Basic Concepts About Cfd Models

8/2/2019 Basic Concepts About Cfd Models

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Basic Concepts about CFD Models 1

Summer School in Heat and Mass Transfer

Lappeenranta University of Technology August 2010

Lecture Notes on

BASIC CONCEPTS ABOUT CFD MODELS

Prof. Walter AMBROSINI

University of Pisa, Italy

NOTICE: This material was personally prepared by Prof. Ambrosini specifically for this Course and is freely distributed to its

attendees or to anyone else requesting it. It has not the worth of a textbook and it is not intended to be an official publication. It

was conceived as the notes that the teacher himself would take of his own lectures in the paradoxical case he could be both

teacher and student at the same time (sometimes space and time stretch and fold in strange ways). It is also used as slides to be

projected during lectures to assure a minimum of uniform, constant quality lecturing. As such, the material contains reference

to classical textbooks and material whose direct reading is warmly recommended to students for a more accurate

understanding. In the attempt to make these notes as original as feasible and reasonable, considering their purely educational

purpose, most of the material has been completely re-interpreted in the teachers own view and personal preferences about

notation. Requests of clarification, suggestions, complaints or even sharp judgements in relation to this material can be directly

addressed to Prof. Ambrosini at the e-mail address:

[email protected]

MAIN SOURCES AND REFERENCE TEXTBOOKS:

N.E. Todreas, M. S. Kazimi Nuclear Systems I, Taylor & Francis, 1990. D.J. Tritton Physical Fluid Dynamics, Oxford Science Publications, 2nd Edition, 1997. H.K. Veersteg and W. Malalasekera An introduction to computational fluid dynamics, Pearson, Prentice Hall,

1995.

D.C. Wilcox Turbulence Modeling for CFD, 2nd Edition, DCW Industries, 1998. M. Van Dyke, An Album of Fluid Motion, The Parabolic Press, Stanford, CA, 1998.


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GENERAL REMARKS ON TURBULENT FLOW

Stability of laminar flow

In the boundary layer, the velocity profile evaluated by Blasius mayundergo unstable behaviour

In order to study stability of a nonlinear system by analytical means themethodology of linear stability analysis is often adopted. This has the

objective todetermine the stability conditions consequent to infinitesimalperturbations

In the present case, the methodology is applied as follows:the Blasius flow field is assumed for a given value of the thickness of

the boundary layer and of the free stream velocity w an infinitesimal perturbation of the pressure and velocity fields is

considered, having the form

( ) ( )tikxyWw xx += exp

( ) ( )tikxyWw yy += exp

( ) ( )tikxyPp += exp

where k is a real (wave) number and ir i + is complex;

the perturbation is introduced in the flow equations and termshaving order higher then the first are neglected (i.e., those termsincluding the product of more perturbations) thus linearising theequations

the asymptotic behaviour of the system at t is thus considered forvarying values of physical parameters (e.g., = wRe )

x

Turbulence

Buffer

Laminar Sublayer

Laminar Boundary Layer Turbulent Boundary Layer


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In the present case, it is found that the transient evolution is oscillatoryin time ( 0i ). According to the value ofRe and the frequency of the

perturbation (i.e., of i ) we may have:

0r the system is asymptotically stable <

0r the system is in a condition of marginal stability =

0r the system is asymptotically unstable >

Marginal stability conditions ( 0=r ) identify therefore a boundaryseparating stable and unstable conditions

As it can be noted, a value of the boundary layer Reynolds numberexists below which no perturbation is amplified

For greater values of the Reynolds number, it is predicted that somesmall perturbation may evolve in an unstable way ( 0>r ), giving rise to

the Tollmien-Schlichting waves

These waves have been observed in experiments in which perturbationshave been purposely introduced by different means

As a consequence of the nonlinear phenomena which are not consideredin the linear stability analysis actually the wave amplitude does notincrease indefinitely

firstly these waves become three-dimensionalthen their structure becomes more and more complicated giving rise

to eddies having a great geometrical complexity

2w

i

= wRe 1510critRe

Stable

Stable

Unstable


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In the case of a laminar flow of a fluid in a pipe, it is believed that thePoiseuille flow is actually stable for any value of the Reynolds number

The well known transition to turbulence demonstrated by OsborneReynolds in a celebrated experiments is therefore explained with the

instability of the boundary layer during its growth

These depends also on the way in which the fluid enters the duct (e.g.,on the shape of the inlet section) and on the level of disturbances (e.g.,vibrations in the whole system)

The result is a great variability of the Reynolds number at whichtransition occurs

( ) 2000 10mD critcrit

w DRe

=

Generally for Re = 4000 the flow is assumed turbulent Jets are also prone to transition to turbulence; this

occurs at values of the Reynolds number in the orderof 10 (opposed to 10

3for boundary layers)

In the case of external motion over cylinders in crossflow, the first phenomenon leading to a turbulent

wake is the appearance of von Karman vortex streets

(Re 40) whose frequency of detachment follows the

rule2.0= wfDSt

(St = Strouhal number); for Re > 300 theflow becomes irregular and a turbulent

wake is finally formed

TransitiondevL

Boundary LayerPotential flow core


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STATISTICAL TREATMENT OF TURBULENT FLOW

(according to Reynolds)

Turbulent flow is characterised by the chaotic fluctuation of variables(velocity, pressure, temperature, etc.) around mean values that may bealso variable (more slowly) in time

A description of the instantaneous behaviour of the fluid is of limitedinterest for engineering purposes

It is therefore preferable to describe the change in time or space ofaverage values, adopting a statistical treatment for fluctuations

The average value of the intensive variable c is therefore defined by therelationship

( ) ( )2

2

1 t t

t tc t c d

t

+

=

and the instantaneous value of c is decomposed in the summation of theaverage and thefluctuating value, having a zero time average

( ) ( ) ( )c t c t c t = + and ( )2

20

t t

t tc d

+

=

The time interval adopted in averaging t must be chosen long enoughto filter the turbulent fluctuations, but short enough to avoid

jamming the longer term variation of average quantities

The extent of fluctuations can be quantified by their quadraticaverages: 2c

c

t


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As a particular case, let us consider the quantities2

iw turbulence intensity for the i-th velocity component

222

zyx

www ++ turbulence intensity

ji ww ),,,( zyxji = double correlation

Turbulent intensity is strictly related to theturbulent kinetic energy( )2 2 2

1

2x y zk w w w = + +

Balance equations in terms of averaged quantities

In the case of turbulent flow, the local and instantaneous formulation ofbalance equation

( ) ( ) c cc c w J t

+ + =

(

1)

would be conveniently expressed in terms of average quantities

In this aim, the time averaging operator is applied to both sides of theabove equation, obtaining

( ) ( ) c cc c w J t

+ + =

where the linearity of the integral operator has been used

As a consequence of the assumptions on t it is( ) ( )c

tc

t

Moreover of a stationary reference frame, it is( ) ( ) cc JJwcwc

==

Therefore, it is( ) ( ) c cc c w J

t

+ + =

(1) As in previous Units, please consider that a different convention for the sign of the surface flux is adopted than in the

textbook by Todreas and Kazimi


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It is now introduced the assumption that each variable can bedecomposed in an average and a fluctuating component

+= www =

+= 000 uuu

taking into account that the average of each fluctuating component iszero

It is:( ) ( )( )[ ] c

tc

tc

tc

tcc

tc

t

+

+

+

=++

=

Considering that

cc = and 0== cc

it is

( ) ct

ct

ct

+

=

For the advection term, it is also:( ) ( )( )( )wwccwc +++=

wcwcwcwcwcwcwcwc +++++++=

where

wcwc

= 0=== wcwcwc

wcwcwcwcwcwc

===

It is therefore:

( ) =wc

+wc

wcwcwcwc +++

It is then useful to putwcwcwcwcJ

tc +++=

It can be recognised that this flux, though it is obtained by an

advection term, it is conveniently expressed as a sort ofdiffusive termoriginating from turbulence

In analogy with the above, it is also:( )( ) ccccc +=++=


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Making use of this definition, the general balance equation becomes( ) ( ) ( ) tc c c cc c w J c J

t t

+ + = +

where, besides the terms depending on the averaged variables, terms

due to the presence of fluctuations appear

Assuming that density fluctuations are negligible (or that the fluid isincompressible) it is obtained

( ) wcJct

tcc ==

==

,00

Therefore the balance equation in terms of averaged variablesbecomes

( ) ( ) c cc c w J c wt

+ + =

which is formally similar to the original local and instantaneousformulation, except for the appearance of the term

wc

The presence of this term remembers that, though the equation isexpressed in terms of averaged variables, turbulent fluctuations play a

major role in the transport of the extensive property C

This becomes even clearer, by writing( ) ( ) ( )tc c cc c w J J

t

+ + + =

from which it can be noted that the equations in terms of averagequantities can be formally treated as the local instantaneous ones byadopting an appropriate definition of the effective flux term

eff t

c c cJ J J= +

taking into account both themolecular and theturbulent transfers:

( ) ( )effc cc c w J

t + + =


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In fact, the transfer of momentum and energy in a turbulent flowoccurs due to molecular and also turbulent phenomena; in fact:

regions with larger or smaller velocity exchange fluid with each othergiving rise to a net transport of momentum

regions with higher and lower temperature exchange fluid with eachother giving rise to a thermal energy transfer

This occurs also in the presence of azero net fluid motion (zero mean

advection), i.e., even if ( ) 0= wc ,since even in such a case it may be

0 wc

This justifies the choice to define wc as a term with of superficial flux

having a seemingly diffusive nature

Therefore, making the usual choicesfor c , cJ

and c the averaged mass,

momentum and energy equations are

obtained

mass( ) 0=+

wt

momentum( ) ( ) ( ) ( )wwgIpwww

t+=+

energy( ) ( ) ( )

++

+=+

wuwgqwIpqwuu

t

000

Increasin Tem erature

Heat Flux


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MOMENTUM TRANSFER IN TURBULENT FLOW

Eddy Viscosity

As it was seen, momentum equation in terms of averaged quantities is( ) ( ) ( ) ( )wwgIpwww

t+=+

According to the above treatment the term w w can be interpretedas the turbulent contribution to the superficial flux of momentum, that

is to thedeviatoric stress tensor

This contribution takes the name ofReynolds stress tensor (it is a tensoras a result of the dyadic product of fluctuating velocity)

wwRe =

its meaning can be understood withreference to the figure, obtained in

analogy to the one already seen for heattransfer

The effective value of the deviatorictensor is therefore given by the

summation of the molecular and the

turbulent contributions

wwReeff =+=

The momentum equation thereforebecomes

( ) ( ) ( ) gpwwwt

Re

+++=+

The evaluation of the Reynolds stress tensor may be performed usingtheBoussinesq assumption in similarity with the case of laminar flow

( ) ( )Re , , 23ji

T iji j i j

j i

www w kx x

= = +

where T is the eddy diffusivity for momentum transfer or eddy viscosity

Increasing velocity

Momentum


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It is therefore possible to define:( ) [ ]

,

jieff T

i jj i

ww

x x

= + +

(the diagonal term is collapsed with pressure)

T has the same units of , i.e., [ ]sm2 but, unlike the kinematicviscosity, it is not a thermo-physical property of the fluid, since it depends

also on the flow field

Different kinds of turbulent flows must be distinguished to modelturbulent viscosity:

isotropic turbulence: the quantities characterising turbulence do notdepend on the reference frame orientation at a given location

homogeneous turbulence: the quantities characterising turbulence donot change in space

homogeneous isotropic turbulence: it is an ideal conditions sharingthe characteristic of the two previous cases; sit can be obtained, e.g.,

in wind tunnel downstream an appropriate mesh

wall turbulence: the turbulent motion is affected by the presence of asolid wall; turbulence is in this case non homogeneous andanisotropic

free turbulence: it is the case of turbulent flow which is not directlyaffected by a material boundary (e.g., in jets and wakes)

Concerning the effect of walls on turbulence, it is interesting to analysethe following classical plots for flow between two parallel plates

The plots show that, in the considered case:

max,2

zx ww

x

max,2

zz ww

025.0

050.0

075.0

100.0

125.0

x

effzx,

tzx,


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turbulent intensity is in the order of some percentage of themaximum axial velocity (up to 10% along the flow direction)

close to the wall turbulence intensity along z is greater than along x(anisotropic turbulence)

close to the wall in turbulence intensity undergoes considerablechanges, reaching a maximum at some distance from it

turbulence intensity in the centre of the channel has comparablevalues in both directions; moreover, the change along x is milder

(there is a tendency towards homogeneous isotropic turbulence)

the effective shear stress linearly increases with x, being zero in thecentreline, as it can be shown by a force balance between shear stress

and pressure drop

the turbulent shear stress is zero close to the wall (in the viscoussublayer where turbulence is zero) but it becomes rapidly equal to

the total shear stress at some distance from it

Different algebraic models have been defined to evaluate eddyviscosity. Some are quoted hereafter; a more complete discussion of

turbulent transport equations will be provided later on.

Turbulent viscosity according to BoussinesqIt is the basic assumption for isotropic turbulence models proposedback in 1877, consisting in the definition of a turbulent dynamic

viscosity t to be defined locally; it is the assumption at the basis of

many models

y

wxttyx

=

Prandtl Mixing Length Theory(1925)Prandtl assumed a definition having the form

y

w

y

wl xx

tyx

=

2

where l is themixing length, i.e, the distance to be covered by eddies

to produce the observed shear stress

The model has an analogy with kinetic theory of gases, in whichmolecular viscosity is the result of an average molecular velocitymultiplied by a mean free path


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Prandtl assumed that l was linearly dependent on the distancefrom the wall: kyl = .

Similarity assumption by von Karman (1930)On the basis of dimensional considerations von Karman proposed

the following formulation

( )

y

w

yw

ywk x

x

xtyx

=

222

322

where 2k is a universal constant whose value for turbulent flow in

pipes is about 0.36-0.40

Deissler empirical relationship (1955)To deal with the region close to a wall Deissler proposed thefollowing empirical formulation

( )[ ]y

wywnywn xxx

tyx

= 22 exp1

On the basis of velocity distribution in pipes, it can be obtained

124.0n

Velocity distribution in turbulent flow

In turbulent flow, it is often considered a velocity scale characteristic ofthe region close the wall, saidshear or friction velocity

ww =

It has been found experimentally that the turbulent intensity is scaled as

w

The velocity profile close to a wall can be therefore described as afunction of three quantities

( ) ( ) ,, ywFywz = In dimensionless form

+++ = yFywz

where

( ) ( )w

ywyw zz =

++ and ywy =+


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are respectively the dimensionless (universal) velocity and the

dimensionless distance from the wall

Close to the wall (in the laminar sublayer, 85 +y ) it is possible toassume that the velocity profile is linear

( ) yyw wz=

therefore

( ) ( ) +++ ==== yywywywyyw zwz

2

or+++ = yywz (close to the wall)

Far from the wall (well beyond the laminar sublayer, 30>+y ) viscosityplays a less and less relevant role; therefore, velocity variations do notdepend on

( )ywFy

wz ,=

(far from the wall)

and we can assume

yK

w

y

wz =

where K is called von Karman constant and it is found experimentally

that it as a value of 0.41

Therefore, at sufficient distance from the wall, the velocity profile has alogarithmic form

( ) CyK

wyw

yK

w

y

wz

z +==

ln

in dimensionless terms, it is

( ) [ ]AyK

ywz +=+++ ln

1


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With main reference to the three zones in which the boundary layer issubdivided (laminar sublayer, transition region and turbulent zone) it istherefore possible to describe the universal velocity profile as follows

+++ = yywz 5+

y

+++ += yywz ln00.505.3 305 +y

Note that: K141.015.2

The validity of the logarithmic profile ceases in the external part of theboundary layer (velocity defect layer) In flow inside circular cross section ducts, the velocity profile is

approximated by apower law

( )n

z

n

zzR

rw

R

ywrw

=

= 1max,max,

in which n has appropriate values. A frequent choice is the so-calledone-seventh power law profile (n= 71 )

( )71

max, 1

=

R

rwrw zz

0

5

10

15

20

25

0.1 1 10 100 1000

y+

w+


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It is noted that the velocity profile in turbulent flow in a circular duct isflatter than in the case of laminar flow; in fact:

in laminar flow( )

=

2

max, 1

R

rwrw zz and max,zz w.w 50= (

2)

in turbulent flow with the one-seventh power lawmax,zmax,z

R

max,zz w.wdrrR

rw

Rw 8170

60

4921

1

0

71

2==

=

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Dimensionless Radial Coordinate

LocalVelocity/AverageVe

locity

Laminar Flow

Turbulent Flow

Moreover, at the same flow rate, the friction pressure drops are larger

in turbulent flow than in laminar flow

(2) The overbar on the velocity symbol from now on takes again the meaning of a cross section average.


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HEAT TRANSFER IN TURBULENT FLOW

Eddy Diffusivity

Resuming the energy equation in terms of averaged variables, it is( ) ( ) ( )

++

+=+

wuwgqwIpqwuu

t

000

From this equation it is possible to obtain the equation of thermalenergy balance for a compressible flow in steady state conditions

( )

++= wTCqqwTC pp

It can be noted that the vectorwTCp

represents the contribution of turbulent flow to heat transfer

wTCqq peff +=

In analogy with the Fourier law for heat conduction, it is possible towrite

{ }p p Tj j

TC T w C

x

=

whereT

represents turbulent thermal diffusivity

[ ]sm

2 . It is

therefore

{ } [ ]eff p T jj

Tq C

x

= +

where the molecular and turbulent contribution appear and T is the

time-averaged temperature

In relation to the nature of T considerations similar to those made forT hold; in particular, it does not depend on the fluid properties, but

also on the flow field


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Since eddies giving rise to turbulent momentum transport are the sametransporting energy, it is reasonable to assume that

t1T

T

Pr turbulent Prandtl number

= =

This holds with acceptable approximation for fluid with molecularPrandtl number close to 1. In this case, it is therefore

T T =

For liquid metals it is T T < and the relationship by Dwyer holds1.4

max

1.821

Pr

T

T T

=

where max indicates the maximum value in a channel


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BASIC CONCEPTS ABOUT COMPUTATIONAL

MODELLING OF TURBULENT FLOWS

LENGTH SCALES IN TURBULENCE

In turbulent flow an energy cascade occurs representing a transfer ofturbulence kinetic energy (per unit mass), identified byk, from large tosmaller eddies. In particular:

o large eddies receive energy by the average flow field at themacroscopic scales characterising it

o small eddies, on the other hand, are mainly responsible forturbulence kinetic energy dissipation

oit can be reasonably assumed that small eddies are in anequilibrium state in which they receive from large eddies thesame rate of energy they dissipate (universal equilibrium theory

by Kolmogorov, 1941)

Motion at the smaller scales involved in turbulence phenomena isgoverned by the following variables:

o turbulence kinetic energy dissipation per unit time2 3

dk dt m s = =

o kinematic viscosity 2m s = By dimensionally combining the above variables, it is possible to

determine theKolmogorov length, time and velocity scales

( ) [ ]

1 43

2 31 4

3

2

m sm

s m

= =

( ) [ ]

1 22 3

1 2

2

m ss

s m

= =

( )

1 42 2

1 4

3

m m m

s s s

= =

v

The length scale is generally much smaller that the mean free pathsof molecules; therefore, turbulent flow is essentially a continuum

phenomenon


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Nevertheless,this length scale is many orders of magnitude smaller thanthat of lager eddies, whose size is in the order of the length of the bodies

which generated them

The length scale characterising large eddies is identified by l and ameasure of it is said the integral turbulence length scale, representingthe distance over which a fluctuating component of velocity keeps

correlated, i.e., such that the mean ( ) ( )1 2i iw r w r

is not negligible for a

distance between the two points in the order of l . It is >>l .

Both on an experimental and on a dimensional basis it was possible toestablish the relation between , k ed l applicable for high Reynoldsnumber turbulence (see later). This relationship has the form

3 2k

l

Therefore, considering the definition of it is:

( )

( ) ( )( )

1 4 3 4 3 43 2 3 4 1 21 4 1 23 4

1 4 3 4 3 4 3 43T

k k kRe

= = = = =

l l ll l l l

where1 2

T

kRe

lis the turbulence Reynolds number.

Concerning the energy distribution at the different length scales, aspectral distribution originating from a Fourier series decomposition isused

( )E d turbulent kinetic energy between and d = +

with

( ) ( )2 2 20

1

2x y z

k w w w E d

= + + = .

In this distribution the wave

number is related to the

wavelength, , by therelationship 2 = .

The figure shows thequalitative trend of theturbulent energy spectrum in

bi-logarithmic scale

Energy

Containing

Eddies

InertialSubrange

Viscous

Range

( )E

-1l

-1

( ) 2 3 5 3E C =


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Three regions appear:1.the one of lengths comparable with large eddies, where turbulence

takes energy form the mean flow;2.on the other side, at small values of the wave number, the region of

viscous dissipation;3.the intermediate region, where transfer of energy by inertialmechanisms dominates; in this region, as it has been verified by

experiments, the spectrum is proportional to 2 3 5 3 (the Kolmogorov-5/3 law)

DIRECT NUMERICAL SIMULATION (DNS)

It is virtually the most accurate method to model turbulent flow. It is

based on considering that the Navier-Stokes equations include all the

relevant information needed to predict turbulence behaviourDirect Numerical Simulation DNS does not require special

constitutive models for dealing with turbulence; it involves the transientsolution of the Navier-Stokes equatons, which model instability

phenomena giving rise to eddies; for incompressible flow it is:

w =

(continuity equation)

gpwDt

wD

+= 2 (Navier-Stokes equations)

In this light,DNS can be thought as a source of data having the same

worth of experimental ones:

making use of accurate numerical techniques (for instance, spectral orpseudo-spectral methods), it allows to reproduce with reasonable

accuracy phenomena as the onset of turbulence and its characteristics;

it allows to obtain more detailed data than any experiment will ever beable to provide.

However, beware:

Nothing can really substitute experience!!!

The main problem involved in DNS is that the direct solution of

Navier-Stokes equations should be sufficiently accurate over the wholerange of involved lengths

This results in a formidable computational problem, since all the

involved lengths scales should be adequately resolved (from the


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Kolmogorov microscale, , to the integral length scale, being in the order

of the size of the duct or the flow surrounded object):

an estimate of the number of equally spaced nodes necessary in thispurpose in a duct having an height H is available (Wilcox 1998 book)

and is in the order 106 109 increasing with ( )9 4Re , where

( )2Re w H = and ww = ;

similarly, the time step should be in the order of the time scale , givingrise to a very large number of time advancements;

For these reasons, DNS is presently an interesting tool for research, under

continuous development, but its applications are limited by the present

computer capabilities.

LARGE EDDY SIMULATION (LES)

In the attempt to overcome the problem of resolving the small scales ofturbulence, LES methods have been proposed, having the followingcharacteristics:

the large turbulence scales are directly solved as in DNS; the smaller scales are treated with subgrid models (SGS SubGrid

Scale).

In some relevant cases, the LES technique allowed to obtain results similarto those of DNS with a computational effort in the order of some

percentage in terms of required number of nodes and time advancements.

A key point in LES is the choice of a technique to filter the smallscales; different options are available:

volume-average box filter( )

( )

( )( )

1, ,

i i

V r

w r t w r t dV

V r

=

where it is

( ) { }2 2, 2 2, 2 2V r x x x x x y y y y y z z z z z + + +

(V is a parallelepiped box, having sides , ,x y z around r

); in this

case, iw is theresolvable-scale filtered velocity, representing the velocity

scale which can be resolved numerically


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Obviously, it is:

i i iw w w= +

formally similar to the relationships applicable in the case of RANS on

the basis of time averages that, in this case, is based on the selected

spatial averaging process; 3 x y z = is said the filter width and iw

and thesubgrid-scale velocity

filter functionsin this case filter functions ( ),G r r

are introduced; they give

( ) ( ) ( )( )

, , ,i i

V r

w r t G r r w r t dV =

and satisfy the obvious normalization condition:( )

( )

, 1V r

G r r dV =

There are different possible choices:

o volume-average box filter( )

( ) ( )1 ,,

0,

V r r V r G r r

otherwise

=

o Gaussian filter( )

3 2 2

2 2

6, exp 6

r rG r r

=

ofilters based on the Fourier transform (spectral methods)once the velocity field is expressed in terms of wave number (i.e.,the reciprocal of a length scale) it is possible to impose that the filtercuts all the components characterised by a wave number greater than

a threshold max 2 = ; an example of such technique is the followingFourier cutoff filter:

( )( )

( )

( )

( )

( )

( )

( )

sin sin sin1,

x x y y z zG r r

V r x x y y z z

=


25/38


Once the resolvable scales and the subgrid scales have been defined,the Navier-Stokes equations, making use of the Einstein notation (therepeated index in a term implies summation over all the applicable values

of such index), can be written in averaged form:

0i

i

w

x

=

(continuity)

21i ji i

j i k k

w ww wp

t x x x x

+ = +

(momentum)

The average appearing as an argument of the derivative in the secondterm at the LHS can be decomposed as follows:

( )( )i j i i j j i j i j j i i jw w w w w w w w w w w w w w = + + = + + +

or (note that in general: w w )

( )

ijijij

i j i j i j i j i j j i i j

R SGS Reynolds stressC cross term stressL Leonard stress

w w w w w w w w w w w w w w

== =

= + + + +

TheLeonard stress is often implicitly represented by the truncation error

of the numerical scheme, if it is a second order one, otherwise it must be

directly evaluated. It is also possible to show that

( )2

ij i jL w w

Nevertheless, by adopting the notation:

( )ij i j i jw w w w =

or, alternatively, putting

1

3ij ij kk ij

Q Q

=

1

3kk ij

P p Q = + ij ij ijQ C R= +

we have finally an equation having the form:

1i ji iij

j i j j

w ww wP

t x x x x

+ = + +

The above relationship shows thatthe fundamental problem in LES is

the determination of a model for thesubgrid stresses, ij .


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Smagorinsky in 1963 proposed a relatively successful subgrid model

based on the definition of an eddy viscosity, T such that

2ij T ij

S =

with

( )2

2T S ij ijC S S = 1

2

jiij

j i

wwS

x x

= +

where C is the Smagorinsky coefficient representing a parameter to be

adjusted for the particular problem to be dealt with; values in the range

0.10 to 0.24 have been adopted for typical problems.

In some more recent dynamic subgrid scale models C is updated at

each advancement.The LES models require particular care in imposing the boundary

conditions, being virtually suitable for the use beyond the viscous

boundary layer, at large Reynolds number.

LES models are promising for design applications.


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REYNOLDS AVERAGED NAVIER-STOKES EQUATIONS (RANS)

This approach is basically the one above introduced as statisticaltreatment of turbulence and is one of the most generally used incommercial CFD codes.

Turbulent intensity is strictly related to theturbulent kinetic energy

( )2 2 21

2x y zk w w w = + +

The time (Reynolds) averaging applied to the Navier-Stokes equations

leads to the following expression:

( ) ( ) ( ) ( )wwgIpwwwt

+=+

wheret

Rew w = =

is theReynolds stress tensor.This tensor is the main quantity to be

simulated in turbulence flows by the RANS

approach, since it represents the additionalmomentum flux due to turbulence.

The Boussinesq approximation allowsmaking use of the concept of eddy viscosity,

T ,

for evaluating this stress in similarity withformulations adopted for laminar flow

2jRe i

ij T ij T

j i

wwS

x x

= = +

Different models have been proposed tocalculate this stress. They can be distinguished in the following

categories:

1.Algebraic models (orzero-equation models, already dealt with above)2.One-equation models3.Two-equation models

The complexity of these models is greater the larger is the number ofequations (i.e., partial differential equations, PDEs) that must beadded to the averaged mass, energy and momentum balance equations

(RANS); in particular:

Increasing velocity

Momentum Flux


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ono additional PDE is added in algebraic models;oone or two PDEs are added in one-equation and two-equation

models.

Stress transport models, on the other hand, do not make use of theBoussinesq approximation, defining transport equations for each of thesix independent components of the turbulent stress tensor

With respect to algebraic models, models with one or more equationsallow specify the transport of kinetic energy, so that the previous and

upstream history of the flow is accounted for in addition to local

conditions

An important distinction between turbulence models is anyway the onebetweencomplete and incomplete models:

o completeness of the model is related to its capability toautomatically define a characteristic length of turbulence

o in a complete model, therefore, only the initial and boundaryconditions are specified, with no need to define case by case

parameters depending on the particular considered flow

Algebraic ModelsPrandtl mixing length theory (1925)

As we already saw, Prandtl assumed that the turbulent stress tensor couldbe defined by

2t x xyx mix

w wl

y y

=

where mixl is the mixing length; the model is similar to the one for

molecular viscosity in which kinematic viscosity is a interpreted as the

product of a mean molecular velocity by a length (the mean free path).

It is an incomplete model, since the mixing length is different

according to the particular flow (boundary layers, jets, wakes, ).In the case of a wall, Prandtl assumed mixl to be linearly dependent on

the distance from the wall, by a law having the form mixl Cy= , with C and

empirical constant. In the case of a jet or of the mixing between two

streams at different velocity (mixing layer) mixl is proportional to the

width of the jet or of the mixing layer, i.e., to the width of the zone in


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which velocity is sufficiently different from the one of the freeunperturbed stream.

Notwithstanding its simplicity, the mixing length model provides

reasonable results in a reasonable number of conditions, after beingreasonably tuned for the particular flow

Some of the variants to the model have been:

the introduction by Van Driest (1956) of a damping function0

01 26y A

mixl y e A

+ + + = =

0.41 von Karman constant = improving the behaviour of the Reynolds stress at

0y+ , in agreement with theoretical predictions ( yx y );

a modification introduced by Clauser (1956) in order to improve therepresentation of turbulent viscosity in the defect layer;

the introduction of two different formulations for turbulent viscosity inthe inner layer and the outer layer (two-layer models by Cebeci-Smith, 1967, and Baldwin-Lomax, 1978);

the introduction of an ordinary differential equation to define turbulentviscosity in the outer layer in two-layer models (1/2 equation models by

Johnson and King, 1985, and Johnson and Coakley, 1990)Algebraic models, anyway, though they have some attractiveness for

their simplicity, require being dressed over the particular flow to be

predicted, requiring a considerable degree of tuningIn this light, they must be considered incomplete, in the above specified

meaning of this word.

Partial Differential Equation ModelsA look to the stress transport equationsThough the stress transport models do not fall in the considered category(they are actually beyond the Boussinesq approximation), they are the

starting point to understand the derivation of the turbulence kineticenergy equation

Following the treatment for an incompressible fluid (v. Wilcox, 1998),

it is:

the general component of the Navier-Stokes equation can be written as


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( )2

0 ( , , , )i i ii k

k i k k

w w wpN w w i k x y z

t x x x x

= + + = =

(

3)

considering the identity( ) ( ) ( )0 , , ,i j j iN w w N w w i j x y z + = =

and applying to it the time-averaging operator, it is:

( ) ( ) ( )0 , , ,i j j iN w w N w w i j x y z + = = () the same techniques and assumptions adopted in deriving the RANS

equations lead now to equations for each stress tensor component; forinstance, consider the transient term in the Navier-Stokes equations:

( ) ( )j j j ji i i ij i j j i i

w w w ww w w ww w w w w w

t t t t t t

+ + + = + + +

00

j j j ji i i ij j i i j j i i

w w w ww w w ww w w w w w w w

t t t t t t t t

==

= + + + = + + +

( ) ( )i ji jj ijij i

w ww wwww w

t t t t t

= + = = =

where, on the contrary of the notation adopted up to now, from here on ij

identifies the specific Reynolds stress tensor, defined as

ij i jw w = (differing from the usual notation ij i jw w = ).

By proceeding in a similar way, term by term, from () it is:

2ij ij j j j iji i i

k ik jk i j k

k k k k k j i k k

w w ww w w p pw w w w

t x x x x x x x x x

+ = + + + + +

This equation shows the typical difficulties encountered when trying

to close the turbulence equations. In fact:

the application of the time-averaging operator to the Navier-Stokesequations makes the Reynolds stress tensor to appear as a tensor of

correlation between two fluctuating velocity components ( i jw w );

the derivation of transport equations for the Reynolds stress tensormakes higher order correlation terms to appear: ( i j kw w w ).

(3) The Einsteins notation is again adopted.


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This endless process can be therefore closed only including closure

laws for the unknown terms at some stage. In the Reynolds stress

transport equations the unknown terms became a lot:

10 unknown terms having the form i j kw w w 6 unknown terms having the form ji

j i

ww p p

x x

+

6 unknown terms having the form 2 jik k

ww

x x

The turbulence kinetic energy equation

The turbulence kinetic energy equation can be now obtained by taking the

trace of the equations for the specific transport of Reynolds stress tensorcomponents (i.e., taking the summation of the diagonal terms). In fact:

( )2 2 2 2ii i i x y zw w w w w k = = + + = Its classical form is:

1 1

2

ji ij ij i i j j

j j k k j j

unsteady turbulentdissipation pressureconvective production molecularterm transport

diffusionterm diffusion

ww wk k kw w w w p w

t x x x x x x

+ = +

where the various terms are:

unsteady term: as in every balance equation, it represents the localchange rate of the quantity to be conserved;

convective (or advective) term: it represents the turbulence kineticenergy transport due to the mean fluid motion;

production term: it represents the transfer of energy from the meanflow per unit time; the Reynolds stress appearing in it is evaluated by:

2 223 3

jiij T ij ij T ij

j i

wwS k kx x

= = +

where T is the turbulent diffusivity of momentum (eddy viscosity);

dissipation term: it represents the rate at which the turbulence kineticenergy is converted into thermal internal energy; on the basis of

dimensional considerations, it is defined as:


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i i

k k

w w

x x

=

and is approximated by relationships having the form3 2k

l

molecular diffusion term: it represents the diffusive transport due toprocesses occurring at a molecular level;

turbulent transport term: it represents the contribution to the kineticenergy transport due to the velocity turbulent fluctuations;

pressure diffusion term: it is the term due to the correlation existingbetween pressure and velocity fluctuations.

Turbulent and pressure diffusion transport terms are sometimes groupedtogether and represented with a single term:

1 1

2

Ti i jj

k j

kw w w p w

x

+

in which k is a parameter correlating turbulent diffusivity of momentum

to that of turbulence kinetic energy. It is therefore:

i Tj ij

j j j k j

wk k kw

t x x x x

+ = + +

One-Equation ModelsPrandtl (1945) proposed to express dissipation rate as

3 2

D

kC =

l

However, in this way, the integral turbulence length scale must be defined,for instance, on the basis of approaches similar to those adopted for themixing length theory.

The one-equation model by Prandtl takes therefore the form3 2

i Tj ij D

j j j k j

wk k k k w Ct x x x x

+ = + + l

A further closure equation is defined for the turbulent viscosity2

1 2

T D

kk C

= =l

More complex models have been proposed later on, though they referto similar expressions.


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In general, one-equation models are incomplete, since the turbulencelength scale must be defined on a case by case basis; complete versions are

anyway available which specify independently this length (e.g., Baldwin-

Barth, 1990).

Two-equation models

As we saw, one-equation models, though they introduce the transportequation for turbulence kinetic energy, are generally incomplete, since

they do not define explicitly the turbulence length scaleIn order to solve this problem, different two-equation approaches

have been proposed:

Kolmogorov in 1942 proposed that a new equation for the transport ofthespecific dissipation rate,

1s = , dimensionally related to the other

quantities by the relationships:1 2

T k k k l

Chou in 1945 proposed the introduction of an exact equation for ,related to the other quantities by

2 3 2

T k k k l

Zeierman and Wolfstein in 1986 proposed an equation for the transportof the product ofk and the turbulence dissipation time, , which is

essentially the reciprocal of Kolmogorovs ; it is:1 2

T k k k l

From these proposals the so-called k , k and k k where

obtained. Other proposed models where the k k l (Rotta, 1951).A short description of the k and k models follows, since they

were the ones that received the greatest attention up to the present time.

k Model

Kolmogorov defined as the rate of dissipation of energy in unit volume

and unit time. He underlined its relation with the turbulence length scale,defining as a mean frequency given by

1 2c k = l

wherec is a constant.Most of considerations by Kolmogorov in relation to and its

transport equation were based on dimensional reasoning; in his workthere is no formal derivation of the equation for .


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Wilcox (1998) proposed in the following way the possible steps ofKolmogorovs reasoning in identifying as a variable whose transport

evaluation is needed:

also basing on the Boussinesq approximation, it is reasonable to assumethat eddy viscosity is proportional to the turbulent kinetic energy:

Tk ;

as 2T m s = and 2 2k m s = , their ratio has the dimension of a time; similarly, 2 3m s = and then [ ]1k s = we can therefore close from a dimensional point of view the

relationships between the different quantities by defining a variablehaving the dimension of a time or of its reciprocal.

Then, to define an equation for we can assume that the essential

terms that it must contain must represent the time rate of change,convection (advection) diffusion, dissipation, dispersion and production

The equation, in the form proposed by Kolmogorov, was:

2

j T

j j j

wt x x x

+ = +

From the original formulation by Kolmogorov, the k model was

subjected to different developments. The Wilcox (1998) version is the

following:

( )* *ij ij Tj j j j

wk k kw kt x x x x

+ = + +

( )2ij ij Tj j j j

ww

t x k x x x

+ = + +

with additional formulations for the appearing constants.

For dissipation, turbulent viscosity and the turbulence characteristic

length scale in this model it is:*k = T k =

1 2k =l

The coefficients appearing in the above equations are all defined on

the basis of laws which do not include any arbitrary assumption ob the

relevant parameters (v. Wilcox, 1998, Sect. 4.3.1): the model is therefore

complete.


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k Model

It is the most often used turbulence model. The so-called standard k

model was presented in a fundamental paper by Jones and Launder

(1972).Launder and Sharma in 1974 made retuning of the model, so also

their paper is often taken as reference.

Unlike the equation for , the transport equation for may beobtained by a rigorous process based on the Navier-Stokes equations

( )2

0 ( , , , )i i ii k

k i k k

w w wpN w w i k x y z

t x x x x

= + + = =

by developing the following identity:

( )2 0i ij j

wN w

x x

=

The development is relatively complex and leads to an equation including

at the RHS the following terms: production of dissipation, dissipation of

dissipation, molecular diffusion of dissipation and turbulent transport of

dissipation.

The equations of thestandardk modelare:

i Tj ij

j j j k j

wk k kw

t x x x x

+ = + +

2

1 2i T

j ij

j j j j

ww C Ct x k x k x x

+ = + +

where2

TC k = ( )C k =

3 2C k =l

and the constants are given by:

1 1.44C = 2 1.92C = 0.09C = 1k = 1.3 =

As it is seen, also this model iscomplete.

In summary: by two-equation models, after evaluating the couple k or k , theeddy viscosity

T is evaluated:2

T C k = or T k =

allowing to calculate the Reynolds stress tensor, by the Boussinesq

approximation


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2 22

3 3

ji

ij T ij ij T ij

j i

wwS k k

x x

= = +

when accepting the Reynolds analogy between heat and momentumtransfer, a prescribed value of the turbulent Prandtl number (often close

to unity) allows for the calculation of the thermal eddy diffusivity

1TtT

Pr

=

necessary to evaluate the turbulent contribution in energy averagedequations

Concluding remarks

It can be noted that also the equations of two-equation models can beput in the general conservation form

( ) ( )jj j j

w St x x x

+ = +

to be discretised with the same numerical techniques adopted for general

balance equations and described in the first part of this lecture

It is quite difficult to catch turbulent phenomena close to the wall,because of the sharp gradients of turbulence intensity, that are difficult to

be described with enough detail

max,2

zx ww

x

z

max,2

zz ww

025.0

050.0

075.0

100.0

125.0

x

z

effzx,

tzx,


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This is the reason why the application of k and k turbulencemodels close to the wall requires attention, because standard models

cannot be integrated up to the wall, where turbulence is damped in the

buffer and laminar sublayer regions

In this regard,two possible choices are presently available:o the use of wall functions, adopting the well known logarithmic

form of the velocity profile to

obtain the appropriateboundary conditions to beimposed in the first node close

to the wall; in this case, thefirst node must be put at alarge enough value of y+ (e.g.,

greater than 30)

with

ww = ( ) ( )w

ywyw zz =

++ ywy =+

o as an alternative, low Reynolds number models must be used, inwhich corrections aiming at a better evaluation of the viscous

effects close to the wall are introduced (by damping functions).

In this case, the first node close to the wall must be put at 1y+ < ,

well within the laminar sublayer: a very refined mesh is necessaryat the walls

For a compressible fluid, the averaging process to be adopted is the so-called Favre averaging, consisting in averaging the different variables

using density as the weight; for instance for velocity it is:2

2

1 1 t ti i

t tw w dt

t

+

=

On the basis of this definition, it is possible to define the conservationequations averaged according to Favre for mass, energy andmomentum as the equations for the Reynolds stress tensor components

and of turbulence kinetic energy

The latter is given by:( ) ( )

1

2

i ij ij ij i j i i j i

j j j i i

w uPk w k t u u u u p u u p

t x x x x x

+ = + +

where

0

5

10

15

20

25

0.1 1 10 100 1000

y+

w+


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22

3

kij ij ij

k

wt s

x

=

p P p= + i i iw w w= +

The last two terms appearing in the k equation are pressure work andpressure dilatation.

Basic Concepts About Cfd Models

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