Turbulence 2

8/9/2019 Turbulence 2

1/28


2/28

Turbulence

We require that this relationship depend only on the properties of the material, not of the motion

e.g. : Newtonian fluid Relation between stress & strain rate is linear

+

=

=

i

j

j

iijijkkij

v

ijx

u

x

ueee

2

1;

3

12)(

)ijij eF=

Newtonian :( )

= ijkkijv

ij ee 1

2

Viscosity (independent of the flow,dependent, on the fluid)

(6EA)EA9:ij

(6EA)EA9:ije

Most general linear relation

=klijklij

Symmetry reduces unknowns from 81 to 27

All but (2) (1 in practice) can be eliminate for isotropic fluid Newtonian

Turbulence & Turbomachinery Lab. Hanyang Univ. 2


3/28

Turbulence

Incom ressible

Derivative of density following fluid material pt. is zero

0=Dt

D

0or0 =

=

i

i

x

uur

+= ijkkijijij eep 1

2

+

=

k

k

k

kkk

x

u

x

ue

2

1

0 incompressilbe.0

+

+=

i

j

j

iijij

x

u

x

up for incompressible.

2

+

+

=

+

=

i

j

j

i

ji

ij

ji

i

x

u

x

u

xx

P

exxDt

if =constant throughout flow ,

0= i

i

x

u

0,1 2

=

+

= iiiuuPDu



4/28

Turbulence

Re nolds decom osition

Impossible to approach turbulence deterministically ?split flow into a mean and a random part (fluctuation part)

=~ iii

Instantaneous mean fluctuation

VVV~~ pPp +=

ijijij +=

deal with only constant density case

0

~

== R

: no densit fluctuation

( )

VV

0iii

uU

uUx

+

=+

ijij

jij

jjiix

pxx

uut

+

++

=

+++

Average these instantaneous equations to get equations governing the mean flow

( ) ( ) ( )t

UuU

tuU

tuU

t

iiiiiii

=+

=+

=+



5/28

Turbulence

ii xx

p

=

+

( ) )()()( VijVijVij Tx

Tx

=+

: gradient of mean pressure

: divergence of mean viscous stress

( ) ( )

j

i

j

j

j

ji

j

j

j

i

j

j

i

j

j

i

j

j

i

j

j

i

j

j

ii

jj

x

uu

x

Uuu

x

U

x

UU

x

uu

x

Uu

x

uU

x

UU

x

uUuU

+

+

+

=

+

+

+

=

++

0 0

Product of fluctuating quantities can not be assumed zero, they may or may not be zero. e.g.) 02 u

Look at continuity equation

=

=

0

i

i

i

i

j

j

ii

i

x

U

xxx

=

0 mean satisfies same continuity equation as does instantaneous

at equat on oes uctuat on sat s y

0~

=

=

i

i

ux

( ) 0~ =

ii Uux

i

ix0=

i

i

x

u even fluctuation satisfy same continuity equation



6/28


7/28


8/28

Turbulence

ibleincompress)(2 jiET ijtR

ij =

turbulent viscosity

The medium in this case is not the fluid, but the flow itself

Turbulent flow are flows !!

Keep going in spite of obvious inconsistence



9/28

Turbulence

Can we derive a dynamical equation for Reynolds stress ?

Can we get equation for velocity fluctuation itself ?

Instantaneous

V~~~~ )(

ij

V

ij

j

ij

ij

ij

i exxx

uu

t

u ~2~where)equationStokesNavier(~

=

+

=

+

321

ij

ij

ij

ij

xuu

xuU

xUu

xUU

+

+

+

Mean

ji

jj

V

ij

ij

j

j

i uuxx

T

x

P

x

UU

t

U

+

=

+

)(

11

j

i

jx

uu

442

Recall

pPpuUu iii

+=+=

~~

Subtract mean from instantaneous to get equation for fluctuation

j

V

ij

ij

i

j

j

i

j

j

i

j

j

i

j

i

xx

P

x

uu

x

uu

x

Uu

x

uU

t

u

+

=

+

+

+

)(

11

Fluctuation


Reynolds stress


10/28

Turbulence

Suppose infinitesimal disturbanceRemaining equation for fluctuation is linear & Reynolds stress is gone (2nd order)

Fluctuation :V

ijiii PUuu

Uu

+

=

+

+

)(11

linearjijj xxxxt

Mean :

j

V

ij

ij

i

j

i

x

T

x

P

x

UU

t

U

+

=

+

)(

11

No Reynolds stress

Equations for infinitesimal disturbance are linear & closed (well-posed)

Equations for mean flow is just the unchanged original viscous equations

- presence of infinitesimal disturbances does not change base flow (laminar solution)

Fluctuation equation can easily be transformed into Orr-Sommerfeld equation

Superimpose disturbance

, , ...

If infinitesimal disturbance grows, where does additional energy come from?

Develop energy equation for fluctuation 22211mean uc ua on ne c nergy per un mass

To 2nd order ( multiply linearized equation by )

32122

uuuuu ii ==

iu

edisturbancmalinfinitesiofEnergyKinetic1111

)(V

ij

iii

iiiii uP

uU

uuuuUuu

+

=

+

+


jjjj xxxxt


11/28

Turbulence

Energy equation for turbulent fluctuation

( momentum of fluctuation ) ( velocity ) = ( kinetic energy of fluctuation )

{

volumeunitperEnergyKinetic2~~~~

momentum

== uuuu

volumeun tper

+

=

+

j

ij

j

ii

j

ij

j

i

ij

ij

ii

x

Uu

x

uu

x

uu

x

u

x

P

x

uU

t

uu

2

21

321 0 continuity

+

=

+

=

j

i

ij

i

i

j

j

i

jj

ij

x

u

xx

u

x

u

x

u

xx

e

2

2

2

22222 q

kuuuuu =++==Let2

iji

iji

j

ii

ij

Uuu

uuu

x

uu

puqU

q

+

+

=

+

022

2

2

22

Reynolds stress

orjj

j

ij

i

ij

x

eu

2

2

iii uPpuPu 1

0 continuity

=

=

j

jiji

puxxxx

term where net effect on energy is zero

2


==

j

jj

j

j

i

ji uqxx

ux

uu22

since continuity


12/28

Turbulence

2

2

222

2

22 iii

i

j

i

i

ix

u

x

q

x

u

x

uu

xx

u

xu

x

uu

=

=

=

( ) ( ) ( )22222 ijijijj

i

ijiji

jj

ij

i eeuxx

ueeu

xx

eu

=

=

)0( =+ ijijijij ee

Summary

+

=

+

j

i

j

i

j

i

ji

j

jj

jj

jx

u

x

u

x

Uuu

q

xuqpu

xx

qU

t

q

22

11222

222

form 1

form 1

ijij

j

i

jiijijj

jj

jee

x

Uuueuuqpu

xx

qU

t

q

22

2

1122 222

+

=

+



13/28

Turbulence

True dissi ation

111

2

22

uuuuuu

ee ijij

0422

221

+

+

=

+

+

= Lxxxxxx ijij

Can only remove kinetic energy

Dissipation removes fluctuation kinetic energy and sends it to internal energy.

ijijeu2

.

- due to deformation work

Note difference with which acts to accelerate (increase Kinetic Energy) of neighboringu y ac on o v scous s ress.



14/28

Turbulence

Pseudo dissipation

Commonly the dissipation in the sense that synthetic

( )( ) ( )

++=++=

22

2D

D

eeeee

x

u

x

u

ijijijijijijijijijij

j

i

j

i

0

+

+

+

=+= LL2

1

1

1

44D

xxee ijijijij

in general D

Note : D0 always

Why is the real dissipation and not Dwhen both clearly remove Kinetic Energy from flow ?

0

0

=

ij

ije

u

r

1

0

0

=

ij

ije

ur

. .

distortionbut no rotationr

rigid body rotationfluid particle has no shape changer



15/28

Turbulence

ee 2 cannot de end on rotation directl

[ ]ijijijij

j

i

j

i eex

u

x

u +=

=D

2

Dissipation of energy is due only to distortion ( i. e. strain rates) not rotation

is real dissipation of energy

If turbulent is isotropic (almost never is) thenD

= and

115

=

ix

u

because all derivatives are interrelated

But at very high turbulent Re, the small scale turbulence is nearly isotropic local isotropy !At high turbulent Re, we shall see that dissipation occurs only at smallest scales of motion

Therefore in all turbulent flow at hi h turbulent Re 43 ~ul

= Dand15

2

1

ix

u

For turbulence modeling usually use form of Kinetic Energy equation involving D and and

treat D as if it were the dissipation.

O.K. at hi h turbulent Re

( )

jj x

q

x

22

,



16/28


17/28

Turbulence

Taylors frozen field hypothesis

suppose I can imagine turbulence being swept a fixed probe at velocity in such awaythat the turbulence does not change as it passes the probe

cU

cU

frequency : cUf =

1x

spatial disturbance : wavelength

CU

If disturbance is convected along 1x

t

then

tUx c =

1

1

-

Taylors frozen field hypothesis

- only good when turbulence intencity is less then 10%

c

rms

U

u

- differentiate to get

- square & ave. to get

t

u1

2

1

u

uppose ave pro e o measure a xe po n1 tu


t


18/28

Turbulence

Using Taylor frozen field hypothesis (if applicable)

2

1

2

2

1

1 1

=

t

u

Ux

u

c

If isotro a lies or locall isotro hi h turb. Re

:

2

1

2

115

=

t

u

Uc

.

high )10( 4>=

ulRl



19/28


20/28

Turbulence

Production

Dissi ation

= 00 VV ji

ji dVu

u

uudV = rate at which turb. KE is produced in 0V

massunitperproduction=

=0V

dV rate at which energy is dissipated in 0V

+

+

0

)(2

2

~~~

22

v

V

ijijj

j

uuudV

euuqpux

dV

r

0

2V

Since integrand is a divergence, we can apply the divergence them (since simply-connected region)

rrr

x=

vector field

( )xnn ~~~ =

+=

+

00

22

22

22

A

jijijj

V

ijijj

j

dsneuuqpueuuqpux

dV

~~~~ r

= dsnAdVArrr

snuuqup2

+

[ ] is evaluated on surface enclosing0A 0V


0 0


21/28

Turbulence

njj upnpunuprrr

==

0=nu if surface is at rest ( Kinematic B.C )

K.B.C 0=Dt

DF0=+

Fu

t

F r

0= nu on surface

02

2 jjnuq

on surface by kinematic B.C

ji

V

ijjiji nuneu)(2 =

ur

is identically zero on surface

- normal component due to K.B.C- tangential component due to no-slip

entire integral is exactly zero !

Therefore net effect of turbulent transport term (divergence) is zero :

on surface0)( j

V

iji nzu

ey ont ncrease or ecrease .

They only move energy around.

jpu - moves energy by action of pressure forces HLH

juq2

2- carries energy from region of higher energy to region of lower energy

i

V

ijijiueu)(

2 = - accelerate adjacent region of flow due to viscous stress

q2


jx - s m ar to a ove


22/28

Turbulence

Di ression on model for turbulencek- forget whole Reynolds stress eq.

- use eddy viscosity

2

kul =

LLLLk

2

3

2

2

2

2

2

~][~1

~][~][

2

2

1qk KE

e

ijijee 2= dissipation

+=

=

VUu

Euu

e

ijeji

2

xy

write & close eqn.s for &k

DP

+

=

+

jjj

kuqpu

kU

k2

1

jjj

Model transport term using a gradient-diffusion approach

- assume energy is moved down gradients in energy by these terms i.e. from high to low

Simplest model

j

jjx

kuqpu

22

1

[ transport term ]j

ex

D+

=

+

= ijeji

EECk

Ck

Uk

Eeuu

1


e

j

e

jj xxxt


23/28

Turbulence

Derive KE eq. for mean motion

ii PUu 1

ji

j

ij

jjj

jixxxxt

ijiji

jiijiijijj EExUuuEUUuuPU

xQ

xUQ

t 221

21

21 22 + +=+

KE eq. for mean flow

2

3

2

2

2

1 UUUUUQ ji ++==(when )

EEE ijij =2 : dissipation of KE to heat due to mean motion ( deformation )

j

iji

x

Uuu

is only term which occurs in both equs for mean and fluctuation , but with opposite sign!

Effect of this term is take energy from mean and put it into flucuation , or vice versa

iU

Almost always in engineering flow has apposite sign asjiuuj

x

0

j

iji

x

Uuu almost always 0P

mean fluctuation

j

iji

x

Uuu

PU

uu iji

is called turbulent production term


j


24/28

Turbulence

-

KE eq. For fluctuation

+

=

+ i

jiijij

jq

j

q uuueuuq

puu 2

1 22222

jjj

3-dimensionality ! ( tendency to isotropic )

222uuu +

2

3

2

2 uu

3217.0 uuu ==

Fluctuation mom.

j

ij

j

ij

j

ijij

jij

ij

i

x

Uuu

uux

uuexx

p

x

uUt

u

+

=

+

)2(

1

for i=1

+

=

+

j

j

j

j

j

jij

jj

jx

Uu

u

uu

x

uue

xx

p

x

uU

t

uu 111

1

111 )2(

1

or div. (no net effect on closed sys.)

j

j

j

jjj

jj

jx

ue

x

Uuu

x

upeuuu

xpu

xu

xUu

t

+

+

+

=

+

11

11

1

111

2

11

1

2

1

2

1 21

22

11

2

1

2

1

1P u


+

jjxxx

v1


25/28

Turbulence

j

j

j

jjj

jj

jx

ue

x

Uuu

x

upeuuu

xpu

xu

xUu

t

+

+

+

=

+

2222

2

222222

1

2222 21

22

11

2

1

2

1i=2 :

i=3 :

j

j

j

jjj

jj

jx

ue

xuu

x

upeuuu

xpu

xu

xUu

t

+

+

+

=

+

33

33

3

333

2

33

3

2

3

2

3 22222

continuity

+

=

3

3

2

2

1

1

x

u

x

u

x

u

+

=

3

3

2

2

1

1 11

x

u

x

up

x

up

The rote of the pressure-strain rate term in the eq.s for the individual compents of KE is to attempt to distribute

the energy among them, more or less equally.

This course a tendency of turbulence to return-to-isotropy when left alone (isotropy : )2

3

2

2

2

1 uuu ==



26/28

T b l


27/28

Turbulence

Uall derivatives of averaged quantities (except ) are identically zero

i.e.

2x( etcuuxx

u

02

1, 3

2

1

32

2

21

=

jx

ue

x

up

11

1

1

averages of products of fluctuations are not, in general ,zero even if flow is homogeneous !

write comp. of KE equs for the fluctuation.

ijij

uUu

eey

Uuv

=

=

111

20

0

P

j

j

j

j

uu

x

ue

x

up

xyx

+=

33

22

2

2

1

1

1

21

0

j

j

j

j

x

ue

y

Uuv

x

up

x

up

xe

xp

+

=

=

11

3

3

2

2

3

3

21

0

/


Turbulence


28/28

Turbulence

At very high turb. Re , the dissipation tends to be equally distributed between components( local isotropy of small scale )

11 Uuu

3

1

10

30

2

2

32

+=

+

=

x

up

yuv

xp

xp

3

1

10

3

3

+=

x

up

In fact it is the action of the ressure-strain rate terms which is res onsible for the local isotro ofthe small scales where the dissipation occurs

Production Dissipation in many shear flows, even where other terms are present.


Documents

Turbulence 2