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4 TURBULENCE MODELS

4.1 Introduction

For practical computations, turbulent flows are commonly computed using the

NavierStokes equations in an averaged form (e.g., Reynolds or Favre averaging). The

averaging process gives rise to new unknown terms representing the transport of mean

momentum and heat flux by fluctuating quantities. These undetermined terms are the

Reynolds stresses or heat fluxes and they lead to the well known closure problem for

turbulent flow computations (Hinze 1959). In order to determine these quantities,

turbulence models are required which consist of a set of algebraic or differential

equations.

Several onepoint turbulence models have been developed which can bebroadly classified into the following two categories:

(a) eddy viscosity models which are based on the assumption that the Reynolds

stresses are a local property of the mean flow and are related to the mean flow

gradients via a turbulent viscosity (e.g., Launder and Spalding 1974), and

(b) Reynolds stress models which assume that the Reynolds stresses are dependent

variable quantities which can be solved directly from their own transport equations

(which are derived from the NavierStokes equations), along with some modelling

equations (e.g., Launder et al.1973).

The derivation of these models has been largely based on intuition, empirical

correlation, and to some extent, constraints set forth by physical realizability (Lumley

1978). More recently, turbulence models have been developed with more rigor and

mathematical formalism based on the Renormalization Group (RNG) theory (Yakhot

and Orszag 1986, Speziale and Thangam 1992).


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Among the eddy viscosity models, a further classification can be made based

on the number of differential equations utilized. Thus, zero, onehalf, one and

twoequation eddy viscosity models have been developed. Among these, the ke

model is one of the most widely used for practical applications. The standard ke model

is only valid in fully turbulent regions and, for wallbounded flows, requires additional

modelling of nearwall regions. Within the family of ke models, various techniques

have been employed to resolve the nearwall flow structure for wallbounded flows.

These include: (a) wall functions (Jones and Launder 1972), (b) lowReynolds number

modifications (e.g., Chien 1982), and (c) algebraic or oneequation models in the wall

region (e.g., Rodi 1991). The oldest and perhaps the most common of these is the wall

functions approach, and although it is not accurate in some flow situations (such as flow

separation), it has been shown to yield satisfactory results in complex flowfields (Shyy

et al. 1997). Moreover, the wall functions approach is very beneficial in complex

threedimensional geoemtries due to a large saving in the total number of grid points

required (and hence a saving in computer memory and CPU time) compared to the

other two approaches (Viegas and Rubesin 1983). Though the concept of wall

functions is long established (Jones and Launder 1972, Launder and Spalding 1974),

its implementation in finite volume algorithms employing generalized bodyfitted

coordinates is not well documented and requires careful consideration. Sondak and

Pletcher (1995) have presented a framework for estimating wall shear stresses in

curvilinear coordinates which involves several steps to incorporate the wall stresses

into the Cartesian stress tensor. The present work provides a consistent framework for

the implementation of wall functions in finite volume algorithms for threedimensional

geometries using nonorthogonal bodyfitted coordinates with a staggered

arrangement of velocity components and scalar variables. The method presented

includes the implementation of not only the wall shear stresses in the momentum

equations but also the source tems in the ke equations.

4.2 Modelling Reynolds Stresses via Eddy Viscosity

The Reynoldsaveraged equations are obtained by replacing the various

dependent variables with their mean and fluctuating components (e.g., uinstantaneous=u + u) in the NavierStokes equations and time averaging the equations (see Section

1.2). This leads to additional unknowns in the form of Reynolds stresses (see Eq. (1.7)),

which need to be modelled. Eddy viscosity models are based on the assumption that

the Reynolds stresses are a local property of the mean flow and are related to the mean

flow gradients via a turbulent viscosity as shown below:


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* ruiuj + mtuixj )ujxi

(4.1)where mt is some turbulent viscosity which needs to be modeled.

The turbulent viscosity mt is the vehicle through which the time and length scaleeffects of turbulent flows are introduced into the equations of the mean flow. Thus,

modeling mtrequires specification of local length and time scales (or equivalently local

velocity and length scales). The ke models provide the velocity scale via the modeled

turbulent kinetic energy (k) and the length scale via a combination of k and the rate of

viscous dissipation of turbulent kinetic energy (e). Dimensional analysis yields the

turbulent viscosity as

mt+Cm fm r k2

e(4.2)

where Cm and fm are proportionality constants to be defined later. For the standard ke

model,fm =1.0.

4.3 The ke Model with Wall Functions

Two major issues related to wall functions are considered in this section. One

is the implementation of the wall shear stress in the momentum equations and the other

is the implementation of the source terms in the turbulent kinetic energy equation. The

estimation of the wall shear stress using the assumptions of Couette flow and local

equilibrium near a noslip boundary is well established. However, for bodyfitted grids,a correct estimation of the tangential velocity at the nearwall nodes using the local

surface curvature and its resolution into components in the Cartesian coordinates is

required; we attempt to provide a general framework for obtaining these components.

Since we employ a staggered arrangement for storing the velocity components and the

scalar variables, interpolations of velocity components are required to compute the

tangential velocity at the nearwall nodes for the k and e equations. It is demonstrated

in this study that, near a noslip wall, interpolations consistent with the assumption of

logarithmic variation of nearwall velocity are necessary in order to yield wall shear

stress which is frameinvariant. For the estimation of nearwall turbulence quantitiesusing wall functions, several different techniques are available. The dissipation rate (e)

at nearwall nodes is most often assigned assuming local equilibrium, instead of

solving the equation governing it. The tubulent kinetic energy (k) is either assigned

(Patel et al. 1985, Sondak and Pletcher 1995) or computed using the usual governing

equation (Launder and Spalding 1974). In the latter approach, the source terms in the


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kequation (production and dissipaton of k) need to be appropriately estimated using

the assumptions underlying wall functions. It is this step which has been dealt with in

different ways by different researchers. In the context of finite volume algorithms, we

evaluate three methods for estimating the source terms in the kequation that are most

common in the literature. Turbulent flow through a straight channel, in horizontal and

tilted orientations, is computed to validate and evaluate the performance of these

implementations.

4.3.1 The ke Equations

The transport equations for k and e, after the modelling assumptions are

incorporated, can be expressed using indicial notation in Cartesian coordinates, as

follows:

trf ) xj ruj f + xjm ) mtsf fxj) R1 ) R2 (4.3)where f 5 k or e, with

R1 +

P for the k*eqn.C1 e P

kfor the e*eqn.

(4.4)

and

R2 +

*r e 5 * Cm r 2 k*mt k for the k*eqn.*C2 r

e2

k5 *C2 r e *k* e for the e*eqn.

(4.5)

Note that P is the generation (production) of kfrom the mean flow shear stresses:

P+ tijuixj

+ mtuixj )ujxi @ uixj + mt@ R (4.6)

whereR in the expanded form can be written as:

R + 2ux

2 ) vy2

) wz2

) uy ) vx

2

) uz ) wx2

) vz ) wy2

(4.7)


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P ut

n^

c

x

y

YP

rs

rP

h

Yvis

ueuw

vn

s

n

Note: Y is the local coordinatenormal to the wall

(edge of viscoussublayer)

Figure 26. Nomenclature for nearwall treatment.

purpose, as will be discussed in the next section. The wall shear stress used in the

mean flow momentum equations needs to be appropriately estimated, for which (a) thevelocity component tangential to solid walls, and (b) its derivative along the direction

normal to the walls, are required. We formulate the procedure to estimate these

quantities in this section.

4.3.3.1 TANGENTIAL VELOCITY

Let us use the notation c1,c2,c35 (c,h, z) for the curvilinear coordinatesystem used in the present algorithm. For a surface ci + constant(for an illustration,

see Fig. 26) the unit normal vector can be represented as

n^ +ci ci

(4.12)

with components:


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utx + u* un^1) vn^2) wn^3 n^1uty + v* un^1) vn

^2) wn

^3 n^2

utz + w* un^1) vn^2) wn

^3 n^3

(4.16)

These components can be further expressed in terms of contravariant velocities asgiven in Table 1.

4.3.3.2 DERIVATIVE OF A FUNCTION F NORMAL TO A PLANEci +constant

The expression for derivatives of a function normal to a coordinate surface can

be obtained from the base vectors (Thompson et al. 1985). We first need to define the

covariant and contravariant base vectors for a surface to facilitate the expression of the

derivative of a function normal to that surface. Covariant base vectors are tangent to

coordinate lines and the covariant base vector for a coordinate line along which civaries is given by the following:

a~i + r~

ci(i + 1, 2, 3) (4.17)

(where r~ +xi^)yj

^)zk

^) with the following components:

a~1 +xc i^)yc j

^)zc k

^

a~2 +xh i^)yh j

^)zh k

^

a~3 +xz i^)yz j

^)zz k

^

(4.18)

Contravariant base vectors are normal to coordinate surfaces and for a surface on

which ci is a constant, the contravariant base vector is given by

a~i +

~ci (i + 1, 2, 3) (4.19)

+ 1J

a~j a~

k (i,j, k cyclic) (4.20)

(where Jis the Jacobian) and its components can be written as

a~1 + 1

Ja~2 a

~3 +

1Jf11 i^)f12 j^)f13 k^

a~2 + 1

Ja~3 a

~1 +

1Jf21 i^)f22 j^)f23 k^

a~3 + 1

Ja~

1 a~

2 +1Jf31 i^)f32 j^)f33 k^

(4.21)


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The contravariant metric tensor is given by

gij + a~ i @ a~j (i,j + 1, 2, 3) (4.22)

and it components are given by

gij + 1J2

q11 q12 q13

q21 q22 q23

q31 q32 q33

(4.23)

Finally, the derivative of a quantity Fnormal to a coordinate surface ci + const.is given by

Fn (i) +

1

gii3

j+1

gij Fcj(i + 1, 2, 3) (4.24)

and its expanded form for the three families of surfaces is given in Table 1.

4.3.3.3 TRANSFORMATION OF STRESS IN COMPONENTS PARALLELAND NORMAL TO THE WALL

Consider a surface defined by c + constant and take the derivative of thexcomponent of the tangential velocity, utx, in a direction normal (denoted by Y) to this

surface:

utxY+

uY*

YUf11q11 (4.25)

+ 1J q11

q11uc) q12uh) q13uz* YUf11q11 (4.26)For a node next to the wall, the second term in Eq. (4.26) can be neglected. Thus, for

example, for the viscous term in the umomentum equation, given by Eq. (NO TAG),

we obtain:

mtJq11uc) q12uh) q13uz + q11 @ mt

utxY + q11

@ twallx (4.27)

In a similar manner, one can express the viscous terms in the momentum equations

for all wall boundaries in a given block of the domain as shown in Table 1.


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4.3.4 Wall Shear Stress

For a fully developed turbulent flow near a noslip wall, the normalized nearwall

tangential velocity, assuming a twolayer structure (viscous sublayer followed by the

log layer), can be written as follows (White 1974):

u) + Y)

1k

logEY)Y)v 11.63Y)u 11.63 (4.28)

where

u)+ ut

utY)+

r YP utm ut +

twallr

(4.29)The von Karman constant k has the value 0.4187. The quantity Eis assigned the value

9.793 for smooth walls. Note that Yis the coordinate normal to the wall as shown in Fig.26 and the viscous sublayer is assumed to extend upto Y+ of 11.63. The above form

has problems near flow separation since it becomes singular as ut approaches zero.

To alleviate this problem, one resorts to the following assumptions:

(a) Couette flow,

(b) local equilibrium between production and dissipation, and

(c) constant stress layer near the wall.

From these, the wall shear stress can be related to turbulent kinetic energy as

twall

r

+ Cm k (4.30)

This form avoids the singularity problem near separation since twall now never

becomes zero (k is not zero at separation though ut is). Substituting Eq. (4.30) fully into

the right hand side and partly into the left hand side of Eq. (4.28), we get the following:

twall + tvisY)

u)(4.31)

where

tvis + m ut

YP (4.32)

Y)+r C14

mk12 YP

m(4.33)

The above expression for the shear stress (in the log layer) can also be expressed as


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twall +r k C14

mk12

log(EY))ut (4.34)

The components of wall shear stress along x, y and zdirections which are

implemented in the u, v and wmomentum equations, respectively, are given by

twallx +m Y)

YP u) u

tx twally +

m Y)

YP u) u

ty twallz +

m Y)

YP u) u

tz (4.35)

where the components of tangential velocity are as given in Table 1.

4.3.5 Interpolation Procedure for Estimating Tangential Velocity on aStaggered Grid

Since a staggered grid is employed in the present algorithm to store the velocity

components and the scalar variables, interpolations of the velocity components are

required to estimate the tangential velocity at the scalar nodes near a wall (e.g., node

P in Fig. 26). Referring to Fig. 26, the ucomponent at node P is obtained by using ue

and uw whereas the vcomponent at node P is obtained using vn (vs being zero at a

noslip wall). The ucomponent can be obtained by a linear interpolation of ue and uw.

One could use the same procedure for the vcomponent which yields vP + vn2.However, as will be demonstrated later, this leads to an estimation of the tangential

velocity at node P (u

t

P) which is not independent of the orientation of the wall boundary(i.e., it is not rotationinvariant) if the wall function treatment is employed. The reason

is that wall functions assume a nonlinear velocity variation which is not compatible with

linear twopoint averaging. Hence, it is necessary to use an interpolation procedure

which is consistent with the assumptions underlying wall functions. Specifically, since

a logarithmic variation of velocity is assumed in the direction normal to the wall if

Y)P u 11.63, the vcomponent at node P is estimated as follows:

vP + vn @logEY)

P

logEY)n (4.36)

where Y)n + 2Y)P . If Y)P t 11.63, linear interpolation is used because the node is within

the laminar sublayer. As will be demonstrated later, this procedure yields a tangential

velocity (and hence a wall shear stress) which is independent of the local orientation

of the wall boundary.


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4.3.6 Source terms (Pre) in the kEquation at a NearWall Node

The nearwall values of turbulent kinetic energy can be obtained by assigning

the value of kassuming that the rate of production is equal to the rate of dissipation of

k:

k+u2t

Cm(4.37)

where ut can be obtained in an iterative manner (Sondak and Pletcher 1995) at each

step. Another popular approach is to solve the kequation at the nearwall nodes after

estimating the production and dissipation terms consistent with wall function

assumptions (Launder and Spalding 1974). An advantage of this approach is that in

conditions far removed from equilibrium, all the terms in thekequation can play a role,

even though the assumptions underlying wall functions may no longer be valid. It is this

approach which has been adopted in the present algorithm. With this approach,

however, different implementations have been employed in the context of finite volume

methods. For example, one method of estimating the net source term has been to

obtain its value at the node P and multiplying it by the control volume height (effectively

assuming the net source term to be constant over the control volume), e.g. TASCflow

(1995). This method is termed SOURCE1 in this study. A second method is to perform

the integration over the entire control volume height (upto the north face denoted by n

in Fig. 26), along with the assumptions of local equilibrium and constant stress in the

log layer, e.g., Launder (1988), Lien and Leschziner (1994). This method is labelled

SOURCE2. A third approach follows the original implementation proposed by Launder

and Spalding (1974) which is an average rate of production minus dissipation obtained

by integrating upto the nearwall point P. This procedure has been employed by several

workers to represent the net production minus dissipation over the entire control

volume surrounding the node P (e.g., Shi and Ribando 1992). We label this method as

SOURCE3 in the present work. We next present the three different methods,

mentioned above, of estimating the source terms in the kequation.

4.3.6.1 NODAL VALUE (SOURCE1)

Production Term

The production term consistent with wall function assumptions is given by

P+ twallutY

(4.38)

Using Eq. (4.32), the tangential velocity gradient can be written as


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dut

dY+

twallm

du)

dY)(4.39)

Substituting Eq. (4.31) and Eq. (4.39) into Eq. (4.38), we get

P+t2

vis

m Y)

u)2

du)

dY) (4.40)

where du)dY) is computed from Eq. (4.28). For the log layer, the above expressiontakes the following form:

P+t2

wall

k C14m r YP k12

(4.41)

which is multiplied by the volume of the nearwall control volume.

Dissipation Term

For conditions of local equilibrium, P+ re and twall + r Cm k(see Eq. (4.30)),and using Eq. (4.41), we get the following for the dissipation term at the node P:

* re + *r C34m k

32

k YP(4.42)

4.3.6.2 INTEGRATED VALUE (SOURCE2)

Production Term

In this formulation, the average value of the production term in the nearwall

control volume is used which is obtained by integrating over the nearwall controlvolume and dividing by the height of the control volume (Launder 1988). Consistent with

the wall function assumption, the integration is carried out normal to the wall and the

lateral variation is assumed constant:

P+ 1YnYn

0

twalldut

dY. dY (4.43)

Using the assumptions that (a) there is no production in the viscous sublayer, and (b)

twall is constant in the log layer,

the integration in Eq. (4.43) is conducted between the limits Y+ Yvis and Y+ Yn, andyields the following:

P+twallYn

utn * utvis (4.44)


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Using the relation between twall and utgiven by Eq. (4.34) and the fact that Yn + 2 YP

(Fig. 26), the above is finally expressed as follows:

P+t2

wall

k C14

m

r Yn k12log2 Y)P

Y)vis

(4.45)It should be noted that, as pointed out by Launder (1988), an erroneous

expression for the production term has been commonly used, which is obtained by the

wrong assumption that the turbulent shear stress equals the wall stress even in the

viscous sublayer. Such an assumption, after performing the integration in Eq. (4.43)

from Y+ 0 to Y+ Yn, yields:

P+twallYn

utn (4.46)

The above expression typically yields a generation rate too large by a factor of four (aspointed out by Launder 1988).

Dissipation Term

Upon integrating the dissipation term in the kequation, across the control

volume, we get

* r e + *r

YnYn

0

e.dY+ *2 m k

Yv Yn*

C34m r k

32

k Ynlog2 Y)P

Y)vis

(4.47)

In the present implementation, the first term on the right hand side of Eq. (4.47) isneglected.

4.3.6.3 INTEGRATED VALUE (SOURCE3)

Production Term

A third method of estimating the source terms is also prevalent in the literature,

in which the production term is written as:

P+twallYP

utP

(4.48)

This form is obtained by performing the integration in Eq. (4.43) from the wall to the

node P, assuming constant twall in that region. Though this procedure is not consistent

with the finite volume method (which dictates that the integration be performed across

the entire height of the nearwall control volume), it is commonly adopted by several

researchers (e.g., Shi and Ribando 1992). This form was originally proposed by


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Launder and Spalding (1974) and is appropriate if one assumes the node P to be the

upper bound of the nearwall control volume. Note that the above can also be

expressed, using Eq. (4.34), as

P+

t2wall

k C14m r YP k12 logEY

)P (4.49)

Dissipation Term

Consistent with the estimation of the production term in this method, given

above, the dissipation term is estimated as follows:

* r e + *r

YPYP

0

e.dY+ *r C34m u

)P

k32

YP(4.50)

4.3.6.4 COMPARISON BETWEEN THE ABOVE THREE METHODS

For the sake of comparison, let us define the following:

C*1 +twall

2

k C14m r YP k12

C*2 +r C34m k

32

k YPC* + C*1 * C*2 (4.51)

Then the various forms of the source terms for the kequation can be summarized as

shown in Table 2. Under the assumption of local equilibrium, twall +

r Cm

k(see Eq.

(4.30)), and we have C*1 + C*2 which implies P + re, which is consistent with localequilibrium. However, in conditions far removed from equilibrium, such as regions in the

vicinity of separation and reattachment, C*1 0 C*2. Near flow separation, for example,twall vanishes faster than kand thus the production term will vanish faster than the

dissipation term. Of course, the very premise of the applicability of wall functions is in

doubt under conditions of nonequilibrium, but within these limitations, it appears that

the particular method of computing source terms in the kequation may impact the

estimation of nearwall estimation of turbulence quantities in regions of flow far

removed from equilibrium.

4.3.7 e at a NearWall Node

At a nearwall node, the eequation is not solved; instead the nearwall e is

specified depending on its location in the boundary layer:


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C*

2@ log2Y)PY)

vis C* @ logEY)P C*

SOURCE1 SOURCE2 SOURCE3

(P* re)

For y)P + 40 C* 0.96 C* 5.8 C*

TABLE 2.

Note: C* is defined in Eq. (4.51).

e

+

2 m k

r Yvis2

e +C34m

k32

k YP

viscous sublayer (linear velocity profile)

law of the wall layer (logarithmic velocity profile)

(4.52)

4.3.8 Test Case

Turbulent flow in a straight channel with an inlet Reynolds number of 14,544 is

used to aid in the assessment of the alternative nearwall treatments discussed above.

For quick reference, the figures to be discussed in the following are summarized here:

Fig. 27. computation using linear interpolation for vcomponent for estimation of

tangential velocity at the nearwall node.

Fig. 28. computation using loglaw interpolation for vcomponent for estimation of

tangential velocity at the nearwall node.

Fig. 29. comparison between the different implementations (SOURCE1, SOURCE2

and SOURCE3) of production and dissipation terms in the kequation for the

channel in horizontal orientation.

Fig. 30. comparison between the different implementations (SOURCE1, SOURCE2and SOURCE3) of production and dissipation terms in the kequation for the

channel in an inclined orientation (30 degrees).

Fig. 31. effect of an erroneous (inconsistent) estimation of the generation term in the

ke model, namely, Eq. (4.46) instead of Eq. (4.44).


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The channel geometry used for the computation has an aspect ratio of 50:1. The

direct numerical simulation data of Kim et al. (1987) for the same geometry is used for

comparison. The Reynolds number based on inlet height, inlet velocity and laminar

viscosity is 14,544. A uniform velocity profile is assigned at the inlet and the turbulent

kinetic energy profile is assigned by assuming an inlet turbulent intensity of 3%. Profiles

of normalized turbulent kinetic energy k)+ ku2t, normalized turbulent shear stressuv) + uvu2t and u) are plotted against DNS data for comparison.

We first illustrate the deficiency of the linear interpolation procedure for the

vcomponent of velocity (vn) for estimating the tangential velocity utP. It can be seenfrom Fig. 27 that such an interpolation yields a different result for the same physical flow

but with the local coordinate system rotated by 30. This difference is expected to be

more pronounced for flows with (a) even higher angles of rotation, (b) higher Reynolds

number, and (c) grids which have a large spacing for the nearwall nodes. The

logarithmic interpolation given by Eq. (4.36), on the other hand, produces nearly

rotationinvariant profiles as shown in Fig. 28. It is interesting to note that the procedure

outlined in Section 4.3.4 for the computation of wall shear stress (including the

logarithmic interpolation for the vcomponent of velocity and estimation of tangential

velocity) yields just as good results on the coarse (51x33) grid as on the fine (145x61)

grid. For all subsequent computations, the logarithmic interpolation procedure for the

nearwall vcomponent has been adopted. It should be pointed out that the results in

Fig. 27 and 28 are obtained using SOURCE2 implementation of the source terms in

kequation, though this has no bearing on the velocity interpolation procedure.

Figures 29 and 30 show, for the horizontal and tilted channels respectively, the

profiles predicted by the three implementations of the source term, namely, SOURCE1,

SOURCE2 and SOURCE3. All three implementations yield nearly identical profiles. For

the flow through the straight channel in which the condition of local equilibrium in the

nearwall region holds, all three implementations provide a balance between

production and dissipation of turbulent kinetic energy and hence yield the same

magnitude for k at the nearwall nodes. If this balance is not maintained, erroneous

values of nearwall turbulence quantities will result. For example, if, instead of Eq.

(4.46), we employ Eq. (4.44) for the estimation of the generation term in the method

SOURCE2 (which implies that there is production even in the viscous sublayer and that

the turbulent shear stress equals wall shear stress even in the viscous sublayer) we

obtain the production rate which is too large, and it yields wrong values of turbulence

quantities as shown in Fig. 31(a) and (b).


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Figure 27. Linear interpolation: comparison between horizontal and tilted (30_) channel computations

using linear interpolation for the nearwall vcomponent of velocity (51x33 grid). (a) k+, (b) uv+.

DNS data

Horizontal channel

Tilted channel

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

y+

k+

(a) Linear interpolation: k+ on 51x33 grid

DNS data

Horizontal channel

Tilted channel

0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

1.2

y+

uv+

(b) Linear interpolation: uv+ on 51x33 grid


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DNSd

ata

horizontal

tilted(30deg)

0

50

100

150

200

250

300

350

400

00.51

1.52

2.53

3.54

4.55

y+

k+

k+

on51x33grid

DNSd

ata

horizontal

tilted

(30

deg)

0

50

100

150

200

250

300

350

400

00.51

1.52

2.53

3.54

4.55

y+

k+

k+

on145x61

grid

DNSd

ata

horizontal

tilted

(30

deg)

0

50

100

150

200

250

300

350

400

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

y+

uv+

uv+

on145x61

grid

Figure 28. Logarithmic interpolation: comparison between horizontal and tilted channel computation

using logarithmic interpolation for the nearwall vcomponent of velocity. (a) k+, (b) uv+ on 51x33

grid; (c) k+, (d) uv+ on 145x61 grid.

DNSd

ata

horizontal

tilted(30deg)

0

50

100

150

200

250

300

350

400

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

y+

uv+

uv+

on51x33

grid

(a)

(b)

(c)

(d)


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108

DNS data

SOURCE1

SOURCE2

SOURCE3

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

y+

k+

(a) k+ on 51x33 grid horizontal channel

DNS data

SOURCE1

SOURCE2

SOURCE3

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y+

uv+

(b) uv+ on 51x33 grid horizontal channel

DNS data

SOURCE1

SOURCE2

SOURCE3

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

y+

u+

(c) u+ on 51x33 grid horizontal channel

Figure 29. Comparison of the three implementations of nearwall source terms in the kequation for

horizontal channel using 51x33 grid.


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Figure 30. Comparison of the three implementations of the nearwall source terms in the kequation

tilted channel using 51x33 grid.

DNS data

SOURCE1

SOURCE2

SOURCE3

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

y+

k+

(a) k+ on 51x33 grid tilted channel

DNS data

SOURCE1SOURCE2

SOURCE3

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y+

uv+

(b) uv+ on 51x33 grid tilted channel

DNS data

SOURCE1

SOURCE2

SOURCE3

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

y+

u+

(c) u+ on 51x33 grid tilted channel


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Figure 31. Effect of an erroneous estimation of the generation term in the ke model; specifically,

Eq. (4.46), instead of Eq. (4.44) is used in the method SOURCE2.

DNS: k+

Present: k+

DNS: uv+

Present: uv+

0 50 100 150 200 250 300 350 400 4500

1

2

3

4

5

6

y+

k+

oruv+

(d) SOURCE2 integration over entire cell horizontal channel

DNS: k+

Present: k+

DNS: uv+

Present: uv+

0 50 100 150 200 250 300 350 400 4500

1

2

3

4

5

6

y

k+

oruv+

(d) SOURCE2 integration over entire cell tilted channel

(a) Horizontal Channel

(b) Tilted Channel


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4.3.9 Summary of Wall Function Treatment

A consistent implementation of wall functions for finite volume algorithms was

presented in the sections above. The proper estimation of shear stress for curved walls

using bodyfitted coordinates was detailed. It was shown that, when the velocity

components are arranged in a staggered manner, the interpolations used to estimatethe tangential velocities near the walls must be consistent with the key assumption

underlying wall functions, namely, a logarithmic profile for velocity. Finally, various

choices for implementing the production and dissipation terms (source terms) in the ke

model were discussed, and the implementation which is consistent with the finite

volume method was identified.

4.4 LowReynolds Number ke Models

The lowReynolds number models integrate the governing equations all the wayto the wall and thus obviate the need to make any assumptions about the nature of

turbulence or the velocity profile near solid walls. A wide variety of lowReynolds ke

models can be found in literature (e.g., see Patel et al. 1985). We have implemented

two such models, namely, those proposed by Chien (1982) and Nagano and Tagawa

(1990). Chiens model is one of the first to incorporate nearwall modifications in the

standard ke model. We have chosen Nagano and Tagawas model with an application

to wall heat transfer in mind, since it has been developed for boundary layer flows with

heat transfer (Youssef et al. 1992). This model has been shown to accurately predict

the nearwall limiting behavior of turbulence and the effect of adverse pressuregradient on shear layers. It has been validated for various kinds of wall turbulent shear

flows, e.g., a pipe flow, a flatplate boundary layer, a diffuser flow, a relaminarizing flow,

etc. (Nagano and Tagawa 1990). Also, Youssef et al. (1992) have used this model in

conjunction with a twoequation model for heat transfer (which models the transport

equations for the variance of temperature and its dissipation rate), and have reported

satisfactory results for heat transfer in turbulent boundary layers with different types of

wall thermal conditions. The model can be summarized as follows:

4.4.1 Chiens Model

The equations and the various constants used in the model are as follows:

D(rk)

Dt+ xim) mtsk kxi) P* re*

2mk

y2(4.53)


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D(re)

Dt+ xi

m) mtse kxi) C1f1Pek* C2f2re2

k* 2m e

y2e(*0.5y

)) (4.54)

fm + 1 * e(*0.115y)) (4.55a)

f2 + 1.0 * 0.22e*Ret6

2

(4.55b)

Ret+rk2

me (4.55c)

Cm + 0.09 C1 + 1.35 C2 + 1.8 f1 + 1.0 sk+ 1.4 se + 1.3

The eddy viscosity is computed from Eq. (4.2). The boundary conditions at a noslip wall

are k=0 and e=0.

4.4.2 Nagano and Tagawas Model

This model has been developed for boundary layer flows with heat transfer. The

equations and the various constants used in the model are as follows:

D(rk)

Dt+ xi

m) mtsk kxi) P* re (4.56)D(re)

Dt

+

xim)

mtsek

xi) C1f1P

e

k

* C2f2re2

k

(4.57)

f2 +

1 * 0.3exp* Ret

6.52@ 1 * exp* y)

62 (4.58a)

fm + 1 * exp* y)262

@1 ) 4.1

Re34t

(4.58b)

Ret+ rk2

me (4.58c)

Cm + 0.09 C1 + 1.45 C2 + 1.9 f1 + 1.0 sk+ 1.4 se + 1.3

The eddy viscosity is computed from Eq. (4.2). The boundary conditions at a solid wall

are as follows:


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k+ ky

+ 0 , e + 2m

r ky

2

w

(4.59)

4.5 References

Anderson, D.A., Tannehill, J.C. and Pletcher, R.H. 1984, Computational Fluid Mechanics andHeat Transfer, Hemisphere, New York.

Chien, K.Y. 1982, Predictions of Channel and BoundaryLayer Flows with a LowReynoldsNumber Turbulence Model, AIAA J., Vol. 20, pp 3338.

Hinze, O. 1959, Turbulence, McGrawHill, New York.

Jones, W.P. and Launder, B.E. 1972, The Prediction of Laminarization with a TwoEquationModel of Turbulence, Int. J. Heat Transfer, Vol. 15, pp 301314.

Kim, J., Moin, P. and Moser, R. 1987, Turbulence Statistics in Fully Developed Channel Flowat Low Reynolds Number, J. Fluid Mech., Vol. 177, pp 133166.

Launder, B.E. 1988, On the Computation of Complex Heat Transfer in Complex TurbulentFlows, Int. J. Heat Transfer, Vol. 110, pp 11121128.

Launder, B.E. and Spalding, D.B. 1974, The Numerical Computation of Turbulent Flows,Comp. Meth. Appl. Mech. Eng., Vol. 3, pp 269289.

Launder, B.E., Reece, G. and Rodi, W. 1973, Progress in the Development of aReynoldsStress Turbulence Closure, J. Fluid Mech., Vol. 68, pp 537566.

Lien, F.S. and Leschziner, M.A. 1994, A General NonOrthogonal Collocated Finite VolumeAlgorithm for Turbulent Flow at All Speeds Incorporating SecondMoment

TurbulenceTransport Closure, Part 1: Computational Implementation, Comp. Meth. Appl.Mech. Eng., Vol. 114, pp 123148.

Lumley, J.L. 1978, Computational Modeling of Turbulent Flows, In Advances in AppliedMechanics (ed. C.S. Yih), Vol. 18, pp 123, Academic Press, New York.

Nagano, Y. and Tagawa, M. 1990, An Improved ke Model for Boundary Layer Flows, J.Fluids. Engg., Vol. 112, pp 3339.

Patel, V.C., Rodi, W. and Scheurer 1985, Turbulence Models for NearWall and Low ReynoldsNumber Flows: A Review, AIAA J., Vol. 23, pp 13081319.

Rodi, W. 1991, Experience with TwoLayer Models Combining the ke Model with aOneEquation Model Near the Wall, AIAA Paper 910216.

Shi, Q. and Ribando, R.J. 1992, Numerical Simulations of Viscous Rotating Flows Using aNew PressureBased Method, Comp. Fluids, Vol. 21, pp 475489.

Shyy, W., Thakur, S.S., Ouyang, H., Liu, J. and Blosch, E. 1997, Computational Techniquesfor Complex Transport Phenomena, Cambridge University Press, New York.

Sondak, D.L. and Pletcher, R.H. 1995, Application of Wall Functions to GeneralizedNonorthogonal Curvilinear Coordinate Systems, AIAA J., Vol. 33, pp 3341.


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Speziale, C.G. and Thangam, S. 1992, Analysis of an RNGbased Turbulence Model forSeparated Flows, Int. J. Eng. Sci., Vol. 30, pp 13791388.

TASCflow Theory Documentation 1995 , Advanced Scientific Computign Ltd., Waterloo,Ontario, Canada.

Thompson, J.F., Warsi, Z.U.A. and Mastin, C.W. 1985, Numerical Grid Generation, Elsevier,

New York.

Viegas, J.R. and Rubesin, M.W. 1983, WallFunction Boundary Conditions in the Solution ofthe NavierStokes Equations for Complex Compressible Flows, AIAA Paper 831694.

White, F.M. 1974 , Viscous Fluid Flow, McGraw Hill, New York.

Yakot, V. and Orszag, S.A. 1986, Renormalization Group Analysis of Turbulence. I. BasicTheory, J. Sci. Computing, Vol. 68, pp 151179.

Youssef, M.S., Nagano, Y. and Tagawa, M. 1992 A TwoEquation Heat Transfer Model forPredicting Turbulent Thermal Fields under Arbitrary Wall Thermal Conditions, Int. J. HeatMass Trans., Vol. 35, pp 30953104.

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