Estimation of the Accuracy Obtained from CFD for industrial applications Prepared by Imama Zaidi

Sources of Uncertainty Computational Grid –Grid Spacing –Grid Topology Numerical Approximation User Errors Post Processing Turbulence Modelling Flow complexity (multi-phase flow, combustion… etc)

Problem Definition Rayleigh Number Ra= Reference Velocity V 0 =

Part I Laminar Flow

Flow inside Cavity

Types of Grids tested Square grid Polyhedral Skewed Butterfly type

Numerical Schemes tested Convection schemes: Second Order Upwind First Order Upwind Central Differencing Scheme Note: Runs are carried out in steady state mode

Richardson Extrapolation Grid independent solution Order of convergence Exact solution Targets: Nu and Mass flow across ½ section =>

Second Order Upwind Square grid

First Order Upwind Square grid

Central Differencing Scheme n=2.33 Ra=10 3

Effect of Order of Convergence Rich. Extrap. Using 10 2, 20, 40 Rich. Extrap. Using 20 2, 402, 802

Effect of Order of Convergence n=1.72 n=1.96

Richardson Extrapolation at Hot Wall

Post Processing Error

20X20 40X40

Example for user input Error Reference velocity is defined as:

Effect of Changing Reference Velocity

Effect of Numerical Scheme

Square Grid

Polyhedral Grid

Skewed Grid

Butterfly Type Grid

Why does error not always decrease? For square grid –Dx.Dt error > Dx2 ? But this is steady state –Residual normalisation? But increasing nb of iteration => no change For skewed grid –Error = constant, whatever h. –Need to test other “gradient reconstruction” methods for non-orthogonality

Part II Turbulent Flow

Flow inside Cavity

Type of Mesh

Low Reynolds Number Model Standard Low Reynolds Number Model (Lein et all) Abe Nagano Kondoh (ANK)Low Reynolds Model (Abe et all) V2f Model(Durbin et all) Model(Mentor) Spalart Allamaras (SA) Model (Baldwin et all)

Y+ from 40*40 Grid

Y+ from 80*80 Grid

Y+ from 160*160 Grid

Error From Different Turbulence Models For Nu Kω-sst model

Spalart Allmaras

V2f Model

K  -sst Model

Near Wall Grid Dependence Kw-sst Model Grid 80X80 and changing near-wall cell  x

Near Wall Grid Dependence V2f Model

Conclusions For distorted grids (Skewed Mesh), the refinement does not guarantee the accuracy The higher the order of scheme is, the higher will be the accuracy. Richardson extrapolation theory tested for laminar flow, seems to be in good agreement with the results for order of convergence nearly equals to 2

Conclusions K  SST model compared to the other models tested, seems more dependent on the grid refinement near the wall and grid independence is not reached even with 160x160 grid.