Luís Eça

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 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. Estimation of the numerical uncertainty of any integral or local flow quantity.Least squares fits to power series expansions to handle noisy data.Excellent results obtained for manufactured solutions.Consistent results obtained for practical CFD calculations.Reduces to the well known Grid Convergence Index for well-behaved data sets.

A manufactured solution for a two-dimensional steady wall-bounded incompressible turbulent flow

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.

Verification of RANS solvers with manufactured solutions

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.

Code Verification, Solution Verification and Validation: An Overview of the 3rd Lisbon Workshop

This paper presents an overview of the 3 Workshop on CFD Uncertainty Analysis, dedicated to Verification (of Code and Solution) and Validation, held in Lisbon in October 2008. The Workshop proposed three different exercises for incompressible Reynolds-averaged Navier-Stokes (RANS) flow solvers: Code Verification with a Manufactured Solution, resembling a near-wall turbulent flow; Solution Verification for the flow over a backward facing step; Validation with a simplified version of the procedure proposed by the ASME VV the grid density required to attain the “asymptotic range” in RANS solutions is significantly higher than what is used nowadays in calculations for engineering purposes; it is nearly impossible to recognize that a given solution is in the “asymptotic range” from a single evaluation of the observed order of accuracy. As a consequence, in many practical RANS solutions it is safer to assume that the data are not in the “asymptotic range” (instead of blaming the uncertainty estimator for being too conservative). The Validation procedure tested in the Workshop is clearly a step forward compared to the “standard” graphical comparison between experiments and numerical predictions.

Manufactured solutions for steady-flow Reynolds-averaged Navier–Stokes solvers

This paper presents manufactured solutions (MSs) 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 MSs are obtained from the blending of the basic two-dimensional MSs 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.

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.

Flow Past a Circular Cylinder: A Comparison Between RANS and Hybrid Turbulence Models for a Low Reynolds Number

Several offshore applications deal with highly unsteady and detached flows, dominated by three dimensional effects. On such conditions, the usage of scale-resolving simulation (SRS) turbulence models has increased due to the well-known limitations of common RANS models. However, some of these offshore applications, such as flows past cylinders or raisers, present highly complex non-turbulent phenomena which, if not properly resolved, may pollute the outcome of any turbulence model. Therefore, it is crucial to mimic the flow conditions of the problem, the physical settings, and fulfil the numerical requirements of such problems to obtain reliable and accurate predictions. This paper assesses RANS and hybrid turbulence models, focusing on the dependence of the numerical predictions on the physical settings. To this end, the flow past a circular cylinder at a Reynolds number of 3900 is simulated using RANS, DDES and XLES models. The obtained results reveal a large dependence on the grid spatial resolution and physical settings, in particular on the computational domain width and boundary conditions. A substantial improvement of RANS predictions is found when a 3D computational domain is used. As expected, the hybrid models, DDES and XLES, lead to a better agreement with the experiments.Copyright

THE PROS AND CONS OF WALL FUNCTIONS

Simulations of viscous flows based on the Reynolds-Averaged Navier-Stokes (RANS) equations have become an engineering tool used on a daily basis. One of the main goals of such calculations is to determine friction forces, which are a consequence of the shear-stress at solid walls.In RANS (and other more sophisticated mathematical models), there are two main approaches for the determination of the shear-stress at a wall: direct application of the no-slip condition, i.e. the velocity gradient is determined directly at the surface; wall functions which determine the shear-stress at the wall from semi-empirical equations applicable up to the outer edge of the so-called “wall layer/log layer”. Although the first option is physically preferable, its numerical requirements may lead to iterative convergence problems and/or excessive calculation times. Therefore, especially at high Reynolds numbers, it is not unusual to use the latter approach.In this paper we discuss the advantages and disadvantages of wall-function boundary conditions. To this end we have calculated the flow around a flat plate, conventional and laminar airfoils and a circular cylinder. The influence of the location where wall functions are applied (distance to the wall) and the effect of the Reynolds number (ranging from model to full scale applications) are discussed. Griding requirements for wall-function boundary conditions are also addressed. The results obtained with wall functions are compared with those obtained from the direct application of the no slip at the wall.The results obtained in this study show that the use of wall functions in viscous flow calculations may be justifiable or completely unacceptable depending on the flow conditions. Furthermore, it is also shown that wall-function boundary conditions also require clustering of grid nodes close to the wall, but obviously less demanding than the direct application of no slip condition.Copyright

On the Numerical Prediction of the Flow Around Smooth Circular Cylinders

The numerical prediction of the flow around a smooth cylinder is one the classical test cases of Computational Fluid Dynamics (CFD). Different mathematical models have been used to address this statistically periodic flow. Namely, ensemble-averaged Navier-Stokes equations (URANS); partially-averaged Navier-Stokes equations (PANS); space filtered Navier-Stokes equations (large eddy-simulation LES or variational multi-scale VMS) and direct solution of the Navier-Stokes equation (DNS). Although all these models deal with turbulence in a very different way, all of them require a numerical solution and so they all require a careful control of the numerical uncertainty. We present an overall view of the values of the average drag coefficient (one of the most simple flow quantities that we could select) that have been published in the open literature, which shows a worrying spread of data. Therefore, it is logical to wonder if all these results are obtained with negligible numerical errors/uncertainties, especially when the scatter in the data also applies to results obtained with the same mathematical model. In this paper, we present Solution Verification exercises for the simplest model of those mentioned above: URANS. The calculations are performed at different Reynolds numbers and with different iterative convergence criteria using the ReFRESCO solver. The two-equation SST k–ω eddy-viscosity turbulence model is used in all the calculations performed in this study. The results presented show that numerical (iterative and discretization) errors may have a strong impact in the predictions and that misleading apparent convergence may be obtained with careless iterative convergence criteria. Furthermore, it is shown that grids with similar numbers of cells but different space distributions may lead to significantly different numerical uncertainties.Copyright

CODE VERIFICATION OF REFRESCO WITH A STATISTICALLY PERIODIC MANUFACTURED SOLUTION

Lu´is Ec¸aIST-ULMechanical Engineering DepartmentAv. Rovisco Pais 1, 1049-001 LisboaPortugalEmail: luis.eca@ist.utl.ptGuilherme VazMARINRD discretization errors (space and time) and the determina-tion of the observed order of (space and time) convergence.The availability of an exact solution allows the determina-tion of the numerical error and so the effects of iterative and dis-cretization errors can be addressed. The paper presents grid andtime refinement studies with different (iterative) convergence cri-teria and demonstrates that grid and time resolution are stronglyconnected when attempts are made to minimize the numericaluncertainty in the calculation of unsteady flows.The paper also addresses error estimation based on powerseries expansions in the calculation of unsteady (space and timedependent) flows. Simultaneous grid and time refinement is com-pared to grid refinement with fixed time step and time refinementwith fixed grid. The advantages and limitations of both optionsare discussed in the context of Code Verification (error evalua-tion) and Solution Verification (error estimation).INTRODUCTIONPapers presented at previous and present OMAE Confer-ences illustrate that the use of Computational Fluid Dynamics(CFD) tools in practical Engineering problems has become com-mon practice. But, considering the complexity of the flow prob-lems being handled, sooner or later the question about the trust-worthiness of CFD must arise. Is the CFD-result physically re-alistic or is it heavily biased by numerical errors of one kind oranother. That such question would arise has been anticipated andserious efforts have already been made to establish a methodol-ogy for estimating the confidence level of a numerical simula-tion. This is the realm of what is today generally denominatedby Verification and Validation (V&V) [1,2].The first step to check the trustworthiness of any CFD codeis Code Verification [1,2], i.e. the demonstration that the code isfree of errors. To achieve such goal, discretization errors, i.e. thedifference between the numerical and the exact solution must beevaluated. For the most applied mathematical model in CFD, i.e.the Reynolds-Averaged Navier-Stokes (RANS) equations, exactsolutions are as a rule not available. Fortunately, the Method ofthe Manufactured Solutions (MMS) [3, 4] offers an alternative.In the MMS, a continuum solution is first constructed, i.e. onespecifies all unknowns by mathematical functions. In general,this constructed solution will not satisfy the governing equationsbecause of the arbitrary nature of the choice. But by adding anappropriate source term, which removes any imbalance causedby the choice of the continuum solution, the governing equationsare forced to become a model for the constructed solution.In statistically steady flows, time-averaging is applied to theflow properties and to the conservation principles. As a conse-quence, time derivatives of the (mean) dependent variables van-ish and so (as for any steady flow) the RANS equations requirespatial discretization techniques only. In the open literature sev-eral examples of Code Verification exercises can be found forsteady (statistically or not) flow solvers, as for example [5,6,7,8].In all these examples, the correctness of the code is demonstratedwith numerical solutions obtained in a succession of systemati-cally refined grids. The main goal of the procedure is to demon-strate that the discretization error