Grid Refinement Study Research Articles

Variational stokes

We propose a novel unsteady Stokes solver for coupled viscous and pressure forces in grid-based liquid animation which yields greater accuracy and visual realism than previously achieved. Modern fluid simulators treat viscosity and pressure in separate solver stages, which reduces accuracy and yields incorrect free surface behavior. Our proposed implicit variational formulation of the Stokes problem leads to a symmetric positive definite linear system that gives properly coupled forces, provides unconditional stability, and treats difficult boundary conditions naturally through simple volume weights. Surface tension and moving solid boundaries are also easily incorporated. Qualitatively, we show that our method recovers the characteristic rope coiling instability of viscous liquids and preserves fine surface details, while previous grid-based schemes do not. Quantitatively, we demonstrate that our method is convergent through grid refinement studies on analytical problems in two dimensions. We conclude by offering practical guidelines for choosing an appropriate viscous solver, based on the scenario to be animated and the computational costs of different methods.

Sixth Drag Prediction Workshop Results Using FUN3D with k-kL-MEAH2015 Turbulence Model

The Common Research Model wing/body configuration is investigated with the k-kL-MEAH2015 turbulence model implemented in FUN3D. This includes results presented at the Sixth Drag Prediction Workshop and additional results generated after the workshop with a nonlinear quadratic constitutive relation variant of the same turbulence model. The workshop-provided grids are used, and a uniform grid refinement study is performed at the design condition. A large variation between results with and without a reconstruction limiter is exhibited on “medium” grid sizes, indicating that the medium grid size is too coarse for drawing conclusions in comparison with experiment. This variation is reduced with grid refinement. At a fixed angle of attack near design conditions, the quadratic constitutive relation variant yielded decreased lift and drag compared with the linear eddy-viscosity model by an amount that was approximately constant with grid refinement. The k-kL-MEAH2015 turbulence model produced wing–root junction flow behavior consistent with wind-tunnel observations.

Fluid–structure interaction solver for compressible flows with applications to blast loading on thin elastic structures

In this study, we report the development and application of a fluid–structure interaction (FSI) solver for compressible flows with large-scale flow-induced deformation of the structure. The FSI solver utilizes a partitioned approach to strongly couple a sharp interface immersed boundary method-based flow solver with an open-source finite-element structure dynamics solver. The flow solver is based on a higher-order finite-difference method using a Cartesian grid, where it employs the ghost-cell methodology to impose boundary conditions on the immersed boundary. Higher-order accuracy near the immersed boundary is achieved by combining the ghost-cell approach with a weighted least squares error method based on a higher-order approximate polynomial. We present validations for two-dimensional canonical acoustic wave scattering on a rigid cylinder at a low Mach number and for flow past a circular cylinder at a moderate Mach number. The second order spatial accuracy of the flow solver was established in a grid refinement study. The structural solver was validated according to a canonical elastostatics problem. The FSI solver was validated based on comparisons with published measurements and simulations of the large-scale deformation of a thin elastic steel panel subjected to blast loading in a shock tube. The solver correctly predicted the oscillating behavior of the tip of the panel with reasonable fidelity and the computed shock wave propagation was qualitatively consistent with the published results. In order to demonstrate the fidelity of the solver and to investigate the coupled physics of the shock–structure interaction for a thin elastic plate, we employed the solver to simulate a 6.4 kg TNT blast loading on the thin elastic plate. The initial conditions for the blast were taken from previously reported field tests. Using numerical schlieren, the shock front propagation, Mach reflection, and vortex shedding at the tip of the plate were visualized during the impact of the shock wave on the plate. We discuss the coupling between the nonlinear dynamics of the plate and blast loading. The plate oscillates under the influence of blast loading and the restoration of elastic forces. The time-varying displacement of the tip of the plate is the superimposition of two dominant frequencies, which correspond to the first and second modes of the natural frequency of a vibrating plate. The effects of the material properties and length of the plate on the flow-induced deformation are briefly discussed. The proposed FSI solver is a versatile computational tool for simulating the impact of a blast wave on thin elastic structures and the results presented in this study may facilitate the design of thin structures subjected to realistic blast loadings.

USM3D Simulations for the F-16XL Aircraft Configuration

Numerical simulations of flow around the F-16XL are presented as a contribution to the Cranked Arrow Wing Aerodynamics Project International II. The NASA Tetrahedral Unstructured Software System with its USM3D solver is used to perform numerical simulations. This computational fluid dynamics suite is developed and maintained by the NASA Langley Research Center. The CRISP CFD package is used to provide error estimates and grid adaptation for validation of USM3D results. A subsonic high-angle-of-attack case flight condition 25 is computed and analyzed. Three turbulence models are used in the calculations: the one-equation Spalart–Allmaras, the two-equation shear-stress transport, and the turbulence models. Computational results and surface-static-pressure profiles are presented and compared with flight data. Solution validation is performed using formal grid-refinement studies, the solution of error-transport equations, and adaptive mesh refinement. The current study shows that the USM3D solver coupled with CRISP CFD can be used to provide valuable data in predicting vortex-flow physics on a complex configuration.

Effect of grid type and refinement method on CFD-DEM solution trend with grid size

Identification and quantification of numerical error is complicated when Computational Fluid Dynamics (CFD) is coupled with Discrete Element Method (DEM) to model particle-fluid interactions. The presence of solid particles in the model disrupts the typical monotonic convergence of CFD with grid size, due to the changing distribution of calculated void fraction. Quantification of numerical error is crucial to the development of CFD-DEM as an analysis and design tool, but investigations of this error have been limited. Guidelines are needed for model design to permit the quantification of numerical error, and subsequent grid refinement to minimize error. The present study investigates the solution trend and relative numerical error for different computational grid types over a range of grid sizes. Simulations of flow through a stationary particle bed are conducted using computational grids for columns with circular and square cross-sections. A hybrid rectangular and cylindrical hex mesh is used for the circular cross-section column, while an initially cubic hex mesh is applied to the square cross-section column. Non-uniform, uniform, and one-dimensional refinements are individually performed on grids for both types of columns. The grid refinement study procedure by Celik et al. (2008) is applied to the solution trend of simulations using each uniform refinement method, and the resulting average grid convergence index (GCI) is shown in several cases to be a good predictor of the relative numerical error.

Extension of a Regularization Based Time-adaptive Numerical Method for a Degenerate Diffusion-Reaction Biofilm Growth Model to Systems Involving Quorum Sensing

We extend a regularization based numerical method for a highly degenerate partial differential equation that describes biofilm growth to systems of PDEs describing biofilms with several particulate substances. The example for which we develop the method is a quorum sensing biofilm which consists of down- and up-regulated biomass fractions. We carry out computational studies to assess the effect of the regularization parameter, a grid refinement study and report briefly on parallel performance of our code under OpenMP on desktop workstations.

Investigation of Oscillating-Foil Power Generation in Constrained Flow

This paper presents a numerical study of a 2D pitching and plunging flat plate foil operating in the vicinity of near side walls. The efects of the side wall proximity and the stroke reversal fraction are compared to previously published water tunnel experimental results and the resulting vortex structures are used to comment on the efect of the constrained flow on overall power generation. Three side wall distances are defned by the closest approach of the trailing edge of the foil during the flapping cycle, including dw = 0.1c, dw = 0.5c, and dw = 1.0c as well as ‘Free Flow’ case done with tunnel walls at a distance of 10.0c. The stroke reversal fraction determines how quickly the foil pitches at the top and bottom of each fapping stroke and varies in increments of 0.1 from ∆TR = 0.1 to ∆TR = 0.5 (the sinusoidal pitching case). The foil pivots about its 0.44 chord point and has a plunge amplitude of 1.05c, a phase angle between pitch and plunge of ϕ = 90°, and a pitching amplitude of 73°. All cases are run at a Reynolds Number of 10,000 and a non dimensional frequency of k = 0.8 The flow solver is Fluent v14.5, run with second order spatial discretization and first order time stepping using a dynamically layered mesh to plunge the airfoil. Grid refinement and turbulence model studies are performed. The results are compared to previously published water tunnel experiments that use direct force measurements as well as PIV to visualize the flow patterns. Overall, the efect of the side walls is small compared to the effect of the reversal fraction or kinematic parameters, however it is demonstrated that the lift on the foil is increased near the wall due to an increase of the local velocity over the foil as the wall distance decreases. However, this changes the positions of the vortices during the rest of the flapping cycle and may or may not increase the overall efficiency depending on the choice of kinematic parameters. The highest increase in efficiency occurs for the sinusoidal motion (∆TR = 0.5) at a wall distance of dw = 0.5, in which the overall power extraction efciency increases to η = 14.2% as compared to the free flow efficiency of, = 11.1%, a relative increase of 28%. These trends are similar to those shown by the previously published experimental results.

Verification and Validation exercises for the flow around the KVLCC2 tanker at model and full-scale Reynolds numbers

This paper presents the quantification of numerical and modelling errors for the solution of the flow around the KVLCC2 tanker at model-scale Reynolds number. Numerical errors are also quantified for full-scale Reynolds number simulations to address the numerical accuracy of the prediction of scale-effects. The calculations are performed with the solver ReFRESCO using fourteen distinct Reynolds-Averaged Navier-Stokes (RANS) equations models. The quantities of interest for the Validation exercises at model-scale are the resistance coefficient and the velocity and turbulence kinetic energy fields at the propeller plane. Modelling errors are estimated using the ASME V&V20 procedure which requires numerical and experimental data with their respective uncertainties. Numerical uncertainties are dominated by the contribution of the discretization error, which is determined by grid refinement studies. Scale-effects are also assessed for the wake-fraction and form-factor.The outcome shows that quantifying modelling errors is not a trivial exercise that depends on the quality and details of simulations and experiments. Nonetheless, it is also evident that a quantitative evaluation of modelling errors is more reliable than traditional graphical comparisons of simulations and experiments. Full-scale results show scale-effects larger than numerical uncertainties that are illustrated for the form-factor and wake-fraction.

Computational investigations and grid refinement study of 3D transient flow in a cylindrical tank using OpenFOAM

The study of physic fluid for a liquid draining inside a tank is easily accessible using numerical simulation. However, numerical simulation is expensive when the liquid draining involves the multi-phase problem. Since an accurate numerical simulation can be obtained if a proper method for error estimation is accomplished, this paper provides systematic assessment of error estimation due to grid convergence error using OpenFOAM. OpenFOAM is an open source CFD-toolbox and it is well-known among the researchers and institutions because of its free applications and ready to use. In this study, three types of grid resolution are used: coarse, medium and fine grids. Grid Convergence Index (GCI) is applied to estimate the error due to the grid sensitivity. A monotonic convergence condition is obtained in this study that shows the grid convergence error has been progressively reduced. The fine grid has the GCI value below 1%. The extrapolated value from Richardson Extrapolation is in the range of the GCI obtained.

Source Term Discretization Effects on the Steady-State Accuracy of Finite Volume Schemes

Source terms often arise in Computational Fluid Dynamics to describe a variety of physical phenomena, including turbulence, chemical reactions, and certain methods used for code verification, such as the method of manufactured solutions. While much has already been published on the treatment of source terms, here we follow an uncommon approach, designing compatible source term discretizations in terms of spatial truncation error for finite volume schemes in multiple dimensions. In this work we examine the effect of source term discretization on three finite volume flux schemes applied to steady flows: constant reconstruction, linear reconstruction, and a recently published third-order flux correction method. Three source term discretization schemes are considered, referred to as point, Galerkin, and corrected. The corrected source discretization is a new method that extends our previous work on flux correction to equations with source terms. In all cases, computational grid refinement studies confirm the compatibility (or lack thereof) of various flux-source combinations predicted through detailed truncation error analysis.

High Order Numerical Methods for the Dynamic SGS Model of Turbulent Flows with Shocks

AbstractSimulation of turbulent flows with shocks employing subgrid-scale (SGS) filtering may encounter a loss of accuracy in the vicinity of a shock. This paper addresses the accuracy improvement of LES of turbulent flows in two ways: (a) from the SGS model standpoint and (b) from the numerical method improvement standpoint. In an internal report, Kotov et al. ( “High Order Numerical Methods for large eddy simulation (LES) of Turbulent Flows with Shocks”, CTR Tech Brief, Oct. 2014, Stanford University), we performed a preliminary comparative study of different approaches to reduce the loss of accuracy within the framework of the dynamic Germano SGS model. The high order low dissipative method of Yee & Sjögreen (2009) using local flow sensors to control the amount of numerical dissipation where needed is used for the LES simulation. The considered improved dynamics model approaches include applying the one-sided SGS test filter of Sagaut & Germano (2005) and/or disabling the SGS terms at the shock location. For Mach 1.5 and 3 canonical shock-turbulence interaction problems, both of these approaches show a similar accuracy improvement to that of the full use of the SGS terms. The present study focuses on a five levels of grid refinement study to obtain the reference direct numerical simulation (DNS) solution for additional LES SGS comparison and approaches. One of the numerical accuracy improvements included here applies Harten's subcell resolution procedure to locate and sharpen the shock, and uses a one-sided test filter at the grid points adjacent to the exact shock location.

On code verification of RANS solvers

This article discusses Code Verification of Reynolds-Averaged Navier Stokes (RANS) solvers that rely on face based finite volume discretizations for volumes of arbitrary shape. The study includes test cases with known analytical solutions (generated with the method of manufactured solutions) corresponding to laminar and turbulent flow, with the latter using eddy-viscosity turbulence models. The procedure to perform Code Verification based on grid refinement studies is discussed and the requirements for its correct application are illustrated in a simple one-dimensional problem. It is shown that geometrically similar grids are recommended for proper Code Verification and so the data should not have scatter making the use of least square fits unnecessary. Results show that it may be advantageous to determine the extrapolated error to cell size/time step zero instead of assuming that it is zero, especially when it is hard to determine the asymptotic order of grid convergence. In the RANS examples, several of the features of the ReFRESCO solver are checked including the effects of the available turbulence models in the convergence properties of the code. It is shown that it is required to account for non-orthogonality effects in the discretization of the diffusion terms and that the turbulence quantities transport equations can deteriorate the order of grid convergence of mean flow quantities.

Code verification for unsteady 3-D fluid–solid interaction problems

This paper describes a procedure to synthesize Manufactured Solutions for Code Verification of an important class of Fluid–Structure Interaction (FSI) problems whose behaviors can be modeled as rigid body vibrations in incompressible fluids. We refer this class of FSI problems as Fluid–Solid Interaction problems, which can be found in many practical engineering applications. The methodology can be utilized to develop Manufactured Solutions for both 2-D and 3-D cases. We demonstrate the procedure with our numerical code. We present details of the formulation and methodology. We also provide the reasonings behind our proposed approach. Results from grid and time step refinement studies confirm the verification of our solver and demonstrate the versatility of the simple synthesis procedure. In addition, the results also demonstrate that the modified decoupled approach to verify flow problems with high-order time-stepping schemes can be employed equally well to verify code for multi-physics problems (here, those of the Fluid–Solid Interaction) when the numerical discretization is based on the Method of Lines.

Large eddy simulations of isothermal confined swirling flow in an industrial gas-turbine

The paper describes the results of a computational study of the strongly swirling isothermal flow in the combustion chamber of an industrial gas turbine. The flow field characteristics are computed using large eddy simulation in conjunction with a dynamic version of the Smagorinsky model for the sub-grid-scale stresses. Grid refinement studies demonstrate that the results are essentially grid independent. The LES results are compared with an extensive set of measurements and the agreement with these is overall good. The method is shown to be capable of reproducing the observed precessing vortex and central vortex cores and the profiles of mean and rms velocities are found to be captured to a good accuracy. The overall flow structure is shown to be virtually independent of Reynolds number.

Large-eddy simulation of turbulent flow past wind turbines/farms: the Virtual Wind Simulator (VWiS)

A large-eddy simulation framework, dubbed as the Virtual Wind Simulator (VWiS), for simulating turbulent flow over wind turbines and wind farms in complex terrain is developed and validated. The wind turbines are parameterized using the actuator line model. The complex terrain is represented by the curvilinear immersed boundary method. The predictive capability of the present method is evaluated by simulating two available wind tunnel experimental cases: the flow over a stand-alone turbine and an aligned wind turbine array. Systematic grid refinement studies are carried out, for both single turbine and multi-turbine array cases, and the accuracy of the computed results is assessed through detailed comparisons with wind tunnel experiments. The model is further applied to simulate the flow over an operational utility-scale wind farm. The inflow velocities for this case are interpolated from a mesoscale simulation using a Weather Research and Forecasting (WRF) model with and without adding synthetic turbulence to the WRF-computed velocity fields. Improvements on power predictions are obtained when synthetic turbulence is added at the inlet. Finally the VWiS is applied to simulate a yet undeveloped wind farm at a complex terrain site where wind resource measurements have already been obtained. Good agreement with field measurements is obtained in terms of the time-averaged streamwise velocity profiles. To demonstrate the ability of the model to simulate the interactions of terrain-induced turbulence with wind turbines, eight hypothetical turbines are placed in this area. The computed extracted power underscores the significant effect of site-specific topography on turbine performance. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical analysis of water impact forces using a dual-time pseudo-compressibility method and volume-of-fluid interface tracking algorithm

An implicit algorithm based on a dual-time pseudo-compressibility method is developed to compute water impact forces on bodies. Flow fields of incompressible viscous fluids are solved using unsteady Reynolds-averaged Navier–Stokes equations. Pseudo-time derivatives are introduced into the equations to improve computational efficiency. A second-order volume-of-fluid interface tracking algorithm is developed in a generalized curvilinear coordinate system to track the interface between the two phases in the computational domain. A grid refinement study of the dam-break flow is performed as a validation, and the obtained solutions agreed well with the experimental data and with the results of other numerical simulations. Numerical analysis of water impact forces on a hemisphere, two cones, and a wedge through free falling in one degree of freedom is then performed. Free surface deformation, pressure coefficients, impact velocities, and vertical accelerations during impact are compared with available experimental data and theoretical results. Good agreement with these results is obtained.

NUMERICAL MODELING OF CAPILLARY–GRAVITY WAVES USING THE PHASE FIELD METHOD

In this paper, we present a numerical model based on the widely used finite element formulation to analyze in detail the effect of surface active agents on capillary–gravity wave parameters such as phase velocity and wave amplitude. Moreover, the effect of a physicochemical parameter, which is the ratio of surface concentration to surface tension is also considered. For a number of fluid samples covering a range of concentrations from 0 to 0.01 molar, the phase speed of waves propagating on the surface of the liquid is found to decrease monotonically as the concentration of the solution considered is increased up to a limit of 0.004 molar. This is attributed to the corresponding increase in capillary number. It is shown numerically that the Marangoni effects contribute to the interfacial dynamics for fluid with physicochemical parameter value greater than 0.5. Moreover, a grid refinement study shows accuracies and convergence orders of the numerical model.

Reprint of “CFD simulations of circulating fluidized bed risers, part I: Grid study”

In this work, a detailed grid refinement study was carried out for two well-documented circulating fluidized bed (CFB) systems with the focus on grid convergence of 2D numerical simulations. It is demonstrated that the grid convergence of numerical simulations depends on the flow field variable chosen for verification. For axial pressure gradient, this study shows that no general rule for grid size is available to guarantee the grid-independent results. In addition, the inlet and outlet configuration used in the 2D simulations shows a significant impact on the grid convergence. A 3D grid study is also presented with the intent to probe the differences between 2D and 3D numerical simulations with respect to the grid convergence. For the case considered in this study, the 3D simulation demonstrates better grid convergent behavior than the 2D simulation with comparable grid sizes.

A procedure for the estimation of the numerical uncertainty of CFD calculations based on grid refinement studies

This paper offers a procedure for the estimation of the numerical uncertainty of any integral or local flow quantity as a result of a fluid flow computation; the procedure requires solutions on sys...

Summary of Data from the Fifth Computational Fluid Dynamics Drag Prediction Workshop

Results from the Fifth AIAA Computational Fluid Dynamics Drag Prediction Workshop are presented. As with past workshops, numerical calculations are performed using industry-relevant geometry, methodology, and test cases. This workshop focused on force/moment predictions for the NASA Common Research Model wing-body configuration, including a grid refinement study and an optional buffet study. The grid refinement study used a common grid sequence derived from a multiblock topology structured grid. Six levels of refinement were created, resulting in grids ranging from to hexahedra, a much larger range than is typically seen. The grids were then transformed into structured overset and hexahedral, prismatic, tetrahedral, and hybrid unstructured formats all using the same basic cloud of points. This unique collection of grids was designed to isolate the effects of grid type and solution algorithm by using identical point distributions. This study showed reduced scatter and standard deviation from previous workshops. The second test case studied buffet onset at using the medium grid ( nodes) from the sequence described earlier. The prescribed alpha sweep used finely spaced intervals through the zone where wing separation was expected to begin. Some solutions exhibited a large side of body separation bubble that was not observed in the wind-tunnel results. An optional third case used three sets of geometry, grids, and conditions from the Turbulence Model Resource website prepared by the Turbulence Model Benchmarking Working Group. These simple cases were intended to help identify potential differences in turbulence model implementation. Although a few outliers and issues affecting consistency were identified, the majority of participants produced consistent results.