This paper presents the assessment of the performance of nine discretization uncertainty estimates based on grid refinement studies including methods that use grid triplets and others that use a largest number of data points, which in this study was set to five. The uncertainty estimates are performed for the dataset proposed for the 2017 ASME Workshop on Estimation of Discretization Errors Based on Grid Refinement Studies including functional and local (boundary and interior) flow quantities from the two-dimensional flows of an incompressible fluid over a flat plate and the NACA 0012 airfoil. The data were generated with a Reynolds-averaged Navier-Stokes (RANS) solver using three eddy-viscosity turbulence models with double precision and sufficiently tight iterative convergence criteria to ensure that the numerical error is dominated by the discretization error. The use of several geometrically similar grid sets with different near-wall cell sizes for the same flow conditions lead to a wide range of convergence properties for the selected flow quantities, which enables the assessment of the numerical uncertainty estimators in conditions that are representative of the so-called practical applications.The evaluation of uncertainty estimates is based on the ratio of the uncertainty estimate over the "exact error" that is obtained from an "exact solution" obtained from extra grid sets significantly more refined than those used to generate the Workshop data. Although none of the methods tested fulfilled the goal of bounding the exact error 95 times out of 100 that was tested, the results suggest that the methods tested are useful tools for the assessment of the numerical uncertainty of practical numerical simulations even for cases where it is not possible to generate data in the "asymptotic range."

Two workshops were held at the ASME V&V Symposiums of 2017 and 2018 dedicated to Iterative Errors in Unsteady Flow Simulations. The focus was on the effect of iterative errors on numerical simulations performed with implicit time integration, which require the solution of a nonlinear set of equations at each time-step. The main goal of these workshops was to create awareness to the problem and to confirm that different flow solvers exhibited the same trends. The test case was a simple two-dimensional, laminar flow of a single-phase, incompressible, Newtonian fluid around a circular cylinder at the Reynolds number of 100. A set of geometrically similar multiblock structured grids was available and boundary conditions to perform the simulations were proposed to the participants. Results from seven flow solvers were submitted, but not all of them followed exactly the proposed conditions. One set of results was obtained with adaptive grid and time refinement using triangular elements (CADYF) and another used a compressible flow solver with a dual time stepping technique and a Mach number of 0.2 (DLR-Tau). The remaining five submissions were obtained with five different incompressible flow solvers (ansys cfx 14.5, pimplefoam, refresco, saturne, star ccm+ v12.06.010-R8) using implicit time integration in the proposed grids. The results obtained in this simple test case showed that iterative errors may have a significant impact on the numerical accuracy of unsteady flow simulations performed with implicit time integration. Iterative errors can be significantly larger (one to two orders of magnitude) than the residuals and/or solution changes used as convergence criteria at each time-step. The Courant number affected the magnitude of the iterative errors obtained in the proposed exercise. For the same iterative convergence criteria at each time-step, increasing the Courant number tends to increase the iterative error.

This article discusses numerical errors in unsteady flow simulations, which may include round-off, statistical, iterative, and time and space discretization errors. The estimation of iterative and discretization errors and the influence of the initial condition on unsteady flows that become periodic are discussed. In this latter case, the goal is to determine the simulation time required to reduce the influence of the initial condition to negligible levels. Two one-dimensional, unsteady manufactured solutions are used to illustrate the interference between the different types of numerical errors. One solution is periodic and the other includes a transient region before it reaches a steady-state. The results show that for a selected grid and time-step, statistical convergence of the periodic solution may be achieved at significant lower error levels than those of iterative and discretization errors. However, statistical convergence deteriorates when iterative convergence criteria become less demanding, grids are refined, and Courant number increased.For statistically converged solutions of the periodic flow and for the transient solution, iterative convergence criteria required to obtain a negligible influence of the iterative error when compared to the discretization error are more strict than typical values found in the open literature. More demanding criteria are required when the grid is refined and/or the Courant number is increased. When the numerical error is dominated by the iterative error, it is pointless to refine the grid and/or reduce the time-step. For solutions with a numerical error dominated by the discretization error, three different techniques are applied to illustrate how the discretization uncertainty can be estimated, using grid/time refinement studies: three data points at a fixed Courant number; five data points involving three time steps for the same grid and three grids for the same time-step; five data points including at least two grids and two time steps. The latter two techniques distinguish between space and time convergence, whereas the first one combines the effect of the two discretization errors.

The Volume-of-Fluid (VoF) model with interface-capturing scheme, available in many commercial and open-source CFD packages, may be widely used for simulations of free-surface hydrodynamics but is certainly not used without difficulties. While practitioners have come to expect conservation of mass and momentum, unconditional stability, second-order of accuracy and good iterative convergence, none of these qualities are evident when a free surface gets involved. In this paper, we investigate how these qualities are affected by the discontinuous nature of the volume fraction, which requires specialized numerical techniques for advection, for the interpolation of material properties, for the hydrostatic balance and for the pressure-weighted interpolation used in finite-volume methods with co-located variables. We find that conservation properties and good iterative convergence can still be attained, but not second-order of accuracy and unconditional stability. Code verification is presented to substantiate these findings, with cases ranging from one to three dimensions, from uniform to locally refined grids and from the stand-alone volume fraction equation to the complete set of mass, momentum and volume fraction equations. For the latter, a novel manufactured solution based on a sinusoidal wave in deep water is introduced. These tests permit not only to quantitatively assess the results for comparison with theoretical predictions but also ensure that our findings are characteristic of the VoF method and not of a particular implementation. (C) 2018 Elsevier Ltd. All rights reserved.

This paper presents grid refinement studies for statistically steady, two-dimensional (2D) flows of an incompressible fluid: a flat plate at Reynolds numbers equal to 107, 108, and 109 and the NACA 0012 airfoil at angles of attack of 0, 4, and 10 deg with Re = 6 × 106. Results are based on the numerical solution of the Reynolds-averaged Navier–Stokes (RANS) equations supplemented by one of three eddy-viscosity turbulence models of choice: the one-equation model of Spalart and Allmaras and the two-equation models k – ω SST and k−kL. Grid refinement studies are performed in sets of geometrically similar structured grids, permitting an unambiguous definition of the typical cell size, using double precision and an iterative convergence criterion that guarantees a numerical error dominated by the discretization error. For each case, different grid sets with the same number of cells but different near-wall spacings are used to generate a data set that allows more than one estimation of the numerical uncertainty for similar grid densities. The selected flow quantities include functional (integral), surface, and local flow quantities, namely, drag/resistance and lift coefficients; skin friction and pressure coefficients at the wall; and mean velocity components and eddy viscosity at specified locations in the boundary-layer region. An extra set of grids significantly more refined than those proposed for the estimation of the numerical uncertainty is generated for each test case. Using power law extrapolations, these extra solutions are used to obtain an approximation of the exact solution that allows the assessment of the performance of the numerical uncertainty estimations performed for the basis data set. However, it must be stated that with grids up to 2.5 (plate) and 8.46 (airfoil) million cells in two dimensions, the asymptotic range is not attained for many of the selected flow quantities. All this data is available online to the community.

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.

Eca and Hoekstra [1] proposed a procedure for the estimation of the numerical uncertainty of CFD calculations based on the least squares root (LSR) method. We believe that the LSR method has potential value for providing an extended Richardson-extrapolation solution verification procedure for mixed monotonic and oscillatory or only oscillatory convergent solutions (based on the usual systematic grid-triplet convergence condition R). Current Richardson-extrapolation solution verification procedures [2–7] are restricted to monotonic convergent solutions 0 < R < 1. Procedures for oscillatory convergence simply either use uncertainty estimate based on average maximum minus minimum solutions [8,9] or arbitrarily large factors of safety (F S ) [2]. However, in our opinion several issues preclude the usefulness of the presented LSR method: five criticisms follow.

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 systematically refined grids. The error is estimated with power series expansions as a function of the typical cell size. These expansions, of which four types are used, are fitted to the data in the least-squares sense. The selection of the best error estimate is based on the standard deviation of the fits. The error estimate is converted into an uncertainty with a safety factor that depends on the observed order of grid convergence and on the standard deviation of the fit. For well-behaved data sets, i.e. monotonic convergence with the expected observed order of grid convergence and no scatter in the data, the method reduces to the well known Grid Convergence Index. Examples of application of the procedure are included.

This paper discusses Verification and Validation for Computational Fluid Dynamics (CFD) applications. It emphasizes the differences between the mathematical problem of “solving the equations right” (Verification) and the science/engineering activity of “solving the right equations” (Validation). A clear distinction is made between Code Verification (error evaluation) and Solution Verification (error estimation) and procedures based on grid refinement studies are presented and discussed for both activities. The paper presents examples of Code Verification for Reynolds-Averaged Navier–Stokes solvers using the Method of the Manufactured Solutions; Solution Verification exercises including the KVLCC2 tanker at model and full scale Reynolds number and two examples of the application of the American Society of Mechanical Engineers (ASME) V&V 20 Validation procedure. This paper is written to stimulate a conscientious approach to CFD in marine applications.

SUMMARYThis paper discusses the estimation of discretization errors on the basis of power series expansions for grid sets that are not geometrically similar, that is, grids not exhibiting a constant grid refinement ratio for the entire computational domain. Simple test cases with structured and unstructured grids are used to demonstrate that reliable error estimates on the basis of power series expansions can be made if the grids are refined systematically. However, if the grid refinement ratio is not constant in the complete domain, the definition of the typical cell size is not obvious, and the observed order of accuracy may not be equal to the expected theoretical order of the discretization. Some alternatives for the definition of the typical cell size are tested. In these tests, the error estimation does not show a significant effect of the definition of the typical cell size even for some cases with data sets clearly outside the ‘asymptotic range’. For non‐geometrically similar grids, the best estimates of the observed order of accuracy are obtained with the typical cell size on the basis of the mode of the cell size (the cell size that occurs more often in a given grid). Copyright © 2012 John Wiley & Sons, Ltd.

This paper presents manufactured solutions (MS's) for code verification of incompressible flow solvers based on the Reynolds-averaged Navier–Stokes (RANS) equations. The proposed solutions mimic statistically steady, two-dimensional or three-dimensional near-wall turbulent flows in a simple domain (rectangle or rectangular box) at a given Reynolds number. The proposed analytical functions cover the mean flow quantities and the dependent variables of several eddy-viscosity turbulence models. Namely, the undamped eddy-viscosity of the Spalart and Allmaras and Menter one-equations models, from the one (SKL) and two-equation (KSKL) models proposed by Menter, the turbulence kinetic energy and the turbulence frequency included in two-equation k − ω models. A basic flow field resembling a turbulent flat plate flow is constructed with the turbulence quantities defined from ‘automatic wall functions’ that are supposed to reproduce more or less the normal behaviour of these variables. Alternative flow fields are constructed superposing a perturbation flow field that creates a ‘recirculation zone’. However, the near-wall solution of the basic flow is kept to avoid zero friction at the wall. Three-dimensional MS's are obtained from the blending of the basic two-dimensional MS's in the transverse direction. All flow fields satisfy mass conservation, i.e. mean velocity fields are divergence-free. The source functions required for the balancing of momentum and turbulence quantities transport equations and all the dependent variables and their derivatives are available in Fortran 90 modules.

This paper presents a numerical study on quantifying the effect of wall function boundary conditions on the calculation of ship viscous flows based on the RANS equations at model and full scale Reynolds numbers. Grid refinement studies are performed for four eddy-viscosity models with and without wall function boundary conditions. In the latter case, the conditions are imposed at varying distance to the wall. The predicted friction and pressure resistance coefficients are dependent on the location where the wall functions are applied. The magnitude of this influence depends on the Reynolds number and/or the turbulence model, reaching in some cases changes exceeding 10%. It is also shown that wall function boundary conditions may have a strong influence on the predicted wake field, especially at model scale Reynolds number.

This paper presents a study on the numerical requirements of including sand-grain wall roughness effects in the SST k–ω eddy-viscosity model. Three implementations are tried: two retain the direct application of the no-slip condition at the wall, the third is based on a wall function formulation. In the first two options the roughness effect is introduced via a change in wall boundary conditions, either for ω only or for k and ω. The two-dimensional flow along a finite flat plate is adopted to assess the numerical accuracy of the three approaches. The computed results are also compared with semi-empirical formula available in the open literature. It is demonstrated that sand-grain roughness effects can be simulated with acceptable numerical uncertainties with all three options, but the numerical settings to achieve that goal differ significantly.

This paper presents for the simple flow over a flat plate the near-wall profiles of mean flow and turbulence quantities determined with seven eddy-viscosity turbulence models: the one-equation turbulence models of Menter and Spalart & Allmaras; the k - omega two-equation model proposed by Wilcox and its TNT. BSL and SST variants and the k - root kL two-equation model. The results are obtained at several Reynolds numbers ranging from 10(7) to 2.5 x 10(9). Sets of nine geometrically similar Cartesian grids are adopted to demonstrate that the numerical uncertainty of the finest grid predictions is negligible.The profiles obtained numerically have relevance for the application of so-called 'wall function' boundary conditions. Such wall functions refer to assumptions about the flow in the viscous sublayer and the 'log law' region. It turns out that these assumptions are not always satisfied by our results, which are obtained by computing the flow with full near-wall resolution. In particular, the solution in the 'log-law' region is dependent on the turbulence model and on the Reynolds number, which is a disconcerting result for those who apply wall functions. Copyright (C) 2009 John Wiley & Sons, Ltd.

This article presents a study on the estimation of the numerical uncertainty based on grid refinement studies with the method of manufactured solutions. The availability of an exact solution and the convergence of the numerical solution to machine accuracy allow the determination of the exact error and of the distinct contributions of the iterative and discretization errors. The study focuses on three different problems of error/uncertainty evaluation (the uncertainty is in this case the error multiplied by a safety factor): the estimation of the iterative error/uncertainty; the influence of the iterative error on the estimation of the discretization error/uncertainty, and the overall numerical error/uncertainty as a combination of the iterative and discretization errors. The results obtained in this study show that it is possible to obtain a reliable iterative error estimator based on a geometric-progression extrapolation of the L∞ norm of the differences between iterations. In order to obtain a negligible influence of the iterative error on the estimation of the discretization error, the iterative error must be two to three orders of magnitude smaller than the discretization error. If the iterative error is non-negligible it should be added, simply arithmetically, to the discretization error to obtain a reliable estimate of the numerical error; combining by RMS is not conservative.

This article presents a study on the accuracy of the numerical determination of the friction and pressure resistance coefficients of ship hulls. The investigation was carried out for the KVLCC2 tanker at model- and full-scale Reynolds numbers. Gravity waves were neglected, i.e., we adopted the so-called double-model flow. Single-block grids with H–O topology were adopted for all the calculations. Three eddy viscosity models were employed: the one-equation eddy viscosity and the two-equation \( k \text{-}\!\sqrt k L \) models proposed by Menter and the TNT version of the two-equation k-ω model. Verification exercises were performed in sets of nearly geometrically similar grids with different densities in the streamwise, normal, and girthwise directions. The friction and pressure resistance coefficients were calculated for different levels of the iterative error and for computational domains of different size. The results show that on the level of grid refinement used, it is possible to calculate the viscous resistance coefficients in H–O grids that do not match the ship contour with a numerical uncertainty of less than 1%. The differences between the predictions of different turbulence models were larger than the numerical uncertainty; however, these differences tended to decrease with increases in the Reynolds number. The pressure resistance was remarkably sensitive to domain size and far-field boundary conditions. Either a large domain or the application of a viscous–inviscid interaction procedure is needed for reliable results.

This paper presents a study on the numerical calculation of the friction resistance coefficient of an infinitely thin plate as a function of the Reynolds number. Seven eddy-viscosity models have been selected: the one-equation turbulence models of Menter and Spalart–Allmaras; the k-ω two-equation model proposed by Wilcox and its TNT, BSL and SST variants and the \( k \text{-}\!\sqrt k L \) two-equation model. The flow has been computed at 14 Reynolds numbers in sets of seven geometrically similar Cartesian grids to allow a reliable estimation of the numerical uncertainty. The effect of the computational domain size has been reduced to negligible levels (below the numerical uncertainty). And the same holds for the iterative and round-off errors. In the finest grids of each set, the numerical uncertainty of the friction resistance coefficient is always below 1%. Special attention has further been given to the solution behaviour in the laminar-to-turbulent transition region. Curve fits have been applied to the data obtained at the 14 Reynolds numbers and the numerical friction lines are compared with four proposals from the open literature: the 1957 ITTC line, the Schoenherr line and the lines suggested by Grigson and Katsui et al. The differences between the numerical friction lines obtained with the seven turbulence models are smaller than the differences between the four lines proposed in the open literature.

This paper discusses code verification of Reynolds-Averaged Navier Stokes (RANS) solvers with the method of manufactured solutions (MMS). Examples of manufactured solutions (MSs) for a two-dimensional, steady, wall-bounded, incompressible, turbulent flow are presented including the specification of the turbulence quantities incorporated in several popular eddy-viscosity turbulence models. A wall-function approach for the MMS is also described. The flexiblity and usefulness of the MS is illustrated with calculations performed in three different exercises: the calculation of the flow field using the manufactured eddy-viscosity; the calculation of the eddy-viscosity using the manufactured velocity field; the calculation of the complete flow field coupling flow and turbulence variables. The results show that the numerical performance of the flow solvers is model dependent and that the solution of the complete problem may exhibit different orders of accuracy than in the exercises with no coupling between the flow and turbulence variables.

This paper presents a manufactured solution (MS), resembling a two-dimensional, steady, wall-bounded, incompressible, turbulent flow for RANS codes verification. The specified flow field satisfies mass conservation, but requires additional source terms in the momentum equations. To also allow verification of the correct implementation of the turbulence models transport equations, the proposed MS exhibits most features of a true near-wall turbulent flow. The model is suited for testing six eddy-viscosity turbulence models: the one-equation models of Spalart and Allmaras and Menter; the standard two-equation k–ε model and the low-Reynolds version proposed by Chien; the TNT and BSL versions of the k–ω model.

This paper presents manufactured solutions (MSs) for some well-known eddy-viscosity turbulence models, viz. the Spalart & Allmaras one-equation model and the TNT and BSL versions of the two-equation k-omega model. The manufactured flow solutions apply to two-dimensional, steady, wall-bounded, incompressible, turbulent flows. The two velocity components and the pressure are identical for all MSs, but various alternatives are considered for specifying the eddy-viscosity and other turbulence quantities in the turbulence models.The results obtained for the proposed MSs with a second-order accurate numerical method show that the MSs for turbulence quantities must be constructed carefully to avoid instabilities in the numerical solutions. This behaviour is model dependent: the performance of the Spalart & Allmaras and k-omega models is significantly affected by the type of MS. In one of the MSs tested, even the two versions of the k-omega model exhibit significant differences in the convergence properties. Copyright (c) 2006 John Wiley & Sons, Ltd.