1.  Eca, Luis; Klaij, Christiaan; Vaz, Guilherme; Hoekstra, Martin; Pereira, Filipe: On code verification of RANS solvers. In: Journal of Computational Physics, 310 , pp. 418439, 2016. (Type: Journal Article  Abstract  Links  BibTeX) @article{2016JCPEcaKlaijVazPereiraHoekstra, title = {On code verification of RANS solvers}, author = {Luis Eca and Christiaan Klaij and Guilherme Vaz and Martin Hoekstra and Filipe Pereira}, url = {http://www.refresco.org/download/2016jcpecaklaijvazpereirahoekstra_codeverification/}, year = {2016}, date = {20160113}, journal = {Journal of Computational Physics}, volume = {310}, pages = {418439}, abstract = {This article discusses Code Verification of ReynoldsAveraged 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 eddyviscosity 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 onedimensional 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 nonorthogonality 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.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This article discusses Code Verification of ReynoldsAveraged 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 eddyviscosity 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 onedimensional 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 nonorthogonality 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. 
2.  Pereira F., Eca L; G., Vaz: On the order of grid convergence of the hybrid convection scheme for RANS codes. Congreso de Métodos Numéricos en Ingeniería (CNMI), Wageningen, The Netherlands, 2013. (Type: Conference  Abstract  Links  BibTeX) @conference{2013CNMIPereira, title = {On the order of grid convergence of the hybrid convection scheme for RANS codes}, author = {Pereira F., Eca L. and Vaz G.}, url = {http://www.refresco.org/download/ontheorderofgridconvergenceofthehybridconvectionschemeforranscodes/ }, year = {2013}, date = {20130628}, booktitle = {Congreso de Métodos Numéricos en Ingeniería (CNMI)}, address = {Wageningen, The Netherlands}, abstract = {This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends firstorder upwind with the secondorder centraldifference scheme. Although this approach was proposed forty years ago, it is still available in many CFD commercial solvers. One could naively assume that the order of grid convergence of this scheme would change linearly with the blending parameter. Therefore, the aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, twodimensional, incompressible, turbulent, nearwall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddyviscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are secondorder accurate. For flows dominated by convection, the Hybrid scheme remains firstorder accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends firstorder upwind with the secondorder centraldifference scheme. Although this approach was proposed forty years ago, it is still available in many CFD commercial solvers. One could naively assume that the order of grid convergence of this scheme would change linearly with the blending parameter. Therefore, the aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, twodimensional, incompressible, turbulent, nearwall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddyviscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are secondorder accurate. For flows dominated by convection, the Hybrid scheme remains firstorder accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear. 
3.  Pereira, Filipe; Eca, Luis; Vaz, Guilherme: ON THE ORDER OF GRID CONVERGENCE OF THE HYBRID CONVECTION SCHEME FOR RANS CODES. SEMNI Bilbao, Spain, 2013. (Type: Conference  Abstract  Links  BibTeX) @conference{2013CMNI_PereiraEcaVaz_MMS+ReFRESCO, title = {ON THE ORDER OF GRID CONVERGENCE OF THE HYBRID CONVECTION SCHEME FOR RANS CODES}, author = {Filipe Pereira and Luis Eca and Guilherme Vaz}, url = {http://www.refresco.org/download/2013cmni_pereiraecavaz_mmsrefresco/}, year = {2013}, date = {20130625}, address = {Bilbao, Spain}, organization = {SEMNI}, abstract = {This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends firstorder upwind with the secondorder centraldifference scheme. Although this approach was proposed forty years ago, it is still available in many CFD commercial solvers. One could naively assume that the order of grid convergence of this scheme would change linearly with the blending parameter. Therefore, the aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, twodimensional, incompressible, turbulent, nearwall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddyviscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are secondorder accurate. For flows dominated by convection, the Hybrid scheme remains firstorder accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear.}, keywords = {}, pubstate = {published}, tppubtype = {conference} } This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends firstorder upwind with the secondorder centraldifference scheme. Although this approach was proposed forty years ago, it is still available in many CFD commercial solvers. One could naively assume that the order of grid convergence of this scheme would change linearly with the blending parameter. Therefore, the aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, twodimensional, incompressible, turbulent, nearwall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddyviscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are secondorder accurate. For flows dominated by convection, the Hybrid scheme remains firstorder accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear. 
2016 

Eca, Luis; Klaij, Christiaan; Vaz, Guilherme; Hoekstra, Martin; Pereira, Filipe On code verification of RANS solvers Journal Article Journal of Computational Physics, 310 , pp. 418439, 2016. Abstract  Links  BibTeX  Tags: Code Verification, Manufactured solutions, Numerical error, Order of grid convergence, RANS @article{2016JCPEcaKlaijVazPereiraHoekstra, title = {On code verification of RANS solvers}, author = {Luis Eca and Christiaan Klaij and Guilherme Vaz and Martin Hoekstra and Filipe Pereira}, url = {http://www.refresco.org/download/2016jcpecaklaijvazpereirahoekstra_codeverification/}, year = {2016}, date = {20160113}, journal = {Journal of Computational Physics}, volume = {310}, pages = {418439}, abstract = {This article discusses Code Verification of ReynoldsAveraged 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 eddyviscosity 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 onedimensional 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 nonorthogonality 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.}, keywords = {Code Verification, Manufactured solutions, Numerical error, Order of grid convergence, RANS}, pubstate = {published}, tppubtype = {article} } This article discusses Code Verification of ReynoldsAveraged 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 eddyviscosity 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 onedimensional 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 nonorthogonality 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.  
2013 

Pereira F., Eca L; G., Vaz On the order of grid convergence of the hybrid convection scheme for RANS codes Conference Congreso de Métodos Numéricos en Ingeniería (CNMI), Wageningen, The Netherlands, 2013. Abstract  Links  BibTeX  Tags: Hybrid convection scheme, Method of, Order of grid convergence, RANS @conference{2013CNMIPereira, title = {On the order of grid convergence of the hybrid convection scheme for RANS codes}, author = {Pereira F., Eca L. and Vaz G.}, url = {http://www.refresco.org/download/ontheorderofgridconvergenceofthehybridconvectionschemeforranscodes/ }, year = {2013}, date = {20130628}, booktitle = {Congreso de Métodos Numéricos en Ingeniería (CNMI)}, address = {Wageningen, The Netherlands}, abstract = {This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends firstorder upwind with the secondorder centraldifference scheme. Although this approach was proposed forty years ago, it is still available in many CFD commercial solvers. One could naively assume that the order of grid convergence of this scheme would change linearly with the blending parameter. Therefore, the aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, twodimensional, incompressible, turbulent, nearwall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddyviscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are secondorder accurate. For flows dominated by convection, the Hybrid scheme remains firstorder accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear.}, keywords = {Hybrid convection scheme, Method of, Order of grid convergence, RANS}, pubstate = {published}, tppubtype = {conference} } This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends firstorder upwind with the secondorder centraldifference scheme. Although this approach was proposed forty years ago, it is still available in many CFD commercial solvers. One could naively assume that the order of grid convergence of this scheme would change linearly with the blending parameter. Therefore, the aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, twodimensional, incompressible, turbulent, nearwall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddyviscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are secondorder accurate. For flows dominated by convection, the Hybrid scheme remains firstorder accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear.  
Pereira, Filipe; Eca, Luis; Vaz, Guilherme ON THE ORDER OF GRID CONVERGENCE OF THE HYBRID CONVECTION SCHEME FOR RANS CODES Conference SEMNI Bilbao, Spain, 2013. Abstract  Links  BibTeX  Tags: Hybrid convection scheme, Method of Manufactured Solutions, Order of grid convergence, RANS @conference{2013CMNI_PereiraEcaVaz_MMS+ReFRESCO, title = {ON THE ORDER OF GRID CONVERGENCE OF THE HYBRID CONVECTION SCHEME FOR RANS CODES}, author = {Filipe Pereira and Luis Eca and Guilherme Vaz}, url = {http://www.refresco.org/download/2013cmni_pereiraecavaz_mmsrefresco/}, year = {2013}, date = {20130625}, address = {Bilbao, Spain}, organization = {SEMNI}, abstract = {This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends firstorder upwind with the secondorder centraldifference scheme. Although this approach was proposed forty years ago, it is still available in many CFD commercial solvers. One could naively assume that the order of grid convergence of this scheme would change linearly with the blending parameter. Therefore, the aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, twodimensional, incompressible, turbulent, nearwall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddyviscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are secondorder accurate. For flows dominated by convection, the Hybrid scheme remains firstorder accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear.}, keywords = {Hybrid convection scheme, Method of Manufactured Solutions, Order of grid convergence, RANS}, pubstate = {published}, tppubtype = {conference} } This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends firstorder upwind with the secondorder centraldifference scheme. Although this approach was proposed forty years ago, it is still available in many CFD commercial solvers. One could naively assume that the order of grid convergence of this scheme would change linearly with the blending parameter. Therefore, the aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, twodimensional, incompressible, turbulent, nearwall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddyviscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are secondorder accurate. For flows dominated by convection, the Hybrid scheme remains firstorder accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear. 