Lecture Objectives

• Review for exam – Course overview– Example questions from previous exams

• Discuss midterm project

Course Masterial Review

• Navier-Stokes equations and RANS

• Turbulence modeling

• Discretization and equation solver

• Boundary conditions

• Numerical properties

Conservation Equations Navier Stokes Equations

0z

v

y

v

x

v zyx

)(Sz

vμ

y

vμ

x

vμ

y

p)

z

vv

y

vv

x

vv

τ

vρ( yM2

y2

2

y2

2

y2

yz

yy

yx

y

TTgρ

xM2x

2

2x

2

2x

2x

zx

yx

xx S

z

vμ

y

vμ

x

vμ

x

p)

z

vv

y

vv

x

vv

τ

vρ(

zM2z

2

2z

2

2z

2z

zz

yz

xz S

z

vμ

y

vμ

x

vμ

z

p)

z

vv

y

vv

x

vv

τ

vρ(

qΦz

Tk

y

Tk

x

Tk)

z

TV

y

TV

x

TV

τ

T(ρc

2

2

2

2

2

2

zyxp

})y

v

z

v()

x

v

z

v()

x

v

y

v()

z

v()

y

v()

x

v{( 2zy2zx2yx2z2y2x

Reynolds Averaged Navier Stokes equations

xTy

ty

ty

tx

zx

yx

xx S]

z

V)μμ[(

z]

y

V)μμ[(

y]

x

V)μμ[(

xx

P)

z

VV

y

VV

x

VV

τ

Vρ(

yTy

ty

ty

ty

zy

yy

xy S]

z

V)μμ[(

z]

y

V)μμ[(

y]

x

V)μμ[(

xx

P)

z

VV

y

VV

x

VV

τ

Vρ(

zTy

ty

ty

tz

zz

yz

xz S]

z

V)μμ[(

z]

y

V)μμ[(

y]

x

V)μμ[(

xx

P)

z

VV

y

VV

x

VV

τ

Vρ(

]z

v)μμ(

y

v)μμ(

x

v)μμ[(

zSS z

ty

tx

tztzzTz

sSSimilar is for STy and STx

0z

V

y

V

x

V zyx

Momentum:

Continuity:

4 equations 5 unknowns → We need to model

1)

2)

3)

4)

tμ

Modeling of Turbulent Viscosity

μtμ

Fluid property – often called laminar viscosity

Flow property – turbulent viscosity

......

-k

-k

-k

Re

3

2

1Re

-k

Eq.

Two

Eq.-One

TKEM

constantMVM

μon based Models

t

t

fk

kl

l

Curvature

Buoyancy

Low

Layer

Layer

Layer

bounded

wall

Free

High

lengthmixing

MVM: Mean velocity modelsTKEM: Turbulent kinetic energy equation models

LES: Large Eddy simulation modelsRSM: Reynolds stress models

Additional models:

Discretization and equation solver

Step1: solve V* from momentum equations

Step2: introduce correction P’ and express V = V* + f(P’)

Step3: substitute V into continuity equation solve P’ and then V

Step4: Solve T , k , e equations

Guess p*

end

Converged (residual check)

yes

no

p=p*

Discretization of RANS SIMPLE algorithm

Boundary Conditions

– Surfaces (wall functions)• Velocity • Temperature• Concentration

– Inlets and outlets• Supply diffusers and • Outlets

Laminar sub-layer

Velocity profile

Surface cell

momentum sources

Numerical properties that affect the solutions

• Grid distribution- Grid dependent/independent solution

• Differencing scheme- Numerical diffusion

• Convergence - Residual and Number of iterations

• Stability - Relaxation

• Tests the solution - Check the conservation of mass, concentration, and energy

Exam

• Open book, open notes • 120 minutes • 25% of your final grade• Comprehensive • Several short problems and • Several questions

Examples are posted on the course website !

Example of short questions

(yes no or very short answer) 1) Reynolds stresses describe the property of fluid or

flow?

2) What is the difference between the shear stresses and Reynolds stresses?

3) Is the relaxation is necessary for the system of linear equations?

4) Is the SIMPLE algorithm explicit method for solving of system of equations?

5) Why do we use QUICK discretization scheme?

6) How many Reynolds stresses we have in two dimensional flow?

Explanation questions

• Describe the difference between explicit and implicit method. Which are the advantages and disadvantages of both?

• Explain how the dispersion of contaminant in the boundary layer depends on velocity field.

• Describe how the temperature field affects the airflow. Identify the term in conservation equation that link energy and velocity equation, and describe which assumption we used to get this term.

Project 1

T1=30C

T2=20C

outletinlet

outlet

outletinlet

inlet

T1

T2

Pat a) Numerical diffusion The purpose of this project part is to analyze how mesh size and orientation affects the accuracy of result.

Pat b) Learn how to: 1) Model: geometry, heat sources, concentration sources, diffusers, 2) Select important simulation parameters 3) Generate appropriate mesh4) Check the results 5) Present the results