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

A new numerical method for particle tracking (Lagrangian particle advection) on 2-D unstructured grids with triangular cells is presented and tested. This method combines key attributes of published methods, including streamline closure for steady flows and local mass conservation (uniformity preservation). The subgrid-scale velocity reconstruction is linear, and this linear velocity field is integrated analytically to obtain particle trajectories. A complete analytic solution to the 2-D system of ordinary differential equations (ODEs) governing particle trajectories within a grid cell is provided. The analytic solution to the linear system of locally mass-conserving constraints that must be enforced to obtain the coefficients in the ODEs is also provided. Numerical experiments are performed to demonstrate that the new method has substantial advantages in accuracy over previously published methods and that it does not suffer from unphysical particle clustering. The method can be used not only in particle-tracking applications but also as part of a semi-Lagrangian advection scheme. This article is protected by copyright. All rights reserved.

In this work we propose a novel approach to model order reduction for incompressible fluid flows that focuses on the spatio-temporal description of the stresses on the surface of a body, i.e. of the wall shear stress and of the wall pressure. The spatial representation of these two variables is given by a compact set of “wall basis functions”, i.e. elementary basis functions defined on the wall. In this paper, these are derived using the well-known Proper Orthogonal Decomposition, to represent optimally the fluctuation energy of the pressure and shear stress. On the other hand, the functional structure of the dynamic model is derived from first principles using the vorticity form of the Navier-Stokes equations, yielding a set of nonlinear ordinary differential equations for the time-varying amplitudes of the wall shear stress basis functions. Coefficients of this model are then identified from simulation data. To complete the system, we show that the surface pressure distribution, i.e. the time-varying amplitudes of the wall pressure basis functions, can be derived from a quadratic model of the wall shear stress temporal coefficients, stemming from the Poisson equation for the pressure. This further step is crucial for the correct representation of the aerodynamic forces. As a paradigmatic example, we present our approach for the modelling of the free dynamics of the separated flow around a circular cylinder in the laminar regime, at *R**e* = 200. Further implications and potentialities of the proposed approach are discussed. This article is protected by copyright. All rights reserved.

This work presents an approximate Riemann solver to the transient isothermal drift flux model. The set of equations constitutes a non-linear hyperbolic system of conservation laws in one space dimension. The elements of the Jacobian matrix **A** are expressed through exact analytical expressions. It is also proposed a simplified form of **A** considering the square of the gas to liquid sound velocity ratio much lower than one. This approximation aims to express the eigenvalues through simpler algebraic expressions. A numerical method based on the Gudunov's fluxes is proposed employing an upwind and a high order scheme. The Roe linearization is applied to the simplified form of **A**. The proposed solver is validated against three benchmark solutions and two experimental pipe flow data.

In this paper we propose a model based on a new contravariant integral form of the Fully Non-linear Boussinesq Equations (FNBE) in order to simulate wave transformation phenomena, wave breaking and nearshore currents in computational domains representing the complex morphology of real coastal regions. The above-mentioned contravariant integral form, in which Christoffel symbols are absent, is characterized by the fact that the continuity equation does not include any dispersive term. A procedure developed in order to correct errors related to the difficulties of numerically satisfying the metric identities in the numerical integration of FNBE on generalized boundary-conforming grids is presented. The Boussinesq equation system is numerically solved by a hybrid finite volume-finite difference scheme. The proposed high-order upwind Weighted Essentially Non Oscillatory (WENO) finite volume scheme involves an exact Riemann solver and is based on a genuinely two dimensional reconstruction procedure which uses a convex combination of biquadratic polynomials. The wave breaking is represented by discontinuities of the weak solution of the integral form of the Non-linear Shallow Water Equations (NSWE).

The capacity of the proposed model to correctly represent wave propagation, wave breaking and wave induced currents is verified against test cases present in literature. The results obtained are compared with experimental measures, analytical solutions or alternative numerical solutions. This article is protected by copyright. All rights reserved.

Global linear stability analysis combined with computational fluid dynamics (CFD) is considered useful for understanding the physics of fluid flows. However, the numerical techniques of global linear stability analysis for compressible flows have not been well established in comparison with those for incompressible flows. In this study, we develop and assess a set of appropriate numerical techniques required to conduct a global linear stability analysis for compressible flows. For the eigensystem analysis, the Arnoldi method combined with time integration is in effect to preserve the memory (RAM) size of the computer. The compact difference scheme is used for the CFD analysis from the viewpoints of computing accurate global modes and saving memory by reducing the number of grid points to obtain the necessary spatial resolution. To assess the proposed method, two-dimensional compressible flow problems, including regularized cavity flow, flow around a square cylinder, and the compressible mixing layer, are analyzed, and it is confirmed that the proposed method can obtain accurate mode shapes, growth rate, and frequency of the corresponding global modes. In addition, influences and an appropriate formulation of the outflow boundary conditions are investigated. Results reveal that the outflow boundary causes spurious unstable modes in the global linear stability analysis, and the radiation and outflow boundary condition and the extension of the computational domain with grid stretching keep the spurious unstable modes to a minimum. This article is protected by copyright. All rights reserved.

Four dimensional variational data assimilation (4DVAR) is frequently used to improve model forecasting skills. This method improves a model consistency with available data by minimizing a cost function measuring the model-data misfit with respect to some model inputs and parameters. Associated with this type of method, however, are difficulties related to the coding of the adjoint model, which is needed to compute the gradient of the 4DVAR cost function. Proper orthogonal decomposition (POD), is a model reduction method which can be used to approximate the gradient calculation in 4DVAR. In this work, two ways of using POD in 4DVAR are presented, namely Model reduced 4DVAR (MR-4DVAR) and reduced adjoint 4DVAR (RA-4DVAR). Both techniques employ POD to obtain a reduced order approximation of the forward linear tangent operator. The difference between the two methods lies in the treatment of the forward model. MR-4DVAR performs minimization entirely in the POD reduced space, thereby achieving very low computational costs, but sacrificing accuracy of the end result. On the other hand, the RA-4DVAR uses POD to approximate only the adjoint model. The main contribution of this study is a comparative performance analysis of these 4DVAR methodologies on a nonlinear finite element shallow water model. The sensitivity of the methods to perturbations in observations and the number of observation points is examined. The results from twin experiments suggest that the RA-4DVAR method is easy to implement, computationally efficient and provides a robust approach for achieving reasonable results in the context of variational data assimilation. This article is protected by copyright. All rights reserved.

Predicting unsteady flows and aerodynamic forces for large displacement motion of micro-structures require transient solution of Boltzmann equation with moving boundaries. For the inclusion of moving complex boundaries for these problems, three immersed boundary method (IBM) flux formulations (Interpolation, Relaxation and Interrelaxation) are presented. These formulations are implemented in a 2-D FVM solver for ES-BGK equations using unstructured meshes. For the verification, a transient analytical solution for free molecular 1-D flow is derived and results are compared with the IB-ES-BGK methods. In 2-D, methods are verified with the conformal, non-moving finite volume method (FVM) and it is shown that the interrelaxation flux formulation gives an error less than the interpolation and relaxation methods for a given mesh size. Furthermore, formulations applied to a thermally-induced flow for a heated beam near a cold substrate show that Interrelaxation formulation gives more accurate solution in terms of heat flux. As a 2-D unsteady application, IB/ES-BGK methods are used to determine flow properties and damping forces for impulsive motion of microbeam due to high inertial forces. IB/ES-BGK methods are compared to Navier-Stokes solution at low Knudsen numbers and it is shown that velocity slip in the transitional rarefied regime reduces the unsteady damping force. This article is protected by copyright. All rights reserved.

This contribution is concerned with the numerical modeling of an isolated red blood cell (RBC), and more generally of phospholipid membranes. We propose an adaptive Eulerian finite element approximation, based on the level set method, of a shape optimization problem arising in the study of RBCs. We simulate the equilibrium shapes that minimize the elastic bending energy under prescribed constraints of fixed volume and surface area. An anisotropic mesh adaptation technique is used in the vicinity of the cell membrane to enhance the robustness of the method. Efficient time and spatial discretizations are considered and implemented. We address in detail the main features of the proposed method and finally we report several numerical experiments in the two-dimensional and the three-dimensional axisymmetric cases. The effectiveness of the numerical method is further demonstrated through numerical comparisons with *semi-analytical* solutions provided by a reduced order model. This article is protected by copyright. All rights reserved.

Three kinds of two-level consistent splitting algorithms for the time-dependent Navier-Stokes equations are discussed. The basic technique of two-level type methods for solving the nonlinear problem is first to solve a nonlinear problem in a coarse-level subspace, then to solve a linear equation in a fine-level subspace. Hence the two-level methods can save a lot of work compared with the one-level methods. The approaches to linearization are considered based on Stokes, Newton and Oseen corrections. The stability and convergence demonstrate that the two-level methods can get the optimal accuracy with the proper choice of the coarse and fine mesh scales. Numerical examples show that Stokes correction is the simplest, Newton correction has the best accuracy while Oseen correction is preferable for the large Reynolds number problems and the long-time simulations among the three methods This article is protected by copyright. All rights reserved.

This paper proposes a second order accuracy in time fully discrete finite element method for the Oldroyd fluids of order one. This new approach is based on a finite element approximation for the space discretization, the Crank-Nicolson/Adams-Bashforth scheme for the time discretization and the trapezoid rule for the integral term discretization. It reduces the nonlinear equations to an almost unconditionally stable and convergent systems of linear equations that can be solved efficiently and accurately. Here, the numerical simulations for *L*^{2},*H*^{1} error estimates of the velocity and *L*^{2} error estimates of the pressure at different values of viscoelastic viscosities *α*, different values of relaxation time *λ*_{1}, different values of null viscosity coefficient *μ*_{0} are shown. In addition, two benchmark problems of Oldroyd fluids with different solvent viscosity *μ* and different relaxation time *λ*_{1} are simulated. All numerical results perfectly match with the theoretical analysis and show that the developed approach gives a high accuracy to simulate the Oldroyd fluids under a large time step. Furthermore, the difference and the connection between the Newton fluids and the viscoelastic Oldroyd fluids are displayed. This article is protected by copyright. All rights reserved.

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two-dimensional unsteady Navier-Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier-Stokes equations are first discretized (in space) on a staggered grid as in the MAC (Marker And Cell) scheme. The discretized Navier-Stokes equations form an index 2 system of DAEs (Differential Algebraic Equations), which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas (BDFs).

The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. This article is protected by copyright. All rights reserved.

In this study, the Nervier–Stokes equations for incompressible flows, modified by the artificial compressibility method, are investigated numerically. To calculate the convective fluxes, a new high-accuracy characteristics-based (HACB) scheme is presented in this paper. Comparing the HACB scheme with the original characteristic-based method, it is found that the new proposed scheme is more accurate and has faster convergence rate than the older one. The second order averaging scheme is used for estimating the viscose fluxes, and spatially discretized equations are integrated in time by an explicit fourth-order Runge–Kutta scheme. The lid driven cavity flow and flow in channel with a backward facing step have been used as benchmark problems. It is shown that the obtained results using HACB scheme are in good agreement with the standard solutions. Copyright © 2015 John Wiley & Sons, Ltd.

In this study, the Nervier–Stokes equations for incompressible flows, modified by the artificial compressibility method, are investigated numerically. To calculate convective fluxes a new high-accuracy characteristics-based scheme (HACB) based on improved characteristics speeds is presented in this paper. Comparing the HACB scheme with the original characteristic-based method, it is found that the new proposed scheme is more accurate and has faster convergence rate than the older one.

This paper develops methods for interface-capturing in multiphase flows. The main novelties of these methods are as follows: (a) multi-component modelling that embeds interface structures into the continuity equation; (b) a new family of triangle/tetrahedron finite elements, in particular, the P_{1}DG-P_{2}(linear discontinuous between elements velocity and quadratic continuous pressure); (c) an interface-capturing scheme based on compressive control volume advection methods and high-order finite element interpolation methods; (d) a time stepping method that allows use of relatively large time step sizes; and (e) application of anisotropic mesh adaptivity to focus the numerical resolution around the interfaces and other areas of important dynamics. This modelling approach is applied to a series of pure advection problems with interfaces as well as to the simulation of the standard computational fluid dynamics benchmark test cases of a collapsing water column under gravitational forces (in two and three dimensions) and sloshing water in a tank. Two more test cases are undertaken in order to demonstrate the many-material and compressibility modelling capabilities of the approach. Numerical simulations are performed on coarse unstructured meshes to demonstrate the potential of the methods described here to capture complex dynamics in multiphase flows. Copyright © 2015 John Wiley & Sons, Ltd.

We present theory and apply an interface-capturing method for multiphase flow problems in 2D and 3D with emphasis on the use of adaptive unstructured finite element meshes. The method is mass-conserving and able to ensure that key forces such as buoyancy and hydrostatic pressure are exactly balanced. In addition, arbitrary numbers of phases with arbitrary equations of state can be modelled as demonstrated here.

A lattice Boltzmann model for the fractional sub-diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher-order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub-diffusion equation with the second-order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.

Lattice Boltzmann method can be used to implement the numerical simulation of the fractional sub-diffusion equation. The results by the lattice Boltzmann model are compared with the results of classical method. This figure is a snapshot of the three-dimensional example by Lattice Boltzmann method.

The smoothed finite element method (SFEM), which was recently introduced for solving the mechanics and acoustic problems, uses the gradient smoothing technique to operate over the cell-based smoothing domains. On the basis of the previous work, this paper reports a detailed analysis on the numerical dispersion error in solving two-dimensional acoustic problems governed by the Helmholtz equation using the SFEM, in comparison with the standard finite element method. Owing to the proper softening effects provided naturally by the cell-based gradient smoothing operations, the SFEM model behaves much softer than the standard finite element method model. Therefore, the SFEM can significantly reduce the dispersion error in the numerical solution. Results of both theoretical and numerical experiments will support these important findings. It is shown clearly that the SFEM suits ideally well for solving acoustic problems, because of the crucial effectiveness in reducing the dispersion error. Copyright © 2015 John Wiley & Sons, Ltd.

The smoothed finite element method model behaves much softer than the standard finite element method model and hence can significantly reduce the dispersion error in the numerical solution. Results of both theoretical and numerical experiments will support these important findings.

Using variable-size particles in the *moving particle semi-implicit method* (MPS) could lead to inaccurate predictions and/or numerical instability. In this paper, a variable-size particle moving particle semi-implicit method (VSP-MPS) scheme is proposed for the MPS method to achieve more reliable simulations with variable-size particles. To improve stability and accuracy, a new gradient model is developed based on a previously developed MPS scheme that requires *no surface detection* MPS. The dynamic particle coalescing and splitting algorithm is revised to achieve dynamic multi-resolution. A cubic spline function with additional function is employed as the kernel function. The effectiveness of the VSP-MPS method is demonstrated by three verification examples, that is, a hydrostatic pressure problem, a complicated free surface flow problem with large deformation, and a dynamic impact problem. The new VSP-MPS scheme with variable-size particles is found to have balanced efficiency and accuracy that is suitable for simulating large systems with complex flow patterns. Copyright © 2015 John Wiley & Sons, Ltd.

Proposed VSP-MPS method with variable resolution minimizes the particle clustering to improve the computational stability and efficiency with high accuracy. Particles with different sizes have identical effect radii to meet Newton's third law, and new gradient model and additional weight function based on a cubic spline function are proposed to balance the overstated gradient between large and small particles. A five-step article splitting and coalescing algorithm is proposed with random distribution scheme to reduce the particle clustering, and multiple particle sizes could be found in each single resolution area which is new from similar schemes.

Efficient and profitable oil production is subject to make reliable predictions about reservoir performance. However, restricted knowledge about reservoir rock and fluid properties and its geometrical structure calls for history matching in which the reservoir model is calibrated to emulate the field observed history. Such an inverse problem yields multiple history-matched models, which might result in different predictions of reservoir performance. Uncertainty quantification narrows down the model uncertainties and boosts the model reliability for the forecasts of future reservoir behaviour. Conventional approaches of uncertainty quantification ignore large-scale uncertainties related to reservoir structure, while structural uncertainties can influence the reservoir forecasts more significantly compared with petrophysical uncertainty.

Quantification of structural uncertainty has been usually considered impracticable because of the need for global regridding at each step of history matching process. To resolve this obstacle, we develop an efficient methodology based on Cartesian cut cell method that decouples the model from its representation onto the grid and allows uncertain structures to be varied as a part of history matching process. Reduced numerical accuracy due to cell degeneracies in the vicinity of geological structures is adequately compensated with an enhanced scheme of a class of locally conservative flux continuous methods (extended enriched multipoint flux approximation method or extended EMPFA).

The robustness and consistency of the proposed hybrid Cartesian cut cell/extended EMPFA approach are demonstrated in terms of true representation of geological structures influence on flow behaviour. Significant improvements in the quality of reservoir recovery forecasts and reservoir volume estimation are presented for synthetic model of uncertain structures. Copyright © 2015 John Wiley & Sons, Ltd.

To quantify the structural uncertainty, an efficient methodology based on Cartesian cut cell method is developed that decouples the model from its representation onto the grid. Reduced numerical accuracy due to cell degeneracies is adequately compensated with extended enriched multipoint flux approximation method. Method robustness and consistency are verified with correct impression of flow behaviour from variations in geological structures. Significant improvements in the quality of reservoir recovery forecasts and reservoir volume estimation are presented for uncertain structures.

This work constitutes a fraction of a more extensive effort, which ultimate objective is the development of advanced aeroacoustics hybrid methods. Within this framework, we here focus on the interpolation step, on which generally rely all coupling processes that link altogether the various stages constituting any given hybrid method. In that regard, previous works by the present authors had revealed the intrinsic limitations and subsequent side effects (e.g., signal degradation) that weight on usual high-order interpolation schemes, whether the latter are of centered or noncentered nature, as well as optimized in an acoustic sense or not. Based on the outcomes of such study, here, a novel optimization technique for interpolation schemes is proposed. Such a technique, which is designed hereafter as the *interpolation by parts (IBP)*, allows interpolating accurately a given signal, while minimizing its possible degradation. As a result, compared with its standard counterpart, any *IBP*-optimized interpolation scheme exhibits improved characteristics, such as a spurious modes generation that is greatly reduced (up to a 99*%* factor). Such improved characteristics are here validated on the basis of three test cases (of 1*D*, 2*D* and 3*D* nature), which illustrates the potentialities offered by the IBP optimization technique. Copyright © 2015 John Wiley & Sons, Ltd.

In the present work, a novel optimization technique for interpolation processes is proposed. Such a technique, which is hereafter designed as the interpolation by parts, allows interpolating accurately a given signal, while minimizing its possible degradation in terms of spurious modes generation. As a result, compared with its standard counterpart, any interpolation by parts-optimized interpolation scheme exhibits improved characteristics, such as a generation of spurious modes that is greatly reduced (up to a 99% factor).

We present methods for computing either the level set function or volume fraction field from the other at second-order accuracy. Both algorithms are optimal in that O(*N*) computations are needed for *N* total grid points and both algorithms are easily parallelized. This work includes a novel interface reconstruction algorithm in three dimensions that requires a smaller local block of volume fractions than existing algorithms. A compact local solver leads to better algorithm portability and efficiency: for example, fewer restrictions must be imposed on an adaptive mesh, and fewer grid cells must be communicated between processors in a parallel implementation. We also present a fast sweeping method for computing a unique approximation of the signed distance function to a piecewise linear interface. All of the numerical examples confirm second-order accuracy on both uniform and tree-based adaptive grids. Copyright © 2015 John Wiley & Sons, Ltd.

We present methods for computing either the level set function or volume fraction field from the other at second order accuracy. This work includes a novel interface reconstruction algorithm in three dimensions that requires a smaller local block of volume fractions than existing algorithms. All of the numerical examples confirm second order accuracy on both uniform and tree-based adaptive grids.

Problems in the characteristic-wise flux-split based finite difference method when compressible flows with contact discontinuities or material interfaces are computed were presented and analyzed. The current analysis showed the following: (i) Even with the local characteristic decomposition technique, numerical errors could be caused by point-wise flux vector splitting (FVS) methods, such as the Steger–Warming FVS or the van Leer FVS. Therefore, the Lax–Friedrichs type FVS method is required. (ii) If the isobars of a material are vertical lines, the combination of using the local characteristic decomposition and the global Lax–Friedrichs FVS can avoid velocity and pressure oscillations of contact discontinuities in this material for weighted essentially non-oscillatory (WENO) schemes. (iii) For problems with material interfaces, the quasi-conservative approach can be realized using characteristic-wise flux-split based finite difference WENO schemes if nonlinear WENO schemes in genuinely nonlinear characteristic fields can be guaranteed to be the same and the decomposition equation representing material interfaces is discretized properly. Copyright © 2015 John Wiley & Sons, Ltd.

It can be theoretically proven that if the isobars of a material are vertical lines, the combination of using the local characteristic decomposition and the global Lax-Friedrichs flux vector splitting (FVS) can avoid velocity and pressure oscillations of contact discontinuities in this material for finite difference weighted essentially non-oscillatory schemes. However, even with the local characteristic decomposition technique, numerical errors could be caused by point-wise FVS methods or the highly nonlinear equation of state of the material.

An innovative inflow/outflow boundary treatment has been proposed to be used in smoothed particle hydrodynamics (SPH). Among other strategies, it involves the use of extended regions at open boundary sections and a procedure to enforce the mass continuity constraint, as well as to minimize outflow reflections. This methodology has been coupled with a modified ‘particle shifting’ algorithm, so that the robustness of the method could be ensured at high Reynolds number regimes. Confined flow around a square cylinder with an open outflow has been selected as the flow problem to analyze the performance of the new method. Detailed comparisons with data available in the literature for a variety of mesh-based methods have been made for two different values of the blockage ratio *β*, namely for *β* = 1/4 and 1/8, and a range of supercritical Reynolds numbers. The results obtained with the present implementation of truly incompressible SPH have demonstrated numerical accuracy comparable with that of other methods, as well as the success of the open boundary treatment. A direct comparison with previously published SPH results for a distinct blockage ratio, namely for *β* = 1/5, has also revealed that a major improvement has been achieved by the use of the method described in this paper. Copyright © 2015 John Wiley & Sons, Ltd.

Together with a modified particle shifting algorithm, an innovative inflow/outflow boundary treatment has been tested in truly incompressible smoothed particle hydrodynamics simulations of the flow around a square obstacle in plane channel with open outflow. A detailed study of the selected benchmark problem has been conducted up to a Reynolds number of 625, and extensive comparisons with reference data were carried out. The results have demonstrated that the proposed improvements lead to increased robustness and accuracy of the SPH method.

We investigate the performance of a meshless method for the numerical simulation of depth-averaged turbulence flows. The governing equations are shallow water equations obtained by depth averaging of the full Reynolds equations including bed frictions, eddy viscosity, wind shear stresses, and Coriolis forces. As a double-phase closure turbulence model, we consider the depth-averaged *k*-*ϵ* model. A truly meshless numerical method based on radial basis functions is employed to obtain an accurate approximation to the solution of the model. We validate the algorithm on a linear shallow water problem where analytical solutions are available. Numerical results are also compared with experimental data for a backward-facing flow problem. Furthermore, we test the method on a practical problem by simulating tidal flows in the Strait of Gibraltar. The main focus is to examine the performance of the meshless method for complex geometries with irregular bathymetry. The obtained results demonstrate its ability to capture the main flow features. Copyright © 2015 John Wiley & Sons, Ltd.

These are velocity fields at different times for tidal waves in the Strait of Gibraltar using the turbulent shallow water flow conditions. The results are displayed for the semidiurnal M2, S2, and N2 tidal waves as well as the diurnal K1 tidal wave. Using the tidal conditions, the flow exhibits a recirculating zone with different orders of magnitudes near the Caraminal Sill (interface separating the Atlantic Ocean and Mediterranean Sea). These results demonstrate the capabilities of the proposed meshless method to solve turbulent shallow water flows in irregular domains with complex topography.

We aim at quantifying the impact of state uncertainties in shape optimization. This provides confidence bounds for the optimal solution. The approach is presented for inverse designs where the target is assumed uncertain. No sampling of a large dimensional space is necessary, and the approach uses what is already available in a deterministic gradient-based inversion algorithm. Our proposal is based on the introduction of directional quantile-based extreme scenarios knowing the probability density function of the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using the gradient of the functional by an adjoint method. The paper goes beyond inverse design and shows how to apply the method to general optimization problems. The ingredients of the paper are illustrated on a model problem with the Burgers equation and on the optimization of the shape of an aircraft. Overall, the computational complexity is comparable with the deterministic case. Copyright © 2015 John Wiley & Sons, Ltd.

We investigate the impact of state uncertainties in shape optimization to provide the covariance matrix of the optimal shape. The approach is enough efficient to be applied directly during adjoint-based designs of full aircrafts.

In the context of numerical simulations of multiphysics flows, accurate tracking of an interface and consistent computation of its geometric properties are crucial. In this paper, we investigate a level set technique that satisfies these requirements and ensures local third-order accuracy on the level set function (near the interface) and first-order accuracy on the curvature, even for long-time computations. The method is developed in a finite differences framework on Cartesian grids. As in usual level set strategies, reinitialization steps are involved. Several reinitialization algorithms are reviewed and mixed to design an accurate and fast reinitialization procedure. When coupled with a time evolution of the interface, the reinitialization procedure is performed only when there are too large deformations of the isocontours. This strategy limits the number of reinitialization steps and shows a good balance between accuracy and computational cost. Numerical results compare well with usual level set strategies and confirm the necessity of the reinitialization procedure, together with a limited number of reinitialization steps. Copyright © 2015 John Wiley & Sons, Ltd.

We investigate a level set technique that ensures an accurate and consistent computation of geometric properties, even for long time simulations. As usual level set strategies, it is based on a necessary reinitialization procedure that we designed to be fast and accurate. The coupling with time evolution of the interface is carried out with a controlled number of reinitialization steps, performed only when necessary, so as to keep the high-order accuracy and the consistency of the method.

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in ‘flat’ domains. For example, in numerical weather and climate prediction, an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods, which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by Börm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219–234]. We extend the analysis to three dimensions under slightly weakened assumptions and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this, we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one-dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment, which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20 480 cores is demonstrated on the HECToR supercomputer. Copyright © 2015 John Wiley & Sons, Ltd.

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. We describe an algorithmically optimal tensor-product geometric multigrid preconditioner for the pressure correction equation in global atmospheric flow models and proof the convergence of the algorithm analytically. We discuss an efficient and massively parallel implementation on a tensor product grid representing a thin spherical shell and demonstrate the performance and scalability of the solver for different atmospheric model problems.

A refined *r*-factor algorithm for implementing total variation diminishing (TVD) schemes on arbitrary unstructured meshes, referred to henceforth as a face-perpendicular far-upwind interpolation scheme for arbitrary meshes (FFISAM), is proposed based on an extensive review of the existing *r*-factor algorithms available in the literature. The design principles, as well as the respective advantages and disadvantages, of the existing algorithms are first systematically analyzed before presenting the FFISAM. The FFISAM is designed to combine the merits of various existing *r*-factor algorithms. The performance of the FFISAM, implemented in 10 classical TVD schemes, is evaluated against four two-dimensional pure-advection benchmark test cases where analytical solutions are available. The numerical results clearly show that the FFISAM leads to a better overall performance than the existing algorithms in terms of accuracy and convergence on arbitrary unstructured meshes for the 10 classical TVD schemes. Copyright © 2015 John Wiley & Sons, Ltd.

The key findings:

- Several existing
*r*-factor algorithms available in the literature are reviewed in detail for extending TVD schemes to arbitrary unstructured grids, and the respective advantages and disadvantages of these algorithms are also revealed and analyzed. - A refined
*r*-factor algorithm (FFISAM) is proposed based on the review. The FFISAM is designed to overcome several inherent drawbacks of the existing algorithms while preserve their attractive attributes. Numerical results show that the FFISAM leads to a better performance in terms of accuracy and convergence.

Bouncing jets are fascinating phenomenon occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non-Newtonian fluids when the jets fall in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier–Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is carried out with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing. Copyright © 2015 John Wiley & Sons, Ltd.

We investigate numerically fascinating bouncing jet phenomenon. Left: Laboratory Experiment of the Kaye effect taken at the High Speed Fluid Imaging Laboratory of S. Thoroddsen at King Abdullah University of Science and Technology. Right: Numerical simulation of the Kaye effect using our adaptive finite element method.

Two-phase immiscible fluids in a two-dimensional micro-channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd-B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro-channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level-set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two-phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.

A technique for simulating a two-phase Newtonian/viscoelastic flow in a micro-channels network is proposed. This technique involves a level-set approach for tracking the interface between the two fluids, a penalization method for dealing with the geometry, a numerical contact angle model based on the Cox relation and the Oldroyd-B model for modeling the viscoelastic fluid.

Genetic algorithm (GA) is a widely used method for numerical optimisation owing to their good global search ability; however, their local search ability has an obvious shortcoming. To improve local search ability, this paper introduces a simplex method and combines it with a GA to form an improved genetic algorithm (IGA). In the IGA, at each generation of the original GA, high-fitness individuals are selected as vertices of a simplex, and then a one-dimensional search within the simplex is conducted to obtain the most-fit individuals while replacing the inferior ones. Typical test functions show that the IGA can effectively improve the optimisation effect over that of the original GA. To further verify the IGA's practicability, an aspirated compressor profile is optimised with profile, suction flow rate and suction flow location as coupled design parameters. The results again show that the IGA has a better optimising effect than the GA. In addition, it is also verified that coupling the profile and suction flow parameters results in a design that outperforms the uncoupled design; therefore, designing an aspirated compressor blade by arranging suction flow on a conventional blade without considering suction flow is not a good method. Copyright © 2015 John Wiley & Sons, Ltd.

To improve search ability of genetic algorithm (GA), this paper combines it with a simplex method to form an improved genetic algorithm (IGA). Typical test functions and optimisation of aspirated compressor profiles show that the IGA can effectively improve the optimisation effect over that of the original GA. In the figure, the IGA evolution of three different combinations of the two optimisation methods (denoted as OptProfile-1, OptProfile-2 and OptProfile-3) excelled the GA (denoted as OptProfile-0).

We present a new non-intrusive model reduction method for the Navier–Stokes equations. The method replaces the traditional approach of projecting the equations onto the reduced space with a radial basis function (RBF) multi-dimensional interpolation. The main point of this method is to construct a number of multi-dimensional interpolation functions using the RBF scatter multi-dimensional interpolation method. The interpolation functions are used to calculate POD coefficients at each time step from POD coefficients at earlier time steps. The advantage of this method is that it does not require modifications to the source code (which would otherwise be very cumbersome), as it is independent of the governing equations of the system. Another advantage of this method is that it avoids the stability problem of POD/Galerkin. The novelty of this work lies in the application of RBF interpolation and POD to construct the reduced-order model for the Navier–Stokes equations. Another novelty is the verification and validation of numerical examples (a lock exchange problem and a flow past a cylinder problem) using unstructured adaptive finite element ocean model. The results obtained show that CPU times are reduced by several orders of magnitude whilst the accuracy is maintained in comparison with the corresponding high-fidelity models. Copyright © 2015 John Wiley & Sons, Ltd.

The figures displayed above(left) show the velocity solutions of the lock exchange problem at time instances 30 seconds. The solutions compare the predictions from RBF/POD model with full and standard POD model using 32 POD basis functions. Velocity solution of a point in the domain is shown on the right.

Modeling and visualization of a rainwater overland flow is an important tool for a risk assessment, preparation, evacuation planning, and real-time forecasting of flood warning. The objective of this research is to develop a numerical software to visualize the rainwater overland flow based on a finite volume method for shallow water equations and in combination with the dynamically adaptive tree grid technique and the dynamic domain-defining method. The obtained simulations for several experiments were tested and compared with results in literature, both theoretical and experimental results. The comparisons with non-adaptive grids show that the developed algorithm for simulation is very efficient and has a potential for practical usages, in terms of computational times and accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

Numerical software for simulation and visualization of the rainwater overland flow is developed in this work. The algorithm is developed based on a finite volume method for shallow water equations and in combination with the dynamically adaptive tree grid technique and the dynamic domain-defining method. The obtained simulations for several experiments were tested and compared with results in literature, both theoretical and experimental results. The comparisons with non-adaptive grids show that the developed algorithm for simulation is very efficient in terms of computational times and accuracy.

The computational efficiency of existing hydrocodes is expected to suffer as computer architectures advance beyond the traditional parallel central processing unit (CPU) model . Concerning new computer architectures, sources of relative performance degradation might include reduced memory bandwidth per core, increased resource contention due to concurrency, increased single instruction, multiple data (SIMD) length, and increasingly complex memory hierarchies. Concerning existing codes, any performance degradation will be influenced by a lack of attention to performance in their design and implementation.

This work reports on considerations for improving computational performance in preparation for current and expected changes to computer architecture. The algorithms studied will include increasingly complex prototypes for radiation hydrodynamics codes, such as gradient routines and diffusion matrix assembly (e.g., in ). The meshes considered for the algorithms are structured or unstructured meshes. The considerations applied for performance improvements are meant to be general in terms of architecture (not specifically graphical processing unit (GPUs) or multi-core machines, for example) and include techniques for vectorization, threading, tiling, and cache blocking. Out of a survey of optimization techniques on applications such as diffusion and hydrodynamics, we make general recommendations with a view toward making these techniques conceptually accessible to the applications code developer. Published 2015. This article is a U.S. Government work and is in the public domain in the USA.

We present considerations for improving computational performance of hydrodynamics codes in preparation for current and expected changes to computer architecture. The algorithms studied include prototypes such as gradient routines, and the considerations applied for performance improvements include techniques for vectorization, threading, tiling, and cache blocking. Out of a survey of optimization techniques on applications such as diffusion and hydro, we make general recommendations with a view toward making these techniques conceptually accessible to the applications code developer.

We present a new computational method by extending the immersed boundary (IB) method with a geometric model based on parametric radial basis function (RBF) interpolation of the Lagrangian structures. Our specific motivation is the modeling of platelets in hemodynamic flows, although we anticipate that our method will be useful in other applications involving surface elasticity. The efficacy of our new RBF-IB method is shown through a series of numerical experiments. Specifically, we test the convergence of our method and compare our method with the traditional IB method in terms of computational cost, maximum stable time-step size, and volume loss. We conclude that the RBF-IB method has advantages over the traditional IB method and is well-suited for modeling of platelets in hemodynamic flows. Copyright © 2015 John Wiley & Sons, Ltd.

We present an application of our radial basis function (RBF)-based parametric geometric model to the simulation of platelets in hemodynamic flows by the immersed boundary (IB) method. We test the convergence, area conservation and energy-dissipation properties, and time-step restrictions of our new RBF-IB method on a fluid–structure interaction problem. We compare the computational cost of the RBF-IB method with that of the traditional IB method for platelet simulations, and present the results of a platelet aggregation simulation accomplished with the RBF-IB method.

The Navier–Stokes–Korteweg (NSK) system is a classical diffuse-interface model for compressible two-phase flow. However, the direct numerical simulation based on the NSK system is quite expensive and in some cases even not possible. We propose a lower-order relaxation of the NSK system with hyperbolic first-order part. This allows applying numerical methods for hyperbolic conservation laws and removing some of the difficulties of the original NSK system. To illustrate the new ansatz, we first present a local discontinuous Galerkin method in one and two spatial dimensions. It is shown that we can compute initial boundary value problems with realistic density ratios and perform stable computations for small interfacial widths. Second, we show that it is possible to construct a semi-discrete finite-volume scheme that satisfies a discrete entropy inequality. Copyright © 2015 John Wiley & Sons, Ltd.

The Navier-Stokes-Korteweg (NSK) system is a diffuse-interface model for compressible two-phase flow. However, the direct numerical simulation based on the NSK system is quite expensive. We propose a relaxed lower-order approximation of the NSK system with hyperbolic first-order part. This allows to remove some of the difficulties for the NSK system. To illustrate the new ansatz, we present a local discontinuous Galerkin method. Additionally, we construct a semi-discrete finite-difference scheme that satisfies a discrete entropy inequality.

A moment-of-fluid method is presented for computing solutions to incompressible multiphase flows in which the number of materials can be greater than two. In this work, the multimaterial moment-of-fluid interface representation technique is applied to simulating surface tension effects at points where three materials meet. The advection terms are solved using a directionally split cell integrated semi-Lagrangian algorithm, and the projection method is used to evaluate the pressure gradient force term. The underlying computational grid is a dynamic block-structured adaptive grid. The new method is applied to multiphase problems illustrating contact-line dynamics, triple junctions, and encapsulation in order to demonstrate its capabilities. Examples are given in two-dimensional, three-dimensional axisymmetric (*R*–*Z*), and three-dimensional (*X*–*Y*–*Z*) coordinate systems. Copyright © 2015 John Wiley & Sons, Ltd.

A moment-of-fluid method is presented to study incompressible flows involving more than two materials. Various multiphase problems, including problems illustrating contact-line dynamics, triple junctions, and encapsulation, are studied using the new method in order to demonstrate its capabilities.

This paper presents an application of the advancing reduction technique for 2D hybrid mesh generation (triangles + quadrilaterals). Based on an initial rectangle mesh (RM) covering the whole domain, the advancing reduction technique coarsens the base RM in a marching way from the boundary to the interior of the domain so that different zones of sub-RMs with different edge lengths are recognized. These sub-RMs are connected to each other with the so-called transition layers which consist of the transition triangles and quadrilaterals. As demonstrated by examples, the proposed method is simple, efficient, and easy to implement. Copyright © 2015 John Wiley & Sons, Ltd.

In this study, a new hybrid mesh (triangles + quadrilaterals) generation method is developed. The proposed method is grid-based and starts with an initial Rectangle Mesh (RM) covering the whole domain. Marching from the boundaries, the so-called advancing reduction technique is used to automatically decompose the whole domain into multiple zones of RM with different edge lengths. The transition layers, consisting of elements of both triangles and quadrilaterals, connect different sub-RMs to each other. In the proposed algorithm, local mesh refinement can be easily fulfilled by the transition layers.

A space and time third-order discontinuous Galerkin method based on a Hermite weighted essentially non-oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower-upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third-order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third-order discontinous Galerkin methods, and less computing time than the lower-order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems. Copyright © 2015 John Wiley & Sons, Ltd.

A space and time third-order discontinuous Galerkin method based on a Hermite weighted essentially non-oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. Numerical results indicate that this method is able to deliver the designed third-order accuracy of convergence in both space and time while requiring less computing time than the lower-order discontinuous Galerkin methods to achieve the same level of temporal accuracy for computing unsteadyflow problems.

A methodology for improved robustness in the simulation of high void fraction free surface polydisperse bubbly flows in curvilinear overset grids is presented. The method is fully two-way coupled in the sense that the bubbly field affects the continuous fluid and vice versa. A hybrid projection approach is used in which staggered contravariant velocities at cell faces are computed for transport and pressure–velocity coupling while the momentum equation is solved on a collocated grid arrangement. Conservation of mass is formulated such that a strong coupling between void fraction, pressure, and velocity is achieved within a partitioned approach, solving each field separately. A pressure–velocity projection solver is iterated together with a predictor stage for the void fraction to achieve a robust coupling. The implementation is described for general curvilinear grids detailing particulars in the neighborhood to overset interfaces or a free surface. A balanced forced method to avoid the generation of spurious currents is extended for curvilinear grids. The overall methodology allows simulation of high void fraction flows and is stable even when strong packing forces accounting for bubble collisions are included. Convergence and stability in one-dimensional (1D) and two-dimensional (2D) configurations is evaluated. Finally, a full-scale simulation of the bubbly flow around a flat-bottom boat is performed demonstrating the applicability of the methodology to complex problems of engineering interest. Copyright © 2015 John Wiley & Sons, Ltd.

A methodology for improved robustness in the simulation of high void fraction free surface polydisperse bubbly flows in curvilinear overset grids is presented. An additional set of staggered velocities defined at cell faces computed from fast to evaluate algebraic equations satisfy mass conservation exactly while momentum is still solved on a collocated grid. The methodology enables the computation of complex configurations of polydispersed bubbly flows around ships involving motions with dynamic overset and free surface at very high hold-ups.

This paper investigates some important numerical aspects for the simulation of model rocket combustors. Precisely, (1) a new high-order discretization technique (multi-dimensional limiting process (MLP), low diffusion, and MLP^{ld}) is presented and compared with conventional second-order schemes with different flux limiters. (2) Time accurate unsteady Reynolds-averaged Navier–Stokes (RANS) simulations are performed to assess possible improvements in comparison with steady-state RANS simulations. (3) Fully 3D simulations of an axisymmetric rocket combustor are compared with 2D axisymmetric ones. All studies are based on the Penn State preburner combustor experiment, which uses gaseous oxygen and hydrogen. This comprehensive study offers unique insight into how the mentioned numerical influence factors change the flow field, flame, and wall heat fluxes in the model rocket combustor. Because wall heat fluxes are known from the experiment only, numerical results are compared with LES of other authors, too. It will be shown that the high-order spatial discretization significantly improves the agreement with measured wall heat fluxes at low additional computational cost. In general the transition from simple to more complex numerical approaches steadily improves the qualitative agreement between simulation and experiment. Copyright © 2015 John Wiley & Sons, Ltd.

A new high order discretization technique (MLP^{ld} multi-dimensional limiting process, low diffusion) is presented and compared to conventional second order schemes with different flux limiters. Time accurate URANS (unsteady RANS) simulations (two- and three-dimensional) of an axisymmetric rocket combustor are performed to assess possible improvements in comparison to steady-state RANS (Reynolds-averaged Navier-Stokes) simulations. The high order spatial discretization significantly improves the agreement with measured wall heat fluxes at low additional computational cost.

A model formulated in terms of conserved variables is proposed for its use in the study of internal ballistic problems of pyrotechnical mixtures and propellants. It is a transient two-phase flow model adapted from the non-conservative Gough model. This conversion is mathematically attractive because of the wide range of numerical methods for this kind of systems that may be applied. We propose the use of the AUSM+, AUSM + up and Rusanov schemes as an efficient alternative for this type of two-phase problem. A splitting technique is applied, which solves the system of equations in several steps. A second-order approach based on Monotonic Upstream-Centred Scheme for Conservation Laws (MUSCL) is also used. Some tests are used to validate the code, namely a shock wave test, a contact discontinuity problem and an internal ballistics problem. In this last case, one-dimensional numerical results are compared with experimental data of 155-mm gunshots. Copyright © 2015 John Wiley & Sons, Ltd.

A transient two-phase flow model adapted from the non-conservative Gough model and formulated in terms of conserved variables is presented for the study of internal ballistic problems. We propose the use of the AUSM+, AUSM + up and Rusanov schemes as an efficient alternative for this type of problems.

We perform direct numerical simulation of three-dimensional turbulent flows in a rectangular channel, with a lattice Boltzmann method, efficiently implemented on heavily parallel general purpose graphical processor units. After validating the method for a single fluid, for standard boundary layer problems, we study changes in mean and turbulent properties of particle-laden flows, as a function of particle size and concentration. The problem of physical interest for this application is the effect of water droplets on the turbulent properties of a high-speed air flow, near a solid surface. To do so, we use a Lagrangian tracking approach for a large number of rigid spherical point particles, whose motion is forced by drag forces caused by the fluid flow; particle effects on the latter are in turn represented by distributed volume forces in the lattice Boltzmann method. Results suggest that, while mean flow properties are only slightly affected, unless a very large concentration of particles is used, the turbulent vortices present near the boundary are significantly damped and broken down by the turbulent motion of the heavy particles, and both turbulent Reynolds stresses and the production of turbulent kinetic energy are decreased because of the particle effects. We also find that the streamwise component of turbulent velocity fluctuations is increased, while the spanwise and wall-normal components are decreased, as compared with the single fluid channel case. Additionally, the streamwise velocity of the carrier (air) phase is slightly reduced in the logarithmic boundary layer near the solid walls. Copyright © 2015 John Wiley & Sons, Ltd.

We study direct numerical simulation of turbulent flows coupling with particle Lagrangian tracking method to see the effects of particles on modifying the turbulent field. We have found that turbulent Reynolds stress and the production of turbulent kinetic energy are decreased because of the particle effects, and streamwise component of turbulent velocity fluctuations is increased, while the spanwise and wall-normal components are decreased.

We describe a semi-implicit volume-of-fluid free-surface-modelling methodology for flow problems involving violent free-surface motion. For efficient computation, a hybrid-unstructured edge-based vertex-centred finite volume discretisation is employed, while the solution methodology is entirely matrix free. Pressures are solved using a matrix-free preconditioned generalised minimum residual algorithm and explicit time-stepping is employed for the momentum and interface-tracking equations. The high resolution artificial compressive (HiRAC) volume-of-fluid method is used for accurate capturing of the free surface in violent flow regimes while allowing natural applicability to hybrid-unstructured meshes. The code is parallelised for solution on distributed-memory architectures and evaluated against 2D and 3D benchmark problems. Good parallel scaling is demonstrated, with almost linear speed-up down to 6000 cells per core. Finally, the code is applied to an industrial-type problem involving resonant excitation of a fuel tank, and a comparison with experimental results is made in this violent sloshing regime. Copyright © 2015 John Wiley & Sons, Ltd.

We present a free-surface modelling methodology for flow problems involving violent liquid–gas sloshing. The solution methodology is entirely matrix-free and parallel. The code is applied to an industrial-type problem involving resonant excitation of a fuel tank, comparing well with experimental results in this highly dynamic sloshing regime.

In this paper, pressure-based and density-based methods are studied at different flow speeds. The methods are intended for steady flows, and the goal is to find as general an approach as possible to cover different Mach number regimes. The solution methods are based on a finite-volume approach. Various forms of inviscid fluxes are applied and connected with either a pressure-based or density-based implicit solution. For this purpose, a new pressure-correction method is developed that can be applied for incompressible and for compressible flows. Another option is a standard density-based approximate factorization method. In both cases, a convergence is accelerated with a Full Approximation Scheme (FAS) multigrid approach. Sample problems in the range of *M**a* = 0…6 are simulated using different approaches, and their efficiency and accuracy are compared. On the basis of the quality of the solutions, recommendations are made. © 2015 The Authors. *International Journal for Numerical Methods in Fluids* published by John Wiley & Sons Ltd.

Pressure-based and density-based methods are studied at different flow speeds in order to find a general approach to cover different Mach number regimes. Various forms of inviscid fluxes are applied and connected with either a pressure-based or density-based implicit solution, and a new pressure-correction method is developed that can be applied for incompressible and for compressible flows. Sample problems at different Mach numbers are simulated using different approaches, and their efficiency and accuracy are compared.

A virtual-characteristic approach is developed for thermo-flow with finite-volume methodology in which a multidimensional characteristic (MC) scheme is applied along with artificial compressibility. To obtain compatibility equations and pseudo-characteristics, energy equation is taken into account in the MC scheme. With this inherent upwinding of convective fluxes, no artificial viscosity is required even at high Reynolds numbers. Another remarkable advantage of the MC scheme lies in its faster convergence rate with respect to the averaging scheme that is found to exhibit substantial delays in convergence. As benchmarks, forced and mixed convections in a cavity and in flow over cylinder and between parallel plates are examined for a wide range of Reynolds, Grashof, and Prandtl numbers. The MC and averaging schemes are applied for simulation purposes. Results show the better performance of the MC scheme in forced and mixed convections. Results confirm the robustness of the MC scheme in terms of accuracy and convergence. Copyright © 2015 John Wiley & Sons, Ltd.

A virtual-characteristic approach is developed for thermo-flow with finite-volume methodology in which a multidimensional characteristic (MC) scheme is applied along with artificial compressibility. With this inherent upwinding of convective fluxes, no artificial viscosity is required even at high Reynolds numbers. Another remarkable advantage of the MC scheme lies in its faster convergence rate with respect to the averaging scheme that is found to exhibit substantial delays in convergence.

Aircraft holding around busy airports may be requested to sustain as much as 45 min of icing before landing or being diverted to another airport. In this paper, a three-dimensional mesh deformation scheme, based on a structural frame analogy, is proposed for the numerical simulation of ice accretion during extended exposure to adverse weather conditions. The goal is to provide an approach that is robust and efficient enough to delay or altogether avoid re-meshing while preserving (enforcing) nearly orthogonal elements at the highly distorted ice surface. Robustness is achieved by suitably modifying the axial and torsional stiffness components of the frame elements in order to handle large and irregular grid displacements typical of in-flight icing. Computational efficiency is obtained by applying the mesh displacement to an automatically selected small subset of the entire computational domain. The methodology is validated first in the case of deformations typical of fluid-structure interaction problems, including wing bending, a helicopter rotor in forward flight, and the twisting of a high-lift wing configuration. The approach is then assessed for aero-icing on two swept wings and compared against experimental measurements where available. Copyright © 2015 John Wiley & Sons, Ltd.

Findings: A mesh deformation technique that associates the edges of the mesh to structural frame elements endowed with suitable elastic properties is capable of delaying the need to re-mesh in the presence of large irregular ice growth by enforcing zero or minimal skewness near the wall and limiting the degradation of near-wall elements.

Acoustic waves (or oscillating flows) cause both periodic flow and steady streaming around an obstacle. The nonlinear characteristic of such flows further induces acoustic radiation forces exerting on the surface of the obstacle, which is efficient to levitate or manipulate small partials. Two-dimensional lattice Boltzmann methods are applied to simulate the flows around cylinders in acoustic standing waves with moderate viscosity. Multiple relaxation time model coupled with far-field absorbing condition is applied. Our results show recirculating leading order flow in the Stokes layer with a characteristic velocity predicted by potential flow. The consequent Reynolds stresses induce two kinds of patterns of first-order steady streaming, namely, single and double layer streamings, respectively, according to the strength of the viscous effects. Compared with the theoretical analysis, the lattice Boltzmann simulation is accurate for both radiation forces and flow fields. Both numerical results and scaling analysis show that the viscous drag is linearly proportional to the thickness of the penetration depth, which is coincident with the low viscous cases. Copyright © 2015 John Wiley & Sons, Ltd.

Lattice Boltzmann method coupled with absorbing boundary conditions was applied to simulate the acoustic flows around a cylinder in moderately viscous fluids. The viscous flow near the cylinder was modelled as curved lid-driven flow. The scaling analysis, which was based on extremely low viscous model, shows that the viscous components of the acoustic lifting force are linearly proportional to the thickness of the viscous penetrating thickness.

In this paper, we propose a high-order finite volume hybrid kinetic Weighted Essentially Non-Oscillatory (WENO) scheme for inviscid and viscous flows. Based on the WENO reconstruction technique, a hybrid kinetic numerical flux is introduced for the present method, which includes the mechanisms of both the free transfer and the collision of gas molecules. The collisionless free transfer part of the hybrid numerical flux is constructed from the conventional kinetic flux vector splitting treatment, and the collision contribution is considered by constructing an equilibrium gas state and calculating the corresponding numerical flux at the cell interface. The total variation diminishing Runge–Kutta methods are used for the temporal integration. The high-order accuracy and good shock-capturing capability of the proposed hybrid kinetic WENO scheme are validated by many numerical examples in one-dimensional and two-dimensional cases. Copyright © 2015 John Wiley & Sons, Ltd.

A high-order finite volume hybrid kinetic Weighted Essentially Non-Oscillatory scheme is introduced, which uses a simple hybrid kinetic flux at the cell interface to include the effects of both the free transfer and the collision of gas molecules. It is numerically demonstrated that in general, the proposed hybrid kinetic flux performs better than the conventional Kinetic Flux Vector Splitting (free transfer)flux. The high-order accuracy and good shock-capturing capability of the newly proposed scheme are validated by many 1D and 2D numerical tests.

A semi-implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free-surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple *z*-layer discretization is adopted. Semi-implicit finite difference equations for velocities and a finite volume approximation for the free-surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free-surface equation yields a single, well-posed, mildly nonlinear system. This system is efficiently solved by a nested Newton-type iterative method that yields simultaneously the pressure and a non-negative fluid volume throughout the computational grid. The time-step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free-surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.

The governing differential equations are approximated by semi-implicit finite difference and finite volume methods. The discrete free-surface equation yields a reduced *mildly nonlinear* system from which *both**free-surface* and *fluid volumes* are determined *simultaneously*. Exact mass balance is guaranteed everywhere, in presence of wetting and drying, in pressurized flow conditions, in inhomogeneous porous medium, and during free-surface transition. The proposed method solves coupled surface–subsurface flows and simplifies to either a surface or a subsurface discrete model as particular cases.

When dealing with high-order numerical methods, an adequate treatment of curved surfaces is required not only to guarantee that the expected high-order is maintained in the vicinity of surfaces but also to avoid steady-state convergence issues. Among the variety of high-order surface treatment techniques that have been proposed, the ones employing NURBS (non-uniform rational B-splines) to describe curved surfaces can be considered superior both in terms of accuracy and compatibility with computer-aided design softwares. The current study describes in detail the integration of NURBS-based geometry description in a high-order solver based on the discontinuous Galerkin formulation. Particularly, this work also discusses how and why NURBS curves of very high order can be employed within standard NURBS-based boundary treatment techniques to yield reduced implementation complexity and computational overhead. Theoretical estimates are provided along with numerical experiments in order to support the proposed approach. Minding engineering applications in the context of compressible aerodynamics, additional simulations are addressed as numerical examples to illustrate the advantages of using higher-order NURBS in practical situations. Copyright © 2015 John Wiley & Sons, Ltd.

When dealing with high-order numerical methods, an adequate treatment of curved surfaces is required not only to maintain the high-order near surfaces but also to avoid steady-state convergence issues or non-physical phenomena. This study describes how non-uniform rational B-splines (NURBS) data is incorporated in a discontinuous Galerkin setting. We discuss how and why higher-order NURBS can be employed within standard NURBS-based boundary treatment techniques to reduce computational overhead and implementation complexity.

The method of regularized Stokeslets (MRS) is a numerical approach using regularized fundamental solutions to compute the flow due to an object in a viscous fluid where inertial effects can be neglected. The elastic object is represented as a Lagrangian structure, exerting point forces on the fluid. The forces on the structure are often determined by a bending or tension model, previously calculated using finite difference approximations. In this paper, we study spherical basis function (SBF), radial basis function (RBF), and Lagrange–Chebyshev parametric models to represent and calculate forces on elastic structures that can be represented by an open curve, motivated by the study of cilia and flagella. The evaluation error for static open curves for the different interpolants, as well as errors for calculating normals and second derivatives using different types of clustered parametric nodes, is given for the case of an open planar curve. We determine that SBF and RBF interpolants built on clustered nodes are competitive with Lagrange–Chebyshev interpolants for modeling twice-differentiable open planar curves. We propose using SBF and RBF parametric models within the MRS for evaluating and updating the elastic structure. Results for open and closed elastic structures immersed in a 2D fluid are presented, showing the efficacy of the RBF–Stokeslets method. Copyright © 2015 John Wiley & Sons, Ltd.

We present a comparison of spherical basis function (SBF) and radial basis function (RBF) parametric models for the modeling of open elastic curves immersed in viscous fluids. We present convergence results on static test problems using parametric collocation at Chebyshev and Mapped Chebyshev nodes. We then extend the method of regularized Stokeslets (MRS) with the SBF and RBF geometric models and present the result of time-dependent fluid-structure interaction simulations of both open and closed elastic structures.

In the paper, discontinuous Galerkin method is applied to simulation of incompressible free round turbulent jet using large eddy simulation with eddy viscosity approach. The solution algorithm is based on the classical projection method, but instead of the solution of the Poisson equation, a parabolic equation is advanced in pseudo-time, which provides the pressure field ensuring the proper pressure–velocity coupling. For time and pseudo-time integration, explicit Runge–Kutta method is employed. The computational meshes consist of hexahedral elements with flat faces. Within a given finite element, all flow variables are expressed with modal expansions of the same order (including velocity and pressure). Discretisation of the viscous terms in the Navier–Stokes equations and Laplacian in the Poisson equation is stabilised with mixed finite element approach. The correctness of the solution algorithm is verified in a commonly used test case of laminar flow in 3D lid-driven cavity. The results of computations of the free jet are compared with experimental and numerical reference data, the latter obtained from the high-order pseudospectral code. The statistics of centerline flow velocity – mean velocity and its fluctuations – show satisfactory agreement with the reference data. Copyright © 2015 John Wiley & Sons, Ltd.

The discontinuous Galerkin method is applied to simulation of incompressible free round turbulent jet using Vreman model for large eddy simulation with eddy viscosity approach. The solution algorithm is based on the classical projection method. For time integration, explicit Runge–Kutta method is employed. The computational meshes consist of hexahedral elements with flat faces. All flow variables are expressed with modal expansions of the same order. The statistics of centerline flow velocity show satisfactory agreement with the reference data.

This paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity, and pressure, for the Brinkman problem with nonstandard boundary conditions. The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive equation relating the aforementioned unknowns and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic, which, thanks to the Lax–Milgram theorem, and besides proving the well-posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart–Thomas elements of any order
for the velocity field, and piecewise continuous polynomials of degree *k* + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual-based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators and showing the behavior of the associated adaptive algorithms, are reported. Copyright © 2015 John Wiley & Sons, Ltd.

We propose and analyze a mixed finite element method for the Brinkman problem, augmented with least-squares terms arising from the constitutive equation and from the incompressibility condition. Both the continuous and discrete augmented formulations are well posed. A priori error estimates and the corresponding convergence rates are established, and we derive two reliable and efficient residual-based a posteriori error estimators. Several numerical results illustrate the performance of the method and confirm the properties of the estimators.

This work proposes an innovative numerical method for simulating the interaction of fluid with irregularly shaped stationary structures based on Cartesian grids. Instead of prescribing an artificial force to enforce the no-slip boundary condition at the solid–fluid interface, this work imposes two boundary velocities, referred to as the solid and mass-conserving boundary velocities, to satisfy the no-slip boundary condition and mass conservation in the ghost cells around the immersed solid boundary. Both the traditional level set method [41] and the hybrid particle level set method [45] were used to represent the solid boundary and the complex free-surface evolution, respectively. Consequently, the boundary velocities close to the immersed solid boundary can be determined in terms of the level set function and the neighboring fluid velocity. The projection method is further modified to incorporate the solid and mass-conserving boundary velocities into the solution algorithm. A series of numerical experiments were conducted to demonstrate the feasibility of the proposed method. They involved uniform flow past a stationary circular cylinder and the propagation of water waves over a submerged trapezoidal breakwater. Comparisons between the numerical results and experimental data showed very good agreement in all cases of interest. Copyright © 2015 John Wiley & Sons, Ltd.

Two boundary velocities are introduced to satisfy the no-slip boundary condition and the continuity equation around the immersed solid boundary. The first, called the solid boundary velocity, is determined from the no-slip boundary condition, while the second, called the mass-conserving boundary velocity, is determined from the continuity equation around the solid boundary. The conventional projection method is further modified to incorporate the solid and mass-conserving boundary velocities into the solution algorithm.