1.  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. 
2.  Eca, Luis; Hoekstra, Martin; Vaz, Guilherme: Manufactured solutions for steadyflow Reynoldsaveraged Navier–Stokes solvers. In: International Journal of Computational Fluid Dynamics, 26 (5), pp. 313332, 2012. (Type: Journal Article  Abstract  Links  BibTeX) @article{2012_IJCFD_MSsteady_Eca_et_al, title = {Manufactured solutions for steadyflow Reynoldsaveraged Navier–Stokes solvers}, author = {Luis Eca and Martin Hoekstra and Guilherme Vaz}, url = {http://www.refresco.org/download/2012_ijcfd_mssteady_eca_et_al/}, doi = {10.1080/10618562.2012.717617}, year = {2012}, date = {20120605}, journal = {International Journal of Computational Fluid Dynamics}, volume = {26}, number = {5}, pages = {313332}, abstract = {This paper presents manufactured solutions (MS’s) for code verification of incompressible flow solvers based on the Reynoldsaveraged Navier–Stokes (RANS) equations. The proposed solutions mimic statistically steady, twodimensional or threedimensional nearwall 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 eddyviscosity turbulence models. Namely, the undamped eddyviscosity of the Spalart and Allmaras and Menter oneequations models, √kL from the one (SKL) and twoequation (KSKL) models proposed by Menter, the turbulence kinetic energy and the turbulence frequency included in twoequation kw 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 nearwall solution of the basic flow is kept to avoid zero friction at the wall. Threedimensional MS’s are obtained from the blending of the basic twodimensional MS’s in the transverse direction. All flow fields satisfy mass conservation, i.e. mean velocity fields are divergencefree. 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.}, keywords = {}, pubstate = {published}, tppubtype = {article} } This paper presents manufactured solutions (MS’s) for code verification of incompressible flow solvers based on the Reynoldsaveraged Navier–Stokes (RANS) equations. The proposed solutions mimic statistically steady, twodimensional or threedimensional nearwall 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 eddyviscosity turbulence models. Namely, the undamped eddyviscosity of the Spalart and Allmaras and Menter oneequations models, √kL from the one (SKL) and twoequation (KSKL) models proposed by Menter, the turbulence kinetic energy and the turbulence frequency included in twoequation kw 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 nearwall solution of the basic flow is kept to avoid zero friction at the wall. Threedimensional MS’s are obtained from the blending of the basic twodimensional MS’s in the transverse direction. All flow fields satisfy mass conservation, i.e. mean velocity fields are divergencefree. 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. 
2013 

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.  
2012 

Eca, Luis; Hoekstra, Martin; Vaz, Guilherme Manufactured solutions for steadyflow Reynoldsaveraged Navier–Stokes solvers Journal Article International Journal of Computational Fluid Dynamics, 26 (5), pp. 313332, 2012. Abstract  Links  BibTeX  Tags: incompressible flow, Method of Manufactured Solutions, Turbulence Models @article{2012_IJCFD_MSsteady_Eca_et_al, title = {Manufactured solutions for steadyflow Reynoldsaveraged Navier–Stokes solvers}, author = {Luis Eca and Martin Hoekstra and Guilherme Vaz}, url = {http://www.refresco.org/download/2012_ijcfd_mssteady_eca_et_al/}, doi = {10.1080/10618562.2012.717617}, year = {2012}, date = {20120605}, journal = {International Journal of Computational Fluid Dynamics}, volume = {26}, number = {5}, pages = {313332}, abstract = {This paper presents manufactured solutions (MS’s) for code verification of incompressible flow solvers based on the Reynoldsaveraged Navier–Stokes (RANS) equations. The proposed solutions mimic statistically steady, twodimensional or threedimensional nearwall 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 eddyviscosity turbulence models. Namely, the undamped eddyviscosity of the Spalart and Allmaras and Menter oneequations models, √kL from the one (SKL) and twoequation (KSKL) models proposed by Menter, the turbulence kinetic energy and the turbulence frequency included in twoequation kw 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 nearwall solution of the basic flow is kept to avoid zero friction at the wall. Threedimensional MS’s are obtained from the blending of the basic twodimensional MS’s in the transverse direction. All flow fields satisfy mass conservation, i.e. mean velocity fields are divergencefree. 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.}, keywords = {incompressible flow, Method of Manufactured Solutions, Turbulence Models}, pubstate = {published}, tppubtype = {article} } This paper presents manufactured solutions (MS’s) for code verification of incompressible flow solvers based on the Reynoldsaveraged Navier–Stokes (RANS) equations. The proposed solutions mimic statistically steady, twodimensional or threedimensional nearwall 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 eddyviscosity turbulence models. Namely, the undamped eddyviscosity of the Spalart and Allmaras and Menter oneequations models, √kL from the one (SKL) and twoequation (KSKL) models proposed by Menter, the turbulence kinetic energy and the turbulence frequency included in twoequation kw 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 nearwall solution of the basic flow is kept to avoid zero friction at the wall. Threedimensional MS’s are obtained from the blending of the basic twodimensional MS’s in the transverse direction. All flow fields satisfy mass conservation, i.e. mean velocity fields are divergencefree. 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. 