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Abstract: In this paper, we introduce a general principle for constructing robust and accurate viscous discretization, which is applicable to various discretization methods, including finite-volume, residual-distribution, discontinuous-Galerkin, and spectral-volume methods. The principle is based on a hyperbolic model for the viscous term. It is to discretize the hyperbolic system by an advection scheme, and then derive a viscous discretization from the result. A distinguished feature of the proposed principle is that it automatically introduces a damping term into the resulting viscous scheme, which is essential for effective high-frequency error damping and, in some cases, for consistency also. In this paper, we demonstrate the general principle for the diffusion equation on uniform grids in one dimension and unstructured grids in two dimensions, for node/cell-centered finite-volume, residual-distribution, discontinuous-Galerkin, and spectral-volume methods. Numerical results are presented to verify the accuracy of the derived diffusion schemes and to illustrate the importance of the damping term for highly-skewed typical viscous grids. [Copyright &y& Elsevier]

Highlights • A vp-MPM is proposed for weakly compressible problems. • A slope limiter is employed to suppress spurious pressure oscillations. • The vp formulation is incorporated into the ICFEMP to suppress pressure oscillation. • The vp-MPM has much less extra variables than a Hu-Washizu principle based method. Abstract The weakly compressible material point method (WCMPM) suffers from volumetric-locking and numerical oscillation in modeling fluid flow and fluid-structure interaction problems. In this paper, a v-p formulation of the material point method (vp -MPM) is proposed for weakly compressible problems based on a two-field variational principle. As only the velocity v and the pressure p are the independent variables, the v - p formulation has much less extra variables than those based on the Hu-Washizu multi-field variational principle which takes the velocity, strain and stress as independent variables. The pressure is assumed independently in the control volume of each gird node. Spurious pressure oscillation reduces but still occurs at the interface of discontinuity due to large pressure gradient difference across the interface. Therefore, a slope limiter is employed to suppress the oscillation and the general interpolation functions are used to eliminate the cell-crossing error. In order to extend the method to the fluid-structure interaction problems, the v-p formulation is incorporated into the improved coupled finite element material point method. Several numerical examples are presented to validate the vp -MPM. [ABSTRACT FROM AUTHOR]

Highlights • The discrete adjoint mixing-plane is formulated and implemented on a legacy CFD code. • Automatic differentiation is used along hand-differentiation of high-level routines. • Adjoint-based sensitivities show good agreement with finite-differences. • Coupling between rows is evidenced by the computed sensitivities. • Multi-row aero-thermal problems can be efficiently handled. Abstract The use of high-fidelity computational fluid dynamics (CFD) tools in turbomachinery design has seen a continuous increase as a result of computational power growth and numerical methods improvement. These tools are often used in optimization environments, where gradient-based optimization algorithms are the most common due to their efficiency. In cases where the optimization contains a large number of design variables, the adjoint approach for calculating the gradients is beneficial, as it provides a way of obtaining function sensitivities with a computational cost that is nearly independent of the number of design variables. The interaction between adjacent blade rows is of utmost importance for the performance of multistage turbomachines. The most commonly used method to address these effects (i.e. coupling in the simulation of multiple rows) is the mixing-plane treatment, that has become a standard industrial tool in the design environment. In this paper, the formulation and implementation of an adjoint solver for multistage turbomachinery applications are presented, namely the adjoint counterpart of the mixing-plane formulation used in the direct solver. The solver is developed using the discrete ADjoint approach, where the partial derivatives required for the assembly of the adjoint system of equations are obtained using automatic differentiation tools. The sensitivity of several performance metrics relative to neighbor blade/hub row geometry and boundary conditions are shown to highlight the physical coupling in multi-row turbomachines. The verification of the adjoint multistage solver against the finite-difference approach is performed successfully with relative differences below 1 %. [ABSTRACT FROM AUTHOR]

Highlights • Seventh order finite volume compact-WENO scheme is proposed for solving 1D and 2D hyperbolic conservation laws. • The parametrized maximum principle preserving and the parametrized positivity satisfying flux limiter flux limiter is coupled with the scheme. • The one dimensional HLLC flux function is used to compute the interface fluxes. • High-resolution properties with compact stencil. Abstract In this paper, we construct a new seventh-order finite volume compact weighted essentially non-oscillatory (WENO) scheme for solving one and two dimensional hyperbolic conservation laws. We extend the main idea of fifth-order finite volume compact-WENO scheme which comes from the original WENO schemes to construct the present seventh-order scheme, where fourth-order compact stencils will be combined with nonlinear weights to get seventh-order finite volume compact-WENO scheme. The parametrized maximum principle preserving flux limiter is coupled with the seventh-order compact-WENO scheme for solving scalar hyperbolic conservation laws and the parametrized positivity satisfying flux limiter is applied to the new scheme for solving compressible Euler equations to maintain positive density and pressure. The HLLC (Harten, Lax and van Leer) flux function is used to compute the interface fluxes in the finite volume formulation. A number of 1D and 2D numerical tests are performed to demonstrate the efficiency and high-resolution properties of the proposed high-order compact scheme. [ABSTRACT FROM AUTHOR]

Highlights • A new grid system is developed to detect the areas needing high-resolution mesh. • The grid system can locally refine the grid for surface water flow simulation. • The method could accelerate the model by about 3 times without accuracy loss. • The grid system is very suitable for integrating into the shallow water flow model. • The new grid system based model could be used for efficient and accurate prediction. Abstract Aiming at improving the computational efficiency without accuracy losses for surface water flow simulation, this paper presents a structured but non-uniform grid system incorporated into a Godunov-type finite volume scheme. The proposed grid system can detect the key topographic features in the computational domain where high-resolution mesh is in need for reliably solving the shallow water equations. The mesh refinement is automatically carried out in these areas while the mesh in the rest of the domain remains coarse. The criterion determining the refinement is suggested by a dimensionless number with a fixed value of 0.2 after sensitivity analysis. Three laboratory and field-scale test cases are employed to demonstrate the performance of the model for flow simulations on the new non-uniform grids. In all of the tests, the grid system is shown to successfully generate high-resolution mesh only in those areas with abruptly changing topographic features that dominate the flooding processes. To produce numerical solutions of similar accuracy, the non-uniform grid based model is able to accelerate by about two times comparing with the fine uniform grid based counterpart. [ABSTRACT FROM AUTHOR]

Highlights • A direct-forcing IB projection method is proposed for solving FSI problems. • It is a two-stage approach with a prediction-correction process. • A virtual force is introduced and distributed on the immersed solid bodies. • This method allows larger time step than traditional direct-forcing IB methods. • Accuracy of the velocity field is super-linear in space in 1, 2 and maximum norms. Abstract In this paper, we propose a simple and novel direct-forcing immersed boundary (IB) projection method in conjunction with a prediction-correction (PC) process for simulating the dynamics of fluid-solid interaction problems, in which each immersed solid object can be stationary or moving in the fluid with a prescribed velocity. The method is mainly based on the introduction of a virtual force which is distributed only on the immersed solid bodies and appended to the fluid momentum equations to accommodate the internal boundary conditions at the immersed solid boundaries. More specifically, we first predict the virtual force on the immersed solid domain by using the difference between the prescribed solid velocity and the computed velocity, which is obtained by applying the Choi–Moin projection scheme to the incompressible Navier–Stokes equations on the entire domain including the portion occupied by the solid bodies. The predicted virtual force is then added to the fluid momentum equations as an additional forcing term and we employ the same projection scheme again to correct the velocity field, pressure and virtual force. Although this method is a two-stage approach, the computational cost of the correction stage is rather cheap, since the associated discrete linear systems need to be solved in the correction stage are same with that in the prediction stage, except the right-hand side data terms. Such a PC procedure can be iterated to form a more general method, if necessary. The current two-stage direct-forcing IB projection method has the advantage over traditional one-stage direct-forcing IB projection methods, consisting of the prediction step only, by allowing much larger time step, since traditional methods generally request quite small time step for flow field relaxed and adjusted to the solid body movement even using implicit scheme. Numerical experiments of several benchmark problems are performed to illustrate the simplicity and efficient performance of the newly proposed method. Convergence tests show that the accuracy of the velocity field is super-linear in space in all the 1-norm, 2-norm, and maximum norm. We also find that our numerical results are in very good agreement with the previous works in the literature and one correction at each time step appears to be good enough for the proposed PC procedure. [ABSTRACT FROM AUTHOR]

Highlights • Generalization of the 2014 version of the ZIBC combining RANS/LES and the IB method. • Extended presentation of the Zonal Immersed Boundary Conditions approach. • First RANS/LES and IBC application to a full space launcher at high Reynolds number. • ZDES/IBC is highly validated for both the statistical field and the spectral content. Abstract One of the next foreseen challenges in CFD consists in the capability to simulate quantitatively the spectral content of the turbulent flow around realistic geometries. In this context, the present work focuses on a new methodology named ZIBC standing for Zonal Immersed Boundary Conditions enabling to account for complex configurations at high Reynolds numbers. The numerical strategy allowing the coupling between a turbulence modeling method (e.g. RANS, URANS, ZDES, LES or DNS) and IBC (Immersed Boundary Conditions) is detailed. In this paper, the modeling method retained is the Zonal Detached Eddy Simulation (ZDES) which has reached a high level of maturity on turbulent separated flow simulations. This methodology is applied to a full space launcher configuration to assess its capability to return the interactions between the technological details, modeled with IBC, and the simplified afterbody, modeled with a body-fitted (BF) approach consisting in classical no-slip boundary conditions, in the turbulent flow field surrounding the main stage of the space launcher afterbody. The proposed method is thoroughly assessed on a realistic geometry of the European Ariane 5 launcher and the ZIBC simulation is successfully compared with the available experiments. [ABSTRACT FROM AUTHOR]

This paper presents a two phases flow model combined with a high order finite volume solver on unstructured mesh. The solver is highly conservative and preserves the sharpness of the interface without any reconstruction. Special care has been taken for boundedness preservation, as a high order scheme does not guaranty the boundedness of the volume fraction. The efficiency of the method is demonstrated with two numerical experiments: the simple advection test and the interaction between the shock and a bubble. Although experiments have been carried out with fine mesh, it is also demonstrated that the method allows satisfactory results to be obtained with coarse mesh. [ABSTRACT FROM AUTHOR]

Abstract: A two-dimensional finite volume unstructured mesh method (FVUM) based on a triangular mesh is developed for modeling density-dependent flow and transport through saturated–unsaturated porous media. The combined flow and transport model can handle a wide range of real-word problems, including the simulation of flow alone, contaminant transport alone, and combined flow and transport. Saltwater intrusion problems and instability caused by denser water on the top were investigated in this paper. Because the fundamental mechanism causing saltwater intrusion most likely is caused by density-induced convection and dispersion, the developed model is used to assess the interplay between density-driven flow and subsurface media through which the saltwater intrusion occurs. The mathematical formulation of the model is comprised of fluid flow and solute transport equations, coupled by fluid density. In the specific case of saltwater intrusion and unstable brine transport problems, this set of governing equation is non-linear and requires iterative methods to solve them simultaneously. Three case studies, which include a wide range of physical conditions, are used for verification of presented model and comparison with previously published solutions from other researchers presents encouraging agreements. [Copyright &y& Elsevier]

Abstract: This paper is focused on numerical investigation of subsonic flow separation over a NACA0012 airfoil with a 6° angle of attack and flow separation control with vortex generators. The numerical simulations of three cases including an uncontrolled baseline case, a controlled case with passive vortex generator, and a controlled case with active vortex generator were carried out. The numerical simulation solves the three-dimensional Navier–Stokes equations for compressible flow using a fully implicit LU-SGS method. A fourth-order finite difference scheme is used to compute the spatial derivatives. The immersed-boundary method is used to model both the passive and active vortex generators. The characteristic frequency that dominates the flow is the natural frequency of separation in the baseline case. The introduction of the passive vortex generator does not alter the frequency of separation. In the case with active control, the frequency of the sinusoidal forcing was chosen close to the natural frequency of separation. The time- and spanwise-averaged results were used to examine the mean flow field for all three cases. The passive vortex generators can partially eliminate the separation by reattaching the separated shear layer to the airfoil over a significant extent. The size of the averaged separation zone has been reduced by more than 80%. The flow control with active vortex generator is more effective and the separation zone is not visible in the averaged results. The three-dimensional structures of the flow field have also been studied. [Copyright &y& Elsevier]

This paper investigates the applicability of the stencil-adaptive finite difference method for the simulation of two-dimensional unsteady incompressible viscous flows with curved boundary. The adaptive stencil refinement algorithm has been proven to be able to continuously adapt the stencil resolution according to the gradient of flow parameter of interest [Ding H, Shu C. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J Comput Phys 2006;214:397–420], which facilitates the saving of the computational efforts. On the other hand, the capability of the domain-free discretization technique in dealing with the curved boundary provides a great flexibility for the finite difference scheme on the Cartesian grid. Here, we show that their combination makes it possible to simulate the unsteady incompressible flow with curved boundary on a dynamically changed grid. The methods are validated by simulating steady and unsteady incompressible viscous flows over a stationary circular cylinder. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss a new class of schemes, the residual distribution schemes, adapted to compressible flow problems. They can be seen as a link between pure finite element methods such as the streamline diffusion method and the high order upwind method finite volume schemes. In fact they borrow ideas from both classes and this results in very accurate compact schemes. Up to now, they are mainly adapted to triangular type meshes, but can handle steady and unsteady problems. Since the philosophy is quite different form standard schemes, we will provide a full description of the schemes and many numerical illustrations. Some still unsolved issues will also be discussed. [Copyright &y& Elsevier]

Abstract: This paper is concerned with solving the viscous and inviscid shallow water equations. The numerical method is based on second-order finite volume–finite element (FV–FE) discretization: the convective inviscid terms of the shallow water equations are computed by a finite volume method, while the diffusive viscous terms are computed with a finite element method. The method is implemented on unstructured meshes. The inviscid fluxes are evaluated with the approximate Riemann solver coupled with a second-order upwind reconstruction. Herein, the Roe and the Osher approximate Riemann solvers are used respectively and a comparison between them is made. Appropriate limiters are used to suppress spurious oscillations and the performance of three different limiters is assessed. Moreover, the second-order conforming piecewise linear finite elements are used. The second-order TVD Runge–Kutta method is applied to the time integration. Verification of the method for the full viscous system and the inviscid equations is carried out. By solving an advection–diffusion problem, the performance assessment for the FV–FE method, the full finite volume method, and the discontinuous Galerkin method is presented. [Copyright &y& Elsevier]

The discontinuous Galerkin method has been developed and applied extensively to solve hyperbolic conservation laws in recent years. More recently Wang et al. developed a class of discontinuous Petrov–Galerkin method, termed spectral (finite) volume method [J. Comput. Phys. 78 (2002) 210; J. Comput. Phys. 179 (2002) 665; J. Sci. Comput. 20 (2004) 137]. In this paper we perform a Fourier type analysis on both methods when solving linear one-dimensional conservation laws. A comparison between the two methods is given in terms of accuracy, stability, and convergence. Numerical experiments are performed to validate this analysis and comparison. [Copyright &y& Elsevier]