M. Hoekstra

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.

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.

CFD Calculations for Free-Surface-Piercing Low Aspect Ratio Circular Cylinder With Solution Verification and Comparison With Experiments

The flow around free-surface piercing, low aspect-ratio circular cylinder is investigated by means of unsteady Reynolds averaged Navier-Stokes (URANS) calculations together with verification procedures and comparison with small-scale experimental and Particle Image Velocimetry results. A two-phase interface capturing model is used to handle the free-surface flow, together with k-ω SST turbulence model. We investigate physical and modeling aspects of this problem in order to gain more knowledge about the interaction of free-surface and free-end effects so that this mechanism is better understood and taken into account when modeling the problem in engineering-applied situations, such as the vortex induced motion of spars, tension-leg platforms and semi-submersibles.The case herein presented is a captive, low aspect-ratio cylinder (L/D = 2.0) with flow velocity corresponding to Reynolds and Froude numbers (both based on diameter) of Re = 4.3 × 104 and FnD = 0.31, respectively. We will show that appreciable free-surface effects are perceived on the flow, but with dominance of free-end effects, at least in terms of forces. Furthermore, we investigate different boundary conditions that would represent this free-surface problem to show that the separation of viscous and free-surface effects is not valid in this instance. Therefore, the interaction between viscous and free-surface effects is also tangentially investigated. In order to support our conclusions, we will show forces with uncertainty estimation and field variables obtained with different modeling strategies, unveiling physical and numerical aspects of this problem.Copyright

Verification of calculations of the potential flow around two-dimensional foils

An example of verification of calculations using methods based on grid-refinement studies and Richardson extrapolation is presented. The study is performed for numerical simulations of the potential flow around two-dimensional lifting foils by one low-order and two higher-order panel methods based on Morinos perturbation potential formulation. Flows with known analytical solutions have been selected to assess the reliability of the uncertainty estimations. The grid-convergence index method is the basis for these estimations. Two options are compared: estimating the uncertainty of the solution on a specific grid and of the solution found by extrapolation to zero grid cell size