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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



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



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

¿



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



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.



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,



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


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




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



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



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



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



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



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 ¿



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



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



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



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

Ã µ

´ ¾ º ¾ µ

whereÃ 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

Ã µ

Ã

Í

È µ

Í

Í

´ ¾ º ¾ µ

½



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

Ã µ

Ã

Í

È µ

Í

Í

Í

Ù

Ù

µ

´ ¾ º ¿ ¿ µ

where

Ã 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

Ã 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



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



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



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



¿ 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 ¿



¿ 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



¿ 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



¿ 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



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



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



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



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



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



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

¿ ¿



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.

¿



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].

¿




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

¿




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

¿



¿

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).

¿



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

¿



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



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



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



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.

¿



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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kompendium_turb davidson

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