Lecture Objectives:- Define turbulence

– Solve turbulent flow example– Define average and instantaneous velocities

- Define Reynolds Averaged Navier Stokes equations

Fluid dynamics and CFD movies

• http://www.youtube.com/watch?v=IDeGDFZSYo8

• http://www.dlr.de/en/desktopdefault.aspx/tabid-6225/10237_read-26563/

• http://www.youtube.com/watch?v=oOGXEfgKttM

• http://www.youtube.com/watch?v=IFeSZZ49vAs

• http://www.youtube.com/watch?v=o53ghmaSFY8

x

Flow direction

line l

y

Point A

Point B

The figure below shows a turbulent boundary layer due to forced convection above the flat plate. The airflow above the plate is steady-state.

Consider the points A and B above the plate and line l parallel to the plate.

HW problem

Point A

a) For the given time step presented on the figure above plot the velocity Vx and Vy along the line l.b) Is the stress component xy lager at point A or point B? Why?

c) For point B plot the velocity Vy as function of time.

Method for solving of Navier Stokes (conservation) equations

• Analytical- Define boundary and initial conditions. Solve the partial

deferential equations.- Solution exist for very limited number of simple cases.

• Numerical - Split the considered domain into finite number of

volumes (nodes). Solve the conservation equation for each volume (node).

x

v

x

v xx

Infinitely small difference finite “small” difference

Numerical method

• Simulation domain for indoor air and pollutants flow in buildings

Solve p, u, v, w, T, C 3D spaceSplit or “Discretize”into smaller volumes

Capturing the flow properties

nozzle

Eddy ~ 1/100 in

Mesh (volume) should be smaller than eddies ! (approximately order of value)

2”

Mesh size for direct Numerical Simulations (DNS)

Also, Turbulence is 3-D phenomenon !

~2000 cells

~1000

For 2D wee need ~ 2 million cells

Mesh size

• For 3D simulation domain

3D space (room)

5 m

4 m

2.5 mMesh size 0.1m → 50,000 nodes

Mesh size 0.01m → 50,000,000 nodes

Mesh size 0.001m → 5 ∙1010 nodes

Mesh size 0.0001m → 5 ∙1013 nodes

supply

exhaustjet

jet

Indoor airflow

turbulent

The question is: What we are interested in: - main flow or - turbulence?

We need to model turbulence!

Reynolds Averaged Navier Stokes equations

First Methods on Analyzing Turbulent Flow

- Reynolds (1895) decomposed the velocity field into a time average motion and a turbulent fluctuation

- Likewise

stands for any scalar: vx, vy, , vz, T, p, where:

)z,y,(x,vz)y,(x,V)z,y,(x,v 'xxx

,

d

1

Time averaged component

Vx

vx’

From this class We are going to make a difference between large and small letters

Averaging Navier Stokes equations

, ρ ρ ρ 'vVv xxx

, pP p

'vVv yyy

'TT T

'vVv zzz

Substitute into Navier Stokes equations

Continuity equation:

0z

'v

y

'v

x

'v

z

V

y

V

x

V

z

)'vV(

y

)'vV(

x

)'vV(

z

v

y

v

x

v zyxzyxzzyyxxzyx

0z

'v

y

'v

x

'v

z

V

y

V

x

V zyxzyx

Average whole equation:

Instantaneous velocity

Averagevelocity

fluctuationaround averagevelocity

0z

'v

y

'v

x

'v

z

V

y

V

x

V zyxzyx

Average of average = average Average of fluctuation = 0

0 0 0

0z

V

y

V

x

V zyx

Average

time

'0'

2121221121 '')')('(

0'

Time Averaging Operations

divdiv

)( graddivgraddiv

)( )( )( '2

'12121 divdivdiv

Example: of Time Averaging

)vdiv(gradμ z

v

y

v

x

vx2

x2

2x

2

2x

2

) vv(vv) vv(z

vv

y

vv

x

vv xxx

xz

xy

xx

divdivdiv

=0 continuity

x2x

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(

xMxxx S)vdiv(gradμ

x

p))vdiv(v

τ

vρ(

Write continuity equations in a short format:

kvjvivv zyx

Short format of continuity equation in x direction:

Averaging of Momentum Equation

xxxx S)vdiv(gradμ

x

p))vdiv(v

τ

vρ(

xxxx S)vdiv(gradμ

x

p)vdiv(v ρ

τ

vρ

τ

Vρ

τ

Vρ

τ

)v'V(ρ

τ

)v'V(ρ

τ

vρ xxxxxxx

averaging

0

z

vv

y

vv

x

vv

)k)vvjvviv(v()k)vjvi(vv()vv(

'z

'x

'y

'x

'x

'x

'z

'x

''x

'x

'x

'z

''x

'x

''x

yy divdivdiv

z

vv

y

vv

x

vv)V(V )vv()V(V )vv(

'z

'x

'y

'x

'x

'x

x''

xxx

divdivdivdiv

)V div(grad)V div(grad)vdiv(grad xxx

Time Averaged Momentum Equation

x

'z

'x

'y

'x

'x

'x

2x

2

2x

2

2x

2x

zx

yx

xx S

z

vvρ

y

vvρ

x

vvρ

z

Vμ

y

Vμ

x

Vμ

x

P)

z

VV

y

VV

x

VV

τ

Vρ(

x2x

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(

Instantaneous velocity

Average velocities

Reynolds stresses For y and z direction:

y

'z

'y

'y

'y

'x

'y

2

y2

2

y2

2

y2

yz

yy

yx

y Sz

vvρ

y

vvρ

x

vvρ

z

Vμ

y

Vμ

x

Vμ

x

P)

z

VV

y

VV

x

VV

τ

Vρ(

z

'z

'z

'y

'z

'x

'z

2z

2

2z

2

2z

2z

zz

yz

xz S

z

vvρ

y

vvρ

x

vvρ

z

Vμ

y

Vμ

x

Vμ

x

P)

z

VV

y

VV

x

VV

τ

Vρ(

Total nine

Time Averaged Continuity Equation

Time Averaged Energy Equation

0z

v

y

v

x

v zyx

Instantaneous velocities

Averaged velocities

0z

V

y

V

x

V zyx

qΦz

Tk

y

Tk

x

Tk)

z

TV

y

TV

x

TV

τ

T(ρc

2

2

2

2

2

2

zyxp

Instantaneous temperatures and velocities

Averaged temperatures and velocities

qΦz

vTρ

y

vTρ

x

vTρ

z

Tk

y

Tk

x

Tk)

z

TV

y

TV

x

TV

τ

T(ρc

'z

''y

''x

'

2

2

2

2

2

2

zyxp

Reynolds Averaged Navier Stokes equations

0z

V

y

V

x

V zyx

x

'z

'x

'y

'x

'x

'x

2x

2

2x

2

2x

2x

zx

yx

xx S

z

vvρ

y

vvρ

x

vvρ

z

Vμ

y

Vμ

x

Vμ

x

P)

z

VV

y

VV

x

VV

τ

Vρ(

Reynolds stresses total 9 - 6 are unknown

y

'z

'y

'y

'y

'x

'y

2

y2

2

y2

2

y2

yz

yy

yx

y Sz

vvρ

y

vvρ

x

vvρ

z

Vμ

y

Vμ

x

Vμ

x

P)

z

VV

y

VV

x

VV

τ

Vρ(

z

'z

'z

'y

'z

'x

'z

2z

2

2z

2

2z

2z

zz

yz

xz S

z

vvρ

y

vvρ

x

vvρ

z

Vμ

y

Vμ

x

Vμ

x

P)

z

VV

y

VV

x

VV

τ

Vρ(

same

Total 4 equations and 4 + 6 = 10 unknowns

We need to model the Reynolds stresses !

Modeling of Reynolds stressesEddy viscosity models

)vρv(xx

vvρ '

x'x

'x

'x

Is proportional to deformation 'j

'ivρv

Boussinesq eddy-viscosity approximation

ρk3

2

x

V2μvvρ x

txx

i

j

j

i

x

V

x

V

x

V

y

Vμvvρvvρ yx

txyyx

Average velocity

x

V

z

Vμvvρvvρ zx

txzzx

z

V

y

Vμvvρvvρ yz

tzyyz

ρk3

2V2μvvρ y

tyy

y

ρk3

2V2μvvρ z

tzz

z

k = kinetic energy of turbulence

2

vv

2

vv

2

vvk

'z

'z

'y

'y

'x

'x

Substitute into Reynolds Averaged equations

tμ Coefficient of proportionality

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: