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 WENO 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. This article is protected by copyright. All rights reserved.

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. This article is protected by copyright. All rights reserved.

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 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. Due to these properties, it is expected to be widely applied to all speed flow studies. This article is protected by copyright. All rights reserved.

The Finite Volume discretization of non-linear 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 enables 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. This article is protected by copyright. All rights reserved.

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. In order to examine and quantify the amount of false diffusion associated with the often-recommended and versatile second-order upwind (SOU) 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

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. This article is protected by copyright. All rights reserved.

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.

In this paper, we propose a monolithic approach for reduced-order modeling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition–Galerkin method. Parameters of the problem are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time. We provide a detailed description of the parametrized formulation of the multiphysics problem in its components, together with some insights on how to obtain an offline–online efficient computational procedure through the approximation of parametrized nonlinear tensors. Then, we present the monolithic proper orthogonal decomposition–Galerkin method for the online computation of the global structural displacement, fluid velocity, and pressure of the coupled problem. Finally, we show some numerical results to highlight the capabilities of the proposed reduced-order method and its computational performances. Copyright © 2016 John Wiley & Sons, Ltd.

We propose a monolithic approach for reduced order modeling of parametrized fluid-structure interaction problems based on a proper orthogonal decomposition—Galerkin method. Parameters are related to constitutive properties of the fluid or structural problem, or to geometrical parameters related to the domain configuration at the initial time. The parametrized formulation of the multiphysics problem, and efficient offline—online computational procedure, are introduced. Several numerical results highlight the capabilities of the proposed reduced order method and its computational performances.

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.

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.

In this article, we present a higher-order finite volume method with a ‘Modified Implicit Pressure Explicit Saturation’ (MIMPES) formulation to model the 2D incompressible and immiscible two-phase flow of oil and water in heterogeneous and anisotropic porous media. We used a median-dual vertex-centered finite volume method with an edge-based data structure to discretize both, the elliptic pressure and the hyperbolic saturation equations. In the classical IMPES approach, first, the pressure equation is solved implicitly from an initial saturation distribution; then, the velocity field is computed explicitly from the pressure field, and finally, the saturation equation is solved explicitly. This saturation field is then used to re-compute the pressure field, and the process follows until the end of the simulation is reached. Because of the explicit solution of the saturation equation, severe time restrictions are imposed on the simulation. In order to circumvent this problem, an edge-based implementation of the MIMPES method of Hurtado and co-workers was developed. In the MIMPES approach, the pressure equation is solved, and the velocity field is computed less frequently than the saturation field, using the fact that, usually, the velocity field varies slowly throughout the simulation. The solution of the pressure equation is performed using a modification of Crumpton's two-step approach, which was designed to handle material discontinuity properly. The saturation equation is solved explicitly using an edge-based implementation of a modified second-order monotonic upstream scheme for conservation laws type method. Some examples are presented in order to validate the proposed formulation. Our results match quite well with others found in literature. Copyright © 2016 John Wiley & Sons, Ltd.

In this article, we present a higher-order finite volume method with a ‘Modified Implicit Pressure, Explicit Saturation’ formulation to model the 2D incompressible and immiscible two-phase flow of oil and water in heterogeneous and anisotropic porous media. Our higher-order formulation produces very accurate solutions with a sharp front resolution and less grid orientation effects than the traditional first-order upwind method at a reasonable computational cost.

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 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.

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.

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.

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

For the 2D Burgers equation with large Reynolds number (turbulent flow case), we have developed the proper orthogonal decomposition/discrete empirical interpolation method-reduced order model and provided detailed solution. A flow calibration with Tikhonov regularization serving as closure model is also carried out in order to recover the turbulent closure. The computational results exhibit considerable agreement with the real high-fidelity model.

We present a spectral-element discontinuous Galerkin thermal lattice Boltzmann method for fluid–solid conjugate heat transfer applications. Using the discrete Boltzmann equation, we propose a numerical scheme for conjugate heat transfer applications on unstructured, non-uniform grids. We employ a double-distribution thermal lattice Boltzmann model to resolve flows with variable Prandtl (*P**r*) number. Based upon its finite element heritage, the spectral-element discontinuous Galerkin discretization provides an effective means to model and investigate thermal transport in applications with complex geometries. Our solutions are represented by the tensor product basis of the one-dimensional Legendre–Lagrange interpolation polynomials. A high-order discretization is employed on body-conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Thermal and hydrodynamic bounce-back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. As a result, our scheme does not require tedious extrapolation at the boundaries, which may cause loss of mass conservation. We compare solutions of the proposed scheme with an analytical solution for a solid–solid conjugate heat transfer problem in a 2D annulus and illustrate the capture of temperature continuities across interfaces for conductivity ratio *γ* > 1. We also investigate the effect of Reynolds (*R**e*) and Grashof (*G**r*) number on the conjugate heat transfer between a heat-generating solid and a surrounding fluid. Steady-state results are presented for *R**e* = 5−40 and *G**r* = 10^{5}−10^{6}. In each case, we discuss the effect of *R**e* and *G**r* on the heat flux (i.e. Nusselt number *N**u*) at the fluid–solid interface. Our results are validated against previous studies that employ finite-difference and continuous spectral-element methods to solve the Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.

This graphic shows isotherms for Gr = 10^{6} in a horizontal annulus using the proposed spectral-element discontinuous Galerkin thermal lattice Boltzmann method. Using the discrete Boltzmann equations for nearly incompressible, thermal flows, the spectral-element discontinuous Galerkin thermal lattice Boltzmann method is able to solve fluid–solid conjugate heat transfer applications on unstructured, non-uniform grids. Bounce-back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. This scheme does not require tedious extrapolation at the boundaries that may cause loss of mass conservation.

This paper presents a novel mass conservative, positivity preserving wetting and drying treatment for Godunov-type shallow water models with second-order bed elevation discretization. The novel method allows to compute water depths equal to machine accuracy without any restrictions on the time step or any threshold that defines whether the finite volume cell is considered to be wet or dry. The resulting scheme is second-order accurate in space and keeps the C-property condition at fully flooded area and also at the wet/dry interface. For the time integration, a second-order accurate Runge–Kutta method is used. The method is tested in two well-known computational benchmarks for which an analytical solution can be derived, a C-property benchmark and in an additional example where the experimental results are reproduced. Overall, the presented scheme shows very good agreement with the reference solutions. The method can also be used in the discontinuous Galerkin method. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents a new numerical method for the shallow water equations. The article describes computing of the bed slope source term, which is well balanced not only in the flooded domain but also in the wet/dry interface. The scheme is capable to compute the flow of the water depth approaching zero value without loss of the accuracy. Moreover, the resulting scheme is also mass conservative.

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.

In this study, a first attempt has been made to introduce mesh adaptivity into the ensemble Kalman fiter (EnKF) method. The EnKF data assimilation system was established for an unstructured adaptive mesh ocean model (Fluidity, Imperial College London). The mesh adaptivity involved using high resolution mesh at the regions of large flow gradients and around the observation points in order to reduce the representativeness errors of the observations. The use of adaptive meshes unavoidably introduces difficulties in the implementation of EnKF. The ensembles are defined at different meshes. To overcome the difficulties, a supermesh technique is employed for generating a reference mesh. The ensembles are then interpolated from their own mesh onto the reference mesh. The performance of the new EnKF data assimilation system has been tested in the Munk gyre flow test case. The discussion of this paper will focus on (a) the development of the EnKF data assimilation system within an adaptive mesh model and (b) the advantages of mesh adaptivity in the ocean data assimilation model. Copyright © 2016 John Wiley & Sons, Ltd.

The adaptive mesh ensemble Kalman filter data assimilation system was established and tested in this work. The unstructured mesh was adapted with respect to both the state variable and the observation locations. The conservative mesh generation technique ‘supermesh’ was adopted to deal with the different meshes on which the ensembles were defined. It is proved that the adaptive mesh ensemble Kalman filter data assimilation system had a positive effect on the model results.

A novel high-order finite volume scheme using flux correction methods in conjunction with structured finite differences is extended to low Mach and incompressible flows on strand grids. Flux correction achieves a high order by explicitly canceling low-order truncation error terms across finite volume faces and is applied in unstructured layers of the strand grid. The layers are then coupled together using a source term containing summation-by-parts finite differences in the strand direction. A preconditioner is employed to extend the method to low speed and incompressible flows. We further extend the method to turbulent flows with the Spalart–Allmaras model. Laminar flow test cases indicate improvements in accuracy and convergence using the high-order preconditioned method, while turbulent body-of-revolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we address a number of challenges associated with the computation of low-speed and incompressible flows through the use of automated strand grid generation, unique high-order methods, and preconditioning. We explore a preconditioned flux correction method for unstructured layers of the strand grid coupled together using a source term containing summation-by-parts finite differences in the strand direction. Laminar flow test cases indicate dramatic improvements in accuracy and convergence using the high-order preconditioned method, while turbulent body-ofrevolution flow results show improvements in only some cases, perhaps because of dominant errors arising from the turbulence model itself.

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

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

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

This paper presents a stabilized finite element method (FEM) for the acoustic perturbation equations (APE) at low Mach numbers. The proposed stabilized formulation allows one to retain all convective and reaction terms in the APE and to deal with acoustic waves propagating in solenoidal mean flows with non-uniform convection and shear. The numerical examples reveal the contributions of the various terms in the APE and the importance not to neglecting them in many aeroacoustic problems.

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

Data assimilation algorithms derived from optimal estimation theory hinge on the background error covariance. Numerical analyses show that the background error correlation length scale is about 75 km for the streamfunction (left), even with a model at a horizontal resolution of 2 km. With this correlation scale, the spectral power density of the background errors are virtually zero for scales smaller than 150 km, which is twice the correlation length scale (right). Thus, data assimilation algorithms are unable to correct background errors at least for horizontal scales smaller than the twice correlation length scale. A multiscale variational data assimilation scheme is suggested to estimate distinct scales separately for high-resolution models.

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

This study is the first to formulate equations for sound power sensitivity on structural surfaces based on an adjoint operator approach and equations for sound power sensitivity on arbitrary closed surfaces around the radiator based on the direct differentiation approach. Discontinuous higherorder boundary elements are developed for the acoustic domain to achieve higher accuracy in the coupling analysis.

Rhie–Chow interpolation is a commonly used method in CFD calculations on a co-located mesh in order to suppress non-physical pressure oscillations arising from chequerboard effects. A fully parallelized smoothed-interface immersed boundary method on a co-located grid is described in this paper. We discuss the necessity of modifications to the original Rhie–Chow interpolation in order to deal with a locally refined mesh. Numerical simulation with the modified scheme of Choi shows that numerical dissipation due to Rhie–Chow interpolation introduces significant errors at the immersed boundary. To address this issue, we develop an improved Rhie–Chow interpolation scheme that is shown to increase the accuracy in resolving the flow near the immersed boundary. We compare our improved scheme with the modified scheme of Choi by parallel simulations of benchmark flows: (i) flow past a stationary cylinder; (ii) flow past an oscillating cylinder; and (iii) flow past a stationary elliptical cylinder, where Reynolds numbers are tested in the range 10–200. Our improved scheme is significantly more accurate and compares favourably with a staggered grid algorithm. We also develop a scheme to compute the boundary force for the direct-forcing immersed boundary method efficiently. Copyright © 2016 John Wiley & Sons, Ltd.

A fully parallelized smoothed-interface immersed boundary method on a co-located grid is described. An improved Rhie–Chow interpolation scheme is proposed to increase the accuracy in resolving the flow near the immersed boundary. It is validated by benchmark test results of flow past a stationary/oscillating cylinder.

We propose a novel fitted finite element method for two-phase Stokes flow problems that uses piecewise linear finite elements to approximate the moving interface. The method can be shown to be unconditionally stable. Moreover, spherical stationary solutions are captured exactly by the numerical approximation. In addition, the meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations, a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments for our numerical method, which demonstrate the accuracy and robustness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

We propose an unconditionally stable fitted finite element method for two-phase Stokes flow that uses piecewise linear finite elements to approximate the moving interface. Our numerical approximation captures exactly spherical stationary solutions. Moreover, the meshes describing the discrete interface in general do not deteriorate in time. Therefore, it is not necessary to smooth or to remesh them during the numerical simulations. We present several numerical experiments which demonstrate the accuracy and robustness of the proposed algorithm.

This paper presents an efficient procedure for overcoming the deficiency of weighted essentially non-oscillatory schemes near discontinuities. Through a thorough incorporation of smoothness indicators into the weights definition, up to ninth-order accurate multistep methods are devised, providing weighted essentially non-oscillatory schemes with enhanced order of convergence at transition points from smooth regions to a discontinuity, while maintaining stability and the essentially non-oscillatory behavior. We also provide a detailed analysis of the resolution power and show that the solution enhancements of the new method at smooth regions come from their ability to render smoothness indicators closer to uniformity. The new scheme exhibits similar fidelity as other multistep schemes; however, with superior characteristics in terms of robustness and efficiency, as no logical statements or mapping function is needed. Extensions to higher orders of accuracy present no extra complexity. Numerical solutions of linear advection problems and nonlinear hyperbolic conservation laws are used to demonstrate the scheme's improved behavior for shock-capturing problems. Copyright © 2016 John Wiley & Sons, Ltd.

A new class of multistep WENO methods is presented through using new modified nonlinear weights. The weights definition takes into account the novel extra information on the regularity of the solution and renders smoothness indicators closer to uniformity so as to increase the resolution power when approximating smooth solutions. This new method provides WENO schemes with enhanced order of convergence at transition points while maintaining stability and the ENO behavior.

Numerical methods based on geometrical multiscale models of blood flows solve for averaged flow statistics on a network of vessels while providing more detailed information about fluid dynamics in a specific region of interest. In such an approach, a 3D model based on the Navier–Stokes equations posed in a domain with rigid walls is often used to describe blood flow dynamics in the refined region. While ignoring elasticity effects in 3D models is plausible in certain applications and saves computational time significantly, coupling such models with 1D flow models may result in non-physiological phenomena in the computed solutions. Thus, the immediate coupling conditions based on continuity of normal stresses, flow rate, pressure, or a combination of thereof do not account for the inconsistency between elasticity effects in the 1D model and the non-compliance of the 3D model. In this paper, we introduce and study an auxiliary absorbing 0D model, which is placed at the interface between 1D and 3D models. A virtual device mimics the effect of the 3D model compliance and hence reduces pressure wave reflection and instabilities caused by the inconsistency. The absorbing model is developed from basic mechanical principles. As a result, parameters of the 0D model can be designed based on hemodynamic data. We analyze the stability of the geometrical multiscale model and perform several numerical experiments to assess its computational efficiency. Copyright © 2016 John Wiley & Sons, Ltd.

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

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas-kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non-manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.

Coupling gas-kinetic scheme with Spalart–Allmaras turbulence model, a finite volume method is introduced for the solution of turbulent flow. To organize the unstructured mesh data structure efficiently, a non-manifold hybrid mesh data structure is extended for polygonal cells. The adaptive mesh refinement technique is also adopted to reduce computational cost and improve the efficiency of meshes. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flow around a NACA 0012 airfoil.

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

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

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

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

No abstract is available for this article.

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

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

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

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

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

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

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

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

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