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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:
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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Basic Concepts about CFD Models 2
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Basic Concepts about CFD Models 3
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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Basic Concepts about CFD Models 4
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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Basic Concepts about CFD Models 5
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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Basic Concepts about CFD Models 6
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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Basic Concepts about CFD Models 7
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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Basic Concepts about CFD Models 8
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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Basic Concepts about CFD Models 9
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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Basic Concepts about CFD Models 10
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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Basic Concepts about CFD Models 11
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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Basic Concepts about CFD Models 12
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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Basic Concepts about CFD Models 13
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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Basic Concepts about CFD Models 14
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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Basic Concepts about CFD Models 15
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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Basic Concepts about CFD Models 16
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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Basic Concepts about CFD Models 17
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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Basic Concepts about CFD Models 18
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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Basic Concepts about CFD Models 19
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 CFD Models 20
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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Basic Concepts about CFD Models 21
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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Basic Concepts about CFD Models 22
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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Basic Concepts about CFD Models 23
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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Basic Concepts about CFD Models 24
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
=
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Basic Concepts about CFD Models 25
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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Basic Concepts about CFD Models 26
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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Basic Concepts about CFD Models 27
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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Basic Concepts about CFD Models 28
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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Basic Concepts about CFD Models 29
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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Basic Concepts about CFD Models 30
( )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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Basic Concepts about CFD Models 31
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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Basic Concepts about CFD Models 32
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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Basic Concepts about CFD Models 33
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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Basic Concepts about CFD Models 34
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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Basic Concepts about CFD Models 35
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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Basic Concepts about CFD Models 36
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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Basic Concepts about CFD Models 37
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.