Model order reduction (MOR) of the 2D Burgers equation is investigated. The mathematical formulation of POD/DEIM reduced order model (ROM) is derived based on the Galerkin projection and discrete empirical interpolation method (DEIM) from the existing high fidelity implicit finite difference full model. For validation we numerically compared the POD ROM, POD/DEIM and the full model in two cases of *R**e* = 100 and *R**e* = 1000, respectively. We found that the POD/DEIM ROM leads to a speed-up of CPU time by a factor of *O*(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of *R**e* = 1000 has an accuracy with error *O*(10^{−3}) versus *O*(10^{−4}) in the case of *R**e* = 100 when compared to the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated reduced order model (ROM) exhibit considerable agreement with the high-fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.

In this study, a first attempt has been made to introduce mesh adaptivity into the ensemble Kalman fiter method (EnKF). The EnKF data assimilation system was established for an unstructured adaptive mesh ocean model (Fluidity, Imperial College London). The mesh adaptivity involved using high resolution mesh at the regions of large flow gradients and around the observation points in order to reduce the representativeness errors of the observations. The use of adaptive meshes unavoidably introduces difficulties in the implementation of EnKF. The ensembles are defined at different meshes. To overcome the difficulties, a supermesh technique is employed for generating a reference mesh. The ensembles are then interpolated from their own mesh onto the reference mesh. The performance of the new EnKF data assimilation system has been tested in the Munk gyre flow test case. The discussion of this paper will focus on (a) the development of the EnKF data assimilation system within an adaptive mesh model; and (b) the advantages of mesh adaptivity in the ocean data assimilation model. This article is protected by copyright. All rights reserved.

The spatial resolutions of numerical atmospheric and oceanic circulation models have steadily increased over the past decades. Horizontal grid spacing down to the order of 1 km is now often used to resolve cloud systems in the atmosphere and sub-mesoscale circulation systems in the ocean. These fine resolution models encompass a wide range of temporal and spatial scales, across which dynamical and statistical properties vary. In particular, dynamic flow systems at small scales can be spatially localized and temporarily intermittent. Difficulties of current data assimilation algorithms for such fine resolution models are numerically and theoretically examined. An analysis shows that the background error correlation length scale is larger than 75 km for streamfunctions and is larger than 25 km for water vapor mixing ratios, even for a 2-km resolution model. A theoretical analysis suggests that such correlation length scales prevent the currently used data assimilation schemes from constraining spatial scales smaller than 150 km for streamfunctions and 50 km for water vapor mixing ratios. These results highlight the need to fundamentally modify currently used data assimilation algorithms for assimilating high-resolution observations into the aforementioned fine resolution models. Within the framework of four-dimensional variational data assimilation, a multiscale methodology based on scale decomposition is suggested and challenges are discussed.

A numerical model based on the Smoothed Particle Hydrodynamics (SPH) method is developed to simulate depth-limited turbulent open channel flows over hydraulically rough beds. The 2D Lagrangian form of the Navier-Stokes (N-S) equations are solved, in which a drag-based formulation is used based on an effective roughness zone near the bed to account for the roughness effect of bed spheres and an improved Sub-Particle-Scale (SPS) model is applied to account for the effects of turbulence. The SPS model is constructed based on the mixing-length assumption rather than the standard Smagorinsky approach to compute the eddy-viscosity. A more robust in/out-flow boundary technique is also proposed to achieve stable uniform flow conditions at the inlet and outlet boundaries where the flow characteristics are unknown. The model is applied to simulate uniform open channel flow over a rough bed composed of regular spheres and validated by experimental velocity data. To investigate the influence of the bed roughness on different flow conditions, data from 12 experimental tests with different bed slopes and uniform water depths are simulated and a good agreement has been observed between the model and experimental results of the streamwise velocity and turbulent shear stress. This shows that both the roughness effect and flow turbulence should be addressed in order to simulate the correct mechanisms of turbulent flow over a rough bed boundary and that the presented SPH model accomplishes this successfully.

This paper presents a novel mass conservative, positivity preserving wetting and drying treatment for Godunov-type shallow water models with second order bed elevation discretization. The novel method allows to compute water depths equal to machine accuracy without any restrictions on the time step or any threshold which defines whether the finite volume cell is considered to be wet or dry. The resulting scheme is second order accurate in space and keeps the C-property condition at fully flooded area but also at the wet/dry interface. For the time integration is used a second order accurate Runge-Kutta method. The method is tested in two well-known computational benchmarks for which an analytical solution can be derived, a C-property benchmark and in an additional example where the experimental results are reproduced. Overall the presented scheme shows very good agreement with the reference solutions. The method can also be used in the discontinuous Galerkin method. This article is protected by copyright. All rights reserved.

Efficient transport algorithms are essential to the numerical resolution of incompressible fluid flow problems. Semi-Lagrangian methods are widely used in grid based methods to achieve this aim. The accuracy of the interpolation strategy then determines the properties of the scheme. We introduce a simple multi-stage procedure which can easily be used to increase the order of accuracy of a code based on multi-linear interpolations. This approach is an extension of a corrective algorithm introduced by Dupont & Liu (2003, 2007). This multi-stage procedure can be easily implemented in existing parallel codes using a domain decomposition strategy, as the communications pattern is identical to that of the multi-linear scheme. We show how a combination of a forward and backward error correction can provide a third-order accurate scheme, thus significantly reducing diffusive effects while retaining a non-dispersive leading error term. Copyright © 2016 John Wiley & Sons, Ltd.

The acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non-uniform mean flow. As for any hybrid computational aeroacoustics approach, a first computational fluid dynamics simulation is carried out from which the mean flow characteristics and acoustic sources are obtained. In a second step, the APE are solved to get the acoustic pressure and particle velocity fields. However, resorting to the finite element method (FEM) for that purpose is not straightforward. Whereas mixed finite elements satisfying an appropriate inf-sup compatibility condition can be built in the case of no mean flow, i.e., for the standard wave equation in mixed form, these are difficult to implement and their good performance is yet to be checked for more complex wave operators. As a consequence, strong simplifying assumptions are usually considered when solving the APE with FEM. It is possible to avoid them by resorting to stabilized formulations. In this work, a residual-based stabilized FEM is presented for the APE at low Mach numbers, which allows one to deal with the APE convective and reaction terms in its full extent. The key of the approach resides in the design of the matrix of stabilization parameters. The performance of the formulation and the contributions of the different terms in the equations are tested for an acoustic pulse propagating in sheared solenoidal mean flow, and for the aeolian tone generated by flow past a two-dimensional cylinder. Copyright © 2016 John Wiley & Sons, Ltd.

The complete interaction between the structural domain and the acoustic domain needs to be considered in many engineering problems, especially for the acoustic analysis concerning thin structures immersed in water. This study employs the finite element method to model the structural parts and the fast multipole boundary element method to model the exterior acoustic domain. Discontinuous higher order boundary elements are developed for the acoustic domain to achieve higher accuracy in the coupling analysis. Structural–acoustic design sensitivity analysis can provide insights into the effects of design variables on radiated acoustic performance, and thus is important to the structural–acoustic design and optimization processes. This study is the first to formulate equations for sound power sensitivity on structural surfaces based on an adjoint operator approach, and equations for sound power sensitivity on arbitrary closed surfaces around the radiator based on the direct differentiation approach. The design variables include fluid density, structural density, Poisson's ratio, Young's modulus, and structural shape/size. A numerical example is presented to demonstrate the accuracy and validity of the proposed algorithm. Different types of coupled continuous and discontinuous boundary elements with finite elements are used for the numerical solution, and the performances of the different types of finite element/continuous and discontinuous boundary element coupling are presented and compared in detail. Copyright © 2016 John Wiley & Sons, Ltd.

Numerical methods based on geometrical multiscale models of blood flows solve for averaged flow statistics on a network of vessels while providing more detailed information about fluid dynamics in a specific region of interest. In such an approach, a 3D model based on the Navier-Stokes equations posed in a domain with rigid walls is often used to describe blood flow dynamics in the refined region. While ignoring elasticity effects in 3D models is plausible in certain applications and saves computational time significantly, coupling such models with 1D flow models may result in non-physiological phenomena in the computed solutions. Thus, the immediate coupling conditions based on continuity of normal stresses, flow rate, pressure, or a combination of thereof do not account for the inconsistency between elasticity effects in the 1D model and the non-compliance of the 3D model. In this paper we introduce and study an auxiliary absorbing 0D model, which is placed at the interface between 1D and 3D models. A virtual device mimics the effect of the 3D model compliance and hence reduces pressure wave reflection and instabilities caused by the inconsistency. The absorbing model is developed from basic mechanical principles. As a result, parameters of the 0D model can be designed based on hemodynamic data. We analyse the stability of the geometrical multiscale model and perform several numerical experiments to assess its computational efficiency. This article is protected by copyright. All rights reserved.

This paper presents an efficient procedure for overcoming the deficiency of WENO schemes near discontinuities. Through a thorough incorporation of smoothness indicators into the weights definition, up to ninth-order accurate multistep methods are devised, providing WENO schemes with enhanced order of convergence at transition points from smooth regions to a discontinuity, while maintaining stability and the ENO behavior. We also provide a detailed analysis of the resolution power, and show that the solution enhancements of the new method at smooth regions come from their ability to render smoothness indicators closer to uniformity. The new scheme exhibits similar fidelity as other multistep schemes, however with superior characteristics in terms of robustness and efficiency, as no logical statements or mapping function is needed. Extensions to higher orders of accuracy present no extra complexity. Numerical solutions of linear advection problems and nonlinear hyperbolic conservation laws are used to demonstrate the scheme's improved behavior for shock-capturing problems. This article is protected by copyright. All rights reserved.

Rhie-Chow interpolation is a commonly used method in CFD calculations on a co-located mesh in order to suppress non-physical pressure oscillations arising from chequer-board effects. A fully parallelised smoothed-interface immersed boundary method on a co-located grid is described in this paper. We discuss the necessity of modifications proposed by Choi [1] to the original Rhie-Chow interpolation in order to deal with a locally refined mesh. Numerical simulation with the modified scheme of Choi shows that numerical dissipation due to Rhie-Chow interpolation introduces significant errors at the immersed boundary. To address this issue we develop an improved-Rhie-Chow interpolation scheme which is shown to increase the accuracy in resolving the flow near the immersed boundary. We compare our improved scheme with the modified scheme of Choi by parallel simulations of benchmark flows i/ flow past a stationary cylinder, ii/ flow past an oscillating cylinder and iii/ flow past a stationary elliptical cylinder, where Reynolds numbers are tested in the range 10 - 200. Our improved scheme is significantly more accurate and compares favourably with a staggered grid algorithm. We also develop a scheme to compute the boundary force for the direct-forcing immersed boundary method efficiently. This article is protected by copyright. All rights reserved.

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart-Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of gas-kinetic scheme (GKS) is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique (AMR) is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non-manifold hybrid mesh data (NHMD) structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. This article is protected by copyright. All rights reserved.

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

Anisotropic diffusion phenomenon in fluids is simulated using smoothed particle hydrodynamics (SPH). A new SPH approximation for diffusion operator, named anisotropic SPH approximation for anisotropic diffusion (ASPHAD), is derived. Basic idea of the derivation is that anisotropic diffusion operator is first approximated by an integral in a coordinate system in which it is isotropic. The coordinate transformation is a combination of a coordinate rotation and a scaling in accordance with diffusion tensor. Then, inverse coordinate transformation and particle discretization are applied to the integral to achieve ASPHAD. Noting that weight function used in the integral approximation has anisotropic smoothing length, which becomes isotropic under the inverse transformation. ASPHAD is general and unique for both isotropic and anisotropic diffusions with either constant or variable diffusing coefficients. ASPHAD was numerically examined in some cases of isotropic and anisotropic diffusions of a contaminant in fluid, and the simulation results are very consistent with corresponding analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.

A new smoothed particle hydrodynamics (SPH) approximation for diffusion operator, named anisotropic SPH approximation for anisotropic diffusion (ASPHAD), is derived. ASPHAD is general and unique for both isotropic and anisotropic diffusions with either constant or variable diffusin coefficients. Numerical examinations in some cases of isotropic and anisotropic diffusions of a contaminant in fluid show a very good consistence with corresponding analytical solutions.

We present a new closure model for single fluid, multi-material Lagrangian hydrodynamics and its application to high-order finite element discretizations of these equations . The model is general with respect to the number of materials, dimension and space and time discretizations. Knowledge about exact material interfaces is not required. Material indicator functions are evolved by a closure computation at each quadrature point of mixed cells, which can be viewed as a high-order variational generalization of the method of Tipton . This computation is defined by the notion of partial non-instantaneous pressure equilibration, while the full pressure equilibration is achieved by both the closure model and the hydrodynamic motion. Exchange of internal energy between materials is derived through entropy considerations, that is, every material produces positive entropy, and the total entropy production is maximized in compression and minimized in expansion. Results are presented for standard one-dimensional two-material problems, followed by two-dimensional and three-dimensional multi-material high-velocity impact arbitrary Lagrangian–Eulerian calculations. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

We present a closure model that evolves material properties at quadrature point level. The method is general with respect to the number of materials, dimension and space and time discretizations.Material volumes are evolved by imposing partial pressure equilibration, and internal energy exchange between materials is determined by considerations of the expected behavior of the entropy production. Results are presented for standard one-dimensional two-material problems, followed by two-dimensional and three-dimensional multi-material arbitrary Lagrangian-Eulerian high-velocity impacts.

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

A fourth-order finite-volume method for solving the Navier-Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. Shown here, a Mach reflection problem is solved to demonstrate the effectiveness of the mapped algorithm on a non-rectangular physical domain. AMR patches on the finest mesh level are outlined.

A variable-fidelity aerodynamic model based on proper orthogonal decomposition (POD) of an ensemble of computational fluid dynamics (CFD) solutions at different parameters is presented in this article. The ensemble of CFD solutions consists of two subsets of numerical solutions or snapshots computed at two different nominal orders of accuracy or discretization. These two subsets are referred to as the low-fidelity and high-fidelity solutions or data, whereby the low fidelity corresponds with computations made at the lower nominal order of accuracy or coarser discretization. In this model, the relatively inexpensive low-fidelity data and the more accurate but expensive high-fidelity data are considered altogether to devise an efficient prediction methodology involving as few high-fidelity analyses as possible, while obtaining the desired level of detail and accuracy. The POD of this set of variable-fidelity data produces an optimal linear set of orthogonal basis vectors that best describe the ensemble of numerical solutions altogether. These solutions are projected onto this set of basis vectors to provide a finite set of scalar coefficients that represent either the low-fidelity or high-fidelity solutions. Subsequently, a global response surface is constructed through this set of projection coefficients for each basis vector, which allows predictions to be made at parameter combinations not in the original set of observations. This approach is used to predict supersonic flow over a slender configuration using Navier–Stokes solutions that are computed at two different levels of nominal accuracy as the low-fidelity and high-fidelity solutions. The numerical examples show that the proposed model is efficient and sufficiently accurate. Copyright © 2016 John Wiley & Sons, Ltd.

A variable-fidelity aerodynamic model based on proper orthogonal decomposition (POD) of an ensemble of computational fluid dynamics (CFD) solutions at variable fidelity and at different parameters is presented in this article. This approach is used to predict supersonic flow over a slender configuration using Navier–Stokes solutions that are computed at two different levels of nominal accuracy as the low-fidelity and high-fidelity solutions. The numerical results show that the proposed model is efficient and sufficiently accurate.

This paper presents an efficient method to simulate the reactive flow for general equation of states with the compressible fluid model coupled with reactive rate equation. The important aspect is to deal with mixture of different phases in one cell, which will inevitably happen in the Eulerian method for reactive flow. Physical variables such as the pressure,velocity, and speed of sound in each cell need to be reconstructed for the Harten-Lax-Leer-Contact (HLLC) Riemann solver, which will result in nonlinear algebra equations, and these reconstructed variables are used to obtain the flux. Numerical examples of stable and unstable detonations with different equation of states demonstrate the accuracy and efficiency of this method. Copyright © 2016 John Wiley & Sons, Ltd.

Pressure distribution for steady problem of detonation with Cochran-Chan equation of state, with the variables reconstruction for mixing fluids, we obtain an accurate convergence solution.

The blood flow model maintains the steady-state solutions, in which the flux gradients are non-zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted essentially non-oscillatory (WENO) schemes to this model with such well-balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well-balanced property, and keep good resolution for smooth and discontinuous solutions. Copyright © 2016 John Wiley & Sons, Ltd.

A high-order well-balanced finite difference weighted essentially non-oscillatory scheme is designed for the blood flow model. The scheme preserves the well-balanced property and achieves high-order accuracy for smooth solutions. In addition, the scheme possesses sharp shock transition.

In this paper, we propose a new methodology for numerically solving elliptic and parabolic equations with discontinuous coefficients and singular source terms. This new scheme is obtained by clubbing a recently developed higher-order compact methodology with special interface treatment for the points just next to the points of discontinuity. The overall order of accuracy of the scheme is at least second. We first formulate the scheme for one-dimensional (1D) problems, and then extend it directly to two-dimensional (2D) problems in polar coordinates. In the process, we also perform convergence and related analysis for both the cases. Finally, we show a new direction of implementing the methodology to 2D problems in cartesian coordinates. We then conduct numerous numerical studies on a number of problems, both for 1D and 2D cases, including the flow past circular cylinder governed by the incompressible Navier–Stokes equations. We compare our results with existing numerical and experimental results. In all the cases, our formulation is found to produce better results on coarser grids. For the circular cylinder problem, the scheme used is seen to capture all the flow characteristics including the famous von Kármán vortex street. Copyright © 2016 John Wiley & Sons, Ltd.

A class of efficient higher order accurate finite difference schemes is developed for parabolic and elliptic PDEs with discontinuous coefficients and singular source terms. Clubbing a recently developed HOC methodology with special interface treatment renders the schemes at least a second order spatial accuracy. Apart from 1D problems, the 2D extension of the schemes works with equal ease on problems in polar and Cartesian grids. Excellent results are obtained including the famous von Kármán vortex street for flow past circular cylinder.

A least-squares finite element model with spectral/*hp* approximations was developed for steady, two-dimensional flows of non-Newtonian fluids obeying the Carreau–Yasuda constitutive model. The finite element model consists of velocity, pressure, and stress fields as independent variables (hence, called a mixed model). Least-squares models offer an alternative variational setting to the conventional weak-form Galerkin models for the Navier–Stokes equations, and no compatibility conditions on the approximation spaces used for the velocity, pressure, and stress fields are necessary when the polynomial order (*p*) used is sufficiently high (say, *p* > 3, as determined numerically). Also, the use of the spectral/*hp* elements in conjunction with the least-squares formulation with high *p* alleviates various forms of locking, which often appear in low-order least-squares finite element models for incompressible viscous fluids, and accurate results can be obtained with exponential convergence. To verify and validate, benchmark problems of Kovasznay flow, backward-facing step flow, and lid-driven square cavity flow are used. Then the effect of different parameters of the Carreau–Yasuda constitutive model on the flow characteristics is studied parametrically. Copyright © 2016 John Wiley & Sons, Ltd.

A mixed least-squares finite element model with spectral/hp approximations was developed for steady, two-dimensional flows of non-Newtonian fluids obeying the Carreau-Yasuda constitutive model. The mixed least-squares finite element model developed herein has advantages over the weak-form Galerkin model in eliminating any type of locking. In addition, there are no compatibility restrictions placed between velocity, pressure, and stress approximation spaces for sufficiently higher-order polynomials. Also, a combination of spectral/hp approximation functions and least-squares model yields accurate results with spectral convergence.

In this paper, we propose a numerical algorithm for time-dependent convection–diffusion–reaction problems and compare its performance with the well-known numerical methods in the literature. Time discretization is performed by using fractional-step *θ*-scheme, while an economical form of the residual-free bubble method is used for the space discretization. We compare the proposed algorithm with the classical stabilized finite element methods over several benchmark problems for a wide range of problem configurations. The effect of the order in the sequence of discretization (in time and in space) to the quality of the approximation is also investigated. Numerical experiments show the improvement through the proposed algorithm over the classical methods in either cases. Copyright © 2016 John Wiley & Sons, Ltd.

We present a numerical algorithm to get the approximate solution of time-dependent convection–diffusion–reaction problems, especially in the case of small diffusion. The numerical method is based on fractional-step *θ*-scheme in time combined with bubble-based finite element methods in space. We further compare the proposed algorithm with two different stabilized methods on several benchmark problems. Numerical experiments illustrate the good performance of the proposed method even on coarse meshes as compared with the others.

In this paper, we show how the spectral formulation of Baker, Meiron and Orszag can be used to solve for waves on water of infinite depth confined between two flat, vertical walls, and also how it can be modified to take into account water of finite depth with a spatially varying bottom. In each case, we use Chebyshev polynomials as the basis of our representation of the solution and filtering to remove spurious high-frequency modes. We show that spectral accuracy can be achieved until wave breaking, plunging or wall impingment occurs in two model problems. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we show how the spectral formulation of Baker, Meiron and Orszag can be used to solve for waves on water of infinite depth confined between two flat, vertical walls and also how it can be modified to take into account water of finite depth with a spatially varying bottom, in each case using Chebyshev polynomials as the basis for the solution.

Numerical modeling of multiphase flow generally requires a special procedure at the solid wall in order to be consistent with Young's law for static contact angles. The standard approach in the lattice Boltzmann method, which consists of imposing fictive densities at the solid lattice sites, is shown to be deficient for this task. Indeed, fictive mass transfer along the boundary could happen and potentially spoil the numerical results. In particular, when the contact angle is less than 90 degrees, the deficiencies of the standard model are major. Various videos that demonstrate this behavior are provided (Supporting Information). A new approach is proposed and consists of directly imposing the contact angle at the boundaries in much the same way as Dirichlet boundary conditions are generally imposed. The proposed method is able to retrieve analytical solutions for static contact angles in the case of straight and curved boundaries even when variable density and viscosity ratios between the phases are considered. Although the proposed wetting boundary condition is shown to significantly improve the numerical results for one particular class of lattice Boltzmann model, it is believed that other lattice Boltzmann multiphase schemes could also benefit from the underlying ideas of the proposed method. The proposed algorithm is two-dimensional, and the D2Q9 lattice is used. Copyright © 2016 John Wiley & Sons, Ltd.

A numerical approach for modeling Young's law for static contact angles in a multiphase lattice Boltzmann method is proposed. The contact angle at the boundaries is imposed in much the same way as Dirichlet boundary conditions are generally imposed. The proposed method is able to retrieve analytical solutions for static contact angles in the case of straight and curved boundaries even when variable density and viscosity ratios between the phases are considered.

We propose a fully conservative high-order upwind multi-moment method for the conservation equation. The proposed method is based on a third-order polynomial interpolation function and semi-Lagrangian formulation and is a variant of the constrained interpolation profile conservative semi-Lagrangian scheme with third-order polynomial function method. The third-order interpolation function is constructed based on three constraints in the upwind cell (two boundary values and a cell average) and a constraint in the downwind cell (a cell center value). The proposed method shows fourth-order accuracy in a benchmark problem (sine wave propagation). We also propose a less oscillatory formulation of the proposed method. The less oscillatory formulation can minimize numerical oscillations. These methods were validated through scalar transport problems, and compressible flow problems (shock tube and 2D explosion problems). Copyright © 2016 John Wiley & Sons, Ltd.

We proposed a new type of CIP-CSL schemes (conservation equation solver based on a multi-moment concept). The figures show numerical results of a scalar transport problem by CSL2, CSL3D, CSL3, and CSL3DL (*N* = 200 and *t* = 16). The proposed formulations (CSL3D and CSL3DL) are superior to existing CIP-CSL schemes (such as CSL2 and CSL3).

An advanced hybrid lumped parameter code for the simulation of Pulsating Heat Pipes is developed. Being able to simulate transient operative conditions and removing common physical simplified assumptions, it represents a step forward with respect to the present models of passive two-phase systems. Mass, momentum and energy balances account for the thermal and fluid-dynamics phenomena. Heterogeneous and homogeneous phase changes are directly integrated. In addition, a fitting correlation for the wall/vapour heat transfer coefficient is implemented and tuned against experimental data in order to evaluate the influence of the liquid film on conjugate heat transfer. The resulting numerical tool have been validated against experimental data achieved testing a copper pulsating heat pipe during the 58th ESA Parabolic Flight Campaign in several operative conditions and transient gravity levels. The predicted results show very good matching with the actual thermo-physical behaviour of the system. Copyright © 2016 John Wiley & Sons, Ltd.

This work proposes an advanced hybrid lumped parameter code for the simulation of two-phase passive thermal systems named pulsating heat pipes. Even if lumped parameter models are not unusual for such devices, for the first time, transient operative conditions are simulated by removing physical simplified assumptions and embedding phase changes. Advanced numerical technique guaranties stabilization of the model and fast simulations allowing extended sensitivity analysis and device designs. Validation shows very good matching with the actual thermo-physical behaviour of the system.

In this paper, we introduce a shock-capturing artificial viscosity technique for high-order unstructured mesh methods. This artificial viscosity model is based on a non-dimensional form of the divergence of the velocity. The technique is an extension and improvement of the dilation-based artificial viscosity methods introduced in Premasuthan *et al.*, and further extended in Nguyen and Peraire . The approach presented has a number attractive properties including non-dimensional analytical form, sub-cell resolution, and robustness for complex shock flows on anisotropic meshes. We present extensive numerical results to demonstrate the performance of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, pressure stability through the suppression of high-frequency pressure oscillations in the moving particle semi-implicit (MPS) method is presented. To obtain a stable pressure field, we improve the free-surface particle search algorithm. Pressure stability follows from the suppression of high-frequency pressure oscillations due to a correction in the Laplacian operator of the Poisson pressure equation and from the correction of the pressure gradient operator. The three proposed modifications are applied gradually and compared with the MPS method to show the improvements in the hydrostatic pressure and dam-breaking problems. To validate the suppression of the high-frequency numerical pressure oscillations, modified MPS methods with and without a removable wall are compared with published dam-breaking experiment pressure measurements. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, pressure stability through the suppression of high-pressure oscillations in the moving particle semi-implicit (MPS) is presented. To validate the suppression of the high-frequency pressure oscillations, modified MPS methods with and without a removable wall are compared with dam-breaking experiment pressure measurements.

In this paper, the thermal load on an actively cooled lobed strut injector for scramjet (supersonic combustion ramjet) applications is investigated numerically. This requires coupled simulations of the strut internal and external flow fields together with the heat conduction in the solid injector body. In order to achieve a fast mixing, the lobed strut is positioned at the channel axis to inject hydrogen into the core of a Mach 3 air stream. There it is exposed to the extremely high temperatures of the high speed flow. While the external air and hydrogen flows are supersonic, the strut internal hydrogen flow is mainly subsonic, in some regions at very low Mach numbers. To enable a simulation of the internal flow field which ranges from very low to very high Mach numbers (approximately Mach 2.25 at the nozzle exit), a preconditioning technique is employed. The compressible finite-volume scheme uses a spatially fourth order multi-dimensional limiting process discretization, which is used here for a first time to simulate a geometrically and fluid mechanically highly complex problem. It will be demonstrated that besides its high accuracy the multi-dimensional limiting process scheme is numerically stable even in case of demanding practical applications. The coupled simulation of the lobed strut injector delivers unique insight into the flow phenomena inside and outside the strut, the heat fluxes, the temperature distribution in the solid material, the required hydrogen mass flux with respect to cooling requirements and details concerning the conditions at the exit of the injector. Copyright © 2016 John Wiley & Sons, Ltd.

The paper investigates the flow field inside and outside a hydrogen stut injector coupled with heat transfer in the solid. Simulations use a fourth order MLP (multi-dimensional limiting process) discretization and an all-Mach number preconditioning. It will be shown that the chosen high order scheme achieves excellent results at low-additional cost.

The approximation of reduced linear evolution operator (propagator) via dynamic mode decomposition (DMD) is addressed for both linear and nonlinear events. The 2D unsteady supersonic underexpanded jet, impinging the flat plate in nonlinear oscillating mode, is used as the first test problem for both modes. Large memory savings for the propagator approximation are demonstrated. Corresponding prospects for the estimation of receptivity and singular vectors are discussed. The shallow water equations are used as the second large-scale test problem. Excellent results are obtained for the proposed optimized DMD method of the shallow water equations when compared with recent POD-based/discrete empirical interpolation-based model reduction results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

We propose a framework for dynamic mode decomposition (DMD) with application of the reduced Schmid operator instead of the classic DMD approach. Instead of storing the total operator matrix, the proposed technique ensures computer memory and computing time (CPU) savings of about several orders of magnitude. We emphasize an excellent behavior of the Schmid operator by considering two numerical experiments: the case of a 2D supersonic underexpanded jet on a plate and the problem of the 2D shallow water equations.

An improved high-order accurate WENO finite volume method based on unstructured grids for compressible multi-fluids flow is proposed in this paper. The third-order accuracy WENO finite volume method based on triangle cell is used to discretize the governing equations. To have higher order of accuracy, the P1 polynomial is reconstructed firstly. After that, the P2 polynomial is reconstructed from the combination of the P1. The reconstructed coefficients are calculated by analytical form of inverse matrix rather than the numerical inversion. This greatly improved the efficiency and the robustness. Four examples are presented to examine this algorithm. Numerical results show that there is no spurious oscillation of velocity and pressure across the interface and high-order accurate result can be achieved. Copyright © 2016 John Wiley & Sons, Ltd.

An improved high-order accurate WENO finite volume method based on unstructured grids for compressible multi-fluids flow is proposed in this paper. The reconstructed coefficients are calculated by analytical form of inverse matrix rather than the numerical inversion. This greatly improved the efficiency and the robustness. Numerical results show that there is no spurious oscillation of velocity and pressure across the interface and high-order accurate result can be achieved.

A hybrid particle-mesh method was developed for efficient and accurate simulations of two-phase flows. In this method, the main component of the flow is solved using the constrained interpolated profile/multi-moment finite volumemethod; the two-phase interface is rendered using the finite volume particle (FVP) method. The effect of surface tension is evaluated using the continuum surface force model. Numerical particles in the FVP method are distributed only on the surface of the liquid in simulating the interface between liquid and gas; these particles are used to determine the density of each mesh grid. An artificial term was also introduced to mitigate particle clustering in the direction of maximum compression and sparse discretization errors in the stretched direction. This enables accurate interface tracking without diminishing numerical efficiency. Two benchmark simulations are used to demonstrate the validity of the method developed and its numerical stability. Copyright © 2016 John Wiley & Sons, Ltd.

A hybrid particle-mesh method was developed for efficient and accurate simulations of two-phase flows. In this method, the main component of the flows is solved by the constrained interpolated profile/multi-moment method, whereas the interface between two phases is rendered by the finite volume particle method. Two benchmark simulations demonstrated the efficiency and accuracy of the method developed.

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid-structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM-based coupling for low- as well as high-Reynolds-number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous-dominated and convection-dominated flows. Residual-based fluid stabilizations in the interior of the fluid subdomains and accompanying face-oriented fluid and ghost-penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous-dominated and convection-dominated flow problems independent of the interface position. Challenging two-dimensional and three-dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at *R**e* = 10000 and the flow interacting with a three-dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.

Complex high-Reynolds-number fluid-structure interaction applications require high mesh resolution in the boundary layer region around moving/rotating obstacles. For this purpose, a powerful technique based on the extended finite element method is proposed to embed arbitrary fluid patches into a background fluid discretization using cut elements. Stabilization techniques are provided to guarantee stable and accurate weak imposition of interface coupling conditions based on Nitsche's method independent of the patch location.

In this paper, a local mesh refinement (LMR) scheme on Cartesian grids for large-eddy simulations is presented. The approach improves the calculation of ghost cell pressures and velocities and combines LMR with high-order interpolation schemes at the LMR interface and throughout the rest of the computational domain to ensure smooth and accurate transition of variables between grids of different resolution. The approach is validated for turbulent channel flow and flow over a matrix of wall-mounted cubes for which reliable numerical and experimental data are available. Comparisons of predicted first-order and second-order turbulence statistics with the validation data demonstrated a convincing agreement. Importantly, it is shown that mean streamwise velocities and fluctuating turbulence quantities transition smoothly across coarse-to-fine and fine-to-coarse interfaces. © 2016 The Authors International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd

This paper introduces and validates a local mesh refinement approach for simulations of turbulent flows in complex domains. The method features high-order interpolation schemes at the fine-coarse mesh interfaces and uses up to fourth-order central differencing schemes for convective and diffusive fluxes. It is shown that the local mesh refinement method is able to predict accurately first-order and second-order statistics of two challenging flows, a turbulent channel flow and the flow over a matrix of cubes. The method offers significant savings of computational resources due to the placement of very fine meshes into critical areas, for instance around the cubes, while for the rest of the domain, coarser meshes are employed.

Discontinuous Galerkin (DG) methods allow high-order flow solutions on unstructured or locally refined meshes by increasing the polynomial degree and using curved instead of straight-sided elements. However, one of the currently largest obstacles to applying these methods to aerodynamic configurations of medium to high complexity is the availability of appropriate higher-order curved meshes.

In this article, we describe a complete chain of higher-order unstructured grid generation and higher-order DG flow solution applied to a turbulent flow around a three-dimensional high-lift configuration. This includes (i) the generation of an appropriately coarse straight-sided mesh; (ii) the evaluation of additional points on the computer-aided design geometry of the curved-wall boundary for defining a piecewise polynomial boundary representation; (iii) a higher order mesh deformation to translate the curvature from the wall boundary into the interior of the computational domain; and (iv) the description of a DG discretization, which is sufficiently stable to allow a flow computation on the resulting curved mesh. Finally, a fourth-order flow solution of the Reynolds-averaged Navier–Stokes and *k*-*ω* turbulence model equations is computed on a fourth-order unstructured hybrid mesh around the three-dimensional high-lift simulation of wing-flow noise generation configuration. Copyright © 2016 John Wiley & Sons, Ltd.

This paper includes a description of a complete chain of unstructured curvilinear grid generation and higher order Discontinuous Galerkin flow solution applied to a turbulent flow around a 3D high-lift configuration. A fourth order flow solution of the RANS and k-w turbulence model equations is computed on a fourth order unstructured hybrid (mixed-element) mesh around the 3D high-lift SWING configuration. A highly resolved flow solution is obtained featuring a complex vortex system.

We investigate implicit large eddy simulation of the Taylor–Green vortex, Comte-Bellot–Corrsin experiment, turbulent channel flow and transitional and turbulent flow over an SD7003 airfoil using the high-order unstructured correction procedure via reconstruction (CPR) scheme, also known as the flux reconstruction scheme. We employ P1 (second-order) to P5 (sixth-order) spatial discretizations. Results show that the CPR scheme can accurately predict turbulent flows without the addition of a sub-grid scale model. Numerical dissipation, concentrated at the smallest resolved scales, is found to filter high-frequency content from the solution. In addition, the high-order schemes are found to be more accurate than the low-order schemes on a per degree of freedom basis for the canonical test cases we consider. These results motivate the further investigation and use of the CPR scheme for simulating turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.

We perform simulations of turbulent flows using the correction procedure via reconstruction (CPR) scheme. Our results demonstrate that the CPR scheme can be used for implicit large eddy simulation, without the addition of an explicit sub-grid scale model. We find that the high-order schemes are generally more accurate than the low-order schemes on a per degree of freedom basis.

Accurately characterizing the forces acting on particles in fluids is of fundamental importance for understanding particle dynamics and binding kinetics. Conventional asymptotic solutions may lead to poor accuracy for neighboring particles. In this paper, we develop an accurate boundary integral method to calculate forces exerted on particles for a given velocity field. We focus our study on the fundamental two-bead oscillating problem in an axisymmetric frame. The idea is to exploit a correspondence principle between the unsteady Stokes and linear viscoelasticity in the Fourier domain such that a unifying boundary integral formulation can be established for the resulting Brinkman equation. In addition to the dimension reduction vested in a boundary integral method, our formulation only requires the evaluation of single-layer integrals, which can be carried out efficiently and accurately by a hybrid numerical integration scheme based on kernel decompositions. Comparison with known analytic solutions and existing asymptotic solutions confirms the uniform third-order accuracy in space of our numerical scheme. Copyright © 2016 John Wiley & Sons, Ltd.

We develop an accurate boundary integral method to calculate forces exerted on particles in unsteady Stokes flow and linear viscoelastic fluids. Our numerical method is third-order accurate uniformly in space and corrects the error due to the poles at the axis of symmetry.

The influence of mesh motion on the quality of large eddy simulation (LES) was studied in the present article. A three-dimensional, turbulent pipe flow (*R**e*_{τ}=360) was considered as a test case. Simulations with both stretching and static meshes were carried out in order to understand how mesh motion affects the turbulence statistics. The spatial filtering of static and moving mesh direct numerical simulation (DNS) data showed how an ideal LES would perform, while the comparison of DNS cases with static and moving meshes revealed that no significant numerical errors arise from the mesh motion when the simulation is fully resolved. The comparison of the filtered fields of the DNS with a moving mesh with the corresponding LES fields revealed different responses to mesh motion from different numerical approaches. A straightforward test was applied in order to verify that the moving mesh works consistently in LES: when the mesh is stretched in the streamwise direction, the moving mesh results should be in between the two extremal resolutions between which the mesh is stretched. Numerical investigations using four different LES approaches were carried out. In addition to the Smagorinsky model, three implicit LES approaches were used: linear interpolation (non-dissipative), the Gamma limiter (dissipative), and the scale-selective discretisation (slightly dissipative). The results indicate that while the Smagorinsky and the scale-selective discretisation approaches produce results consistent with the resolution of the non-static mesh, the implicit LES with linear interpolation or the Gamma scheme do not. Copyright © 2016 John Wiley & Sons, Ltd.

The effect of mesh motion on the outcome of a large eddy simulation was studied in the present article. A turbulent pipe flow (*Re _{t}* = 360) was used as a test case. The results of this study indicate that runtime mesh deformation can have a noticeable effect on the velocity and energy statistics of a large eddy simulation.

Particle-based CFD methods are powerful approaches to investigate free surface, multiphase flows, and fluid structure interaction problems because of their ability of tracking moving fluid interface even with huge deformations or fragmentation and merging. However, many fluid interface particle detection techniques are simple to implement but with low accuracy or provide relatively good detection results at complicated implementation cost or higher computational time. In case of incompressible flow simulation methods solving the Poisson equation of pressure, such as the moving particle semi-implicit method, boundary particles detection techniques' accuracy affects precision and stability of pressure computation and interaction between fluid phases. In the present work, a new fluid interface particle detection technique is proposed to improve the accuracy of the boundary particles detection and keep the implementation easy. Denominated as the neighborhood particles centroid deviation technique, it is a two-criteria technique based on the particle number density and the neighborhood particles weighted geometric center deviation. Compared with other techniques, the proposed neighborhood particles centroid deviation technique shows the best results by eliminating false interface particles inside the fluid domain and keeping the interface particles layer thin and regular. As a result, relatively stable pressure time histories and more consistent pressure and velocity fields are achieved. Copyright © 2016 John Wiley & Sons, Ltd.

Particle-based CFD methods are powerful approaches to investigate fluid flows with huge deformations or fragmentation and merging because of their ability of tracking moving interfaces. However, many fluid interface particle detection techniques are simple to implement but with low accuracy or provide relatively good detection results at complicated implementation or higher computational time. Besides a review of the main available techniques, in this paper, a new technique is proposed to improve the accuracy while keeping implementation easy and with low computational cost.

Large-eddy simulation (LES) consists in explicitly simulating the large scales of the fluid motion and in modeling the influence of the smallest scales. Thanks to the steady growth of computational resources, LES can now be used to simulate realistic systems with complex geometries. However, when LES is used in such complex geometries, an adequate mesh has to be determined to perform valid LES. In this work, a strategy is proposed to assess the quality of a given mesh and to adapt it locally. Two different criteria are used as mesh adaptation criteria. The first criterion is defined to ensure a correct discretization of the mean field, whereas the second criterion is defined to ensure enough explicit resolution of turbulent scales motions. The use of both criteria is shown in canonical flow cases. As a second part of this work, a numerical strategy for mesh adaptation in high-performance computing context is proposed by coupling the flow solver, YALES2, and the remeshing library, MMG3D, for massively parallel computations. This coupling enables an efficient and parallel remeshing of grids alleviating any memory or performance issues encountered in sequential tools. This strategy is finally applied to the simulation of the isothermal flow in a complex meso-combustor to demonstrate the applicability of the adaptation methodology to complex turbulent flows. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a novel mesh adaptation strategy in the context of Large Eddy Simulation (LES). Two mesh quality criteria are defined, one for the discretization of the mean field and the other for the turbulent kinetic energy resolution. A parallel mesh adaptation strategy, based on these criteria, is proposed and applied to the simulation of the turbulent iso-thermal flow in a complex meso-scale combustor. It shows a large improvement in the quality of the results with a moderate over-cost.

We have investigated a collocation methodology for the numerical simulation of Fujiwhara interactions between cyclone scale vortices. The method is validated by computing the rotational period (t*) of the Fujiwhara interaction, as well as by simulating concentric eyewal patterns and barotropic instability of tropical cyclones. Numerical simulation Fujiwhara interactions at moderately high Reynolds numbers, such as for , show that the kinetic energy of cyclones is consolidated into larger scales with a concurrent enstrophy cascade.

A three-dimensional numerical model is presented for the simulation of unsteady non-hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics-based scheme, which simulates sub-critical and super-critical flows. Three-dimensional velocity components are considered in a collocated arrangement with a *σ*-coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the *σ* coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non-hydrostatic equations enables the model to simulate short-period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.

A numerical model was presented for the simulation of three-dimensional unsteady non-hydrostatic shallow water flows. A sigma coordinate system with a collocated arrangement of three-dimensional velocity components was used to simulate the variation of water depth during the time steps. The numerical results showed that the model is capable of producing non-oscillatory and accurate results.

In this paper, we propose new energy dissipative characteristic numerical methods for the approximation of diffusive Oldroyd-B equations that are based either on the finite element or finite difference discretization. We prove energy stability of both schemes and illustrate their behavior on a series of numerical experiments. Using both the diffusive model and the logarithmic transformation of the elastic stress, we are able to obtain methods that converge as mesh parameter is refined. Copyright © 2015 John Wiley & Sons, Ltd.

We propose new energy dissipative characteristic schemes for the diffusive Oldroyd-B equations, which are based either on the finite element or finite difference discretization. Using both the diffusive model and the logarithmic transformation of the elastic stress, we are able to obtain methods that converge as mesh parameter is refined.

Central moment lattice Boltzmann method (LBM) is one of the more recent developments among the lattice kinetic schemes for computational fluid dynamics. A key element in this approach is the use of central moments to specify the collision process and forcing, and thereby naturally maintaining Galilean invariance, an important characteristic of fluid flows. When the different central moments are relaxed at different rates like in a standard multiple relaxation time (MRT) formulation based on raw moments, it is endowed with a number of desirable physical and numerical features. Because the collision operator exhibits a cascaded structure, this approach is also known as the cascaded LBM. While the cascaded LBM has been developed sometime ago, a systematic study of its numerical properties, such as the accuracy, grid convergence, and stability for well-defined canonical problems is lacking, and the present work is intended to fulfill this need. We perform a quantitative study of the performance of the cascaded LBM for a set of benchmark problems of differing complexity, viz., Poiseuille flow, decaying Taylor–Green vortex flow, and lid-driven cavity flow. We first establish its grid convergence and demonstrate second-order accuracy under diffusive scaling for both the velocity field and its derivatives, that is, the components of the strain rate tensor, as well. The method is shown to quantitatively reproduce steady/unsteady analytical solutions or other numerical results with excellent accuracy. The cascaded MRT LBM based on the central moments is found to be of similar accuracy when compared with the standard MRT LBM based on the raw moments, when a detailed comparison of the flow fields are made, with both reproducing even the small scale vortical features well. Numerical experiments further demonstrate that the central moment MRT LBM results in significant stability improvements when compared with certain existing collision models at moderate additional computational cost. Copyright © 2015 John Wiley & Sons, Ltd.

A comparative numerical study of the cascaded MRT LBM, which is based on central moments, and the standard MRT LBM, which is based on raw moments, is presented. For example, this figure shows that the streamlines in the cavity flow for Re = 5000 computed using the cascaded LBM is in excellent agreement with those of the standard MRT LBM. Furthermore, substantial improvement in the numerical stability is also achieved with the cascaded LBM.

We present a parameter-free stable maximum-entropy method for incompressible Stokes flow. Derived from a least-biased optimization inspired by information theory, the meshfree maximum-entropy method appears as an interesting alternative to classical approximation schemes like the finite element method. Especially compared with other meshfree methods, e.g. the moving least-squares method, it allows for a straightforward imposition of boundary conditions. However, no Eulerian approach has yet been presented for real incompressible flow, encountering the convective and pressure instabilities. In this paper, we exclusively address the pressure instabilities caused by the mixed velocity-pressure formulation of incompressible Stokes flow. In a preparatory discussion, existing stable and stabilized methods are investigated and evaluated. This is used to develop different approaches towards a stable maximum-entropy formulation. We show results for two analytical tests, including a presentation of the convergence behavior. As a typical benchmark problem, results are also shown for the leaky lid-driven cavity. The already presented information-flux method for convection-dominated problems in mind, we see this as the last step towards a maximum-entropy method capable of simulating full incompressible flow problems. Copyright © 2015 John Wiley & Sons, Ltd.

We present a parameter-free stable maximum-entropy method for incompressible Stokes flow. Stable and stabilized velocity-pressure formulation that already exist for other computational methods are investigated and evaluated in order to develop different approaches towards a stable maximumentropy scheme. The method's excellent performance is shown by results for two analytical tests, including a presentation of the convergence behavior, and for the leaky lid-driven cavity as a typical benchmark problem.

In this paper, we propose for the first time a linearly coupled, energy stable scheme for the Navier–Stokes–Cahn–Hilliard system with generalized Navier boundary condition. We rigorously prove the unconditional energy stability for the proposed time discretization as well as for a fully discrete finite element scheme. Using numerical tests, we verify the accuracy, confirm the decreasing property of the discrete energy, and demonstrate the effectiveness of our method through numerical simulations in both 2-D and 3-D. Copyright © 2015 John Wiley & Sons, Ltd.

In the study of the phase-field model for the moving contact line problem, a linear and energy stable numerical scheme was proposed in the paper for solving the Navier-Stokes-Cahn-Hilliard system subject to a new set of complex boundary conditions. Performance and visualization were provided in the work.

A hybrid time stepping scheme is developed and implemented by a combination of explicit Runge–Kutta with implicit LU-SGS scheme at the level of system matrix. In this method, the explicit scheme is applied to those grid cells of blocks that have large local time steps; meanwhile, the implicit scheme is applied to other grid cells of blocks that have smaller allowable local time steps in the same flow field. As a result, the discretized governing equations can be expressed as a compound of explicit and implicit matrix operator. The proposed method has been used to compute the steady transonic turbulent flow over the RAE 2822 airfoil. The numerical results are found to be in excellent agreement with the experimental data. In the validation case, the present scheme saved at least 50% of the memory resources compared with the fully implicit LU-SGS. Copyright © 2015 John Wiley & Sons, Ltd.

We proposed a combination of explicit Runge–Kutta with implicit LU-SGS scheme at the level of system matrix. The combination makes the discretized governing equations expressed as a compound of explicit and implicit matrix operator. Numerical results show that the convergence rate of the present scheme is almost the same as that of the LU-SGS implicit scheme for the same splitting grid and the same CFL number. In the numerical case, the present scheme saved 50% of the memory resources compared with the fully implicit LU-SGS.

In this paper, we propose an interfacial pressure correction algorithm for smoothed particle hydrodynamics (SPH) simulation of multiphase flows with large density ratios. This correction term is based on the assumption of small deformation of the interface, and derived from perturbation expansion analysis. It is also proven to be applicable in cases with complex interfaces. This correction algorithm helps to overcome the discontinuities of the pressure gradient over the interfaces, which may cause unphysical gap between different phases. This proposed correction algorithm is implemented on a recent multiphase SPH model, which is based on the assumption of pressure continuity over the interfaces. The coupled dynamic solid boundary treatment is used to simulate solid walls; and a cut-off pressure is applied to avoid negative particle pressure, which may cause computational instabilities in SPH. Three numerical examples of air–water flows, including sloshing, dam breaking, and water entry, are presented and compared with experimental data, indicating the robustness of our pressure correction algorithm in multiphase simulations with large density ratios. Copyright © 2015 John Wiley & Sons, Ltd.

In this work, we propose an interfacial pressure correction algorithm for SPH simulation of multiphase flows with large density ratios. This correction algorithm helps to improve the stability of the original model, especially for long-time problems.

A discontinuous Galerkin nonhydrostatic atmospheric model is used for two-dimensional and three-dimensional simulations. There is a wide range of timescales to be dealt with. To do so, two different implicit/explicit time discretizations are implemented. A stabilization, based upon a reduced-order discretization of the gravity term, is introduced to ensure the balance between pressure and gravity effects. While not affecting significantly the convergence properties of the scheme, this approach allows the simulation of anisotropic flows without generating spurious oscillations, as it happens for a classical discontinuous Galerkin discretization. This approach is shown to be less diffusive than usual spatial filters. A stability analysis demonstrates that the use of this modified scheme discards the instability associated with the usual discretization. Validation against analytical solutions is performed, confirming the good convergence and stability properties of the scheme. Numerical results demonstrate the attractivity of the discontinuous Galerkin method with implicit/explicit time integration for large-scale atmospheric flows. Copyright © 2015 John Wiley & Sons, Ltd.

A stabilization, based upon a reduced-order discretization of the gravity term, ensures the balance between pressure and gravity effects. Validation with a stability analysis and numerical experiments confirming the correct convergence rate.

We consider a Leray model with a deconvolution-based indicator function for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-resolved meshes. For the implementation of the model, we adopt a three-step algorithm called *evolve–filter–relax* that requires (i) the solution of a Navier–Stokes problem, (ii) the solution of a Stokes-like problem to filter the Navier–Stokes velocity field, and (iii) a final relaxation step. We take advantage of a reformulation of the evolve–filter–relax algorithm as an operator-splitting method to analyze the impact of the filter on the final solution versus a direct simulation of the Navier–Stokes equations.

In addition, we provide some direction for tuning the parameters involved in the model based on physical and numerical arguments. Our approach is validated against experimental data for fluid flow in an idealized medical device (consisting of a conical convergent, a narrow throat, and a sudden expansion, as recommended by the U.S. Food and Drug Administration). Numerical results are in good quantitative agreement with the measured axial components of the velocity and pressures for two different flow rates corresponding to turbulent regimes, even for meshes with a mesh size more than 40 times larger than the smallest turbulent scale. After several numerical experiments, we perform a preliminary sensitivity analysis of the computed solution to the parameters involved in the model. Copyright © 2015 John Wiley & Sons, Ltd.

We adopt a Leray model for the simulation of fluid flows at moderately high Reynolds numbers and provide directions for tuning the parameters involved in the model. For the implementation, we consider a three-step algorithm that we reformulate as an operator-splitting method. For the first time, this method is applied to a realistic problem of practical interest (flow in a nozzle). We carry out a successful validation against experimental measurements.

The *r*-ratio is a parameter that measures the local monotonicity, by which a number of high-resolution and TVD schemes can be formed. A number of *r*-ratio formulations for TVD schemes have been presented over the last few decades to solve the transport equation in shallow waters based on the finite volume method (FVM). However, unlike structured meshes, the coordinate directions are not clearly defined on an unstructured mesh; therefore, some *r*-ratio formulations have been established by approximating the solute concentration at virtual nodes, which may be estimated from different assumptions. However, some formulations may introduce either oscillation or diffusion behavior within the vertex-centered (VC) framework. In this paper, a new *r*-ratio formulation, applied to an unstructured grid in the VC framework, is proposed and compared with the traditional *r*-ratio formulations. Through seven commonly used benchmark tests, it is shown that the newly proposed *r*-ratio formulation obtains better results than the traditional ones with less numerical diffusion and spurious oscillation. Moreover, three commonly used TVD schemes—SUPERBEE, MINMOD, and MUSCL—and two high-order schemes—SOU and QUICK—are implemented and compared using the new *r*-ratio formulation. The new *r*-ratio formulation is shown to be sufficiently comprehensive to permit the general implementation of a high-resolution scheme within the VC framework. Finally, the sensitivity test for different grid types demonstrates the good adaptability of this new *r*-ratio formulation. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a new r-ratio formulation for TVD schemes for vertex-centered finite volume method on an unstructured mesh. Through several numerical tests as shown later (concentration along x-axis in test of a spot moving in a rotating flow field), the new formulation has higher accuracy and less oscillations than traditional formulations, while it demonstrates a good adaptability to different unstructured meshes.

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

A robust numerical solver based on the pseudo-compressibility Navier–Stokes model and the volume-of-fluid interface tracking method was developed for three-dimensional simulation of complex, free surface, and water impact flows. The proposed method is implemented using a generalized curvilinear coordinate system to facilitate complex, arbitrary simulations of the flows in practical problems. Several example computations concerning the numerical resolution, stability, and important physical characteristics of hydraulic and hydrodynamic problems exhibit a very good agreement with experimental and numerical data published in the literature.

The main contribution of this work is to classify the solution region including data extrema for which high-order non-oscillatory approximation can be achieved. It is performed in the framework of local maximum principle (LMP) and non-conservative formulation. The representative uniformly second-order accurate schemes are converted in to their non-conservative form using the ratio of consecutive gradients. Using the local maximum principle, these non-conservative schemes are analyzed for their non-linear LMP/total variation diminishing stability bounds which classify the solution region where high-order accuracy can be achieved. Based on the bounds, second-order accurate hybrid numerical schemes are constructed using a shock detector. The presented numerical results show that these hybrid schemes preserve high accuracy at non-sonic extrema without exhibiting any induced local oscillations or clipping error. Copyright © 2015 John Wiley & Sons, Ltd.

Using a local maximum principle, the solution region of hyperbolic scalar conservation law is classified into sub-regions where at least second-order non-oscillatory approximation can be achieved. Nonlinear stability bounds are given, which ensure for non-occurrence of induced oscillations by second-order schemes. Using these bounds, second-order accurate hybrid numerical schemes are constructed with the help of a shock detector, which can preserve high accuracy at non-sonic extrema without exhibiting any induced local oscillations or clipping error.

In this work, we present a high-order discontinuous Galerkin method (DGM) for simulating variable density flows at low Mach numbers. The corresponding low Mach number equations are an approximation of the compressible Navier–Stokes equations in the limit of zero Mach number. To the best of the authors'y knowledge, it is the first time that the DGM is applied to the low Mach number equations. The mixed-order formulation is applied for spatial discretization. For steady cases, we apply the semi-implicit method for pressure-linked equation (SIMPLE) algorithm to solve the non-linear system in a segregated manner. For unsteady cases, the solver is implicit in time using backward differentiation formulae, and the SIMPLE algorithm is applied to solve the non-linear system in each time step. Numerical results for the following three test cases are shown: Couette flow with a vertical temperature gradient, natural convection in a square cavity, and unsteady natural convection in a tall cavity. Considering a fixed number of degrees of freedom, the results demonstrate the benefits of using higher approximation orders. Copyright © 2015 John Wiley & Sons, Ltd.

We present a high-order discontinuous Galerkin method for simulating variable density flows at low Mach numbers. The solver is based on the lowMach number equations, which are an approximation of the compressible Navier–Stokes equations in the limit of zero Mach number. Numerical tests for Couette flow with a vertical temperature gradient and natural convection in enclosed cavities confirm the high accuracy of the method.

When a block factorisation is used to precondition the saddle-point equations of the discrete Stokes problem, the stability that this gives for the relaxation of residual errors may not be conserved in the coarse-grid approximations (CGA) of algebraic multi-grid (AMG) solvers. If the same first-order interpolation is used in the inter-grid transfer operators for the scalar and the vector fields, the conditioning degrades with each coarsening step until eventually a critical coarsening is reached beyond which residual errors are no longer damped and will become divergent with any further coarsening. It is shown that by introducing the same block pre-conditioner as an integral part of the coarsening algorithm, stable smoothing can be maintained at all levels of the CGA. The pre-conditioning need only be applied at preselected grid levels, one immediately before the critical threshold and others beyond that level if required. Excessive complexity in the CGA is thereby avoided. The method is purely algebraic and may be used for both classical AMG solvers and for smoothed-aggregation AMG solvers. It should be applicable to other coupled vector and scalar fields in science and engineering that involve second-order (block-diagonal) and first-order (block-off-diagonal) discrete difference operators. Copyright © 2015 John Wiley & Sons, Ltd.

A method of stabilisation is proposed for fully-implicit, algebraic multi-grid, solutions of saddle-point, coupled-field, problems. Mesh-independent convergence is demonstrated for (A) smoothed-aggregation AMG and (B) classical AMG. The reduction/convergence factors, *ρ*, are independent of both the mesh resolution, *Q*, and the degree of coarsening, * χ*, in the AMG coarse-grid approximations.

We present a new reference smoothness indicator for third-order weighted essentially non-oscillatory scheme to recover its design-order convergence at critical points. This reference smoothness indicator, which involves both the candidate and global smoothness indicators in the weighted essentially non-oscillatory framework, is devised according to a sufficient condition on the weights for third-order convergence. The recovery of design-order is verified by standard tests. Meanwhile, numerical results demonstrate that the present reference smoothness indicator produces sharper representation of the discontinuity owing to the combined effects of larger weight assignment to the discontinuous stencils and convergence rate recovery. Copyright © 2015 John Wiley & Sons, Ltd.

A new reference smoothness indicator *τ*_{NP} is devised for the third-order weighted essentially non-oscillatory-NP3 scheme to recover its design-order convergence at critical points, by considering the nonlinear combination of the candidate and global smoothness indicators. The good matching between the numerical solutions of weighted essentially non-oscillatory-NP3 and third-order upwind scheme in solving the smooth extremum problem forcefully confirmed the recovery of design-order accuracy. Meanwhile, standard tests also verified the benefit of *τ*_{NP} in producing sharper representation of the discontinuity.

The accuracy of numerical simulations of free-surface flows depends strongly on the computation of geometric quantities like normal vectors and curvatures. This geometrical information is additional to the actual degrees of freedom and usually requires a much finer discretization of the computational domain than the flow solution itself. Therefore, the utilization of a numerical method, which uses standard functions to discretize the unknown function in combination with an enhanced geometry representation is a natural step to improve the simulation efficiency. An example of such method is the NURBS-enhanced finite element method (NEFEM), recently proposed by Sevilla *et al*. The current paper discusses the extension of the spatial NEFEM to space-time methods and investigates the application of space-time NURBS-enhanced elements to free-surface flows. Derived is also a kinematic rule for the NURBS motion in time, which is able to preserve mass conservation over time. Numerical examples show the ability of the space-time NEFEM to account for both pressure discontinuities and surface tension effects and compute smooth free-surface forms. For these examples, the advantages of the NEFEM compared with the classical FEM are shown. Copyright © 2015 John Wiley & Sons, Ltd.

The paper at hand discusses the application of the recently proposed NURBS-enhanced finite element method (NEFEM) to free-surface flow simulations. In this context the 2D spatial NEFEM formulation is extended into the framework of space-time methods and a suitable kinematic rule for the NURBS motion in time is derived. The performance of the space-time NEFEM is compared to the standard FEM and the ability to preserve mass conservation over time is confirmed.

In this paper, we consider edge-based reconstruction (EBR) schemes for solving the Euler equations on unstructured tetrahedral meshes. These schemes are based on a high-accuracy quasi-1D reconstruction of variables on an extended stencil along the edge-based direction. For an arbitrary tetrahedral mesh, the EBR schemes provide higher accuracy in comparison with most second-order schemes at rather low computational costs. The EBR schemes are built in the framework of vertex-centered formulation for the point-wise values of variables.

Here, we prove the high accuracy of EBR schemes for uniform grid-like meshes, introduce an economical implementation of quasi-one-dimensional reconstruction and the resulting new scheme of EBR family, estimate the computational costs, and give new verification results. Copyright © 2015 John Wiley & Sons, Ltd.

The paper considers edge-based reconstruction (EBR) schemes for solving the Euler equations on unstructured tetrahedral meshes. These are vertex-centered schemes for the point-wise values that exploit a high-accuracy quasi-one-dimensional reconstruction of variables on an extended stencil along the edge-based direction. We prove the high accuracy of EBR schemes for uniform grid-like meshes, introduce their economical implementation, and compare them with the polynomial-based finite-volume schemes and flux corrector method.

The representation of geometries as buildings, flood barriers or dikes in free surface flow models implies tedious and time-consuming operations in order to define accurately the shape of these objects when using a body fitted numerical mesh. The immersed boundary method is an alternative way to define solid bodies inside the computational domain without the need of fitting the mesh boundaries to the shape of the object. In the direct forcing immersed boundary method, a solid body is represented by a grid of Lagrangian markers, which define its shape and which are independent from the fluid Eulerian mesh. This paper presents a new implementation of the immersed boundary method in an unstructured finite volume solver for the 2D shallow water equations. Moving least-squares is used to transmit information between the grid of Lagrangian markers and the fluid Eulerian mesh. The performance of the proposed implementation is analysed in three test cases involving different flow conditions: the flow around a spur dike, a dam break flow with an isolated obstacle and the flow around an array of obstacles. A very good agreement between the classic body fitted approach and the immersed boundary method was found. The differences between the results obtained with both methods are less relevant than the errors because of the intrinsic shallow water assumptions. Copyright © 2015 John Wiley & Sons, Ltd.

An immersed boundary method for use on unstructured meshes is proposed, with particular focus on its application to depth averaged shallow water models. Moving least-squares is used to generate the interpolation functions. The method is applied to the flow around a spur dike, a dam break with an isolated obstacle and the flow around an array of obstacles, and results are compared with simulations using classic body fitted meshes and experimental data. Good agreement is found between the numerical methods.

Two-dimensional flows past a stationary circular cylinder near a plane boundary are numerically simulated using an immersed interface method with second-order accuracy. Instead of a fixed wall, a moving wall with no-slip boundary is considered to avoid the complex involvement of the boundary layer and to focus only on the shear-free wall proximity effects for investigating the force dynamics and flow fields. To analyze the convergence and accuracy of our implementation, numerical studies have been first performed on a simple test problem of rotational flow, where the second order of convergence is confirmed through numerical experiments and an optimal range of relative grid-match ratio of Lagrangian to Eulerian grid sizes has been recommended. By comparing the force quantities and the Strouhal number, the accuracy of this method has been demonstrated on the flow past a stationary isolated cylinder. The cylinder is then put in proximity to the wall to investigate the shear-free wall proximity effects in the low Reynolds number regime (20≤*R**e*≤200). The gap ratio, *e*/*D*, where *e* denotes the gap between the cylinder and the moving wall and *D* denotes the diameter of the cylinder, is taken from 0.10 to 2.00 to determine the critical gap ratio, (*e*/*D*)_{critical}, for the alternate vortex shedding, where the fluid forces, flow fields and the streamwise velocity profiles are studied. One of the key findings is that the (*e*/*D*)_{critical} for the alternate vortex shedding decreases as the Reynolds number increases. We also find that, in this low Reynolds number regime, the mean drag coefficient increases and peaks at *e*/*D* = 0.5 with the increase of *e*/*D* and keeps decreasing gently from *e*/*D* = 0.5 to *e*/*D* = 2.0, while the mean lift coefficient decreases monotonically with the increase of *e*/*D*. New correlations are then proposed for computing force coefficients as a function of *R**e* and *e*/*D* for a cylinder in the vicinity of a moving plane wall. Copyright © 2015 John Wiley & Sons, Ltd.

Immersed interface method has been employed to study the shear-free wall proximity effects in the low Reynolds number regime (20≤Re≤200), where the hydrodynamic forces and the critical gap ratio for vortex shedding suppression are investigated. We have found that the mean drag coefficient, *C*_{D}, increases and peaks at ^{e}/_{D}=0.5 with the increase of ^{e}/_{D} and keeps decreasing gently from ^{e}/_{D}=0.5 to 2.0, while the mean lift coefficient, *C*_{L}, decreases monotonically with the increase of ^{e}/_{D} . With the consistent trends of force coefficients, new correlations have been proposed for the lift and drag coefficients as a function of ^{e}/_{D} and *R**e*.

An improved incompressible smoothed particle hydrodynamics (ISPH) method is presented, which employs first-order consistent discretization schemes both for the first-order and second-order spatial derivatives. A recently introduced wall boundary condition is implemented in the context of ISPH method, which does not rely on using dummy particles and, as a result, can be applied more efficiently and with less computational complexity. To assess the accuracy and computational efficiency of this improved ISPH method, a number of two-dimensional incompressible laminar internal flow benchmark problems are solved and the results are compared with available analytical solutions and numerical data. It is shown that using smaller smoothing lengths, the proposed method can provide desirable accuracies with relatively less computational cost for two-dimensional problems. Copyright © 2015 John Wiley & Sons, Ltd.

A consistent Incompressible smoothed particle hydrodynamics (ISPH) method is presented. The method employs first-order consistent discretization schemes for both the first-order and secondorder spatial derivatives and benefits from a robust boundary condition implementation. It is shown that for the range of two-dimensional incompressible laminar internal flow problems studied in this work, the proposed algorithm is more accurate and computationally more efficient compared with its standard ISPH counterpart.

The kernel gradient free (KGF) smoothed particle hydrodynamics (SPH) method is a modified finite particle method (FPM) which has higher order accuracy than the conventional SPH method. In KGF-SPH, no kernel gradient is required in the whole computation, and this leads to good flexibility in the selection of smoothing functions and it is also associated with a symmetric corrective matrix. When modeling viscous incompressible flows with SPH, FPM or KGF-SPH, it is usual to approximate the Laplacian term with nested approximation on velocity, and this may introduce numerical errors from the nested approximation, and also cause difficulties in dealing with boundary conditions. In this paper, an improved KGF-SPH method is presented for modeling viscous, incompressible fluid flows with a novel discrete scheme of Laplacian operator. The improved KGF-SPH method avoids nested approximation of first order derivatives, and keeps the good feature of ‘kernel gradient free’. The two-dimensional incompressible fluid flow of shear cavity, both in Euler frame and Lagrangian frame, are simulated by SPH, FPM, the original KGF-SPH and improved KGF-SPH. The numerical results show that the improved KGF-SPH with the novel discrete scheme of Laplacian operator are more accurate than SPH, and more stable than FPM and the original KGF-SPH. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, an improved kernel gradient free (KGF)-smoothed particle hydrodynamics (SPH) method with a novel discrete scheme of Laplacian operator is presented for modeling viscous, incompressible fluid flows. Improved KGF-SPH method avoids nested approximation of first-order derivatives, and keeps the good feature of ‘KGF’. The 2D incompressible flows of lid-driven shear cavity, both in Euler frame and Lagrangian frame, are simulated by SPH, FPM, original and improved KGF-SPH. As shown in the figure (pressure coefficient profiles along upper wall for a lid-driven shear cavity problem), improved KGF-SPH with the novel discrete scheme of Laplacian operator is more accurate and stable than SPH, FPM and original KGF-SPH.

An integrated finite element method (FEM) is proposed to simulate incompressible two-phase flows with surface tension effects, and three different surface tension models are applied to the FEM to investigate spurious currents and temporal stability. A Q2Q1 element is adopted to solve the continuity and Navier–Stokes equations and a Q2-iso-Q1 to solve the level set equation. The integrated FEM solves pressure and velocity simultaneously in a strongly coupled manner; the level set function is reinitialized by adopting a direct approach using interfacial geometry information instead of solving a conventional hyperbolic-type equation. In addition, a consistent continuum surface force (consistent CSF) model is utilized by employing the same basis function for both surface tension and pressure variables to damp out spurious currents and to estimate the accurate pressure distribution. The model is further represented as a semi-implicit manner to improve temporal stability with an increased time step. In order to verify the accuracy and robustness of the code, the present method is applied to a few benchmark problems of the static bubble and rising bubble with large density and viscosity ratios. The Q2Q1-integrated FEM coupled with the semi-implicit consistent CSF demonstrates the significantly reduced spurious currents and improved temporal stability. The numerical results are in good qualitative and quantitative agreements with those of the existing studies. Copyright © 2015 John Wiley & Sons, Ltd.

In finite element method, a consistent continuum surface force model is introduced by employing the same basis function for both surface tension and pressure variables to damp out spurious currents and to estimate the accurate pressure distribution. The model is further represented as a semi-implicit manner to improve temporal stability with an increased time step. The Q2Q1 integrated FEM coupled with the semi-implicit consistent CSF demonstrates the significantly reduced spurious currents and improved temporal stability.

An embedded formulation for the simulation of immiscible multi-fluid problems is proposed. The method is particularly designed for handling gas–liquid systems. Gas and liquid are modeled using the Eulerian and the Lagrangian formulation, respectively. The Lagrangian domain (liquid) moves on top of the fixed Eulerian mesh. The location of the material interface is exactly defined by the position of the boundary mesh of the Lagrangian domain. The individual fluid problems are solved in a partitioned fashion and are coupled using a Dirichlet–Neumann algorithm. Representation of the pressure discontinuity across the interface does not require any additional techniques being an intrinsic feature of the method. The proposed formulation is validated, and its potential applications are shown. Copyright © 2015 John Wiley & Sons, Ltd.

Gas-liquid systems can be efficiently modeled in a partitioned embedded fashion adopting fixed mesh (Eulerian) approach for the gas and mesh-moving one (Lagrangian) for the liquid. The interface tracking and flow variables' discontinuity across the interface becomes thus a natural feature of the method. Surface tension is applied on the mesh-defined boundary of the liquid phase. The main strength of the approach is the full control over the coupling strength between the sub-domains, allowing for both weak and fully coupled algorithms.

A high-order difference method based multiphase model is proposed to simulate nonlinear interactions between water wave and submerged coastal structures. The model is based on the Navier–Stokes equations using a constrained interpolation profile (CIP) method for the flow solver, and employs an immersed boundary method (IBM) for the treatment of wave–structure interactions. A more accurate interface capturing scheme, the volume of fluid/weighed line interface calculation (VOF/WLIC) scheme, is adopted as the interface capturing method. A series of computations are performed to verify the application of the model for simulations of fluid interaction with various structures. These problems include flow over a fixed cylinder, water entry of a circular cylinder and solitary waves passing various submerged coastal structures. Computations are compared with the available analytical, experimental and other numerical results and good agreement is obtained. The results of this study demonstrate the accuracy and applications of the proposed model to simulate the nonlinear flow phenomena and capture the complex free surface flow. Copyright © 2015 John Wiley & Sons, Ltd.

Numerical simulations of water wave interaction with coastal structures are performed by an improved CIP-based Cartesian grid method, in which a more accurate interface capturing scheme is combined for the free surface capturing and the immersed boundary method for the solid boundary treatment. Computations are compared with available analytical, experimental and other numerical results. The present model proves to be a promising tool for simulation of coastal engineering applications with acceptable accuracy.

The mass-conserving level-set (MCLS) method is a hybrid level-set (LS)/volume of fluid (VoF) based, interface capturing algorithm that combines the mass conserving properties of the VoF, with the benefits of having an explicit description of the interface of the LS method. The efficiency of the method is a result of the fact that the LS formulation allows evaluation of the VoF-field and VoF-fluxes without reconstruction of the interface in each cell. We present the extension of the MCLS method from its original formulation for Cartesian quadrilateral control volumes to triangular control volumes for optimal geometrical flexibility. The LS field is discretized using a second order discontinuous Galerkin method. After each time-step, a mass-conserving correction is imposed based on the simultaneously convected VoF field. This convection step is performed with a second-order Eulerian–Lagrangian approach, combined with a ‘clipping’ algorithm to project the advected field from the Lagrangian to the Eulerian grid. The MCLS method is shown to be accurately mass conserving and shows second order convergence for three different test cases. Copyright © 2015 John Wiley & Sons, Ltd.

We present the extension of the MCLS method toward unstructured triangular grids for two phase flow. The VoF function and the inverse function derived for a triangular mesh are very simple, robust and efficient to evaluate. Our approach is significantly more efficient and robust than the original MCLS formulation. Numerical experiments indicate the LS field converges with second order accuracy in space and mass is conserved up to machine precision.

In this paper, we develop a new hybrid Euler flux function based on Roe's flux difference scheme, which is free from shock instability and still preserves the accuracy and efficiency of Roe's flux scheme. For computational cost, only 5*%* extra CPU time is required compared with Roe's FDS. In hypersonic flow simulation with high-order methods, the hybrid flux function would automatically switch to the Rusanov flux function near shock waves to improve the robustness, and in smooth regions, Roe's FDS would be recovered so that the advantages of high-order methods can be maintained. Multidimensional dissipation is introduced to eliminate the adverse effects caused by flux function switching and further enhance the robustness of shock-capturing, especially when the shock waves are not aligned with grids. A series of tests shows that this new hybrid flux function with a high-order weighted compact nonlinear scheme is not only robust for shock-capturing but also accurate for hypersonic heat transfer prediction. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we develop a very robust hybrid flux function to overcome the shock instability in hypersonic flow simulation with high-order methods. Multidimensional dissipation and entropy correction based on local flow field are introduced to enhance the robustness and resolution of the hybrid flux function. A series of tests shows that this new hybrid flux function with a fifth-order weighted compact nonlinear scheme (WCNS) is not only robust for shock-capturing but also accurate for hypersonic heat transfer prediction.

No abstract is available for this article.

]]>We optimized the Arbitrary accuracy DErivatives Riemann problem (ADER) - Discontinuous Galerkin (DG) numerical method using the CUDA-C language to run the code in a graphic processing unit (GPU). We focus on solving linear hyperbolic partial–differential equations where the method can be expressed as a combination of precomputed matrix multiplications becoming a good candidate to be used on the GPU hardware. Moreover, the method is arbitrarily high order involving intensive work on local data, a property that is also beneficial for the target hardware. We compare our GPU implementation against CPU versions of the same method observing similar convergence properties up to a threshold where the error remains fixed. This behavior is in agreement with the CPU version, but the threshold is slightly larger than in the CPU case. We also observe a big difference when considering single and double precisions where in the first case, the threshold error is significantly larger. Finally, we did observe a speed-up factor in computational time that depends on the order of the method and the size of the problem. In the best case, our novel GPU implementation runs 23 times faster than the CPU version. We used three partial–differential equation to test the code considering the linear advection equation, the seismic wave equation, and the linear shallow water equation, all of them considering variable coefficients. Copyright © 2015 John Wiley & Sons, Ltd.

In the figure, we show a convergence test considering the 2D linear elastic wave equation. We compare double precision and single precision of the graphics processor unit implementation against the CPU code SeisSol for different orders from second (P1) to sixth (P5). In the vertical axis, the error level is obtained using the L_2 norm. The figure on the left depicts the error against mesh size, while on the right, the horizontal axis represents computational time.

Currently, the majority of computational fluid dynamics (CFD) codes use the finite volume method to spatially discretise the computational domain, sometimes as an array of cubic control volumes. The Finite volume method works well with single-phase flow simulations, but two-phase flow simulations are more challenging because of the need to track the surface interface traversing and deforming within the 3D grid. Surface area and volume fraction details of each interface cell must be accurately accounted for, in order to calculate for the momentum exchange and rates of heat and mass transfer across the interface. To attain a higher accuracy in two-phase flow CFD calculations, the intersection marker (ISM) method is developed. The ISM method is a hybrid Lagrangian–Eulerian front-tracking algorithm that can model an arbitrary 3D surface within an array of cubic control volumes. The ISM method has a cell-by-cell remeshing capability that is volume conservative and is suitable for the tracking of complex interface deformation in transient two-phase CFD simulations. Copyright © 2015 John Wiley & Sons, Ltd.

The intersection marker method is a novel approach for modelling an arbitrary 3D surface within an array of cubic control volumes. Intersection marker's novelty lies in its ability to remesh the interface on a cell-by-cell basis whilst maintaining surface continuity and local volume conservation without the use of permanent surface markers.

An unstructured, shock-fitting algorithm, originally developed to simulate steady flows, has being further developed to make it capable of dealing with unsteady flows. The present paper discusses and analyses the additional features required to extend to unsteady flows, the steady algorithm. The properties of the unsteady version of this novel, unstructured shock-fitting technique, are tested by reference to the inviscid interaction between a vortex and a planar shock: a comparative assessment of shock-capturing and shock-fitting is made for the same test problem. Copyright © 2015 John Wiley & Sons, Ltd.

An unstructured, shock-fitting algorithm, originally developed to simulate steady flows, has been further developed to make it capable of dealing with unsteady flows. The present paper discusses and analyses the additional features required to extend to unsteady flows the steady algorithm.