Lecture Objectives:

• Define 1) Reynolds stresses and 2) K-ε turbulence models

• Analyze General CFD (Transport) Equation– Discretization of Transport Equation

• Introduce Boundary Conditions

Modeling of Turbulent Viscosity

μtμ

Fluid property – often called laminar viscosity

Flow property – turbulent viscosity

......

-k-k-k

Re

321

Re

-k

Eq.Two

Eq.-One

TKEM

constantMVM

μon based Models

t

t

fkkll

CurvatureBuoyancyLow

LayerLayerLayer

boundedwall

Free

High

lengthmixing

MVM: Mean velocity modelsTKEM: Turbulent kinetic energy equation models

LES: Large Eddy simulation modelsRSM: Reynolds stress models

Additional models:

One equation models:Prandtl Mixing-Length Model (1926)

yVl ρ x2

tμ

-Two dimensional model-Mathematically simple

-Computationally stable

-Do not work for many flow types

Characteristic length (in practical applications: distance to the closest surface)

l

Vx

y

x

There are many modifications of Mixing-Length Model:- Indoor zero equation model: t = 0.03874 V l

Air velocity Distance to the closest surface

Kinetic energy and dissipation of energyKolmogorov scaleEddy breakup and decay to smaller length scales where dissipation appear

andk

model

Two equation turbulent model

Kinetic energy Energy dissipation

εkC ρ

2

μtμ

constant

We need to model

Two additional equations:

ρ-2k grad/μμdiv))Vdiv(kτkρ( μt ijijt EE

kinetic energy

k

ρ2-2 grad/μμdiv))Vdiv(τ

ρ(2

2211t

fCEEfC ijijt

dissipation

-k

From dimensional analysis

Reynolds Averaged Navier Stokes equations

xy

ty

ty

tx

zx

yx

xx S]

zV

)μμ[(z

]y

V)μμ[(

y]

xV

)μμ[(xx

P)z

VVy

VVx

VVτ

Vρ(

yy

ty

ty

ty

zy

yy

xy S]

zV

)μμ[(z

]y

V)μμ[(

y]

xV

)μμ[(xy

P)z

VV

yV

Vx

VV

τV

ρ(

zz

tz

tz

tz

zz

yz

xz S]

zV)μμ[(

z]

yV)μμ[(

y]

xV)μμ[(

xzP)

zVV

yVV

xVV

τVρ(

0z

Vy

Vx

V zyx

Momentum:

Continuity:

1)

2)

3)

4)

S)graddiv(ΓVdiv ρτ

ρ eff,

General format:

General CFD Equation

S)graddiv(ΓVdiv ρτ

ρ eff,

Values of , ,eff and S

Equation ,eff S

Continuity 1 0 0

x-momentum V1 + t -P/x+Sx

y-momentum V2 + t -P/y-g(T∞-Twall)+Sy

z-momentum V3 + t -P/z+Sz

T-equation T /l + t/t ST

k-equation k (+ t)/k G- +GB

-equation (+ t)/ [ (C1G-C2)/k] +C3GB(/k)

Species C (+ t)/c SC

Age of air + t

t =Ck2/ , G= t (Ui/xj +Uj/xi) Ui/xj , GB=-g(/CP)( t/T,t) T/ xi

C1=1.44, C2=1.92, C3=1.44, C=0.09 , t=0.9, k =1.0, =1.3, C=1.0

Discretization - Computers do not solve a partial differential equation (computers do only 1 + 1, 1 + 0, and, 0 + 1 )

- Convert partial differential equation into algebraic equations

obtain finite number of numerical values instead of continuous solution

- Available methods: - finite volume method

- finite difference method- finite element method

Finite Volume Method- Conservation of for the finite volume

w e

wel

hn

s

P EW

x

x x

- Finite volume is a fixed space in the flow domain with imaginary boundaries that allow the fluid to flow in and out.

- Integral conservation of the quantities such as mass, momentum and energy.

Divide the whole computation domain into sub-domains

One dimension:

General Transport Equation -3D problem steady-state

W E

N

S

H

L

P

Equation for node P in the algebraic format:

fΦaΦaΦaΦaΦaΦaΦa LLHHNNSSWWEEPP

S)gradΓ(div)Vdiv( ρ eff,

A 1D example of discretization of general transport equation

SgraddivΓVdiv ρτ

ρ eff,

Φeff Φ,x S)xΦ(Γ

xρΦρV

x

Steady state 1dimetion (x):

Point W and E represent the cell center of the west and east neighbors of cell P and w, e the neighboring surfaces. Integrating with Gaussian theorem on this control volume gives:

ew

P EW

x

xw xe

dAndVdivAV

e

wΦ

weff Φ,

eeff Φ,wxex dxS}

xΦΓ

xΦΓ{ρΦρVΦρV

To obtain the equations for the value at point P, assumptions are used to convert the surface values to the center values.

0dxS}xΦΓ

xΦΓ{ΦρVΦρV

e

wΦ

weff Φ,

eeff Φ,wxex

Steady–state 1D example

ew

P EW

x

xw xeWw ΦΦ If Vx > 0,

If Vx < 0, Ee ΦΦ

X direction

Pe ΦΦ

Pw ΦΦ

and

and

Diffusion term:

Source term: xSdxS Φ

e

wΦ

Assumption:Source is constant over the control volume

a)

b)

c)

I)

Convection term - Upwind-scheme:

WxPxwxex ΦρVΦρVΦρVΦρV

PxExwxex ΦρVΦρVΦρVΦρV

x

ΦΦ2ΦΓ x

ΦΦx

ΦΦΓxΦΓ -

xΦΓ

:thenx

ΦΦΓxΦΓ and

xΦΦΓ

xΦΓ

WPEeff Φ,

w

WP

e

PEeff Φ,

weff Φ,

eeff Φ,

w

WPeff Φ,

weff Φ,

e

PEeff Φ,

eeff Φ,

When mesh is uniform:X = xe = xw

1D example

Φeff Φ,x S)xΦ(Γ

xΦρV

x

0xSx

ΦΦ2ΦΓΦΦρV ΦWPE

eff Φ,P)W(or E)P(or x

ΦWEPeff Φ,

P)W(or E)P(or x SΦΦΦ2

xxΓ

ρ ΦΦx

ρV

After substitution a), b) and c) into I):

same

We started with partial differential equation:

and developed algebraic equation:

We can write this equation in general format:

fcΦbΦaΦ WPE

Equation coefficients

Φ Unknowns

General Transport Equation -3D problem steady-state

W E

N

S

H

L

PEquation in the algebraic format:

fΦaΦaΦaΦaΦaΦaΦa LLHHNNSSWWEEPP

Wright this equation for each discretization volumeof your discretization domain

x =

FΦA

7-diagonal matrix

60,000 cells (nodes)N=60,000

60,000 elements

60,0

00 e

lem

ents

This is the system for only one variable ( )Φ When we need to solve p, u, v, w, T, k, , C

system of equation is larger

S)gradΓ(div)Vdiv( ρ eff,

Boundary conditionsin CFD application in indoor airflow

Real geometry

Model geometry

Where are the boundary Conditions?

CFD ACCURACY

Depends on airflow in the vicinity ofBoundary conditions

1) At air supply device 2) In the vicinity of occupant 3) At room surfaces

Detailed modeling- limited by computer power

Surface boundaries

Wall surface

W use wall functions to model the micro-flow in the vicinity of surfaceUsing relatively large mesh (cell) size.

0.01-20 mmfor forced convection

thickness

Airflow at air supply devices

Complex geometry - Δ~10-4mWe can spend all our computing power for one small

detail

momentum sources

Diffuser jet properties

High Aspiration diffuser

D

L

D

L

How small cells do you need? We need simplified models for diffusers

Peter V. Nielsen

Simulation of airflow in In the vicinity of occupantsHow detailed should we make the geometry?

AIRPAK Software