Hierarchical Cartesian Grids Research Articles

Numerical simulation of rime ice accretion on airfoil using rigid sphere model

A 2D/3D numerical model for rime ice shape prediction is developed by combining the rigid sphere model (RSM) and computational fluid dynamics (CFD), and its characteristics are investigated. In the RSM, the ice accretion is modeled based on first principles with little physical modeling. The ice particle on an airfoil is described as a rigid sphere, and the ice shape is represented as a collection of rigid spheres. We use a non-body-fitted hierarchical Cartesian grid-based CFD solver so that complex ice shape represented by RSM can be handled. The tree data of the hierarchical Cartesian grid is also used for fast search of droplet positions. In addition, multiple ice particles are modeled as one large particle to reduce the number of trajectory calculations. The sensitivity of several parameters in the calculation is studied in 2D. Subsequently, 3D ice accretion analysis on NACA0012 airfoil using the RSM is performed for the first time. The simulation results are compared with those of the wind-tunnel experiments conducted at NASA. Although the RSM is a simple ice accretion model, reliable results are obtained in both 2D and 3D calculations.

High-order immersed boundary method for inviscid flows applied to flux reconstruction method on a hierarchical Cartesian grid

We propose novel high-order immersed boundary methods for simulating inviscid flows on hierarchical Cartesian grids using the flux reconstruction (FR) method. A high-order distribution of physical quantities derived from the FR method is utilized to interpolate flow variables to maintain the same order near the wall as the interior domain. In particular, the wall’s curvature effect is investigated in depth. We validate the accuracy through basic 2-D and 3-D simulations of inviscid flows. Consequently, the same accuracy as the order of the FR method (up to the fourth order) is achieved, including in the near-wall region.

FFVHC-ACE: Fully Automated Cartesian-Grid-Based Solver for Compressible Large-Eddy Simulation

This study presents a fully automated Cartesian-grid-based compressible flow solver, named FrontFlow/Violet Hierarchical Cartesian for Aeronautics based on Compressible-flow Equations (FFVHC-ACE), for large-eddy simulation (LES) and its aeronautical applications. FFVHC-ACE enables high-fidelity LES of high-Reynolds-number flows around complex geometries by adopting three key numerical methods: hierarchical Cartesian grids, wall modeling in LES, and the kinetic-energy and entropy preserving (KEEP) scheme. The hierarchical Cartesian grids allow fully automated grid generation for complex geometries in FFVHC-ACE, and high-fidelity LES of high-Reynolds-number flows around complex geometries is realized by the wall modeling and the KEEP scheme on the non-body-fitted hierarchical Cartesian grids. We apply FFVHC-ACE to wall-modeled LES around high-lift aircraft configurations at wind-tunnel-scale Reynolds number and real-flight Reynolds number , demonstrating the capability of FFVHC-ACE for fully automated grid generation and high-fidelity simulations around complex aircraft configurations. The computed flowfield and aerodynamic forces at the wind-tunnel-scale Reynolds number agree well with the experimental data provided in the past AIAA High Lift Prediction Workshop (Rumsey et al., Journal of Aircraft, Vol. 56, No. 2, 2019, pp. 621–644). Furthermore, the wall-modeled LES at the real-flight Reynolds number shows good agreement of the lift coefficient with flight-test data.

UTCartによる民間航空機巡航形態の実機レイノルズ数定常流れ場解析

To verify the applicability of UTCart (a compressible flow solver based on the hierarchical Cartesian grid method) to high Reynolds number flows, a flow field analysis around a commercial airliner at cruise was conducted. The analysis is performed in the range of Reynolds number from that in general wind tunnel tests to the flight Reynolds number, and the reproducibility of the Reynolds number effect is verified through comparison with the results in the high Reynolds number wind tunnel tests. The simulations were able to reproduce the Reynolds number effects (lift coefficient increase and drag coefficient decrease, etc.) observed in the wind tunnel tests.

Application of immersed boundary based turbulence wall modeling to the Ahmed body aerodynamics

This paper applies a recently developed immersed boundary-turbulence wall modeling approach to turbulent flows over a generic car geometry, known as the Ahmed body, under massive flow separation within a lattice Boltzmann solver. Although the immersed boundary method combined with hierarchical Cartesian grid offers high flexibility in automatic grid generation around complex geometries, the near-wall solution is significantly deteriorated compared to the body-fitted simulation, especially when coupled to wall models for turbulent flows at high Reynolds number. Enhanced wall treatments have been proposed in the literature and validated for attached flow configurations. In this work, the Ahmed body with a slant surface of angle 35° is considered where the flow separates massively over the slant surface and the vertical base. The large eddy simulation is performed with a Reynolds stress constraint near-wall. The eddy viscosity is computed dynamically by taking into account the actually resolved Reynolds stresses. It approaches the mixing length eddy viscosity in attached boundary layers and returns to the subgrid eddy viscosity in detached boundary layers. An explicit equilibrium wall model has also been proposed to accelerate the calculation. Comparison with the no-slip boundary condition on the separated surfaces shows that the near-wall treatments with the equilibrium wall model operate reasonably well on both attached and detached boundary layers.

Aerodynamic Prediction of Aircraft High Lift Configuration Using Hierarchical Cartesian Grid and Immersed Boundary Method

In this study, the steady RANS simulation around CRM-HL is conducted to investigate the prediction capability of the hierarchical Cartesian grid flow solver to predict the aerodynamic performance of an aircraft high lift configuration. To calculate the turbulent boundary layer, the immersed boundary method and the wall function are used. Three types of grids are used, and their grid numbers are about 100M, 200M, and 400M. The calculated aerodynamic coefficients agreed well with the wind tunnel test results in the low angle of attack region. On the other hand, the calculated aerodynamic coefficients showed a discrepancy in the aerodynamics coefficients to the wind tunnel test at high angle of attack. In addition, grid convergence could not be confirmed for CD and CM at high angle of attack. However, the RANS method used in this paper is not a model which can capture physical phenomena related to separation. Hence, reasonable results for RANS were obtained semi-automatically through automatic grid generation.

Unsteady aerodynamic simulations by the lattice Boltzmann method with near-wall modeling on hierarchical Cartesian grids

Near-wall modeling for large-eddy simulation (LES) based on the lattice Boltzmann method (LBM) is applied to the unsteady aerodynamic simulations around the tandem cylinders and the 30P30N three-element high-lift airfoil using hierarchical Cartesian grids. The near-wall modeling of the inner turbulent boundary layer is essential for LES to accurately calculate the high Reynolds number wall-bounded turbulent flow with a reasonable calculation cost. Therefore, near-wall modeling for LES has been studied within the framework of the Navier–Stokes equations. However, LBM-based modeling is still in its early developing stage because of the LBM’s calculation algorithm using distribution functions on a non-body fitted Cartesian grid, and further developments are needed regarding the applicability to flow simulations around arbitrary complicated shapes that appear in the actual engineering field. In this study, aerodynamic simulations around tandem cylinders are conducted using the proposed near-wall modeling that reproduces the profiles of the turbulent boundary layer in turbulent channel flow simulations on a non-body-fitted Cartesian grid. The same benchmark problem is also calculated using the non-slip wall boundary condition (interpolated bounce-back scheme) for comparison. Those calculation results are validated through the comparisons with the available experimental measurements, and the effectiveness of the proposed near-wall modeling to high Reynolds number wall-bounded turbulent flow simulations around objects with curvilinear surfaces is demonstrated. Followed by the tandem cylinders, the aerodynamic simulation around the 30P30N three-element high-lift airfoil is also conducted in order to demonstrate the applicability to more complicated shapes. The simulation result shows that the proposed method can stably calculate the relatively complicated configuration such as the high-lift airfoil, and the potential of the proposed method to unsteady aerodynamic simulations around arbitrary complicated shapes is demonstrated.

Simulating unsteady flows in a compressor using immersed boundary method with turbulent wall model

Computations of unsteady turbulent flows in turbomachine are still challenging due to the complexity of geometry and relative motion between rotor and stator. A potential solution is using the immersed boundary method (IBM) for its advantages in simulating moving boundary problems, but it loses efficiency with increasing Reynolds number. In the present work, to accurately simulate unsteady flows in a compressor, the well-established turbulent wall model is adopted with the hierarchical Cartesian grid in IBM to alleviate the requirements of grid spacing for turbulent boundary layer simulation. First, verification of the flow solver on two-dimensional (2D) stationary cases of the non-inclined/inclined flat plate and zero attack NACA 0012 were conducted. Furthermore, the Tamaki et al. wall model which had been only validated in stationary problems was first extended to solve the moving boundary problem, i.e. plunging airfoil. Finally, the new immersed boundary solver employed Tamaki et al. wall model was used to simulate the classic rotor/stator interaction. The unsteady normal forces on stator blade of a low-speed compressor were examined with two different sizes of axial gap, i.e. 10% chord and 30% chord. With the large gap, the predicted phase and magnitude of the unsteady aerodynamic force agree with the measurements well. With the more challenging small gap, although the predicted phase of the peak force is lagged slightly with the measurements, the magnitude of the peak force is captured. The agreements between numerical results and measurements indicate the ability of the developed IBM solver based on Tamaki et al. wall model and hierarchical Cartesian grid in dealing with simulations of unsteady flows in turbomachine.

An RBF-Based h-Adaptive Cartesian Grid Refinement Method for Arbitrary Single/Multi-Body Hull Modeling and Reconstruction

Complex single/multi-body structures are generally found in ship and ocean engineering. They have the smooth, sharp, concave, and convex surface features in common. Precise modeling of the structures is the basis of numerical simulation. However, the most widely used explicit modeling method requires considerable manual operations. The result is also difficult to reproduce. Therefore, this paper presents a Radial basis function (RBF) based hierarchical (h-) adaptive Cartesian grid method. The RBF is introduced for arbitrary implicit modeling over the Cartesian framework. Thanks to its natural properties, the RBF is easy to use, highly automated, and only needs a set of scatter points for modeling. The block-based h-adaptive mesh refinement (AMR) combined with the RBF aims to enhance the local grid resolution. It accelerates the calculation of intersecting points compared with the uniform Cartesian grid. The accuracy, efficiency, and robustness of the proposed method are validated by the simulation of the 3D analytical ellipsoidal surface and the unclosed conic surface. To select suitable parameters, we thoroughly investigated the uncertainty factors including sample points, RBF functions, and h-AMR grids. The simulation results of the single/multi-body Wigley hull and KCS hull forms verified the proper selection of the factors and the feasibility of our method to solve practical problems.

Determination of residual stresses in processes with multiple thermal loads

Abstract Numerical models are used to identify correlations between internal material loads and material modifications. Such correlations are called Process Signatures, which are based on the idea that different processes inducing equal internal material loads produce the same material modifications. Here, machining processes involving repeated thermal loads at high frequency as they occur in laser ablation or electric discharge machining are considered. As an example for the material modifications, the resulting residual stress distribution is determined as a function of the temperature. First, the temperature field in the work piece is computed for about 1000 consecutive heat pulses generated by the laser or electric discharges. An in-house finite-volume method is used to solve the time dependent enthalpy equation on a hierarchical Cartesian grid, which allows a high degree of parallelization such that high-performance computing hardware can be used. The resulting temperature fields and the heating rates for several load numbers are used to determine the residual stress state in the material. This is done in a second step by finding an appropriate model for the thermal loads, which provides a boundary condition for a finite element simulation of the surface near modification process. In this step, phase transformation during heating (austenite formation) and cooling (martensitic formation) are taken into account. Finally, the displacements and residual stresses are calculated by the phase and temperature distributions determined in the previous step. The results are compared with measurements and enable the determination of a Process Signature component describing the relation between repeated heat loads and the resulting residual stresses in the surface layer of the work piece.

A Discontinuous Galerkin Time-Domain Method With Dynamically Adaptive Cartesian Mesh for Computational Electromagnetics

A discontinuous Galerkin time-domain (DGTD) method based on dynamically adaptive Cartesian meshes (ACM) is developed for a full-wave analysis of electromagnetic fields in dispersive media. Hierarchical Cartesian grids offer simplicity close to that of structured grids and the flexibility of unstructured grids while being highly suited for adaptive mesh refinement (AMR). The developed DGTD-ACM achieves a desired accuracy by refining non-conformal meshes near material interfaces to reduce stair-casing errors without sacrificing the high efficiency afforded with uniform Cartesian meshes. Moreover, DGTD-ACM can dynamically refine the mesh to resolve the local variation of the fields during propagation of electromagnetic pulses. A local time-stepping scheme is adopted to alleviate the constraint on the time-step size due to the stability condition of the explicit time integration. Simulations of electromagnetic wave diffraction over conducting and dielectric cylinders and spheres demonstrate that the proposed method can achieve a good numerical accuracy at a reduced computational cost compared with uniform meshes. For simulations of dispersive media, the auxiliary differential equation (ADE) and recursive convolution (RC) methods are implemented for a local Drude model and tested for a cold plasma slab and a plasmonic rod. With further advances of the charge transport models, the DGTD-ACM method is expected to provide a powerful tool for computations of electromagnetic fields in complex geometries for applications to high-frequency electronic devices, plasmonic THz technologies, as well as laser-induced and microwave plasmas.

A fully coupled hybrid computational aeroacoustics method on hierarchical Cartesian meshes

Abstract Hybrid computational fluid dynamics (CFD) – computational aeroacoustics (CAA) schemes are the standard method for aeroacoustics simulations. This approach requires the exchange of information between the CFD and the CAA step, which is usually accomplished by storing acoustic source data. This data exchange procedure, however, poses two problems when such hybrid methods are used for large-scale problems with O ( 10 9 ) degrees of freedom: On the one hand, the required disk space becomes large and reaches hundreds of terabytes for a single simulation. On the other hand, the parallel scalability of the overall numerical scheme is limited by the available I/O bandwidth, which typically peaks between 5,000 and 10,000 cores. To avoid these problems, a highly scalable direct-hybrid scheme is presented, in which both the flow and the acoustics simulations run simultaneously. That is, all data between the two solvers is transferred in-memory, avoiding the restrictions of the I/O subsystem. Both solvers operate on a joint hierarchical Cartesian grid, which enables efficient parallelization and dynamic load balancing and inherently supports local mesh refinement. To demonstrate the capabilities of the new scheme, the aeroacoustic field of a co-rotating vortex pair is computed. The results show that the direct-hybrid method is able to efficiently predict the acoustic pressure field and that it is suitable for highly parallel simulations. Furthermore, in comparison to the hybrid method with data exchange via disk I/O, the novel approach shows superior performance when scaling to thousands of cores.

A numerical method for multiphysics simulations based on hierarchical Cartesian grids

A numerical method for multiphysics simulations based on hierarchical Cartesian grids

A mixed Lagrangian–Eulerian method for non-linear free surface flows using multigrid on hierarchical Cartesian grids

This article describes a two-dimensional finite difference method for the numerical simulation of fully non-linear irrotational water waves. The computational domain is discretised in an Eulerian fashion using hierarchical Cartesian meshes, whilst the free surface location is explicitly tracked using a Lagrangian approach. The accuracy of the method is strongly dependent on the quality of the approximation to the free surface velocities and a novel method to compute these is proposed. A multigrid strategy is implemented to take advantage of the hierarchical nature of the grids, incorporating an efficient technique to generate the coarser grids. The method is verified through simulations of quasi-linear low-amplitude waves and through a comparative study using an asymmetric sloshing wave. The non-linear behaviour of waves of moderate amplitude is also simulated.

Using hierarchical Cartesian grids with multigrid acceleration

AbstractThis paper describes the use of a multigrid strategy on hierarchical Cartesian grids for the simulation of small amplitude inviscid waves in a tank. Quadtree grid generation by recursive subdivision is described and the grid storage system discussed along with the associated grid operations, such as neighbour finding and identification of empty cells. A multigrid strategy using a novel global approach is discussed with respect to its application to quadtree grids. A comparative study is presented of the performance of multigrid parameters, such as number of iterations at each level, grid visiting schedules and different prolongation operators. It is shown that quadtree grids are ideally suited to a multigrid strategy when the solution of the elliptic Laplace's equation is sought, with savings of the order of 90% in CPU time with respect to a non‐multigrid solution. Copyright © 2006 John Wiley & Sons, Ltd.