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

V0=

k

THCg P

32

THg

Part I

Laminar Flow

Flow inside Cavity

Types of Grids testedSquare grid

Polyhedral

SkewedButterfly type

_ . _h Control Vol N cells

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 =>

)ln(

)ln(

2

1

32

21

hhXXXX

n

1)(3

2

233

nt

hh

XXXX

Second Order UpwindSquare grid

First Order UpwindSquare grid

Central Differencing Scheme

n=2.33

Ra=103

Effect of Order of ConvergenceRich. Extrap. Using 102, 20, 40

Rich. Extrap. Using 202, 402, 802

Effect of Order of Convergence

n=1.72

n=1.96

Richardson Extrapolation at Hot Wall

Post Processing Error

20X2040X40

Example for user input Error

THgV 0Reference 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)

sstk

k

k

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.