Cartesian Solver Research Articles

AN ANALYSIS OF CYLINDRICAL BORIS SOLVER FOR TYPICAL PARAMETERS OF TWO-DIMENSIONAL PENNING ION SOURCE SIMULATION

The cylindrical Boris solver is analyzed for typical two-dimensional Penning ion source simulation parameters. The analysis comprises the solver's accuracy and stability, especially for the latter simulation stages, typically after about 30 μs. The simulation is done for two cases; the first is a gyration simulation with a homogenous magnetic field, and the second uses the same setup as the Penning simulation. Several investigated quantities to determine the error are the radial position, axial position, and velocity magnitude (or kinetic energy). The error is calculated by comparing the result with the reference result from the exact solver with an incredibly small time step width, dt = 10-15 s. The result shows a discrepancy between cylindrical and cartesian Boris solvers. The velocity magnitude of the particle decays as time goes on for the cylindrical Boris solver, especially when the particle is close to the z-axis, an error not found on the cartesian solver. For typical Penning simulation parameters, the trajectory of individual particles is way off the reference trajectory. However, the mean position is relatively close to the reference compared to the dimension of the simulation domain. The kinetic energy is also relatively accurate, with a similar slow decay related to the deteriorating non-axial velocity components previously observed in the first case. Thus, for the simultaneous simulation of millions of particles, there should not be any significant observable difference in actual Penning simulation compared to Penning simulation with reference time step width.

Heat flux predictions for hypersonic flows with an overset near body solver on an adaptive block-structured Cartesian off-body grid

The manual volume mesh generation process for body-fitted CFD codes can be a time-intensive process, particularly for complex flight vehicles. The work shown here presents a hybrid body-fitted/Cartesian overset CFD solver capable of automatic volume mesh generation capabilities and accurate surface heating predictions. The solver uses a body-conformal near body solver for capturing boundary layer effects and an off-body Cartesian solver with adaptive mesh refinement for shock and wake tracking within the CHAMPS CFD code. The formulated near body Cartesian solver is the first of its kind to be applied within the context of low and high enthalpy, hypersonic viscous flows. The developed solver is validated on a Mach 9.47 Mars Science Lander and a Mach 6 Orion Crew Exploration Vehicle in a low-enthalpy flow at various non-zero angles of attack followed by a Mach 25.2 AS-202 capsule at a 17.8-degree angle of attack for Earth re-entry. Validation is shown based on grid convergence studies as well as good agreement to available numerical and experimental data in consideration of surface heating and surface pressure predictions.

Validation and analysis of a coupled fluid-ablation framework for modeling low-temperature ablator

Ablation of certain highly volatile materials, such as camphor, naphthalene, and dry ice, can be achieved at relatively mild hypersonic conditions in unheated wind tunnels. This provides a convenient way to test fundamental aspects of the ablation process, develop measurement techniques, and validate numerical simulations. In this study, we develop a coupled framework between hypersonic flow and material response solvers and validate the numerical approach against recent experiments conducted by the von Karman Institute of Fluid Dynamics. The flow environment is modeled with an overset near body - Cartesian solver developed within CHAMPS. The solver is equipped with capabilities for automatic mesh generation, adaptive mesh refinement (AMR), and an interpolation algorithm for exchanging boundary conditions with external solvers. The material domain is modeled with a network of one-dimensional rays, incorporating a heat conduction solver, several types of surface ablation models, and a coupled system of surface balance equations required for coupling with the flow solver. The material environment in the simulation aims to closely match the experimental design of the ablating sample, which included a thin layer of camphor applied on top of a copper holder. By taking into account the cooling effect introduced by the back structure, we were able to achieve close agreement with the experimental data for the stagnation point recession and the overall shape change of the geometry. The obtained results are also compared to uncoupled equilibrium-based and steady-state (coupled) solutions, highlighting the importance of the coupled approach for modeling ablation problems. In addition, we explore the effects of different transport properties on the ablation rate of the material, highlighting a strong dependence on the diffusivity of the ablating species. Finally, using the flexibility of the developed algorithm, we explore the effect of the iterative scheme and coupling frequency between the solvers on the accuracy of the solution and the overall duration of the simulation.

Cost vs Accuracy: DNS of turbulent flow over a sphere using structured immersed-boundary, unstructured finite-volume, and spectral-element methods

We report a comparative study of three numerical solvers for the direct numerical simulation of the flow over a sphere at Re = 3700. A high-order spectral-element code (Nek5000), a general purpose, unstructured finite-volume solver (OpenFOAM) and an in-house Cartesian solver using the immersed-boundary method (IBM) are employed for the analysis; results are compared against previous numerical and experimental data. Numerical results show that Nek5000 and the IBM code operate within a similar computational performance range, in terms of cost-vs-accuracy analysis, for both global parameters as well as local flow features. On the other hand, OpenFOAM needed a significantly higher number of degrees of freedom (and, overall, a higher cost) to match some of the basic features of the flow, such as the length of the recirculation bubble forming downstream the sphere. For the finest grid resolutions, the three codes are in good agreement for most of the analyzed flow metrics. Overall, our results suggest that high-order methods and second-order, energy-conserving approaches based on the IBM may be both viable options for high-fidelity scale-resolving simulations of turbulent flows with separation.

Computation of the uncollided-flux moments with advanced MOC and QP methods

The growing need to solve radiation protection and shielding problems in reasonable computational times has led to an increasing interest in the use of deterministic approaches. Codes of this type commonly use the discrete-ordinates method. In cases of propagation of point sources in low scattering media, which are typical of shielding studies, the approximation leads to a set of numerical artefacts known as ray effect. One of the most commonly used techniques to mitigate this effect is the calculation of the uncollided flux, which is the most affected flux component, with a fine transport operator. In this paper we propose four different approaches: the method of characteristics (MOC), the trajectory-splitting MOC (TS-MOC), the Monte Carlo MOC (MC-MOC) and the method based on quadrature points (QP). They have been implemented inside the Cartesian solver IDT. This paper contains the formal description of the four numerical methods and their verification over a set of benchmark problems. An analysis of the accuracy and the computation performances of the methods is also given. Finally, a comparison of the four methods on the Kobayashi benchmark problem is proposed.

On the Analytical Inversion of Solver Matrices Used in Numerical Approximations for the Diffusion Equation

The goal of this paper is to introduce an analytical approach for the inversion of nxn solver matrices, which are typically used in Finite Difference Method approximations. In the present case, they are used to solve the Diffusion Equation numerically, since in many physics and engineering fields, partial differential equations cannot be solved analytically. The method presented in this work is primarily formulated for cylindrical coordinates, which are often used in Gas Release Experiments as those described in [8]. However, it is possible to introduce a generalized method, which also allows solutions for Cartesian solvers. The advantage of having the explicit inverse is considerable, since the computational effort is reduced. In this paper we also carry out an investigation on the eigenvalues of the backward and forward solver matrix in order to determine an optimal range for the discretization parameters.

Prediction Capability of Cartesian Cut-Cell Method with a Wall-Stress Model Applied to High Reynolds Number Flows

The Cartesian cut-cell method is one of the most promising methods for computational fluid dynamics due to its sharp interface treatment. However, the Cartesian cut-cell method and other Cartesian mesh solvers have difficulty with concentrating grid to boundary layers. The wall-modelling of shear stress is one of the most effective methods to reduce computational grids in boundary layers. This study investigated the applicability of a wall-stress model to the Cartesian cut-cell method. In the numerical simulations of the flow around a triangular column, Cartesian cut-cell simulation with the wall-stress model adequately predicted the drag coefficient. In the numerical simulations of the flow around a 30P30N high-lift airfoil configuration, the Cartesian cut-cell simulation with the wall-stress model adequately predicts the lift coefficient. The intermittent vortex structure of the outer layer of the turbulent boundary layer was observed on the suction side of the main element and the flap. The Cartesian cut-cell method with a wall-stress model is useful for predicting high Reynolds number flows at R e ∼ 10 6 .

An immersed boundary method on Cartesian adaptive grids for the simulation of compressible flows around arbitrary geometries

In this article, we present an immersed boundary method for the simulation of compressible flows of complex geometries encountered in aerodynamics. The immersed boundary methods allow the mesh not to conform to obstacles, whose influence is taken into account by modifying the governing equations locally (either by a source term within the equation or by imposing the flow variables or fluxes locally, similarly to a boundary condition). A main feature of the approach which we propose is that it relies on structured Cartesian grids in combination with a dedicated HPC Cartesian solver, taking advantage of their low memory and CPU time requirements but also the automation of the mesh generation and adaptation. Turbulent flow simulations are performed by solving the Reynolds-averaged Navier–Stokes equations or by a Large-Eddy simulation approach, in combination with a wall function at high Reynolds number, to mitigate the cell count resulting from the isotropic nature of Cartesian cells. The objective of this paper is to demonstrate that this automatic workflow is fast and robust and enables to get quantitative aerodynamics results on geometrically complex configurations. Results obtained are in good agreement with classical body-fitted approaches but with a significant reduction of the time of the whole process, that is a day for RANS simulations, including the mesh generation.

Time-Spectral Method for Overlapping Meshes

Overset and Cartesian solvers typically employ conventional time-marching schemes to simulate unsteady flows. Temporal pseudospectral schemes have demonstrated the ability to dramatically reduce the computational effort required to resolve the important subclass of time-periodic phenomena. Incorporating the time-spectral method within these approaches is desirable, but direct application is infeasible. Relative motion introduces dynamic blanking of spatial nodes which move interior to solid bodies; the solution at such nodes is therefore undefined over specific intervals of time. This proves problematic for the conventional time-spectral method because it expands the temporal variation at every node as an infinitely supported Fourier series. An extension of the time-spectral method is presented where dynamically blanked nodes are handled in an alternative manner; the solution through intervals of consecutively unblanked time samples are represented with barycentric rational interpolants. The Fourier- and rational interpolant-based differentiation operators are applied in tandem, providing a hybrid time-spectral scheme capable of consistently resolving relative motion on overlapping meshes. The hybrid scheme is applied to relevant cases in two and three spatial dimensions and the results demonstrate that the hybrid scheme mirrors the performance of the conventional time-spectral method and monotonically converges to analogous high-resolution, time-accurate simulations with increasing temporal modes.

Comparison of the blocked-off and embedded boundary methods in radiative heat transfer problems in 2D complex enclosures at radiative equilibrium

Radiative heat transfer in participating media at radiative equilibrium in two-dimensional complex geometries will be investigated using two Cartesian boundary treatments, i.e. the blocked-off method and the embedded boundary method. The main advantages of Cartesian formulation are to simplify grid generation and using efficient Cartesian solvers for complicated problems. Angular and spatial discretization of the radiative transfer equation are performed using the discrete ordinates method and the finite volume method, respectively. The accuracy of both methods in solution of radiative heat transfer problems in irregular geometries are verified by comparison with benchmark solutions from literatures. Then, in order to investigate all features of the blocked-off and embedded boundary methods, two radiative problems with complex enclosures that contain gray absorbing and emitting medium at radiative equilibrium are examined. The results obtained from the two methods are compared with each other as well as with results obtained by body-fitted grid system. It has been shown that for a medium at radiative equilibrium with Cartesian formulation, the embedded boundary method is the method of choice, especially for calculations near the complex boundaries. It should be noted that for optically thick media, both blocked-off and embedded boundary methods show poor performance.

Coupled radiative-conductive heat transfer problems in complex geometries using embedded boundary method

This study is aimed to analyze combined radiative and conductive heat transfer in two-dimensional irregular geometries using the embedded boundary treatment in Cartesian coordinate system. The main advantage of Cartesian formulation is to simplify grid generation and using efficient Cartesian solvers for problems with complex geometries. The discrete ordinates method is used for angular discretization of radiative transfer equation (RTE) in a participating medium and the finite volume method is used for spatial discretization of RTE and the energy equation. First, the required equations to implement the embedded boundary method in combined conductive-radiative problems are developed. Then, this method is employed to solve several coupled conductive-radiative problems associated with the complex geometry. To assess the accuracy and investigate various characteristics of the embedded boundary method, the results are compared with the other methods for analyzing of irregular geometries, i.e. the blocked-off method and the body-fitted treatment. The results showed that the embedded boundary method is an efficient approach for investigation heat transfer problems of combined conduction-radiation in complex enclosures including inclined or curved boundaries using Cartesian grid system. Especially for calculations near the complex boundaries, this method has high priority over the blocked-off method.

An octree-based solution-adaptive Cartesian grid generator and Euler solver for the simulation of three-dimensional inviscid compressible flows

Cartesian grid generation methods are especially designed algorithms to generate automatic grids for complex geometries and to simulate flows around such geometries regardless of the body shape. Cartesian grids are generated by constructing an octree-based data structure for the purpose of connecting the Cartesian cells to each other. Entire algorithm is implemented in object-oriented FORTRAN programming language. Some special Cartesian algorithms, namely, Ray-Casting method and cut-cell adaptation are used around three-dimensional closed bodies. The flow field around the solid body is obtained by employing Euler equations which are discretised by using finite volume method. Validation of the numerical results is accomplished by comparison with the experimentally obtained data from the flow around ONERA M6 wing. Employing the solution adaptation techniques, pressure coefficients and contours of the flow around the wing have verified and captured two shock waves (weak leading edge shock and midchord shock) by the developed grid-generator-with-eULER-solver-for-3D-applications (GeULER3D) code.

Efficient Adaptive Cartesian Vorticity Transport Solver for Vortex-Dominated Flows

An efficient solver for the velocity―vorticity form of the Navier―Stokes equations on adaptive Cartesian grids is presented. The excessive numerical dissipation common to most grid-based Navier―Stokes solvers is avoided by solving the fluid dynamic equations in vorticity conservation form. Additionally, an adaptive Cartesian solver is employed to efficiently capture and preserve vorticity on-the-fly as the flow develops. For practical purposes, this solver would be used in the wake region and coupled with a full Navier―Stokes solver in the near-body region, thus allowing vorticity to be accurately generated and then convected in the wake region with minimal dissipation. The adaptive Cartesian framework allows for the rapid evaluation of the velocity field using a fast-summation technique based on the Cartesian Treecode method. The implementations of both the solution algorithm and velocity calculation are described in detail. Results are presented for vortex convection applications that show good agreement with the analytical solution, and the accuracy of the scheme is verified numerically using a series of increasingly fine grids. Additionally, fully three-dimensional flow in the presence of a vortex ring is investigated and results are shown to be in very close agreement with published data.

Flow simulations in arbitrarily complex cardiovascular anatomies – An unstructured Cartesian grid approach

Image guided computational fluid dynamics is attracting increasing attention as a tool for refining in vivo flow measurements or predicting the outcome of different surgical scenarios. Sharp interface Cartesian/Immersed-Boundary methods constitute an attractive option for handling complex in vivo geometries but their capability to carry out fine-mesh simulations in the branching, multi-vessel configurations typically encountered in cardiovascular anatomies or pulmonary airways has yet to be demonstrated. A major computational challenge stems from the fact that when such a complex geometry is immersed in a rectangular Cartesian box the excessively large number of grid nodes in the exterior of the flow domain imposes an unnecessary burden on both memory and computational overhead of the Cartesian solver without enhancing the numerical resolution in the region of interest. For many anatomies, this added burden could be large enough to render comprehensive mesh refinement studies impossible. To remedy this situation, we recast the original structured Cartesian formulation of Gilmanov and Sotiropoulos [Gilmanov A, Sotiropoulos F. A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies. J Comput Phys 2005;207(2):457–92] into an unstructured Cartesian grid layout. This simple yet powerful approach retains the simplicity and computational efficiency of a Cartesian grid solver, while drastically reducing its memory footprint. The method is applied to carry out systematic mesh refinement studies for several internal flow problems ranging in complexity from flow in a 90° pipe bend to flow in an actual, patient-specific anatomy reconstructed from magnetic resonance images. Finally, we tackle the challenging clinical scenario of a single-ventricle patient with severe arterio-venous malformations, seeking to provide a fluid dynamics prospective on a clinical problem and suggestions for procedure improvements. Results from these simulations demonstrate very complex cardiovascular flow dynamics and underscore the need for high-resolution simulations prior to drawing any clinical recommendations.

Nonboundary Conforming Methods for Large-Eddy Simulations of Biological Flows

In the present paper a computational algorithm suitable for large-eddy simulations of fluid/structure problems that are commonly encountered in biological flows is presented. It is based on a mixed Eurelian-Lagrangian formulation, where the governing equations are solved on a fixed grid, which is not aligned with the body surface, and the nonslip conditions are enforced via local reconstructions of the solution near the solid interface. With this strategy we can compute the flow around complex stationary/moving boundaries and at the same time maintain the efficiency and optimal conservation properties of the underlying Cartesian solver. A variety of examples, that establish the accuracy and range of applicability of the method are included.