Unstructured Hexahedral Grids Research Articles

Influence of the spatial grid type on the result of calculating the neutron fields in the nuclear power plant shielding

The paper considers the influence of the spatial grid type on the result of solving the equation of neutron transport in the nuclear power plant (NPP) shielding. Neutron fields were calculated in a realistic model of a liquid metal cooled fast neutron tank reactor with an integral equipment layout. Structured cubic and unstructured hexahedral grids (pmsnsys and FRIGATE codes) and unstructured tetrahedral and prismatic grids (RADUGA T code) are used. Limiting values of the group fluxes averaged over the material zones for refined grids have been obtained. It has been shown that the calculation results depend on the type of approximation for the curvilinear inner boundaries between the material zones rather than on the grid cell type (cube, hexahedron, tetrahedron, prism). Using “toothed” approximations for curvilinear boundaries leads to an increase in the area of the boundaries, as well as to the neutron flux refraction condition arising on them. These effects lead to an upward bias in the transport equation solution, and for all energy groups. Conclusion. When solving an equation of neutron transport in the NPP shielding by a grid technique, it is necessary to use grids other than leading to “toothed” approximations of the inner boundaries. Tetrahedral or prismatic grids, or grids of arbitrary hexahedrons can be recommended, as well as composite grids in which cubic cells are located inside the material zone, and hexahedron cells are located near the zone boundary.

Aerothermal Optimization of Turbine Squealer Tip Geometries With Arbitrary Cooling Injection

Abstract The optimization of the turbine rotor tip geometry remains a vital opportunity to create more efficient and durable engines. Balancing the aerodynamic and thermal aspects, while maintaining the mechanical integrity, is key to reshape one of the most vulnerable and life-determining parts of the entire turbine. The ever-increasing turbine gas temperatures, combined with the difficulty of cooling the tip and the aerodynamically penalizing nature of the overtip leakage vortex, make the design of the tip a truly multidisciplinary challenge. While many earlier efforts focused on uncooled geometries or studied the aerothermal impact with a fixed cooling configuration, the current paper presents the outcome of a multi-objective optimization where both the squealer rim geometry and the cooling injection pattern were allowed to vary simultaneously. This study explores a significantly wider design space, seeking a further synergistic aerothermal benefit through the combination of a quasi-fully arbitrary cooling arrangement, with mutating squealer rim structures. Specifically, the current manuscript presents the results of over 330 cooled and uncooled squealer tip geometries. The high-pressure turbine tip was automatically altered using a novel parametrization strategy adopting a maximum of 40 design variables to vary the squealer rim structures, as well as the size and location of the various cooling holes. The aerodynamic and thermal characteristics of every design were evaluated through Reynolds-averaged Navier–Stokes computational fluid dynamics (CFD) simulations with the k–ω shear-stress transport (SST) model for the turbulence closure, adopting an unstructured hexahedral grid typically containing more than 8 × 106 cells. A multi-objective differential evolution algorithm was used to obtain a Pareto front of designs which maximize the aerodynamic efficiency, while minimizing the overtip thermal loads. Eventually, a detailed investigation and robustness study was performed on a set of prime squealer geometries, to further investigate the aerodynamic flow topology and the effect of various cooling injection schemes on the heat transfer patterns.

Three-dimensional high-order least square-based finite difference-finite volume method on unstructured grids

The least square-based finite difference-finite volume (LSFD-FV) method has been developed and successfully applied to solve various two-dimensional flow problems with a high-order of accuracy. In this paper, the extension of the LSFD-FV method to the three-dimensional (3D) case on unstructured grids is presented. Different from other existing high-order methods, the LSFD-FV method combines the good features of the least square-based finite difference (LSFD) scheme for derivative approximation and the finite volume (FV) discretization for conservation of physical laws. Within each control cell, a high-order polynomial resultant from a Taylor series expansion is used to approximate the solution in the FV discretization of governing equations, where the derivatives are approximated by the LSFD scheme. As a result, the mesh-free nature of the LSFD scheme endows the LSFD-FV method with the flexibility on unstructured grids. Additionally, the straightforward algorithm of the LSFD scheme and the direct utilization of the Taylor series polynomial make the LSFD-FV method user-friendly and easy to implement. Furthermore, the inviscid and viscous fluxes are simultaneously evaluated by lattice Boltzmann and gas kinetic flux solvers in this work, which avoids additional and complicated treatment for the viscous discretization. Accuracy studies on unstructured hexahedral and tetrahedral grids validate the third-order of accuracy of the 3D LSFD-FV method. Various 3D incompressible and compressible numerical experiments are also presented. The results show that the proposed method enjoys the higher accuracy than the conventional second-order method on the same mesh. When the same accuracy is achieved, the present high-order method has superior computational efficiency to the second-order counterpart.

Simulation of Air\u2013Water Interface Effects for High-speed Planing Hull

High-speed planing crafts have successfully evolved through developments in the last several decades. Classical approaches such as inviscid potential flow–based methods and the empirically based Savitsky method provide general understanding for practical design. However, sometimes such analyses suffer inaccuracies since the air–water interface effects, especially in the transition phase, are not fully accounted for. Hence, understanding the behaviour at the transition speed is of fundamental importance for the designer. The fluid forces in planing hulls are dominated by phenomena such as flow separation at various discontinuities viz., knuckles, chines and transom, with resultant spray generation. In such cases, the application of potential theory at high speeds introduces limitations. This paper investigates the simulation of modelling of the pre-planing behaviour with a view to capturing the air–water interface effects, with validations through experiments to compare the drag, dynamic trim and wetted surface area. The paper also brings out the merits of gridding strategies to obtain reliable results especially with regard to spray generation due to the air–water interface effects. The verification and validation studies serve to authenticate the use of the multi-gridding strategies on the basis of comparisons with simulations using model tests. It emerges from the study that overset/chimera grids give better results compared with single unstructured hexahedral grids. Two overset methods are investigated to obtain reliable estimation of the dynamic trim and drag, and their ability to capture the spray resulting from the air–water interaction. The results demonstrate very close simulation of the actual flow kinematics at steady-speed conditions in terms of spray at the air–water interface, drag at the pre-planing and full planing range and dynamic trim angles.

A preconditioned lattice Boltzmann flux solver for steady flows on unstructured hexahedral grids

The lattice Boltzmann flux solver (LBFS), first introduced by Shu et al. (2014) on structured meshes, allows fluid flow problems to be solved on unstructured meshes discretised by the finite volume method. The solver calculates the macroscopic fluxes at the cell interfaces from a local reconstruction of the lattice Boltzmann solution. In this paper the LBFS is extended to three-dimensional unstructured hexahedral meshes and a preconditioned lattice Boltzmann flux solver (PLBFS) is presented. The PLBFS involves applying the preconditioning technique proposed by Guo (2004) to the LBFS and is achieved by modifying the equilibrium distribution function used to calculate the macroscopic fluxes at the cell interface. When the PLBFS is applied to steady flow problems, it is shown that convergence is significantly accelerated and the accuracy of predictions with unstructured grids is greatly improved when compared to the LBFS. This paper also introduces a strategy for choosing the optimal value of preconditioning factor with unstructured hexahedral meshes.

A collocated-grid spectral difference method for compressible flows

Among various high-order numerical methods for compressible flows, the spectral difference method is very promising and attractive due to its simple formulation, easy implementation and high efficiency, but most of the so far developed spectral difference methods are based on a staggered grid arrangement and thus suffer from extra interpolation operations between solution points and flux points. In order to further improve computational efficiency, we propose a collocated-grid spectral difference method on unstructured hexahedral grids by removing the interior flux points. The proposed method holds all the advantages of spectral difference methods while involving the solution points and only the interface flux points at each element. In the proposed method, the spatial terms are calculated by a weighted average of two proposed schemes: a spectral difference-like scheme and a discontinuous Galerkin spectral element-like scheme. The collocated-grid spectral difference method is firstly analysed with a Fourier technique in terms of dissipation, dispersion and stability for the one-dimensional linear advection equation. Then, the proposed method is further verified with analytic solutions of Euler vortex propagation problem. The numerical accuracy of order P+1 is verified for a reconstruction polynomial of order P and the best weights are obtained for the enhanced computational efficiency. Finally, the capability of the proposed method in handling curved boundary and turbulent flow is demonstrated with two typical problems: flow over NACA0012 airfoil and decay of Taylor Green vortex. The proposed method bridges the spectral difference and discontinuous Galerkin spectral element method in the framework of differential formulation, and promises a higher computational efficiency over them.

On a robust and accurate localized artificial diffusivity scheme for the high-order flux-reconstruction method

The high-order flux reconstruction (FR) method on unstructured hexahedral grids is coupled with the localized artificial diffusivity (LAD) scheme for aiming at accurate simulation of shock-turbulence interaction. In order to overcome known robustness issues particularly with the high-order (r>0) derivative formulation, important properties of the artificial bulk viscosity (ABV) profile affecting the solution are investigated first. For the purpose of comparison, an approximated Gaussian filter and modified restriction-prolongation (RP) filters are developed and tested for the present FR-LAD approach. Then, we propose a multidimensional RP filter on unstructured quadrilateral and hexahedral grids to address the issues on non-Cartesian grids. It is shown that a simple extension of a one-dimensional RP filter for two-dimensional grids may result in insufficient smoothing of ABV with r=2, and that the proposed multidimensional RP filter can provide smooth ABV with both r=0 and r=2. The proposed FR-LAD scheme is tested for typical shock-related problems, including the 1D shock tube, 1D shock-entropy wave interaction, 2D steady shock flows, and 2D shock-vortex interaction. The FR-LAD scheme has favorable properties of subcell shock capturing with the length scale of O(h/p) and superior preservation of high-order accuracy for smooth flows. Finally, LES of an overexpanded supersonic jet is performed to demonstrate its capability for practical applications. The proposed FR-LAD scheme can be an attractive candidate for LES or DNS of compressible flows involving shocks, contact discontinuities, turbulence, and their interactions on unstructured meshes.

Correction to: The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations

An open-source code that implements the entropy stable discontinuous Galerkin scheme with Legendere–Gauss–Lobatto collocation (DGSEM) on curved unstructured hexahedral grids for compressible Navier–Stokes equations (NSE) is available at github.com/project-fluxo.

Japan Aerospace Exploration Agency Studies for the Fifth AIAA Drag Prediction Workshop

This paper summarizes studies using the multiblock structured grid solver UPACS and an unstructured grid solver FaSTAR by the Japan Aerospace Exploration Agency for the fifth AIAA Computational Fluid Dynamics Drag Prediction Workshop test cases and follow-on problems. Computations are performed on a series of multiblock structured grids provided by the Workshop committee. For FaSTAR, the grids are treated as unstructured hexahedral grids. Other sets of multiblock structured grids and unstructured grids generated at the Japan Aerospace Exploration Agency are also used for computations. The following is mainly discussed: 1) grid-convergence studies for Spalart–Allmaras and shear-stress transport turbulence models using UPACS and far-field drag analyses using postprocessing software called TEBUNCO, cooperatively developed at the Japan Aerospace Exploration Agency and the University of Naples “Federico II”; 2) investigation of side-of-body flow-separation behavior at high angles of attack using UPACS with the Spalart–Allmaras or shear-stress transport model, and the change with and without a nonlinear quadratic constitutive relation model; and 3) validation of FaSTAR with the automatic hexahedral grid generator HexaGrid.

CABARET method on unstructured hexahedral grids for jet noise computation

A new scalable high-resolution CABARET MILES method coupled with a porous Ffowcs Williams–Hawking formulation for far-field noise modelling is applied for the computation of flow and noise from a high-speed turbulent jet, which corresponds to a converging axi-symmetric nozzle and static isothermal conditions of the Jet Exhaust Aerodynamics and Noise experiment. Computational simulation results are provided for a range of grid resolutions. The results for simple means, turbulent kinetic energy, and fourth-order velocity correlations are compared with the experimental data and with a reference LES-Smagorinsky calculation from the literature. Far-field-noise predictions are obtained based on several integral control surfaces, which are used for consistency of the model. The far-field noise spectra are also compared with the experiment. The directivity of the far field noise predicted is analysed with the Azimuthal Decomposition Technique and compared with jet noise data from the TsAGI experimental database.

Vortical flow simulations by a high-order accurate unstructured hexahedral grid method

Vortical flows over a delta wing are numerically simulated by unstructured high-order accurate methods using flux reconstruction (FR) approach by Huynh. The order of accuracy of the scheme for a two-dimensional vortex convection is first tested on various types of grids and with some flux correction functions. Cares for calculating the derivative of the flux polynomial are shown. The flow over a delta wing is then simulated by the fourth order (polynomial degree 3, or p3) and third order (p2) FR as well as the structured MUSCL approach for the comparison. The p3 FR shows the highest suction peak due to the leading-edge vortices and the p2 results are closer to the p3 regardless of the smaller degrees of freedom than the case of MUSCL calculation.

Finding Free Surface of Supercritical Flows - Numerical Investigation

A three-dimensional numerical model (STAR-CCM+) has been used to compute the free surface for different cases with supercritical flow. The program solves the Navier-Stokes equations on an unstructured hexahedral grid using the SIMPLE method and the k-epsilon turbulence model. The location of the free water surface has been computed with the volume of fluid method. Two cases from literature have been used for validation: one with a channel expansion and another with a channel junction. In both cases, oblique standing waves occurred. The geometry of the waves were well reproduced by the numerical model. The third case had a more complex geometry, modelling a physical laboratory study of a natural river geometry upstream of the dam of a hydropower plant. The case was unusual in that supercritical flow and oblique standing waves occurred upstream of the dam. Also for this case the numerical model gave reasonable results for the location of the free water surface.

Application of the multigrid approach for solving 3D Navier-Stokes equations on hexahedral grids using the discontinuous Galerkin method

The Galerkin method with discontinuous basis functions is adapted for solving the Euler and Navier-Stokes equations on unstructured hexahedral grids. A hybrid multigrid algorithm involving the finite element and grid stages is used as an iterative solution method. Numerical results of calculating the sphere inviscid flow, viscous flow in a bent pipe, and turbulent flow past a wing are presented. The numerical results and the computational cost are compared with those obtained using the finite volume method.

Development and Verification of a High-Fidelity Computational Fluid Dynamics Model of Canine Nasal Airflow

The canine nasal cavity contains a complex airway labyrinth, dedicated to respiratory air conditioning, filtering of inspired contaminants, and olfaction. The small and contorted anatomical structure of the nasal turbinates has, to date, precluded a proper study of nasal airflow in the dog. This study describes the development of a high-fidelity computational fluid dynamics (CFD) model of the canine nasal airway from a three-dimensional reconstruction of high-resolution magnetic resonance imaging scans of the canine anatomy. Unstructured hexahedral grids are generated, with large grid sizes ((10-100) x 10(6) computational cells) required to capture the details of the nasal airways. High-fidelity CFD solutions of the nasal airflow for steady inspiration and expiration are computed over a range of physiological airflow rates. A rigorous grid refinement study is performed, which also illustrates a methodology for verification of CFD calculations on complex unstructured grids in tortuous airways. In general, the qualitative characteristics of the computed solutions for the different grid resolutions are fairly well preserved. However, quantitative results such as the overall pressure drop and even the regional distribution of airflow in the nasal cavity are moderately grid dependent. These quantities tend to converge monotonically with grid refinement. Lastly, transient computations of canine sniffing were carried out as part of a time-step study, demonstrating that high temporal accuracy is achievable using small time steps consisting of 160 steps per sniff period. Here we demonstrate that acceptable numerical accuracy (between approximately 1% and 15%) is achievable with practical levels of grid resolution (approximately 100 x 10(6) computational cells). Given the popularity of CFD as a tool for studying flow in the upper airways of humans and animals, based on this work we recommend the necessity of a grid dependence study and quantification of numerical error when presenting CFD results in complicated airways.

Non-Newtonian effects of blood flow in complete coronary and femoral bypasses

A numerical investigation of non-Newtonian steady blood flow in a complete idealized 3D bypass model with occluded native artery is presented in order to study the non-Newtonian effects for two different sets of physiological parameters (artery diameter and inlet Reynolds number), which correspond to average coronary and femoral native arteries. Considering the blood to be a generalized Newtonian fluid, the shear-dependent viscosity is evaluated using the Carreau–Yasuda model. All numerical simulations are performed by an incompressible Navier–Stokes solver developed by the authors, which is based on the pseudo-compressibility approach and the cell-centred finite volume method defined on unstructured hexahedral computational grid. For the time integration, the fourth-stage Runge–Kutta algorithm is used. The analysis of numerical results obtained for the non-Newtonian and Newtonian flows through the coronary and femoral bypasses is focused on the distribution of velocity and wall shear stress in the entire length of the computational model, which consists of the proximal and distal native artery and the connected end-to-side bypass graft.

Numerical simulation of non‐Newtonian blood flow in bypass models

AbstractThe article presents the numerical investigation of non–Newtonian effects of steady blood flow in complete idealized 3–D bypass models, whose native artery is either coronary or femoral with average physiological parameters. Considering the blood to be a generalized Newtonian fluid, the shear–dependent viscosity is described by two well–known macroscopic non–Newtonian models (the Carreau–Yasuda model and the modified Cross model). The results were obtained by own developed computational software based on the pseudo–compressibility approach and on the cell–centred finite volume method defined on unstructured hexahedral grids. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Time-Marching Method for Computations of High-Speed Compressible Flow on Structured and Unstructured Grid

The development of Computational Fluid Dynamics (CFD) compressible codes for 2D structured quadrilateral grid and 3D unstructured hexahedral grid is described. The high-speed flow in a nozzle blade cascade is predicted numerically by solving the 2D/3D Euler Equations in a coupled manner. The new finite volume CFD solvers employ second-order accurate central differencing scheme for spatial discretization and multi-stage Runge-Kutta technique for temporal integration with flow variables stored at the vertices. Artificial dissipations with pressure sensors are introduced to control solution stability and capture shock discontinuity. The predictions have been compared with experimental measurements and good agreement has been found.

Validation Case Results for 2D and 3D MHD Simulations

A consistent and conservative scheme designed by Ni et al. for the simulation of MHD flows with low magnetic Reynolds number has been implemented into a 3D parallel code of HIMAG based on solving the electrical potential equation. The scheme and code are developed on an unstructured collocated mesh, on which velocity (u), pressure (p), and electrical potential (φ) are located in the cell center, while current fluxes are located on the cell faces. The calculation of current fluxes is performed using a conservative scheme, which is consistent with the discretization scheme for the solution of electrical potential Poisson equation. The Lorentz force is calculated at cell centers based on a conservative formula or a conservation interpolation of the current density. We validate the numerical methods, and the parallel code by simulating 2D fully developed MHD flows with analytical solutions existed and 3D MHD flows with experimental data available. The validation cases are conducted with Hartmann number from 100 to 104 on rectangular grids and/or unstructured hexahedral and prism grids.

An arbitrary Lagrangian–Eulerian discretization of MHD on 3D unstructured grids

We present an arbitrary Lagrangian–Eulerian (ALE) discretization of the equations of resistive magnetohydrodynamics (MHD) on unstructured hexahedral grids. The method is formulated using an operator-split approach with three distinct phases: electromagnetic diffusion, Lagrangian motion, and Eulerian advection. The resistive magnetic induction equation is discretized using a compatible mixed finite element method with a second order accurate implicit time differencing scheme which preserves the divergence-free nature of the magnetic field. At each discrete time step, electromagnetic force and heat terms are calculated and coupled to the hydrodynamic equations to compute the Lagrangian motion of the conducting materials. By virtue of the compatible discretization method used, the invariants of Lagrangian MHD motion are preserved in a discrete sense. When the Lagrangian motion of the mesh causes significant distortion, that distortion is corrected with a relaxation of the mesh, followed by a second order monotonic remap of the electromagnetic state variables. The remap is equivalent to Eulerian advection of the magnetic flux density with a fictitious mesh relaxation velocity. The magnetic advection is performed using a novel variant of constrained transport (CT) that is valid for unstructured hexahedral grids with arbitrary mesh velocities. The advection method maintains the divergence-free nature of the magnetic field and is second order accurate in regions where the solution is sufficiently smooth. For regions in which the magnetic field is discontinuous (e.g. MHD shocks) the method is limited using a novel variant of algebraic flux correction (AFC) which is local extremum diminishing (LED) and divergence preserving. Finally, we verify each stage of the discretization via a set of numerical experiments.

Adaptive layered Cartesian cut cell method for the unstructured hexahedral grids generation

适应分层的笛卡儿的切割房间方法被介绍从复杂 CAD 模型解决未组织的具有六面的各向异性的笛卡儿的格子产生的困难。基于放松的 AVL 树合并算法的顶点被调查构造拓扑的结构因为立体声平版印刷术(STL ) 文件，和有特殊特征控制策略的一个基于拓扑学的自我适应的分层的切算法向前被带。在凸的壳的帮助下，一个新 points-in-polygon 方法被采用改进笛卡儿的切割房间方法。Byintegrating 自我适应的分层的切算法和改进笛卡儿的切割房间方法，适应分层的笛卡儿的切割房间方法获得复杂 CAD 模型 inSTL 文件的卷数据并且产生未组织的具有六面的各向异性的笛卡儿的格子。