Hybrid Unstructured Grids Research Articles

Three-dimensional compact multi-resolution weighted essentially non-oscillatory reconstruction under the Arbitrary Lagrange–Euler framework

A third-order compact multi-resolution weighted essentially non-oscillatory (CMR-WENO) reconstruction method for three-dimensional (3D) hybrid unstructured grids is developed using the Arbitrary Lagrange–Euler framework. The finite volume method is used to discretize the governing equations, and some turbulent and moving boundary problems are simulated. Only one compact center stencil comprising the neighboring cells of each control cell is required to construct the polynomials in the algorithm. As a result, the number of stencils and stencil cells is significantly reduced when compared with the traditional WENO scheme. This simplifies the code and improves the robustness of the algorithm. By ensuring the cell average and first-order derivatives are consistent with that in stencil cells an over-determined system of equations can be used to reconstruct the polynomials. This system can then be solved using the compact least squares method to avoid an ill-conditioned coefficient matrix. Furthermore, a coupled implicit iteration strategy is used to solve for the unknown coefficients, so no extra determination is required for the derivatives of each control cell. The final interpolation function for discontinuities in the flow field is obtained using CMR-WENO to nonlinearly combine polynomials of different orders, which further improves the stability of the algorithm. The CMR-WENO can be implemented on 3D hybrid unstructured grids and can be used to simulate complex problems such as those involving turbulence and moving boundaries. Finally, the algorithm presented here is verified to be third-order accurate and to exhibit good robustness when used on several representative numerical examples.

A high-performance computing applied to composition reservoir simulation using distributed memory and 3D hybrid unstructured grids

A high-performance computing applied to composition reservoir simulation using distributed memory and 3D hybrid unstructured grids

Three-dimensional high-order finite-volume method based on compact WENO reconstruction with hybrid unstructured grids

In the present work, a three-dimensional (3D) high-order finite-volume method, based on the compact simple weighted essentially non-oscillatory (WENO) reconstruction for hybrid unstructured grids, is developed. For each control cell, its stencils only involve neighboring cells. By ensuring that the various orders of the derivatives of the conservative variables are conserved over the stencils, an over-determined system of equations is formed, and the polynomials corresponding to each stencil are obtained by solving this system of equations using the least-squares method. The artificially determined weight coefficients, and the smoothing factors based on the smoothness of the flow field, are used to determine the nonlinear weights, which nonlinearly combine the polynomials of each stencil to obtain the final high-order reconstruction polynomial. Consequently, the conservative variables on the interface can be interpolated and the sphere-function based discrete gas-kinetic scheme (DGKS) is used to locally solve the Boltzmann equation. Lastly, the fluxes are calculated using the mesoscopic perspective based on the moment relations. The present algorithm has the following advantages: the stencils only involve the neighboring cells of each control volume, making the algorithm compact and simpler, especially for the operation of the boundary cells; the unknown coefficients are solved by the least-squares method to avoid the coefficient matrix being ill-conditioned; the simple WENO scheme artificially chooses the value to guarantee positive weights, in addition, the algorithm can be applied to 3D hybrid unstructured grids, making it more adaptable. In terms of flux calculation, DGKS not only inherits the advantages of the traditional gas-kinetic schemes (GKS) but also greatly simplifies the flux calculation expressions. Several cases are simulated to verify the correctness and robustness of the algorithm, and the accuracy test shows the algorithm is third-order accurate.

Numerical study of trailing-edge noise reduction mechanism of wind turbine with a novel trailing-edge serration

The serration assembly installed at the trailing-edge of wind turbines can reduce the turbulent boundary-layer trailing-edge noise. This work studies the aeroacoustic response and noise reduction capability of four different trailing-edge serrations, involving a novel Multi-Flapped-Serration with Iron-Shaped Edges (MFS-Iron) presented here. These four serrations are retrofitted to S809 and NACA643418 airfoils. The Large-eddy simulation (LES) method and Ffowcs Williams and Hawkings integration (FW-H) methods are respectively employed to predict the unsteady flow field and far-field noise based on optimized structured and unstructured hybrid grids. From the numerical simulation results, we explore the trailing-edge noise reduction mechanism. The results show that the proposed MFS-Iron can effectively realize noise reduction under appropriate design parameters and optimize the aerodynamic layout. When using the MFS-Iron serration, the cross-flow intensity between the serration gap is significantly reduced, and the original two groups of symmetrical horseshoe vortex are mixed into a single vortex, which has a positive impact on the suppression of high-frequency noise. By further studying the Lamb vector divergence vortex dynamics process near the surface of serrated trailing-edge, it is found that the Lamb vector divergence is directly related to the intensity of the noise source.

Efficacy of line-based explicit and compact high-order finite difference schemes for hybrid unstructured grids

This study enables the use of explicit and compact high-order finite-difference schemes with a line-based solver to solve conservation laws on unstructured grids. A quadrilateral subdivision process is used to identify unique line structures (also known as Hamiltonian loops) before formulating stencil-based discretizations. To demonstrate the methodology for canonical flows represented by the Navier–Stokes equations, up to sixth-order spatial discretization and a maximum of tenth-order low-pass filters are implemented along with the Hamiltonian loops. The filter restores the benefits of the high-order approach even in the presence of abrupt grid discontinuities that cause abrupt changes in loop curvature. Isentropic vortex convection, lid-driven cavity, double periodic shear layer, and inviscid and viscous flow past a cylinder and NACA 0012 airfoil are all used to demonstrate the formulation. The ability of some schemes, particularly the fourth-order explicit, to nearly achieve the formal order of accuracy and successfully predict the flow following available results in the literature is one of the key findings. Compact schemes have been found to be more sensitive to loop curvature than explicit schemes.

Comparison of Finite-Volume and Methods for Large-Eddy Simulation of Combustor Port Flow

High-order accurate methods provide significant accuracy/cost benefits for well-resolved and geometrically simple scale-resolving turbulent flow simulations. However, the benefit on under-resolved unstructured grids for complex industrial geometries is unclear. The purpose of this work is to contribute to understanding of the benefit of high-order schemes for large-eddy simulations (LES) in practical applications. A crucial requirement is a high-order solver that supports hybrid unstructured grids. In this context, accuracy and cost of a high-order spectral/ Nektar++ solver and a standard second-order finite-volume OpenFoam solver are systematically compared for performing LES (with subgrid scale treatments) on two configurations. The first configuration is Taylor–Green vortex, which shows that the accuracy benefit of fifth-order LES depends upon the mesh type and the level of under-resolution. The second configuration is geometrically more complex and represents the dilution port flows of gas turbine combustors. Specifically, the flow consists of crossflow radial jets impinging onto each other, providing a rich variety of features and thus making the case challenging. It is found that, for a given cost, fifth-order LES reproduces the unsteady flow features significantly better and offers moderate improvements in the mean quantities. Moreover, results from a finer more costly OpenFoam simulation suggest that, for a similar accuracy, fifth-order LES could be cheaper.

Wind Tunnel Installation Effects on Japan Aerospace Exploration Agency’s Standard Model

This paper investigates the interference effects of wind tunnel floor and walls on the Japan Aerospace Exploration Agency’s High-Lift Configuration Standard Model (JSM) as an extension to a paper published in the Journal of Aircraft, Vol. 56, No. 3, May–June 2019, pp. 1080–1098. An unstructured hybrid grid generator, MEGG3D, was applied to JSM wing–body and wing–body–nacelle–pylon configurations to generate grids for free-air flow conditions. These grids were then reused in an investigation of the interference effects of Japan Aerospace Exploration Agency’s low-speed wind tunnel walls to better compare the computational results, both with each other and with experiment. The geometry of the wind tunnel used in our computational simulations is described in detail so that they can be reproduced. A Reynolds-averaged Navier–Stokes flow solver for unstructured hybrid grids, the Tohoku University Aerodynamic Simulation (TAS) code, was used for the simulations. The computational results indicated that the standoff and test section floor boundary layer significantly affected the aerodynamics of the JSM at high angles of attack. Better agreement with experiment was observed for the wing–body configuration in the aerodynamic characteristics near maximum lift if the effects of the standoff and floor boundary layer were considered.

A new 3D OpenFoam solver with improved resolution for hyperbolic systems on hybrid unstructured grids

Although the accuracy and the robustness of discretized schemes solving hyperbolic systems have been greatly improved over the last two decades, devising a low-dissipative and non-oscillatory method remains a challenge. Especially, it is algorithmically complicated and computationally expensive to extend and apply high-resolution non-oscillatory schemes to 3D hybrid unstructured grids. Therefore, in this work we develop accurate, practical, and robust 3D hybrid unstructured OpenFoam solvers which can be applied in wide industrial applications. Different from existing 3D high order numerical solvers of which reconstruction schemes are based on polynomials of high degree, the proposed 3D scheme employs a polynomial function and a hyperbolic tangent basis function as two candidates for reconstruction functions. The MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted as the polynomial function. The multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used as the non-polynomial reconstruction candidate. With these reconstruction candidates, a multi-stage boundary variation diminishing (BVD) algorithm which significantly minimizes numerical dissipation is designed on 3D unstructured grids to select the final reconstruction function. With this accurate and robust reconstruction scheme, then we developed new solvers for simulating inviscid high speed compressible single-phase flow and two-phase flow via the implementation of the proposed scheme and algorithm into the open-source computational fluid dynamics (CFD) software, OpenFOAM. The solver with the proposed low-dissipative scheme, named as EulerEquationFoam-M-TQ-BVD, is constructed with the density-based solver for inviscid high speed compressible flow simulations. To simulate compressible multi-material flow with moving material interfaces, a solver named as FiveEquationFoam-M-TQ-BVD is developed based on mechanical-equilibrium diffusive interface model and MieGrüneisen equation of state. The performance of new solvers is evaluated through wide range benchmark tests and are compared with existing high order schemes such as the third-order WENO (Weighted Essentially Non-Oscillatory) scheme. The new solvers produce superior solution across critical points, contact discontinuities and material interfaces. Moreover, in comparison with WENO schemes, the proposed solver enjoys algorithmic simplicity and greater flexibility in 3D hybrid unstructured grids. Thus, this work provides accurate, practical and robust numerical solvers for high speed compressible single- and multi-material flow simulations.

MHD_NX: код идеальной МГД-устойчивости на двумерных неструктурированных сетках

The code developed by the authors for calculating the MHD stability of two-dimensional plasma configurations with an arbitrary magnetic field topology is presented. The choice of an appropriate version of the finite element method on hybrid unstructured computational grids and test calculations are discussed. Examples of using the code for modeling the plasma stability in Galatea traps and in divertor configurations of tokamak taking into account the plasma beyond the separatrix are given.

Development of a phase change solver for concentrated energy beam applications

A phase-change computational fluid dynamics (CFD) solver module was developed for the numerical solution of melting, solidification and evaporation of any substrate under a concentrated energy beam (CEB). The solver module was developed as a part of general purpose CFD solver AnuPravaha. It is a finite volume method-based CFD solver capable of numerical solution of various fluid flow and heat transfer problems over the hybrid unstructured grid. The phase change module for CEB applications was developed in 2-D planer, 2-D axis-symmetric and 3-D framework. A fixed grid enthalpy-porosity technique was used to capture the phase change phenomena. CEB was implemented as a surface heat flux boundary conditions. Various associated phenomena like Marangoni flow and natural convection was studied for laminar flow regime.The solver for turbulent flow is currently getting developed. The numerical solver was validated for melting of copper and melting and evaporation of tin under a 270o bend pencil Electron Beam Gun (EBG) evaporator. The numerical solutions from the solver were found to be in very good agreement with experimental results with less than 5% deviation. A case study was carried out using the validated solver for maximizing the molten pool fraction in the evaporator system for laser based purification processes. It was observed that for the particular geometry of the evaporator, the e-beam power and crucible thickness variation did not assist significantly in maximizing the molten pool fraction. However, the variation in the aspect ratio improves the molten pool fraction significantly. Studies like these can be utilized for designing new evaporator cavity with optimized parameters.

Arbitrary high-order non-oscillatory scheme on hybrid unstructured grids based on multi-moment finite volume method

In this article, we propose a novel high-order multi-moment finite volume method (MMFVM) for solving linear and nonlinear hyperbolic systems on unstructured grids. Different from the previous versions of volume-average/point-value multi-moment (VPM) scheme, the present scheme is developed by using a new reconstruction procedure based on the constrained least-square method which can be naturally extended to arbitrary high-order of accuracy in two and three dimensions. The so-called VPM-CLS (VPM based on constrained least-square method) scheme substantially increases the solution accuracy on a compact stencil which largely simplifies the computations. In order to eliminate the numerical oscillations associated with high-order schemes, we propose a new limiting projection approach named AOD (adaptive order detection) to choose the optimal degree of reconstruction polynomial for trouble cells in presence of discontinuous solutions. The resulting reconstruction method, i.e. VPM-AOD that combines VPM-CLS scheme with AOD limiting projection technique is able to attain 5th-order accuracy on the stencil of all immediately adjacent cells and achieve essentially non-oscillatory and less-dissipative numerical results for discontinuous solutions. The numerical methods are extensively verified by various benchmark tests for the Euler equations of compressible gas dynamics. Numerical results demonstrate the high solution accuracy and excellent capabilities of proposed schemes to handle both complex physics and geometries.

The Physical Influence Scheme applied to staggered unstructured grids for solving fluid flow problems

In this article, a finite volume formulation for solving the Navier-Stokes equations using unstructured hybrid grids and a staggered arrangement of variable pressure and velocity is presented. In this manner, a tight spatial pressure-velocity coupling is ensured without compromising geometrical flexibility. A second contribution of this work lays upon proposing a suitable interpolation function for the convected velocities in the momentum equations. For this purpose, we employ the so-called “Physical Influence Scheme (PIS),” which fully incorporates all physical phenomena involved, namely convection, diffusion, and pressure effects. Each term corresponding to each physical phenomenon is carefully addressed. We present numerical experiments that show the suitability of the PIS for the staggered finite volume formulation and highlight the importance of including diffusion and pressure effects (in addition to convection) into the interpolation function.

A high‐order solver for simulating vortex‐induced vibrations using the sliding‐mesh spectral difference method and hybrid grids

SummaryWe present a high‐order solver for simulating vortex‐induced vibrations (VIVs) at very challenging situations, for example, VIVs of a row of very closely placed objects with large relative displacements. This solver works on unstructured hybrid grids by employing the high‐order tensor‐product spectral difference method for quadrilateral grids and the Raviart‐Thomas spectral difference method for triangular grids. To deal with the challenging situations where a traditional conforming moving mesh is incapable, we split a computational domain into nonoverlapping subdomains, where each interior subdomain encloses an object and moves freely with respect to its neighbors. A nonuniform sliding‐mesh method that ensures high‐order accuracy is developed to deal with sliding interfaces between subdomains. A monolithic approach is adopted to seamlessly couple the fluid and solid vibration equations. Moreover, the solver is parallelized to further improve its efficiency on distributed‐memory computers. Through a series of numerical tests, we demonstrate that this solver is high‐order accurate for both inviscid and viscous flows and has good parallel efficiency, making it ideal for VIV studies.

Numerical Investigations of S-Shaped Air Inlet for Embedded Engines

In this paper, compressible Reynolds-averaged Navier–Stokes computations are conducted for an S-shaped duct, in a clean configuration and also with a single row of vortex generators. The numerical simulations are performed using a hybrid unstructured grid. The influence of including the geometric representation of the probe instrumentation is investigated by considering the rake fairing geometry and also the geometry of the complete rake with the probes. Some aspects of the probe modeling are also discussed. A direct comparison of the numerical simulation results with the experimental values is performed for the wall static pressure along the duct, total pressure recovery, and distortion indices at the aerodynamic interface plane. Three different eddy-viscosity turbulence models are evaluated: the one-equation Spalart–Allmaras, and the two-equation realizable and shear stress transport. A virtual 200-probe rake is also proposed, beyond the standard 40-probe rake, to enrich the total pressure and distortion indices comparisons. An overall good agreement is observed among experimental results and the numerical wall static pressure and the total pressure topology at the aerodynamic interface plane. Some discrepancies for the distortion indices are verified and discussed in detail.

An enhanced AUSM $$^{+}$$ + -up scheme for high-speed compressible two-phase flows on hybrid grids

An enhanced $$\hbox {AUSM}^+$$ -up scheme is presented for high-speed compressible two-phase flows using a six-equation two-fluid single-pressure model. Based on the observation that the $$\hbox {AUSM}^+$$ -up flux function does not take into account relative velocity between the two phases and thus is not stable and robust for computation of two-phase flows involving interaction of strong shock waves and material interfaces, the enhancement is in the form of a volume fraction coupling term and a modification of the velocity diffusion term, both proportional to the relative velocity between the two phases. These modifications in the flux function obviate the need to employ the exact Riemann solver, leading to a significantly less expensive yet robust flux scheme. Furthermore, the Tangent of Hyperbola for INterface Capturing (THINC) scheme is used in order to provide a sharp resolution for material interfaces. A number of benchmark test cases are presented to assess the performance and robustness of the enhanced $$\hbox {AUSM}^+$$ -up scheme for compressible two-phase flows on hybrid unstructured grids. The numerical experiments demonstrate that the enhanced $$\hbox {AUSM}^+$$ -up scheme along with THINC scheme can efficiently compute high-speed two-fluid flows such as shock–bubble interactions, while accurately capturing material interfaces.

A robust and efficient finite volume method for compressible inviscid and viscous two-phase flows

A robust and efficient density-based finite volume method is developed for solving the six-equation single pressure system of two-phase flows at all speeds on hybrid unstructured grids. Unlike conventional approaches where an expensive exact Riemann solver is normally required for computing numerical fluxes at the two-phase interfaces in addition to AUSM-type fluxes for single-phase interfaces in order to maintain stability and robustness in cases involving interactions of strong pressure and void-fraction discontinuities, a volume-fraction coupling term for the AUSM+-up fluxes is introduced in this work to impart the required robustness without the need of the exact Riemann solver. The resulting method is significantly less expensive in regions where otherwise the Riemann solver would be invoked. A transformation from conservative variables to primitive variables is presented and the primitive variables are then solved in the implicit method in order for the current finite volume method to be able to solve, effectively and efficiently, low Mach number flows in traditional multiphase applications, which otherwise is a great challenge for the standard density-based algorithms. A number of benchmark test cases are presented to assess the performance and robustness of the developed finite volume method for both inviscid and viscous two-phase flow problems. The numerical results indicate that the current density-based method provides an attractive and viable alternative to its pressure-based counterpart for compressible two-phase flows at all speeds.

Reconstructed discontinuous Galerkin methods for linear advection–diffusion equations based on first-order hyperbolic system

Newly developed reconstructed Discontinuous Galerkin (rDG) methods are presented for solving linear advection–diffusion equations on hybrid unstructured grids based on a first-order hyperbolic system (FOHS) formulation. Benefiting from both FOHS and rDG methods, the developed hyperbolic rDG methods are reliable, accurate, efficient, and robust, achieving higher orders of accuracy than conventional DG methods for the same number of degrees-of-freedom. Superior accuracy is achieved by reconstruction of higher-order terms in the solution polynomial via gradient variables introduced to form a hyperbolic diffusion system and least-squares/variational reconstruction. Unsteady capability is demonstrated by an L-stable implicit time-integration scheme. A number of advection–diffusion test cases with a wide range of Reynolds numbers, including boundary layer type problems and unsteady cases, are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous grids, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the linear advection–diffusion equations.

Improvement of the computational performance of a parallel unstructured WENO finite volume CFD code for Implicit Large Eddy Simulation

In this paper the assessment and the enhancement of the computational performance of a high-order finite volume CFD code is presented. Weighted Essentially Non-Oscillatory (WENO) schemes are considered to be from the most computationally expensive numerical frameworks, in the context of high-resolution schemes particularly on hybrid unstructured grids. The focus of this study is to assess the computational bottlenecks of the solver for the WENO schemes for Implicit Large Eddy Simulation (ILES) and optimise the performance and efficiency through a series of code modifications e.g. formula rewriting, reduction of number operations, inclusion of linear systems libraries, non-blocking communications amongst others. The code is assessed on five different HPC systems; significant speed-up is achieved ranging from 1.5 to 8.5, with very high-order schemes benefiting the most. Good scalability is also obtained up to 104 number of cores, demonstrating viability and affordability of WENO type schemes for scale resolving simulations.

Portable Solution for Modeling Compressible Flows on All Existing Hybrid Supercomputers

A variant of a numerical algorithm for simulating viscous gasdynamic flows on unstructured hybrid grids and its software implementation for heterogeneous computations is described. The system of Navier–Stokes equations is approximated by the finite-volume method of an increased approximation order with the values of the variables being defined at the mass centers of the grid elements. The distributed software implementation of the numerical algorithm is adapted to running on hybrid computer systems of various architectures. Comparative implementations were created using the MPI, OpenMP, CUDA, and OpenCL software models permitting the use of multicore processors and various types of accelerators, including NVIDIA and AMD graphics processors, and Intel Xeon Phi multicore coprocessors. The data exchange between MPI processes and between processors and accelerators is carried out simultaneously with the execution of calculations (both in MPI + OpenMP mode and when using CUDA or OpenCL). The indicators of parallel efficiency and performance on systems with different types of computing devices are studied in detail. In the tests, up to 260 GPUs were successfully used.

Toward efficient and accurate interface capturing on arbitrary hybrid unstructured grids: The THINC method with quadratic surface representation and Gaussian quadrature

A novel interface capturing scheme is proposed to compute moving interface on arbitrary hybrid unstructured grids. Different from conventional volume of fluid (VOF) schemes that require complicated geometric manipulations for interface reconstructions, the present method can be viewed as a hybrid geometric/algebraic type VOF approach, i.e. the interface is implictly retrieved from an algebraic function that effectively makes use of the geometrical information, such as normal direction and curvature of the interface. Unlike previous versions of the THINC (tangent of hyperbola interface capturing) method, the interface is represented by a quadratic surface for grid cells of arbitrary shapes in this scheme. The Gaussian quadrature is used to estimate the integration of the cell-wise multi-dimensional hyperbolic tangent reconstruction function which is then used to retrieve the interface from the volume fraction value of the target cell. The Gaussian quadrature is also used to compute the numerical fluxes from the reconstruction function. Numerical accuracy for reconstruction and flux computation can be effectively improved by increasing quadrature points. The whole solution procedure follows the finite volume method for advection transport, and is thus simple and easy to use for cells of arbitrary shapes. As verified in the benchmark tests in this paper, the presented scheme, so-called THINC/QQ (THINC method with quadratic surface representation and Gaussian quadrature) scheme, shows significantly improved geometrical fidelity of interface representation particularly for curved surface. Despite algorithmic simplicity, the solution quality of THINC/QQ is comparable to other existing VOF methods with PLIC geometrical interface reconstructions, and thus an accurate and efficient VOF scheme of great practical significance for unstructured grids.