NATO AVT-RTG-349 was dedicated to the validation of computational fluid dynamics (CFD) methods based on the Reynolds-Averaged Navier-Stokes (RANS) equations and statistical turbulence models for non-equilibrium turbulent-boundary-layer flows at high Reynolds numbers. This paper describes and discusses the errors and uncertainties arising in the comparison of RANS simulation results with experimental data from wind-tunnel experiments. These errors and uncertainties are associated with the CFD grid and the discretization, the physical modelling, the measurement accuracy, and the differences in the flow conditions between the experimental facility and the computational set-up. The results show the need for a grid-convergence study using systematically-refined families of CFD grids. The two major sources of errors are the RANS turbulence model and the uncertainty originating from the differences between the computational set-up and the wind-tunnel. Then two possible paths for future research are described: future CFD mesh generation, and future validation experiments at high Reynolds numbers.

This works aims at evaluating the ability of assessing drifting of a drowning body at sea using Computational Fluid Dynamics (CFD). This paper is focused on the evaluation of the numerical accuracy of the CFD simulations performed with the Simerics-MP flow solver, i.e., solution verification. The main goal of the paper is to illustrate the challenges found when using a commercial CFD package. The study of the drift of a drowning body at sea involves the simulation of incompressible, two-phase (air and water) flows around a bluff body and so the simplest mathematical model to tackle it is the Reynolds-averaged Navier-Stokes equations using ensemble average to define the mean flow and to average mass and momentum balances. This leads to unsteady flow problems that require integration in space and time. Therefore, numerical errors include contributions from the round-off, iterative, discretization (time and space) and statistical errors. Two simple situations to address the difficulties of estimating numerical uncertainties in the simulation of the drift of a drowning body at sea have been selected: (i) current induced drag forces, no wind and no waves; (ii) wind induced drag forces, no current and no waves. Grid/time refinement studies are performed for the two selected conditions to determine numerical uncertainties of the three selected quantities of interest: the resistance/drag force and the trim (pitch) and sinkage (heave) of the drowning body. It is clear that “default settings” cannot be used for such exercise and that there are small details that may have a significant influence on the estimated uncertainties. Nonetheless, the results show that it is not easy to obtain negligible numerical uncertainties in this type of simulations and so the planned comparison of simulations and experiments to determine the modelling error of CFD will be affected by a significant validation uncertainty.

Recently, experimental measurements of boundary-layers under non-equilibrium pressure gradients have been performed at Virginia Tech [1].The pressure gradient imposed to the boundary-layers is induced by a rectangular wing with a NACA 0012 section placed in the middle of a wind tunnel at angles of attack ranging between -10 and 12 degrees.A comparison of simulations performed by several RANS flow solvers using different turbulence models with the experimental data has been reported in [2].The study reported in [2] assumed a two-dimensional geometry and the proposed computational domain included a tilted top wall to take into account the displacement thickness of the side walls boundary-layers.The Reynolds number based on the velocity of the incoming flow, airfoil chord and fluid kinematic viscosity is = 2 × 10 , which for most naval applications corresponds to model scale.

Self-rectifying impulse turbines have two sets of guide vanes symmetrically located on each rotor side. Aerodynamic losses resulting from the inherent misalignment between the rotor outflow and the outlet guide vanes penalize these turbines’ efficiency. This paper presents the aerodynamic design of a radial guide-vane system for self-rectifying biradial impulse turbines and the relation between guide-vane flow deflection and blockage, rotor blade angle and turbine efficiency in design conditions. The system comprises two concentric rows of constant-thickness vanes. The design method solves a multi-objective optimization problem that maximizes deflection while minimizing outflow blockage, returning a Pareto optimal set of guide vanes. A subset of these results is used to configure and assess multiple turbine geometries. Data are obtained numerically with a RANS solver. The new guide-vane system improves flow deflection over reference designs based on aerofoil-section vanes with identical blockage. A turbine efficiency of 68.2% is achieved for a guide-vane deflection of 67.3°, blockage factor of 0.61 and rotor blade angle of 35°. The new design leads to an estimated reduction of 58% on the stagnation pressure losses in the outlet guide vanes.

A challenge that faces high-order methods for industrial applications generally is turbulence modeling at high Reynolds numbers. Large eddy simulation is studied extensively for high-order methods, nevertheless, its computational cost is enormous for industrial applications. The hybrid LES/RANS compromises the computational cost and the modeling error, however, solving the Reynolds-averaged Navier-Stokes equations is a resilient task for high-order methods, due to the non-smooth profiles of the turbulence quantities. Taking into account the complexity with high-order methods and the fairly large modeling errors of the RANS modeling, low-order methods has proved to be more pragmatic. For instance, in the discontinuous Galerkin framework, the polynomial approximation for these quantities leads to large oscillations that obstructs the non-linear solver. To use the high-order methods for industrial cases it is essential to have reliable implementations of two-equation turbulence models in RANS formulations. In this paper, a RANS discretization based on hybridizable discontinuous Galerkin is presented for the standard, TNT, BSL and SST versions of the k − ω model for applications at Reynolds numbers up to 109. A particular focus is given to the treatment of the specific rate of turbulence dissipation ω in the high-order framework. The complexity increases with these types of models as the value of ω goes to infinity at solid walls. Additionally, with minor modifications to the numerical flux definition, the turbulence model formulation can be solved by discontinuous Galerkin method as well. The results show remarkable improvements regarding the error magnitudes and non-linear convergence rate (iterative error) compared to second-order finite volume based solvers.

One of the major advantages of Computational Fluid Dynamics (CFD) is the ability to simulate flows at full-scale Reynolds numbers, which for naval applications can reach values of 109. For this range of Reynolds numbers, the Reynolds-averaged Navier-Stokes (RANS) equations are still the most common mathematical model. The averaging procedure used to derive the RANS equations generates the Reynolds-stresses that are determined by a turbulence model. In the last 50 years several turbulence models have been proposed in the open literature ranging from the initial eddy-viscosity algebraic methods to Reynolds-stress models that solve 7 additional transport equations. The assessment of the modeling error of RANS simulations requires the existence of experimental data and so most of the reported exercises of modeling error assessment are performed at model-scale Reynolds numbers. Furthermore, many of these studies rely on simple graphical comparisons without the quantification of experimental, numerical and input uncertainties. In this paper we apply the V&V20 validation metrics to quantitatively assess the modeling error of RANS using six different turbulence models in the simulation of the flow around the KVLCC2 at model-scale Reynolds number, for which experimental data are available at six cross-sections of the stern and near-wake region including the propeller plane. It is illustrated that graphical comparisons are less conclusive than the quantitative assessment of the modeling error taking into account uncertainties of experiments and simulations. The same validation metrics are applied to quantify the differences between simulations performed with the same models at full-scale Reynolds number. The results show quantitatively that differences between the results obtained with the six turbulence models at full-scale Reynolds number are much smaller than those observed at model-scale Reynolds number. In the absence of experimental data at full-scale, such exercise suggests that the role of the turbulence model in full-scale flows around ships is not as important as in model-scale simulations.

In engineering simulations involving turbulent fluid flows, the Reynolds-averaged Navier-Stokes (RANS) equations are still the most common mathematical model. The RANS equations require the use of a turbulence model to calculate the Reynolds stresses generated by the averaging of the momentum equations. Nowadays, the most popular turbulence models require the solution of additional transport equations that can range from one to seven equations. In this paper we illustrate the difficulties in attaining and identifying the so-called asymptotic range in grid refinement studies performed for the numerical solution of the RANS equations in the flow over a flat plate. Three turbulence models are tested: two-equation, eddy-viscosity, k—ω SST and k-kL turbulence models and the seven-equation Reynolds stress model SSG/LRR—ω. The three turbulence models are tested with second and first-order upwind schemes applied to the convective terms of the turbulence models transport equations. The results show that even in this simple flow, attaining the asymptotic order of grid convergence requires unreasonable levels of grid refinement. Furthermore, even in strictly geometrical similar grids, the observed order of grid refinement can be extremely sensitive to the discretization schemes used in the turbulence model and to any disturbances in the data. An alternative and more efficient way to address the quality of an error estimation based on a single term expansion is to determine the change of the estimate of the exact solution with grid refinement.

The determination of the transverse tip deflection of an elastic, hollow, tapered, cantilever, box beam under a uniform loading applied over half the length of the beam presented in the V&V10.1 standard is used to compare the application of the validation procedures presented in the V&V10.1 and V&V20 standards. Both procedures aim to estimate the modeling error of the mathematical/computational model used in the simulations taking into account the variability of the modulus of elasticity of the material used in the beam and the rotational flexibility at the clamped end of the beam. The paper discusses the four steps of the two error quantification procedures: (1) characterization of the problem including all the assumptions and approximations made to obtain the experimental and simulation data; (2) selection of the validation variable; (3) determination of the different quantities required by the validation metrics in the two error quantification procedures; (4) outcome of the two validation procedures and its discussion. The paper also discusses the inclusion of experimental, input, and numerical uncertainties (assumed or demonstrated to be negligible in V&V10.1) in the two validation approaches. This simple exercise shows that different choices are made in the two alternative approaches, which lead to different ways of characterizing the modeling error. The topics of accuracy requirements and validation comparisons (model acceptance/rejection) for engineering applications are not addressed in this paper.

Roughness effects are one of the main challenges of the prediction of ship resistance using traditional model tests and extrapolation procedures. Computational Fluid Dynamics (CFD) can play an important role in the improvement of empirical correlations. Nowadays, most CFD RANS solvers use an equivalent sand-grain roughness height to model roughness effects. Therefore, the simulation of roughness effects includes two main challenges: estimate the equivalent sand-grain roughness height that corresponds to a given average roughness height typically used to characterize the roughness of ships; include sand-grain roughness effects in the most accurate RANS turbulence models for the simulation of ship flows, as for example the $$k-\omega $$ SST eddy-viscosity model. In this work, the flows around different geometries (flat plate, submarine and two ships) at full scale Reynolds numbers ( $$10^8$$ to $$10^9$$ ) are simulated with RANS solvers using the $$k-\omega $$ SST eddy-viscosity model. Roughness effects are included in the k and $$\omega $$ boundary conditions for values of the sand-grain roughness height covering hydraulically smooth and fully rough surfaces. It is shown that with the proper scaling, the increase of the friction resistance coefficient with the sand-grain roughness height is equivalent for the four geometries tested. Conversion of average roughness height to sand-grain roughness is assessed by comparing CFD results with Bowden and Davison and Townsin et al. empirical correlations. Results of the simulations show the best agreement with the Townsin et al. correlation with a small variation of the ratio between average roughness and sand-grain roughness heights.

This paper discusses the application of the Area Metric to the quantification of modeling errors. The focus of the discussion is the effect of the shape of the two distributions on the result produced by the Area Metric. Two different examples that assume negligible experimental and numerical errors are presented: the first case has experimental and simulated quantities of interest defined by normal distributions that require the definition of a mean value and a standard deviation; the second example is taken from the V&V10.1 ASME Standard that applies the Area Metric to quantify the modeling error of the tip deflection of a loaded hollow tapered cantilever beam simulated with the static Bernoulli-Euler beam theory. The first example, shows that relatively small differences between the mean values are sufficient for the area metric to be insensitive to the standard deviation. Furthermore, the example of the V&V10.1 ASME Standard produces an Area Metric equal to the difference between the mean values of experiments and simulations. Therefore, the error quantification is reduced to a single number that is obtained from a simple difference of two mean values. This means that the Area Metric fails to reflect a dependence for the difference in the shape of the distributions representing variability. The paper also presents an alternative version of the Area Metric that does not filter the effect of the shape of the distributions by utilizing a reference simulation that has the same mean value of the experiments. This means that the quantification of the modeling error will have contributions from the difference in mean values and from the shape of the distributions.

The goal of this paper is to summarize and clarify the scope and interpretation of the validation procedure presented in the V&V20-2009 ASME Standard. In V&V20-2009, validation is an assessment of the model error, without regard to the assessment satisfying validation requirements. Therefore, validation is not considered as a pass/fail exercise. The purpose of the validation procedure is the estimation of the accuracy of a mathematical model for specified validation variables (also known as quantities of interest, system responses or figures of merit) at a specified validation point for cases in which the conditions of the actual experiment are simulated. The proposed procedure can be applied to variables defined by a scalar. For the sake of clarity, the paper reiterates the development and assumptions behind the V&V20-2009 procedure that requires the knowledge of the experimental values D and simulation values S at the set point and an estimate of the experimental, numerical and parameter uncertainties. The difference E between S and D is the centre of the interval that should contain the model error (with a certain degree of confidence) and the width of the interval is obtained from the validation uncertainty that is a consequence of the combination of the experimental, numerical and parameter uncertainties. The paper presents the alternatives to address parameter uncertainty and expands upon the interpretation of the final result. The paper also includes two examples demonstrating the application of the V&V20-2009 validation procedure including one problem from V&V10.1-2012 on solid mechanics.

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.

The paper presents a numerical study on the aerodynamics of a biradial turbine for wave energy conversion. The turbine is symmetrical with respect to a plane perpendicular to its axis of rotation. The two inlet/outlet rotor openings are axially offset from each other facing the radial direction and two rows of radial-flow movable guide-vanes surround the rotor. The existence of a counter-rotating eddy in the relative flow between rotor blades affects both the flow incidence angle at the rotor inlet and the outlet flow angle. A model was used to predict the incidence at the rotor inlet and the slip at the outlet, and the resulting data was employed to redesign the rotor inlet/outlet blade metal angle for the original guide-vane set. Numerical results for the new geometry were compared with data for the original turbine. The rotor and diffuser losses were reduced especially at operating conditions away from the peak efficiency point, resulting in higher average turbine efficiency.

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 Reynolds-Averaged Navier Stokes equations supplemented by eddy-viscosity models are still the most common mathematical model for the simulation of wall-bounded viscous flows at high Reynolds numbers. Viscous flow simulations around complex geometries are more accurate with the direct application of the no-slip condition at walls, i.e. avoiding the use of wall functions. Determination of the shear-stress at the wall from its definition requires the use of near-wall cells which are typically requested to present a non-dimensional height of y(+)similar or equal to 1. However, the effect of this widespread rule of thumb on the numerical accuracy of the solutions has not been established and it may be dependent on the turbulence model choice. In this paper, we discuss the numerical accuracy of viscous flow simulations performed with a RANS solver supplemented by three eddy-viscosity models in the calculation of statistically steady, incompressible flows: the one-equation model of Spalart & Allmaras and the two-equation models SST k-omega and k-root kL. The selected test cases are: a flat plate at Reynolds numbers Re equal to 10(7), 10(8) and 10(9); the NACA 0012 airfoil at angles of attack of 0, 4 and 10 degrees with Re = 6 x 10(8) and the KVLCC2 tanker at model and full scale Reynolds numbers, Re = 4.6 x 10(8) and Re = 2.03 x 10(9). Calculations are performed in sets of geometrically similar grids with different sizes of the near-wall cells height to assess the influence of this choice on the numerical uncertainty of the solutions. The results show that the numerical accuracy of the simulations is dependent on the selected turbulence model. For y(+) < 1. there is no significant effect of the near-wall cell height on the numerical uncertainty of the selected flow quantities for the Spalart & Allmaras and k-root kL models and so y(+)similar or equal to 1 is sufficient to obtain numerical uncertainties of friction resistance coefficients smaller than 1% . On the other hand, with the SST k-omega model, y(+) similar or equal to 1 leads to numerical uncertainties of friction resistance coefficients larger than 5% for all the present test cases. To attain the same uncertainty of the other turbulence models with the SST k-omega, one must use y(+)similar or equal to 0.1. (C) 2018 Elsevier Ltd. All rights reserved.

This study employs Partially-Averaged Navier-Stokes (PANS) equations to simulate the flow around a smooth circular cylinder at Reynolds number 3900. It intends to evaluate the importance of discretization and modelling errors on the accuracy of this mathematical model. Furthermore, the study addresses the effect of the physical resolution, or fraction of turbulence kinetic energy being modelled f(k), on the predictions accuracy. To this end, Validation exercises are carried out using five different values of f(k) which range from typical values for well resolved Scale-Resolving Simulations (f(k) <= 0.25) to Reynolds-Averaged Navier-Stokes equations (f(k) = 1.00). Naturally, these exercises require the evaluation of numerical errors, i.e. Verification studies. Consequently, and taking advantage of the ability of PANS to enable the distinction between discretization and modelling errors, spatial and temporal grid refinement studies are carried out to assess the magnitude of the discretization error, as well as its dependence on f(k). The outcome confirms the ability of PANS, in combination with f(k) < 0.50, to substantially decrease the modelling error when compared to f(k) = 1.00. However, the reduction of fk tends to increase the model dependence on the spatial and temporal resolution. It is demonstrated that similarly to the effect of the spatial and temporal grid resolution on the magnitude of the numerical error, the modelling error diminishes with the physical resolution (f(k) -> 0). The convergence of the predictions with f(k) is also illustrated.

The objective of this work is to investigate the challenges encountered in Scale-Resolving Simulations (SRS's) of turbulent wake flows driven by spatially-developing coherent structures. SRS's of practical interest are expressly intended for efficiently computing such flows by resolving only the most important features of the coherent structures and modelling the remainder as stochastic field. The success of SRS methods depends upon three important factors: i) ability to identify key flow mechanisms responsible for the generation of coherent structures; ii) determine the optimum range of resolution required to adequately capture key elements of coherent structures; and iii) ensure that the modelled part is comprised nearly exclusively of fully-developed stochastic turbulence. This study considers the canonical case of the flow around a circular cylinder to address these three key issues. It is first demonstrated using experimental evidence that the vortex-shedding instability and flow-structure development involves four important stages. A series of SRS computations of progressively increasing resolution are performed. An a priori basis for locating the origin of the coherent structures development is proposed and examined. The criterion is based on the fact that the coherent structures are generated by the Kelvin-Helmholtz (KH) instability. The most important finding is that the key aspects of coherent structures can be resolved only if the effective computational Reynolds number exceeds the critical value of the KH instability in laminar flows. Finally, a quantitative criterion assessing the nature of the unresolved field based on the strain-rate ratio of mean and unresolved fields is examined. The two proposed conditions and underlying rationale offer a quantitative basis for develop "good practice" guidelines for SRS of complex turbulent wake flows with coherent structures.

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.