Grid Convergence Study Research Articles

The hybridized Discontinuous Galerkin method for Implicit Large-Eddy Simulation of transitional turbulent flows

We present a high-order Implicit Large-Eddy Simulation (ILES) approach for transitional aerodynamic flows. The approach encompasses a hybridized Discontinuous Galerkin (DG) method for the discretization of the Navier–Stokes (NS) equations, and a parallel preconditioned Newton-GMRES solver for the resulting nonlinear system of equations. The combination of hybridized DG methods with an efficient solution procedure leads to a high-order accurate NS solver that is competitive to alternative approaches, such as finite volume and finite difference codes, in terms of computational cost. The proposed approach is applied to transitional flows over the NACA 65-(18)10 compressor cascade and the Eppler 387 wing at Reynolds numbers up to 460,000. Grid convergence studies are presented and the required resolution to capture transition at different Reynolds numbers is investigated. Numerical results show rapid convergence and excellent agreement with experimental data. In short, this work aims to demonstrate the potential of high-order ILES for simulating transitional aerodynamic flows. This is illustrated through numerical results and supported by theoretical considerations.

Development of a reduced chemical mechanism targeted for a 5-component gasoline surrogate: A case study on the heat release nature in a GCI engine

Gasoline Compression Ignition (GCI) is a promising engine operating mode that can reduce maximum pressure rise rate (MPRR) without knock tendency and better control the combustion phasing compared to the Homogeneous Charge Compression Ignition (HCCI) by using a late direct-injection (DI). In this study, a 107-species reduced mechanism and a 207-species skeletal mechanism were developed using the Computer Assisted Reduction Mechanism (CARM) and validated under engine conditions for a newly developed 5-component surrogate for a Haltermann 437 certification gasoline (AKI=93). Then, 3D computational fluid dynamics (CFD) simulations with an optimized grid size determined by a grid convergence study were performed with the 107-species reduced mechanism and the 5-component certification gasoline surrogate. Two experimental boosted GCI cases with similar, moderate MPRR and heat release parameters, but different second DI timings (−52° aTDC and −5° aTDC), were validated and analyzed. For the −52° aTDC DI case, the combustion can be interpreted as a partially sequential auto-ignition due to the competition between the charge cooling effect and the equivalence ratio (ϕ)-sensitive effect of the stratified mixture, which is responsible for mitigating the MPRR. For the −5° aTDC DI case, the combustion can be decoupled into a partially sequential auto-ignition and a subsequent non-premixed combustion by the DI fuel near top dead center in the compression stroke. The MPRR is relaxed through the slow, mixing-limited combustion between the injected fuel and the premixed mixture.

Benchmark computations for flows around a stationary cylinder with high Reynolds numbers by RANS-overset grid approach

The aim of this paper is to evaluate the accuracy, stability and efficiency of the overset grid approach coupled with the RANS (Reynolds Averaged Navier-Stokes) model via the benchmark computations of flows around a stationary smooth circular cylinder. Two dimensional numerical results are presented within a wide range of Reynolds numbers (6.31×104∼7.57×105) including the critical flow regime. All the simulations are carried out using the RANS solver pimpleFoam provided by OpenFOAM, an open source CFD (Computational Fluid Dynamics) toolkit. Firstly, a grid convergence study is performed. The results of the time-averaged drag and lift force coefficients, root-mean square value of lift force coefficient and Strouhal number (St number) are then compared with the experimental data. The velocity, vorticity fields and pressure distribution are also given. One main conclusion is that the numerical solutions in regard to a fixed cylinderare not deteriorated due to the implementation of the overset grid. Furthermore, it can be an appealing approach to facilitate simulations of Vortex Induced Vibrations (VIV), which involves grid deformation. The present study is a good start to implement the overset grid to solve VIV problems in the future.

Grid convergence study of a cyclone separator using different mesh structures

In a cyclone design, pressure drop and collection efficiency are two important performance parameters to estimate its implementation viability. The optimum design provides higher efficiencies and lower pressure drops. In this paper, a grid independence study was performed to determine the most appropriate mesh to simulate the two-phase flow in a Stairmand cyclone. Computational fluid dynamic (CFD) tools were used to simulate the flow in an Eulerian-Lagrangian approach. Two different mesh structure, one with wall-refinement and the other with regular elements, and several mesh sizes were tested. The grid convergence index (GCI) method was applied to evaluate the result independence. The CFD model results were compared with empirical correlations from bibliography, showing good agreement. The wall-refined mesh with 287 thousand elements obtained errors of 9.8% for collection efficiency and 14.2% for pressure drop, while the same mesh, with regular elements, obtained errors of 8.7% for collection efficiency and 0.01% for pressure drop.

Design optimization of a three-dimensional diffusing S-duct using a modified SST turbulent model

This paper examines design optimization of a three-dimensional diffusing S-duct using a modified k–ω shear stress transport (SST) turbulent model as a turbulence prediction method. According to based-Reynolds-stress and based-separation ideas, a robust solution procedure for the model is described and a grid convergence study is presented. An automated system employs this model to design an S-duct. The aerodynamic performances of the optimal duct are investigated in on-design and off-design conditions. It is shown that the sensitivity of the modified model with respect to shape variations allows its use in the design system. Using a multi-objective optimization strategy, this design system significantly improves aerodynamic performance of the S-duct and has low computation cost and excellent design efficiency. The centerline's curvature and the cross-sectional area ratio become reasonable to avoid overexpansion of the optimal duct. Compared with the original design, the flow distortion coefficient of the optimal duct is reduced by 16.3% and the total pressure recovery factor is increased by 1.1% in on-design condition.

A hybrid LES-RANS study on square cylinder unsteady heat transfer

Flow passing a heated square cylinder is investigated using a hybrid LES-RANS approach on unstructured grids at a moderate Reynolds number of 22,050. The implicit sub-grid scale model is apply to LES, and two turbulence models, the Spalart-Allmaras model and the SST k-ω model, are employed for near-wall RANS. Combined with LES, both models present good predictions of the time- and phase-averaged velocity profiles on a 4-million-cell grid. Results of the LES-SST approach agree better with the experimental data especially at locations close to the cylinder surface and this leads to improved surface convective heat transfer compared to LES-SA. Grid convergence study shows that grid resolution in the near-wall region and on the cylinder surfaces is important in resolving the unsteady convective heat transfer. Results of velocity field and surface heat transfer from the fine grid with 8 million cells compare favourably with the experimental data and show significant improvement over that of the medium and coarse grids. Analysis of turbulent statistics is performed by means of energy spectra and anisotropy invariants of the Reynolds stress tensor. Proper orthogonal decomposition (POD) is used to identify the vortex shedding phases. It is shown that the POD based phase-averaging produces more accurate velocity profiles than the conventional pressure-signal based method.

HLLC-type Riemann solver with approximated two-phase contact for the computation of the Baer–Nunziato two-fluid model

The computation of compressible two-phase flows with the Baer–Nunziato model is addressed. Only the convective part of the model that exhibits non-conservative products is considered and the source terms of the model that represent the exchange between phases are neglected. Based on the solver proposed by Tokareva & Toro [1], a new HLLC-type Riemann solver is built. The key idea of this new solver lies in an approximation of the two-phase contact discontinuity of the model. Thus the Riemann invariants of the wave are approximated in the “subsonic” case. A major consequence of this approximation is that the resulting solver can deal with any Equation Of State. It also allows to bypass the resolution of a non-linear equation based on those Riemann invariants. We assess the solver and compare it with others on 1D Riemann problems including grid convergence and efficiency studies. The ability of the proposed solver to deal with complex Equations Of State is also investigated. Finally, the different solvers have been compared on challenging 2D test-cases due to the presence of both material interfaces and shock waves: a shock–bubble interaction and underwater explosions. When compared with others, the present solver appears to be accurate, efficient and robust.

Computational analysis of nozzle geometry variations for subsonic turbulent jets

Large-eddy simulations (LES) of turbulent hot jets emanating from realistic helicopter engine nozzle configurations at a Reynolds number of Re = 7.5 × 105 and a Mach number of M=0.341 are conducted. The numerical method is based on hierarchically refined Cartesian meshes. The nozzle wall boundaries are resolved by a conservative cut-cell method. Three nozzle geometries of increasing complexity are considered, i.e., the flow fields of a clean geometry without any built-in components, a nozzle with a centerbody, and a nozzle with a centerbody plus struts are computed. The numerical method is validated by solutions for a single, a coaxial, and a chevron nozzle jet problem. A grid convergence study shows that the essential flow characteristics due to the intricacy of the nozzle geometry are well resolved. The results evidence that the flow field in the region 35 nozzle radii downstream of the exit is dominated by flow structures induced by the geometry. Compared to the clean geometry, the other two configurations show enhanced turbulent mixing. The centerbody and centerbody-plus-strut nozzle configurations reveal a spectral peak in the near nozzle exit region at St=0.15 which is caused by the wake flow of the centerbody.

Numerical simulation of long-period fluid temperature fluctuation at a mixing tee for the thermal fatigue problem

Thermal fatigue cracks may be initiated at mixing tees where high and low temperature fluids flow in and mix. According to a previous study, damage by thermal fatigue depends on the frequency of the fluid temperature fluctuation near the wall surface. Structures have the time constant of structural response that depends on physical properties of the structure and the gain of the frequency response tends to become maximum at the frequency lower than the typical frequency of fluid temperature fluctuation. Hence the effect of the lower frequency, that is, long-period temperature fluctuation is important for the thermal fatigue assessment. The typical frequency of fluid temperature fluctuation is about St=0.2 (nearly 6Hz), where St is Strouhal number and means non-dimensional frequency. In the experimental study by Miyoshi et al. (2014), a longer-period fluctuation than St=0.2 was also observed. Results of a fluid–structure coupled analysis by Kamaya et al. (2011) showed this long-period temperature fluctuation causes severer damage to piping. In the present study, a large eddy simulation was carried out to investigate the predictive performance of the long-period fluid temperature fluctuation more quantitatively. Numerical simulation was conducted for the WATLON experiment which was the water experiment of a mixing tee performed at the Japan Atomic Energy Agency. Four computational grids were used to confirm grid convergence. In the short time (9s) simulations, tendencies of time-averaged and fluctuated velocities could be followed. Time-averaged temperature distributions were also reproduced, although overestimation appeared near the wall. The fluid temperature fluctuation intensity near the wall surface could be predicted qualitatively, while the peak value was overestimated. From the engineering viewpoint, it was concluded that the numerical simulation provided results that were conservative and on the side of safety. From the grid convergence study, the coarsest grid for which grid convergence was almost attained was selected and the simulation was continued until 100s. Frequency analysis of the fluid temperature fluctuation showed that the long-period fluctuation appeared as well as the well-known typical frequency (St=0.2). This indicated that numerical simulation could reproduce the long-period temperature fluctuation at the mixing tee.

Grid-Convergence of Reynolds-Averaged Navier–Stokes Solutions for Benchmark Flows in Two Dimensions

A detailed grid-convergence study has been conducted to establish reference solutions corresponding to the one-equation linear eddy-viscosity Spalart–Allmaras turbulence model for two-dimensional turbulent flows around the NACA 0012 airfoil and a flat plate. The study involved the three widely used codes CFL3D (NASA), FUN3D (NASA), and TAU (DLR, The German Aerospace Center), as well as families of uniformly refined structured grids that differed in the grid density patterns. Solutions computed by different codes on different grid families appeared to converge to the same continuous limit but exhibited strikingly different convergence characteristics. The grid resolution in the vicinity of geometric singularities, such as a sharp trailing edge, was found to be the major factor affecting accuracy and convergence of discrete solutions; the effects of this local grid resolution were more prominent than differences in discretization schemes and/or grid elements. The results reported for these relatively simple turbulent flows demonstrated that CFL3D, FUN3D, and TAU solutions were very similar on the finest grids used in the study, but even those grids were not sufficient to conclusively establish an asymptotic convergence order.

Large-Eddy Simulation of Time Evolution and Instability of Highly Underexpanded Sonic Jets

High-pressure jet injection into quiescent air is a challenging fluid dynamics problem in the field of aerospace engineering. Although plenty of experimental, theoretical, and numerical studies have been conducted to explore this flow, there is a dearth of literature detailing the flow evolution and instability characteristics, which is vital to the mixing enhancement design and jet noise reduction. In this paper, a density-based solver for compressible supersonic flow, astroFoam, is developed based on the OpenFOAM library. Large-eddy simulations of highly underexpanded jets with nozzle pressure ratios from 5.60 to 11.21 at a Reynolds number around are carried out with a high-resolution grid. A grid-convergence study has been conducted to confirm the fidelity of the large-eddy simulation results. The large-eddy simulation results have also been validated against available literature data in terms of the time-averaged near-field properties of underexpanded jets. The turbulent transition processes are revealed based on the instantaneous flow features and are quantitatively resolved according to the jet penetration and maximum width. The vorticity analysis is conducted to understand the turbulent transition mechanism, and it is found that the vortex stretching term plays a leading role on the distortion of the vortex rings in the near field of the jets. The dominant instability modes of jets, visualized by helicity, are quantitatively revealed based on the spectrum and relative phase of pressure fluctuation. The single helical modes corresponding to a phase angle close to with the helices are dominant for nozzle pressure ratios of 5.60 and 7.47, whereas the complex and multiple helices for the other two higher nozzle pressure ratios are due to the superposition of the single and double helical modes. In addition, the performance of the coarse mesh and different subgrid-scale models on capturing the dominant instability characteristics in large-eddy simulation of underexpanded jets is investigated.

Poisson–Boltzmann model for protein–surface electrostatic interactions and grid-convergence study using the PyGBe code

Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson–Boltzmann equation. We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson–Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(NlogN) scaling, with further acceleration on hardware via multi-threaded execution on gpus. It computes solvation and surface free energies, providing a framework for studying the effect of electrostatics on adsorption. We derived an analytical solution for a spherical charged surface interacting with a spherical dielectric cavity, and used it in a grid-convergence study to build evidence on the correctness of our approach. The study showed the error decaying with the average area of the boundary elements, i.e., the method is O(1/N), which is consistent with our previous verification studies using PyGBe. We also studied grid-convergence using a real molecular geometry (protein GB1D4′), in this case using Richardson extrapolation (in the absence of an analytical solution) and confirmed the O(1/N) scaling. With this work, we can now access a completely new family of problems, which no other major bioelectrostatics solver, e.g. APBS, is capable of dealing with. PyGBe is open-source under an MIT license and is hosted under version control at https://github.com/barbagroup/pygbe. To supplement this paper, we prepared “reproducibility packages” consisting of running and post-processing scripts in Python for replicating the grid-convergence studies, all the way to generating the final plots, with a single command.

Turbulence Model Verification and Validation in an Open Source Environment

In this paper, the verification of two low-Reynolds-number turbulence models in an open source environment is reported. The two models are the Spalart-Allmaras model without the ft2 term and the k – ω SST model using vorticity in production estimation. Grid convergence is achieved in all verification cases, while reasonable agreements with other codes are found. The two turbulence models are also validated through comparison with some well-known validation cases, the results of which show good agreement with experimental data. The differences between the results obtained from incompressible and compressible codes are also discussed. The impact of wall distance estimation on the performance of turbulence modelling is also evaluated through grid convergence studies. Within the realm of search algorithms for wall distance estimation, a three-level search approach is found to be necessary in order to achieve correct and converged results, especially for skewed meshes. It is shown, through a 2D bump flow grid convergence study, that the use of a one-level search approach can not only lead to a maximum friction coefficient deviation of 3.8%, but also cause an inconsistent convergence behaviour.

Tracking a Tip Vortex with Adaptive Vorticity Confinement and Hybrid RANS-LES

The prediction of coherent vortices with standard RANS solvers suffers especially from discretisation and modelling errors which both introduce numerical diffusion. The adaptive Vorticity Confinement (VC) method targets to counteract one part of the discretisation error: the one due to the discretisation of the convection term. This method is applied in conjunction with a hybrid RANS-LES turbulence model to overcome the overprediction of turbulence intensity inside vortex cores which is a typical deficiency of common RANS solvers. The third main source for numerical diffusion originates from the spatial discretisation of the solution domain in the vicinity of the vortex core. The corresponding error is analysed within a grid convergence study. A modification of the adaptive VC method used in conjunction with a high-order discretisation of the convection term is presented and proves to be superior. The simulations of a wing tip vortex flow are validated in terms of vortex velocity profiles using the results of a wind tunnel experiment performed by Devenport and colleagues (1996). Besides, the results are compared with another numerical study by Wells (2009) who uses a Reynolds Stress turbulence model. It turns out that the application of the modified adaptive VC method on the one hand reinforces the tip vortex, and on the other hand accelerates the axial flow which leads to a slight degradation compared to the experimental results. The result of Wells is more accurate close to the wing, but the result obtained here is superior further downstream as no excessive diffusion of the tip vortex occurs.

Large-eddy simulation of the flow past a circular cylinder at sub- to super-critical Reynolds numbers

Large-eddy simulation of turbulent flow past a circular cylinder at sub- to super-critical Reynolds numbers is performed using a high-fidelity orthogonal curvilinear grid solver. Verification studies investigate the effects of grid resolution, aspect ratio and convection scheme. Monotonic convergence is achieved in grid convergence studies. Validation studies use all available experimental benchmark data. Although the grids are relatively large and fine enough for sufficiently resolved turbulence near the cylinder, the grid uncertainties are large indicating the need for even finer grids. Large aspect ratio is required for sub-critical Reynolds number cases, whereas small aspect ratio is sufficient for critical and super-critical Reynolds number cases. All the experimental trends were predicted with reasonable accuracy, in consideration the large facility bias, age of most of the data, and differences between experimental and computational setup in particular free stream turbulence and roughness. The largest errors were for under prediction of turbulence separation.

Dynamic overset grids in OpenFOAM with application to KCS self-propulsion and maneuvering

An implementation of the dynamic overset grid technique into naoe-FOAM-SJTU solver developed by using the open source code OpenFOAM is presented. OpenFOAM is attractive for ship hydrodynamics applications because of its high quality free surface solver and other capabilities, but it lacks the ability to perform large-amplitude motions needed for maneuvering and seakeeping problems. The implementation relies on the code Suggar to compute the domain connectivity information (DCI) dynamically at run time. Several Suggar groups can be used in multiple lagged execution mode, allowing simultaneous evaluation of several DCI sets to reduce execution time and optimize the exchange of data between OpenFOAM and Suggar processors. A towed condition of the KRISO Container Ship (KCS) are used for static overset tests, while open-water curves of the KP505 propeller and self-propulsion and zig-zag maneuvers of the KCS model are exercised to validate the dynamic implementation. For self-propulsion the ship model is fitted with the KP505 propeller, achieving self-propulsion at Fr=0.26. All self-propulsion factors are obtained using CFD results only, including those from open-water curves, towed and self-propulsion conditions. Computational results compare well with experimental data of resistance, free-surface elevation, wake flow and self-propulsion factors. Free maneuvering simulations of the HSVA KCS model appended with the HSVA propeller and a semi-balanced horn rudder are performed at constant self-propulsion propeller rotational speed. Results for a standard 10/10 zig-zag maneuver and a modified 15/1 zig-zag maneuver show good agreement with experimental data, even though relatively coarse grids are used. Grid convergence studies are performed for the open-water propeller test and bare hull KCS model to further validate the implementation of the overset grid approach.

Exergy-Based Performance Assessment of the NASA Common Research Model

Aircraft have evolved into extremely complex systems that require adapted methodologies and tools for efficient design processes. A theoretical formulation based on exergy management has been recently proposed by Arntz et al. for assessing the aerothermopropulsive performance of future aircraft configurations. The present article focuses on the validation of its numerical implementation in a FORTRAN code for the postprocessing of Reynolds-averaged Navier–Stokes flow solutions. The flow around the wing-body NASA Common Research Model is assessed in terms of anergy destruction. A 2 MW work potential associated with the lift-induced vortices is identified in the wake of the airplane. Subsequently, a six-level grid convergence study enables determining the robustness and accuracy of the exergy postprocessing code. The introduction and calibration of a numerical correction allows to account for the spurious numerical vortex dissipation and to obtain an accuracy similar to the traditional near-field drag method. Finally, the postprocessing code is validated for drag prediction against computational fluid dynamics and experimental wind-tunnel data.

Design of adaptive transonic laminar airfoils using the [formula omitted] transition model

This paper examines the design of adaptive laminar airfoils in the transonic flow regime using the γ-Re˜θt model as a transition prediction method. A robust solution procedure for the transition model is described and a grid convergence study is presented. Predictions from the model are then compared to an extensive set of experimental results containing transition measurements on a morphing airfoil. It is shown that the sensitivity of the γ-Re˜θt transition model with respect to shape variations allows its use in a design framework. Using a multi-point optimization strategy, the model is then used to design transonic airfoils suitable for a range of flow conditions. The potential improvement in drag associated with the implementation of adaptive airfoil technology is then carefully quantified using single-point optimization with the multi-point designed airfoil as a reference. Structural design constraints are taken into account to obtain a realistic estimate of possible drag reduction through airfoil morphing using current wing construction technology. Airfoil drag improvements of up to 4.6% are obtained through morphing of the section of upper surface located between the wing spars with respect to the baseline multi-point design in flow conditions typical of large business jet operation.

Cut-cell method based large-eddy simulation of tip-leakage flow

The turbulent low Mach number flow through an axial fan at a Reynolds number of 9.36 × 105 based on the outer casing diameter is investigated by large-eddy simulation. A finite-volume flow solver in an unstructured hierarchical Cartesian setup for the compressible Navier-Stokes equations is used. To account for sharp edges, a fully conservative cut-cell approach is applied. A newly developed rotational periodic boundary condition for Cartesian meshes is introduced such that the simulations are performed just for a 72° segment, i.e., the flow field over one out of five axial blades is resolved. The focus of this numerical analysis is on the development of the vortical flow structures in the tip-gap region. A detailed grid convergence study is performed on four computational grids with 50 × 106, 250 × 106, 1 × 109, and 1.6 × 109 cells. Results of the instantaneous and the mean fan flow field are thoroughly analyzed based on the solution with 1 × 109 cells. High levels of turbulent kinetic energy and pressure fluctuations are generated by a tip-gap vortex upstream of the blade, the separating vortices inside the tip gap, and a counter-rotating vortex on the outer casing wall. An intermittent interaction of the turbulent wake, generated by the tip-gap vortex, with the downstream blade, leads to a cyclic transition with high pressure fluctuations on the suction side of the blade and a decay of the tip-gap vortex. The disturbance of the tip-gap vortex results in an unsteady behavior of the turbulent wake causing the intermittent interaction. For this interaction and the cyclic transition, two dominant frequencies are identified which perfectly match with the characteristic frequencies in the experimental sound power level and therefore explain their physical origin.

Japan Aerospace Exploration Agency Studies for the Second High-Lift Prediction Workshop

This paper summarizes Japan Aerospace Exploration Agency studies using a structured grid solver UPACS and an unstructured grid solver TAS-code for the second AIAA CFD High-Lift Prediction Workshop test cases and follow-on problems. Computations are performed for the grid convergence study using the configuration without any slat tracks and flap track fairings on a series of multi-block structured grids and a series of unstructured grids provided by the High Lift Prediction Workshop committee. In this paper, the following are mainly discussed: 1) grid convergence verification study of the turbulence models using a two-dimensional bump case, 2) grid convergence studies for Spalart–Allmaras and shear stress transport turbulence models using UPACS, 3) change of grid convergence and predicted flow fields with and without a nonlinear quadratic constitutive relation model for Spalart–Allmaras and shear stress transport turbulence models using UPACS including evaluation at higher angles of attack, and 4) comparison studies using different solvers UPACS and TAS on different grid systems with and without the quadratic constitutive relation model for the Spalart–Allmaras model.