Internal Report 00/4
On the Wall Boundary Condition for Turbulence Models
JONAS BREDBERG
Department of Thermo and Fluid Dynamics
HALMERS NIVERSITY OF ECHNOLOGY
C ¢¡ U T £¥¤§¦©¨ £ ¦©¦¥ ¥ On the Wall Boundary Condition for Turbulence Models
JONAS BREDBERG Department of Thermo and Fluid Dynamics Chalmers University of Technology
ABSTRACT This report explains and discuss two main boundary condi- tions for turbulence models: the integrating and the wall fun- ction approach. The report thoroughly derives the law-of-the- wall for both momentum and thermal field. The deviation of
these laws from DNS-data is discussed. The Wilcox ¢¡¤£ tur- bulence model constants are derived on the basis of the law- of-the-wall and experiments. Using a simplistic treatment, the low-Reynolds number modifications (damping functions)
to the ¥¡¤£ model are explained. Included in the report is also a guideline on the implemen- tation of wall functions, both the standard and the Chieng- Launder two-layer variant. A new boundary condition for turbulence models combining wall function and integration approach is presented.
Keywords: boundary condition, wall function, turbulence model
ii Contents
Nomenclature iv
1 Introduction 1
2 Near-wall Physics 1 2.1 Buffer Layer ...... 1 2.2 Viscous sub-layer ...... 1 2.3 Inertial Sub-layer ...... 2
3 Law-of-the-Wall 3 3.1 Momentum ...... 3 3.2 Energy/Temperature ...... 4 3.3 Turbulence ...... 6 3.4 Turbulence Model Constants ...... 7
3.4.1 Coefficient, ¢¡ ...... 7
3.4.2 Coefficient, ...... 7
3.4.3 Coefficients, £¥¤¦£§¡ ...... 7
3.4.4 Coefficient, ¨ ...... 7 3.5 HRN Turbulence Model ...... 8
4 Modelling Near-wall Turbulence 8 4.1 LRN-models: Integration Method ...... 8
4.1.1 The damping functions ...... 9 ¢¡ 4.1.2 © modification, ...... 9
4.1.3 Production-to-dissipation rate ...... 10 ¡ 4.1.4 Rationale behind the and coefficients 10 4.2 HRN-models: Wall Functions ...... 11 4.2.1 Standard wall functions ...... 11 4.2.2 Launder-Spalding methodology ...... 11 4.2.3 Improvements of the near-wall represen- tation, Chieng-Launder model ...... 12 4.2.4 Modification to the Chieng-Launder model 13 4.2.5 Extensions to the Chieng-Launder model 14 4.3 New Model: Blending Integrating and Wall Function Boundary Conditions ...... 14 4.3.1 Blending function ...... 15
4.3.2 HRN part: Simplified Chieng-Launder ¡¤£ model applied to a -type turbulence model ...... 15
4.3.3 LRN part: Standard ¥¡¤£ ...... 16
4.3.4 The blending ¥¡¤£ turbulence model . . . 16
5 Modelling Near-wall Heat Transfer 18 5.1 Integration Method ...... 18 5.2 Wall Function Method ...... 19 5.3 New Blending Model ...... 19
iii Nomenclature Greek Symbols
Latin Symbols
¡ +,¨- . /¢
Thermal diffusivity
¡£¢
Turbulence model constant ¡
¤ ¡ ¡ ¡£¢
Turbulence model coefficients
¥¤ ¡§¦©¨ ¢
© Specific heat
¤ ¡ ¡£¢
Turbulence model coefficients ¡
¡ ¡£¢
© Various constants
¡ ¡£¢
¨ Turbulence model coefficient
k
¡ ¡£¢
© Length-scale constant "¢
Characteristic length scale ¡
l
© ¡£¢
Turbulence model coefficient ¡ "¢
Cell size ¡
£ ¥¨ ¡ ¡£¢
© Skin friction coefficient,
¡ ¨4=m¢
R Dissipation rate
¡ ¢
Hydrualic diameter n
¡ ¡£¢
! Normalized wall distance
¡ "¢
Rib-size o
#
¡£¢
Van Karman constant ¡
p
9 E
¡£¢
Turbulence model constant ¡
$
¤q
¨ ¡ +,¨ G ¢
Thermal conductivity, ©
¡£¢
Damping function ¡
$
¡ ¨4.¢
M Kinematic viscosity
7
% ¤'& ¤)( ¡ ¡£¢
¤ Functions
*
q 4-¨ :¢
Dynamic viscosity ¡ "¢
Channel half-height ¡
*
=
¡ c ¨- ¢
Density +,¨- . /¢
Heat transfer coefficient ¡
0
£ 4 Z\[:¢
Specific dissipation rate ¡ "¢
Channel height ¡
1
£ ¡ ¡ ¡£¢
£¥¤ Turbulence model coefficients
32 ¡ ¡£¢
¤ Tensor indices,
¡ 4m¢ Time
streamwise: 1 (U) 7
¡ ¨- G:¢
Shear stress
8 7
wall normal: 2 (V) X
¡ ¨ G:¢ Wall shear,
spanwise: 3 (W)
¡ ¨45¢ Turbulent kinetic energy
6 Superscripts
¡ ¢ Length scale !
78 Effective value
r?s
8
¡ ¡£¢
Nusselt number X 7
9 Normalized value using :
s t
_ _ _
¡ ¨ :¢
Static pressure X
9
¨
s
t
8
¡ "¢
Rib-pitch X
9<;
u¨
s t
8
¡ ¨4=5¢
Turbulent production, X
@
U U ¨M
8?>@A8?>
B
s t
8
B.C C?D
¨
X
O¨
9 E
s t
8©v
¡ ¡£¢
Prandtl number X
R RWMO¨
9 E
s t
8
¡ ¡£¢
Functional Prandtl number X
@
£ £MO¨
F
s t
8
¡ +,¨- G5¢
Heat flux X
©¨
HJI H ; HLK
s t
8?w 8 w 8
> >
¤ ¤ ¡ ¡£¢
Turbulence model constants X
¨
H 0
!
r
¨M ¡ ¡£¢
Reynolds number, V
H
¡
!N Normalized value using :
v
¡ ¡£¢ t
Turbulent Reynolds number, V
[)x
N
U ¡ U © O¨MyP ¡zRQ
O¨P £MQ ¨ PSRMOQ M ¨M v
, , t
V
H
!T
¡ ¡:[)x
U U ¨MyP ¡¤£Q
¡£¢
Viscous sub-layer Reynolds ¡
TV T
¨WM
number, U
H 8
!X X
¡£¢ ¡ Subscripts
based Reynolds f
8
X M
U Non-modified value
number, ]
@
Y
B
4 Z\['¢ ¡ First interior node
Strain-rate tensor, {
@ @
]
B^ B
C C?D C C?D
¨ ¨ Q
¨OP Bulk value
_
© |
¢ ¡ Centerline value
Temperature 0/H 7
>
`
¢ ¡ HRN (logarithmic) part
Fluctuating temperature 6
_
X /¢
¡ Laminar quantity
Friction temperature, H 7
_ _ 8
X F
|
¡ Qacb ¨ P LRN (viscous) part
@
}
¡ /¨4.¢
Velocity North face value
~
@
>
8 /¨4.¢
¡ Node value
Fluctuating velocity `
@
8d> 8d>
B Turbulent quantity
¡ G¨45¢
Reynolds stresses _ 8
X Thermal quantity
w
.¥¨ ¡ ¨W4.¢
Friction velocity, e
8 8
]gfWfhfji
X
X Viscous sub-layer value
¡ ¨W ¡ ¡£¢
Normalized value:
Wall value
D "¢ Streamwise coordinate ¡
Quantity based on the
¡ "¢ U Wall normal coordinate friction velocity
iv 1 Introduction 2 Near-wall Physics
One of the most common engineering problems is com- The near-wall region may be sub-divided into three dif- puting turbulent flows that are influenced by an adja- ferent areas, Tennekes and Lumley [31]:
cent wall. Examples of this are flows in turbomachinery,
s
f¢¡ ¡¤£
around vehicles, and in pipes. The main two effects of a viscous sub-layer U
wall are:
s
£¥¡ ¡§¦hf
U
buffer layer
Damping the wall normal components, making the
s
¦hf¥¡ ¡ fWf
turbulent flow anisotropic. inertial sub-layer U Increasing the production of turbulence through the shearing mechanism in the flow. The turbulence is negligible in the viscous sub-layer, while the viscous effects are small in the inertial sub- The wall gives rise to a boundary layer, where the ve- layer. In the buffer layer, however, both turbulent and locity changes from the no-slip condition at the wall to viscous effects are of importance, see Sahay and Sreeni- its free stream value. The variation is usually largest in vasan [29] and Fig. 1. the near-wall region, and hence the strongest gradients are found here. Similarly, for heat transfer applications, 1.5 there exists a thermal boundary layer with equally large Total shear gradients. Because both heat transfer and friction are Turbulent shear computed using gradients of the dependent variable, it Laminar shear is very important to accurately capture this near-wall variation. 1 The standard method is to apply a very fine mesh close to the wall, to resolve the flow. This method is called the integration method, which necessitates an LRN (low- ¨ © Reynolds-number) type of turbulence model. At higher Reynolds numbers, the region under the wall influence 0.5 diminishes. However, because it is equallyPSfragimportantreplacementsto accurately capture the near-wall gradients, a large num- ber of nodes are then necessary. From an engineering point of view, this becomes inconceivable and a function that bridge the near-wall region is instead introduced, 0
thereby significantly reducing the computational requi- 0 10 20 30 40 50 rements. The anticipation is that this can be done with ©
only a small deterioration in the results. This latter ap- Figure 1: Laminar and turbulent shear in the near wall
t
H ¦ £ proach is denoted the wall function method, which uses region. Channel flow, DNS-data, !X [25]. an HRN (high-Reynolds-number) type of turbulence mo- del.
2.1 Buffer Layer
The maximum turbulent production occurs in the buffer
s t
] layer at roughly U , slightly dependent on the Rey- nolds number. Due to large variations in the different turbulence source terms, see Figs. 2 and 3, the model- ling becomes very difficult. Today there exists no general method for applying a turbulence model, with the first computational interior node located in the buffer layer. Instead of trying to model the behaviour in the buffer layer, the common practise is to place the first near-wall node in either the viscous sub-layer (LRN-models), or in the inertial sub-layer (HRN-models). A viscous appro- ach is valid in the former case, while a turbulent ap- proach is more correct in the latter, for the first interior computational cell.
2.2 Viscous sub-layer In the viscous sub-layer the following asymptotic re- lation for the velocity, temperature, turbulence kinetic
1 0.25 10 , DNS 0.2 9 Production , Model Dissipation
0.15 , DNS 8
, Model
0.1 , DNS 7
, Model
¡ 0.05 , DNS
6 ¥ PSfrag replacements
, Model
¦ ¤
¡£¢ 0 5
−0.05 4
−0.1 3
PSfrag replacements −0.15 2
−0.2 1
−0.25 0
0 10 20 30 40 50 0 2 4 6 8 10
©
©
_
¤ ¤ ¥¤ R
Figure 2: Production and dissipation in the -equation.
t H
¦ £ Figure 4: Near-wall variation in . Channel flow,
!
X
t
H ¦ £ Channel flow, DNS-data, [25]. DNS-data, !X [25].
0.25 20
0.2 Viscous diffusion 18 Turbulent transport Pressure-diffusion 16 0.15 Sum of diffusion terms 14 0.1
12 ¦¨§ ©
0.05 r 10
0 8 PSfrag replacements 6 −0.05 4
−0.1 2
PSfrag replacements 0 0 10 20 30 40 50
10 20 30 40 50 60 70 80 90 100
©
s U
Figure 3: Viscous diffusion, turbulent transport and _
¤ ¤ ¥¤ R
Figure 5: Variation in in the inertial sub-layer.
t
H
¦ £
!
pressure diffusion terms in the -equation. Channel X
t
H
¦ £
!-X Channel flow, DNS-data, [25]. Labels as in flow, DNS-data, [25]. Fig. 4
energy and dissipation rate is valid:
the dissipation rate, while the turbulent kinetic energy
s s
t
U (1) should be solved and not set a priori.
s s
t
_ 9 E
U (2)
s s
t
© U
(3)
[ s
t 2.3 Inertial Sub-layer © R (4)
The inertial sub-layer is the region observed from
s
s t
¦hf
P Q where the variables are normalised according to the around U and outwards, where the assumed va- nomenclature. riations are, see e.g. Wilcox [36]:
The constants used in the relation for the turbulence ]
t t
f ] f
s s
t
© ©
quantities in Fig. 4 are and , which gi- ^
U
[ o (5)
ves close agreement in the wall vicinity. As can be seen ]
s s
t
_
from the figure, the relations for the velocity, tempera- ^
U o (6)
ture and dissipation rate are a fair approximation, even
]
s t
]f
s t
up to U , although the model for turbulent kinetic s
(7)
U
©
energy yields a strong over-estimation for larger va- e
s t
lues. In light of this, a plausible turbulence model would M
R 8
be to assume a variation in both the velocity profile and XWo (8) U
2
t t t
¡
f ] f f £ £
o where , © and are justified 3 Law-of-the-Wall
by DNS-data. For air, Kays and Crawford [20] give the
t t t
¡£¢ 9 E
f ¦
o¨ N o following values: ¨ and in The variation in the different parameters in the loga- the relation for the temperature. The inertial sub-layer rithmic region, postulated in the previous section, is ba- is also denoted the logarithmic (log) region, due to the sed on the law-of-the-wall and its consequences. Be- above characteristic profiles, which feature a logarith- low, the law-of-the-wall for both momentum and tem- mic behaviour. perature are derived. The latter is an extension of the The models assumed for the inertial sub-layer are former, based on the modified Reynolds analogy. The as- compared with DNS-data in Fig. 5. There is excellent sumed variation in the turbulence quantities can only
agreement for both the velocity and temperature rela- be proven, see section 3.3 and 3.4, in conjunction with a
s¥¤
¦Wf
¡ £ tions beyond U . The turbulent quantities are ho- specific turbulence model. In this paper the Wilcox wever less accurate, and the assumed constant values turbulence model is used. The constants included are
for the turbulence kinetic energy are not correct. At derived in section 3.4.
s t ]gfhf
U there is a descrepancy of around 25% when using either Eq. 7 or 8 as compared with DNS-data. 3.1 Momentum Assuming that a Couette-like flow prevails, the following simplifications can be made to the momentum equations:
1D flow, with variation only in the wall-normal
$ rGt
direction: PAUQ .
Fully developed flow with zero gradients in the
t
f
C C?D
streamwise direction; ¨ , apart from a
~
D
¤)UQ pressure gradient: P .
Negligible convection.
D
¡JQ PSU ¡JQ The streamwise P and the wall-normal momen-
tum equation then reduces to:
§
¦ ¦
9
C
t
8 w
f
> >©¨
^
¦ q ¦
¡
¡ (9)
C?D
U U
¦
9
C
t
w w
f
> >
^
¦
¡
¡ (10)
C
U U
Because Eq. 10 is now an ordinate differential, it can
easily be integrated to yield:
t
9 w w 9
> >
^
D D
P ¤%UQ ¡ P Q
(11)
t
w w
f
> >
C?D C ¨ Inserting this into Eq. 9, assuming that ,
the integration yields:
§
¦ ¦ ¦
9
t ¦ ¦
8 w
f
¨
> >
^
¦ ¦ ¦
q
¡ U ¡ U
D
U U
¦ ¦
9
t
8 w
f
> >
^
¦ ¦
q
¡ U ¡z ¡ .
D U
(12)
t * Q
At the centreline PAU – of a channel – the shear
8 w
> >
C C stress, , and the velocity gradient, u¨ , are zero and the pressure gradient can consequently be
established as:
¦
9
]
t
¦ * g
¡ (13) D
The shear stress variation can thus be written as:
¦
t
8 w
] U
> >
q ¦ *
¡ ¡
(14)
U
turbulent laminar
3
8 w > Note that > is negative and hence the left-handside 25 (LHS) of Eq. 14 – i.e. the total shear stress – linearly decreases from the wall to the centre of the channel. DNS Linear This is of course in excellent agreement with DNS-data 20 Logarithmic
for fully developed flow, see Fig. 1. 8 If Eq. 14 is normalised with the friction velocity, X ,
and a length-scale based on wall quantities: 15
8
X
s
s¡
U
U (15) M 10
The relation becomes:
¦ s s
t
8©w
] U
s
¦
H ¡
¡ PSfrag replacements s
!X (16) 5 U
where the turbulent Reynolds number is defined as:
t *
H 8
!X X
¨M . s H 0
!-X 0 1 2
For a large ratio of ¨U , one can assume that the 10 10 10 s
total shear is constant. Hence the velocity and shear §©¨
PAU Q
s
stress can be written as a function of U only:
s s $
t Figure 6: U-velocity profile with linear and log-law ap-
t
H
¦ £
!
X
PAU Q
(17) proximation. Channel flow, DNS-data, [25].
s
t
8©w
s
dPAU Q ¡ (18)
These relations are called the ’law-of-the-wall’ expres- where is an integration constant, which is given a va-
£ £ £ sed in inner variables, as proposed by Prandtl [27]. lue of ¡ based on experimental data. The logarith- If the inertial sub-layer is instead approached from mic law or log-law for the velocity is thus:
the core of the flow, and the equation is normalised with
]
s s
t
£
the mean flow length scale: £ ^
¡
PAU Q
f ]
(25)
U n
* (19)
The velocity profile using this relation and the linear re-
s t s U
and the friction velocity, the following relation is given: lation, , is compared with DNS-data in Fig. 6.
s t
]h]
U
¦ s
] The intercept between the two laws is found at ,
t
8?w
]
s
n
¦
H ¡
¡ located in the buffer layer, which is, as is obvious from
! n X (20)
the figure, not accurately modelled by either of the equa- H For large !X the first term on the LHS can be neglected tions. and thus the relation reduces to:
0.01 1
t
8©w
s
n
( P Q ¡ (21) Pr =ν /α → Note however that this does not give a relation for t t t itself. On the basis of purely dimensional grounds, it is possible to establish a relation for the velocity pro- file, similar to Eq. 17, although as a function of the wall ← α distance in outer variables only, van Karman [33]: 0.005 t 0.5
t ← ν
¡,£¢¥¤
n t
8
& P Q
X (22)
s
s
where ¢¦¤ is the centreline velocity and is the de- fect velocity. This relation is called the ’velocity-defect law’, Tennekes and Lumley [31] 0 0
Somewhere in the flow it is plausiblePSfragto assumereplacementsthat 0 100 200 300 400 both relations for the -velocity hold. Differentiating s Eqs. 17 and 22 and adding a proportional constant as U
done by Clauser [9] gives: Figure 7: Diffusivities and turbulent Prandtl number for
t
H
¦ £
!X
$
¦ ¦
]
s air. Channel flow, DNS-data [25], [18].
t t
&
n
¦ ¦
U
s o
n (23)
U
¡ ¡
f f
o where is the von Karman constant, equal to ¡ .
Integrating the latter relation gives: 3.2 Energy/Temperature
]
s
$
t
^ An inspection of the structure of the governing equation
PSU Q o (24) for energy and momentum reveals that the near-wall
4
s
@ 9 E
N ¥§¦ mechanisms would be similar only if the Prandtl num- U bers (both the molecular and turbulent) are unity. The Air: 0.7 13.2
definition of the turbulent Prandtl number is: Water: 5.9 7.55
¦ ¦
_
9 E 8 w w
> > > >
`
N
¦ ¦
¨ (26)
s
U U
Table 1: Critical U values, air and water
The turbulent Prandtl number and the turbulent mo- N
mentum diffusivity, M , and turbulent thermal diffusi-
N N N
M
vity, , are plotted in Fig. 7. The definitions of and Re-arranging and integrating once more gives:
§
¦
_ F
are: N
t
q q
8 w ¨
> >
^
¦
9 E 9 E
¡
N
N
¤
© U
¦ ¦
M (27)
u¨ U
¦ (33)
F
w
¦ t
_
U
> ` >
9 E 9 E
¡
N
N N
©¨
^
¦ ¦
_
q q
©¥¤ ¨
(28) ¨
¨ U
Reynolds, in an attempt to describe the heat exchange On introducing normalised values: 8
process, proposed a model which is based on similarities X
s U
between the momentum and energy transfer. Reynolds’ U (34)
M
N
_ _ 8 M
analogy presupposes that the two diffusivities, and X
s
_
¤
P ¡ Q © N
, are equal and hence, from Eq. 26, that the turbulent
F (35)
t
9 E
] N
Prandtl number is unity, . For a fluid with a unit
t
9 E ] molecular Prandtl number, , this would result in and re-arranging, the equation yields:
identical temperature and velocity profiles. The validity s
of the Reynolds analogy can be observed in Fig. 7. As ¦
s
t
_
U
9 E
N
9 E 9 E
] N N (36)
noted the assumption of a constant can only be ac- ^
P M ¨MOQ)¨
ceptable in the inertial sub-layer, and with a value less ¨
9 E ¢
f
N¡
s s than unity, . _ This gives the variation of as a function of U , alt- Although the validity of the Reynolds analogy is limi- hough in an integrated form, which is not preferable ted, there still exists a strong similarity between the ve- when making comparisons with the law-of-the-wall for locity field and the temperature field, which implies that the momentum equation. The formulation could ho- there could exist a corresponding law-of-the-wall for the wever be simplified if it were possible to divide the near- thermal field. wall region into a laminar and a turbulent layer. In With the same assumptions as for the momentum the case of the momentum equation, the velocity profile equation, i.e. 1D fully developed flow field and ther- follows approximate a linear relation in the viscous sub- mal field, the energy (temperature) equation in the wall- layer (laminar) region and a logarithmic relation in the
normal direction can be simplified to: inertial sub-layer (turbulent) region. If the thermal fi-
§
¦ ¦ ¦
_
]
F
t t
w
f
¨ >
>§` eld behaved similarly, the above integral can be divided
¦ ¦ ¦ ¡
(29)
¤
U U U © and integrated successfully. The location of this parti-
tion depends on the molecular Prandtl number, which
¥¤
where © is the specific heat of the fluid and is the for a value of unity would be identical to that of the mo-
s t (viscous) thermal diffusivity. ]W]
mentum equation, i.e. U . For a lower value of the
t
9 E
c] Note that the pressure gradient, contrary to the momen- Prandtl number, e.g. air ( f ), this critical value
tum equation, does not appear in this equation. If a non- will be higher, with the opposite being true for a high
t
9 E
negligible pressure gradient were present, much of the Prandtl number fluid, e.g. water ( £ ). Examples similarity between the momentum and energy equation s of these cross-over locations, or critical U values, are gi-
would be lost.
w
> > ` ven in table 1, see also Kays and Crawford [20]. Taking Assuming that the turbulent heat flux, , can be the value for air, the two integrals would be:
described with the turbulent thermal diffusivity and the s
turbulent Prandtl number, the relation can be re-written ¦
s [ = s
t ¦
_ 9 E
U
^
9 E
U N
as: N (37)
P3M ¨MQ'¨
¢ §
¦ ¦
_
N
[ =
t
q q
f
¨
^
¦ ¦
9 E 9 E
N (30)
U¤£ U where it was assumed that the turbulent part could be neglected in the first integral and the laminar part in the When integrating this equation from the wall to the
centre of the channel, the heat flux is found as: second integral. The first integral is easily integrated to
§ ¦
_ yield the linear-law for the temperature as:
N
t
q q
¨
^
s
s
t
_ 9 E
¦
9 E 9 E ©
N (31)
U
U (38)
[%=
where the integration constant, © , is a function of the
However the second part can only be integrated if the
9 E N heat flux applied at the wall: N M ¨M
F variation in and is known. In the inertial sub-
t
¡
© (32) layer the turbulent Prandtl number can with acceptable
N
¤
© ¨M accuracy be assumed constant. The ratio M , or the
5 turbulent Reynolds number, can be approximated using model (see section 3.5) reduces in the inertial sub-layer
the mixing-length theory, which gives: to:
N
s
t t
8
M
o o
N X
§
¦ ¦
M U
U (39)
t
f
¨
M
N
¦ ¦
M (42) U
Adopting this and evaluating the integral yields: U
§ §
¦ ¦ ¦
t
f
¨ ¨
N N
^
§
¦ s s
9 E ¡ ¡
¦ ¦ ¦
M ¡ £ £ M
N (43)
s
t t
_ 9 E
U U
¨
U U U
N
] ¦
s
o o
§
(40) §
¦ ¦ ¦
U
[%=
t
f £
¨ ¨
N
[%=
^
¦ ¦ ¦
¡ £ £ M
¨ (44)
U U
Thus the law-of-the-wall for the thermal field, or the U
t
temperature log-law, is, adding Eq. 38 and 40 N
M (45)
£
§
s
9 E
N
s s
t t
_ 9 E
] ¦ U f £ ¦
¨
^ ^
PSU Q
] ¦
o where the laminar viscosity in the diffusion terms is (41) neglected. Note that the neglected pressure gradient in
where the constants are found by using properties for air the momentum equation (Eq. 42) is a severe limitation
t t
¢ 9 E 9 E
f f £
( and N ) and the van Karman constant, in the following derivation. This makes the wall func-
t
¡
f ] o . tion approach even more questionable for the turbulent The two formulations, the linear-law and the log-law, quantities than for the velocity field. are compared with DNS-data in Fig. 8. Similar to the The above equations can be satisfied in the inertial momentum (velocity) model, the agreement is less ac- sub-layer with the following set of wall functions, see curate in the buffer layer where neither equation gives Wilcox [36]:
acceptable performance. For air, the cross-over between
s t
] ¦
8 8
U X
the two laws is found at , and hence the lami- X
t
U
^
o
nar thermal layer extends further out than the laminar (46) M
momentum layer. 8
X
t
V (47)
¡
20 8
X
t
£ o
18 DNS V (48)
U
¡ t
Linear 8
o
N X U 16 Logarithmic M (49)
14
12 These relations can be proven by re-inserting them into
the above transport equations. Using Eqs. 49 and 46 in
s _ 10 Eq. 42 yields:
8
¢
¦ ¦
8 8
X X
t t
8
U
f
o
X
^
¦ ¦
6 U
o
£
U M
PSfrag replacements U
§
¦ ¦
8 ]
4 X (50)
t t t
8 8
f
¨
XWo
X
¦ ¦
U
o
U U 2 U
0 0 1 2
10 10 10 Thus these wall functions satisfy the momentum equa- s
U tion. Eqs. 46, 47, 48 and 49 are substituted into Eq. 43 which gives:
Figure 8: Temperature profile with linear and log-law
t
H
¦ £
approximation. Channel flow, DNS-data, !WX [18].
¢
¦
8 8 8 8
X X X
X
t
8
f U
XWo
^
¡
¦
U ¡
o
o
V V
U M £
U
¡ ¡
¢ §
¦ ¦
8
X
t
8
¨ XWo
3.3 Turbulence ^
¡
¦ ¦
£ U
V
£
U U
There is a number of different approaches to the turbu- ¡
§
8 8 8 8
]
X
X X X
t t t
8
f = = f =
¨
o X
lent law-of-the-wall, or log-laws, where the derivations ^
U ¡ ¡
o o o o
U U U below mirror that of Wilcox [36]. If 1D fully developed U flow is assumed, and the pressure influence is neglected (51)
1
¡ £ (even in the streamwise direction) the Wilcox HRN
1Note that the log-law may be derived for any turbulence model.
¡£¢¥¤ The turbulent kinetic energy equation is also fulfilled. The Wilcox is used however, because it will later be used in the description of blending turbulence model. Turning to the dissipation rate equation, Eq. 44, and
6
using the wall functions, Eqs. 46, 47, 48 and 49, gives: from Eq. 47 and the measured relation of above, Eq. 53, §
¢ we can write the coefficient as:
¦
8 8 8
X X X
t
U
f
¨
8
^
¦
¡ ¨
X
o
V o
t t
8
f ¦ f ¦
£
U M X
U
¡
V
§ ¢ §
¦ ¦
8 8
]
¡ X
X (55)
t
8
¨ ¨
t
f f
XWo
^
¦ ¦
¨ U ¡ £
¡
o
V o
£
U U U
U
¡
§
¦
8 8
]
X
X
t
8
¡
¨
Xho
^
¦
¡ £ U
o
V o
U U U
¡
¡ 3.4.2 Coefficient,
§ §
8 8
]
X
X
t t
¨ ¨
^ By studying how grid-turbulence dies out, it is possible
¨ ¡ £
o
V
U U
¡
¡ to determine the coefficient, . In decaying turbulence,
§
t t
£
f
¨ the turbulence equations simplify to:
o
^
¨ ¡
V
¦
¡
¡
t
§
V
¡
¦
¡ £
t
(56)
¨
¡
o
¡ ¨
¦
£
¡
t
£
¦
£ ¡ (57)
(52)
To satisfy the last relation, it is necessary to establish Solving this equation system gives:
o
I I
¡
¨ £
an identity between coefficients , , , and .
Z ¡'x
©¨ (58)
3.4 Turbulence Model Constants Measurements [32] indicate that the turbulence kinetic energy decays as:
0.3
t
] £ ,f f
}
Z
©¨ ¤
(59)
t t
£ f f f f
0.25 ]
g¨ ¡ and hence the ratio ¢¡ . With
from above, the correct value for would be:
0.2
t
f f f f
¢
¡
£ ¤
¢ 0.15
¡
¡ £
The turbulence model by Wilcox [34] gives this
t t
¡
¦ f f f £ ¨ coefficient the value of .
0.1 £§¡ 0.05 3.4.3 Coefficients, £¥¤ PSfrag replacements There is no consensus within the turbulence modelling 0 community about the values for these coefficients. Diffe- 0 20 40 60 80 100 s rent Schmidt numbers are arrived at depending on the U set of experimental data used. In recent years there has
Figure 9: Shear stress to turbulent kinetic energy. also been a greater tendency to introduce modifications
t
H ¦ £ Channel flow, DNS-data, !-X [25]. for these coefficients, and the values used in turbulence models thus differs. The two Schmidt numbers for the
¢¡¤£ model [34] are optimised to:
] t
3.4.1 Coefficient, ¢¡
£
Experiments have shown that the shear stress and the ]
t ¡
turbulent kinetic energy is related as £
f ¦
¦¥
(53) ¨ 3.4.4 Coefficient, in the logarithmic region. The validity of this approx-
imation is shown in Fig. 9. If the shear stress can be The final coefficient, ¨ , is set using the deduced relation
assumed to be constant and the laminar part is neglec- from the specific dissipation rate equation, Eq. 52. With
t
¡
f ] o
ted, then: and the above constants, ¨ is calculated as:
¡
t t
8 f f £ f ] f £
o
N
X
^
t t t
£
f £ £ ¦
¦¥ .
(54)
¨ ¡ f f ¡
f f V
V (60)
§
¡
¡
t
8
£
X
.¥¨ ¡ £ ¨ ¨
where the friction velocity is defined as e . In the Wilcox model is given the value of
£ £ £ Now, using the relation for the turbulent kinetic energy f .
7 3.5 HRN Turbulence Model 4 Modelling Near-wall Turbu- On the basis of the above derived turbulence constants lence
it is possible to write the Wilcox ¥¡¤£ model as follows: t
With respect to wall treatment, there are two different N
M (61) categories of turbulence models. The first are the LRN- £
models, which use a refined mesh close to wall in order ¢
to resolve all the important physics. The second met-
C C
t
9
;
N ^
^ hod employs the HRN-models, which bridge the near-
¡ ¡
¡ £ £ M Q
P3M (62)
B B
C?D C?D £ wall region using wall functions. The latter approach is
of course less demanding of computer resources, but a ¢
significant amount of information is lost.
C C
t
9
;
£ £ £
N
^ ^
¨ P M £ M Q £
¡ (63)
B B C?D
C?D The wall function method is based on the previously
£
mentioned ’law-of-the-wall’ and is strictly valid only in with the constants as derived earlier: the inertial sub-layer (log-layer), 1D fully developed flow,
where the pressure gradient can be neglected. The situ-
t t t t t
¡
£ ]gfhf ¦ f ] ]
¡ ¡
¨ ¨ ¤§ ¨ ¤ ¨ ¨W ¤¢£ ¨ ¤§£ ations in which all these simplifications are fulfilled are relatively few, and the predictions made using wall fun- ction or HRN turbulence models are thus generally less accurate than those that apply the integrated method of LRN turbulence models. The reason why there is still an interest in wall func- tions is the need to reduce computational requirements, as well as the better numerical stability and convergence speed. If it were possible to improve the predictions using wall functions, more problems of engineering in- terest could be solved with academic quality, resulting in fewer safety margins and less need of experimental validation when designing new products. The previous section derived the commonly used wall functions based on the law-of-the-wall, while this section will focus on alternatives to these wall functions. However, before embarking on strategies in con- structing wall functions, it is beneficial to study the near-wall physics from a modelling point of view. Atten- tion is first directed towards the LRN-models and their damping functions.
4.1 LRN-models: Integration Method
It is generally implied that a turbulence model that can integrated toward the wall is denoted a LRN turbulence model. This is usually the same as including damping functions for certain terms in the turbulence equations. The damping functions are introduced to represent the viscous effects near a wall. Successfully devised dam- ping functions should reproduce the correct asymptotic
behaviour in the limit of a wall. The exception to the
¡ £ rule that ’damping function LRN’ is the HRN turbulence model by Wilcox (1988) [34]. This model can, like the LRN version [35], be integrated to the wall but without the need to employ damping functions. In this paper the identity of damping function and LRN is used for these models as well. This nomenclature thus helps
to distinguish them, with the ’88 model somewhat erro-
¡ £ neously denoted the HRN and the ’93 model the
LRN ¢¡¤£ model.
¡ £ The Wilcox HRN model was given in the previous
8
¡¤£ ¡ £ section and the LRN model [35] is: part of the buffer layer, while the HRN and LRN
formulations become similar in the inertial sub-layer.
t
t H
f
!N
N
¡
M (64) In the limit of , i.e. at the wall, the coefficients
£
H H
; !
N for the LRN and HRN formulation give:
^
t
¨ ¢¡
¡
H H
;
] !
N (65) ^
¨ LRN HRN LRN/HRN
¢¡ 0.025 1 0.025
¢
§¡
C C
t
9<;
0.025 0.09 0.28
N
^ ^
¡ ¡
¡ £ P3M £ M Q
(66)
B B
C?D
C?D 2.22 0.55 4
£
¢ H HuI v
£ ]
!gN 0.075 0.075 1
^
t
¨ P ¨ Q
¡
P Q
¡
H H I
]gfhf ]
v 0.055 0.55 0.1 !
!N (67)
^
P ¨ Q
Thus, in the near -wall region, the production terms (first
¢
¡
C C
t
£ £ 9<;
£ terms on the RHS of Eqs. 66 and 68) of the LRN
N
^ ^
¡ £ P3M £ M Q
¡ £
(68) £
B B C©D
C?D model are reduced as compared with the HRN
£
N
¡
H HLK
M
] £
!gN model, through the reduced eddy-viscosity, , via . In
^
t
¨
H HJK
] addition, the dissipation term (the second term on the
!gN (69)
^
¨
¡ RHS) of the -equation, Eq. 66, is also reduced, through
where the turbulent Reynolds number is defined as §¡ .
t t
H 9
;
!
N
O¨ P3M £Q
and the turbulent production as
@ @
Y
N B
B 0.1
C C©D
¨ hM . The coefficients involved are:
DNS
¢ ¤
0.09 ¡
t t t t t
¡
] ]gf ¦ ¦ f ] ]
¡ ¡
¨ ¤ <¨ ¤ ¨ ¤ £ ¨W ¤ £ ¨W
t t t
HJI ¢ H ; HJK
]gf
0.08
¤ ¨ ¤
0.07
¦ £
¡ £ This model reduces to the HRN , see section , if
the damping functions, Eqs. 65, 67 and 69, are set equal 0.06
] ]gfWf £
¨ ¨
© to , and , respectively. 0.05
4 0.04
0.03
¤¦¥¨§ © ¥¨§ ¡£¢
3.5 DNS: ¡£¢
¡£¢ ¤
¤¥¨§ © ¥£§ ¡£¢
¡£¢ PSfrag replacements 0.02
¤¦¥¨§ ¥¨§
© ¡
DNS: ¡
¡£¢ ¤ ¥¨§
3 ¤¦¥£§
© ¡
¡ 0.01
©
¤¥¨§ ¥¨§ ¨¢
DNS: ¨¢
¡ ¢ ¤
©
¤¥¨§ ¥¨§
¢ 2.5 ¢ 0
0 20 40 60 80 100 s s
’
2 U
PSfrag replacements
t
H
¦ £
!-X
© ¡ £ 1.5 Figure 11: -coefficient, LRN and DNS [25].
1
¢¡ 0.5 4.1.2 © modification,
0 The most important damping function is the damping
0 5 10 15 20 25 30 N M
s of the turbulence kinetic viscosity, . This is done by
U
© ©
adding a damping function to , which reduces to
©
an effective according to:
¡ £
Figure 10: Damping functions in Wilcox models.
§ w
Channel flow, DNS-data, [25]. 8
> >
$
t t
¨
© ©
(70)
4.1.1 The damping functions Measurements and DNS show that the assumed relation
t
8 w
f ¦
> >
The near-wall behaviour of the three important con- of ¨ is valid only in the inertial sub-layer. The
¤ §¡
stants ¢¡ and are shown in Fig. 10. The intro- ratio is reduced in the near-wall region, as shown in Fig.
¡ R © duction of the wall viscous effects, via the damping fun- 9. For a model the effective is computed as:
ctions in LRN-model can observed in the figure. The re- N
t
M R
t
H
¦ £
!WX
©
sults are based on both a priori DNS-data, (71)
¡ £
[25], and also on a computation made using the
N
¡ £ M R model. Using a formulation, both and in the relation
As can be seen in the figure the damping functions are must be substituted with the appropriate definitions. In
¡ £ R most effective in the viscous sub-layer and in the inner the case of the LRN turbulence model, is equal to
9
¡ £
, while the turbulent viscosity is defined as in Eq. 2
© DNS
64. The effective thus becomes:
1.8 LRN
t HRN
¡ ¡
©
(72) 1.6
f
f 1.4 ¢¡
where ¢¡ and the second part (non ) of perform
$
¡ R
the role of the damping function, , found in 1.2
¥
© turbulence models. Any deviation from the correct ¢
1 ¡ is primarily modelled with the damping function, . Note however that the turbulence equations are strongly 0.8 coupled, and thus a change to the damping functions in
0.6
£ ¡
either or would also require a modificationPSfrag replacementsof .
© ¡ £ The variation in the computed using the LRN 0.4 is compared with DNS-data in Fig. 11. The agreement is good, especially as compared to other models, see e.g. 0.2 Patel et al. [26]. 0
0 5 10 15 20 25 30
© 4.1.3 Production-to-dissipation rate Figure 12: Production-to-dissipation ratio, HRN and
Another important parameter in a turbulence model is
¡ £
LRN formulation of the . Channel flow, DNS-data
t
H
¦ £
the correct modelling of the production-to-dissipation !X 9 ; at .
rate, ¨R . By comparing the HRN and LRN formulation
¡ £ of the model, it is possible to show the effect these damping functions have on the production-to-dissipation Decrease the coefficient for the dissipation term in
ratio. the -equation. From Eq. 66, and assuming 1D flow, the production-
to-dissipation rate is: Increase the coefficient for the production term in
R £
¡ the secondary equation ( or ).
§
¦
9
;
t
¨
¡ ¡
¦
£
Decrease the coefficient for the destruction term in
£ U
R the secondary equation.
¡
§
8 8
X X
t
¨
¡ R
¡£¢
¡ (73)
The last method is commonly employed in the mo-
o
o
U
U¥¤ ¡
e dels. By properly balancing the dissipation rate equa-
H H I v©¨
]
H H ;
!N !
!N
^ ^
tion, a desirable level of is predicted that matches the
P ¨ Q
t
¨ Q P ¡
H H ; ¢ H HuI
] £ ]
v
R
!N !gN ^
^ computed . Because the production of dissipation is au-
P ¨ Q§¦¡ ¨ P ¨ Q ¢ N
tomatically decreased through the reduction of M , the
¡ ¢¡
where is the value of in the logarithmic region, simplest correction is to reduce the destruction term in
t
f f ¡ the dissipation rate equation. This is usually done in a
. The log-laws, Eqs. 46 and 48, were used in the $
¦ ¦
¡ R
U £ u¨ model by introducing a damping function , see
relations for and , respectively. The approximate
identity is correct only in the inertial sub-layer for which Patel et al. [26].
s ¡ the log-laws are valid. In the near-wall region, U There is more to the damping functions than balan- ]f , the above relation may only be used qualitatively. cing the equations, however. As previously discussed, it The a priori computed production-to-dissipation ratios is also important to predict an accurate production-to- dissipation ratio, which puts an additional demand on
for the HRN and LRN formulation are compared with t
R ¢¡ £ ¡ £ DNS-data in Fig. 12. the actual values of and . In a mo-
The LRN formulation, similar to the DNS-data, incre- del, where the dissipation term is a function of both £
and , it is unrealistic to believe that merely modifying
ases the production in the buffer layer, which results in 9¥;
£f
£ ¨R a higher peak of for the LRN-model as compared the -equation would yield a correct , as well as a with the HRN-model. Slightly dependent on the Rey- correct level of . Hence, apart from modifying the speci-
nolds number, DNS-data give the peak of turbulent ki- fic dissipation rate equation, the dissipation term in the
s t
] £
¡ £
netic energy at around U , close to the maxima of -equation is also modified. In the LRN model these
9 ;
the ¨R -ratio given in Fig. 12. two aspects are accomplished through both an increase
of production of £ (increasing ), and a reduction of the
¡
dissipation term in the -equation (decreasing ). Note ¢¡
4.1.4 Rationale behind the and coefficients
¡JR
however that, contrary to the model, the destruction
¡ P R ¤ £Q As was previously shown, the damping function, , term in the secondary equation is unchanged. was introduced to reduce the turbulence viscosity in the Considering Patel et al. [26], it seems likely that the
near-wall region. This also effectively reduces the pro- minimum number of damping functions required for an £ duction terms in the - and -equations. In order to pre- Eddy-Viscosity Model (EVM) to correctly predict the tur-
dict the same level of turbulence, the dissipation term bulent kinetic energy, , in the near-wall region is three.
in the -equation must also be reduced. This can be ar- In that paper the only model that accurately reproduces
¡,R ranged in three different ways: the profile of is the Lam-Bremhorst [21] model,
10 which is one of only two models tested employing three The turbulent kinetic energy is set in the first node by
damping functions. iteratively computing the friction velocity from the law- 8
of-the-wall. This iterative process is needed because X 8
appears implicitly in the log-law. Initially X is set from
t
8 v 8 X
4.2 HRN-models: Wall Functions X
¤
§¡:[)x Eq. 47, as e . Using this relation, a new The rationale behind wall function is the reduced com- is found from the log-law:
putational requirement and the increase numerical sta- 8
o
X
s
t t
8
¤ U¤
X s
bility and convergence speed. By adopting a mesh, U #
where ¤ (78)
M
¤
Q P U where the first interior node is located in the inertial
sub-layer, it is possible to use the law-of-the-wall to spe- s U The new value is then used to compute ¤ and re-fed
cify the boundary condition for the dependent variables 8 X
8 into the expression. This process is repeated until £ , and . convergence, with the turbulent kinetic energy finally set as:
4.2.1 Standard wall functions
8
X
t
¤
In its simplest form, the logarithmic laws, Eqs. 46, 47, V (79)
48 are directly applied to the first interior node. These ¡ wall functions are here referred to as the standard wall
For the specific dissipation rate, £ , the friction velocity, 8 function method. The three steps involved in a CFD- X , is substituted by the relation for in the log-law as:
code would then be:
v
8
X
§¡:[)x ¤ -¤
t t t
e e ¤
Solve the momentum equation with a modified wall £
v
o
o o V
V (80)
U¤
U-¤ U-¤
¡:[)x ¡ friction, either through an added source term or via ¡ a modified effective viscosity.
4.2.2 Launder-Spalding methodology Set at the first node iteratively with the use of the law-of-the-wall. The standard wall function method is very limited in its
usage. This is especially true for re-circulating flows,
Set £ with . where the turbulent kinetic energy becomes zero in se-
In a turbulent boundary layer, the strongest velocity parating and re-attachment points, where, by definition, 8 gradient is found close to the wall. With a wall- X is zero. This singular behaviour severely deteriora- function based turbulence model, which utilises a relati- tes the predictions of, for instance, heat transfer in rib- vely coarse mesh, it is impossible to resolve these near- roughened channels. Launder-Spalding [24] proposed a wall gradients. The predicted wall friction would thus modification to the standard wall function method that
be largely in error if a modification is not introduced: involves the following steps:
l
C
¤ t
¤
Solve the momentum equation with a modified wall
l
q q q
. (74)
C
¤
U¡ U U
viscosity.
~ where the subscript is used for the first interior node. Solve the turbulent kinetic energy, with modified in- The necessary modification could either be made tegrated production and dissipation terms.
through
Set R using the predicted . (i) an added source term simulating the correct wall friction or The same reason and method of specifying the wall
viscosity as for the standard wall functions apply in this
8 X (ii) a modified viscosity, an effective viscosity, q , that
case. However, instead of using in the log-law, the
v
t
8
V
X
ensures the correct friction even though the velocity [)x
identity © is utilised:
gradient is erroneous.
v
o
[%x
¤ ¤
© U t
Through the law-of-the-wall: e q
# (81)
]
Q P U
¡
¤
s
t
#
8
P U Q
X o
(75)
U ¡ where ¤ is defined with as:
the wall friction is computed as
v
V
[)x
8
t
U ©
X o
t
¡ U
¤ (82)
#
. s
(76) M
P U Q
8
In the -equation, the integrated production term is de-
X
with . fined from (1D):
For a wall function based model this can either be set
Y£¢
t
C
.¥¤§¦
¦ ¦
9 ;
directly using a source term, , or via a modified N
U U
(83) C
effective wall viscosity [10]: U
8 8
X o X o
t t t
¤ ¤ ¤
U
s s
q q
. #
# (77) In the inertial sub-layer the laminar shear stress,
U¤
¤ ¤
P U Q P U Q
C C
q U J¨ , is negligible and, by assuming that the shear
11 N stress is constant, the turbulent shear stress, , is equal north
to the wall shear stress: PSfrag replacements
t t t
f¢¡
N
^
g ¤
¥§¦
t
N (84)
. P The accuracy of this assumption is shown in Fig. 1. The vs
production can be set using the law-of-the-wall: ¦©¨
C
l l
¦ t ¦ t t t
9
; ¤
q
U U
wall
C
¤
U U
.
v
V (85)
o
[)x
l
t t
¤ ©
¡
#
Eq. 81 Figure 13: Assumed variation of variables in the first
P U Q
¡ ¤
computational node, Chieng-Launder model. vs=edge of
l
¦
C C
£ P u¨ UQ U
viscous sub-layer.
The integration of is equal to , since is constant. To set the value for the dissipation rate, production is Comparing the two formulations, Eqs. 88 and 91, gives:
assumed to be equal to dissipation:
l
t
#
f
t
9 ;
¡
P UQ ¡ P Q P U Q
(92)
R
(86) Eq. 88 Eq. 91
The production can be re-written as:
8 X
C The latter formulation is used in the Launder-Spalding
t t
9 ; 8 w 8
> >
X
¡
o (87) model, with the integrated dissipation rate in the -
C U U equation set as:
where the constant shear stress approximation, Eq. 54, v
t t t
8 w 8
> >
X
=:x ='x'
¤
C C ¦ t
#
©
¡ u¨ U
¥ , and the law-of-the-wall,
¡
8
R U P U Q
¤
o
X o
(93)
¨P U Q
, were used. Integrating the dissipation rate
l U
between the wall and the edge of the first cell, , yi- v
8
X
='x
= ='x'
elds: where in Eq. 91 has been substituted by © .
v
¤ ¤
=:x
8 8
© ©
X
X Note that in [24], is used rather than , although
l
¦ t ¦ t
= = f
R U U ¡ P UQ ¡ P Q ¢
this is most probably a typographical error.
o
o (88)
U The dissipation rate equation is not solved for the first
The mathematical singularity at the wall of this equa- interior node, and R is instead fixed according to:
f
v
P Q 8
tion ( ) necessitates re-consideration. The short-
X
=:x =:x'
¤
t t
= ©
coming of the above derivation is the assumption that R o
o (94)
¤ U the law-of-the-wall is valid all the way from the wall, U which is erroneous. However, by choosing a different relation for the in- which is a direct consequence of substituting the friction
velocity with the log-law for turbulent kinetic energy,
R
v t
tegrated , other approximations could be found. The 8
V
X
[%x
integrated velocity gradient in the production term can, © , into Eq. 86. according to the mean value theorem, be re-written as:
¤ 4.2.3 Improvements of the near-wall representa-
C C
l
¦ t
tion, Chieng-Launder model
U U
(89)
C C
U U
It is evident from the above derivation that there is room
C C U
for a certain value of J¨ . Approximating the velocity for improvement in the Launder-Spalding model, espe-
l l U gradient with discrete values, u¨ , and, assuming cially in its representation of the near-wall region. Chi- that this value could be established in the inertial sub- eng and Launder [7] enhanced the model by introdu- layer, the velocity gradient could be re-written with the cing a more complete model of the near-wall turbulence.
help of the law-of-the-wall: They argued that there exists two distinct regions, the
l 8
X viscous sub-layer and the inertial sub-layer, which have
C
t
#
¡
l l
U Q
P different turbulent structure. The partition line between
o (90)
C
U U U these regions is defined by a viscous sub-layer Reynolds
Hence the integrated dissipation rate can also be written number:
V T
as: T
t
H
U
T
!
¤ ¤
¤ (95)
C
M
¦ t ¦ t ¦ t t
9<; 8 w
¡
> >
R U U U ¡
Eq. 87
C
U The near-wall variation in the variables assumed in the
8 8
X
X
l
t t t
# # 8
¡ =
X Chieng and Launder model is shown in Fig. 13.
¡ ¡ ¡
l
U P U Q P U Q
T
o
Eq. 90 o
U U U The turbulent shear stress is zero for and is (91) linearly interpolated between the wall friction and the
12
¤ t t
#
f ¦ £ f
T
o
U ¡ ¡
shear stress at the edge of the cell for U : where the new constants, and , are
t ¡
f related to the standard values, through the identities:
T
N
v v
t t
# #
U U
o o
[)x [%x
© ¡ ©
¡ and . For local equilibrium flow,
t
8
¤ t
U
X T
N (96)
^
¨ ©
e
U U PS ¡z Q
, and with the above identities, the modified
U law-of-the-wall reduces to the standard formulation. where subscripts } and are used for the value at the The integrated production following the same grounds north edge of the cell and at the wall, respectively. as for the Launder-Spalding model yields (1D):
The turbulent kinetic energy is approximated as:
¦ t
9 ;
U
§
t
¡ U
¨
T T
¢
¢
U U
C C
T
t ¦ ¦
U
N
U
^ ^
¡
U . PS ¡ . Q U
C
(97) C
T
U U £ U
t ¤
¡
T T
^
U U U
T
£
U ¡zU (101)
The dissipation rate is approximated in the region be- T
low U by the viscous sub-layer approximation used in The first part is identical zero, since the turbulent shear
¡ R
numerous turbulence models, e.g. Jones and Laun- stress is assumed to be zero in the viscous sub-layer. The
¤
T U der [15], while the log-law is used for U : second part is split into two, where the first part is easily
integrated:
V
T
C
t t
hM ¡
¢
T
C
¢
R hM U U
¦ t
T
T
C
U U
U P ¡ Q
¤ (102) C
(98) U
v
=:x ='x'
¤ ¤ t
©
T
U U R
o In the second part, the velocity gradient is re-written U¤
using the law-of-the-wall: ¢
The turbulent kinetic energy gradient in the first rela- ¢
C
¦ t
U
tion is approximated using the previously assumed vari-
PS ¡ Q U
C
£
U U
©¨ U©
ation in in the viscous sub-layer, i.e. .
¢
¢
t ¦ t
U ¨
Because the finite volume method commonly applied,
PS ¡ . Q U
V
T (103)
o
£
and the strong variation in these quantities in the near- U
U ¡
wall region, it is advantageous to use integrated rela-
t
.£PS ¡ . Q
T
¡zU Q
tions in the first computational cell. The resulting terms PAU
V
T
o
U ¡ are then incorporated into the -equation, which is sub-
sequently solved. In additional and identical to the The wall friction, , used in these relations is deduced Launder-Spalding model, the wall friction is corrected from the above modified law-of-the-wall, Eq. 100.
and the dissipation rate set. However, before continuing The integration of the dissipation term is given as: R
with the integration of and , it is proper to define the T
location of the viscous sub-layer. In the Chieng-Launder =:x'
¦ t
^
T
H
R U U
model, is deduced from the assumption that the sub- !T ¢
layer Reynolds number is constant and equal to 20, and ] (104)
p
¤
T T
^ ^ ^
=:x) =:x' [%x' [)x'
P ¡ Q P ¡ Q
hence: ¦
© £
f
t
M
T
t p
£ £
U T
V (99)
© where . The constant, , depends on the turbu-
lent kinetic energy via variable ¤ : T
However, since is unknown, the problem is not clo-
¢
T
V V
¤ ¤
V V
^
p"t ¤
¡ Q'¨ P Q f
sed. is approximated by extrapolating the slope of P
§©¨
¤ ¤
='x'
V V
T ¤ T ¤
V V
^
from the computed ’s in the two first near wall nodes. £
P ¡ Q.P Q
¢ (105)
V V
It should be noted however that this methodology is not T
p"t
¡ f
¥¡¦ ¥£¦
¤ ¤
Z\[ =:x) Z [
¡ ¡ Q
numerically satisfying. It can be shown using DNS-data P
V ¤ V ¤
t s t
H
f ]h] £
T
!
£
¡ ¡
U that is found at . The profile of
around this value can be seen in Figs. 4 and 5. The max-
s t £
] with U
ima of is found at . Thus the computed slope
s t
¦Wf
t
P ¤ ¡ Q
U
of using nodes 1 and 2, located at say and ¤
s t
]fWf
U¤ ¤ ¡
(106)
U
¤
¡zU Q becomes negative, which may be of concern in PSU terms of numerical stability. In the Chieng-Launder model, the common practise of 4.2.4 Modification to the Chieng-Launder model
estimating the friction velocity with the turbulent kine-
t
H
f
T ! tic energy is taken one step further, with the law-of-the- The assumption of is not valid for all types of
wall modified as: flows. Using DNS-data on fully developed channel flows,
t s t
H
f ]W] £
T
!
§
U
] is equivalent to , which is the loca-
V T V T
s t s
t
#
U
¨
U
¡ tion of the cross-over of the linear law, , and the
o
s t s #
(100) ]
o
¨ M
¡
¨ P U Q
log-law, . Thus, for equilibrium flows,
13
t
H
f
T !
is a good partition value, since the two formula- viscous sub-layer in order to accurately predict the wall tions become a laminar and a turbulent approximation, friction and heat transfer. To be able to significantly im- respectively. prove the results, it is necessary to mimic the LRN met- However, in a subsequently paper by Johnson and hod of resolving the viscous sub-layer, especially in non- Launder [14], it was found that the heat transfer predic- equilibrium regions. Thus the wall function approach tion could be improved by letting the viscous sub-layer of bridging the viscous sub-layer seems unattainable, thickness, i.e. the sub-layer Reynolds number, vary ac- apart from regions in which the flow is in equilibrium. cording to the turbulence level: Hence the only possible path to reducing the computa-
T tional demands and maintain accuracy would be a tur-
t
H H H
¡ W
!T !T !T
¡ ©
T (107) bulence model that adopts itself to the flow, employing
a LRN type model where necessary and switching to a
t
f £
© © h where © , and is the linear extrapolated HRN type model otherwise. While such artificial intel- turbulent kinetic energy at the wall. ligence may sound incredible it is possible to achieve it Ciofalo and Collins [8] used an approach however through a smartly devised model. where they proposed to let the sub-layer thickness vary, The second problem is associated with the underlying dependent on the turbulence intensity. physics of the wall functions, i.e. the law-of-the-wall. When employing wall function, it is necessary to apply 4.2.5 Extensions to the Chieng-Launder model the first computational node in the inertial sub-layer. This fact is less of a problem in equilibrium flow but it Amano et al. ([2], [3]) devised two different models ba- is accentuated in regions of flow separation2, where the sed on the same zonal principle as the Chieng-Launder s wall shear stress and hence U decreases. For these re- model. In both models, the dissipation rate equation is gions, the size of the first cell must be enlarged to ensure solved using the layered approach rather than allowing that the cell node is kept within the logarithmic region. it to be set as in the Chieng-Launder model. The results Hence the mesh generation becomes a delicate business were only marginally improved however, and hence the using wall functions for re-circulating flows. two-layered approach was extended in the second model
Both of the above explained problems are solved with
¤ ¥¤ R to a three-layered version. Here the parameters, the two-layer Chieng-Launder model. The CL-model, were allowed to vary in accordance with measured data however gives rise to additional complexity. Through for the viscous sub-layer, the buffer layer and the iner- prolonged mathematical operations the viscous sub- tial sub-layer. layer and the inertial sub-layer are modelled simultane- ously, although the author encountered numerical dif- 4.3 New Model: Blending Integrating ficulties using this approach. If the two regionscould and Wall Function Boundary Condi- be separate mathematically, a numerically simpler and tions more robust wall function could be constructed. Such a turbulence model will be presented here. A less The physical treatment improved, w hen progressing complete derivation is given in Bredberg et al. [6], where from the Launder-Spalding model to the more advanced predictions of heat transfer in rib-roughened channels models, although at the expense of increased numerical are included. The reader is referred to the paper for a complexity. The modifications by Johnson and Launder discussion of the accuracy and for comparisons between [14] and Ciofalo and Collins [8] showed that the impro- the model and other turbulence models. vements with the zonal approach were not particularly Dependent on some flow quantity, the model will eit- great if a variable sub-layer thickness was not included. her employ a wall function or use the integrating method The further refinement by Amano et al. enhanced the as a wall boundary condition, and hence greatly simp- predictions only marginally, with a significantly increa- lify the numerical scheme as compared to the Chieng- sed complexity. Launder model. The new model will, like the wall fun- Thus, here, instead of further optimising the layered ction, approach reducing the computational demand – models, a new strategy will be adopted. by coarsening the mesh, when possible, and still be able Two main problems with wall-functions must be add- to produce accurate results through the use of the LRN ressed to make such an effort worthwhile: part of the model, when necessary. The physical conse- The negligence of the physics in the viscous sub- quence of separating and re-circulating regions can ac- layer, tually be utilised with such an adopting turbulence mo-
del. In non-equilibrium nodes where the physics reduces The necessity of locating the first computational s the U value, the hybrid model automatically switches to node in the inertial sub-layer the LRN mode, producing accurate results even for a re- The first problem has been discussed in numerous pa- lative coarse mesh. Thus even a uniform mesh could be pers, see e.g. Launder [22]. The conclusion made in sufficient for complex re-circulating flow, and, as a re- these studies is that the standard wall function formu- sult, reduces the workload of mesh-generation substan- lation (HRN-models) are indisputably inferior to the in- tially. tegrating formulation (LRN-models). This is particu- larly true for non-equilibrium flows, where it is of cri- 2Although the log-law in itself is rather questionable in such cir- tical importance to include the variation found in the cumstances.
14 4.3.1 Blending function improves predictions for non-equilibrium flows. This is utilised by the Launder-Spalding model and subsequent The critical part of a hybrid turbulence model is the con- models. In addition the HRN part of the new model as- struction of the function that effectively changes the mo- sumes a variation of the near-wall turbulence similar to del into either LRN or HRN mode. It is numerically what is done in the Chieng and Launder model. Ho- undesirable to adopt a function that abruptly switches wever, since the viscous sub-layer is treated separately between the two different modes. It is also physically from the HRN part in the new model, a simplified model
questionable, because the flow is only strictly laminar
¡ £ is sufficient here. The model is also adopted to a - in the immediate near-wall region – enabling an LRN type turbulence model.
model to be used – while only in the inertial sub-layer The assumed variation of the dependent variables for can the flow be treated as fully turbulent. The latter is this model are shown in Fig. 14. Both the turbulent of course a requirement for using a wall function-based kinetic energy and the shear stress have lost their li- HRN-model, see Fig. 1. In the buffer layer it is thus most near variation in the inertial sub-layer, as compared to appropriate to employ a smoothing function that blends the Chieng-Launder model. This of course simplifies the the two formulation together. One of the simplest mat- treatment immensely. The specific dissipation is assu- hematical functions that accomplishes this is the expo- ] med to vary either according to the log-law, ¨U , or the
nential function, which can be designed to operate from ]
¨U ] f linear-law, . to dependent on some flow quantity. On the grounds As was stated above, only the region above the viscous of the above discussion, it would be most beneficial to s sub-layer is of interest in this section. The integrated
base the exponential function on U . However owing to production of turbulent kinetic energy in the HRN re-
¡ ¡ T
numerical issues, it is disadvantageous to include any
U U gion, i.e. from U , is found as:
wall distance, and hence the blending function is ins-
¢ ¢
C
¦ t ¦
9 8 w
;
> > T
tead constructed using the turbulent Reynolds number,
t
H
U U P ¡ Q
N
!N (110)
C
M ¨M
as: U
$
t
¢
! ~
D
¢¡3x Z (108) The last identity can be used only if it can be assumed that the shear stress is constant and that the laminar
where © is a tunable constant. shear stress is negligible in the inertial sub-layer, see The model is then constructed to linearly combine the Fig. 1 and the discussion in connection with Eq. 84.
LRN and HRN formulations using this blending func- The wall shear stress in the new model is set similar
$ $
t t
H
f ] !
tion. Since N , and hence at the wall, is to the Chieng-Launder model (see Eq. 100) as:
$
]
¡ Q
multiplied with the LRN part and P is multiplied
V
T
o
t
¡
¤
with the HRN part:
. #
V
T (111)
P ¨WMQ U¤
¡
$ $
r t r r¤§
]
¤£
¦¥ ¦¥
^
P ¡ Q
¤
# ¡
£
(109) f
o
¡ h
where ¡ and are set to and , respectively, in
r
¡ £ where is some turbulent quantity. The LRN and HRN order to comply with the equilibrium values of the parts needed in this model are treated separately below. turbulence model. The a priori result using this equa-
tion is compared with the turbulent shear stress predic-
¡ £ ted using the LRN model in Fig. 15. Note that
PSfrag replacements 1.2
¦©¨
, new model
, LRN
1 ¦©¨
0.8
£
[% Figure 14: Assumed variation of variables first interior ¨ © 0.6 node, new model. 0.4
PSfrag replacements
4.3.2 HRN part: Simplified Chieng-Launder mo- 0.2
¡ £
del applied to a -type turbulence model 8 The logarithmic equations based on X , Eqs. 25 and 41 0 0 10 20 30 40 50
are unsuitable in separated flows. This applies especi-
©
ally for heat transfer since the predicted Nusselt num-
N
ber becomes zero in separation and re-attachment points Figure 15: The accuracy of the assumption, . using these equations. In contrast, experiments indicate Channel flow.
a maximum level of heat transfer at or in the vicinity of
8
V X
these points (Launder [23]). Substituting with the turbulent production is negligible in the near-wall
15 region; hence any discrepancy comparing to the Chieng- 10
DNS ¢¥¤ Launder model is a consequence of the simplistically as- ¡ 9 LRN sumed variation in the turbulent shear stress. At this New, sum moment it may perhaps be of interest to compare the 8 New, LRN part turbulent shear stress given by DNS-data, Fig. 1, with New, HRN part 7
the assumed variations of the two models. The assump-
T ¢
tion of a constant turbulent shear stress beyond U (new 6
¥§¦ ¤£ model) is perhaps not any worse than the linear varia- ¢
¢ 5 tion from the wall shear to edge of the cell in the Chieng- Launder model. Note especially in FigPSfrag. 1 thatreplacementsthe (lami- ¡ 4 nar) wall shear is higher than any given turbulent shear, 3 and hence the linear increase in the turbulent shear from the wall shear is not physically founded. 2 The integrated dissipation rate of the turbulent kine- 1 tic energy is based on the logarithmic law, Eq. 48:
0
¢ ¢ ¢
V 0 20 40 60 80 100
¦ t ¦ t ¦ t
s
¡ ¡
R U £ U U
v
o
U
U
¡:[%x
§
v
£ £ £
=:x'
¤ t
U §¡)=:x
¨ Figure 16: Integrated production term, -equation.
t
H
¦ £
!
X
o T
Channel flow, DNS-data, [25]. U (112) 1
where was assumed constant. 0 Contrary to the Chieng-Launder model, the dissipa- −1 tion rate is not set using the log-law. Rather, the mean −2
integrated value from the log-law is used: DNS ¢¥¤
LRN ¡ ]
¢ −3
¢
¤
e ¦ t
t New, sum
T
£ U
¦
¥ v T
o −4 New, LRN part
Z
U ¡ U
¢ U :[%x ¡ (113) New, HRN part
T −5 £
¡©¨
¤
PSU ¨U Q
t e
PSfrag replacements
v T
o −6
PSU ¡ U Q
¡'[)x
−7 £
4.3.3 LRN part: Standard ¥¡ −8
¡ £ The LRN part of the model is identical to the Wilcox −9 [34] (HRN) turbulence model. The model was shown in −10
Eqs. 61- 63, and is shown again here for clarity: 0 20 40 60 80 100
s
U
t
N
M
£ Figure 17: Integrated dissipation term, -equation.
¢
C C
t
9 ;
N
^ ^
¡ ¡
¡ £ P M £ M Q
B B
C?D C?D
£
¥¡¤£ ¢
4.3.4 The blending turbulence model
C C
t
£ 9<; £ £
N
^ ^
¡ £ ¨ P M £ M Q
¡ £
B B C?D
C?D The new blending turbulence model is summarised
£
t below and its performance is found in Bredberg [6].
9¥; et al.
with the production of turbulence computed as
@ @
Y The production and dissipation terms in the turbulent
N
B B
C C©D
¨ hM . kinetic energy equation are:
The model coefficients are: ¤
¦ t
9<;
t t t t t
¡
f £ ] ] f f ¦
U
(116)
¡ ¡
¤ ¨ ¨ ¤§£ ¨W ¤¢£ ¨ ¤§ ¨
§
T
C
>
l
$ $
9
] . ¡ ¨
The asymptotic boundary conditions applicable for the N
^ ^
U M P ¡ Q
C
U
turbulent quantities are: U
¤
f
t
(114) ¦
¡
£ U
(117)
M
£
(115)
§
v
U
=:x'
l
$ $
¤
M U ] §¡)=:x
¨
^
¡
U -¤ P ¡ Q
T T
o
The latter relation is valid only for computational nodes
U PSU ¡ U Q U
s
¤
¡ £
within U (Wilcox [36]). To establish a grid inde- >
pendent solution, 7-10 nodes are required in this range. where 9 is the contribution of the non-primary shear
t ¦ ¦
u¨ U Both requirements can however be eased, especially the stress components ( ). The production term and latter, with negligible deterioration of the predictions, the dissipation term are shown in Fig. 16 and Fig. 17, see e.g. Bredberg et al. [5], [4]. respectively.
16 2 3
DNS
¢ ¤
¡ DNS
¢ ¤ LRN 2.5 LRN ¡ 1.5 New New 2
1
¢
¢ 1.5
¦
¥ ©
¢ 0.5 1
¦¥§¨
¡ £¥¤ 0.5 0 PSfrag replacements PSfrag replacements 0
−0.5 −0.5
−1 −1
0 20 40 60 80 100 0 20 40 60 80 100
s
s
U
U £ Figure 18: Integrated diffusion term, -equation. Figure 20: variation.
2
¡ £
1.5 DNS A comparison with DNS-data and the LRN model
¡ ¢ ¤ LRN is shown in Fig. 20.
1 New
¢ ¦
¥ 0.5 The wall viscosity is set as: ¢
0
¢
¢
V
o T
$ $
t
] ¡mU
¢
^ ^
M M P ¡ Q M
−0.5 # T
V (119)
£
¤
¨WMQ P U
¡
¡
¨
−1
¢ £ ¢
−1.5 $
PSfrag replacements¡ To close the model, the blending function, (Eq. 108),
T T
−2 and the viscous sub-layer values, U and , must be de- T −2.5 fined. The viscous sub-layer thickness, U , is calculated using the definition of the viscous sub-layer Reynolds −3 0 20 40 60 80 100 number and the assumed variation of the turbulent ki-
s netic energy: U
Figure 19: Sum of integrated source terms in -equation.
t
V
T
¤
H
U
!T
t
H
T !T V
T (120)
U MO¨
The third source term on the RHS of the turbu- M
t
H
V
V
T T
!
H
U ¡
MO¨-U T
lent kinetic energy equation, Eq. 66, is the diffusion !
t
V
V T
T (121)
U U ¨ term, which is not modified in the present model, with M the standard relation used. The accuracy using this
simplistic model is shown in Fig. 18.
H
T !
It is interesting to combine the three terms on the The constants, © (in the blending function), and
RHS of the turbulent kinetic energy equation in order are optimised using DNS-data [25]. The combination of
t t
H
]W] ]
T !
to estimate its balance. This is done in Fig. 19. Both and © gave the overall best results, as
the DNS-data and the LRN ¢¡¤£ mode yield perfect ba- compared with DNS-data. The effect of these values is a
lance. The new model is not very well balanced, yielding rather narrow blending region, see Fig. 21. The calcula-
s s
T T
¡¤£ a large negative source in the buffer layer and slowly ted U and using LRN , [35] data (a priori) and increasing the value in the logarithmic region which yi- the selected constants are shown in Fig. 22. elds a large positive source in the off-wall region. This The new model is defined by Eqs. 116, 117, 118 119
latter fact may be the reason why the new model gives a $ and the blending function, : fairly large discrepancy of the predicted in the centre
of a channel.
$
The specific dissipation in the first near-wall node is t
¢¡3x5[ Z (122)
set as:
T
¤
PAU ¨U Q
$ $
e t
M
]
T T
^
U
P ¡ Q £
v and the above routines for the specification of and . T
o (118)
U
PAU ¡zU Q
¤ ¡:[%x
17 1 5 Modelling Near-wall Heat
0.9 Transfer
¦
¨¤£¦¥§¥©¨ 0.8 ¢¡ In contrast to previous sections, this one is written in 0.7 a brief and and abbreviated style. This is not to imply
0.6 that the thermal field is any easier to predict or that the $ 0.5 modelling is any simpler, but is merely a consequence of the lesser efforts made to improve the thermal mo- 0.4 dels. To exemplify this, it can be noted that, in some 0.3 predictions, an RSM turbulence model – which involves seven turbulent transport equations – is combined with 0.2 PSfrag replacements a constant turbulent Prandtl number heat transfer mo- 0.1 del, i.e. a zero-equation isotropic model. The turbulent 0 Prandtl number heat transfer model using the simple
0 5 10 15 20 gradient diffusion hypothesis, or SGDH, relies on the si- s
U milarity of the turbulence and heat transfer fields. The
$ SGDH assumes that they are identical apart from a pro- £ Figure 21: Blending function, , using LRN ¥¡ data portionality factor, the turbulent Prandtl number. The
8 validity of this approximation in fully developed chan- nel flow can be seen in Fig. 7. Using the SGDH ap- 7 proximation, it is also implied that the thermal field is isotropic. With an RSM the individual Reynolds stres- 6 ses are known however, and can be used to predict the
turbulent anisotropic level in the flow. If it is assumed ¦
5 ¨
¡
¨ that the anisotropicity of the heat transfer is identical to > r that of the turbulence, a GGDH, or Generalized Gradi- 4 4 ent Diffusion Hypothesis, model can be employed for the heat transfer prediction. 3 Although a GGDH would be more appropriate than PSfrag replacements 2 a SGDH, the fundamental approximation of an identity between the flow field and thermal field is still used, and 1 should be questioned. Similar to the turbulence field, it is also possible to solve a set of equations for ther- 0
0 10 20 30 40 50 mal quantities. Examples of this approach are the two-
`
N
s
¡ R equation model, , of the Nagano group, Abe et al. U [1] and EAHF (Explicit Algebraic Heat Flux) models of
Figure 22: Comparison of the new models variation of So and Sommer [30] and Dol et al. [11]. Here, however,
t H
viscous sub-layer quantities, with those given at !WT only the simple model based on the turbulent Prandtl
s
]h]
U (constant lines). number is used.
5.1 Integration Method The integrated or LRN method implies that the heat transfer is governed purely by viscous effects in the limit of the wall. The heat transfer is thus estimated using
the Fourier law, such as:
¦ ¦
_ _
t p t
¤-q
©
F
¦ ¦
¡ ¡ 9 E
(123)
U U
p t
9 E
q
¤ ¨
where is the thermal conductivity given as © . 6 The in the heat flux indicates that it is the ’laminar’ or the molecular part. Using the simple gradient diffusion hypothesis (SGDH) for the turbulent region, the ’turbu-
lent’ heat flux is computed as:
¦
_
N
t
q
©¥¤
FmN
¦
9 E ¡
N (124) U
The total heat transfer is then calculated by adding the
t
FgN N F F.N
^
two parts together, .
18 0.5 5.3 New Blending Model 0.45 During the numerical optimisation of the new model it 0.4 was found that the most appropriate blending parame-
ter for the thermal field was the normalised tempera- s
0.35 ture, _ . The non-dimensionalised wall distance in Eq.
v
¡ ¡:[)x
U U¤ ¤ ¨M e 0.3 DNS 125, is however based on ¤ instead of
New model s
U o the commonly used although less satisfying :
0.25
]
t
_
^
¡ ¡ ¡ Q
0.2 PAU
o (128)
0.15 _
¡ has not yet been given explicitly for the LRN re-
t _
PSfrag replacements 0.1 _
P ¡
gion. Introducing normalised variables, ¡
t
_ v v
V V
F
¤
© §¡:[)x O¨ U ¡ U §¡:[)x O¨M 0.05 Qa and , Eq. 123 can be re-written as: 0
0 20 40 60 80 100 _
F
t
¤
U ¡.M © cM ¡
s
F
9 E
¨
U
v v V
V (129)
¤
©
¡:[%x ¡:[)x o
Figure 23: Coefficient, in log-law for temperature.
t
H
¦ £ ! Model Eq. 132 and DNS-data, X , [18]. Re-arranging, the linear law for temperature is given
(see also Eq. 38):
t
_ 9 E
¡ ¡ Using a constant Prandtl number model, it is under- U (130) stood that all wall effects are embedded in the turbu- lence model, and hence any LRN modifications to the The non-dimensionalised temperature in the blending
heat transfer model are normally excluded. However, as model thus becomes: ¢
shown in Fig. 7, the turbulent Prandtl number is also ]
$ $
t 6
9 E _
]
}
^ ^
¡ ¡ ¡
U P ¡ Q PAU Q
¤ ¤ affected by the wall, and there thus exists a number of o (131)
heat transfer models that include some wall modifica- £ $ tions, see e.g. Reynolds [28] and Kays [19]. where is defined as for the turbulence field, see Eq. 108. To achieve accurate results with this heat transfer model in the case of a first node location in the buffer
5.2 Wall Function Method
¨ layer, the values of o and must be modified. This The law-of-the-wall for the temperature was derived in is not unreasonable since constant values are valid only
section 3.2. From that the temperature profile in the in the logarithmic layer. In the new model, we chose o inertial sub-layer is approximated with a logarithmic to modify , which is reduced in the buffer layer, as
equation as given by Eq. 40: found from DNS computations [18] and [17]. Reasonable
o
_ _ _ _ 8 ]
X agreement with the predicted by DNS was found for
s s
t
_
¡ P ¡ Qacb
^
_ Q
PSU the relation:
X F o
t
H
f ¦ ¦ ] £ f ] £
o !N
^
¡ P ¡ ¨ Qa¢
(125) ¡ (132)
t t t
¡£¢ 9 E
¦ f f c]
t
¡ ¢
f
o o with and for air ( ), see see Fig. 23. The expression gives in the lo-
experimental data in Kader and Yaglom [16]. The re- garitmic layer, as indicated by experimental data [12,
t
H ¦ £
ader is also referred to Huang and Bradshaw [12] and 16, 20]. Note that the DNS-data ( !X ) indicate
t
¡
f
o¨
Kays and Crawford [20] for a recent discussions on the a slightly lower value of . There seems to be
topic. Kawamura et al. [18] provide valuable DNS-data a Reynolds number dependency for this coefficient, ho-
t
H
] £Wf
for the temperature equation in fully developed channel wever, as another DNS [17] ( !X ) yields an even
t
f ¦ ¨
flow. The above relation for the wall temperature is not o s
_ lower value, .
commonly used, however, and is instead expressed
s t
8
8 Results obtained using this model are presented in X
with the normalised velocity, u¨ , as: Bredberg et al. [6].
s s
t
_ 9 E 8 9 E
N
^
P Q
(126)
t t
9 E 9 E 9 E
o o
N N
Acknowledgments
¨ ¡ ¨
where , and , [16]. The
9 E
constant, , depends on the Prandtl number, which Funding for the present work has been provided by
according to Jayatillaka [13] is: STEM, the Volvo Aero Corporation and ALSTOM Power
v §
9 E via the Swedish Gas Turbine Center.
='x
t
9 E ¡
]
¨
9 E ¡
N §
¢ (127)
9 E
N
¢
] f f fWf
¨
^
9 E
¡ £
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