A new method to admit large Courant numbers in the numerical simulation of multiphase flow is presented. The governing equations are discretised in time using an adaptive *θ*-method. However, the use of implicit discretisations does not guarantee convergence of the non-linear solver for large Courant numbers. In this work, a double-fixed point iteration method with backtracking is presented that improves both convergence and convergence rate. Moreover, acceleration techniques are presented to yield a more robust non-linear solver with increased effective convergence rate. The new method reduces the computational effort by strengthening the coupling between saturation and velocity, obtaining an efficient backtracking parameter, using a modified version of Anderson's acceleration and adding vanishing artificial diffusion. This article is protected by copyright. All rights reserved.

The Lagrangian Smoothed Particle Hydrodynamics (SPH) method is used to simulate shock waves in inviscid, supersonic (compressible) flow. It is shown for the first time that the fully Lagrangian SPH particle method, without auxiliary grid, can be used to simulate shock waves in compressible flow. The wall boundary condition is treated with ghost particles combined with a suitable repulsive potential function, while corners are treated by a novel “angle sweep” technique. The method gives accurate predictions of the flow field and of the shock angle as compared with the analytical solution. The study shows that SPH is a good potential candidate to solve complex aerodynamic problems, including those involving rarefied flows, such as atmospheric re-entry. This article is protected by copyright. All rights reserved.

In this article a masked bubble strategy is proposed using the front-tracking method when simulation of multi-density bubbles to reduce remarkably the computational cost from both the Ram-Usage and the number of computations at each time step comparing with the regular method. In the masked bubble strategy instead of using full-domain to update the properties at each time step, each bubble is considered as enclosed in the smallest box required to compute the properties based on the Pesken's function, which needs at least two full mesh sizes from both sides of the interface of each bubbles in any directions. To show the performance of the masked bubble strategy in the front-tracking method, we study the multi-denstiy bubbles motion in a curved duct flow induced by a pressure gradient in the absence of gravity. To solve the density Poisson equation, the PARDISO scheme is tested. The comparison of numerical simulations at the same conditions indicates that the PARDISO scheme under the masked bubble strategy considerably reduces the computational time and RAMS usage relative to the regular full-domain method, providing using simulations on finer grid resolutions. This article is protected by copyright. All rights reserved.

The moving particle semi-implicit (MPS) method has been widely applied in free surface flows. However, the implementation of MPS remains limited because of compressive instability occurred when the particles are under compressive stress states. This study proposed an inter-particle force stabilization and consistency restoring MPS (IFS-CR-MPS) method to overcome this numerical stability. For inter-particle force stabilization, a hyperbolic-shaped quintic kernel function is developed with a non-negative and smooth second order derivative to satisfy the stability criterion under compressive stress state. Then a contrastive study is conducted on the contradiction between the common understanding of the conventional MPS hyperbolic-shaped kernel function and its performance. The result shows that the conventional MPS hyperbolic-shaped kernel function can easily cause violent repulsive inter-particle force and then lead to the compressive instability. Therefore, the first order derivative of the modified hyperbolic-shaped quintic kernel function is recommended as the form of the contribution of the neighbor particles to achieve a more stable inter-particle repulsive force. For consistency restoring, the Taylor series expansion and the hyperbolic-shaped quintic kernel are combined to improve the accuracy of the viscosity and pressure calculation. The IFS-CR-MPS algorithm is subsequently verified by the inviscid hydrostatic pressure, jet impacting and viscous droplet impacting problems. These results can be used for choosing kernel function and the contribution of neighbor particles in particle methods.

This is a first attempt to develop the Meshless Local Petrov-Galerkin method with Rankine source solution (MLPG_R method) to simulate multiphase flows. In this paper, we do not only further develop the MLPG_R method to model two-phase flows but also propose two new techniques to tackle the associated challenges. The first technique is to form an equation for pressure on the explicitly identified interface between different phases by considering the continuity of the pressure and the discontinuity of the pressure gradient (i.e., the ratio of pressure gradient to fluid density), the latter reflecting the fact that the normal velocity is continuous across the interface. The second technique is about solving the algebraic equation for pressure, which gives reasonable solution not only for the cases with low density ratio but also for the cases with very high density ratio, such as more than 1000. The numerical tests show that the results of the newly developed two-phase MLPG_R method agree well with analytical solutions and experimental data in the cases studied. The numerical results also demonstrate that the newly developed method has a second-order convergent rate in the cases for sloshing motion with small amplitudes.

This article presents a new nonlinear finite-volume scheme for the nonisothermal two-phase two-component flow equations in porous media. The face fluxes are approximated by a nonlinear two-point flux approximation, where transmissibilities nonlinearly depend on primary variables. Thereby, we mainly follow the ideas proposed by Le Potier combined with a harmonic averaging point interpolation strategy for the approximation of arbitrary heterogeneous permeability fields on polygonal grids. The behavior of this interpolation strategy is analyzed and its limitation for highly anisotropic permeability tensors is demonstrated. Moreover, the condition numbers of occurring matrices are compared with linear finite-volume schemes. Additionally, the convergence behavior of iterative solvers is investigated. Finally, it is shown that the nonlinear scheme is more efficient than its linear counterpart. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a contribution to level-set reinitialization in the context of discontinuous Galerkin finite element methods. We focus on high-order polynomials for the discretization and level set geometries, which are comparable to the element size. In contrast to hyperbolic and geometric reinitialization techniques, our method relies on solving a nonlinear elliptic PDE iteratively. The algorithm is an extension of the work of Basting and Kuzmin (Computing, 2012) to the discontinuous Galerkin method. We critically compare two different variants of the algorithm experimentally in numerical studies. The results demonstrate that the method is stable for nontrivial test cases and shows high-order accuracy. This article is protected by copyright. All rights reserved.

This paper proposes WCNSCUZ, a weighted compact nonlinear scheme (WCNS), which incorporates adapted central difference and low-dissipative weights together with concepts of the adaptive central-upwind sixth-order weighted essentially non-oscillatory scheme (WENOCU) and WENO-Z schemes. The newly developed WCNSCUZ is a high-resolution scheme, because interpolation of this scheme employs a central stencil constructed by upwind and downwind stencils. The smoothness indicator of the downwind stencil is calculated using the entire central stencil, and the downwind stencil is stopped around the discontinuity for stability. Moreover, interpolation of the sixth-order WCNSCUZ shows sufficient accuracy in the smooth region by using low-dissipative weights. The sixth-order WCNSCUZs are implemented with a robust linear difference formulation (RWCNSCU6Z), and the resolution and robustness of this scheme were evaluated. These evaluations showed that RWCNSCU6Z is capable of achieving a higher resolution than the seventh-order classical robust WCNS and can provide a crisp result in terms of discontinuity. Among the schemes tested, RWCNSCU6Z is determined to be robust, and variable interpolation type RWCNSCU6Z (RWCNSCUV6Z) provides a stable computation by modifying the first-order interpolation when negative density or negative pressure arises after nonlinear interpolation.

Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell-centred velocity reconstructions, the standard first-order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable-density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy-preserving scheme is applied to the momentum equations, namely, the symmetry-preserving scheme. Furthermore, a new approach to define the far-neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self-igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable-density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second-order behaviour on unstructured meshes, and the symmetry-preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.

In the paper finite-volume collocated and unstructured discretizations to simulate Low Mach flows are analyzed. The collocated method is shown to be more computationally efficient. Furthermore, a Symmetry-Preserving and Upwinding numerical schemes for face interpolations are studied, focusing on their behavior on unstructured meshes. Tests cases include non-reactive and chemically reactive simulations.

The shock instability phenomenon is a well-known problem for hypersonic flow computation by the shock-capturing Roe scheme. The pressure checkerboard is another well-known problem for low-Mach-number flow computation. The momentum interpolation method (MIM) is necessary for low-Mach-number flows to suppress the pressure checkerboard problem, and the pressure-difference-driven modification for cell face velocity can be regarded as a version of the MIM by subdividing the numerical dissipation of the Roe scheme. In this paper, MIM has been discovered through analysis and numerical tests to have the most important function in shock instability. MIM should be completely removed for nonlinear flows. However, the unexpected MIM is activated on the cell face nearly parallel to the flow for the high-Mach-number flows or low-Mach-number cells in numerical shock. Therefore, MIM should be retained for low-Mach-number flows and be completely removed for high-Mach-number flows and low-Mach-number cells in numerical shock. For such conditions, two coefficients are designed on the basis of the local Mach number and a shock detector. Thereafter, the improved Roe scheme is proposed. This scheme considers the requirement of MIM for incompressible and compressible flows, and is validated for good performance of numerical tests. An acceptable result can also be obtained with only the Mach number coefficient for general practical computation. Therefore, the objective of decreasing rather than increasing numerical dissipation to cure shock instability can be achieved with simple modification. Moreover, the mechanism of shock instability has been profoundly understood, in which MIM plays the most important role, although it is not the only factor. Copyright © 2016 John Wiley & Sons, Ltd.

- The inherent momentum interpolation method (MIM) of the Roe scheme plays the most important role in the shock instability phenomenon.
- Unexpected MIM is activated on the cell faces nearly parallel to high-Mach-number flows and low-Mach-number cells in numerical shock.
- An improved Roe scheme is proposed, which consider the requirement of MIM for incompressible and compressible flows, and can achieve the aim of decreasing numerical dissipation to cure shock instability.

Mathematical modeling and simulation of fluid–structure interaction problems are in the focus of research already for a longer period. However, taking into account also chemical reactions, leading to structural changes, including changes of mechanical properties of the solid phase, is rather new but for many applications is highly important area. This paper formulates a model system for reactive flow and transport in a vessel system, the penetration of chemical substances into the solid wall. Inside the wall, reactions take place that lead to changes of volume and of the mechanical properties of the wall. Numerical algorithms are developed and used to simulate the dynamics of such a mechano-chemical fluid–structure interaction problem. As a proof of concept scenario, plaque formation in blood vessels is chosen. The arbitrary Lagrangian Eulerian approach (ALE) is chosen to solve the systems numerically. Temporal discretization of the fully coupled monolithic model is accomplished by backward Euler scheme and spatial discretization by stabilized finite elements. The numerical approach is verified by numerical tests, and effective methods to maintain mesh qualities under large deformations are described. For realistic system parameters, the simulations show that the plaque formation in blood vessel is a long-time effect. The time scale of the formation is in the simulation of comparable order as in reality. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we formulate a mechano-chemical fluid–structure interaction problem. A plaque formation model is chosen as a proof of concept scenario. The ALE method is chosen to solve the system numerically and delivers very reliable results.

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

A mesh relaxation method which completely removes the Lagrangian behavior of the Boundary Integral mesh was developed, allowing long-term simulations without significant mesh distortion, even for high viscosity ratio drops. The dependence of the mesh on the flowŠs history was monitored by computing the standard deviation of the elements areas along the simulations. An automatic time-step control method, based on the Drop Cauchy-Green tensor was created to accelerate time evolution and save computational time, without loss of accuracy.

In this paper, we present a finite element method for two-phase incompressible flows with moving contact lines. We use a sharp interface Navier–Stokes model for the bulk phase fluid dynamics. Surface tension forces, including Marangoni forces and viscous interfacial effects, are modeled. For describing the moving contact lines, we consider a class of continuum models that contains several special cases known from the literature. For the whole model, describing bulk fluid dynamics, surface tension forces, and contact line forces, we derive a variational formulation and a corresponding energy estimate. For handling the evolving interface numerically, the level-set technique is applied. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space. We apply a Nitsche technique to weakly impose the Navier slip conditions on the solid wall. A unified approach for discretization of the (different types of) surface tension forces and contact line forces is introduced. Results of numerical experiments are presented, which illustrate the performance of the solver. Copyright © 2016 John Wiley & Sons, Ltd.

We present a level-set based finite element method for two-phase incompressible flows with moving contact lines. We use a sharp interface model and consider a class of continuum models for describing the moving contact lines. A general variational formulation and a corresponding energy estimate are derived. The discontinuous pressure is accurately approximated by using a stabilized extended finite element space, and a Nitsche technique is applied to weakly impose the Navier boundary conditions.

A new optimal control problem that incorporates the residual of the Eikonal equation into its objective is presented. The formulation of the state equation is based on the level set transport equation but extended by an additional source term, correcting the solution so as to minimize the objective functional. The method enforces the constraint so that the interface cannot be displaced at least in the continuous setting. The system of first-order optimality conditions is derived, linearized, and solved numerically. The control also prevents numerical instabilities, so that no additional stabilization techniques are required. This approach offers the flexibility to include other desired design criteria into the objective functional. The methodology is evaluated numerically in three different examples and compared with other PDE-based reinitialization techniques. Copyright © 2016 John Wiley & Sons, Ltd.

A new reinitialization technique based on an optimal control approach is presented. The residual of the Eikonal equation is incorporated into the objective functional. The state equation is given by the level set transport equation but augmented by an additional source term. The method is evaluated numerically in three different examples and compared to other PDE-based reinitialization techniques.

In this paper, an accurate semi-implicit rotational projection method is introduced to solve the Navier–Stokes equations for incompressible flow simulations. The accuracy of the fractional step procedure is investigated for the standard finite-difference method, and the discrete forms are presented with arbitrary orders or accuracy. In contrast to the previous semi-implicit projection methods, herein, an alternative way is proposed to decouple pressure from the momentum equation by employing the principle form of the pressure Poisson equation. This equation is based on the divergence of the convective terms and incorporates the actual pressure in the simulations. As a result, the accuracy of the method is not affected by the common choice of the pseudo-pressure in the previous methods. Also, the velocity correction step is redefined, and boundary conditions are introduced accordingly. Several numerical tests are conducted to assess the robustness of the method for second and fourth orders of accuracy. The results are compared with the solutions obtained from a typical high-resolution fully explicit method and available benchmark reports. Herein, the numerical tests are consisting of simulations for the Taylor–Green vortex, lid-driven square cavity, and vortex–wall interaction. It is shown that the present method can preserve the order of accuracy for both velocity and pressure fields in second-order and high-order simulations. Furthermore, a very good agreement is observed between the results of the present method and benchmark simulations. Copyright © 2016 John Wiley & Sons, Ltd.

Herein, we propose an accurate technique to decouple pressure from the momentum equation by incorporating the principle form of the pressure Poisson equation for semi-implicit projection methods. The velocity correction step is redefined, and boundary conditions are introduced accordingly. It is shown that the present method can preserve the order of accuracy for second-order and high-order finite difference simulations. A very good agreement is observed between the results of the present method and the benchmark simulations.

A variational multiscale method for computations of incompressible Navier–Stokes equations in time-dependent domains is presented. The proposed scheme is a three-scale variational multiscale method with a projection-based scale separation that uses an additional tensor valued space for the large scales. The resolved large and small scales are computed in a coupled way with the effects of unresolved scales confined to the resolved small scales. In particular, the Smagorinsky eddy viscosity model is used to model the effects of unresolved scales. The deforming domain is handled by the arbitrary Lagrangian–Eulerian approach and by using an elastic mesh update technique with a mesh-dependent stiffness. Further, the choice of orthogonal finite element basis function for the resolved large scale leads to a computationally efficient scheme. Simulations of flow around a static beam attached to a square base, around an oscillating beam and around a plunging aerofoil are presented. Copyright © 2016 John Wiley & Sons, Ltd.

A projection-based variational multiscale method for computations of incompressible Navier–Stokes equations in time-dependent domains is presented. An arbitrary Lagrangian–Eulerian approach with an elastic mesh moving technique with mesh-dependent stiffness is tailored with the variational multiscale method to handle moving boundaries. Simulations of flow around a static beam attached to a square base, around an oscillating beam, and around a plunging aerofoil are presented.

In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high-order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space-time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the *a posteriori* MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element-local space-time Galerkin finite element predictor on moving curved meshes to obtain a high-order accurate one-step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level *t*^{n} is connected to the new one at *t*^{n + 1} by *straight* edges, hence providing unstructured space-time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature-free integration, in which the space-time boundaries of the control volumes are split into *simplex* sub-elements that yield *constant* space-time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub-elements can be evaluated once and for all *analytically* during a preprocessing step.

We apply the new high-order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.

The arbitrary-Lagrangian-Eulerian ADER MOOD quadrature-free algorithm is a finite volume scheme that achieves high order of accuracy in space by limiting the reconstruction relying on the a posteriori MOOD strategy and reaches the same order of accuracy in time using the ADER approach. Efficiency is furthermore improved by a quadrature-free integration of the numerical fluxes. Hydrodynamics and magnetohydrodynamics equations are considered in multiple space dimensions on unstructured meshes in 2D and in 3D, and the speedup is monitored.

It is well-known that the traditional finite element method (FEM) fails to provide accurate results to the Helmholtz equation with the increase of wave number because of the ‘pollution error’ caused by numerical dispersion. In order to overcome this deficiency, a gradient-weighted finite element method (GW-FEM) that combines Shepard interpolation and linear shape functions is proposed in this work. Three-node triangular and four-node tetrahedral elements that can be generated automatically are first used to discretize the problem domain in 2D and 3D spaces, respectively. For each independent element, a compacted support domain is then formed based on the element itself and its adjacent elements sharing common edges (or faces). With the aid of Shepard interpolation, a weighted acoustic gradient field is then formulated, which will be further used to construct the discretized system equations through the generalized Galerkin weak form. Numerical examples demonstrate that the present algorithm can significantly reduces the dispersion error in computational acoustics. Copyright © 2016 John Wiley & Sons, Ltd.

A novel approach that combines Shepard interpolation and linear shape functions is proposed for reducing the dispersion error in acoustic analysis. For each independent element, the gradient field is formed based on the element itself and its adjacent elements sharing common edges (or faces). Theoretic analysis and numerical results illustrate that the present algorithm performs well in simulating high wavenumber problems.

The simple low-dissipation advection upwind splitting method (SLAU) scheme is a parameter-free, low-dissipation upwind scheme that has been applied in a wide range of aerodynamic numerical simulations. In spite of its successful applications, the SLAU scheme could be showing shock instabilities on unstructured grids, as many other contact resolved upwind schemes. Therefore, a hybrid upwind flux scheme is devised for improving the shock stability of SLAU scheme, without compromising on accuracy and low Mach number performance. Numerical flux function of the hybrid scheme is written in a general form, in which only the scalar dissipation term is different from that of the SLAU scheme. The hybrid dissipation term is defined by using a differentiable multidimensional-shock-detection pressure weight function, and the dissipation term of SLAU scheme is combined with that of the Van Leer scheme. Furthermore, the hybrid dissipation term is only applied for the solution of momentum fluxes in numerical flux function. Based on the numerical test results, the hybrid scheme is deemed to be a successful improvement on the shock stability of SLAU scheme, without compromising on the efficiency and accuracy. Copyright © 2016 John Wiley & Sons, Ltd.

A hybrid upwind scheme is devised for improving the shock stability of SLAU scheme. The dissipation term of SLAU flux function is modified to give a new hybrid flux function. The hybrid flux function is used on the solution of momentum fluxes. In the test cases, the hybrid scheme is showing improvements on the numerical shock stability.

This paper aims to reassess the Riemann solver for compressible fluid flows in Lagrangian frame from the viewpoint of modified equation approach and provides a theoretical insight into dissipation mechanism. It is observed that numerical dissipation vanishes uniformly for the Godunov-type schemes in the sense that associated dissipation matrix has zero determinant if an exact or approximate Riemann solver is used to construct numerical fluxes in the Lagrangian frame. This fact connects to some numerical defects such as the wall-heating phenomenon and start-up errors. To cure these numerical defects, a traditional numerical viscosity is added, as well as the artificial heat conduction is introduced via a simple passage of the Lax–Friedrichs type discretization of internal energy. Copyright © 2016 John Wiley & Sons, Ltd.

This paper aims to reassess the Riemann solver for compressible fluid flows in the Lagrangian frame from the viewpoint of modified equation approach and provides a theoretical insight into dissipation mechanism. It is observed that numerical dissipation vanishes uniformly for the Godunov-type schemes in the Lagrangian frame, which connects to some numerical defects such as the wall-heating phenomenon and start-up errors. To cure the defects, the artificial heat conduction, in addition to the traditional numerical viscosity, is introduced via a passage of the Lax–Friedrichs type discretization of internal energy.

In this paper, we propose a simple robust flux splitting method for all-speed flows with low dissipation. Following the Toro-Vazquez splitting, the inviscid flux is split into the convective and the pressure parts first. Then, we apply the Harten-Lax-van Leer (HLL) algorithm to each parts with low dissipation modification. Here, the modification improves the method with accurately resolving contact discontinuity and high resolution for low speed flows. Several carefully chosen numerical tests are investigated, and the results show that the proposed scheme is capable of resolving contact discontinuity, robust against the shock anomaly and accurate at all-speeds. Because of these properties, it is expected to be widely applied to all-speed flow studies. Copyright © 2016 John Wiley & Sons, Ltd.

We propose a robust flux splitting method with low dissipation for all speed flows. A low-dissipation modification is properly applied to the convection and the pressure parts of the inviscid terms of the Euler equation, respectively, which improves the accuracy obviously without enhancing the computational costs of the method. The results of this study prove that the new method can enhance the accuracy and robustness for solving all-speed flows.

Hybrid models have found widespread applications for simulation of wall-bounded flows at high Reynolds numbers. Typically, these models employ Reynolds-averaged Navier–Stokes (RANS) and large eddy simulation (LES) in the near-body and off-body regions, respectively. A number of coupling strategies between the RANS and LES regions have been proposed, tested, and applied in the literature with varying degree of success. Linear eddy-viscosity models (LEVM) are often used for the closure of turbulent stress tensor in RANS and LES regions. LEVM incorrectly predicts the anisotropy of Reynolds normal stress at the RANS-LES interface region. To overcome this issue, use of non-linear eddy-viscosity models (NLEVM) have started receiving attention. In this study, a generic non-linear blended modeling framework for performing hybrid simulations is proposed. Flow over the periodic hills is used as the test case for model evaluation. This case is chosen due to complex flow physics with simplified geometry. Analysis of the simulations suggests that the non-linear hybrid models show a better performance than linear hybrid models. It is also observed that the non-linear closures are less sensitive to the RANS-LES coupling and grid resolution. Copyright © 2016 John Wiley & Sons, Ltd.

In this study, a generic non-linear blended modeling framework for performing hybrid RANS-LES simulations is proposed and flow over the periodic hills is used as the test case for model evaluation. Analysis of the simulations suggests that the non-linear closures are less sensitive to the RANS-LES coupling method and grid resolution. Also, good agreement has been found for flow statistics compared with the existing experimental data for simulations performed using NSST-Blended at higher Reynolds number.

A new mixed-interpolation finite element method is presented for the two-dimensional numerical simulation of incompressible magnetohydrodynamic (MHD) flows which involve convective heat transfer. The proposed method applies the nodal shape functions, which are locally defined in nine-node elements, for the discretization of the Navier–Stokes and energy equations, and the vector shape functions, which are locally defined in four-node elements, for the discretization of the electromagnetic field equations. The use of the vector shape functions allows the solenoidal condition on the magnetic field to be automatically satisfied in each four-node element. In addition, efficient approximation procedures for the calculation of the integrals in the discretized equations are adopted to achieve high-speed computation. With the use of the proposed numerical scheme, MHD channel flow and MHD natural convection under a constant applied magnetic field are simulated at different Hartmann numbers. The accuracy and robustness of the method are verified through these numerical tests in which both undistorted and distorted meshes are employed for comparison of numerical solutions. Furthermore, it is shown that the calculation speed for the proposed scheme is much higher compared with that for a conventional numerical integration scheme under the condition of almost the same memory consumption. Copyright © 2016 John Wiley & Sons, Ltd.

A high-speed finite element scheme is proposed for simulation of incompressible MHD flows with convective heat transfer. In the proposed scheme, Q2-Q1 elements are used for the interpolations of the velocity, pressure, and temperature, while the electric field and magnetic flux density are interpolated using vector shape functions in the subdivided four-node elements. The robustness of the scheme is investigated using highly distorted meshes in well-known problems, and the results showing the improvement of calculation speed are also presented.

Liquid mixing is an important component of many microfluidic concepts and devices, and computational fluid dynamics (CFD) is playing a key role in their development and optimization. Because liquid mass diffusivities can be quite small, CFD simulation of liquid micromixing can over predict the degree of mixing unless numerical (or false) diffusion is properly controlled. Unfortunately, the false diffusion behavior of higher-order finite volume schemes, which are often used for such simulations, is not well understood, especially on unstructured meshes. To examine and quantify the amount of false diffusion associated with the often recommended and versatile second-order upwind method, a series of numerical simulations was conducted using a standardized two-dimensional test problem on both structured and unstructured meshes. This enabled quantification of an ‘effective’ false diffusion coefficient (*D _{false}*) for the method as a function of mesh spacing. Based on the results of these simulations, expressions were developed for estimating the spacing required to reduce

Numerical simulations were conducted to determine an ‘effective’ false diffusion coefficient (*D _{false}*) for the second-order upwind finite volume method. Expressions for estimating the spacing required to reduce

The finite volume discretization of nonlinear elasticity equations seems to be a promising alternative to the traditional finite element discretization as mentioned by Lee *et al*. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell-centered finite volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency, which are imposed by mean of the Coleman–Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free energy with respect to the left Cauchy–Green tensor. Moreover, the materials being isotropic, the free-energy is function of the left Cauchy–Green tensor invariants, which enable the use of the neo-Hookean model. The hyperelasticity system is discretized using the cell-centered Lagrangian scheme from the work of Maire *et al*. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents the 3D extension of the EUCCLHYD scheme (Explicit Unstructured Cell-Centered Lagrangian HYDrodynamics) [Maire SIAM 2007] for the numerical modeling of the hyperelasticity system at first order in space and time. The constitutive law is derived by means of a Coleman–Noll procedure in the case of isotropic neo-Hookean materials. The scheme is validated on three test cases and is proved to have an inherent 3D nature when shearing is present such as in the oscillating beam problem (refer to Figure 1).

A six degrees of freedom (6DOF) algorithm is implemented in the open-source CFD code REEF3D. The model solves the incompressible Navier–Stokes equations. Complex free surface dynamics are modeled with the level set method based on a two-phase flow approach. The convection terms of the velocities and the level set method are treated with a high-order weighted essentially non-oscillatory discretization scheme. Together with the level set method for the free surface capturing, this algorithm can model the movement of rigid floating bodies and their interaction with the fluid. The 6DOF algorithm is implemented on a fixed grid. The solid-fluid interface is represented with a combination of the level set method and ghost cell immersed boundary method. As a result, re-meshing or overset grids are not necessary. The capability, accuracy, and numerical stability of the new algorithm is shown through benchmark applications for the fluid-body interaction problem. Copyright © 2016 John Wiley & Sons, Ltd.

The paper discusses the implementation of a novel six degree of freedom (6DOF) algorithm in the open-source CFD code REEF3D. The new 6DOF algorithm makes re-meshing or overset grids unnecessary, resulting in a simpler, faster, and more stable algorithm. Several benchmark applications show that the new floating body algorithm can handle even impact scenarios in a weakly coupled manner while maintaining numerical stability and accuracy.

An enhanced goal-oriented mesh adaptation method is presented based on aerodynamic functional total derivatives with respect to mesh nodes in a Reynolds-Averaged Navier-Stokes (RANS) finite-volume mono-block and non-matching multi-block-structured grid framework. This method falls under the category of methods involving the adjoint vector of the function of interest. The contribution of a Spalart–Allmaras turbulence model is taken into account through its linearization. Meshes are adapted accordingly to the proposed indicator. Applications to 2D RANS flow about a RAE2822 airfoil in transonic, and detached subsonic conditions are presented for the drag coefficient estimation. The asset of the proposed method is patent. The obtained 2D anisotropic mono-block mesh well captures flow features as well as global aerodynamic functionals. Interestingly, the constraints imposed by structured grids may be relaxed by the use of non-matching multi-block approach that limits the outward propagation of local mesh refinement through all of the computational domain. The proposed method also leads to accurate results for these multi-block meshes but at a fraction of the cost. Finally, the method is also successfully applied to a more complex geometry, namely, a mono-block mesh in a 3D RANS transonic flow about an M6 wing. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents an enhanced goal-oriented adjoint-based mesh adaptation method based on a scalar indicator for one mesh level only for RANS flows, where the linearization of the Spalart–Allmaras turbulence model is addressed. The adaptation procedure is assessed on standard monoblock and non-conforming multi-block-structured mesh with non-matching interfaces between blocks. The method is efficient for Euler and RANS flows, standard and non-conforming meshes, and transonic and detached subsonic operational flow conditions.

The quasi-steady assumption is commonly adopted in existing transient fluid–solid-coupled convection–conduction (conjugate) heat transfer simulations, which may cause non-negligible errors in certain cases of practical interest. In the present work, we adopt a new multi-scale framework for the fluid domain formulated in a triple-timing form. The slow-varying temporal gradient corresponding to the time scales in the solid domain has been effectively included in the fluid equations as a source term, whilst short-scale unsteadiness of the fluid domain is captured by a local time integration at a given ‘frozen’ large scale time instant. For concept proof, validation and demonstration purposes, the proposed methodology has been implemented in a loosely coupled procedure in conjunction with a hybrid interfacing treatment for coupling efficiency and accuracy. The present results indicate that a much enhanced applicability can be achieved with relatively small modifications of existing transient conjugate heat transfer methods at little extra cost. Copyright © 2016 John Wiley & Sons, Ltd.

A new multi-scale framework in a triple-timing form is adopted to avoid the common quasi-steady flow assumption. Slow temporal variations corresponding to the solid time scales are included in the fluid domain as a source term, whilst short-scale fluid unsteadiness is captured by local time integration. The test case results indicate that a much enhanced applicability can be achieved by relatively small modifications of existing transient conjugate heat transfer methods.

This paper presents a generalization of the incompressible Oldroyd-B model based on a thermodynamic framework within which the fluid can be viewed to exist in multiple natural configurations. The response of the fluid is viewed as a combination of an elastic component and a dissipative component. The dissipative component leads to the evolution of the underlying natural configurations, while the response from the natural configuration to the current configuration is considered elastic and therefore non-dissipative. For an incompressible fluid, it is necessary that both the elastic behavior as well as the dissipative behavior is isochoric. This is achieved by ensuring that the determinant of the stretch tensor associated with the elastic response meets the constraint that its determinant is unity.

A new stabilized mixed method is developed for the velocity, pressure and the kinematic tensor fields. Analytical models for fine scale fields are derived via the solution of the fine-scale equations facilitated by the Variational Multiscale framework that are then variationally embedded in the coarse-scale variational equations. The resulting method inherits the attributes of the classical SUPG and GLS methods, while a significant new contribution is that the form of the stabilization tensors is explicitly derived. A family of linear and quadratic tetrahedral and hexahedral elements is developed with equal-order interpolations for the various fields. Numerical tests are presented that validate the incompressibility of the elastic stretch tensor, show optimal *L _{2}* convergence for the conformation tensor field, and present stable response for high Weissenberg number flows. Copyright © 2016 John Wiley & Sons, Ltd.

A stabilized mixed formulation is developed for a generalized incompressible Oldroyd-B model that is based on a thermodynamically consistent framework for multiple natural configurations. The new method uniformly imposes the incompressibility condition on the elastic stretch tensor and shows optimal L2 convergence for the conformation tensor field on benchmark problems. This figure shows the magnitude of the elastic stretch tensor * B* and its rate of convergence for various element types.

Gas Kinetic Method-based flow solvers have become popular in recent years owing to their robustness in simulating high Mach number compressible flows. We evaluate the performance of the newly developed analytical gas kinetic method (AGKM) by Xuan *et al.* in performing direct numerical simulation of canonical compressible turbulent flow on graphical processing unit (GPU)s. We find that for a range of turbulent Mach numbers, AGKM results shows excellent agreement with high order accurate results obtained with traditional Navier–Stokes solvers in terms of key turbulence statistics. Further, AGKM is found to be more efficient as compared with the traditional gas kinetic method for GPU implementation. We present a brief overview of the optimizations performed on NVIDIA K20 GPU and show that GPU optimizations boost the speedup up-to 40*x* as compared with single core CPU computations. Hence, AGKM can be used as an efficient method for performing fast and accurate direct numerical simulations of compressible turbulent flows on simple GPU-based workstations. Copyright © 2016 John Wiley & Sons, Ltd.

Evaluation of the analytical gas kinetic method developed by Xuan and Xu (2013) is done in its performance to simulate decaying compressible turbulence on graphical processing unit (GPU)s. We find that analytical gas kinetic method results show excellent agreement with high-order accurate direct numerical simulation results. We perform GPU optimizations on NVIDIA K20 GPU, which boosts the speedup up-to 40*x* as compared with CPU computations.

The viscosity plays an important role, and a multiphase solver is necessary to numerically simulate the oil spilling from a damaged double hull tank (DHT). However, it is uncertain whether turbulence modelling is necessary, which turbulence model is suitable; and what the role of compressibility of the fluids is. This paper presents experimental and numerical investigations to address these issues for various cases representing different scenarios of the oil spilling, including grounding and collision. In the numerical investigations, various approaches to model the turbulence, including the large eddy simulation (LES), direct numerical simulation and the Reynolds average Navier–Stokes equation (RANS) with different turbulence models, are employed. Based on the investigations, it is suggested that the effective Reynolds numbers corresponding to both oil outflow and water inflow shall be considered when classifying the significance of the turbulence and selecting the appropriate turbulence models. This is confirmed by new lab tests considering the axial offset between the internal and the external holes on two hulls of the DHT. The investigations conclude for numerically simulating oil spilling from a damaged DHT that when the effective Re is smaller the RANS approaches should not be used and LES modelling should be employed; while when the effective Reynolds numbers is large, the RANS models may be used as they can give similar results to LES in terms of the height of the mixture in the ballast tank and discharge but costing much less CPU time. The investigation on the role of the compressibility of the fluid reveals that the compressibility of the fluid may be considerable in a small temporal-spatial scale but plays an insignificant role on macroscopic process of the oil spilling. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents comparative studies on the turbulence modelling and the role of compressibility on oil spilling from DHTs. It suggests criterion to select the appropriate turbulence model in terms of computational robustness using the effective Reynolds number, considering both oil outflow and water inflow. It also concludes that the compressibility of the fluid may be considerable in a small temporal-spatial scale but plays insignificant role on macroscopic process of the oil spilling.

A new modified Galerkin/finite element method is proposed for the numerical solution of the fully nonlinear shallow water wave equations. The new numerical method allows the use of low-order Lagrange finite element spaces, despite the fact that the system contains third order spatial partial derivatives for the depth averaged velocity of the fluid. After studying the efficacy and the conservation properties of the new numerical method, we proceed with the validation of the new numerical model and boundary conditions by comparing the numerical solutions with laboratory experiments and with available theoretical asymptotic results. Copyright © 2016 John Wiley & Sons, Ltd.

A fully discrete numerical scheme for some fully-nonlinear shallow water equations with wall boundary conditions is developed. Shoaling and reflecting solitary waves are studied in detail. The accuracy and the efficiency of this numerical method is demonstrated while the match between numerical results, experimental data, and theoretical approximations is very satisfactory.

The nonlocal theory of the radiative energy transport in laser-heated plasmas of arbitrary ratio of the characteristic inhomogeneity scale length to the photon mean free paths is applied to define the closure relations of a hydrodynamic system. The corresponding transport phenomena cannot be described accurately using the Chapman–Enskog approach, that is, with the usual fluid approach dealing only with local values and derivatives. Thus, we directly solve the photon transport equation allowing one to take into account the effect of long-range photon transport. The proposed approach is based on the Bhatnagar–Gross–Krook collision operator using the photon mean free path as a unique parameter. Such an approach delivers a calculation efficiency and an inherent coupling of radiation to the fluid plasma parameters in an implicit way and directly incorporates nonequilibrium physics present under the condition of intense laser energy deposition due to inverse bremsstrahlung. In combination with a higher order discontinuous Galerkin scheme of the transport equation, the solution obeys both limiting cases, that is, the local diffusion asymptotic usually present in radiation hydrodynamics models and the collisionless transport asymptotic of free-streaming photons. In other words, we can analyze the radiation transport closure for radiation hydrodynamics and how it behaves when deviating from the conditions of validity of Chapman–Enskog method, which is demonstrated in the case of exact steady transport and approximate multigroup diffusion numerical tests. As an application, we present simulation results of intense laser-target interaction, where the radiative energy transport is controlled by the mean free path of photons. Copyright © 2016 John Wiley & Sons, Ltd.

Nonlocal radiative transport in laser-heated plasmas of arbitrary Knudsen number is a challenging task. We directly solve the photon transport equation based on the Bhatnagar-Gross-Krook(BGK) collision operator, which gives an inherent coupling of radiation to the fluid plasma parameters. Our high-order discontinuous Galerkin scheme of the BGK transport equation and thefluid energy equation gives solutions obeying both limiting cases of transport, i.e. diffusion and free-streaming. As an application, we present simulation results of intense laser-target interaction.

In this paper, a fully discrete high-resolution arbitrary Lagrangian–Eulerian (ALE) method is developed over untwisted time–space control volumes. In the framework of the finite volume method, 2D Euler equations are discretized over untwisted moving control volumes, and the resulting numerical flux is computed using the generalized Riemann problem solver. Then, the fluid flows between meshes at two successive time steps can be updated without a remapping process in the classic ALE method. This remapping-free ALE method directly couples the mesh motion into a physical variable update to reflect the temporal evolution in the whole process. An untwisted moving mesh is generated in terms of the vorticity-free part of the fluid velocity according to the Helmholtz theorem. Some typical numerical tests show the competitive performance of the current method. Copyright © 2016 John Wiley & Sons, Ltd.

Our work is focused on two important technologies in ALE method:

- The untwisted adaptively moving time–space control volume is generated only by the irrotational part of the flow velocity based on the Helmholtz theorem to avoid the mesh tangling.
- The GRP solver with the whole wave configuration is employed for flux computation to ensure the high resolution.

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

We have developed a new divergence preserving method for the reconstruction of the Cartesian components of a vector field from the orthogonal projection of a vector field to the normals to edges in 2D. The new global divergence preserving method is exact for linear vector fields.

Kalman filter is a sequential estimation scheme that combines predicted and observed data to reduce the uncertainty of the next prediction. Because of its sequential nature, the algorithm cannot be efficiently implemented on modern parallel compute hardware nor can it be practically implemented on large-scale dynamical systems because of memory issues. In this paper, we attempt to address pitfalls of the earlier low-memory approach described in and extend it for parallel implementation. First, we describe a low-memory method that enables one to pack covariance matrix data employed by the Kalman filter into a low-memory form by means of certain quasi-Newton approximation. Second, we derive parallel formulation of the filtering task, which allows to compute several filter iterations independently. Furthermore, this leads to an improvement of estimation quality as the method takes into account the cross-correlations between consequent system states. We experimentally demonstrate this improvement by comparing the suggested algorithm with the other data assimilation methods that can benefit from parallel implementation. Copyright © 2016 John Wiley & Sons, Ltd.

Kalman filter is a known sequential algorithm that allows to estimate states of dynamical systems using predicted a-priori information and observed data. However, due to its sequential nature the algorithm cannot be efficiently implemented on parallel systems and suffers from memory and performance issues when dimension of the state space becomes large. In the present paper we alleviate these problems by using certain approximation of the filter and reformulating the classical filtering task to allow for parallelism.

The smoothed-profile method for the motion of solid bodies suspended in a fluid phase is investigated when combined with a high-order spatial discretization. The performance of the combined method is tested for a wide range of flow and geometry parameters as well as for static and for moving particles. Moreover, a sensitivity analysis is conducted with respect to the smoothed-profile function. The algorithm is extended to include thermal effects in Boussinesq approximation. Several benchmark problems are considered to demonstrate the potential of the technique. The implementation of the energy equation is verified by dedicated tests. All simulations are compared with either theoretical, numerical, or experimental data. The results demonstrate the accuracy and efficiency of the smoothed-profile method for non-isothermal problems in combination with a discontinuous finite-element solver for the fluid flow, which allows for a flexible handling of the grid and the order of spectral approximation in each element. Copyright © 2016 John Wiley & Sons, Ltd.

The smoothed-profile method combined with a discontinuous Galerkin finite-element method is investigated for simulating the motion of solid bodies in a fluid phase. Different smoothed-profile functions are compared and the algorithm is extended to include thermal effects. The solver is benchmarked against theoretical and experimental results for several problems.

Determining liquid–vapor phase equilibrium is often required in multiphase flow computations. Existing equilibrium solvers are either accurate but computationally expensive or cheap but inaccurate. The present paper aims at building a fast and accurate specific phase equilibrium solver, specifically devoted to unsteady multiphase flow computations. Moreover, the solver is efficient at phase diagram bounds, where non-equilibrium pure liquid and pure gas are present. It is systematically validated against solutions based on an accurate (but expensive) solver. Its capability to deal with cavitating, evaporating, and condensing two-phase flows is highlighted on severe test problems both 1D and 2D. Copyright © 2016 John Wiley & Sons, Ltd.

This work presents a new method to compute thermochemical equilibrium in liquid–vapor flows. The proposed method is both accurate and fast, in addition of being much easier to code than the usual iterative methods. Through a series of test cases from simple 1D flow configurations to a complex 2D evaporating liquid jet, the solver is proved to successfully cope with cavitation, evaporation, condensation, and boiling.

This paper describes a collocated numerical scheme for multi-material compressible Euler equations, which attempts to suit to parallel computing constraints. Its main features are conservativity of mass, momentum, total energy and entropy production, and second order in time and space. In the context of a Eulerian Lagrange-remap scheme on planar geometry and for rectangular meshes, we propose and compare remapping schemes using a finite volume framework. We consider directional splitting or fully multi-dimensional remaps, and we focus on a definition of the so-called corner fluxes. We also address the issue of the internal energy behavior when using a conservative total energy remap. It can be perturbed by the duality between kinetic energy obtained through the conservative momentum remap or implicitly through the total energy remap. Therefore, we propose a kinetic energy flux that improves the internal energy remap results in this context. Copyright © 2016 John Wiley & Sons, Ltd.

In the context of a Eulerian Lagrange-remap scheme on planar geometry and for rectangular meshes, we propose and compare remapping schemes using a finite volume framework. We consider directional splitting or fully multi-dimensional remaps, and we focus on a definition of the so-called corner fluxes. We also address the issue of the internal energy behavior when using a conservative total energy remap.

In this paper, the newly developed lattice Boltzmann flux solver (LBFS) is developed into a version in the rotating frame of reference for simulation of turbomachinery flows. LBFS is a finite volume solver for the solution of macroscopic governing differential equations. Unlike conventional upwind or Godunov-type flux solvers which are constructed by considering the mathematical properties of Euler equations, it evaluates numerical fluxes at the cell interface by reconstructing local solution of lattice Boltzmann equation (LBE). In other words, the numerical fluxes are physically determined rather than by some mathematical approximation. The LBE is herein expressed in a relative frame of reference in order to correctly recover the macroscopic equations, which is also the basis of LBFS. To solve the LBE, an appropriate lattice Boltzmann model needs to be established in advance. This includes both the determinations of the discrete velocity model and its associated equilibrium distribution functions. Particularly, a simple and effective D1Q4 model is adopted, and the equilibrium distribution functions could be efficiently obtained by using the direct method. The present LBFS is validated by several inviscid and viscous test cases. The numerical results demonstrate that it could be well applied to typical and complex turbomachinery flows with favorable accuracy. It is also shown that LBFS has a delicate dissipation mechanism and is thus free of some artificial fixes, which are often needed in conventional schemes. Copyright © 2016 John Wiley & Sons, Ltd.

A promising lattice Boltzmann flux solver (LBFS) is developed into a version in the rotating frame of reference for simulation of turbomachinery flows. Since the numerical fluxes at the cell interface are evaluated by reconstructing local solution of lattice Boltzmann equation, it has a delicate dissipation mechanism and is thus free of additional artificial fixes. Numerical tests for several typical turbomachinery flows with different complexities demonstrate the accuracy and robustness of the present method.

Hybrid schemes are very efficient for complex compressible flow simulation. However, for most existing hybrid schemes in literature, empirical problem-dependent parameters are always needed to detect shock waves and hence greatly decrease the robustness and accuracy of the hybrid scheme. In this paper, based on the nonlinear weights of the weighted essentially non-oscillatory (WENO) scheme, a novel weighting switch function is proposed. This function approaches 1 with high-order accuracy in smooth regions and 0 near discontinuities. Then, with the new weighting switch function, a seventh-order hybrid compact-reconstruction WENO scheme (HCCS) is developed. The new hybrid scheme uses the same stencil as the fifth-order WENO scheme, and it has seventh-order accuracy in smooth regions even at critical points. Numerical tests are presented to demonstrate the accuracy and robustness of both the switch function and HCCS. Comparisons also reveal that HCCS has lower dissipation and less computational cost than the seventh-order WENO scheme. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a parameter-free weighting switch function is proposed for developing a family of high order hybrid schemes. As an example, a seventh-order hybrid compact-CRWENO scheme (HCCS) is constructed and analyzed with the proposed function. Numerical results show that the new scheme has very low dissipation and maintains the essentially non-oscillation property of the base-CRWENO scheme. As shown in this figure, HCCS is even more efficient than the seventh-order WENO scheme.

Arbitrary Lagrangian–Eulerian finite volume methods that solve a multidimensional Riemann-like problem at the cell center in a staggered grid hydrodynamic (SGH) arrangement have been proposed. This research proposes a new 3D finite element arbitrary Lagrangian–Eulerian SGH method that incorporates a multidimensional Riemann-like problem. Two different Riemann jump relations are investigated. A new limiting method that greatly improves the accuracy of the SGH method on isentropic flows is investigated. A remap method that improves upon a well-known mesh relaxation and remapping technique in order to ensure total energy conservation during the remap is also presented. Numerical details and test problem results are presented. Copyright © 2016 John Wiley & Sons, Ltd.

A finite element arbitrary Lagrangian–Eulerian method that solves a multidirectional Riemann-like problem incorporating two limiting coefficients: the first is based on the minmod limiter and the second is a function of the discrete Mach number is presented. Our approach produces substantially less internal energy errors than the minmod limiter alone for a steel shell implosion as shown in Figure 1. For strong shock problems, the new limiter is more accurate and converges at a higher rate than the quadratic artificial viscosity.

In this paper, we study an interface transport scheme of a two-phase flow of an incompressible viscous immiscible fluid. The problem is discretized by the characteristics method in time and finite elements method in space. The interface is captured by the level set function. Appropriate boundary conditions for the problem of mold filling are investigated, a new natural boundary condition under pressure effect for the transport equation is proposed, and an algorithm for computing the solution is presented. Finally, numerical experiments show and validate the effectiveness of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we propose an interface transport scheme of a two-phase flow for an incompressible viscous immiscible fluid of large-density ratio to model 3D molds filling. A new natural boundary condition under pressure effect for the transport equation is proposed. Finally, numerical validations show the effectiveness of the proposed scheme.

This paper presents an assessment of fast parallel pre-conditioners for numerical solution of the pressure Poisson equation arising in large eddy simulation of turbulent incompressible flows. Focus is primarily on the pre-conditioners suitable for domain decomposition based parallel implementation of finite volume solver on non-uniform structured Cartesian grids. Bi-conjugate gradient stabilized method has been adopted as the Krylov solver for the linear algebraic system resulting from the discretization of the pressure Poisson equation. We explore the performance of multigrid pre-conditioner for the non-uniform grid and compare its performance with additive Schwarz pre-conditioner, Jacobi and SOR(*k*) pre-conditioners. Numerical experiments have been performed to assess the suitability of these pre-conditioners for a wide range of non-uniformity (stretching) of the grid in the context of large eddy simulation of a typical flow problem. It is seen that the multigrid preconditioner shows the best performance. Further, the SOR(*k*) preconditioner emerges as the next best alternative. Copyright © 2016 John Wiley & Sons, Ltd.

Large eddy simulation of non-uniform Cartesian grids and the immersed boundary method is used for incompressible turbulent flow simulations of complex geometry bodies. Acceleration of the pressure Poisson equation's solution is sought. The present work has brought forth two new aspects: (a) a geometric multigrid preconditioner for pressure Poisson equation in an immersed boundary Navier–Stokes solver on stretched grids and (b) efficacy of the simple SOR(k) preconditioner for highly stretched grids.

Decoupled implementation of data assimilation methods has been rarely studied. The variational ensemble Kalman filter has been implemented such that it needs not to communicate directly with the model, but only through input and output devices. In this work, an open multi-functional three-dimensional (3D) model, the coupled hydrodynamical-ecological model for regional and shelf seas (COHERENS), has been used. Assimilation of the total suspended matter (TSM) is carried out in 154 km^{2} lake Säkylän Pyhäjärvi. Observations of TSM were derived from high-resolution satellite images of turbidity and chrolophyll-a. For demonstrating the method, we have used a low-resolution model grid of 1 km. The model was run for a period from May 16 to September 14. We have run the COHERENS model with two-dimensional (2D) mode time steps and 3D mode time steps. This allows COHERENS to switch between 2D and 3D modes in a single run for computational efficiency. We have noticed that there is not much difference between these runs. This is because satellite images depict the derived TSM for the surface layer only. The use of additional 3D data might change this conclusion and improve the results. We have found that in this study, the use of a large ensemble size does not guarantee higher performance. The successful implementation of decoupled variational ensemble Kalman filter method opens the way for other methods and evolution models to enjoy the benefits without having to spend substantial effort in merging the model and assimilation codes together, which can be a difficult task. © 2016 The Authors. International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd.

In ensemble data assimilation, such as the variational ensemble Kalman filter, increasing ensemble size does not always improve the analysis, as is seen in the figures attached that depict the truth and analysis with 10, 30, and 50 ensemble members. This phenomenon emphasizes the property of data assimilation that innovation is often captured in a much lower-dimensional space than the entire state space. Variational ensemble Kalman filter automatically identifies such a low-dimensional space.

A novel method for simulating multi-phase flow in porous media is presented. The approach is based on a control volume finite element mixed formulation and new force-balanced finite element pairs. The novelty of the method lies in (i) permitting both continuous and discontinuous description of pressure and saturation between elements; (ii) the use of arbitrarily high-order polynomial representation for pressure and velocity and (iii) the use of high-order flux-limited methods in space and time to avoid introducing non-physical oscillations while achieving high-order accuracy where and when possible. The model is initially validated for two-phase flow. Results are in good agreement with analytically obtained solutions and experimental results. The potential of this method is demonstrated by simulating flow in a realistic geometry composed of highly permeable meandering channels. © 2016 The Authors International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd

Multi-phase flow through highly permeable underground channels (the flow is from the left). The novel method presented and validated in this paper allows for discontinuous description of pressure and saturation between elements which results in minimal numerical dispersion. In addition, the method is high-order accurate and uses fully unstructured meshes as shown in thefigure.

Estimating river discharge from *in situ* and/or remote sensing data is a key issue for evaluation of water balance at local and global scales and for water management. Variational data assimilation (DA) is a powerful approach used in operational weather and ocean forecasting, which can also be used in this context. A distinctive feature of the river discharge estimation problem is the likely presence of significant uncertainty in principal parameters of a hydraulic model, such as bathymetry and friction, which have to be included into the control vector alongside the discharge. However, the conventional variational DA method being used for solving such extended problems often fails. This happens because the control vector iterates (i.e., approximations arising in the course of minimization) result into hydraulic states not supported by the model. In this paper, we suggest a novel version of the variational DA method specially designed for solving estimation-under-uncertainty problems, which is based on the ideas of iterative regularization.

The method is implemented with SIC^{2}, which is a full Saint-Venant based 1D-network model. The SIC^{2} software is widely used by research, consultant and industrial communities for modeling river, irrigation canal, and drainage network behavior. The adjoint model required for variational DA is obtained by means of automatic differentiation. This is likely to be the first stable consistent adjoint of the 1D-network model of a commercial status in existence.

The DA problems considered in this paper are offtake/tributary estimation under uncertainty in the cross-device parameters and inflow discharge estimation under uncertainty in the bathymetry defining parameters and the friction coefficient. Numerical tests have been designed to understand identifiability of discharge given uncertainty in bathymetry and friction. The developed methodology, and software seems useful in the context of the future Surface Water and Ocean Topography satellite mission. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents a novel version of variational data assimilation (DA) approach designed for solving estimation-under-uncertainty problems. A modified iterative process which guaranties regular convergence is used. This is vitally important if the model domain (i.e. the set of the control/parameter values for which the model solution exist) is bounded. The method is applied for solving discharge estimation problem under uncertainty in bathymetry and the friction coefficient involving SIC^{2}, which is the full Saint-Venant hydraulic network model of a commercial status.

Understanding the impact of the changes in pollutant emission from a foreign region onto a target region is a key factor for taking appropriate mitigating actions. This requires a sensitivity analysis of a response function (defined on the target region) with respect to the source(s) of pollutant(s). The basic and straightforward approach to sensitivity analysis consists of multiple simulations of the pollution transport model with variations of the parameters that define the source of the pollutant. A more systematic approach uses the adjoint of the pollution transport model derived from applying the principle of variations. Both approaches assume that the transport velocity and the initial distribution of the pollutant are known. However, when observations of both the velocity and concentration fields are available, the transport velocity and the initial distribution of the pollutant are given by the solution of a data assimilation problem. As a consequence, the sensitivity analysis should be carried out on the optimality system of the data assimilation problem, and not on the direct model alone. This leads to a sensitivity analysis that involves the second-order adjoint model, which is presented in the present work. It is especially shown theoretically and with numerical experiments that the sensitivity on the optimality system includes important terms that are ignored by the sensitivity on the direct model. The latter shows only the direct effects of the variation of the source on the response function while the first shows the indirect effects in addition to the direct effects. Copyright © 2016 John Wiley & Sons, Ltd.

In the presence of variational data assimilation, the sensitivity analysis must be carried on the optimality system. The sensitivity on the optimality system captures the indirect effects of the variation of the source on the response function, which is not possible with the sensitivity analysis on the model alone.

For simulating freely moving problems, conventional immersed boundary-lattice Boltzmann methods encounter two major difficulties of an extremely large flow domain and the incompressible limit. To remove these two difficulties, this work proposes an immersed boundary-lattice Boltzmann flux solver (IB-LBFS) in the arbitrary Lagragian–Eulerian (ALE) coordinates and establishes a dynamic similarity theory. In the ALE-based IB-LBFS, the flow filed is obtained by using the LBFS on a moving Cartesian mesh, and the no-slip boundary condition is implemented by using the boundary condition-enforced immersed boundary method. The velocity of the Cartesian mesh is set the same as the translational velocity of the freely moving object so that there is no relative motion between the plate center and the mesh. This enables the ALE-based IB-LBFS to study flows with a freely moving object in a large open flow domain. By normalizing the governing equations for the flow domain and the motion of rigid body, six non-dimensional parameters are derived and maintained to be the same in both physical systems and the lattice Boltzmann framework. This similarity algorithm enables the lattice Boltzmann equation-based solver to study a general freely moving problem within the incompressible limit. The proposed solver and dynamic similarity theory have been successfully validated by simulating the flow around an in-line oscillating cylinder, single particle sedimentation, and flows with a freely falling plate. The obtained results agree well with both numerical and experimental data. Copyright © 2016 John Wiley & Sons, Ltd.

- The paper presents an immersed boundary-lattice Boltzmann flux solver in the arbitrary Lagrangian–Eulerian coordinates for simulating solid objects falling freely in unbounded domains;
- A dynamic similarity theory is introduced for the lattice Boltzmann schemes to achieve the incompressible condition;
- The proposed solver and dynamic similarity theory are successfully validated by simulating several challenging benchmark problems, including freely falling plate as shown in the figure.

A method for creating static (e.g., stationary) error covariance of reduced rank for potential use in hybrid variational-ensemble data assimilation is presented. The choice of reduced rank versus full rank static error covariance is made in order to allow the use of an improved Hessian preconditioning in high-dimensional applications. In particular, this method relies on using block circulant matrices to create a high-dimensional global covariance matrix from a low-dimensional local sub-matrix. Although any covariance used in variational data assimilation would be an acceptable choice for the pre-defined full-rank static error covariance, for convenience and simplicity, we use a symmetric Topelitz matrix as a prototype of static error covariance. The methodology creates a square root covariance, which has a practical advantage for Hessian preconditioning in reduced rank, ensemble-based data assimilation. The experiments conducted examine multivariate covariance that includes the impact of cross-variable correlations, in order to have a more realistic assessment of the value of the constructed static error covariance approximation. The results show that it may be possible to reduce the rank of matrix to *O*(10) and still obtain an acceptable approximation of the full-rank static covariance matrix. Copyright © 2016 John Wiley & Sons, Ltd.

A method for creating static error covariance of reduced rank for potential use in hybrid variational-ensemble data assimilation is presented, based on the use of singular value decomposition and circulant matrices. The main benefit of the reduced rank error covariance is in improving the Hessian preconditioning in high-dimensional applications. The results show that it may be possible to reduce the rank of matrix from *O*(10^{5}) to *O*(10) and still obtain an acceptable approximation of the full-rank static covariance matrix.

In several settings, diffusive behavior is observed to not follow the rate of spread predicted by parabolic partial differential equations (PDEs) such as the heat equation. Such behaviors, often referred to as anomalous diffusion, can be modeled using nonlocal equations for which points at a finite distance apart can interact. An example of such models is provided by fractional derivative equations. Because of the nonlocal interactions, discretized nonlocal systems have less sparsity, often significantly less, compared with corresponding discretized PDE systems. As such, the need for reduced-order surrogates that can be used to cheaply determine approximate solutions is much more acute for nonlocal models compared with that for PDEs. In this paper, we consider the construction, application, and testing of proper orthogonal decomposition (POD) reduced models for an integral equation model for nonlocal diffusion. For certain modeling parameters, the model we consider allows for discontinuous solutions and includes fractional Laplacian kernels as a special case. Preliminary computational results illustrate the potential of using POD to obtain accurate approximations of solutions of nonlocal diffusion equations at much lower costs compared with, for example, standard finite element methods. Copyright © 2016 John Wiley & Sons, Ltd.

We present a novel reduced-order approach to the one-dimensional nonlocal anomalous diffusion problem. Results show good convergence of the POD-ROM method at approximating the nonlocal solution in a few different problem settings.

An efficient adjoint sensitivity technique for optimally collecting targeted observations is presented. The targeting technique incorporates dynamical information from the numerical model predictions to identify when, where and what types of observations would provide the greatest improvement to specific model forecasts at a future time. A functional (goal) is defined to measure what is considered important in modelling problems. The adjoint sensitivity technique is used to identify the impact of observations on the predictive accuracy of the functional, then placing the sensors at the locations with high impacts. The adaptive (goal) observation technique developed here has the following features: (i) over existing targeted observation techniques, its novelty lies in that the interpolation error of numerical results is introduced to the functional (goal), which ensures the measurements are a distance apart; (ii) the use of proper orthogonal decomposition (POD) and reduced order modelling for both the forward and backward simulations, thus reducing the computational cost; and (iii) the use of unstructured meshes.

The targeted adaptive observation technique is developed here within an unstructured mesh finite element model (Fluidity). In this work, a POD reduced order modelling is used to form the reduced order forward model by projecting the original complex model from a high dimensional space onto a reduced order space. The reduced order adjoint model is then constructed directly from the reduced order forward model. This efficient adaptive observation technique has been validated with two test cases: a model of an ocean gyre and a model of 2D urban street canyon flows. Copyright © 2016 John Wiley & Sons, Ltd.

An efficient adjoint sensitivity technique for optimally collecting targeted observations is presented. The targeting technique incorporates dynamical information from the numerical model predictions to identify when, where, and what types of observations would provide the greatest improvement to specific model forecasts at a future time. A functional (goal) is defined to measure what is considered important in modelling problems. The adjoint sensitivity technique is used to identify the impact of observations on the predictive accuracy of the functional, then placing the sensors at the locations with high impacts. The adaptive (goal) observation technique developed here has the following features: (1) over existing targeted observation techniques, its novelty lies in that the interpolation error of numerical results is introduced to the functional (goal) which ensures the measurements are a distance apart; (2) the use of proper orthogonal decomposition (POD) and reduced order modelling (ROM) for both the forward and backward simulations, thus reducing the computational cost; and (3) the use of unstructured meshes.

In this paper, we present a Bayesian framework for estimating joint densities for large eddy simulation (LES) sub-grid scale model parameters based on canonical forced isotropic turbulence direct numerical simulation (DNS) data. The framework accounts for noise in the independent variables, and we present alternative formulations for accounting for discrepancies between model and data. To generate probability densities for flow characteristics, posterior densities for sub-grid scale model parameters are propagated forward through LES of channel flow and compared with DNS data. Synthesis of the calibration and prediction results demonstrates that model parameters have an explicit filter width dependence and are highly correlated. Discrepancies between DNS and calibrated LES results point to additional model form inadequacies that need to be accounted for. Copyright © 2016 John Wiley & Sons, Ltd.

We present a Bayesian framework for estimating joint densities for large eddy simulation sub-grid scale model parameters based on canonical forced isotropic turbulence direct numerical simulation data. Posterior densities for sub-grid scale model parameters are then propagated forward through large eddy simulation of channel flow and compared to channel flow direct numerical simulation data.

Numerical oscillation has been an open problem for high-order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods and thus are not able to make full use of the sub-cell information available in the local high-order reconstructions. This paper presents a novel algorithm that introduces a nodal value-based weighted essentially non-oscillatory limiter for constrained interpolation profile/multi-moment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222 (2007), 849–871) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP-CSL-WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the weighted essentially non-oscillatory (WENO) reconstruction that makes use of the sub-cell information available from the local DOFs and is built from the point values at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity, and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the fourth-order accuracy and oscillation-suppressing property of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a novel algorithm that introduces a nodal value-based WENO limiter for CIP/MM FVM as a trial to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP-CSL-WENO4 scheme, which is free of the *ad hoc* TVB ‘trouble cell’ indicator can achieve superior accuracy compared with Eulerian formulation due to its semi-Lagrangian nature. The numerical results of benchmark tests show excellent solution quality compared with other existing schemes.

This paper presents a non-intrusive reduced order model for general, dynamic partial differential equations. Based upon proper orthogonal decomposition (POD) and Smolyak sparse grid collocation, the method first projects the unknowns with full space and time coordinates onto a reduced POD basis. Then we introduce a new least squares fitting procedure to approximate the dynamical transition of the POD coefficients between subsequent time steps, taking only a set of full model solution snapshots as the training data during the construction. Thus, neither the physical details nor further numerical simulations of the original PDE model are required by this methodology, and the level of non-intrusiveness is improved compared with existing reduced order models. Furthermore, we take adaptive measures to address the instability issue arising from reduced order iterations of the POD coefficients. This model can be applied to a wide range of physical and engineering scenarios, and we test it on a couple of problems in fluid dynamics. It is demonstrated that this reduced order approach captures the dominant features of the high-fidelity models with reasonable accuracy while the computation complexity is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a non-intrusive reduced order model (NIROM) for general, dynamic partial differential equations. Based upon proper orthogonal decomposition (POD) and Smolyak sparse grid collocation, the method first projects the unknowns with full space and time coordinates onto a reduced POD basis. Then we introduce a new least squares fitting procedure to approximate the dynamical transition of the POD coefficients between subsequent time steps, taking only a set of full model solution snapshots as the input. Thus, the physics and numerics of the original PDE model are fully transparent to this methodology, and its level of non-intrusiveness is improved compared with existing reduced order models. Furthermore, we take adaptive measures to address the instability issue arising from reduced order iterations of the POD coefficients. This model can be applied to a wide range of physical and engineering scenarios, and we test it on a couple of problems in fluid dynamics. It is demonstrated that this reduced order approach captures the dominant features of the high fidelity models with reasonable accuracy while the computation complexity is reduced by several orders of magnitude.

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. If the model is ‘perfect,’ the optimal solution (analysis) error rises because of the presence of the input data errors (background and observation errors). Then, this error is quantified by the covariance matrix, which can be approximated by the inverse Hessian of an auxiliary control problem. If the model is not perfect, the optimal solution error includes an additional component because of the presence of the model error. In this paper, we study the influence of the model error on the optimal solution error covariance, considering strong and weak constraint data assimilation approaches. For the latter, an additional equation describing the model error dynamics is involved. Numerical experiments for the 1D Burgers equation illustrate the presented theory. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents a generic methodology for assessing the optimal solution error covariance matrix in data assimilation involving imperfect models. We consider both the strong constraint and weak constraint variational data assimilation formulations. The later includes a dynamical model describing the model error evolution. In the first case, the covariance is approximated by the inverse Hessian, whereas in the second case, a special formula has been derived. The theory is verified by numerical tests involving the one-dimensional Burgers' equation.

Three numerical methods, namely, volume of fluid (VOF), simple coupled volume of fluid with level set (S-CLSVOF), and S-CLSVOF with the density-scaled balanced continuum surface force (CSF) model, have been incorporated into OpenFOAM source code and were validated for their accuracy for three cases: (i) an isothermal static case, (ii) isothermal dynamic cases, and (iii) non-isothermal dynamic cases with thermocapillary flow including dynamic interface deformation. Results have shown that the S-CLSVOF method gives accurate results in the test cases with mild computation conditions, and the S-CLSVOF technique with the density-scaled balanced CSF model leads to accurate results in the cases of large interface deformations and large density and viscosity ratios. These show that these high accuracy methods would be appropriate to obtain accurate predictions in multiphase flow systems with thermocapillary flows. Copyright © 2016 John Wiley & Sons, Ltd.

Three numerical methods for multiphase flow with thermocapillary flow were validated for their accuracy by using OpenFOAM: volume of fluid, simple coupled volume of fluid with level set (S-CLSVOF), and S-CLSVOF with density-scaled balanced continuum surface force (CSF) model. Results have shown that the S-CLSVOF method gives accurate results, and S-CLSVOF method with density-scaled balanced CSF model leads to accurate results in the cases of large interface deformations and large density and viscosity ratios.

In this article, we describe a non-intrusive reduction method for porous media multiphase flows using Smolyak sparse grids. This is the first attempt at applying such an non-intrusive reduced-order modelling (NIROM) based on Smolyak sparse grids to porous media multiphase flows. The advantage of this NIROM for porous media multiphase flows resides in that its non-intrusiveness, which means it does not require modifications to the source code of full model. Another novelty is that it uses Smolyak sparse grids to construct a set of hypersurfaces representing the reduced-porous media multiphase problem. This NIROM is implemented under the framework of an unstructured mesh control volume finite element multiphase model. Numerical examples show that the NIROM accuracy relative to the high-fidelity model is maintained, whilst the computational cost is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.

In this article, we describe a non-intrusive reduction method for porous media multiphase flows using Smolyak sparse grids.This is the first attempt at applying such an non-intrusive reduced order modelling (NIROM) based on Smolyak sparse grids to porous media multiphase flows. This NIROM is implemented under the framework of an unstructured mesh control volume finite element multiphase model. Numerical examples show that the NIROM accuracy relative to the high-fidelity model is maintained, whilst the computational cost is reduced by several orders of magnitude.

The figures displayed earlier (left) show the saturation solutions of the high-permeability domain embedded in a low-permeability domain problem at time instances 0.05. The solutions compare the predictions from non-intrusive reduced order model with high-fidelity full model using 36 proper orthogonal decomposition basis functions.

This paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The sparse algorithm developed herein relies on the discrete empirical interpolation method and uses only samples of the nonzero entries of the matrix series. The proposed approach can approximate very large matrices, unlike the current matrix discrete empirical interpolation method, which is limited by its large computational memory requirements. The empirical interpolation indices obtained by the sparse algorithm slightly differ from the ones computed by the matrix discrete empirical interpolation method as a consequence of the singular vectors round-off errors introduced by the economy or full singular value decomposition (SVD) algorithms when applied to the full matrix snapshots. When appropriately padded with zeros, the economy SVD factorization of the nonzero elements of the snapshots matrix is a valid economy SVD for the full snapshots matrix. Numerical experiments are performed with the 1D Burgers and 2D shallow water equations test problems where the quadratic reduced nonlinearities are computed via tensorial calculus. The sparse matrix approximation strategy is compared against five existing methods for computing reduced Jacobians: (i) matrix discrete empirical interpolation method, (ii) discrete empirical interpolation method, (iii) tensorial calculus, (iv) full Jacobian projection onto the reduced basis subspace, and (v) directional derivatives of the model along the reduced basis functions. The sparse matrix method outperforms all other algorithms. The use of traditional matrix discrete empirical interpolation method is not possible for very large dimensions because of its excessive memory requirements. Copyright © 2016 John Wiley & Sons, Ltd.

This paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The method uses only samples of the nonzero entries of the matrix series. The sparse matrix approximation strategy is compared against various existing methods for computing reduced Jacobians in the case of the 1D Burgers and 2D shallow water equations models.

This study presents an improved ghost-cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work, ghost cells are mirrored through the boundary described using a level-set method to farther image points, incorporating a higher-order extra/interpolation scheme for the ghost-cell values. A sensor is introduced to deal with image points near the discontinuities in the flow field. Adaptive mesh refinement is used to improve the representation of the geometry efficiently in the Cartesian grid system. The improved ghost-cell method is validated against four test cases: (a) double Mach reflections on a ramp, (b) smooth Prandtl–Meyer expansion flows, (c) supersonic flows in a wind tunnel with a forward-facing step, and (d) supersonic flows over a circular cylinder. It is demonstrated that the improved ghost-cell method can reach the accuracy of second order in *L*^{1} norm and higher than first order in *L*^{∞} norm. Direct comparisons against the cut-cell method demonstrate that the improved ghost-cell method is almost equally accurate with better efficiency for boundary representation in high-fidelity compressible flow simulations. Copyright © 2016 John Wiley & Sons, Ltd.

We present an improved ghost-cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work, ghost cells are mirrored through the boundary described using a level-set method to farther image points, incorporating a higher-order extrapolation/interpolation scheme for the ghost-cell values. Direct comparisons against the cut-cell method demonstrate that the present method is almost equally accurate with better efficiency for boundary representation in high-fidelity compressible flow simulations.

This work honors the 75th birthday of Professor Ionel Michael Navon by presenting original results highlighting the computational efficiency of the adjoint sensitivity analysis methodology for function-valued operator responses by means of an illustrative paradigm dissolver model. The dissolver model analyzed in this work has been selected because of its applicability to material separations and its potential role in diversion activities associated with proliferation and international safeguards. This dissolver model comprises eight active compartments in which the 16 time-dependent nonlinear differential equations modeling the physical and chemical processes comprise 619 scalar and time-dependent model parameters, related to the model's equation of state and inflow conditions. The most important response for the dissolver model is the time-dependent nitric acid in the compartment furthest away from the inlet, where measurements are available at 307 time instances over the transient's duration of 10.5 h. The sensitivities to all model parameters of the acid concentrations at each of these instances in time are computed efficiently by applying the adjoint sensitivity analysis methodology for operator-valued responses.

The uncertainties in the model parameters are propagated using the above-mentioned sensitivities to compute the uncertainties in the computed responses. A predictive modeling formalism is subsequently used to combine the computational results with the experimental information measured in the compartment furthest from the inlet and then predict optimal values and uncertainties throughout the dissolver. This predictive modeling methodology uses the maximum entropy principle to construct an optimal approximation of the unknown *a priori* distribution for the *a priori* known mean values and uncertainties characterizing the model parameters and the computed and experimentally measured model responses. This approximate *a priori* distribution is subsequently combined using Bayes' theorem with the “likelihood” provided by the multi-physics computational models. Finally, the posterior distribution is evaluated using the saddle-point method to obtain analytical expressions for the optimally predicted values for the parameters and responses of both multi-physics models, along with corresponding reduced uncertainties. This work shows that even though the experimental data pertains solely to the compartment furthest from the inlet (where the data were measured), the predictive modeling procedure used herein actually improves the predictions and reduces the predicted uncertainties for the entire dissolver, including the compartment furthest from the measurements, because this predictive modeling methodology combines and transmits information simultaneously over the entire phase-space, comprising all time steps and spatial locations. Copyright © 2016 John Wiley & Sons, Ltd.

The predictive modeling methodology developed by Cacuci and Ionescu-Bujor (2010) is applied to a spent nuclear fuel dissolver model to obtain best-estimate values for predicted model responses (e.g., acid concentrations) and parameters (e.g., time-dependent inlet boundary conditions), with reduced predicted uncertainties. The adjoint sensitivity analysis methodology for operator-valued responses developed by Cacuci (1981) is used for computing most efficiently the response sensitivities needed for the accompanying uncertainty quantification, data assimilation, and model calibration.

We suggest a new set of equations to employ smoothed particle hydrodynamics (SPH) in a curvilinear space, and we refer to it as curvSPH. In classical SPH, the horizontal and vertical resolution of discretization is supposed to be equal for fluid particles. However, curvSPH makes the horizontal and vertical resolutions independent from each other. This is performed by transformation of physical space into an appropriate computational space with a different scale in horizontal and vertical directions. Solving a problem using SPH in a curvilinear space also provides capability to model curved boundaries as straight lines. In classical SPH, special care is needed to reach a uniform mass distribution along curved boundaries; however, producing uniform mass distribution along a line using curvSPH is straight forward. Different simulations, including simulation of a flip bucket are performed to demonstrate the applicability of the proposed method. Good agreement of results with experimental data and classical SPH confirms the capabilities of curvSPH. Copyright © 2016 John Wiley & Sons, Ltd.

We suggest a new set of equations to employ smoothed particle hydrodynamics (SPH) in a curvilinear space, and we refer to it as curvSPH. The new method makes the horizontal and vertical resolutions independent from each other. It also provides capability to model curved boundaries as straight lines. Different simulations, including simulation of a flip bucket are performed to demonstrate the applicability of the proposed method. Good agreement of results with experimental data and classical SPH confirms the capabilities of curvSPH.

No abstract is available for this article.

]]>A numerical model based on the smoothed particle hydrodynamics method is developed to simulate depth-limited turbulent open channel flows over hydraulically rough beds. The 2D Lagrangian form of the Navier–Stokes equations is 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 model is applied to account for the effect of turbulence. The sub-particle-scale model is constructed based on the mixing-length assumption rather than the standard Smagorinsky approach to compute the eddy-viscosity. A 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 flows 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 smoothed particle hydrodynamics model accomplishes this successfully. © 2016 The Authors International Journal for Numerical Methods in Fluids Published by John Wiley & Sons Ltd

We have significantly improved the turbulence modelling and rough boundary treatment to enable the smoothed particle hydrodynamics method to work in depth-limited open channel uniform flows over a rough bed surface with a robust technique for the inflow and outflow boundaries. The computed velocity and shear stress profiles are found to be in good agreement with the experimental data measured in a laboratory flume with a well-packed bed of uniform-sized spheres.

Hybrid Monte Carlo sampling smoother is a fully non-Gaussian four-dimensional data assimilation algorithm that works by directly sampling the posterior distribution formulated in the Bayesian framework. The smoother in its original formulation is computationally expensive owing to the intrinsic requirement of running the forward and adjoint models repeatedly. Here we present computationally efficient versions of the hybrid Monte Carlo sampling smoother based on reduced-order approximations of the underlying model dynamics. The schemes developed herein are tested numerically using the shallow-water equations model on Cartesian coordinates. The results reveal that the reduced-order versions of the smoother are capable of accurately capturing the posterior probability density, while being significantly faster than the original full-order formulation. Copyright © 2016 John Wiley & Sons, Ltd.

We introduce computationally efficient versions of the hybrid Monte Carlo sampling smoother based on reduced-order approximations of the underlying model dynamics. These reduced versions are capable of accurately capturing the posterior probability density while being significantly faster than the original full-order formulation. The proposed methods are sampling a fully projected posterior and the high-fidelity posterior distribution with approximate gradient using a reduced-order model.

This paper presents a quantitative risk assessment for design and development of a renewable energy system to support decision-making among design alternatives. Throughout the decision-making phases, resources are allocated among exploration and exploitation tasks to manage the uncertainties in design parameters and to adapt designs to new information for enhanced performance. The resource allocation problem is formulated as a sequential decision feedback loop for a quantitative analysis of exploration and exploitation trade-offs. We support decision-making by tracking the evolution of uncertainties, the sensitivity of design alternatives to the uncertainties, and the performance, reliability, and robustness of each design. This is achieved by analyzing the uncertainties in the wind resource, the turbine performance and operation, and the models that define the power curve and wake deficiency. Comparison of the performance, reliability, and robustness of aligned and staggered turbine layouts before and after wind assessment experiments aids in improving micro-siting decisions. The results demonstrate that design decisions can be supported by efficiently allocating resources towards improved estimates of achievable design objectives and by quantitatively assessing the risk in meeting those objectives. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a quantitative risk assessment for design and development of a renewable energy system to support decision-making among design alternatives. Throughout the decision-making phases, resources are allocated amongst exploration and exploitation tasks to manage the uncertainties in design parameters and to adapt designs to new information for enhanced performance. The resource allocation problem is formulated as a sequential decision feedback loop that is guided by global and regional sensitivity analyses.

In this paper, a new vector-filtering criterion for dynamic modes selection is proposed that is able to extract dynamically relevant flow features from dynamic mode decomposition of time-resolved experimental or numerical data. We employ a novel modes selection criterion in parallel with the classic selection based on modes amplitudes, in order to analyze which of these procedures better highlight the coherent structures of the flow dynamics. Numerical tests are performed on two distinct problems. The efficiency of the proposed criterion is proved in retaining the most influential modes and reducing the size of the dynamic mode decomposition model. By applying the proposed filtering mode technique, the flow reconstruction error is shown to be significantly reduced. Copyright © 2016 John Wiley & Sons, Ltd.

We propose a new criterion for dynamic modes selection that is able to extract dynamically relevant flow features from dynamic mode decomposition of time-resolved experimental or numerical data. We employ a novel modes selection criterion in parallel with the classic selection based on modes amplitudes. Numerical tests are performed on two distinct problems. The efficiency of the proposed criterion is proved in retaining the most influential modes and reducing the size of the dynamic mode decomposition model.

This paper constructs an ensemble-based sampling smoother for four-dimensional data assimilation using a Hybrid/Hamiltonian Monte-Carlo approach. The smoother samples efficiently from the posterior probability density of the solution at the initial time. Unlike the well-known ensemble Kalman smoother, which is optimal only in the linear Gaussian case, the proposed methodology naturally accommodates non-Gaussian errors and nonlinear model dynamics and observation operators. Unlike the four-dimensional variational method, which only finds a mode of the posterior distribution, the smoother provides an estimate of the posterior uncertainty. One can use the ensemble mean as the minimum variance estimate of the state or can use the ensemble in conjunction with the variational approach to estimate the background errors for subsequent assimilation windows. Numerical results demonstrate the advantages of the proposed method compared to the traditional variational and ensemble-based smoothing methods. Copyright © 2016 John Wiley & Sons, Ltd.

We introduce an ensemble-based sampling smoother for four-dimensional data assimilation using a Hybrid/Hamiltonian Monte-Carlo approach. The Hybrid/Hamiltonian Monte-Carlo sampling smoother naturally accommodates non-Gaussian errors and nonlinear model dynamics and observation operators. The proposed methodology can provide a consistent and accurate approximation of the posterior distribution in the non-Gaussian data assimilation framework.