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Publication 97/2
An Introduction to TurbulenceModels
Lars Davidson, http://www.tfd.chalmers.se/ lada
Department of Thermo and Fluid Dynamics
CHALMERS U NIVERSITY OF TE C H N O L O G Y
Goteborg, Sweden, January 2003
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Ç Æ Ì Æ Ì Ë ¾
Ó Ò Ø Ò Ø ×
N omenclature 3
1 Turbulence 5
1.1 In trod u ction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Tu r bu len t Sca les . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Vorticity/ Velocity Gradient Interaction . . . . . . . . . . . . .
1.4 En erg y sp ectru m . . . . . . . . . . . . . . . . . . . . . . . . .
2 Turbulence Models 12
2.1 In trod u ction . . . . . . . . . . . . . . . . . . . . . . . . . . . .½ ¾
2.2 Bo ussin esq A ssu m p tio n . . . . . . . . . . . . . . . . . . . . .½
2.3 Alg ebr aic Mod els . . . . . . . . . . . . . . . . . . . . . . . . .½
2.4 Eq u at io n s fo r Kin et ic En er gy . . . . . . . . . . . . . . . . . .½
2.4.1 The Exact
Eq uat ion . . . . . . . . . . . . . . . . . . .½
2.4.2 The Equation for½ ¾ Í
Í
µ
. . . . . . . . . . . . . . .½
2.4.3 The Equation for ½ ¾
Í
Í
µ . . . . . . . . . . . . . . . ½
2.5 The Modelled
Equ ation . . . . . . . . . . . . . . . . . . . .¾ ¼
2.6 O ne Eq u at io n M od els . . . . . . . . . . . . . . . . . . . . . . .¾ ½
3 Tw o-Equation Turbulence Models 22
3.1 The Modelled
Eq uat ion . . . . . . . . . . . . . . . . . . . . .¾ ¾
3.2 Wall Fu n ction s . . . . . . . . . . . . . . . . . . . . . . . . . . .¾ ¾
3.3 Th e
Mod el . . . . . . . . . . . . . . . . . . . . . . . . . .¾
3.4 Th e
Mod el . . . . . . . . . . . . . . . . . . . . . . . . .¾
3.5 Th e
Mod el . . . . . . . . . . . . . . . . . . . . . . . . . .¾
4 Low -Re N umber Turbulence Models 28
4.1 Low-Re
Mod els . . . . . . . . . . . . . . . . . . . . . . .¾
4.2 The Laund er-Sharma Low-Re
Mod els . . . . . . . . . .¿ ¾
4.3 Bound ary Condition for
an d
. . . . . . . . . . . . . . . . .¿ ¿
4.4 The Two-Layer
Mod el . . . . . . . . . . . . . . . . . . .¿
4.5 The low-Re
Mod el . . . . . . . . . . . . . . . . . . . . .¿
4.5.1 The low-Re
Model of Peng et al. . . . . . . . . .¿
4.5.2 The low-Re
Model of Bredberg et al. . . . . . . .¿
5 Reynolds Stress Models 38
5.1 Rey no ld s St ress M od els . . . . . . . . . . . . . . . . . . . . .¿
5.2 Reynolds Stress Models vs. Edd y Viscosity Models . . . . . . ¼
5.3 Cu rv at ure Effects . . . . . . . . . . . . . . . . . . . . . . . . . ½
5.4 A cce le ra tio n a n d Ret ar d at io n . . . . . . . . . . . . . . . . . .
¾
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Æ Ó Ñ Ò Ð Ø Ù Ö ¿
Æ Ó Ñ Ò Ð Ø Ù Ö
Ä Ø Ò Ë Ý Ñ Ó Ð ×
½
¾
×
constants in the Reynolds stress model
¼
½
¼
¾
constants in the Reynolds stress model
½
¾
constants in the modelled
equation
½
¾
constants in the modelled
equation
constant in turbulence model
energy (see Eq. 1.8); constan t in w all fun ctions (see Eq. 3.4)
damping function in pressure strain tensor
turbulent kinetic energy (
½
¾
Ù
Ù
)
Í
instantaneou s (or laminar) velocity inÜ
-direction
Í
instantaneou s (or laminar) velocity inÜ
-direction
Í
time-averaged velocity inÜ
-direction
Í
time-averaged velocity inÜ
-direction
Ù Ú Ù Û shear stressesÙ
fluctuating velocity inÜ
-direction
Ù
¾ normal stress in theÜ
-direction
Ù
fluctuating velocity inÜ
-direction
Ù
Ù
Reynold s stress tensor
Î
instantaneou s (or laminar) velocity inÝ
-direction
Î
time-averaged velocity inÝ
-direction
Ú
fluctuating (or laminar) velocity inÝ
-direction
Ú
¾ normal stress in theÝ
-direction
Ú Û
shear stress
Ï
instantaneou s (or laminar) velocity inÞ
-direction
Ï time-averaged velocity in Þ -directionÛ
fluctuating velocity inÞ
-direction
Û
¾ normal stress in theÞ
-direction
Ö Ë Ý Ñ Ó Ð ×
Æ
boundary layer thickness; half channel height
dissipation
wave nu mber; von Karman constant ( ¼ ½
)
dy nam ic viscosity
Ø
dynamic turbulent viscosity
kinematic viscosity
Ø
kinematic turb ulent viscosity
ª
instantaneous (or laminar) vorticity component inÜ
-direction
¿
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Æ Ó Ñ Ò Ð Ø Ù Ö
ª
time-averaged vorticity comp onent inÜ
-direction
specific d issipation (»
)
fluctuating vorticity component inÜ
-direction
turbulent Prand tl num ber for variable
Ð Ñ
laminar shear stress
Ø Ó Ø total shear stress
Ø Ù Ö
turbulent shear stress
Ë Ù × Ö Ô Ø
centerline
Û
wall
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½ Ì Ù Ö Ù Ð Ò
½ º ½ Á Ò Ø Ö Ó Ù Ø Ó Ò
Almost all fluid flow which we encounter in daily life is turbulent. Typical
examples are flow around (as well as in) cars, aeroplanes and buildings.
The boundary layers and the wakes around and after bluff bodies such as
cars, aeroplanes and bu ildings are turbulent. Also the flow and combustion
in engines, both in piston engines and gas turbines and combustors, are
highly tu rbu lent. Air movemen ts in rooms are also turbu lent, at least along
the w alls w here wall-jets are formed. Hence, wh en w e compute flu id flow
it will most likely be tu rbu lent.
In turbulent flow we usually divide the variables in one time-averaged
part
Í
, which is ind ependent of time (wh en the m ean flow is steady), and
one fluctuating partÙ
so thatÍ
Í · Ù
.
There is no definition on turbu lent flow, but it has a nu mber of charac-
teristic features (see Tenn ekes & Lum ley [41]) such as:
Á º Á Ö Ö Ù Ð Ö Ø Ý . Turbu lent flow is irregular, random and chaotic. Theflow consists of a spectrum of different scales (eddy sizes) where largest
edd ies are of the order of the flow geometry (i.e. bound ary layer thickness,
jet width, etc). At the other end of the spectra we h ave the smallest ed-
dies which are by viscous forces (stresses) dissipated into internal energy.
Even though turbulence is chaotic it is deterministic and is described by
the N avier-Stokes equat ions.
Á Á º « Ù × Ú Ø Ý
. In turbulent flow the diffusivity increases. This means
that th e spreading rate of boun dary layers, jets, etc. increases as th e flow
becomes tu rbulent. The turbu lence increases the exchange of momentu m
in e.g. bound ary layers and redu ces or d elays thereby separation at bluff
bodies such as cylinders, airfoils and cars. The increased d iffusivity alsoincreases the resistance (wall friction) in intern al flow s such as in channels
and pipes.
Á Á Á º Ä Ö Ê Ý Ò Ó Ð × Æ Ù Ñ Ö ×
. Tur bulent flow occurs at high Reynolds
num ber. For example, the transition to turbu lent flow in p ipes occurs that
Ê
³ ¾ ¿ ¼ ¼
, and in boun dary layers atÊ
Ü
³ ½ ¼ ¼ ¼ ¼ ¼
.
Á Î º Ì Ö ¹ Ñ Ò × Ó Ò Ð
. Turb ulent flow is alway s three-dim ensional.
However, when the equations are time averaged we can treat the flow as
two-dimensional.
Î º × × Ô Ø Ó Ò
. Turbu lent flow is dissipative, which means that ki-
netic energy in the small (dissipative) eddies are transformed into internal
energy. The sm all edd ies receive the kinetic energy from slightly larger ed -dies. The slightly larger edd ies receive their energy from even larger edd ies
and so on. The largest eddies extract their energy from the m ean flow. This
process of transferred energy from the largest turbulent scales (eddies) to
the smallest is called cascade process.
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½ º ¾ Ì Ù Ö Ù Ð Ò Ø Ë Ð ×
Î Á º Ó Ò Ø Ò Ù Ù Ñ
. Even though we have small turbulent scales in the
flow they are much larger than the molecular scale and we can treat the
flow as a continuum .
½ º ¾ Ì Ù Ö Ù Ð Ò Ø Ë Ð ×
As mentioned above there are a wide ran ge of scales in turbu lent flow. The
larger scales are of the order of the flow geometry, for examp le the bou nd -
ary layer thickness, with length scale
and velocity scale Í
. These scales
extract kinetic energy from th e mean flow wh ich h as a time scale compara-
ble to the large scales, i.e.
Í
Ý
Ç Ì
½
µ Ç Í µ
The kinetic energy of the large scales is lost to slightly smaller scales w ith
which the large scales interact. Through the cascade process the kinetic en-
ergy is in this way transferred from the largest scale to smaller scales. Atthe smallest scales the frictional forces (viscous stresses) become too large
and the kinetic energy is transformed (dissipated ) into intern al energy. The
dissipation is denoted by
which is energy per un it time and unit mass
( Ñ
¾
×
¿
). The dissipation is proportional to the kinematic viscosity
times the fluctuating velocity gradient up to the power of two (see Sec-
tion 2.4.1). The friction forces exist of course at all scales, bu t they are
larger the smaller the eddies. Thus it is not quite true that ed dies which
receive their kinetic energy from slightly larger scales give aw ay a ll of that
the slightly smaller scales but a small fraction is dissipated. However it is
assumed that most of the energy (say 90 %) that goes into the large scales
is finally d issipated at th e smallest (dissipative) scales.
The smallest scales where dissipation occurs are called the Kolmogo-
rov scales: the velocity scale
, the length scale
and the time scale
. We
assume that these scales are d etermined by viscosity
and dissipation
.
Since the kinetic energy is destroyed by viscous forces it is natural to as-
sume that viscosity plays a part in determining these scales; the larger vis-
cosity, the larger scales. The amoun t of energy that is to be dissipated is
. The more energy that is to be transformed from kinetic energy to inter-
nal energy, the larger the velocity gradients must be. Having assumed that
the dissipative scales are determined by viscosity and dissipation, we can
express
,
an d
in
an d
using dimensional analysis. We write
Ñ × Ñ
¾
× Ñ
¾
×
¿
´ ½ º ½ µ
wh ere below each variable its dim ensions are given. The d imensions of the
left-hand and the right-hand side must be the same. We get two equations,
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½ º ¿ Î Ó Ö Ø Ø Ý » Î Ð Ó Ø Ý Ö Ò Ø Á Ò Ø Ö Ø Ó Ò
one for meters Ñ
½ ¾ · ¾ ´ ½ º ¾ µ
and one for seconds ×
½ ¿ ´ ½ º ¿ µ
which gives ½
. In the same w ay w e obtain the expressions for
an d
so that
µ
½
¿
½
½ ¾
´ ½ º µ
½ º ¿ Î Ó Ö Ø Ø Ý » Î Ð Ó Ø Ý Ö Ò Ø Á Ò Ø Ö Ø Ó Ò
The interaction between vorticity and velocity gradients is an essential in-
gredient to create and maintain turbu lence. Disturbances are amp lified
– the actual process depending on type of flow – and these disturbances,which still are laminar and organized and well defined in terms of phys-
ical orientation and frequency are turned into chaotic, three-dimensional,
random fluctuations, i.e. turbu lent flow by interaction between the vor-
ticity vector and the velocity gradients. Two idealized phenomena in this
interaction p rocess can be ident ified: vortex stretching an d vortex tilting.
In order to gain some insight in vortex shedd ing we w ill stud y an ide-
alized, inv iscid (viscosity equals to zero) case. The equation for instan ta-
neou s vorticity (ª
ª
·
) reads [41, 31, 44]
Í
ª
ª
Í
· ª
ª
Í
´ ½ º µ
where
is the Levi-Civita tensor (it is· ½
if
,
are in cyclic order, ½
if
,
are in a nti-cyclic order, and¼
if any two of
,
are equal), and where
µ
denotes derivation with respect toÜ
. We see that this equation is not
an ordinary convection-diffusion equation but is has an additional term on
the right-hand side which represents amplification and rotation/ tilting of
the vorticity lines. If we w rite it term-by-term it reads
ª
½
Í
½ ½
· ª
¾
Í
½ ¾
· ª
¿
Í
½ ¿
´ ½ º µ
ª
½
Í
¾ ½
· ª
¾
Í
¾ ¾
· ª
¿
Í
¾ ¿
ª
½
Í
¿ ½
· ª
¾
Í
¿ ¾
· ª
¿
Í
¿ ¿
The diagonal terms in this matrix represent vortex stretching. Imagine aÎ Ó Ö Ø Ü
× Ø Ö Ø Ò
slender, cylindr ical flu id element wh ich vorticityª
. We introduce a cylin-
drical coordinate system with theÜ
½
-axis as the cylinder axis andÜ
¾
as the
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½ º ¿ Î Ó Ö Ø Ø Ý » Î Ð Ó Ø Ý Ö Ò Ø Á Ò Ø Ö Ø Ó Ò
ª
½
ª
½
Ù Ö ½ º ½ Î Ó Ö Ø Ü × Ø Ö Ø Ò º
radial coordinate (see Fig. 1.1) so thatª ´ ª
½
¼ ¼ µ
. A positiveÍ
½ ½
will
stretch th e cylinder. From the continu ity equation
Í
½ ½
·
½
Ö
Ö Í
¾
µ
¾
¼
we fin d th at the rad ial derivative of the rad ial velocityÍ
¾
must be negative,
i.e. the radius of the cylinder will decrease. We have neglected the viscosity
since viscous diffusion at high Reynolds number is much smaller than the
turb ulent on e and since viscous dissipation occurs at sm all scales (see p. 6).
Thus th ere are no viscous stresses acting on the cylind rical flu id element
surface w hich means that the rotation momentu m
Ö
¾
ª ´ ½ º µ
remains constant as the rad ius of the fluid element d ecreases (note that also
the circulation » ª Ö
¾ is constant). Equation 1.7 shows that the vortic-
ity increases as the rad ius d ecreases. We see that a stretching/ compressing
will decrease/ increase the rad ius of a slender fluid element and in crease/ d ecrease
its vorticity component aligned with the element. This process will affect
the vorticity components in the other two coordinate directions.
The off-diagonal terms in Eq. 1.6 represent vortex tilting. Again, take aÎ Ó Ö Ø Ü
Ø Ð Ø Ò
slender fluid element with its axis aligned with th eÜ
¾
axis, Fig. 1.2. The ve-
locity grad ientÍ
½ ¾
will tilt the fluid element so that it rotates in clock-wise
direction. The second termª
¾
Í
½ ¾
in line one in Eq. 1.6 gives a contribution
to ª
½ .Vortex stretching and vortex tilting thus qualitatively explains how in-
teraction between vorticity and velocity gradient create vorticity in all three
coordinate directions from a disturbance which initially was well defined
in one coordinate direction. Once this process has started it continues, be-
cause vorticity generated by vortex stretching and vortex tilting interacts
with the velocity field and creates further vorticity and so on. The vortic-
ity and velocity field becomes chaotic and random: turbulence has been
created. The turbu lence is also maintain ed b y th ese processes.
From the d iscussion above we can n ow u nd erstand wh y turbu lence al-
ways m ust be three-dimensional (Item IV on p . 5). If the instantaneous
flow is two-dimensional w e find that all interaction terms between vortic-ity and velocity gradients in Eq. 1.6 vanish. For example if
Í
¿
¼
and all
derivatives with respect toÜ
¿
are zero. If Í
¿
¼
the third line in Eq. 1.6
vanishes, and if Í
¿
¼
the th ird column in Eq. 1.6 disap pears. Finally, the
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½ º Ò Ö Ý × Ô Ø Ö Ù Ñ
ª
¾
ª
¾
Í
½
Ü
¾
µ
Ù Ö ½ º ¾ Î Ó Ö Ø Ü Ø Ð Ø Ò º
remaining terms in Eq. 1.6 will also be zero since
ª
½
Í
¿ ¾
Í
¾ ¿
¼
ª
¾
Í
½ ¿
Í
¿ ½
¼
The interaction between vorticity and velocity gradients will, on av-
erage, create smaller and smaller scales. Whereas the large scales wh ich
interact with the mean flow have an orientation imposed by the mean flow
the small scales will not remember th e structure and orientation of the large
scales. Thu s the small scales w ill be isotropic, i.e independent of direction.
½ º Ò Ö Ý × Ô Ø Ö Ù Ñ
The turbulent scales are distributed over a range of scales w hich extends
from the largest scales wh ich interact with the m ean flow to the smallest
scales w here dissipation occurs. In wave num ber space the energy of ed-
dies from
to ·
can be expressed as
µ ´ ½ º µ
wh ere Eq. 1.8 expresses the contribution from th e scales with w ave nu mber
between
an d ·
to the turbulent kinetic energy
. The dimension
of wave number is one over length; thus we can think of wave number
as p roportional to th e inverse of an eddy’s radius, i.e » ½ Ö
. The totalturbulent kinetic energy is obtained by integrating over the whole wave
number space i.e.
½
¼
µ ´ ½ º µ
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½ º Ò Ö Ý × Ô Ø Ö Ù Ñ ½ ¼
Á
Á
Á
Á
Á
Á
µ
Ù Ö ½ º ¿ Ë Ô Ø Ö Ù Ñ Ó Ö º Á Ê Ò Ó Ö Ø Ð Ö ¸ Ò Ö Ý Ó Ò Ø Ò Ò × º
Á Á Ø Ò Ö Ø Ð × Ù Ö Ò º Á Á Á Ê Ò Ó Ö × Ñ Ð Ð ¸ × Ó Ø Ö Ó Ô × Ð × º
The kinetic energy is the su m of the kinetic energy of the three fluctuat-
ing velocity comp onents, i.e.
½
¾
Ù
¾
· Ú
¾
· Û
¾
½
¾
Ù
Ù
´ ½ º ½ ¼ µ
The spectrum of
is show n in Fig. 1.3. We find region I, II and III w hich
correspond to:
I. In the region w e have the large edd ies wh ich carry most of the energy.
These edd ies interact with the m ean flow and extract energy from th e
mean flow. Their energy is past on to slightly smaller scales. The
edd ies’v elocity and length scales are Í
an d
, respectively.
III. Dissipation range. The eddies are small and isotropic and it is here
that the dissipation occurs. The scales of the eddies are described by
the Kolmogoro v scales (see Eq. 1.4)
II. Inertial subran ge. The existence of this region requires that the Reynolds
nu mber is high (fully tu rbulent flow ). The edd ies in this region repre-
sent the m id-region. This region is a “tran sport” region in th e cascade
process. Energy per time unit (
) is coming from the large eddies at
the lower part of this range and is given off to the d issipation range
at the higher part. The eddies in th is region are independent of both
the large, energy containing edd ies and the edd ies in the d issipation
range. One can argue that the eddies in this region should be char-
acterized by the flow of energy (
) and the size of the eddies½
.
Dimensional reasoning gives
µ Ó Ò × Ø
¾
¿
¿
´ ½ º ½ ½ µ
This is a very important law (Kolmogorov spectrum law or the ¿
law) which states that, if the flow is fully turbulent (high Reynolds
½ ¼
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½ º Ò Ö Ý × Ô Ø Ö Ù Ñ ½ ½
number), the energy spectra should exhibit a ¿
-decay. This of-
ten u sed in exp eriment and Large Edd y Simu lations (LES) and Direct
Nu merical Simu lations (DNS) to verify that the flow is fully turbu -
lent.
As explained on p. 6 (cascade process) the energy dissipated at the smallscales can be estimated using the large scales
Í an d
. The energy at the
large scales lose their energy du ring a time prop ortional to Í
, wh ich gives
Ç
Í
¾
Í
Ç
Í
¿
´ ½ º ½ ¾ µ
½ ½
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½ ¾
¾ Ì Ù Ö Ù Ð Ò Å Ó Ð ×
¾ º ½ Á Ò Ø Ö Ó Ù Ø Ó Ò
When the flow is turbulent it is preferable to decompose the instantaneous
variables (for example velocity comp onents and pressure) into a mean value
and a fluctuating value, i.e.
Í
Í
· Ù
È
È · Ô
´ ¾ º ½ µ
One reason wh y we d ecomp ose the variables is that when w e measure flow
quantities we are usually interested in the mean values rather that the time
histories. Another reason is that when we want to solve the Navier-Stokes
equation numerically it would require a very fine grid to resolve all tur-
bulent scales and it would also require a fine resolution in time (turbulent
flow is always un steady).
The continuity equation and the Navier-Stokes equation read
Ø
· Í
µ
¼ ´ ¾ º ¾ µ
Í
Ø
· Í
Í
µ
È
·
Í
· Í
¾
¿
Æ
Í
´ ¾ º ¿ µ
where µ
den otes derivation with respect toÜ
. Since we are d ealing with
incompressible flow (i.e low Mach num ber) the dilatation term on th e right-
han d side of Eq. 2.3 is neglected so that
Í
Ø
· Í
Í
µ
È
· Í
· Í
µ
´ ¾ º µ
Note that w e here use the term “incompressible” in the sense that d ensity is
independ ent of pressure ( È ¼
) , but it does not m ean that d ensity is
constant; it can be dep end ent on for example temperatu re or concentration.
Inserting Eq. 2.1 into the continu ity equation (2.2) and th e Navier-Stokes
equation (2.4) we obtain the time averaged continuity equation and Navier-
Stokes equ ation
Ø
·
Í
µ
¼ ´ ¾ º µ
Í
Ø
·
Í
Í
¡
È
·
¢
Í
·
Í
µ Ù
Ù
£
´ ¾ º µ
A new term Ù
Ù
appears on the right-hand side of Eq. 2.6 w hich iscalled the Reynolds stress tensor . The tensor is symm etric (for example
Ù
½
Ù
¾
Ù
¾
Ù
½
). It represents correlations between fluctuating velocities. It is an ad-
ditional stress term du e to turbu lence (fluctuating velocities) and it is u n-
known. We need a model forÙ
Ù
to close the equation system in Eq. 2.6.
½ ¾
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¾ º ½ Á Ò Ø Ö Ó Ù Ø Ó Ò ½ ¿
Ý
Ð Ñ
Ø Ù Ö
Ø Ó Ø
Ý
·
³ ½ ¼
Ù Ö ¾ º ½ Ë Ö × Ø Ö × × Ò Ö Û Ð Ð º
This is called t he closure problem: the nu mber of unkn owns (ten: three veloc- Ð Ó × Ù Ö
Ô Ö Ó Ð Ñ
ity components, pressure, six stresses) is larger than the n um ber of equa-
tions (four: the continuity equation and three compon ents of the N avier-
Stokes equations).
For steady, two-dimensional boun dary-layer type of flow (i.e. bound -
ary layers along a flat p late, channel flow, pipe flow, jet and wak e flow, etc.)
where
Î
Í
Ü
Ý
´ ¾ º µ
Eq. 2.6 read s
Í
Í
Ü
·
Î
Í
Ý
È
Ü
·
Ý
Í
Ý
Ù Ú
ß Þ
Ø Ó Ø
´ ¾ º µ
Ü Ü
½
denotes streamwise coordinate, andÝ Ü
¾
coordinate normal to
the flow. Often the pressure gradient
È Ü
is zero.
To the viscous shear stress
Í Ý
on the right-hand side of Eq. 2.8× Ö
× Ø Ö × ×
app ears an add itional turbulent one, a turbu lent shear stress. The total shearstress is thu s
Ø Ó Ø
Í
Ý
Ù Ú
In the wall region (the viscous sublayer, the buffert layer and the logarith-
mic layer) the total shear stress is approximately constant and equal to the
wall shear stress
Û
, see Fig. 2.1. Note that the tot al shear stress is constant
only close to the wall; further away from the wall it decreases (in fully de-
veloped channel flow it decreases linearly by the distance form the wall).
At the wall the turbu lent shear stress vanishes asÙ Ú ¼
, and th e viscous
shear stress attains its w all-stress value
Û
Ù
¾
£
. As we go away from the
wall the viscous stress decreases and turbu lent one increases and at Ý
·
³ ½ ¼
they are ap proximately equal. In the logarithm ic layer th e viscous stress is
negligible compared to the turbulent stress.
In boundary-layer type of flow the turbulent shear stress and the ve-
locity grad ient
Í Ý
have nearly always opposite sign (for a wall jet this
½ ¿
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¾ º ½ Á Ò Ø Ö Ó Ù Ø Ó Ò ½
Ü
Ý
Ú ¼
Ú ¼
Ý
½
Ý
¾
Í Ý µ
Ù Ö ¾ º ¾ Ë Ò Ó Ø Ø Ù Ö Ù Ð Ò Ø × Ö × Ø Ö × × Ù Ú Ò Ó Ù Ò Ö Ý Ð Ý Ö º
is not th e case close to the wall). To get a physical picture of this let u s
study the flow in a boundary layer, see Fig. 2.2. A fluid particle is moving
dow nw ards (particle draw n with solid line) fromÝ
¾
toÝ
½
with (the turbu-
lent flu ctuating) velocityÚ
. At its new location theÍ
velocity is in average
smaller than at its old, i.e.
Í Ý
½
µ
Í Ý
¾
µ
. This means that when the par-
ticle atÝ
¾
(which h as streamw ise velocityÍ Ý
¾
µ
) comes dow n toÝ
½
(where
the streamwise velocity isÍ Ý
½
µ
) is has an excess of streamwise velocity
comp ared to its new environment atÝ
½
. Thus the streamwise fluctuation is
po sitive, i.e.Ù ¼
and the correlation betweenÙ
an dÚ
is negative (Ù Ú ¼
).
If we look at the other particle (dashed line in Fig. 2.2) we reach the
same conclusion. The particle is moving up wa rds (Ú ¼
), and it is bringing
a deficit inÍ
so thatÙ ¼
. Thus, again,Ù Ú ¼
. If we stud y this flow for
a long time and average over time we getÙ Ú ¼
. If we change the sign
of the velocity gradient so that
Í Ý ¼
we w ill find th at the sign of Ù Ú
also changes.Above we have used physical reasoning to show the the signs of
Ù Ú
an d
Í Ý
are opposite. This can also be found by looking at the prod uc-
tion term in th e transp ort equ ation of the Reynold s stresses (see Section 5).
In cases where the shear stress and the velocity gradient have the same
sign (for examp le, in a w all jet) this means that there are other terms in the
transport equation w hich are more important than the produ ction term.
There are different levels of app roximations involved wh en closing the
equation system in Eq. 2.6.
Á º Ð Ö Ñ Ó Ð × º
An algebraic equation is u sed to compu te a tur-
bulent viscosity, often callededdy
viscosity. The Reynolds stress ten-sor is then comp uted u sing an assump tion which relates the Reynolds
stress tensor to th e velocity grad ients and the tu rbulent v iscosity. This
assumption is called the Boussinesq assumption. M odels which are
based on a turbulent (edd y) viscosity are called eddy viscosity m od-
½
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¾ º ¾ Ó Ù × × Ò × Õ × × Ù Ñ Ô Ø Ó Ò ½
els.
Á Á º Ç Ò ¹ Õ Ù Ø Ó Ò Ñ Ó Ð × º
In these mod els a transp ort equation is solved
for a turbulent quantity (usually the turbulent kinetic energy) and
a second tu rbulent qu antity (usually a turbulent length scale) is ob-
tained from an algebraic expression. The turbu lent viscosity is calcu-lated from Boussinesq assumption.
Á Á Á º Ì Û Ó ¹ Õ Ù Ø Ó Ò Ñ Ó Ð × º
These models fall into the class of edd y vis-
cosity m odels. Two transport equations are derived wh ich describe
transport of tw o scalars, for examp le the tu rbulent kinetic energy
and its dissipation
. The Reynolds stress tensor is then compu ted
using an assumption wh ich relates the Reynolds stress tensor to the
velocity gradients and an eddy viscosity. The latter is obtained from
the two transported scalars.
Á Î º Ê Ý Ò Ó Ð × × Ø Ö × × Ñ Ó Ð × º
Here a transport equation is derived for
the Reynolds tensorÙ
Ù
. One transp ort equation has to be add ed fordetermining the length scale of the tu rbulence. Usually an equation
for the d issipation
is used.
Above the different types of turbulence models have been listed in in-
creasing order of complexity, ability to model the turbulence, and cost in
terms of computational work (CPU time).
¾ º ¾ Ó Ù × × Ò × Õ × × Ù Ñ Ô Ø Ó Ò
In eddy viscosity turbulence models the Reynolds stresses are linked to
the velocity grad ients via the tu rbu lent viscosity: this relation is called the
Boussinesq assum ption, w here the Reynolds stress tensor in the tim e aver-aged Navier-Stokes equation is replaced by the turbulent viscosity multi-
plied by the velocity gradients. To show this we introduce this assumption
for the diffusion term at the right-hand side of Eq. 2.6 and make an identi-
fication
¢
Í
·
Í
µ Ù
Ù
£
¢
·
Ø
µ
Í
·
Í
µ
£
which gives
Ù
Ù
Ø
Í
·
Í
µ ´ ¾ º µ
If we in Eq. 2.9 do a contraction (i.e. setting indices
) the right-hand
side gives
Ù
Ù
¾
where
is the tu rbulent kinetic energy (see Eq. 1.10). On the other hand
the continuity equation (Eq. 2.5) gives th at the right-hand side of Eq. 2.9
½
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¾ º ¿ Ð Ö Å Ó Ð × ½
is equal to zero. In order to make Eq. 2.9 valid up on contraction we ad d
¾ ¿ Æ
to the left-hand side of Eq. 2.9 so th at
Ù
Ù
Ø
Í
·
Í
µ ·
¾
¿
Æ
´ ¾ º ½ ¼ µ
Note that contraction of Æ
gives
Æ
Æ
½ ½
· Æ
¾ ¾
· Æ
¿ ¿
½ · ½ · ½ ¿
¾ º ¿ Ð Ö Å Ó Ð ×
In edd y viscosity mod els we w ant an expression for the turbu lent viscosity
Ø
Ø
. The dimension of
Ø
is Ñ
¾
×
(same as
). A tu rbulent velocity Ý
Ú × Ó × Ø Ý
Ñ Ó Ð
scale multiplied with a turbulent length scale gives the correct dimension,
i.e.
Ø
» Í ´ ¾ º ½ ½ µ
Above we have used Í
an d
wh ich are characteristic for the large turbu lent
scales. This is reasonable, because it is these scales which are responsible
for most of the transport by turbulent diffusion.
In an algebraic turbulence mod el the velocity gradient is u sed as a ve-
locity scale and some p hysical length is used as the length scale. In bou nd -
ary layer-type of flow (see Eq. 2.7) we obtain
Ø
¾
Ñ Ü
¬
¬
¬
¬
Í
Ý
¬
¬
¬
¬
´ ¾ º ½ ¾ µ
whereÝ
is the coordinate normal to the w all, and where
Ñ Ü
is the mixing
length, and the model is called the mixing length model. It is an old model
and is hardly used any more. One problem with the model is that
Ñ Ü
is
unknown and m ust be determined.
More modern algebraic models are the Baldwin-Lomax model [2] and
the Cebeci-Smith [6] model which are frequently used in aerodynamics
wh en computing th e flow aroun d airfoils, aeroplanes, etc. For a p resen-
tation and discussion of algebraic turbulence models the interested reader
is referred to Wilcox [46].
¾ º Õ Ù Ø Ó Ò × Ó Ö Ã Ò Ø Ò Ö Ý
¾ º º ½ Ì Ü Ø Õ Ù Ø Ó Ò
The equation for tu rbulent kinetic energy
½
¾
Ù
Ù
is derived from the
Nav ier-Stokes equation w hich reads assum ing steady, incompressible, con-
stan t v iscosity (cf. Eq. 2.4)
Í
Í
µ
È
· Í
´ ¾ º ½ ¿ µ
½
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¾ º Õ Ù Ø Ó Ò × Ó Ö Ã Ò Ø Ò Ö Ý ½
Á º Ó Ò Ú Ø Ó Ò
.
Á Á º È Ö Ó Ù Ø Ó Ò
. The large turbulent scales extract energy from the mean
flow. This term (includ ing the m inus sign) is almost always p ositive.
Á Á Á º
The two first terms representØ Ù Ö Ù Ð Ò Ø « Ù × Ó Ò
by pressure-velocity
fluctuations, and velocity fluctuations, respectively. The last term isviscous diffusion.
Á Î º × × Ô Ø Ó Ò
. This term is responsible for transformation of kinetic
energy at small scales to internal energy. The term (including the
minus sign) is always negative.
In boundary-layer flow the exact
equation read
Í
Ü
·
Î
Ý
Ù Ú
Í
Ý
Ý
Ô Ú ·
½
¾
Ú Ù
Ù
Ý
Ù
Ù
´ ¾ º ¾ µ
Note that the d issipation includes all d erivatives. This is because th e d is-
sipation term is at its largest for small, isotropic scales where the usualbound ary-layer ap proximation that
Ù
Ü Ù
Ý
is not valid.
¾ º º ¾ Ì Õ Ù Ø Ó Ò Ó Ö ½ ¾ Í
Í
µ
The equation for the instantaneous kinetic energyÃ
½
¾
Í
Í
is d erived
from the Navier-Stokes equation. We assume steady, incompressible flow
w ith con stan t v iscosity, see Eq. 2.13. M ultip ly Eq. 2.13 byÍ
so that
Í
Í
Í
µ
Í
È
· Í
Í
´ ¾ º ¾ µ
The term on the left-hand side can be rewritten as
Í
Í
Í
µ
Í
Í
Í
Í
Í
Í
µ
½
¾
Í
Í
Í
µ
½
¾
Í
Í
Í
µ
Í
à µ
´ ¾ º ¾ µ
whereà ½ ¾ Í
Í
µ
.
The first term on the right-han d side of Eq. 2.25 can be wr itten as
Í
È
Í
È µ
´ ¾ º ¾ µ
The viscous term in Eq. 2.25 is rewritten in th e same w ay as th e viscous
term in Section 2.4.1, see Eqs. 2.21 and 2.22, i.e.
Í
Í
Ã
Í
Í
´ ¾ º ¾ µ
Now w e can assemble the transport equation forÃ
by insertin g Eqs. 2.26,
2.27 and 2.28 into Eq. 2.25
Í
à µ
Ã
Í
È µ
Í
Í
´ ¾ º ¾ µ
½
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¾ º Õ Ù Ø Ó Ò × Ó Ö Ã Ò Ø Ò Ö Ý ½
We recognize the usual transport terms due to convection and viscous dif-
fusion. The second term on the right-hand side is responsible for transport
of Ã
by p ressure-velocity interaction. The last term is th e dissipation term
wh ich transforms kinetic energy into internal energy. It is interesting to
compare this term to the dissipation term in Eq. 2.23. Insert the Reynolds
decomposition so that
Í
Í
Í
Í
· Ù
Ù
´ ¾ º ¿ ¼ µ
As th e scales of
Í
is much larger than those of Ù
, i.e. Ù
Í
we get
Í
Í
³ Ù
Ù
´ ¾ º ¿ ½ µ
This shows that the dissipation taking place in the scales larger than the
sma llest ones is negligible (see furth er discussion a t the end of Sub -section 2.4.3).
¾ º º ¿ Ì Õ Ù Ø Ó Ò Ó Ö ½ ¾
Í
Í
µ
The equation for ½ ¾
Í
Í
µ is derived in the same way as th at for ½ ¾ Í
Í
µ .Assume steady, incompressible flow with constant viscosity and multiply
the time-averaged Navier-Stokes equations (Eq. 2.14) so that
Í
Í
Í
µ
Í
È
·
Í
Í
Í
Ù
Ù
µ
´ ¾ º ¿ ¾ µ
The term on the left-hand side and the two first terms on the right-hand
side are treated in the sam e wa y as in Section 2.4.2, and we can wr ite
Í
à µ
Ã
Í
È µ
Í
Í
Í
Ù
Ù
µ
´ ¾ º ¿ ¿ µ
where
à ½ ¾
Í
Í
µ
. The last term is rewritten as
Í
Ù
Ù
µ
Í
Ù
Ù
µ
· Ù
Ù
µ
Í
´ ¾ º ¿ µ
Inserted in Eq. 2.33 gives
Í
à µ
Ã
Í
È µ
Í
Í
Í
Ù
Ù
µ
· Ù
Ù
Í
´ ¾ º ¿ µ
On th e left-hand side we have the u sual convective term. On th e right-han d
side we fin d: tra nsp ort by viscous diffusion, transp ort by pressure-velocity
interaction, viscous d issipation, tran sport by v elocity-stress interaction an d
loss of energy to the fluctuating velocity field, i.e. to
. Note that the last
term in Eq. 2.35 is the same as the last term in Eq. 2.23 but with opp osite
sign: here we clearly can see that the main source term in the
equation
(the prod uction term) appears as a sink term in the
Ã
equation.
It is interesting to comp are the source terms in th e equation for
(2.23),
Ã
(Eq. 2.29) and
Ã
(Eq. 2.35). In theÃ
equation, the dissipation term is
Í
Í
Í
Í
Ù
Ù
´ ¾ º ¿ µ
½
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¾ º Ì Å Ó Ð Ð Õ Ù Ø Ó Ò ¾ ¼
In the
Ã
equation the dissipation term and the negative production term
(representing loss of kinetic energy to the
field) read
Í
Í
· Ù
Ù
Í
´ ¾ º ¿ µ
and in the
equation the p roduction an d the dissipation terms read
Ù
Ù
Í
Ù
Ù
´ ¾ º ¿ µ
The dissipation terms in Eq. 2.36 app ear in Eqs. 2.37 and 2.38. The dissi-
pation of the instantaneous velocity fieldÍ
Í
· Ù
is distributed into
the time-averaged field and the fluctuating field. How ever, as mentioned
above, the dissipation at the fluctuating level is much larger than at the
time-aver aged level (see Eqs. 2.30 and 2.31).
¾ º Ì Å Ó Ð Ð Õ Ù Ø Ó Ò
In Eq. 2.23 a num ber of terms are u nknow n, namely the p roduction term,
the turbulent diffusion term and the dissipation term.
In the prod uction term it is the stress tensor w hich is unknow n. SinceÔ Ö Ó Ù Ø Ó Ò
Ø Ö Ñ
we h ave an expression for this which is used in the N avier-Stokes equation
we u se the same expression in the prod uction term. Equation 2.10 inserted
in th e p rodu ction term (term II) in Eq. 2.23 gives
È
Ù
Ù
Í
Ø
Í
·
Í
¡
Í
¾
¿
Í
´ ¾ º ¿ µ
Note that the last term in Eq. 2.39 is zero for incompressible flow du e to
continuity.
The triple correlations in term III in Eq. 2.23 is mod eled using a grad ientØ Ù Ö Ù Ð Ò Ø
« Ù × Ó Ò
Ø Ö Ñ
law wh ere we assume that is diffused down the grad ient, i.e from region of high
to regions of small
(cf. Fourier’s law for heat flu x: heat is d iffused
from hot to cold regions). We get
½
¾
Ù
Ù
Ù
Ø
´ ¾ º ¼ µ
where
is the turbulent Prandtl nu mber for
. There is no mod el for the
pressure d iffusion term in Eq. 2.23. It is small (see Figs. 4.1 and 4.3) and
thus it is simply neglected.
The dissip ation term in Eq. 2.23 is basically estima ted as in Eq. 1.12. The × × Ô Ø Ó Ò
Ø Ö Ñ
velocity scale is now
Í
Ô
´ ¾ º ½ µ
so that
Ù
Ù
¿
¾
´ ¾ º ¾ µ
¾ ¼
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¾ º Ç Ò Õ Ù Ø Ó Ò Å Ó Ð × ¾ ½
The mod elled
equation can now be assembled and we get
Í
µ
·
Ø
· È
¿
¾
´ ¾ º ¿ µ
We have one constant in the turbulent diffusion term and it will be de-
termined later. The dissipation term contains another u nknow n q uantity,
the turbu lent length scale. An ad ditional transport w ill be derived from
which we can compute
. In the
model, where
is obtained from its
own transport, the dissipation term
¿
¾
in Eq. 2.43 is simp ly
.
For boun dar y-layer flow Eq. 2.43 has the form
Í
Ü
·
Î
Ý
Ý
·
Ø
Ý
·
Ø
Í
Ý
¾
¿
¾
´ ¾ º µ
¾ º Ç Ò Õ Ù Ø Ó Ò Å Ó Ð ×
In one equation mod els a transport equ ation is often solved for the turbu -lent kinetic energy. The unknow n tu rbulent length scale must be given,
and often a n algebraic expression is used [4, 48]. This length scale is, for
example, taken as proportional to the thickness of the boun dary layer, the
width of a jet or a wake. The main disadvantage of this type of model is
that it is not applicable to general flows since it is not possible to find a
general expression for an algebraic length scale.
How ever, some proposals have been m ade w here the turbulent length
scale is computed in a more general way [14, 30]. In [30] a transport equa-
tion for turbu lent viscosity is used.
¾ ½
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¾ ¾
¿ Ì Û Ó ¹ Õ Ù Ø Ó Ò Ì Ù Ö Ù Ð Ò Å Ó Ð ×
¿ º ½ Ì Å Ó Ð Ð Õ Ù Ø Ó Ò
An exact equation for the dissipation can be derived from the Navier-Stokes
equation (see, for instance, Wilcox [46]). How ever, the nu mber of un know n
terms is very large and they involve double correlations of fluctuating ve-
locities, and gradients of fluctuating velocities and pressure. It is better to
derive an
equation based on p hysical reasoning. In the exact equation for
the production term includes, as in the
equation, turbu lent quantities
and and velocity gradients. If we choose to includ eÙ
Ù
an d
Í
in the
produ ction term and only turbu lent qu antities in the dissipation term, we
take, glancing at th e
equ ation (Eq. 2.43)
È
½
Í
·
Í
¡
Í
´ ¿ º ½ µ
× × Ø Ö Ñ
¾
¾
Note that for the production term we haveÈ
½
µ È
. Now we can
write the transport equation for the dissipation as
Í
µ
·
Ø
·
½
È
¾
µ ´ ¿ º ¾ µ
For bound ary-layer flow Eq. 3.2 reads
Í
Ü
·
Î
Ý
Ý
·
Ø
Ý
·
½
Ø
Í
Ý
¾
¾
¾
´ ¿ º ¿ µ
¿ º ¾ Ï Ð Ð Ù Ò Ø Ó Ò ×
The natural w ay to treat wall bound aries is to make the grid sufficiently fine
so that the sharp grad ients prevailing th ere are resolved. Often, wh en com-
pu ting complex three-dimensional flow, that requires too mu ch compu ter
resources. An alternative is to assume that the flow near the wall behaves
like a fully developed turbulent boundary layer and prescribe boundary
cond itions employing wall functions. The assump tion that the flow n ear
the wall has the characteristics of a that in a boundary layer if often not
true at all. However, given a maximum number of nodes that we can af-
ford to use in a computation, it is often preferable to use wall functions
which allows us to use fine grid in other regions where the gradients of the
flow variables are large.
In a fully tu rbulent bound ary layer the p roduction term an d the dissi-
pation term in the log-law region (¿ ¼ Ý
·
½ ¼ ¼
) are much larger than the
¾ ¾
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¿ º ¾ Ï Ð Ð Ù Ò Ø Ó Ò × ¾ ¿
0 200 400 600 800 1000 1200
−20
−10
0
10
20
Ä
Ó
×
×
Ò
Ý
·
È Ö Ó Ù Ø Ó Ò
× × Ô Ø Ó Ò
« Ù × Ó Ò
Ó Ò Ú Ø Ó Ò
Ù Ö ¿ º ½ Ó Ù Ò Ö Ý Ð Ó Ò - Ø Ô Ð Ø º Ò Ö Ý Ð Ò Ò Õ Ù Ø Ó Ò ¿ º
Ê
Æ
³ ¼ ¼ ¸ Ù
£
Í
¼
³ ¼ ¼ ¿ º
other term s, see Fig. 3.1. The log-law w e use can be w ritten as
Í
Ù
£
½
Ð Ò
Ù
£
Ý
´ ¿ º µ
¼ ´ ¿ º µ
Comparing this with the standard form of the log-law
Í
Ù
£
Ð Ò
Ù
£
Ý
· ´ ¿ º µ
we see that
½
´ ¿ º µ
½
Ð Ò
In the log-layer w e can w rite the mod elled
equ ation (see Eq. 2.44) as
¼
Ø
Í
Ý
¾
´ ¿ º µ
wh ere we h ave replaced the d issipation term
¿
¾
by
. In the log-law
region the shear stress Ù Ú
is equal to the w all shear stress
Û
, see Fig. 2.1.
The Boussinesq assu mp tion for the shear stress reads (see Eq. 2.10)
Û
Ù Ú
Ø
Í
Ý
´ ¿ º µ
Using the definition of the wall shear stress
Û
Ù
¾
£
, and inserting Eqs. 3.9,
3.15 into Eq. 3.8 w e get
Ù
¾
£
¾
´ ¿ º ½ ¼ µ
¾ ¿
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¿ º ¾ Ï Ð Ð Ù Ò Ø Ó Ò × ¾
From experiments we h ave that in the log-law region of a boun dary layer
Ù
¾
£
³ ¼ ¿
so that
¼ ¼
.
Ó Ò ¹
× Ø Ò Ø
When w e are using w all functions
an d
are not solved at the nod es
adjacent to the w alls. Instead they are fixed according to th e theory pre-
sented above. The turbulent kinetic energy is set from Eq. 3.10, i.e. º º Ó Ö
È
½ ¾
Ù
¾
£
´ ¿ º ½ ½ µ
where the friction velocityÙ
£
is obtained, iteratively, from the log-law (Eq. 3.4).
IndexÈ
den otes the first interior nod e (adjacent to the w all).
The dissipation
is obtained from observing that production and dis-
sipation are in balance (see Eq. 3.8). The dissipation can thus be w ritten
as º º Ó Ö
È
È
Ù
¿
£
Ý
´ ¿ º ½ ¾ µ
where the velocity gradient in the production term Ù Ú Í Ý
has been
computed from the log-law in Eq. 3.4, i.e.
Í
Ý
Ù
£
Ý
´ ¿ º ½ ¿ µ
For the velocity component parallel to the wall the wall shear stress is º º Ó Ö
Ú Ð Ó Ø Ý
used as a flu x bound ary condition (cf. prescribing h eat flux in the temper-
ature equation).
When th e wall is not p arallel to any v elocity compon ent, it is more con-
venient to prescribe the turbulent viscosity. The wall shear stress
Û
is ob-
tained by calculating the viscosity at the n ode ad jacent to the w all from th e
log-law. The viscosity u sed in m omentum equations is p rescribed at the
nod es adjacent to th e wall (index P) as follows. The sh ear stress at the w all
can be expressed as
Û
Ø È
Í
Ø È
Í
È
where
Í
È
denotes the velocity parallel to the wall and
is the normal
distance to the w all. Using the d efinition of the friction v elocityÙ
£
Û
Ù
¾
£
we obtain
Ø È
Í
È
Ù
¾
£
Ø È
Ù
£
Í
È
Ù
£
SubstitutingÙ
£
Í
È
with the log-law (Eq. 3.4) we finally can w rite
Ø È
Ù
£
Ð Ò
·
µ
where
·
Ù
£
.
¾
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¿ º ¿ Ì Å Ó Ð ¾
¿ º ¿ Ì Å Ó Ð
In the
mod el the mod elled transport equations for
an d
(Eqs. 2.43,
3.2) are solved . The tur bu lent length scale is obtain ed from (see Eq. 1.12,2.42)
¿ ¾
´ ¿ º ½ µ
The tu rbu lent v iscosity is com pu ted from (see Eqs. 2.11, 2.41, 1.12)
Ø
½ ¾
¾
´ ¿ º ½ µ
We have five u nknow n constants
½
¾
an d
, which w e hope
shou ld be u niversal i.e same for all types of flows. Simp le flows are chosen
wh ere the equation can be simplified and w here experimental da ta are used
to determine the constants. The
constant w as d etermined above (Sub-
section 3.2). The
equation in the logarithmic part of a turbu lent bou nd ary
layer was studied where the convection and the diffusion term could be
neglected.
In a similar w ay we can find a value for the
½
constant . We look at th e
½
Ó Ò ¹
× Ø Ò Ø
equation for the logarithm ic part of a turbu lent bound ary layer, where the
convection term is negligible, and utilizing th at prod uction and dissipation
are in balanceÈ
, we can w rite Eq. 3.3 as
¼
Ý
Ø
Ý
ß Þ
·
½
¾
µ
¾
´ ¿ º ½ µ
The dissipation and production term can be estimated as (see Sub-section 3.2)
¿ ¾
´ ¿ º ½ µ
È
Ù
¿
£
Ý
which together withÈ
gives
¿
Ý ´ ¿ º ½ µ
In the logarithm ic layer we have th at Ý ¼
, but from Eqs. 3.17, 3.18 we
find that Ý ¼
. Instead the diffusion term in Eq. 3.16 can be rew ritten
using Eqs. 3.17, 3.18, 3.15 as
Ý
Ø
Ý
¿ ¾
¿
Ý
¾
¾
¾
½ ¾
´ ¿ º ½ µ
¾
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¿ º Ì Å Ó Ð ¾
The constants are determined as in Sub-section 3.3:¬
£
¼ ¼
,
½
,
¾
¿ ¼
,
¾
an d
¾
.
When w all fun ctions are used
an d
are prescribed as (cf. Sub-section 3.2):
Û Ð Ð
¬
£
µ
½ ¾
Ù
¾
£
Û Ð Ð
¬
£
µ
½ ¾
Ù
£
Ý
´ ¿ º ¾ µ
In regions of low tu rbu lence wh en both
an d
go to zero, large nu mer-
ical problems for the
model appear in the
equation as
becomes
zero. The destruction term in the
equation includes
¾
, and this causes
problems as ¼
even if
also goes to zero; they must both go to zero
at a correct rate to avoid problems, and this is often not the case. On the
contrary, no such problems appear in the
equation. If ¼
in the
equation in Eq. 3.24, the tu rbu lent diffusion term simply goes to zero. N ote
that the produ ction term in th e
equation does not includ e
since
½
È
½
Ø
Í
Ü
·
Í
Ü
Í
Ü
½
¬
£
Í
Ü
·
Í
Ü
Í
Ü
In Ref. [35] the
model was used to predict transitional, recirculating
flow.
¿ º Ì Å Ó Ð
One of the most recent p roposals is the
mod el of Speziale Ø Ð º
[39]
wh ere the transport equation for the tu rbulent time scale
is derived. The
exact equation for
is derived from the exact
an d
equations. The
modelled
an d
equations read
Í
µ
·
Ø
· È
´ ¿ º ¾ µ
Í
µ
·
Ø
¾
·
´ ½
½
µ È
·
¾
½ µ
´ ¿ º ¾ µ
·
¾
·
Ø
½
¾
·
Ø
¾
Ø
The constants are:
,
½
an d
¾
are taken from the
mod el, and
½
¾
½ ¿
.
¾
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¾
Ä Ó Û ¹ Ê Æ Ù Ñ Ö Ì Ù Ö Ù Ð Ò Å Ó Ð ×
In the previous section we discussed w all fun ctions w hich are used in order
to reduce the nu mber of cells. However, we must be aware that this is
an approximation which, if the flow near the boundary is important, can
be rather crud e. In many internal flows – w here all bound aries are eitherwalls, symmetry planes, inlet or outlets – the boundary layer may not be
that important, as the flow field is often pressure-determined. For external
flows (for example flow around cars, ships, aeroplanes etc.), however, the
flow conditions in the bound aries are almost invariably important. When
we are predicting heat transfer it is in general no good idea to use wall
functions, because the heat transfer at the walls are very important for the
temperature field in the w hole domain.
When we chose not to use wall functions we thus insert sufficiently
many grid lines near solid boundaries so that the boundary layer can be
adequately resolved. However, when the wall is approached the viscous
effects become m ore important and forÝ
·
the flow is viscous d om-
inating, i.e. the viscous d iffusion is mu ch larger that the tu rbulent one
(see Fig. 4.1). Thus, the tu rbulence m odels presented so far may not be
correct since fully turbulent conditions have been assumed; this type of
models are often referred to as high-Ê
num ber models. In this section
we will d iscuss modifications of high-Ê
number models so that they can
be used all the way down to the wall. These modified models are termed
low Reynolds number models. Please note that “high Reynolds number”
and “low Reynolds number” do not refer to the global Reynolds num ber
(for exampleÊ
Ä
,Ê
Ü
,Ê
Ü
etc.) but h ere we are talking about the local
turbulent Reynolds n um berÊ
Í
formed by a tu rbulent fluctuation
and turbulent length scale. This Reynolds num ber varies throughou t the
comp utational domain and is proportional to the ratio of the turbu lent and
ph ysical viscosity
Ø
, i.e.Ê
»
Ø
. This ratio is of the order of 100 or
larger in fully turbu lent flow and it goes to zero when a w all is app roached.
We start by studying how various quantities behave close to the wall
whenÝ ¼
. Taylor expansion of the flu ctuating velocitiesÙ
(also valid
for the mean velocities
Í
) gives
Ù
¼
·
½
Ý ·
¾
Ý
¾
Ú
¼
·
½
Ý ·
¾
Ý
¾
Û
¼
·
½
Ý ·
¾
Ý
¾
´ º ½ µ
where
¼
¾
are functions of space and time. At the w all we have no-slip, i.e.
Ù Ú Û ¼
which gives
¼
¼
¼
. Furthermore, at the wall
Ù Ü Û Þ ¼
, and the continuity equation gives Ú Ý ¼
so that
¾
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º ½ Ä Ó Û ¹ Ê Å Ó Ð × ¾
0 5 10 15 20 25 30
−0.2
−0.1
0
0.1
0.2
0.3
Ä
Ó
×
×
Ò
Ý
·
È
Ì
·
Ô
Ù Ö º ½ Ð Ó Û Ø Û Ò Ø Û Ó Ô Ö Ð Ð Ð Ô Ð Ø × º Ö Ø Ò Ù Ñ Ö Ð × Ñ Ù Ð ¹
Ø Ó Ò × ¾ ½ º Ê Í
Æ ¼ º Ù
£
Í
¼ ¼ ¼ º Ò Ö Ý Ð Ò Ò
Õ Ù Ø Ó Ò º È Ö Ó Ù Ø Ó Ò È
¸ × × Ô Ø Ó Ò ¸ Ø Ù Ö Ù Ð Ò Ø « Ù × Ó Ò ´ Ý Ú Ð Ó Ø Ý
Ø Ö Ô Ð Ó Ö Ö Ð Ø Ó Ò × Ò Ô Ö × × Ù Ö µ
Ì
·
Ô
¸ Ò Ú × Ó Ù × « Ù × Ó Ò
º Ð Ð
Ø Ö Ñ × Ú Ò × Ð Û Ø Ù
£
º
½
¼
. Equation 4.1 can now be written
Ù
½
Ý ·
¾
Ý
¾
Ú
¾
Ý
¾
Û
½
Ý ·
¾
Ý
¾
´ º ¾ µ
From Eq. 4.2 we imm ediately get
Ù
¾
¾
½
Ý
¾
Ç Ý
¾
µ
Ú
¾
¾
¾
Ý
Ç Ý
µ
Û
¾
¾
½
Ý
¾
Ç Ý
¾
µ
Ù Ú
½
¾
Ý
¿
Ç Ý
¿
µ
¾
½
·
¾
½
µ Ý
¾
Ç Ý
¾
µ
Í Ý
½
Ç Ý
¼
µ
´ º ¿ µ
In Fig. 4.2 DNS-data for th e fully developed flow in a channel is pre-
sented.
º ½ Ä Ó Û ¹ Ê Å Ó Ð ×
There exist a number of Low-Re number
models [32, 7, 10, 1, 27].
When deriving low-Re models it is common to study the behavior of the
terms whenÝ ¼
in the exact equations and require that the correspond-
ing terms in the mod elled equations behave in the same w ay. Let us stud y
¾
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º ½ Ä Ó Û ¹ Ê Å Ó Ð × ¿ ¼
0 20 40 60 80 1000
0.5
1
1.5
2
2.5
3
Ý
·
Ù
¼
Ù
£
Ú
¼
Ù
£
Û
¼
Ù
£
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Ý Æ
Ù
¼
Í
Ú
¼
Í
Û
¼
Í
Ù Ö º ¾ Ð Ó Û Ø Û Ò Ø Û Ó Ô Ö Ð Ð Ð Ô Ð Ø × º Ö Ø Ò Ù Ñ Ö Ð × Ñ Ù Ð ¹
Ø Ó Ò × ¾ ½ º Ê Í
Æ ¼ º Ù
£
Í
¼ ¼ ¼ º Ð Ù Ø Ù Ø Ò Ú Ð Ó Ø Ý
Ó Ñ Ô Ó Ò Ò Ø × Ù
¼
Ô
Ù
¾
º
the exact
equation near the wall (see Eq. 2.24).
Í
Ü
·
Î
Ý
Ù Ú
Í
Ý
ß Þ
Ç Ý
¿
µ
Ô Ú
Ý
Ý
½
¾
Ú Ù
Ù
ß Þ
Ç Ý
¿
µ
·
¾
Ý
¾
Ù
Ù
ß Þ
Ç Ý
¼
µ
´ º µ
The p ressure d iffusion Ô Ú Ý
term is usually neglected, partly because it
is not measurable, and partly because close to the w all it is not important,
see Fig. 4.3 (see also [28]). The modelled equation reads
Í
Ü
·
Î
Ý
Ø
Í
Ý
¾
ß Þ
Ç Ý
µ
·
Ý
Ø
Ý
ß Þ
Ç Ý
µ
·
¾
Ý
¾
ß Þ
Ç Ý
¼
µ
´ º µ
When arriving at that the p roduction term isÇ Ý
µ
we have u sed
Ø
¾
Ç Ý
µ
Ç Ý
¼
µ
Ç Ý
µ ´ º µ
Comparing Eqs. 4.4 and 4.5 we find that the d issipation term in the mod -
elled equation behaves in the same way as in the exact equation when
¿ ¼
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º ½ Ä Ó Û ¹ Ê Å Ó Ð × ¿ ½
0 5 10 15 20 25 30−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Ä
Ó
×
×
Ò
Ý
·
Ì
Ô
Ù Ö º ¿ Ð Ó Û Ø Û Ò Ø Û Ó Ô Ö Ð Ð Ð Ô Ð Ø × º Ö Ø Ò Ù Ñ Ö Ð × Ñ Ù Ð ¹
Ø Ó Ò × ¾ ½ º Ê Í
Æ ¼ º Ù
£
Í
¼ ¼ ¼ º Ò Ö Ý Ð Ò Ò
Õ Ù Ø Ó Ò º Ì Ù Ö Ù Ð Ò Ø « Ù × Ó Ò Ý Ú Ð Ó Ø Ý Ø Ö Ô Ð Ó Ö Ö Ð Ø Ó Ò ×
Ì
¸ Ì Ù Ö Ù Ð Ò Ø
« Ù × Ó Ò Ý Ô Ö × × Ù Ö
Ô
¸ Ò Ú × Ó Ù × « Ù × Ó Ò
º Ð Ð Ø Ö Ñ × Ú Ò
× Ð Û Ø Ù
£
º
Ý ¼
. How ever, both the modelled prod uction and the diffusion term
are of Ç Ý
µ
whereas the exact terms are of Ç Ý
¿
µ
. This inconsistency of
the modelled terms can be removed by replacing the
constant by
where
is a damping function
so that
Ç Ý
½
µ
whenÝ ¼
an d
½
whenÝ
·
¼
. Please note that the term “damping term” in this
case is not correct since
actually is augm enting
Ø
whenÝ
¼
rather
than damping. However, it is common to call all low-Re number functions
for “damping functions”.
Instead of introducing a dam ping function
, w e can choose to solve
for a modified dissipation which is denoted
, see Ref. [25] and Section 4.2.
It is possible to proceed in the same way when deriving damping func-
tions for the
equation [39]. An alternative way is to study the modelled
equation near the wall and keep only the terms wh ich d o not tend to zero.
From Eq. 3.3 we get
Í
Ü
ß Þ
Ç Ý
½
µ
·
Î
Ý
ß Þ
Ç Ý
½
µ
½
È
ß Þ
Ç Ý
½
µ
·
Ý
Ø
Ý
ß Þ
Ç Ý
½
µ
·
¾
Ý
¾
ß Þ
Ç Ý
¼
µ
¾
¾
ß Þ
Ç Ý
¾
µ
´ º µ
where it has been assumed that the production termÈ
has been suitable
modified so thatÈ
Ç Ý
¿
µ
. We find that the only term w hich d o not
vanish at th e wall are the viscous d iffusion term and the d issipation term
¿ ½
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º ¾ Ì Ä Ù Ò Ö ¹ Ë Ö Ñ Ä Ó Û ¹ Ê Å Ó Ð × ¿ ¾
so that close to the wall the dissipation equation reads
¼
¾
Ý
¾
¾
¾
´ º µ
The equation needs to be modified since the diffusion term cannot balance
the destruction term w henÝ ¼
.
º ¾ Ì Ä Ù Ò Ö ¹ Ë Ö Ñ Ä Ó Û ¹ Ê Å Ó Ð ×
There are at least a dozen different low Re
models presented in the
literature. Most of them can be cast in the form [32] (in bou nd ary-layer
form, for convenience)
Í
Ü
·
Î
Ý
Ý
·
Ø
Ý
·
Ø
Í
Ý
¾
´ º µ
Í
Ü
·
Î
Ý
Ý
·
Ø
Ý
·
½
½
Ø
Í
Ý
¾
¾
¾
¾
·
´ º ½ ¼ µ
Ø
¾
´ º ½ ½ µ
· ´ º ½ ¾ µ
Different mod els u se d ifferent dam ping fun ctions (
,
½
an d
¾
) and
different extra terms (
an d
). Many m odels solve for
rather than for
where
is equal to the w all value of
which gives an easy boundary
condition ¼
(see Sub-section 4.3). Other mod els wh ich solve for
u se
no extra source in the
equation, i.e. ¼
.
Below we give some d etails for on e of the m ost pop ular low-Re
models, the Launder-Sharma model [25] which is based on the model of Ä Ù Ò Ö ¹
Ë Ö Ñ
Jones & Laund er [20]. The m odel is given by Eqs. 4.9, 4.10, 4.11 and 4.12
where
Ü Ô
¿
´ ½ · Ê
Ì
¼ µ
¾
½
½
¾
½ ¼ ¿ Ü Ô
Ê
¾
Ì
¡
¾
Ô
Ý
¾
¾
Ø
¾
Í
Ý
¾
¾
Ê
Ì
¾
´ º ½ ¿ µ
¿ ¾
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º ¿ Ó Ù Ò Ö Ý Ó Ò Ø Ó Ò Ó Ö Ò ¿ ¿
The term
was added to match the experimental peak in
aroundÝ · ³
¾ ¼
[20]. The
¾
term is introdu ced to m imic the final stage of decay of turbu-
lence behind a tu rbulence generating grid wh en the exponent in » Ü
Ñ
changes fromÑ ½ ¾
toÑ ¾
.
º ¿ Ó Ù Ò Ö Ý Ó Ò Ø Ó Ò Ó Ö Ò
In many low-Re
models
is the dependent variable rather than
.
The main reason is that the boundary condition for
is rather comp licated.
The largest term in the
equation (see Eq. 4.4) close to the wall, are the
dissipation term and the viscous diffusion term which both are of Ç Ý
¼
µ
so
that
¼
¾
Ý
¾
´ º ½ µ
From this equation we get immediately a boundary condition for
as
Û Ð Ð
¾
Ý
¾
´ º ½ µ
From Eq. 4.14 we can derive alternative bou nd ary cond itions. The exact
form of the dissipation term close to th e w all reads (see Eq. 2.24)
Ù
Ý
¾
·
Û
Ý
¾
µ
´ º ½ µ
where Ý Ü ³ Þ
an dÙ ³ Û Ú
have been assumed. Using
Taylor expansion in Eq. 4.1 gives
¾
½
·
¾
½
· ´ º ½ µ
In the same way we get an expression for the turbulent kinetic energy
½
¾
¾
½
·
¾
½
Ý
¾
´ º ½ µ
so that
Ô
Ý
¾
½
¾
¾
½
·
¾
½
´ º ½ µ
Comp aring Eqs. 4.17 and 4.19 we find
Û Ð Ð
¾
Ô
Ý
¾
´ º ¾ ¼ µ
¿ ¿
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º Ì Ì Û Ó ¹ Ä Ý Ö Å Ó Ð ¿
In the Sharm a-Laun der m odel this is exactly the expression for
in Eqs. 4.12
and 4.13, which means that the boundary condition for
is zero, i.e. ¼
.
In the mod el of Chien [8], the following bou nd ary condition is u sed
Û Ð Ð
¾
Ý
¾
´ º ¾ ½ µ
This is obtained by assuming
½
½
in Eqs. 4.17 and 4.18 so that
¾
¾
½
¾
½
Ý
¾
´ º ¾ ¾ µ
w hich g ives Eq. 4.21.
º Ì Ì Û Ó ¹ Ä Ý Ö Å Ó Ð
Near the walls the one-equation model by Wolfshtein [48], modified by
Chen an d Patel [7], is used. In this mod el the standard equation is solved;the d iffusion term in the
-equation is mod elled using th e edd y viscosity
assum ption. The turbu lent length scales are p rescribed as [15, 11]
Ò ½ Ü Ô ´ Ê
Ò
µ
Ò ½ Ü Ô ´ Ê
Ò
µ
(Ò
is the normal distance from the w all) so that th e dissipation term in the
-equation and the turbulent viscosity are obtained as:
¿ ¾
Ø
Ô
´ º ¾ ¿ µ
The Reynolds nu mberÊ
Ò
and the constants are defined as
Ê
Ò
Ô
Ò
¼ ¼
¿
¼
¾
The one-equation model is used near the walls (forÊ
Ò
¾ ¼
), and the
standard high-Ê
in the remaining p art of the flow. The matching line
could either be chosen along a p re-selected grid line, or it could be d efined
as the cell where the damping function
½ Ü Ô ´ Ê
Ò
µ
takes, e.g., the value¼
. The matching of the one-equation model and the
mod el does not pose any p roblems but gives a smooth distribution of
Ø
an d
across the m atching line.
¿
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º Ì Ð Ó Û ¹ Ê Å Ó Ð ¿
º Ì Ð Ó Û ¹ Ê Å Ó Ð
A model wh ich is being used m ore and more is the
mod el of Wilcox [45].
The standard
mod el can actually be used all the way to the w all with-
out any modifications [45, 29, 34]. One problem is the boundary condition
for
at w alls since (see Eq. 3.25)
¬
£
Ç Ý
¾
µ ´ º ¾ µ
tends to infinity. In Sub-section 4.3 w e d erived bound ary conditions for
by studying the
equation close to the w all. In the same w ay w e can
here use the
equation (Eq. 3.25) close to the w all to derive a bound ary
condition for
. The largest terms in Eq. 3.25 are the viscous diffusion term
and the destruction term, i.e.
¼
¾
Ý
¾
¾
¾
´ º ¾ µ
The solution to this equation is
¾
Ý
¾
´ º ¾ µ
The
equation is norm ally not solved close to the wall but forÝ
·
¾
is
computed from Eq. 4.26, and thus no boundary condition actually needed.
This works w ell in finite volum e method s but w hen finite element method s
are used
is needed at the w all. A slightly different ap proach m ust th en
be used [16].
Wilcox has also p roposed a
mod el [47] which is mod ified for vis-
cous effects, i.e. a tru e low-Re mod el with d amp ing function. He d emon-strates that this model can predict transition and claims that it can be used
for taking the effect of surface roughn ess into accoun t w hich later has b een
confirmed [33]. A m odification of this mod el has been prop osed in [36].
¿
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º Ì Ð Ó Û ¹ Ê Å Ó Ð ¿
º º ½ Ì Ð Ó Û ¹ Ê Å Ó Ð Ó È Ò Ø Ð º
The
mod el of Peng et al. reads [36]
Ø
·
Ü
Í
µ
Ü
·
Ø
Ü
· È
Ø
·
Ü
Í
µ
Ü
·
Ø
Ü
·
½
È
¾
µ ·
Ø
Ü
Ü
Ø
½ ¼ ¾ ¾ Ü Ô
Ê
Ø
½ ¼
¼ ¼ ¾ ·
½ Ü Ô
Ê
Ø
½ ¼
¿
µ
¼ ·
¼ ¼ ¼ ½
Ê
Ø
Ü Ô
Ê
Ø
¾ ¼ ¼
¾
µ
½ · ¿ Ü Ô
Ê
Ø
½
½ ¾
½ · ¿ Ü Ô
Ê
Ø
½
½ ¾
¼ ¼
½
¼ ¾
¾
¼ ¼
¼
¼
½ ¿
´ º ¾ µ
º º ¾ Ì Ð Ó Û ¹ Ê Å Ó Ð Ó Ö Ö Ø Ð º
A new
mod el was recently prop osed by Bredberg et al. [5] wh ich read s
Ø
·
Ü
Í
µ È
·
Ü
·
Ø
Ü
Ø
·
Ü
Í
µ
½
È
¾
¾
·
·
Ø
Ü
Ü
·
Ü
·
Ø
Ü
´ º ¾ µ
The turbu lent viscosity is given by
Ø
¼ ¼ ·
¼ ½ ·
½
Ê
¿
Ø
½ Ü Ô
Ê
Ø
¾
¾
µ
´ º ¾ µ
¿
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º Ì Ð Ó Û ¹ Ê Å Ó Ð ¿
with the turbu lent Reynolds num ber defined asÊ
Ø
µ
. The constants
in the model are given as
¼ ¼
½
½ ½
½
¼
¾
¼ ¼ ¾
½
½
´ º ¿ ¼ µ
¿
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¿
Ê Ý Ò Ó Ð × Ë Ø Ö × × Å Ó Ð ×
In Reynolds Stress Models the Boussinesq assu mp tion (Eq. 2.10) is not u sed
but a partial differential equation (transport equation) for the stress tensor
is derived from the Navier-Stokes equation. This is d one in th e same w ay
as for the
equ ation (Eq. 2.23).Take the Navier-Stokes equation for the instantaneous velocity
Í
(Eq. 2.4).
Subtr act the momentu m equ ation for the time averaged velocity
Í
((Eq. 2.6)
and multiply byÙ
. Derive the same equation with indices
an d
inter-
changed. Add the two equations and time average. The resulting equation
reads
Í
Ù
Ù
µ
ß Þ
Ù
Ù
Í
Ù
Ù
Í
ß Þ
È
·
Ô
Ù
· Ù
µ
ß Þ
Ù
Ù
Ù
·
Ô Ù
Æ
·
Ô Ù
Æ
Ù
Ù
µ
ß Þ
¾ Ù
Ù
ß Þ
´ º ½ µ
where
È
is the produ ction of Ù
Ù
[note thatÈ
½
¾
È
(È
È
½ ½
· È
¾ ¾
·
È
¿ ¿
)];
is the pressure-strain term, wh ich p romotes isotropy of the tu rbu-
lence;
is the dissipation (i.e. transformation of m echanical energy into
heat in the sm all-scale tu rbulence) of Ù
Ù
;
an d
are the convection and diffusion, respectively, of Ù
Ù
.
Note that if we take the trace of Eq. 5.1 and divide by two we get the
equation for the tu rbulent kinetic energy (Eq. 2.23). When taking th e trace
the p ressure-strain term van ishes since
¾
Ô
Ù
¼ ´ º ¾ µ
du e to continuity. Thus the p ressure-strain term in the Reynolds stress
equation does not add or destruct any turbulent kinetic energy it merelyredistributes the energy between the normal components (
Ù
¾ ,Ú
¾ an dÛ
¾ ).
Furtherm ore, it can be show n u sing ph ysical reasoning [19] that
acts to
reduce the large normal stress component(s) and distributes this energy to
the other normal component(s).
¿
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º ½ Ê Ý Ò Ó Ð × Ë Ø Ö × × Å Ó Ð × ¿
º ½ Ê Ý Ò Ó Ð × Ë Ø Ö × × Å Ó Ð ×
We find that there are terms which are unknown in Eq. 5.1, such as the
triple correlationsÙ
Ù
Ù
, the pressure diffusion Ô Ù
Æ
· · Ô Ù
Æ
µ
an d
the pressure strain
, and the d issipation tensor
. From Navier-Stokes
equation we could d erive transport equations for this unkn own quantities
but this would add further unknowns to the equation system (the closure
problem, see p. 12). Instead w e sup ply mod els for the un known terms.
The pressure strain term, which is an imp ortant term since its contribu -
tion is significant, is modelled as [24, 17]
½
·
¾
·
¼
½
·
¼
¾
½
½
Ù
Ù
¾
¿
Æ
¾
¾
È
¾
¿
Æ
È
¼
Ò Ò ½
¾
¼
½
Ù
¾
Ò
¼
× × ½
¼
½
Ù
¾
Ò
¿
¾
¾ Ü
Ò
´ º ¿ µ
The object of the wall correction terms
¼
Ò Ò ½
an d
¼
× × ½
is to take t he effect of
the wall into account. Here we have introdu ced a× Ò
coordinate system,
with×
along the wall andÒ
normal to the wall. Near a w all (the term
“near” may well extend toÝ
·
¾ ¼ ¼
) the norm al stress norm al to the wall is
dam ped (for a w all located at , for examp le,Ü ¼
this mean that the normal
stressÚ
¾ is dam ped ), and the other tw o are augm ented (see Fig. 4.2).
In the literature there are many proposals for better (and more compli-
cated) pressure strain models [40, 23].
The triple correlation in the d iffusion term is often m odelled as [9]Ø Ö Ô Ð
Ó Ö Ö Ð ¹
Ø Ó Ò
×
Ù
Ù
Ñ
Ù
Ù
µ
Ñ
´ º µ
The pressure diffusion term is for tw o reasons comm only neglected. First, it
is not possible to measure this term an d before DNS-da ta (Direct Nu merical
Simulations) were available it was thus not possible to model this term.
Second , from DNS-data is has ind eed b een found to be sm all (see Fig. 4.3).
The dissipation tensor
is assum ed to be isotropic, i.e.
¾
¿
Æ
´ º µ
From the definition of
(see 5.1) we find that the assu mp tion in Eq. 5.5 is
equivalent to assu ming that for small scales (where d issipation occurs) the
¿
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º ¾ Ê Ý Ò Ó Ð × Ë Ø Ö × × Å Ó Ð × Ú × º Ý Î × Ó × Ø Ý Å Ó Ð × ¼
two derivativeÙ
an dÙ
are not correlated for
. This is the same as
assum ing that for small scalesÙ
an dÙ
are not correlated for
which is
a good app roximation since the turbu lence at these small scale is isotropic,
see Section 1.4.
We have given models for all unknown term in Eq. 5.1 and the modelled
Reynold s equ ation read s [24, 17]
Í
Ù
Ù
µ
Ù
Ù
Í
Ù
Ù
Í
·
Ù
Ù
µ
·
×
Ù
Ù
Ñ
Ù
Ù
µ
Ñ
·
¾
¿
Æ
´ º µ
where
should be taken from Eq. 5.3. For a review on RSMs (Reynolds
Stress Models), see [18, 22, 26, 38].
º ¾ Ê Ý Ò Ó Ð × Ë Ø Ö × × Å Ó Ð × Ú × º Ý Î × Ó × Ø Ý Å Ó Ð ×
Whenever non-isotropic effects are important we shou ld consider u sing RSMs.
Note that in a turbu lent bound ary layer the turbu lence is always non-isotropic,
but isotropic edd y viscosity mod els hand le this type of flow excellent as far
as mean flow quantities are concerned. Of course a
mod el give very
poor representation of the normal stresses. Examples where non-isotropic
effects often are important are flows with strong curvature, swirling flows,
flows w ith strong acceleration/ retardation. Below w e present list some ad -
vantages and disadvantages with RSMs and edd y viscosity m odels.
Advantages w ith edd y viscosity m odels:
i) simple due to the use of an isotropic eddy (turbulent) viscosity;
ii) stable via stability-promoting second-order gradients in the mean-
flow equations;iii) wor k reasonably well for a large num ber of engineering flows.
Disadvantages with eddy viscosity models:
i) isotropic, and thu s not good in predicting normal stresses (Ù
¾
Ú
¾
Û
¾ );
ii) as a consequen ce of i) it is un able to accoun t for curvatu re effects;
iii) as a consequ ence of i) it is u nable to accoun t for irrotational strains.
Advantages with RSMs:
i) the produ ction terms need not to be modelled;
ii) thanks to i) it can selectively augment or damp the stresses due tocurvature effects, acceleration/ retardation, swirling flow, buoyancy
etc.
Disadvan tages with RSMs:
¼
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º ¿ Ù Ö Ú Ø Ù Ö « Ø × ½
00
AA
B BUθ
(r)r
Ù Ö º ½ Ù Ö Ú Ó Ù Ò Ö Ý Ð Ý Ö - Ó Û Ð Ó Ò Ö Ó Ò × Ø Ò Ø º Í
Í
Ö µ Í
Ö
¼ º
i) comp lex and d ifficult to implement;
ii) numerically unstable because small stabilizing second-order deriva-
tives in the momentum equations (only laminar diffusion);
iii) CPU consuming.
º ¿ Ù Ö Ú Ø Ù Ö « Ø ×
Curv ature effects, related either to curva ture of th e wall or streamline cur-
vature, are know n to h ave significant effects on the tu rbulence [3]. Both
types of curvature are present in attached flows on curved surfaces, and
in separation regions. The entire Reynolds stress tensor is active in the
interaction process between shear stresses, normal stresses and mean ve-
locity strains. When predicting flows where curvature effects are impor-
tant, it is thu s necessary to u se turbu lence mod els that accurately p redictall Reynolds stresses, not on ly the shear stresses. For a d iscussion of curva-
ture effects, see Refs. [12, 13].
When the streamlines in boundary layer type of flow have a convex
(concave) curvatu re, the turbu lence is stabilized (destabilized), wh ich da mp -
ens (augments) the turbu lence [3, 37], especially the shear stress and the
Reynolds stress norm al to th e w all. Thus convex streamline curvature de-
creases the stress levels. It can b e show n th at it is the exact mod elling of the
prod uction terms in the RSM w hich allows the RSM to respon d correctly to
this effect. The
mod el, in contrast, is not able to respond to streamline
curvature.
The ratio of boundary layer thicknessÆ
to curvature rad iusÊ
is a com-mon par ameter for quant ifying the curvatu re effects on the turbu lence. The
work reviewed by Bradshaw demonstrates that even such small amounts
of convex curvature asÆ Ê ¼ ¼ ½
can have a significant effect on the tur-
bulence. Thompson and Whitelaw [42] carried out an experimental inves-
½
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º ¿ Ù Ö Ú Ø Ù Ö « Ø × ¾
x
ystreamline
Ù Ö º ¾ Ì × Ø Ö Ñ Ð Ò × Û Ò - Ø ¹ Ô Ð Ø Ó Ù Ò Ö Ý Ð Ý Ö × Ö Ð Ó Ò
Ø Ü ¹ Ü × Ö × Ù Ò Ð Ý - Ø Ù Ô Û Ö × ´ Ó Ò Ú Ù Ö Ú Ø Ù Ö µ Ó Û Ò Ø Ó º º
Ò Ô Ô Ö Ó Ò × Ô Ö Ø Ó Ò Ö Ó Ò º
tigation on a configuration simulating the flow near a trailing edge of an
airfoil, where they measuredÆ Ê ³ ¼ ¼ ¿
. They reported a 50 percent de-
crease of Ú
¾ (Reynolds stress in the nor mal d irection to th e wall) owing to
curvature. The reduction of Ù
¾ an d Ù Ú
was also substantial. In addition
they reported significant damping of the turbulence in the shear layer in
the outer part of the separation region.
An illustrative mod el case is curved bou nd ary layer flow. A polar coor-
dinate systemÖ
(see Fig. 5.1)) with
locally aligned with the streamline
is introdu ced. AsÍ
Í
Ö µ
(with Í
Ö ¼
an dÍ
Ö
¼
), the radial
inviscid momentum equation degenerates to
Í
¾
Ö
Ô
Ö
¼ ´ º µ
Here the variables are instantaneous or laminar. The centrifugal force ex-
erts a force in the norm al direction (outward ) on a fluid following the stream-
line, wh ich is balanced by th e pressure grad ient. If the fluid is displaced by
some disturbance (e.g. turbulent flu ctuation) outwards to level A, it en-coun ters a pressure gradient larger than that to wh ich it was accustomed
atÖ Ö
¼
, as Í
µ
Í
µ
¼
, which from Eq. 5.7 gives Ô Ö µ
Ô Ö µ
¼
.
Hence the fluid is forced back toÖ Ö
¼
. Similarly, if the flu id is disp laced
inwards to level B, the pressure gradient is smaller here than atÖ Ö
¼
an d
cannot keep the flu id at level B. Instead the centrifugal force d rives it back
to its original level.
It is clear from th e m odel problem above th at convex curvatu re, when
Í
Ö ¼
, has a stabilizing effect on (turbulent) fluctuations, at least in
the rad ial direction. It is discussed below how the Reynolds stress model
responds to streamline curvature.
Assume that there is a flat-plate bound ary layer flow. The ratio of thenormal stresses
Ù
¾ an d Ú
¾ is typically 5. At oneÜ
station, the flow is
deflected u pw ards, see Fig. 5.2. H ow will this affect th e tur bulence? Let us
stud y the effect of concave streamline curv ature. The p rodu ction term sÈ
owing to rotational strains can be written as
¾
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º ¿ Ù Ö Ú Ø Ù Ö « Ø × ¿
Í
Ö ¼
Í
Ö ¼
Ó Ò Ú Ü Ù Ö Ú Ø Ù Ö × Ø Ð Þ Ò × Ø Ð Þ Ò
Ó Ò Ú Ù Ö Ú Ø Ù Ö × Ø Ð Þ Ò × Ø Ð Þ Ò
Ì Ð º ½ « Ø Ó × Ø Ö Ñ Ð Ò Ù Ö Ú Ø Ù Ö Ó Ò Ø Ù Ö Ù Ð Ò º
Ê Ë Å Ù
¾
Õ È
½ ½
¾ Ù Ú
Í
Ý
´ º µ
Ê Ë Å Ù Ú Õ È
½ ¾
Ù
¾
Î
Ü
Ú
¾
Í
Ý
´ º µ
Ê Ë Å Ú
¾
Õ È
¾ ¾
¾ Ù Ú
Î
Ü
´ º ½ ¼ µ
È
Ø
Í
Ý
·
Î
Ü
¾
´ º ½ ½ µ
As long as the streamlines in Fig. 5.2 are p arallel to the w all, all p ro-
duction is a result of
Í Ý
. However as soon as the streamlines are de-
flected, there are m ore terms resulting from
Î Ü
. Even if
Î Ü
is much
smaller that
Í Ý
it will still contribute non-negligibly toÈ
½ ¾
as Ù
¾ is
much larger than Ú
¾ . Thus the magnitud e of È
½ ¾
will increase (È
½ ¾
is nega-
tive) as
Î Ü ¼
. An increase in the magn itude of È
½ ¾
will increase Ù Ú
,
which in turn will increaseÈ
½ ½
an dÈ
¾ ¾
. This means that Ù
¾ an d Ú
¾ will
be larger and the magnitud e of È
½ ¾
will be further increased, and so on. It
is seen that there is a positive feedback, which continuously increases the
Reynolds stresses. It can be said that the turbulence is × Ø Ð Þ
owing
to concave curv ature of th e streamlines.
How ever, the
model is not very sensitive to streamline curvature
(neither convex nor concave), as the two rotational strains are multiplied
by th e same coefficient (the tu rbulent viscosity).
If the flow (concave cur vatu re) in Fig. 5.2 is a wall jet flow wh ere
Í Ý
¼
, the situation will be reversed: the turbulence will be× Ø Ð Þ
. If the
streamline (and the wall) in Fig. 5.2 is deflected downwards, the situation
will be as follows: th e turbu lence is stabilizing w hen
Í Ý ¼
, and d esta-
bilizing for
Í Ý ¼
.
The stabilizing or destabilizing effect of streamline curvature is thusdep end ent on the type of curvatu re (convex or concave), and wh ether there
is an increase or decrease in momentum in the tangential direction with
radial d istance from its origin (i.e. the sign of
Í
Ö
). For conven ience,
these cases are summarized in Table 5.1.
¿
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º Ð Ö Ø Ó Ò Ò Ê Ø Ö Ø Ó Ò
x1
x2x1
x2
x
y
Ù Ö º ¿ Ë Ø Ò Ø Ó Ò - Ó Û º
º Ð Ö Ø Ó Ò Ò Ê Ø Ö Ø Ó Ò
When th e flow accelerates and / or decelerate the irrotational strains (
Í Ü
,
Î Ý
an d
Ï Þ
) become imp ortant.
In boundary layer flow, the only term which contributes to the pro-
du ction term in the
equation is Ù Ú Í Ý
(Ü
denotes streamwise direc-
tion). Thompson and Whitelaw [42] found that, near the separation p oint
as well as in the separation zone, the production term
Ù
¾
Ú
¾
µ Í Ü
is of equal imp ortance. This was confirmed in pred iction of separ ated flow
using RSM [12, 13].
In pure boundary layer flow the only term which contributes to the
production term in the
an d
-equations is Ù Ú
Í Ý
. Thompson and
Whitelaw [42] found that near the separation point, as well as in the sepa-
ration zone, the produ ction term Ù
¾
Ú
¾
µ
Í Ü
is of equ al import ance.
As the exact form of the production terms are used in second-moment clo-
sures, the prod uction d ue to irrotational strains is correctly accoun ted for.
In the case of stagnation-like flow (see Fig. 5.3), whereÙ
¾
³ Ú
¾ the pro-
du ction d ue to norm al stresses is zero, which is also the results given by
second-moment closure, whereas
models give a large production. Inorder to illustrate this, let us write the production due to the irrotational
strains
Í Ü
an d
Î Ý
for RSM an d
:
Ê Ë Å ¼ È
½ ½
· È
¾ ¾
µ Ù
¾
Í
Ü
Ú
¾
Î
Ý
È
¾
Ø
Í
Ü
¾
·
Î
Ý
¾
µ
If Ù
¾
³ Ú
¾ we getÈ
½ ½
· È
¾ ¾
³ ¼
since
Í Ü
Î Ý
due to continuity.
The production termÈ
in
mod el, however, will be large, since it w ill
be× Ù Ñ
of the two strains.
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Ê Ê Æ Ë
Ê Ö Ò ×
[1] ¸ Ã º ¸ Ã Ó Ò Ó ¸ Ì º ¸ Ò Æ Ò Ó ¸ º
A new turbulence model
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[2] Ð Û Ò ¸ º ¸ Ò Ä Ó Ñ Ü ¸ À º
Thin-layer approximation and alge-
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AL, 1978.
[3] Ö × Û ¸ È º
Effects of streamline curvature on turbulent flow.
Agard ograp h progress n o. 169, AGARD, 1973.
[4] Ö × Û ¸ È º ¸ Ö Ö × × ¸ º ¸ Ò Ø Û Ð Ð ¸ Æ º
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[7] Ò ¸ À º ¸ Ò È Ø Ð ¸ Î º
Near-wall turbu lence mod els for complex
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[8] Ò ¸ Ã º
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[9] Ð Ý ¸ º ¸ Ò À Ö Ð Ó Û ¸ º
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[10] Ú × Ó Ò ¸ Ä º
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[11] Ú × Ó Ò ¸ Ä º
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Prediction of the flow around an airfoil using a
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Ê Ê Æ Ë
[13] Ú × Ó Ò ¸ Ä º
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[14] Ú × Ó Ò ¸ Ä º ¸ Ò Ç Ð × × Ó Ò ¸ º
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Nav ier-Stokes stall p redictions using
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[16] Ñ Ú Ò ¸ È º ¸ Ò Ú × Ó Ò ¸ Ä º
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[17] × Ó Ò ¸ Å º ¸ Ò Ä Ù Ò Ö ¸ º
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[18]À Ò Ð
¸ Ã º
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[19]À Ò Þ ¸ Â º
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[20]Â Ó Ò × ¸ Ï º ¸ Ò Ä Ù Ò Ö ¸ º
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[21]Ã Ñ ¸ Â º
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[34]È Ò ¸ Ë º ¹ À º ¸ Ú × Ó Ò ¸ Ä º ¸ Ò À Ó Ð Ñ Ö ¸ Ë º
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