provided as the input of the cost function. The optimized initial conditions based on background error covariance matrix and observations improve the forecast. The targeted observations determined by using methods such as adjoint sensitivity, observation sensitivity, or singular vectors may further improve the forecast. In this paper, we are proposing a new technique—consisting of a penalized 4D-Var data assimilation method that is able to reduce the forecast error significantly. This technique consists in penalizing the cost function by a forecast aspect defined over the verification domain at the verification time. The results obtained using the penalized 4D-Var method show that the initial condition is optimally estimated, thus resulting in a better forecast by significantly reducing the forecast error over the verification domain at verification time. Copyright © 2011 John Wiley & Sons, Ltd.

The open-source CFD library OpenFoam® contains a method for solving free surface Newtonian flows using the Reynolds averaged Navier–Stokes equations coupled with a volume of fluid method. In this paper, it is demonstrated how this has been extended with a generic wave generation and absorption method termed ‘wave relaxation zones’, on which a detailed account is given. The ability to use OpenFoam for the modelling of waves is demonstrated using two benchmark test cases, which show the ability to model wave propagation and wave breaking. Furthermore, the reflection coefficient from outlet relaxation zones is considered for a range of parameters. The toolbox is implemented in C++, and the flexibility in deriving new relaxation methods and implementing new wave theories along with other shapes of the relaxation zone is outlined. Subsequent to the publication of this paper, the toolbox has been made freely available through the OpenFoam-Extend Community. Copyright © 2011 John Wiley & Sons, Ltd.

In the present paper, a mesh adaptation process for solving the advection equation on a fully unstructured computational mesh is introduced, with a particular interest in the case it implicitly describes an evolving surface. This process mainly relies on a numerical scheme based on the method of characteristics. However, low order, this scheme lends itself to a thorough analysis on the theoretical side. It gives rise to an anisotropic error estimate which enjoys a very natural interpretation in terms of the Hausdorff distance between the exact and approximated surfaces. The computational mesh is then adapted according to the metric supplied by this estimate. The whole process enjoys a good accuracy as far as the interface resolution is concerned. Some numerical features are discussed and several classical examples are presented and commented in two or three dimensions. Copyright © 2011 John Wiley & Sons, Ltd.

This paper examines the magnetohydrodynamic boundary layer flow of Jeffrey fluid due to a rotating disk. The governing partial differential equations are first transformed into the coupled system of ordinary differential equations and then solved by using the homotopy analysis method. The influence of various involved physical parameters on the dimensionless radial and azimuthal velocities is sketched and analyzed. The variation of skin friction coefficients in radial and azimuthal directions is studied for various values of pertinent parameters. Copyright © 2011 John Wiley & Sons, Ltd.

The distributed Lagrange multiplier/fictitious domain method proposed for the direct numerical simulation of particle-laden flows is considered in this work. First, it is demonstrated that improved accuracy is obtained with a coupled numerical scheme, whereby the pressure and the Lagrange multiplier fields enforcing incompressibility and rigid body motion, respectively, are calculated and applied together. However, the convergence characteristics of the iterative solution of the coupled scheme are poor because symmetric but indefinite and poorly conditioned matrices are produced. An analysis is then presented, which suggests that the cause for the matrix pathologies lies in the interaction of the respective matrix operators enforcing incompressibility and rigid body motion. On the basis of this analysis, an alternative formulation is developed for the Lagrange multipliers, being now composed of a set of forces distributed only on the particle boundary together with a set of couples distributed within the particle core. The new formulation is tested with several types of flows with stationary or moving particles under creeping or finite Reynolds number conditions and it is demonstrated that it produces correct results and better conditioned matrices, thus enabling faster and more reliable convergence of the conjugate gradient method. The analysis and tests, therefore, support the expectation that the proposed formulation is promising and worthy of further study and improvement. Copyright © 2011 John Wiley & Sons, Ltd.

We restrict the variational multiscale method to a class of methods we denote by numerical multiscale methods. Numerical multiscale methods are methods obtained by enriching the piecewise linear functions with special local functions. The enrichment provides additional stabilization via terms obtained by static condensation. The resulting methods are improvements of the coarse scale solutions by the approximations of the fine scales emanating from the enrichments.Copyright © 2011 John Wiley & Sons, Ltd.

In this work, a method for the numerical simulation of free surface flows of Newtonian and non-Newtonian flows is developed. The method is based on the employ of three main ingredients. The first is an updated Lagrangian approach for the description of kinematics. This approach is made possible by using natural neighbor Galerkin methods (also known as natural element methods).This allows the use of the method of characteristics to integrate the equations of motion along the nodal path-lines. The second ingredient relies on the use of a second-order time method of characteristics, which has proved to be indispensable for an accurate solution of some problems, even very simple ones. Finally, the third ingredient relies on the use of shape constructors (particularly, α-shapes) to avoid the use of boundary markers or any explicit description of the boundary in general. After the theoretical description of the proposed method, some examples illustrating its performance are given. Copyright © 2011 John Wiley & Sons, Ltd.

A modified front-tracking method was proposed for the simulation of fluid-flexible body interactions with large deformations. A large deformable body was modeled by restructuring the body using a grid adaptation. Discontinuities in the viscosity at the fluid-structure interface were incorporated by distributing the viscosity across the interface using an indicator function. A viscosity gradient field was created near the interface, and a smooth transition occurred between the structure and the fluid. The fluid motion was defined on the Eulerian domain and was solved using the fractional step method on a staggered Cartesian grid system. The solid motion was described by Lagrangian variables and was solved by the finite element method on an unstructured triangular mesh. The fluid motion and the structure motion were independently solved, and their interaction force was calculated using a feedback law. The interaction force was the restoring force of a stiff spring with damping, and spread from the Lagrangian coordinates to the Eulerian grid by a smoothed approximation of the Dirac delta function. In the numerical simulations, we validated the effect of the grid adaptation on the solid solver using a vibrating circular ring. The effects of the viscosity gradient field were verified by solving the deformation of a circular disk in a linear shear flow, including an elastic ring moving through a channel with constriction, deformation of a suspended catenary, and a swimming jellyfish. A comparison of the numerical results with the theoretical solutions was presented. Copyright © 2011 John Wiley & Sons, Ltd.

The objective of this study is to propose a parameter identification of a river current and diffusion coefficients by using the reduced Kalman filter finite element method, which has been previously presented and now extended by the authors. For comparison, the estimation computations of velocity, water elevation, and chemical oxygen demand (COD) concentration are carried out on the basis of nonlinear shallow water flow and compared with the observations carried out at the Teganuma river in Japan. A marked discrepancy in COD concentration is found between the computed and observed results. The correlation factor between the computed and observed results is 0.51. To reduce the discrepancy, the authors believe that the diffusion coefficients should be identified. Assuming that the diffusion coefficient is constant in the entire domain and over the entire total duration, satisfactory results were not obtained. Thus, the computational domain is divided into four subdomains according to the water depth. Assuming that the diffusion coefficients are constant in each subdomain, the identification is carried out. Relatively good, albeit unsatisfactory, results are obtained. The discrepancy between the computed and observed COD concentration has special features. In some time zones, the computed results are larger whereas in other time zones, they are smaller than the observed results. To compensate this discrepancy, we assumed that the diffusion coefficient is a function of COD concentration. The correlation factor is improved to be 0.73. The identified diffusion coefficients are time functions that change cyclically with a period of 24 h. This fact suggests that biological phenomenas occurred in the river. Copyright © 2011 John Wiley & Sons, Ltd.

The Reynolds-averaged Navier–Stokes equations/ k −ε approach is the popular and practical choice for carrying out simulations involving the atmospheric boundary layer. However, despite its widespread use, implementation of this approach is not without its challenges—even when considering the simplest case of horizontally homogeneous conditions. Most notably, the distributions of turbulent kinetic energy and its dissipation rate have proved difficult to maintain near solid boundaries, particularly in wind engineering applications where the near-wall grid is relatively coarse. In this work, the origin of these errors is investigated, and it is shown that by applying appropriate discretization schemes in conjunction with the Richards and Hoxey boundary conditions, truly invariant profiles of all flow properties can be obtained on such grids. Furthermore, with this finding, a wall treatment for coarse grids is proposed that could be implemented for non-homogeneous conditions. All simulations are carried out using OpenFOAM-1.6.x. Copyright © 2011 John Wiley & Sons, Ltd.

The requirements for flux limiter functions preserving total variation diminishing (TVD) are derived based on a 1D nonuniform grid, and a new TVD region is determined to fit arbitrary 1D grids. Some second-order TVD schemes called improved TVD schemes are developed, such as modified Van Leer scheme, modified Van Albada scheme, and modified SUPERBEE scheme. Then, they are extended to 2D grids. Because the element sizes and face positions are taken into account, good behaviors are observed in the implementations in both 1D and 2D cases for pure advection simulation. That is, good conservation, better monotonicity, and higher accuracy are maintained by the improved TVD schemes compared with the present ones deduced from uniform grids, and they keep superiorities even when implemented on poor grids. Among all the improved TVD schemes, the modified SUPERBEE is only recommended for poor 1D grids, but the modified Van Leer scheme can suit both poor 1D and 2D grids. Copyright © 2011 John Wiley & Sons, Ltd.

A numerical scheme is developed for two-dimensional, depth-averaged, shallow water flows based on the DG method. In the shallow water equations, the pressure force term and the bed slope term are combined to eliminate numerical imbalance. The HLLC approximate Riemann solver is employed to calculate the numerical flux for the DG scheme. A slope limiting procedure used for compressible flows is adapted for modeling incompressible two-dimensional flows. A simple treatment for modeling flow over initially dry bed is presented. To validate the scheme, numerical tests are conducted to simulate hydraulic jump, partial dam break, circular dam break, wetting and drying in parabolic bowl, and a real world dam break in the Toce River. Numerical results show that this scheme can accurately model shock waves, wetting and drying, and flows in the channel with varying geometry and bed topography found in natural channels.Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, the lattice Boltzmann method is used to study the Prandtl number effect on flow structure and heat transfer rates in a magnetohydrodynamic flow mixed convection in a lid-driven cavity filled with a porous medium. The right and left walls are at constant but different temperatures (θ_h and θ_c), while the other walls are adiabatic. Gallium and salt water (0.02 < Pr < 13.4) are used as samples of the electroconducting fluids in the cavity. Typical sets of streamlines and isotherms are presented to analyze the flow patterns set up by the competition among the forced flow created by the lid-driven wall, the buoyancy force of the fluid and the magnetic force of the applied magnetic field. Mathematical formulations in the porous media were constructed based on the Brinkman–Forchheimer model, while the multidistribution-function model was used for the magnetic field effect. Numerical results were obtained and the effects of the Prandtl number and the other effective parameters such as Richardson, Hartman, and Darcy numbers were investigated. It was found that the fluid fluctuations within the cavity were reduced by increasing the Hartman number. A similar pattern was observed for the Darcy number reduction. Heat transfer was essentially dominated by the conduction for the low Prandtl number and forced convection dominated as the Prandtl number increased. Also, the average Nusselt number was raised by increasing the Prandtl number. It was discovered that a remarkable heat transfer enhancement of up to 28% could be reached by increasing the Prandtl number (from 0.02 to 13.4) at constant Richardson and Darcy numbers. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we further develop an approach previously introduced in Lassila and Rozza, 2010, for shape optimization that combines a suitable low-dimensional parametrization of the geometry (yielding a geometrical reduction) with reduced basis methods (yielding a reduction of computational complexity). More precisely, free-form deformation techniques are considered for the geometry description and its parametrization, whereas reduced basis methods are used upon a FE discretization to solve systems of parametrized partial differential equations. This allows an efficient flow field computation and cost functional evaluation during the iterative optimization procedure, resulting in effective computational savings with respect to usual shape optimization strategies. This approach is very general and can be applied to a broad variety of problems. In this paper, we apply it to find the optimal shape of aorto-coronaric bypass anastomoses based on vorticity minimization in the down-field region. Blood flows in the coronary arteries are modeled using Stokes equations; afterwards, results have been verified in feedback using Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.

In this work we are concerned with the finite increment calculus (FIC) method. The method has been developed for efficient approximation of advection-diffusion equations with high Péclet numbers. Since the natural application of FIC is within the framework of the FEM, we consider the BVP in a weak sense on finite dimensional spaces. Here we provide a result on existence and uniqueness of the solution as well as an error analysis. Also we propose a choice of the stabilization parameter. We test the method on some troublesome 2D problems. Copyright © 2011 John Wiley & Sons, Ltd.

This paper describes a new variant of hybrid scheme that is constructed by a wave-capturing scheme and a nonoscillatory scheme for flow computations in the presence of shocks. The improved fifth-order upwind weighted essentially nonoscillatory scheme is chosen to be conjugated with the seven-point dispersion-relation-preserving scheme by means of an adaptive switch function of grid-point type. The new hybrid scheme can achieve a better resolution than the hybrid scheme which is based on the classical weighted essentially scheme. Ami Harten's multiresolution analysis algorithm is applied to density field for detecting discontinuities and setting point values of the switch function adaptively. Moreover, the tenth-order central filter is applied in smooth part of the flow field for damping dispersion errors. This scheme can promote overall computational efficiency and yield oscillation-free results in shock flows. The resolution properties and robustness of the new hybrid scheme are tested in both 1D and 2D linear and nonlinear cases. It performs well for computing flow problems with rich structures of weak/strong shocks and large/small vortices, such as the shock-boundary layer interaction problem in a shock tube, which illustrates that it is very robust and accurate for direct numerical simulation of gas-dynamics flows. Copyright © 2011 John Wiley & Sons, Ltd.

The paper details the implementation of the Godunov-type finite volume Arbitrary high order schemes using Derivatives (ADER) scheme for the case of a large source term in the continuity equation of the nonlinear shallow water equations. The particular application is the movement of a bore on a highly permeable slope. The large source term is caused by the infiltration into the initially unsaturated slope material. Infiltration is modelled as vertical downwards piston-like flow with Forchheimer quadratic parameterisation of the resistance law. The corresponding ODE is solved using the fourth-order Runge–Kutta method. The surface and subsurface flow models have been tested by comparison with analytical solutions. Example predictions of surface bore propagation and wetting front propagation are presented for a range of slope permeabilities. The effects of permeability on bore run-up, water depths and velocities are illustrated. The ADER scheme is capable of handling the source term, including the extreme case when this term dominates the volume balance. Copyright © 2011 John Wiley & Sons, Ltd.

In this article, we investigate the influence of heat and mass transfer on the peristaltic flow of magnetohydrodynamic second-order fluid in a channel when the induced magnetic field effects are present. Problem formulation in a wave frame of reference is presented. The governing nonlinear analysis is carried out under the assumption of small wave number. Explicit expressions of the pressure gradient, the stream function, the magnetic force function, the axial induced magnetic field, the current density distribution, the temperature, and the concentration distribution are derived. The effects of embedded parameters are also examined. Copyright © 2011 John Wiley & Sons, Ltd.

We consider a numerical approximation of the classical problem of the turbulent free jet. The same boundary layer equation is used for the laminar jet, but with the introduction of the turbulent viscosity. This viscosity depends on the kinematic momentum and the slenderness parameter and varies in space. Here, the problem under consideration is taken to be singularly perturbed, with the slenderness parameter as the perturbation parameter taking arbitrary values from (0,1]. The effective width of the layer depends on the turbulent viscosity and becomes thinner as the slenderness parameter tends to zero. This singularly perturbed character of the turbulent free jet makes this investigation both unique and exploratory. The problem is solved numerically on a finite subdomain by using boundary conditions obtained from the similarity solution. We construct a robust numerical method on the basis of the piecewise-uniform meshes condensing in the vicinity of the centre of the jet. By numerical experiments, we show that errors for the computed velocity components do not depend on the perturbation parameter. Copyright © 2011 John Wiley & Sons, Ltd.

A hybrid sequential niche algorithm is used for the automated identification of critical points of velocity fields. This method combines an adaptive sequential niche technique with deterministic local optimization to detect critical points: focus, node and saddle points. A particle swarm algorithm performs a global search whereas vortex core identification functions compute the precise location as the extremum of the corresponding function. Once a critical point is found, a rectangular niche is constructed around the point. The particle swarm then proceeds to explore different regions of the velocity field. The process advances sequentially, avoiding areas near previously found critical points by blocking niches obtained from previous steps. The niche size is automatically adjusted each time a search enters inside an existing niche. Vortex core functions are used for critical point identification and calculating its precise location inside each niche. The procedure is validated on particle image velocimetry data obtained with two types of flows, an impinging jet flow and a flow downstream of a model building. The hybrid algorithm proved to be very efficient and robust for automated detection and identification of critical points. It can be used as a first step for studying the time-dependent dynamic behavior of instantaneous velocity fields by tracking topological critical points. This is the first study that uses a multi-modal particle swarm algorithm for critical point identification. Copyright © 2011 John Wiley & Sons, Ltd.

In the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well-known pressure-correction solver. This new reformulation introduces the AUSM⁺ − up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM⁺ − up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test-case. The code is also benchmarked against a very tough test-case for the supersonic and hypersonic flow over a cylinder. Copyright © 2011 John Wiley & Sons, Ltd.

A high-order alternating direction implicit (ADI) method for solving the unsteady convection-dominated diffusion equation is developed. The fourth-order Padé scheme is used for the discretization of the convection terms, while the second-order Padé scheme is used for the diffusion terms. The Crank–Nicolson scheme and ADI factorization are applied for time integration. After ADI factorization, the two-dimensional problem becomes a sequence of one-dimensional problems. The solution procedure consists of multiple use of a one-dimensional tridiagonal matrix algorithm that produces a computationally cost-effective solver. Von Neumann stability analysis is performed to show that the method is unconditionally stable. An unsteady two-dimensional problem concerning convection-dominated propagation of a Gaussian pulse is studied to test its numerical accuracy and compare it to other high-order ADI methods. The results show that the overall numerical accuracy can reach third or fourth order for the convection-dominated diffusion equation depending on the magnitude of diffusivity, while the computational cost is much lower than other high-order numerical methods. Copyright © 2011 John Wiley & Sons, Ltd.

In this work, a new enrichment space to accommodate jumps in the pressure field at immersed interfaces in finite element formulations, is proposed. The new enrichment adds two degrees of freedom per element that can be eliminated by means of static condensation. The new space is tested and compared with the classical P₁ space and to the space proposed by Ausas et al (Comp. Meth. Appl. Mech. Eng., Vol. 199, 1019–1031, 2010) in several problems involving jumps in the viscosity and/or the presence of singular forces at interfaces not conforming with the element edges. The combination of this enrichment space with another enrichment that accommodates discontinuities in the pressure gradient has also been explored, exhibiting excellent results in problems involving jumps in the density or the volume forces. Copyright © 2011 John Wiley & Sons, Ltd.

The spatial discretization of unsteady incompressible Navier–Stokes equations is stated as a system of differential algebraic equations, corresponding to the conservation of momentum equation plus the constraint due to the incompressibility condition. Asymptotic stability of Runge–Kutta and Rosenbrock methods applied to the solution of the resulting index-2 differential algebraic equations system is analyzed. A critical comparison of Rosenbrock, semi-implicit, and fully implicit Runge–Kutta methods is performed in terms of order of convergence and stability. Numerical examples, considering a discontinuous Galerkin formulation with piecewise solenoidal approximation, demonstrate the applicability of the approaches and compare their performance with classical methods for incompressible flows. Copyright © 2011 John Wiley & Sons, Ltd.

The present investigation deals with the three-dimensional flow of an Oldroyd-B fluid over a stretching surface. The governing equations for the three-dimensional flow are developed. Similarity transformations are invoked for the conversion of nonlinear partial differential equations into the coupled system of ordinary differential equations. Computations for the series solution are presented through implementation of homotopy analysis method. The salient features of the involved parameters have been pointed out.Copyright © 2011 John Wiley & Sons, Ltd.

This paper presents a family of High-order finite volume schemes applicable on unstructured grids. The k-exact reconstruction is performed on every control volume as the primary reconstruction. On a cell of interest, besides the primary reconstruction, additional candidate reconstruction polynomials are provided by means of very simple and efficient ‘secondary’ reconstructions. The weighted average procedure of the WENO scheme is then applied to the primary and secondary reconstructions to ensure the shock-capturing capability of the scheme. This procedure combines the simplicity of the k-exact reconstruction with the robustness of the WENO schemes and represents a systematic and unified way to construct High-order accurate shock capturing schemes. To further improve the efficiency, an efficient problem-independent shock detector is introduced. Several test cases are presented to demonstrate the accuracy and non-oscillation property of the proposed schemes. The results show that the proposed schemes can predict the smooth solutions with uniformly High-order accuracy and can capture the shock waves and contact discontinuities in high resolution. Copyright © 2011 John Wiley & Sons, Ltd.

A new stream function–vorticity formulation-based immersed boundary method is presented in this paper. Different from the conventional immersed boundary method, the main feature of the present model is to accurately satisfy both governing equations and boundary conditions through velocity correction and vorticity correction procedures. The velocity correction process is performed implicitly based on the requirement that velocity at the immersed boundary interpolated from the corrected velocity field accurately satisfies the nonslip boundary condition. The vorticity correction is made through the stream function formulation rather than the vorticity transport equation. It is evaluated from the firstorder derivatives of velocity correction. Two simple and efficient ways are presented for approximation of velocity-correction derivatives. One is based on finite difference approximation, while the other is based on derivative expressions of Dirac delta function and velocity correction. It was found that both ways can work very well. The main advantage of the proposed method lies in its simple concept, easy implementation, and robustness in stability. Numerical experiments for both stationary and moving boundary problems were conducted to validate the capability and efficiency of the present method. Good agreements with available data in the literature were achieved. Copyright © 2011 John Wiley & Sons, Ltd.

On the basis of two local Gauss integrations, a stabilized finite element method for transient Navier–Stokes equations is presented, which is defined by the lowest equal-order conforming finite element subspace such as (or ) elements. The best algorithmic feature of our method is using two local Gauss integrations to replace projection operator. The diffusion term in these equations is discretized by using finite element method, and the temporal differentiation and advection terms are treated by characteristic schemes. Moreover, we present some numerical simulations to demonstrate the effectiveness, good stability, and accuracy properties of our method. Especially, the rate of convergence study tells us that the stability still keeps well when the Reynolds number is increasing. Copyright © 2011 John Wiley & Sons, Ltd.

A nonhydrostatic finite volume model is presented to simulate free surface flow in a two-dimensional vertical plane. The algorithm is based on a projection method including the solution of the pressure Poisson equation (PPE). The model is developed in a Cartesian grid in which the size of all the cells in the computational domain, excluding those of the top layer, is constant in time. To simulate the variable water surface, the heights of the top layer cells are variable and proportional to the local water elevation. Taking the layout of the grid system into consideration, a new method is proposed to solve the PPE derived in Cartesian coordinates. In this method, the system of pressure equations is divided into two subsystems, namely a subsystem for the upper layer cells and another for the remaining cells. The coefficient matrix of the former is variable with respect to time, whereas that of the latter remains constant. Therefore, the coefficient matrix of the latter subsystem can be inversed once and saved throughout the simulation. The application of this procedure reduces the computational cost compared with other PPE solvers in certain conditions. The model is applied to simulate a series of numerical tests including strong vertical accelerations and is verified against analytical and experimental results, demonstrating satisfactory performance. Copyright © 2011 John Wiley & Sons, Ltd.

A conservative local interface sharpening scheme has been developed for the constrained interpolation profile method with the conservative semi-Lagrangian scheme, because the conservative semi-Lagrangian scheme does not feature a mechanism to control the interface thickness, thus causing an increase of numerical error with the advance of the time step. The proposed sharpening scheme is based on the conservative level set method proposed by Olsson and Kreiss. However, because their method can cause excessive deformation of the free-surface in certain circumstances, we propose an improvement of the method by developing a local sharpening technique. Several advection tests are presented to assess the correctness of the advection and the improved interface sharpening scheme. This is followed by the validations of dam-breaking flow and the rising bubble flows. The mass of the fluid is exactly conserved and the computed terminal velocity of the rising bubble agrees well with the experiments compared with other numerical methods such as the volume of fluid method (VOF), the front tracking method, and the level set method. Copyright © 2011 John Wiley & Sons, Ltd.

A Q2Q1 (quadratic velocity/linear pressure) finite element/level-set method was proposed for simulating incompressible two-phase flows with surface tension. The Navier–Stokes equations were solved using the Q2Q1 integrated FEM, and the level-set variable was linearly interpolated using a ‘pseudo’ Q2Q1 finite element when calculating the density and viscosity of a fluid to avoid an unbounded density/viscosity. The advection of the level-set function was calculated through the Taylor–Galerkin method, and the direct approach method is employed for reinitialization. The proposed method was tested by solving several benchmark problems including rising bubbles exhibiting a large density difference and the surface tension effect. The numerical results of the rising bubbles were compared with the existing results to validate the benchmark quantities such as the centroid, circularity, and rising velocity. Furthermore, we focused our attention mainly on mass conservation and time-step. We observed that the present method represented a convergence rate between 1.0 and 1.5 orders in terms of mass conservation and provided more stable solutions even when using a larger time-step than the critical time-step that was imposed because of the explicit treatment of surface tension. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we consider the unsteady boundary layer flow of an incompressible viscous fluid produced by periodic motion of an elastic cylinder. The number of independent variables involved in the governing partial differential equations is reduced by using the similarity transformation. The transformed equations are then solved numerically with the help of a finite difference scheme by altering the semi-infinite domain to a finite domain. The numerical results are compared with a previously published work and a good agreement is achieved. The imperative parameters rising in the governing equations because of the effects of oscillations and the curvature are St (Strouhal number), Re (Reynolds number), and ϵ (amplitude of oscillations). The effects of these parameters on the vacillating velocity and the skin friction are discussed through graphs and tables. The large values of Re correspond to small curvature, so for large Reynolds number the solution approaches to that of the stretching flat plate case. Copyright © 2011 John Wiley & Sons, Ltd.

Stability is achieved in most approximate Riemann solvers through ‘flux upwinding’, where the flux at the interface is arrived at by adding a dissipative term to the average of the left and right flux. Motivated by the existence of a collapsed interface state in the gas-kinetic Bhatnagar–Gross–Krook (BGK) method, an alternative approach to upwinding is attempted here; an interface state is arrived at by taking an upwinded average of left and right states, and then the flux is calculated as a function of this ‘collapsed’ interface state. This so called ‘state-upwinding’ approach gives rise to a new scheme called the linearized Riemann solver for the Euler and Navier–Stokes equations. The scheme is shown to be closely associated with the Roe scheme. It is, however, computationally less expensive and gives qualitatively comparable results over a wide range of problems. Most importantly, this scheme is found to preserve stationary contacts while not exhibiting the carbuncle phenomenon which plagues the Roe and other contact-preserving schemes. The scheme is therefore motivated as a new starting point to analyze the origin of the carbuncle phenomenon. Copyright © 2011 John Wiley & Sons, Ltd.

On the basis of the existing density distribution function reconstruction operator, the temperature distribution operator was derived to calculate heat transfer by coupling the lattice Boltzmann method (LBM) with the finite volume method. The present coupling model was validated by two-dimensional natural convection flows with and without an isolated internal vertical plate. The results from the coupling model agree well with those from the pure-finite volume method, pure-LBM and references, and all the physical quantities cross the coupled interface smoothly. On the basis of residual history curves, it is likely that the convergence property and the numerical stability of the present model are better than those of the pure-LBM at fine grid numbers and high Rayleigh numbers. Copyright © 2011 John Wiley & Sons, Ltd.

In this study, a numerical investigation of melting phenomenon with natural convection in a cavity with fin has been performed using enthalpy-based lattice Boltzmann method. The lattice D2Q9 model was applied to determine the density and velocity fields, and the D2Q5 model for the temperature field. The effect of vertical position and length of the fin on the melting rate was studied. The simulations were carried out for Stefan number of 10, Rayleigh number of 10 ⁵ and relative thermal conductivity (k_fin∕k_fluid) ranging from 5 to 30. The obtained results show that the rate of melting increases when the relative thermal conductivity and the length of the fin become greater. We also found that the variation of vertical position of the fin from bottom to middle has an insignificant effect on melting while it causes the increase of full melting time when the fin is mounted on the top of the cavity. Copyright © 2011 John Wiley & Sons, Ltd.

This paper presents an anisotropic mesh adaptation method applied to industrial combustion problems. The method is based on a measure of the distance between two Riemannian metrics called metric non-conformity. This measure, which can be used to build a cost function to adapt meshes comprising several types of mesh elements, provides the basis for a generic mesh adaptation approach applicable to various types of physical problems governed by partial differential equations. The approach is shown to be applicable to industrial combustion problems, through the specification of a target metric computed as the intersection of several Hessian matrices reconstructed from the main variables of the governing equations. Numerical results show that the approach is cost effective in that it can drastically improve the prediction of temperature and species distributions in the flame region of a combustor while reducing computational cost. The results can be used as a basis for pollutant prediction models. Copyright © 2011 John Wiley & Sons, Ltd.

The suitability of Wilcox's 2006 k– ω turbulence model for scramjet flowfield simulations is demonstrated by validation against five test cases that have flowfields representative of those to be expected in scramjets. The five test cases include a 2D flat plate, an axisymmetric cylinder, a backward-facing step, the mixing of a pair of coaxial jets and the interaction between a shock wave and turbulent boundary layer. A generally good agreement between the numerical and experimental results is obtained for all test cases. These tests reveal that despite the turbulence model's sensitivity to freestream turbulence properties, the numerically predicted skin friction agrees with experimental data and theoretical correlations to their degree of uncertainty. The tests also confirm the importance of using a y⁺ value of less than 1 in getting accurate surface heat transfer distributions. In the coaxial jets case, the importance of matching the turbulence intensities at the inflow plane in improving the predictions of the turbulent mixing phenomena is also shown. A review of guidelines with regard to the setting up of grids and specification of freestream turbulence properties for turbulent Reynolds-averaged Navier–Stokes CFD simulations is also included in this paper. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we study the breakup behavior of Newtonian liquid and non-Newtonian liquid jets with an arbitrary variation surface tension imposed along its length. The effect of duty cycle, fluid properties, and the various profiles of the surface tension is investigated. It is shown that the breakup behavior of a jet can be constructed by using the Fourier expansion of the surface tension profile. When the dimensionless wavenumber k is larger than 0.5, the jet breakup behavior is determined by the lowest frequency of the Fourier series expansion of the surface tension profile. As k decreases, higher frequency Fourier modes come to play. In general, for k between, 1∕(n+ 1) and 1∕n,n Fourier modes are needed to determine the jet breakup behavior. The current nonlinear model differs from the existing linear slender jet model in the literature in several ways. While the principle of superposition is valid for the linear model, it is not generally valid for the current nonlinear model. For the linear model, the jet will never break up when the wavenumber is larger than 1. The current model, however, shows clearly that the jet can indeed break up when the wavenumber is larger than 1. Furthermore, the current nonlinear model predicts a breakup time substantially higher than that from the linear model.Copyright © 2011 John Wiley & Sons, Ltd.

Adjoints are used in optimization to speed-up computations, simplify optimality conditions or compute sensitivities. Because time is reversed in adjoint equations with first-order time derivatives, boundary conditions, and transmission conditions through shocks can be difficult to understand. In this article, we analyze the adjoint equations that arise in the context of compressible flows governed by the Euler equations of fluid dynamics. We show that the continuous and discrete adjoints computed by automatic differentiation agree numerically; in particular, the adjoint is found to be continuous at the shocks and usually discontinuous at contact discontinuities by both.Copyright © 2011 John Wiley & Sons, Ltd.

We present numerical results for in-line and cross-flow vibrations of a circular cylinder, which is immersed in a uniform flow and is elastically supported by damper-spring systems to compute vibrations of a rigid cylinder. In the case of a circular cylinder with a low Scruton number, it is well-known that two types of self-excited vibrations appear in the in-line direction in the range of low reduced velocities. On the other hand, a cross-flow vibration of the circular cylinder can be excited in the range of high reduced velocities. Therefore, we compute the flow-induced vibrations of the circular cylinder in the wide range of the reduced velocities at low and high Scruton numbers and discuss about excitation mechanisms in the in-line and cross-flow directions. Copyright © 2011 John Wiley & Sons, Ltd.

A family of flux-continuous, locally conservative, control-volume-distributed multi-point flux approximation (CVD-MPFA) schemes has been developed for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids. These schemes are applicable to the full-tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full-tensor flow approximation. The family of flux-continuous schemes is characterized by a quadrature parameterization. Improved numerical convergence for the family of CVD-MPFA schemes using the quadrature parameterization has been observed for structured and unstructured grids in two dimensions.

The CVD-MPFA family cell-vertex formulation is extended to classical general element types in 3-D including prisms, pyramids, hexahedra and tetrahedra. A numerical convergence study of the CVD-MPFA schemes on general unstructured grids comprising of triangular elements in 2-D and prismatic, pyramidal, hexahedral and tetrahedral shape elements in 3-D is presented.Copyright © 2011 John Wiley & Sons, Ltd.

The influence of aspect ratio in three-dimensional, numerical experiments of separated flows is studied in the case of the backward-facing step at Reynolds numbers 600, 800, and 950. The computational domain is designed as an actual laboratory experiment. The governing equations are the steady state, isothermal, and incompressible Navier–Stokes equations. The expansion ratio of the computational domain is 1:2. The aspect ratio varies from 1:10 to 1:40. The results of the computations focus on the spanwise variations of the length and the strength of the two eddies along the lower and upper wall. It is concluded that both numerical and laboratory experiments should be designed with an aspect ratio of at least 1:20, if only the accuracy of the position of the detachment and the re-attachment points matters. If the accuracy of the shear-stress distributions is also taken into account, then an aspect ratio of at least 1:30 should be chosen. Finally, if the magnitudes of the vortex centers are also considered, then only the aspect ratio of 1:40 qualifies for a realization of two-dimensional flow conditions in the plane of symmetry. This is contrary to the common practice in the field, at least from the side of laboratory experiments, where an aspect ratio of 1:10 is still considered adequate for this purpose. Copyright © 2011 John Wiley & Sons, Ltd.

In this work we introduce a model of the boundary layer equations for a perfect conducting micropolar fluid with stretch, bounded by an infinite vertical flat plane surface of a constant temperature. This model is applied to study the effects of free convection currents on the flow of the fluid in the presence of a constant magnetic field. The state space technique is adopted for the solution of a one-dimensional problem for any set of boundary conditions. The resulting formulation together with the Laplace transform techniques are applied to a thermal shock problem. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results are given and illustrated graphically for the problem. Copyright © 2011 John Wiley & Sons, Ltd.

In-depth-averaged and cross-section-averaged morphodynamic models, based on explicit time integration, it may happen that the computed bed level becomes lower than the top level of a non-erodible layer (e.g. concrete, bedrock or armoured layer). This is a standard pitfall, which has been addressed in different ways. In this paper, we present an original approach for avoiding computation of non-physical bed levels, using an iterative procedure to correct the outward sediment fluxes. The procedure is shown to be computationally efficient while it achieves a high accuracy in terms of mass conservation. We compare our original approach with the existing Struiksma's method and with a reformulation of the problem in terms of mathematical optimization of a linear or nonlinear objective function under linear constraints.

The new procedure has been incorporated into an existing finite volume morphodynamic model. It has been validated with several 1D benchmarks leading to configurations with sediment transport over non-erodible bottom. The computation time has been verified not to increase by more than 15% compared with runs without non-erodible bottom. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, an adaptive refinement strategy based on a node-moving technique is proposed and used for the efficient solution of the steady-state incompressible Navier–Stokes equations. The value of a least squares functional of the residual of the governing differential equation and its boundary conditions at nodal points is regarded as a measure of error and used to predict the areas of poor solutions. A node-moving technique is then used to move the nodal points to the zones of higher numerical errors. The problem is then resolved on the refined distribution of nodes for higher accuracy. A spring analogy is used for the node-moving methodology in which nodal points are connected to their neighbors by virtual springs. The stiffness of each spring is assumed to be proportional to the errors of its two end points and its initial length. The new positions of the nodal points are found such that the spring system attains its equilibrium state. Some numerical examples are used to illustrate the ability of the proposed scheme for the adaptive solution of the steady-state incompressible Navier–Stokes equations. The results demonstrate a considerable improvement of the results with a reasonable computational effort by using the proposed adaptive strategy. Copyright © 2011 John Wiley & Sons, Ltd.

The aim of this paper is to propose an effective anisotropic mesh adaptation procedure for the solution of the shallow water equations. The hyperbolic partial differential equation system is solved via the streamline diffusion FEM, suitably modified by a shock-capturing correction. The proposed adaptation procedure relies on a recovery-based error estimator. In particular, we look for an anisotropic error estimator able to select not only the size but also the shape and the orientation of the mesh elements, with the aim of optimizing the computational advantages yielded by a standard isotropic mesh adaptation strategy. The robustness of the proposed estimator is assessed when either a single physical quantity, meaningful for the problem at hand, or a combination of all the shallow water system components drives the adaptation procedure. For this purpose, steady shallow water problems as well as unsteady configurations are considered. Copyright © 2011 John Wiley & Sons, Ltd.

We have developed a production-level foam processing computational model suitable for predicting the self-expansion of foam in complex geometries. The model is based on a finite element representation of the equations of motion, with the movement of the free surface represented using the level set method. An empirically based time-dependent and temperature-dependent density model is used to encapsulate the complex physics of foam nucleation and growth in a numerically tractable manner. The evolving density drives the dynamics of foam self-expansion. This continuum-level model uses a homogenized description of foam, which does not include the gas explicitly, but allows varying local fields, such as temperature and gas volume fraction, and material models. In addition, material models vary with the location of the level set interface, taking properties of the displaced air phase in the negative level set region and the foam in the positive region. The level set zero describes the location of the interface, where surface forces are applied using the continuous surface force treatment. The variation from foam to gas properties is handled with a diffuse interface method using a smooth Heaviside function and equation averaging. Material model development was guided and populated by careful experiments. Results from the model are compared with temperature-instrumented flow visualization experiments giving the location of the foam front as a function of time for a physically blown, epoxy foam. Good qualitative agreement is seen between simulations and experiments, although some of the subtleties of the filling process are lost to the model. Published 2011. This article is a US Government work and is in the public domain in the USA.

We consider the magnetohydrodynamic flow that is laminar and steady of a viscous, incompressible, and electrically conducting fluid in a semi-infinite duct under an externally applied magnetic field. The flow is driven by the current produced by a pressure gradient. The applied magnetic field is perpendicular to the semi-infinite walls that are kept at the same magnetic field value in magnitude but opposite in sign. The wall that connects the two semi-infinite walls is partly non-conducting and partly conducting (in the middle). A BEM solution was obtained using a fundamental solution that enables to treat the magnetohydrodynamic equations in coupled form with general wall conductivities. The inhomogeneity in the equations due to the pressure gradient was tackled, obtaining a particular solution, and the BEM was applied with a fundamental solution of coupled homogeneous convection–diffusion type partial differential equations. Constant elements were used for the discretization of the boundaries (y  =  0, −a ⩽ x ⩽ a) and semi-infinite walls at x  =  ±a, by keeping them as finite since the boundary integral equations are restricted to these boundaries due to the regularity conditions as y  →  ∞ . The solution is presented in terms of equivelocity and induced magnetic field contours for several values of Hartmann number (M), conducting length (l), and non-conducting wall conditions (k). The effect of the parameters on the solution is studied. Flow rates are also calculated for these values of parameters. Copyright © 2011 John Wiley & Sons, Ltd.

A comparison among three weakly nonlinear approaches for thermo-gravitational instability in a Newtonian fluid layer heated from below is presented. First, the dynamical systems describing the time evolution of the problem from different weakly nonlinear approaches, namely, the Lorenz model, the amplitude equations and the perturbation expansion approaches are obtained. Next, the steady states and their stability, as well as the transient behaviour are obtained from each dynamical system. The similarity and difference among the three models are emphasized. The role of each of the nondimensional groups, the Rayleigh number and the Prandtl number is compared for the three models. The different approaches lead to similar behaviours when the Rayleigh number is just above its critical value and Prandtl number is high. However, only the dynamical system obtained from the amplitude equations is able to reflect the role of the Prandtl number. On the other hand, the amplitude equations and perturbation expansion techniques are not suitable for predicting the uniform oscillatory behaviour observed frequently in Rayleigh–Bénard convection. The novelty of the current work lies in studying the critical differences in the findings of the three popular approaches to investigate weakly nonlinear thermal convection for the first time. Copyright © 2011 John Wiley & Sons, Ltd.

A two-phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre-breaking, overturning and post-breaking processes. The model is based on the Reynolds-averaged Navier–Stokes equations with the k −ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure-velocity coupling and the air-water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep-water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash-up during wave breaking have been captured by the two-phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.

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.

This paper explores the application of SPH to a DNS of decaying turbulence in a two-dimensional no-slip wall-bounded domain. In this bounded domain, the inverse energy cascade, and a net torque exerted by the boundary, results in a spontaneous spin-up of the fluid, leading to a typical end state of a large monopole vortex that fills the domain. The SPH simulations were compared against published results using a high-accuracy pseudo-spectral code. Ensemble averages of the kinetic energy, enstrophy and average vortex wavenumber compared well against the pseudo-spectral results, as did the evolution of the total angular momentum of the fluid. However, although the pseudo-spectral results emphasised the importance of the no-slip boundaries as generators of long-lived coherent vortices in the flow, no such generation was seen in the SPH results. Vorticity filaments produced at the boundary were always dissipated by the flow shortly after separating from the boundary layer. The kinetic energy spectrum of the SPH results was calculated using an SPH Fourier transform that operates directly on the disordered particles. The ensemble kinetic energy spectrum showed the expected k⁻³ scaling over most of the inertial range. However, the spectrum flattened at smaller length scales (initially less than 7.5 particle spacings and growing in size over time), indicating an excess of small-scale kinetic energy.Copyright © 2011 John Wiley & Sons, Ltd.

The influence of partial slip, thermal radiation, chemical reaction and temperature-dependent fluid properties on heat and mass transfer in hydro-magnetic micropolar fluid flow over an inclined permeable plate with constant heat flux and non-uniform heat source/sink is studied. The transverse magnetic field is assumed as a function of the distance from the origin. Also it is assumed that the fluid viscosity and the thermal conductivity vary as an inverse function and linear function of temperature, respectively. With the use of the similarity transformation, the governing system of non-linear partial differential equations are transformed into non-linear ordinary differential equations and are solved numerically using symbolic software MATHEMATICA 7.0 (Wolfram Research, Champaign, IL). The numerical values obtained for the velocity, microrotation, temperature, species concentration, skin friction coefficient and the Nusselt number are presented through graphs and tables for several sets of values of the parameters. The effects of various physical parameters on the flow and heat transfer characteristics are discussed.Copyright © 2011 John Wiley & Sons, Ltd.

A horizontally curvilinear non-hydrostatic free surface model that embeds the second-order projection method, the so-called θ scheme, in fractional time stepping is developed to simulate nonlinear wave motion in curved boundaries. The model solves the unsteady, Navier–Stokes equations in a three-dimensional curvilinear domain by incorporating the kinematic free surface boundary condition with a top-layer boundary condition, which has been developed to improve the numerical accuracy and efficiency of the non-hydrostatic model in the standard staggered grid layout. The second-order Adams–Bashforth scheme with the third-order spatial upwind method is implemented in discretizing advection terms. Numerical accuracy in terms of nonlinear phase speed and amplitude is verified against the nonlinear Stokes wave theory over varying wave steepness in a two-dimensional numerical wave tank. The model is then applied to investigate the nonlinear wave characteristics in the presence of dispersion caused by reflection and diffraction in a semicircular channel. The model results agree quantitatively with superimposed analytical solutions. Finally, the model is applied to simulate nonlinear wave run-ups caused by wave-body interaction around a bottom-mounted cylinder. The numerical results exhibit good agreement with experimental data and the second-order diffraction theory. Overall, it is shown that the developed model, with only three vertical layers, is capable of accurately simulating nonlinear waves interacting within curved boundaries. Copyright © 2011 John Wiley & Sons, Ltd.

This work presents a numerical model designed for the simulation of water-wave impacts on a structure when aeration of the liquid phase is considered. The model is based on a multifluid Navier–Stokes approach in which all fluids are assumed compressible. The numerical method is based on a finite volume algorithm in space and a second order Runge–Kutta method in time. A validation of this model is performed. It shows a good accuracy for acoustic and shock wave propagation in a bubbly liquid and for wave breaking. Copyright © 2011 John Wiley & Sons, Ltd.

In this article, a new methodology for developing discrete geometric conservation law (DGCL) compliant formulations is presented. It is carried out in the context of the finite element method for general advective–diffusive systems on moving domains using an ALE scheme. There is an extensive literature about the impact of DGCL compliance on the stability and precision of time integration methods. In those articles, it has been proved that satisfying the DGCL is a necessary and sufficient condition for any ALE scheme to maintain on moving grids the nonlinear stability properties of its fixed-grid counterpart. However, only a few works proposed a methodology for obtaining a compliant scheme. In this work, a DGCL compliant scheme based on an averaged ALE Jacobians formulation is obtained. This new formulation is applied to the θ family of time integration methods. In addition, an extension to the three-point backward difference formula is given. With the aim to validate the averaged ALE Jacobians formulation, a set of numerical tests are performed. These tests include 2D and 3D diffusion problems with different mesh movements and the 2D compressible Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.

This paper presents a comparison in terms of accuracy and efficiency between two fully nonlinear potential flow solvers for the solution of gravity wave propagation. One model is based on the high-order spectral (HOS) method, whereas the second model is the high-order finite difference model OceanWave3D. Although both models solve the nonlinear potential flow problem, they make use of two different approaches. The HOS model uses a modal expansion in the vertical direction to collapse the numerical solution to the two-dimensional horizontal plane. On the other hand, the finite difference model simply directly solves the three-dimensional problem. Both models have been well validated on standard test cases and shown to exhibit attractive convergence properties and an optimal scaling of the computational effort with increasing problem size. These two models are compared for solution of a typical problem: propagation of highly nonlinear periodic waves on a finite constant-depth domain. The HOS model is found to be more efficient than OceanWave3D with a difference dependent on the level of accuracy needed as well as the wave steepness. Also, the higher the order of the finite difference schemes used in OceanWave3D, the closer the results come to the HOS model. Copyright © 2011 John Wiley & Sons, Ltd.

We implement and evaluate a massively parallel and scalable algorithm based on a multigrid preconditioned Defect Correction method for the simulation of fully nonlinear free surface flows. The simulations are based on a potential model that describes wave propagation over uneven bottoms in three space dimensions and is useful for fast analysis and prediction purposes in coastal and offshore engineering. A dedicated numerical model based on the proposed algorithm is executed in parallel by utilizing affordable modern special purpose graphics processing unit (GPU). The model is based on a low-storage flexible-order accurate finite difference method that is known to be efficient and scalable on a CPU core (single thread). To achieve parallel performance of the relatively complex numerical model, we investigate a new trend in high-performance computing where many-core GPUs are utilized as high-throughput co-processors to the CPU. We describe and demonstrate how this approach makes it possible to do fast desktop computations for large nonlinear wave problems in numerical wave tanks (NWTs) with close to 50/100 million total grid points in double/single precision with 4 GB global device memory available. A new code base has been developed in C++ and compute unified device architecture C and is found to improve the runtime more than an order in magnitude in double precision arithmetic for the same accuracy over an existing CPU (single thread) Fortran 90 code when executed on a single modern GPU. These significant improvements are achieved by carefully implementing the algorithm to minimize data-transfer and take advantage of the massive multi-threading capability of the GPU device. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, the steady flow and heat transfer of a magnetohydrodynamic fluid is studied. The fluid is assumed to be electrically conducting in the presence of a uniform magnetic field and occupies the porous space in annular pipe. The governing nonlinear equations are modeled by introducing the modified Darcy's law obeying the Sisko model. The system is solved using the homotopy analysis method (HAM), which yields analytical solutions in the form of a rapidly convergent infinite series. Also, HAM is used to obtain analytical solutions of the problem for noninteger values of the power index. The resulting problem for velocity field is then numerically solved using an iterative method to show the accuracy of the analytic solutions. The obtained solutions for the velocity and temperature fields are graphically sketched and the salient features of these solutions are discussed for various values of the power index parameter. We also present a comparison between Sisko and Newtonian fluids. Copyright © 2011 John Wiley & Sons, Ltd.

Recently, we developed an explicit a posteriori error estimator especially suited for fluid dynamics problems solved with a stabilized method. The technology is based upon the theory that inspired stabilized methods, namely, the variational multiscale theory. The salient features of the formulation are that it can be readily implemented in existing codes, it is a very economical procedure, and it yields very accurate local error estimates uniformly from the diffusive to the advective regime.

In this work, the variational multiscale error estimator is applied to develop adaptive strategies for the advection–diffusion-reaction equation. The performance of L₁ and L₂ local error norms combined with three strategies to adapt the mesh is investigated. Emphasis is placed on flows with sharp boundary and interior layers but also attention is given to diffusion-dominated flows. Computational results show that the method generates meshes with a smooth transition of the element size, which capture all the flow features. Copyright © 2011 John Wiley & Sons, Ltd.

In this paper, we propose a computational algorithm for the solution of thermally coupled flows in subsonic regime. The formulation is based upon the compressible Navier–Stokes equations, written in nonconservation form. An efficient modular implementation is obtained by solving the energy equation separately and then using the computed temperature as a known value in the momentum-continuity system. If an explicit single-step time integration scheme for the energy equation is used, the decoupling results to be natural.

Integration of the momentum-continuity system is carried out using a semi-explicit method, combining Runge–Kutta and Backward Euler schemes for the momentum and continuity equations, respectively. Implicit treatment of pressure leads to favorable time step estimates even in the low Mach number (Ma ≪ 1) regimes. The numerical dissipation introduced by the Backward Euler scheme ensures absence of the spurious high frequencies in the numerical solution.

The key point of the method is the assumption of linear variation of the temperature within a time step. Combined with a fractional splitting of the momentum-continuity system, it allows to solve the continuity only once per time step. Omitting the necessity of solving for the pressure at every intermediate step of the Runge–Kutta scheme minimizes the computational cost associated to the implicit step and leads to an efficiency close to that of a purely explicit scheme.

The method is tested using two benchmark examples.Copyright © 2011 John Wiley & Sons, Ltd.

The two-dimensional flows past a circular cylinder near a moving wall are simulated by lattice Boltzmann method. The wall moves at the inlet velocity and the Reynolds number ranges from 300 to 500. The influence of the moving wall on the flow patterns is demonstrated and the corresponding mechanism is illustrated by using instability theory. The correlations among flow features based on gap ratio are interpreted. Force coefficients, velocity profile and vortex structure are analyzed to determine the critical gap ratio. Copyright © 2011 John Wiley & Sons, Ltd.

A novel augmented forcing point method is presented to solve the problem of chemical transport in the fluid outside of a collection of suspended cells coupled with chemical reactions on the surfaces of the cells. In this method, the chemical concentrations and the forcing function values are determined simultaneously from an augmented system of equations. The method is more stable and accurate than predictor-corrector-type forcing point methods, yields the same solution as a corresponding ghost cell method with much less computational cost, and provides solutions that are pointwise second-order accurate in space and time, even for closely-spaced cells. Copyright © 2011 John Wiley & Sons, Ltd.

A typical large-scale CFD code based on adaptive, edge-based finite-element formulations for the solution of compressible and incompressible flow is taken as a test bed to port such codes to graphics hardware (graphics processing units, GPUs) using semi-automatic techniques. In previous work, a GPU version of this code was presented, in which, for many run configurations, all mesh-sized loops required throughout time stepping were ported. This approach simultaneously achieves the fine-grained parallelism required to fully exploit the capabilities of many-core GPUs, completely avoids the crippling bottleneck of GPU–CPU data transfer, and uses a transposed memory layout to meet the distinct memory access requirements posed by GPUs. The present work describes the next step of this porting effort, namely to integrate GPU-based, fine-grained parallelism with Message-Passing-Interface-based, coarse-grained parallelism, in order to achieve a code capable of running on multi-GPU clusters. This is carried out in a semi-automated fashion: the existing Fortran–Message Passing Interface code is preserved, with the translator inserting data transfer calls as required. Performance benchmarks indicate up to a factor of 2 performance advantage of the NVIDIA Tesla M2050 GPU (Santa Clara, CA, USA) over the six-core Intel Xeon X5670 CPU (Santa Clara, CA, USA), for certain run configurations. In addition, good scalability is observed when running across multiple GPUs. The approach should be of general interest, as how best to run on GPUs is being presently considered for many so-called legacy codes.Copyright © 2011 John Wiley & Sons, Ltd.

On the basis of the Helmholtz decomposition, a grid-free numerical scheme is provided for the solution of unsteady flow in hydraulic turbines. The Lagrangian vortex method is utilized to evaluate the convection and stretch of the vorticity, and the BEM is used to solve the Neumann problem to define the potential flow. The no-slip boundary condition is satisfied by generating vortex sticks at the solid surface. A semi-analytical regularization technique is applied to evaluate the singular boundary surface integrals of the potential velocity and its gradients accurately. The fast multipole method was extended to evaluate the velocity and velocity gradients induced by the discretized vortex blobs in the Lagrangian vortex method. The successful simulation for the unsteady flow through a hydraulic turbine's runner has manifested the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.

The local domain-free discretization method is extended in this work to simulate fluid–structure interaction problems, the class of which is exemplified by the self-propelled anguilliform swimming of deforming bodies in a fluid medium. Given the deformation of the fish body in its own reference frame, the translational and rotational motions of the body governed by Newton's Law are solved together with the surrounding flow field governed by Navier–Stokes equations. When the body is deforming and moving, no mesh regeneration is required in the computation. The loose coupling strategy is employed to simulate the fluid–structure interaction involved in the self-propelled swimming. The local domain-free discretization method and an efficient algorithm for classifying the Eulerian mesh points are described in brief. To validate the fluid–structure interaction solver, we simulate the ‘lock-in’ phenomena associated with the vortex-induced vibrations of an elastically mounted cylinder. Finally, we demonstrate applications of the method to two-dimensional and three-dimensional anguilliform-swimming fish. The kinematics and dynamics associated with the center of mass are shown and the rotational movement is also presented via the angular position of the body axis. The wake structure is visualized in terms of vorticity contours. All the obtained numerical results show good agreement with available data in the literature.Copyright © 2011 John Wiley & Sons, Ltd.

The experimental and numerical analysis of spheres falling into viscous flows is considered. The physical model is built using a set of silicone and glass spheres falling into oil and water. The rigid-body trajectory of the sphere and the free surface evolution are obtained from videos. The numerical results are obtained using two different finite element codes. The first code uses a fractional step approach with adaptive meshes and time-step sizes whereas the second code uses a monolithic fully coupled fixed-mesh technique. The results exhibit a good comparison between both numerical techniques and with the experiments. Copyright © 2011 John Wiley & Sons, Ltd.

In this study, an immersed boundary vortex-in-cell (VIC) method for simulating the incompressible flow external to two-dimensional and three-dimensional bodies is presented. The vorticity transport equation, which is the governing equation of the VIC method, is represented in a Lagrangian form and solved by the vortex blob representation of the flow field. In the present scheme, the treatment of convection and diffusion is based on the classical fractional step algorithm. The rotational component of the velocity is obtained by solving Poisson's equation using an FFT method on a regular Cartesian grid, and the solenoidal component is determined from solving an integral equation using the panel method for the convection term, and the diffusion term is implemented by a particle strength exchange scheme. Both the no-slip and no-through flow conditions associated with the surface boundary condition are satisfied by diffusing vortex sheet and distributing singularities on the body, respectively. The present method is distinguished from other methods by the use of the panel method for the enforcement of the no-through flow condition. The panel method completes making use of the immersed boundary nature inherent in the VIC method and can be also adopted for the calculation of the pressure field. The overall process is parallelized using message passing interface to manage the extensive computational load in the three-dimensional flow simulations. Copyright © 2011 John Wiley & Sons, Ltd.

In this article, an extension to the total variation diminishing finite volume formulation of the lattice Boltzmann equation method on unstructured meshes was presented. The quadratic least squares procedure is used for the estimation of first-order and second-order spatial gradients of the particle distribution functions. The distribution functions were extrapolated quadratically to the virtual upwind node. The time integration was performed using the fourth-order Runge–Kutta procedure. A grid convergence study was performed in order to demonstrate the order of accuracy of the present scheme. The formulation was validated for the benchmark two-dimensional, laminar, and unsteady flow past a single circular cylinder. These computations were then investigated for the low Mach number simulations. Further validation was performed for flow past two circular cylinders arranged in tandem and side-by-side. Results of these simulations were extensively compared with the previous numerical data. Copyright © 2011 John Wiley & Sons, Ltd.

Mean-flow three-dimensionalities affect both the turbulence level and the coherent flow structures in wall-bounded shear flows. A tailor-made flow configuration was designed to enable a thorough investigation of moderately and severely skewed channel flows. A unidirectional shear-driven plane Couette flow was skewed by means of an imposed spanwise pressure gradient. Three different cases with 8°, 34°and 52°skewing were simulated numerically and the results compared with data from a purely two-dimensional plane Couette flow. The resulting three-dimensional flow field became statistically stationary and homogeneous in the streamwise and spanwise directions while the mean velocity vector V and the mean vorticity vector Ω remained parallel with the walls. Mean flow profiles were presented together with all components of the Reynolds stress tensor. The mean shear rate in the core region gradually increased with increasing skewing whereas the velocity fluctuations were enhanced in the spanwise direction and reduced in the streamwise direction.

The Reynolds shear stress is known to be closely related to the coherent flow structures in the near-wall region. The instantaneous and ensemble-averaged flow structures were turned by the skewed mean flow. We demonstrated for the medium-skewed case that the coherent structures should be examined in a coordinate system aligned with V to enable a sound interpretation of 3D effects. The conventional symmetry between Case 1 and Case 2 vortices was broken and Case 1 vortices turned out to be stronger than Case 2. This observation is in conflict with the common understanding on the basis of the spanwise (secondary) mean shear rate. A refined model was proposed to interpret the structure modifications in three-dimensional wall-flows. What matters is the orientation of the mean vorticity vector Ω relative to the vortex vorticity vector ω_v, that is, the sign of Ω ·ω_v. In the present situation, Ω ·ω_v  >  0 for the Case 1 vortices causing a strengthening relative to the Case 2 vortices. Copyright © 2011 John Wiley & Sons, Ltd.

This paper presents an incompressible SPH (ISPH) technique to simulate multifluid flows. The SPH method is a mesh-free particle modeling approach that can treat free surfaces and multi-interfaces in a simple and efficient manner. The ISPH method employs an incompressible hydrodynamic formulation to solve the fluid pressure that ensures a stable pressure field. Two multifluid ISPH models are proposed following different interface treatments: the coupled ISPH model does not distinguish the different fluid phases and applies the standard ISPH technique across the interface, whereas the decoupled ISPH model first treats each fluid phase separately and then couples the different phases by applying pressure and shear stress continuities across the interface. The two proposed models were used to investigate a gravity underflow with a low density ratio in a Generalized Reservoir Hydrodynamics (GRH) flume and a horizontal lock exchange flow with a high density ratio. Comparisons with data and relevant numerical error analysis indicated that the decoupled model performed well in cases of both low and high density ratios because of the accurate treatment of interface boundaries. Copyright © 2011 John Wiley & Sons, Ltd.

The automatic optimization of flow control devices is a delicate issue because of the drastic computational time related to unsteady high-fidelity flow analyses and the possible multimodality of the objective function. Thus, we experiment in this article the use of kriging-based algorithms to optimize flow control parameters because these methods have shown their efficiency for global optimization at moderate cost. Navier–Stokes simulations, carried out for different control parameters, are used to build iteratively a kriging model. At each step, a statistical analysis is performed to enrich the model with new simulation results by exploring the most promising areas, until optimal flow control parameters are found. This approach is validated and demonstrated on two problems, including comparisons with similar studies: the control of the flow around an oscillatory rotating cylinder and the reduction of the intensity of a shock wave for a transonic airfoil by adding a bump to the airfoil profile. Copyright © 2011 John Wiley & Sons, Ltd.

A population balance system that models the synthesis of urea is studied in this paper. The equations for the flow field, the mass and the energy balances are given in a three-dimensional domain, while the equation for the particle size distribution is given in a four-dimensional domain. This problem is convection-dominated and aggregation-driven. Both features require the application of appropriate numerical methods. This paper presents a numerical approach for simulating the population balance system, which is based on finite element schemes, a finite difference method and a modern method to evaluate convolutIon integrals that appear in the aggregation term. Two experiments are considered and the numerical results are compared with experimental data. Unknown parameters in the aggregation kernel have to be calibrated. For appropriately chosen parameters, good agreements are achieved of the experimental data and the numerical results computed with the proposed method. A detailed study of the computational results reveals the influence of different parts of the aggregation kernel.Copyright © 2011 John Wiley & Sons, Ltd.

One of the techniques available for optimising parameters that regulate dispersion and dissipation effects in finite difference schemes is the concept of minimised integrated exponential error for low dispersion and low dissipation. In this paper, we work essentially with the two-dimensional (2D) Corrected Lax–Friedrichs and Lax–Friedrichs schemes applied to the 2D scalar advection equation. We examine the shock-capturing properties of these two numerical schemes, and observe that these methods are quite effective from the point of being able to control computational noise and having a large range of stability. To improve the shock-capturing efficiency of these two methods, we derive composite methods using the idea of predictor/corrector or a linear combination of the two schemes. The optimal cfl number for some of these composite schemes are computed. Some numerical experiments are carried out in two dimensions such as cylindrical explosion, shock-focusing, dam-break and Riemann gas dynamics tests. The modified equations of some of the composite schemes when applied to the 2D scalar advection equation are obtained. We also perform some convergence tests to obtain the order of accuracy and show that better results in terms of shock-capturing property are obtained when the optimal cfl obtained using minimised integrated exponential error for low dispersion and low dissipation is used. Copyright © 2011 John Wiley & Sons, Ltd.

This article presents a novel shock-capturing technique for the discontinuous Galerkin (DG) method. The technique is designed for compressible flow problems, which are usually characterized by the presence of strong shocks and discontinuities. The inherent structure of standard DG methods seems to suggest that they are especially adapted to capture shocks because of the numerical fluxes based on suitable approximate Riemann solvers, which, in practice, introduces some stabilization. However, the usual numerical fluxes are not sufficient to stabilize the solution in the presence of shocks for large high-order elements. Here, a new basis of shape functions is introduced. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization because of numerical fluxes. Large high-order elements can therefore be used and shocks captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Several numerical examples for transonic and supersonic flows are studied to demonstrate the applicability of the proposed approach. Copyright © 2011 John Wiley & Sons, Ltd.

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. Copyright © 2011 John Wiley & Sons, Ltd.

This paper first applies a flux vector-type splitting method based on the numerical speed of sound for computing incompressible single and multifluid flows. Here, a preconditioning matrix based on Chorin's artificial compressibility concept is used to modify the incompressible multifluid Navier–Stokes equations to be hyperbolic and density or volume fraction-independent. The current approach can reduce eigenvalues disparity induced from density or volume fraction ratios and enhance numerical stability. Also, a simple convection-pressure flux-splitting method with high-order essentially nonoscillatory-type primitive variable extrapolations coupled with monotone upstream-centered schemes for conservation laws-type volume fraction recompressed reconstruction is used to maintain the preservation of sharp interface evolutions in multifluid flow simulations. Benchmark tests including a solid rotation test of a notched two-dimensional cylinder, the evolution of spiral and rotational shapes of deformable circles, a dam breaking problem, and the Rayleigh–Taylor instability were chosen to validate the current incompressible multifluid methodology. An incompressible driven cavity was also chosen to check the robustness of the proposed method on the computation of single fluid incompressible flow problems. Copyright © 2011 John Wiley & Sons, Ltd.

In this work, we studied the peristaltic flow of a Jeffrey-six constant fluid in a uniform tube. The governing equations of the Jeffrey-six constant fluid were simplified by using the assumptions of long wave length and low Reynolds number approximation. The simplified form of equations were solved using the perturbation, homotopy analysis and finite difference methods. The comparison of the three solutions are shown graphically. The variation of pressure rise and frictional forces with the different parameters were also examined numerically. Results are presented at the end of the article. Copyright © 2011 John Wiley & Sons, Ltd.

The boundary layer stretched flow of a Jeffrey fluid subject to the convective boundary conditions was investigated. The governing dimensionless problems were computed by using the homotopy analysis approach. Convergence of the derived solutions was checked and the influence of embedded parameters was analyzed by plotting graphs. It was noticed that the velocity increases with an increase in the Deborah number. Furthermore, it was found that the temperature is also an increasing function of the Biot number. We further found that for fixed values of other parameters, the local Nusselt number increases by increasing the suction parameter and Deborah number. Numerical values of the skin friction coefficient and local Nusselt numbers were computed and examined. Copyright © 2011 John Wiley & Sons, Ltd.

An efficient discontinuous Galerkin formulation is applied to the solution of the linearized Euler equations and the acoustic perturbation equations for the simulation of aeroacoustic propagation in two-dimensional and axisymmetric problems, with triangular and quadrilateral elements. To improve computational efficiency, a new strategy of variable interpolation order is proposed in addition to a quadrature-free approach and parallel implementation. Moreover, an accurate wall boundary condition is formulated on the basis of the solution of the Riemann problem for a reflective wall. Time discretization is based on a low dissipation formulation of a fourth-order, low storage Runge–Kutta scheme. Along the far-field boundaries a perfectly matched layer boundary condition is used. For the far-field computations, the integral formulation of Ffowcs Williams and Hawkings is coupled with the near-field solver. The efficiency and accuracy of the proposed variable order formulation is assessed for realistic geometries, namely sound propagation around a high-lift airfoil and the Munt problem. Copyright © 2011 John Wiley & Sons, Ltd.

The solution for the shallow water equations using smoothed particle hydrodynamics is attractive, being a mesh-free, automatically adaptive method without special treatment for wet–dry interfaces. However, the relatively new method is limited by the variable kernel size or smoothing length being inversely proportional to water depth causing poor resolution at small depths. Boundary conditions at solid walls have also not been well resolved. To solve the resolution problem in small depths, a particle splitting procedure was developed (conveniently into seven particles), which conserves mass and momentum by varying the smoothing length, velocity and acceleration of each refined particle. This improves predictions in the shallowest depths where the error associated with splitting is reduced by one order of magnitude in comparison to other published works. To provide good shock capturing behaviour, particle interactions are treated as a Riemann problem with Monotone Upstream-centred Scheme for Conservation Laws (MUSCL) reconstruction providing stability. For solid boundaries, the recent modified virtual boundary particle method was developed further to enable the zeroth moment to be accurately conserved where the smoothing length of particles is changing rapidly during particle splitting. The resulting method is applied to the one-dimensional and the two-dimensional axisymmetric wet-bed dam break problems showing close agreement with analytical solutions, demonstrating the need for particle splitting. To demonstrate wetting and drying in a more complex case, the scheme is applied to oscillating water in a two-dimensional parabolic basin and produces good agreement with the analytical solution. The method is finally applied to the European Concerted Action on DAm break Modelling dam-break test case representative of realistic conditions and good predictions are made of experimental measurements with a 40% reduction in the computational time when particle splitting is employed. The overall method has thus become quite sophisticated but its generality and versatility will be attractive for various shallow water problems. Copyright © 2011 John Wiley & Sons, Ltd.