The k-ϵ Model, the RNG k-ϵ Model and the Phase-Averaged Model-11

Gautam Biswas

1. IntroductionLaminar flow is well-ordered and is characterized by fluid layers sliding over one another. The flow is well-ordered in the macroscopic sense despite the chaotic motion of molecules at the microscopic level. Turbulent flow is chaotic even at the macroscopic level. Over last two decades, many sophisticated experiments have revealed that the turbulent flows are not as random as it has been believed in the past. Experiments also indicate that there are discrete dramatic events which occur intermittently in wall turbulent flows and that there exists coherence of large structures in free turbulent flows. Coherent structure also exists in the sublayer (Gupta et al., 1971). The following arc the characteristic features of turbulent flow:

1. The irregularity is manifested through complex variations of velocity, temperature, etc. with space and time (fluctuations). The irregular motion is generated due to random fluctuations. It is postulated that the fluctuations inherently come from disturbances (such as, roughness of the solid surface) and they may be either damped out due to viscous damping or may grow by drawing energy from the free stream. At a Reynolds number less than critical, the kinetic energy of flow is not enough to sustain the random fluctuation against the viscous damping and in such cases laminar flow continues to exist. At somewhat higher Reynolds number than the critical Reynolds number, the kinetic energy of flow supports the growth of fluctuations and transition to turbulence is induced.

2. The mixing promotes dissemination of axial momentum in the normal direction and normal momentum in the axial direction which together culminate in more uniform velocity distributions in turbulent duct flows as compared to laminar duct flows (see Figure 1). As such, high transfer rate of momentum, heat, and mass by fluctuating turbulent motion, are practically most important feature of turbulence.

3. Turbulent motion is always three dimensional. Even for a parallel flow, it can be written that the axial velocity component is

u ( y ,t )=u ( y )+u , ( Γ , t )(1)

Figure 1: Comparison of Velocity Profiles in Turbulent Duct Flow for (a) the same mean velocity and (b) the same Pressure Gradient; (i) Laminar (ii) Turbulent.

We use y as a coordinate normal to the predominant flow direction and Γ as any of the three spatial variables.

Even if the bulk motion is parallel, the fluctuation u' being random varies in all directions. Now let us look at the continuity equation

∂ u∂ x

+ ∂u '

∂ x+ ∂ v

∂ y+ ∂ w

∂ y=0

(2)

Since du'/dx ≠0, Equation (2) depicts that y and z components of velocity exist even for the parallel flow if the flow is turbulent. We can write

u ( y ,t )=u ( y )+u' ( Γ ,t )v=0+v ' ( Γ , t )w=0+w ' ( Γ , t ) }(3)

4. Turbulent motion carries vorticity. Turbulent motion carries vorticity which is composed of eddies interacting with each other. At large Reynolds numbers there exists a continuous transport of energy from the free stream to large eddies. From the large eddies a series of increasingly smaller eddies are formed. The smallest eddies dissipate energy and destroy themselves. The smaller eddies are influenced by the strain rate imposed by the large eddies and are continually stretched. In conclusion, it can be said that turbulence consists of a wide spectrum of eddies (Figure 2).

2. Why Do We Need Turbulence Models?

Any flow whether laminar or turbulent, is fully represented by the Navier Stokes equations. The Navier Stokes equations can be solved on a fine enough grid with

Figure 2: Wide Spectrum of Eddy Sizes and Corresponding Fluctuation Frequencies

an exceptionally accurate discretization method so that both the fine scale and large scale aspects of turbulence can be calculated. This is termed as the Direct Numerical Simulation (DNS) of turbulence (Rai and Moin, 1991). The modeling effort and simplifications that are employed in the study of turbulence are the consequences of the difficulty encountered in solving the full Navier Stokes equations on a fine grid. In numerical solutions, where DNS of turbulence is performed, the mesh spacings and the time steps need to be significantly less than those over which appreciable variation of velocity occurs, otherwise details of the evolution will not be correctly reproduced by the numerical prediction. The length-scale-range of the eddies of varying sizes (smallest eddies of the order of mm’s in the domain where the mean velocity is not much greater than 100 m/s) and the time-scale-range of the velocity fluctuations due to the eddying motion cannot be economically resolved by ordinary discretization methods. Therefore the Engineering Problems may be solved using Statistical Calculation Methods. Details of turbulent fluctuations are usually not of interest to engineers anyway. Hence statistical approach is taken and turbulence is averaged out. Statistical quantities are as follows:

ui=ui+u' i , p=p+p ' i ,u= 1t 1

∫t0

t0+t 1

u dt , t 1→ ∞

Introduction of this separation into the Navier-Stokes equations and subsequent averaging leads to the appearance of turbulence correlations (turbulent, or Reynolds stresses). The Reynolds averaged equations are:

∂ ui

∂t+u j

∂ ui

∂ x j=−1

ρ∂ p∂ x i

+ ∂∂ x j

(ν∂ ui

∂ x j−u'

iu'

j)(4)

and

∂ ui

∂ x j=0(5)

The statistical approach has two tasks:

a) to relate the Reynolds stresses u 'i u ' j to the turbulence parameters and to the mean flow field

b) to determine the distribution of the parameters over the flow field

Most approaches employ the eddy-viscosity concept, which is given by

−u'iu

'j=ν t( ∂ ui

∂ x j+

∂u j

∂ x i)−2

3k δij(6)

The symbol vt is the turbulent eddy viscosity which is not a fluidproperty but depends strongly on the state of turbulence. Hence, vt may vary significantly from one point in the flow to another and also from flow to flow.

The term involving the Kronecker delta δ ijin equation (6) is a seemingly unfamiliar addition to the eddy-viscosity expression. It is necessary to make the expression applicable also to normal stresses (when i = j). The first part of (6) involving the velocity gradients would yield the normal stresses

u '2=−2 ν t∂ u∂ x

, v ' 2=−2ν t∂ v∂ y

,w '2=−2 ν t∂ w∂ z

whose sum is zero because of the continuity Equation (5). However, all normal stresses are by definition positive quantities, and their sum is twice the kinetic energy k of the fluctuating motion:

k=12(u'2+v'2+w ' 2)

Inclusion of the second part of the eddy viscosity expression (6) assures that the sum of the normal stresses is equal to 2k. The normal stresses act like pressure forces. The turbulent kinetic energy k is a scalar quantity, the second part of (6) constitutes a pressure. Therefore, when (6) is used to eliminate u 'i u ' j in the momentum Equation (4), this second part can be absorbed by the pressure-gradient term so that in effect the static pressure is replaced as unknown quantity by the pressure p+2/3 k . Therefore the appearance of k in Equation (6) does not necessitate the determination of k. It is the

distribution of the eddy viscosity ut that has to be determined.

3. The k-ϵ ModelThe turbulent viscosity in Equation (6), vt is computed from a velocity scale (k1/2) and a length scale (k3/2/ϵ ) which are predicted at each point in the flow via solution of the following transport equations for turbulent kinetic energy (k) and its dissipation rate (ϵ):

∂ k∂ t

+ui∂ k∂ xi

= ∂∂ x i

( ν t

σ k

∂ k∂x i

)+νt ( ∂ ui

∂ x j+

∂ u j

∂x i) ∂ ui

∂ x j−ϵ (7)

∂ ϵ∂ t

+ui∂ ϵ∂ x i

= ∂∂ xi

( ν t

σϵ

∂ ϵ∂ xi

)+C 1ϵϵk

G−C2 ϵϵ2

k(8)

where G is the generation of k and is given by

G=ν t( ∂ ui

∂ x j+

∂u j

∂ xi) ∂ ui

∂ x j(9)

The turbulent viscosity is then related to k and ϵ by the expression

ν t=C μk2

ϵ(10)

The coefficients Cμ, C1 ϵ, C2 ϵ, σ k and σ ϵ, are constants which have the following empirically derived values

Cμ=0.09 ,C1 ϵ=1.44 , C2 ϵ=1.92 , σk=1.0 , σϵ=1.3

This is the central concept for a family of two equations models (Rodi and Spalding, 1970; Jones and Launder, 1972; Launder and Spalding, 1974) where the equation for turbulent kinetic energy determines the velocity scale. The two equation models are quite successful and have become very popular for engineering applications. With only little modification, they are able to simulate a large variety of flows with reasonably good degree of accuracy. A comprehensive review of different variety of two equation models is available in open literature (Nallasamy, 1987).

Following the philosophy of momentum equation, the thermal energy or species concentration conservation equation can be written as

∂T∂ t +ui

∂T∂ x i

=∂

∂ x i (α ∂ T∂ xi )+S(11)

where T is temperature and S is the source term. Reynolds decomposition suggests:

ui=ui+u' i ,T=T +T '

and the time averaged equation becomes

∂ T∂ t

+ui∂T∂ x i

= ∂∂ x i [α ∂ T

∂ x i−u'

iT']+S (12)

In direct analogy to the turbulent momentum transport, the turbulent heat or mass transport is often assumed to be related to the gradient of the transported quantity

−u'iT

'=αt∂ T∂ x i

(13)

where α t is the turbulent diffusivity of heat or mass. Like eddy viscosity, α t is not a fluid property but depends on the state of the turbulence. In fact, the Reynolds analogy between heat and momentum transport suggests

α t=v t

σ t(14)

The quantity σ t is called turbulent Prandtl or Schmidt number.

Experiments have shown that σ t varies very little across any flow. Its variation from flow to flow is also small. Therefore many models make use of σ tas a constant. For the flow of air, a value of 0.9 may be taken. It should be pointed out that buoyancy and streamline curvature affect σ t .

3.1. Non-dimensionalization of the Governing Equation

Here we shall write bar on the velocity and pressure. All barred quantities are to be considered as time averaged values. Let us consider the following conservation equations:

Continuity equation

∂ u∂ x

+ ∂ v∂ y

+ ∂ w∂ z

=0 (15)

Momentum equations We shall write only x momentum equation and make it dimensionless. Other two momentum equations will be similar

( DuDt )=−1

ρ∂ p∂ x

+ ∂∂ x [2 (ν+ν t )

∂ u∂ x

−23

k ]+∂∂ y [ ( ν+ν t ) ( ∂u

∂ y+ ∂v

∂x )]+∂∂ z [( ν+νt )( ∂ w

∂ x+ ∂u

∂ z )](16)

Energy equation

( DTDt )= ∂

∂ x [( α+αt ) ∂ T∂ x ]+ ∂

∂ y [( α+αt ) ∂ T∂ y ]+ ∂

∂ z [ ( α+α t )∂ T∂ z ](17)

k -equation

( D kDt )= ∂

∂ x ( ν t

σk

∂ k∂ x )+ ∂

∂ y ( ν t

σk

∂k∂ y )+ ∂

∂ z ( ν t

σk

∂ k∂ z )+G−ϵ (18)

ϵ -equation

( DϵDt )= ∂

∂ x ( ν t

σϵ

∂ ϵ∂ x )+ ∂

∂ y ( ν t

σ ϵ

∂ ϵ∂ y )+ ∂

∂ z ( ν t

σϵ

∂ ϵ∂ z )+C1 ϵ

ϵk

G−C2 ϵϵ 2

k(19)

Production by shear

G=ν t( ∂ ui

∂ x j+

∂u j

∂ xi) ∂ ui

∂ x j

or,

G=ν t[2( ∂ u∂ x )

2

+2( ∂ v∂ y )

2

+2( ∂ w∂ z )

2

+( ∂ u∂ y

+ ∂ v∂ x )

2

+( ∂ v∂ z

+ ∂ w∂ y )

2

+( ∂ w∂ x +

∂ u∂ z )

2 ](20)

The dimensionless variables may be defined as

U = uU 0

, V= vU 0

, W = wU 0

, X= xH

,Y= yH

,Z= zH

P=p−p0

ρU 02 , t= t

H /U 0, θ=

T−T ∞

T w−T ∞

k n=k

U 02 , ϵ n=

ϵU 0

3/ H, v t ,n=

v t

v, αt , n=

α t

α

Finally, we get

Continuity equation

∂ U∂ X

+ ∂V∂Y

+ ∂ W∂ Z

=0 (21)

Momentum equation The compact form of the momentum equation may be written as:

DU j

Dt=

−∂(P+2kn

3 )∂ X j

+ 1ℜ ∙ ∂

∂ X i [ (1+ν t ,n )( ∂U i

∂ X j+

∂ U j

∂ X i)](22)

Where

ℜ=U 0 H

ν

Energy equation

DθDt

= 1RePr [ ∂

∂ X (1+αt ,n ) ∂θ∂ X

+ ∂∂ Y (1+α t , n ) ∂θ

∂ Y+ ∂

∂ Z (1+α t ,n ) ∂ θ∂ Z ](23)

k -equation

Dkn

Dt= 1

ℜ∂

∂ X [ ν t , n

σk

∂ kn

∂ X ]+ 1ℜ

∂∂Y [ ν t , n

σk

∂ kn

∂Y ]+ 1ℜ

∂∂ Z [ ν t ,n

σ k

∂ kn

∂ Z ]+Gn−ϵ n(24)

ϵ -equation

Dεn

Dt= 1

ℜ∂

∂ X [ ν t , n

σ ϵ

∂ ϵn

∂ X ]+ 1ℜ

∂∂ Y [ νt ,n

σϵ

∂ ϵ n

∂ Y ]+ 1ℜ

∂∂Z [ νt ,n

σϵ

∂ ϵ n

∂ Z ]+C1 ϵϵ n

knGn−C2 ϵ

ϵ n2

kn(25)

Production

Gn=νt ,nℜ [2( ∂U

∂ X )2

+2( ∂ V∂ Y )

2

+2( ∂W∂ Z )

2

+( ∂ U∂ Y

+ ∂V∂ X )

2

+( ∂ V∂ Z

+ ∂ W∂ Y )

2

+( ∂W∂ X

+ ∂ U∂ Z )

2](26)

Finally v tand α t should be nondimensionalized in the following way:

ν t ,n=Cμ ℜk n

2

ϵ n(27)

and

α t , n=Cμ RePr kn

2

σ t ϵ n(28)

Equations (21), (22), (23), (24), (25) and (26) can be solved in the domain of interest in order to simulate a turbulent flow situation. For a given flow domain, the grid system has to be set and then strategies for the application of boundary conditions are to be worked out. In this event we shall consider turbulent flow in a plane channel and discuss the implementation of turbulence modeling through the k- ε family of eddy viscosity models.

3.2. Grid System Used

Computational domain is divided into a set of rectangular cells (Figure 3) and a staggered grid arrangement is used such that the velocity components are defined at the center of the cell faces to which they are normal (Figure 4). The pressure and temperature arc defined at the center of the cell. In such an arrangement, pressure difference between two adjacent cells is the driving force for the velocity component located between the interface of these cells. The pressure field will accept a reasonable pressure distribution for a correct velocity field. Another important advantage of such a grid system is that transport rates across the faces of the control volumes can be computed without interpolation of velocity components.

4. Boundary Conditions for Turbulent Flow

The inlet conditions for velocity and temperature can be specified using the profiles we want to use. The turbulent kinetic energy kn and its dissipation rate are calculated from the value of turbulence intensity specified at the inlet.

Figure 3: The Computational Domain and the Grid Spacing

The inlet boundary conditions can be specified as following:

¿

Where , uτ ,n=Cμ

14 k

12

k n=1.5 I2

ϵ n (Y )=kn

32 C μ

34

χYforY <( λ

χ )¿

kn

32 Cμ

34

λ Y pforY >( λ

χ ) }(30)

where u τ ,nis non-dimensional friction velocity, Y+ is given by yuτ

ν, I is turbulent

intensity, χ=0.42which is known as von Karman constant, λis a constant prescribing ramp distribution of mixing length in boundary layers and equal to 0.09 and E= 9.0.

Often the turbulence quantities at the inlet of the domain are not known and one is forced to estimate these quantities closest to the actual situation. In general, the inlet turbulence intensity and characteristic length are a function of the flow details and the

geometry at the upstream. A very quiescent upstream flow might yield an inlet turbulence intensity of 2- 3%. If the upstream flow

Figure 4: Three-dimensional Staggered Grid Showing the Locations of the Discretized Variables

involves flow over rough edges or turning, the inlet intensity may be as high as 10 - 15%.

The boundary conditions for the outlet and the confining walls (both no-slip and free-slip) as shown in Figure 5, are as follows:

Outlet:

∂ f∂ X

=0 ; f =(U , V ,W ,θ , kn , ϵ n )(31)

Free-slip (symmetric) wall (plane I, III):

W =0 , ∂ f∂ Z

=0; f = (U ,V ,θ , kn , ϵ n )(32)

Free-slip wall (plane II):

V=0 , ∂ f∂Y

=0 ; f =(U ,W , θ , kn , ϵ n) (33)

no-slip wall (plane IV):

U =V=W =0 , θ=1(34)

The wall functions due to Launder and Spalding (1947) are used to mimic the near wall region in boundary-layers. For Y p

+¿≥ ¿ Equ.(63)

τ w ,nx =

U p Cμ1 /4 kn , p

1/2 χln¿¿

Figure 5: Cross-section of the Computational Domain

τ w ,nx =

W p Cμ1 /4 kn , p

1/2 χln¿¿

where Y p+¿=Y pℜCμ

1/4 kn , p1/ 2¿. The subscript p refers to the first grid point adjacent to the wall.

The production rate of kn and the average dissipation rate of ϵ nover a near wall cell for Equation (24) at point p are computed from the following

Gn , p=(√(τ w, nx )2+(τw ,n

z )2) (√U p+W p2

2 )/Y p(37)

ϵ n , p=1

Y p∫

0

Y p

ϵ dY=Cμ

3 /4 kn , p3 /2

χ Y pln¿

Instead of using Equation (25) near the wall,the ϵ n at point p is computed from

ϵ n , p=Cμ

3 /4 kn , p3/2

χ Y p(39)

In order to calculate the wall functions for temperature, the heat flux at the wall can be expressed in the following way: for Y p

+¿≥ Equ .¿ ¿)

qw ,n=(θw−θ)Cμ

1/ 4 kn , p1/ 2

¿¿

where, qw ,n is the nondimensional heat flux at the wall given by

qw ,n=qw

ρ c p U0 ( Tw−T∞ )(41)

It may be mentioned that Pfnin Equation (40) is the Pee function which may be written as (see Dutta and Acharya, 1993)

Pfn=9[ Pr

σ t−1] [ Pr

σ t ]−14(42)

5. Details of Near-wall Treatment in Connection with the Momentum Equations

A very thin layer next to the wall behaves like a near wall region of laminar flow. The

layer is known as viscous sub-layer and its velocities are such that the viscous forces dominate over the inertia forces. No turbulence exists in it. As it has been stated that in the near wall region, inertial effects are insignificant and we can write

ν ∂2u∂ y2−

∂ u' v '

∂ y=0(43)

which can be integrated as

ν ∂u∂ y

−u' v '=constant

Again, as we know that the fluctuating components vanish near the wall, the shear stress on the wall is purely viscous and it follows

ν ∂u∂ y|

y=0=τw / ρ

or

u−0y−0

=τw

ρ ν=

uτ2

ν

where u τ❑

is the friction velocity ¿√τw / ρ . The friction velocity is chosen as the characteristic velocity or velocity scale. Once u τ

❑ is specified, the structure of the sub layer is specified.

However, from the above mentioned expression, it is possible to write

yuτ

ν= u

uτ(44)

Hence a non-dimensional coordinate may by defined as Y+¿=

yuτ

v¿and we write down

the variation of the nondimensional velocity within the sublayer as

U+¿=Y+¿(45) ¿¿

However, in the turbulent zone, the universal velocity profile may be written as

uuτ

= 1χ [ ln yu τ

ν−l n β]

Where χ is the von-Karman constant. We can also write

U+¿=A1 ln Y+¿+D1( 46)¿ ¿

Where β is absorbed in D1

The constants were determined from experiments. For smooth ducts, it has been observed that,A1=2.5 and D1=5.5, which leads to

U+¿=2.5 ln Y+¿+5.5¿ ¿

or

U+¿=2.5ln ¿¿

where E = 9.0 for smooth walls. Finally, for the turbulent zone, the log-law is defined as

U+¿= 1

χ ln ¿¿

Figure 6 suggests that the log-profile and linear profile meet at the buffer zone. If U+¿ ¿ is substituted by Y+ in Equation (47), we get

Y+¿=1

χ ln ¿¿

Therefore, Y +¿=Eqn .(63)¿ is the location where the log-profile and the linear profile meet.

As we have seen earlier, the velocity is scaled with uτ❑ because the turbulent velocity

scale is presumably related tou τ❑. Next, the theoretical basis of expressing u τ

❑ in terms

of other input parameters related to turbulent flows will be discussed.

The local equilibrium between the production and the rate of dissipation of turbulent kinetic energy near the wall gives rise to

−ρ u' v ' ∂ u∂ y

=ρ ϵ (48)

Just outside the viscous sublayer, the shear stress (still equal to τ w) will be produced entirely by turbulent eddies (Bradshaw, 1971) so that

−ρ u' v '=τ w(49)

From this we can write

u' v '=τw

ρ=ν t

∂ u∂ y

(50)

Invoking ν t=cμ k2/ϵ in the above expression, we get

−u' v '=cμ k2

ϵ∂ u∂ y

(51)

Substituting the value of ∂u∂ y in terms of the dissipation rate and the fluctuating

velocity, we get

(−u ' v ' )=cμ k2 1(−u' v ' )

or

(u' v ')2=cμ k2

Figure 6: Universal Velocity Distribution for Turbulent Flows

or

τ w / ρ2=cμ k2

Invoking uτ √ τw

ρ,we obtain

uτ=c μ1 /4 k 1/2(52)

If uτ and k both are non-dimensionalized by the reference velocity U 0, we shall get

uτ ,n=cμ1/4 kn , p

1/2 (53)

Now Equations (35) to (39) need little more explanation. At a solid boundary, the no-slip condition applies, so that both mean and fluctuating velocities are zero. However, the dissipation rate ϵ is finite and requires special attention. If Y p

+¿≥ 11.63¿, the universal law of the wall can be written as:

U p

+¿= 1χ ln ¿¿

WhereY p

+¿=Y pℜCμ1/4 kn ,p

1 /2¿

We can also writeup

u τ= 1

χln ¿

or

U p U0

Cμ

14 k n

12 U0

= 1χ

ln¿

or

Cμ1/4 kn

1 /2=U p χln ¿¿

or

(Cμ1/4 kn

1 /2)2=U p χ Cμ

1/4 kn1/2

ln ¿¿

or

(u τ ,n)2=

U p χ C μ1/4 kn

1 /2

ln¿¿

or

τ w ,nx =

U p χCμ1 /4 kn

1/2

ln¿¿

which is Equation (35) prescribing the wall function forthe shear stress. Similarly, z component of shear stress is given by

τ w ,nz =

W p χCμ1/4 kn

1/2

ln¿¿

This is the basis of using Equation (36).

The mean generation rate at the near wall point to be used in Equation (24) while applying the equation for the point P is

Gp=1y p∫0

yp {ν t ( ∂u i

∂ x j+

∂ u j

∂ x i )∂ ui

∂ x j }dy

or

Gn , p=¿¿

This is the Equation (37).

Before writing down the mean dissipation rate for Equation (25), we have to find out the dissipation rate at point P. Instead of Equation (25), this will be the representative expression for ϵ at point P.

From local equilibrium, we can write

ϵ p=τw

res

ρ∙ ∂

∂ y(V res )

or

ϵ p=uτ2 ∂

∂ y(V res)

Following the log-law it can be written as

V res

u τ= 1

χln¿

or∂V res

∂ y=

uτ

χy

Finally, ϵ p becomes equal to uτ

3

χ yp

ϵ n , p U 03

H=

(U τ ,n)3U 0

3

χ Y p Hor

ϵ n , p=Cμ

3 /4 kn3/2

χ Y p

which is Equation (39).

Next, we shall derive the expression for average dissipation rate (average over the viscous sub layer where turbulent kinetic energy is completely dissipated). This expression will be used in Equation (24).

ϵ p=1y p∫0

y p

ϵ p dy

We can readily write

ϵ p=1y p∫0

y p Cμ3 /4 k p

3 /2

χydy

or

ϵ p=1y p

{[Cμ

34 k p

32

χln y ]

y0

y p}or

ϵ p=1y p

{Cμ

34 k p

32

χ ( ln y p−ln y0)}or

ϵ p=1y p

{Cμ

34 k p

32

χ ( lny pur

ν −ln β )}

The quantity β can be absorbed in arbitrary constant D2 and we get

ϵ p=1y p

¿

or

ϵ p=1y p

¿

or

U 03 ϵ n , p

H= 1

Y p H¿

Finally

ϵ n , p=Cμ

34 k n , p

32

χ Y pl n¿

which is basically Equation (38) prescribing the wall function for the dissipation rate.

5.1. Prediction of the Skin Friction Coefficient

Since

τ w ,nx =

U p Cμ1 /4 kn , p

1/2 χln¿¿

the local skin friction coefficient may be expressed as

C f , x=τ w

x

1/2(ρ U02)

=2 τw ,nx

So,

C f , x=2 τ w ,nx (54)

6. Details of Wall Function Treatment for the Energy Equation

In turbulent flows, the fluid side heat flux near the wall is assumed to be same in the turbulent zone and it is invariant in y (normal) direction. On a y-normal surface, ∂ T /∂ y is negative for a positive heat flux. We can write

−k t∂ T∂ y

=qw

or

−σ t∂ T∂ y

=qw

ρC p

In analogy with the turbulent viscosity ν t= χ 2 y2( ∂u∂ y

),we can also write

α t= χT2 y2(|−∂ T

∂ y |) .The negative sign appears due to the temperature profile that has

been considered here. This leads to

χT2 y2( ∂ T

∂ y )2

=qw

ρC p

or

∂ T∂ y

=±√ qw

ρ C p∙ 1

χT y(55)

Following the concept of friction velocity, we introduce friction temperatureas

T τ=qw

ρ C pu τ

Plugging this in Equation (55) and considering that negative ∂ T∂ y yields positiveqw, we

get

−∂ T∂ y

=√T τu τ

χT

1y

or

−T=√T τ uτ

χTln y+C

In the above expression, the wall temperature condition cannot be satisfied with a finite constant and we use at y= y0, T=T w based on the assumption that y0 is of the order of the viscous sublayer. This produces

−T=√T τ uτ

χT( ln y−ln y0 )−T w

or

T w−T=√T τ uτ

χT(ln

yu τ

v− ln π )

Where π is any arbitrary constant. From the above expression, it is possible to write

(T w−T )ρ C pu τ

qw= 1

χ C¿

In the above expression πis absorbed in E¿. However, the above equation can be further reduced to

(T w−T )ρ C p√τ w /ρqw

= 1χC

¿

or

T+¿= 1

χC¿ ¿

Equation (56) is known as universal temperature profile for the turbulent flow with χC=0.46. This equation can be compared with the universal velocity profile given by Equation (47).

Jayatilleke (1969) combined Equations (56) and (47) to obtain

T +¿=¿¿

where P is afunction of E and E¿and σ t= χ / χC is effectively the turbulent Prandtl number for heat transport. It is sometimes convenient to rewrite Equation (57) as

(T w−T )ρ C pu τ

qw=¿

{(T w−T ∞ )−(T−T ∞ )} (T w−T ∞ ) ρ Cp u τ ,n U 0

( T w−T ∞ ) qw

=¿

or

(θw−θ)C μ1 /4 k n

1 /2

qw ,n=

σ t

χl n¿

or

qw ,n=(θw−θ)Cμ

1/4 kn1/2

σ t

χln¿¿

This is the same as Equation (40) stated earlier and the parameter Pf ,n has already been explained through Equation (42).

Finally, the prediction parameter from the solution of energy equations is the Nusselt number, Nu(x,z). This can be evaluated in the following way:

N u=hHk

=h(T w−T B)×vα

∙ 1ρC p

∙ 1U0(Tw−T B)

∙U 0 H

ν

Where T B is the average bulk temperature

or

Nu=qw Pr ℜ

ρ Cp U0(T w−T B)

or

N u=qw

ρ Cp U0(T w−T B)∙ 1(θw−θB)

∙Pr ℜ

or

N u=qw ,n Pr ℜ(θw−θB)

(58)

7. Method of Solution

Providing a physical model to represent turbulent transport rates is only one part of the problem of computing turbulent flow. The other lies in the procedures adopted for discretizing and solving the resulting nonlinear partial differential equations describing the convective transport process of interest.

Two widely used numerical methods are being referred for discretizing and solving the non-linear partial differential equations. These are MAC (Marker and Cell) method due to Harlow and Welch (1965) and SIMPLE (Semi Implicit Method for Pressure Linked Equations) method due to Patankar (1980).

The second order terms in the differential equations are discretized by using central differences. The first order convective terms may be discretized by different methods depending on requirements of accuracy and computational speed. Some popular choices are Weighted average scheme (Hirt et al 1975), QUICK scheme (Leonard, 1979) and third order upwind differencing scheme of Kawamura et al (1986).

Both in MAC and SIMPLE methods, velocities for the next time step are calculated using the convective accelerations, viscous diffusions and pressure gradients at the

previous time level. Usually in MAC method this part is executed through an explicit calculation whereas in SIMPLE method the evaluation procedure is implicit. The velocities for the next time step may not necessarily lead to a flow field with zero mass divergence in each cell. This implies that at this stage the pressure distribution is not correct. The corrected pressures are obtained from the solution of a Poisson equation for pressure-correction. The source term of the Poisson equation is obtained through the continuity equation. Interested readers are suggested to refer to Biswas (1995) for further details of the solution algorithm concerning the Poisson equation for pressure-correction. It may be mentioned that Patankar (1980) recommends an alternative SIMPLER algorithm for SIMPLE-Revised, which uses pressure corrections only to change the velocity field. Pressures are then computed from a Poisson equation for pressure. The SIMPLER method reduces the computational time considerably.

8. Remarks on k-ϵ Model and Wall Function Treatment

It is understood that the wall function approach greatly reduces the storage and computational time since it avoids the tricky calculations in the viscous sub layer region. However, it is also well-known that the wall-function approach is used on a rather simple intuition and it is not always consistent with complex physical picture near the wall. Over and above, in k−ϵ family of models the non-local property of turbulence is accounted usually by one turbulent length scale that is dictated by a model equation which is derived on the basis of closure assumptions for most parts of the governing processes. Such models need several empirical coefficients and for obvious reasons such coefficients cannot be universal constants. Despite these limitations, many timesk−ϵ family of eddy viscosity models have produced fairly acceptable results in predicting flows which have predominantly small scale turbulence structures and which can be considered to be interpolates of basic experiments for which the coefficients have been derived. Zhu, Fiebig and Mitra (1993) and Deb, Biswas and Mitra (1995) have computed three dimensional turbulent flows with longitudinal vortices embedded in the boundary layer on a channel wall. Although the behavior of k−ϵ model is not known for such highly vortical flows, comparison between the measurements due to Pauley and Eaton (1988 a,b) and the computed results shows that the interaction of the longitudinal vortices with boundary layer within a turbulent channel flow is captured reasonably well by the numerical simulation. A sample comparison has been shown in Figure 7. Lee, Ryou and Choi (1999) have computed heat transfer in a similar geometrical configuration. They have concluded that the predictions due to the Reynolds Stress Model (Gibson and Launder, 1978) are more accurate than those due to the standard k−ϵ model.

9. The RNG k−ϵ Turbulence Model

The RNG based k−ϵ turbulence model provides with both accuracy and efficiency in the modeling of turbulent flows. This model follows the two equation turbulence modeling framework and has been derived from the original governing equations for fluid flow using mathematical techniques called Renormalization Group (RNG) method due to Yakhot and Orszag (1986). The RNG model provides a more general and fundamental model and is expected to yield improved predictions of near wall

flows, separated flows, flows in curved geometries and flows that are strained by effects such as impingement or stagnation. Time dependent flows with large-scale motions, as in turbulent vortex shedding are well predicted by RNG k−ϵ turbulence model.

9.1. Basic Features of the RNG k−ϵ as Compared to Standard k−ϵ Model

The constants and functions in the RNG model are derived rigorously from first principles with less number of approximations.

Near wall behavior of the flow, when important, can be resolved without the use of wall functions. Thus adhoc damping functions, used in conventional k−ϵ are not required. If the near wall flows are unimportant, however, wall functions can be used with the RNG k−ϵ model.

In RNG k−ϵ model, more terms appear in the dissipation rate in transport equation, including a rate of strain term, which is important for treatment of non-equilibrium effects and flows in rapid distortion limit such as separated flows and stagnation flows.

Figure 7: Vector Plots of Secondary Flow: ℜH

2=67000

In standard k−ϵ model high generation rates are predicted in the regions of high strain rate or streamline curvature (Launder and Spalding, 1974). This excess of turbulence energy creates high level of turbulent viscosity. The high turbulent viscosity tends to over predict mixing and delay separation.

9.2. Basis of the RNG k−ϵ Model

The RNG method is applicable to scale invariant phenomena lacking externally imposed length and time scales. For turbulence, this signifies that the method can describe the small scales which should be statistically independent of the external initial conditions and dynamical forces that create them through different instability

phenomena. The RNG method gives a theory of the Kolmogorov equilibrium range of turbulence, comprising the so-called inertial range of small-scale eddies whose energy spectrum follows the famous Kolmogorov lawE(k ) k−5/3.

9.3. The RNG Theory

The turbulent eddies range from energy containing eddies of size L, the integral scale, down to eddies of size L/ℜ3 /4 , where ℜ=vrms L /v is the Reynolds number. Still smaller eddies have very low energy due to viscous dissipation. Therefore, the accurate solution of the three dimensional Navier-Stokes equations for a turbulent flow needs grid-mesh of the order of ℜ9/4. As it has been discussed earlier, if Re is large, these computational requirements are exceedingly high. The RNG method reduces the computational requirement by eliminating the inertial range eddies from the equations of motion, yielding equations for averaged flow quantities at the integral scale.

Figure 8 shows the scales of effective excitation in turbulence. They range from the low wave number k 0=2π /L(large scale eddies) to high wavenumber viscous cutoff A (corresponding to smallest energy containing eddies). The RNG method removes a narrow band of modes near Λby representing these modes in terms of lower modes in the interval k 0<k< Λ e−l ( l≪1 ) .When this narrow band of modes is removed, the resulting equations of motion for the remaining modes is a modified system of Navier-Stokes equations. The equations are dictated by a modified viscosity. The first band of modes is removed from the dynamics and the process of removal of degrees of freedom is repeated (also see Choudhury,1993). In this way the RNG method enables computation of Navier- Stokes equations on relatively coarser grids at high Reynolds numbers.

9.4. Determination of Effective Viscosity

The scale elimination procedure enables one to develop an equation for the variation of effective viscosity which is given by

dvdl

= Aϵ l3

ν (l)2 (59)

Where A is a constant.

Figure 8: Scales of Effective in Turbulence

Integration of Equation (59) over the eddy length scale l and noting that v = vo when l = ld (it may be mentioned that Id is the Kolmogorov dissipation scale L/ ℜ3 /4) yields

ν ( l )=νo[1+3 A ϵ

4 vo3 ( l4−ld

4 )] for l>ld(60)

Equation (60) gives an interpolation formula for v(l) between the molecular viscosity vo (at dissipation scales) and the high Reynolds number limit of L≫l d . In the high Reynolds number limit it can be shown that Equation (60) produces

νeddy=(0.094 L)2|∇u|(61)

Equation (61) is very close to Prandtl’s mixing length model. Considering that the total kinetic energy contained in the inertial range eddies of scale less than L isk=0.71 ϵ 2/3 L2 /3, Equation (61) becomes

νeddy=Cμk2

ϵ(62)

with Cμ=0.0845. Note that a similar constant in k−ϵ model is equal to 0.09. The RNG k−ϵ model extends the prediction of effective viscosity beyond this high Reynolds number limit, computing the eddy viscosity, νeddy using the interpolation formula. Equation (60) is rearranged in terms of k and ϵ . The simple algebraic form of the relationship between νeddy and k&ϵ is given by

νeddy=νo[1+√Cμ

vo

k√ϵ ]

2

(63)

Equation (63) allows extension of the model to low-Reynolds number and near wall

flows.

9.5. Model Equations

The Reynolds averaged Navier Stokes equations are governing equations for mo-mentum involving turbulent stresses which are modeled in the following way

∂ ui

∂t+ ∂

∂ x j(ui u j )=

−1p

∂ p∂ x i

+ ∂∂ x j [veddy ( ∂ ui

∂ x j+

∂ u j

∂ x i )](64)

The effective viscosity is calculated through Equation (63). Equation (63) requires the values of k and ϵ which are determined from the following transport equations

∂ k∂ t +ui

∂ k∂ xi

=ν t S2−ϵ +

∂∂ x i (α νt

∂ k∂ xi ) (65 )

and

∂ ϵ∂ t

+ui∂ ϵ∂x i

=C ϵ1ϵk

ν t S2−C ϵ 2

ϵ2

k−R+ ∂

∂ x i (α ν t∂ ϵ∂ xi )(66)

The quantity α is inverse Prandtl number for turbulent transport, evaluation procedure of α will be discussed in a subsequent paragraph. The turbulent viscosity is given as ν t=(νeddy−νo) and the term R is given by

R=2 νo S ij

∂u i

∂ x j

∂ ui

∂ x j(67)

This term is expressed in the RNG k−ϵ model as

R=Cμ η3(1−η/η0)

1+ β η3ϵ 2

k(68)

where η=Skϵ and S2=2 Sij Sijis the magnitude of the rate-of-strain. The RNG theory

gives the values of the constants as C ϵ 1=1.42 , C ϵ2=1.68 ,∧α=1.39 . Similar constants in standard k−ϵ model assume the values of C ϵ 2=1.92 ,∧α=1.00 .

Equations (65) and (66) can be applied in low Reynolds number regions (e.g., the near

wall region) because of the dependence of the turbulent viscosity νt on the local Reynolds number through the relationship given in Equation (63). The quantity α is inverse Prandtl number and can be expressed as (Yakhot and Orszag, 1986)

| α−1.3929α 0−1.3929|

0.6321

| α+2.3929α 0+1.3929|

0.3679

=vo

v eddy(69)

with α 0=1. For high-Reynolds number, fully developed turbulence, α=1.3929 and the turbulent Prandtl number,Pr t=0.7179. Thus the RNG theory allows the k and ϵ transport equations to be applied in low Reynolds number regions without recourse to wall functions. Because vt and α vary smoothly with effective Reynolds number from molecular values to fully turbulent values, the RNG k−ϵ transport equations include a natural damping effect in the near wall region.

9.6. Choice of Grid for RNG k−ϵ Model

The RNG model can be applied on a similar grid which uses k−ϵ model. For computational economy, wall functions can be applied in conjunction with RNG k−ϵ model on a grid that does not extend into the viscous sublayer. This approach is particularly suitable if the problem is not dominated by near-wall heat or momentum transfer. Wake flows, massively separated flows are examples of such flows. If the physics of the problem is such that the near wall effects are dominating then the use of wall functions should be avoided. The RNG k−ϵ model in such cases is extended into the viscous sublayer of the turbulent boundary layer. Preferably one should choose to resolve the near wall region down to a y+ value of 4 or 5 to achieve a high degree of accuracy.

10. Phase-Averaged Model of Turbulence

Flow past bluff bodies is a configuration of great importance in engineering ap-plications. The wake of bluff objects is unsteady, the near-wake being strongly periodic and driven by the shed vortices. The nuances of the wake structure depend on the interaction between the periodic motion and the random turbulent fluctuations. Such complex flow structures are seen in quite a few practical situations. The present model is aimed at solving the problems, which have both large scale and small scale structures. At high Reynolds numbers, three-dimensional random turbulent fluctuations are superimposed on the unsteady periodic motion. The random motion represents small scales of turbulence and can be simulated by a stochastic model. In the wake of a bluff-body flow, the time varying component ϕ (for example, velocity and pressure) may be written as the combination of global mean component ϕ , a periodic component ~ϕand a random component ϕ ' (Hussain, 1983). This can be written mathematically as (Figure 11.9; after Bosch and Rodi (1998))

ϕ ( x i , t )=ϕ ( x i )+~ϕ ( x i , t )+ϕ ' ( xi , t )=⟨ ϕ ⟩ ( xi , t )+ϕ' ( xi , t ) (70)

Together, the time mean and the periodic part are called the phase-averaged or ensemble-averaged component ⟨ ϕ ⟩ (t), which is resolevd in the numerical calculation. Based on this idea, one can adopt the following viewpoint (Bosch and Rodi 1996): The Reynolds averaged Navier-Stokes equations determine the phase- averaged velocity and pressure (quantities inside ' ⟨ ⟩ 'in Equation (70)). The eddy viscosity arising from the Boussinesq approximation can now be associated with the random fluctuations (quantities marked by ' in Equation 70). The eddy viscosity in turn can be determined by the transport equations: one for the turbulent energy level k and one for the rate of energy dissipation ϵ . This has come to be called as the k−ϵ family of models.

In the present study, three high Reynolds number versions of the k−ϵ model have been used to determine the eddy viscosity and hence the Reynolds stress tensor ⟨ ui ' u j ' ⟩ .All the three models relate eddy viscosity ⟨ v t ⟩ to turbulent kinetic energy ⟨ k ⟩ and the rate of dissipation ⟨ ϵ ⟩. The three models used are (1) standard k−ϵ (Launder and Spalding, 1974), (2) Kato-Launder (hereafter referred as KaLa) modified k−ϵ (Kato and Launder, 1993) and (3) RNG k−ϵ (Yakhot and Orszag, (1986); Yakhot et al., (1992)).

Figure 9: Triple decomposition of a turbulent, unsteady signal.

10.1. Phase-Averaged Flow EquationsAt high Reynolds numbers, the wake of a square cylinder can be visualized as the superposition of the three-dimensional (3D) turbulent fluctuations over a two-dimensional (2D) flow field. This can also be viewed as a three-dimensional unsteady periodic flow field. An instantaneous quantity ϕ can be, therefore, described by the summation of the phase-averaged value and the stochastic fluctuation as expressed in

Equation (70). Assuming incompressible flow, the phase- averaged continuity and momentum equations can be written as (Saha, Biswas and Muralidhar, 1999)

∂ ⟨ui ⟩∂ xi

=0 (71)

∂ ⟨ui ⟩∂ t

+∂ [ ⟨ui ⟩ ⟨u i ⟩ ]

∂ x i= 1

ρ∂ ⟨ p ⟩∂ xi

+ ∂∂ x i

[ ν∂ ⟨u i ⟩∂ x i

− ⟨ui ' u j ' ⟩](72)

To enforce closure, the Reynolds stress tensor, ⟨ ui ' u j ' ⟩ in the momentum equations are to be suitably modeled. The alternatives available for evaluating ⟨ ui ' u j ' ⟩ are far greater compared to the choices for the numerical procedure. Three turbulence models of the k−ϵ type have been taken up for comparison in the present analysis.

10.2. Turbulence Models for Phase-aver aged EquationsAs enumerated earlier, three different high Reynolds number versions of the two- equations model that have been used in the present simulation are : (1) the standard k−ϵ (2) KaLa and (3) RNG k−ϵ . All three models relate the turbulent viscosity ⟨ v t ⟩ to the turbulent kinetic energy ⟨ k ⟩and its rate of dissipation ⟨ ϵ ⟩.

Table 1: Model Parameters

Models Cμ C ϵ1 C ϵ2 σ k σ ϵ β0 η0 Standard k−ϵ 0.09 1.44 1.92 1.0 1.3 - -and KaLaRNG k−ϵ 0.0845 1.42 1.68 0.7179 0.0179 0.012 4.38

In the standard k−ϵ model, the transport equations for ⟨ k ⟩ and ⟨ ϵ ⟩ are

∂ ⟨k ⟩∂t

+∂ [ ⟨ui ⟩ ⟨k ⟩ ]

∂ x i= ∂

∂ x i [ ⟨ν t ⟩σk

∂ ⟨k ⟩∂ x i ]+Pk−⟨ ϵ ⟩(73)

∂ ⟨ϵ ⟩∂t

+∂ [ ⟨ ui ⟩ ⟨ϵ ⟩ ]

∂ xi= ∂

∂ xi [ ⟨ νt ⟩σ ϵ

∂ ⟨ϵ ⟩∂ x i ]+C ϵ 1 Pk

⟨ ϵ ⟩⟨k ⟩

−C ϵ2⟨ϵ ⟩ 2

⟨k ⟩(74)

Where the production term is

Pk=C μ ⟨ϵ ⟩ S2 , S=⟨k ⟩⟨ϵ ⟩ √ 1

2 [ ∂ ⟨ui ⟩∂ x j

+∂ ⟨ u j ⟩∂x i ]

2

(75)

The KaLa model is similar to the standard k−ϵ , except that the turbulence production term in Equation (75) is replaced by

Pk=C μ ⟨ϵ ⟩ S Ω , Ω=⟨k ⟩⟨ϵ ⟩ √ 1

2 [ ∂ ⟨ui ⟩∂ x j

−∂ ⟨u j ⟩∂ x i ]

2

(76)

The quantity Ω is related to the average rotation of a fluid element. In the simple shear flow context, S and Ω are equal. However, in stagnation flows, Ω =0 and S >0. This leads to the desired reduction of the production of kinetic energy near the forward stagnation point of the bluff objects. This has an important effect of lowering eddy viscosity in the boundary-layers and permits vortices to be shed from the rear side.

In the RNG k−ϵ model, Pk is given by Equation (75) and Equation (74) is augmented on the right hand side by an extra strain-rate term R given by

R=−Cμ η3(1−

ηη0 ) ⟨ϵ ⟩ 2

(1+β0 η3 ) ⟨k ⟩(77)

Where the quantity η is given by

η=⟨k ⟩⟨ϵ ⟩ [( ∂ ⟨ui ⟩

∂ x j+

∂ ⟨ u j ⟩∂ x i

) ∂ ⟨ui ⟩∂ x j ]

12(78)

The eddy viscosity ⟨ v t ⟩ for the standard k−ϵ and KaLa models is determined from the expression

⟨ v t ⟩=Cμ⟨ k ⟩2

⟨ϵ ⟩(79)

For the RNG k−ϵ the eddy viscosity expression is

⟨ ν t ⟩=v [1+( Cμ

ν )12 ⟨k ⟩

⟨ϵ ⟩12 ]

2

(80)

The parameters for each of the above models appearing in Equations (73)-(80) are giving in Table 1

10.3. Numerical Method and Computational Domain

The differential equations (71) and (72) have been solved on a staggered grid by using a modified version of the MAC algorithm of Harlow and Welch (1965) (also see Hoffman and Benocci, (1994)). The physical problem considered is flow past a cylinder of square cross-section, placed centrally in a channel. The computational domain for this geometry is presented in Figure 11.10. A uniform mesh with 386x98 cells has been used. The obstacle surface, and the top and bottom surfaces are treated as no-slip boundaries. At the inlet, the flow enters with a uniform velocity uav and the

prescribed turbulence intensity (I=√( ui' 2

2 )/ua v ) at the inlet is set to 10%. For the all the

computations, the eddy viscosity is specified as v t

v=10at the inflow plane (Bosch and

Rodi, 1996). The value of ⟨ ϵ ⟩ is specified using the Equations (79) or (80). Wall function treatment has been used at all the solid boundaries for the standard k−ϵ and the KaLa models. In contrast, no such treatment has been adopted for the RNG k−ϵ model. The RNG k−ϵ model has been tested on finer grid sizes in order to see the effect of avoiding the wall function treatment. The time-averaged drag coefficients were seen to change by less than 1.5% for the finest grid used. During wall function treatment, the first grid points from the wall have fallen in the range of 10 ≤ y+¿ ≤30 ¿. At the outlet, the convective boundary condition due to Orlanski (1976) has been used. This condition may be stated as

∂Ψ i

∂ t+uc

∂Ψ i

∂ x=0

where,Ψ can take the values of ui,k and ϵ . The convective velocity,uc is the streamwise celerity of the vortices leaving the outflow plane. The time-step used for the present simulation for all the three models is 4% of the time period of vortex shedding. The time-averaged quantities have been obtained by integrating the instantaneous field over a long period of not less than 40 shedding cycles, but without including the initial transients.

10.4. Comparison with Experiments

Figure 11 shows a comparison of the time-averaged stream wise velocity profiles at different downstream locations, namely (x = 0,1 and 5) for the three turbulence models. The Reynolds number of interest in this simulation is 21400. The definition of Reynolds number is based on the obstacle dimension B and given byℜ=ua v B /v. The experimental data of Lyn et al. (1995) has also been plotted in this figure. At the axial location of x = 0, the comparison is extremely

Figure 10: Two-dimensional channel flow with built-in obstacle

good except very near the obstacle surface. The discrepancy could be due to inadequate positional accuracy in the experiments on the one hand and the use of wall functions on the other. At the other two axial locations, the comparison reveals minor differences. A higher blockage due to the channel walls may also be responsible for the differences between the numerical simulation and experiments. The overall model predictions of the x-component of velocity however are sensibly close.

The numerically obtained time-averaged transverse component of velocity profiles at two different axial locations (x=0 and 5) have been compared with the experiments of Lyn et al. (1995) in Figure 12. At x=0, the three models show good agreement with the experiments though the numerical values are higher in regions near the cylinder. Once again, this can be attributed to the finite blockage due to the channel walls in the computation as against the infinite medium in the case of experiments. At x=5, the results of the standard k−ϵ and KaLa models are closer to the experimental measurements. However, those of the RNG k−ϵ differ significantly.

Figure 13 shows the time-averaged x-component of velocity variation along the centerline y=0. Among the three, the KaLa model displays better agreement with experiments in the region immediate downstream of the obstacle. This could be because the KaLa model produces just the right amount of turbulent kinetic energy and helps in retaining the periodic fluctuations. The periodic fluctuations strongly influence the momentum transfer between the instantaneous and time-averaged velocity components in the near-wake. This indirectly affects the centerline recovery of the time-averaged stream wise component of the fluid velocity.

The comparison between the computations and the experiment with regard to the time-averaged kinetic energy variation along the centerline y=0 is taken up next (Figure 14). The kinetic energy plotted in Figure 14 is that of the

Figure 11: Time-averaged stream wise velocity profiles at: (a) x = 0, (b) x = 1.0 and (c) x = 5

Figure 12: Time-averaged transverse velocity profile at: (a) x = 0, (b) x = 5.

Total velocity fluctuation ui(¿~ui+u ' i) and is defined for the plane channel flows as (Hadid et al 1992)

kT=34

(u2+v2)

using the approximation w2=12

(u2+ v2)In experiments, u and v can be directly

measured. In computations one can employ the formula

u2=(~u+u ')2=~u2+u ' 2+2~u u '

Here ~u can be determined from the computed velocity ⟨u ⟩by subtracting u, the quantity u '2 is associated with kinetic energy k of the k−ϵ model and the third term is limited by the upper bound

~uu '<√~u2 √k

Figure 13: Time-averaged streamwise velocity recovery along the centerline y=0

Figure 14: Streamwise variation of total time-averaged kinetic energy along the centerline y=0

In the present study, the unbiased estimate~u u '=1

2 √~u2k

has been utilized

An examination of Figure 14 shows that the KaLa model predicts the peak value of kT in good agreement with the experiments. The peak value of kT due to the RNG k−ϵ model shows the greatest departure. The shift in the location of the peak could be due to the presence of boundary walls in the computations. A similar trend is to be seen in earlier studies as well (Hadid et al, 1992), but the shift in the present study is seen to be smaller, and hence represents a closer agreement with experiments.

A detailed comparison in terms of velocity profiles reveals that Kato-Launder model to have the closest agreement with the experiments. This is further reinforced in the centerline profiles of the kinetic energy of the total fluctuations.

11. ConclusionsAmong the number of turbulence models in existence, the standard k−ϵ model is most popular and applicable to many complex flows of engineering importance. The model is computationally economical and accuracy is reasonable. Quite a few improvements on the k−ϵ model have been proposed in the recent past. The RNG k−ϵ and Kato-Launder k−ϵ are two such improved models. The relative performance of these models have been described together with the detailed implementation strategies.

The Phase-averaged method leads to the usual RANS (Reynolds Averaged Navier Stokes) equations except that the time-averages are replaced by the ensemble averages. Further, it describes two kinds of fluctuations viz., the fluctuations due to the Periodicity of the mean flow and the other is the usual Stochastic fluctuations. As such, the phase-averaged method is also termed as unsteady RANS, in which the complete spectrum of the stochastic motion is simulated by the turbulence model. The use of the Kato-Launder modification on the standard k−ϵ equation has brought about a major improvement, which is confirmed through the performance analysis of different eddy viscosity models. A number of other modifications have also been proposed to modify the eddy viscosity models. These are described below. Speziale (1987) has proposed a non-linear k−ϵ model, adding nonlinear terms to the Boussinesq approximation of turbulent stresses. Speziale introduced quadratic terms of mean velocity gradients and transport effects of the strain tensor into the stress-strain relationship. This model predicts the secondary flows in a square duct and yields more accurate predictions for separated turbulent flows past a backward facing step. Basically, the nonlinear model equations were developed to account for the nonisotropic behavior of the turbulent stresses, the tensorially invariant eddy viscosity structure of the two- equation model was utilized. Myong and Kasagi (1990) ans Shih et. al (1993) have also proposed quadratic nonlinear k−ϵ models. Although these models employed similar expressions in the stress-strain relation, the cofficients of

their nonlinear terms are different from one another. Among these recently proposed nonlinear models, the Myong and Kasagi (1990) model can be applied all the way to the wall; all the other models are coupled with the wall functions. The model deals with the turbulence anisotropy and reproduces a variety of complex flows influenced by anisotropic Reynolds stresses. Durbin (1995) has shown that the k−ϵ−ν2 model can be applied successfully to complex separated flows. In his model, ν2 is interpreted as a scalar, which handles accurately both the massive and smooth separations

References

Biswas, G., 1995, Solution of Navier-Stokes Equations for Incompressible Flows Using MAC and SIMPLE Algorithms, in Computational Fluid Flow and Heat Transfer, ed. K. Muralidhar and T. Sundararajan, Narosa Publishing House, India.

Bradshaw, P., 1971, An Introduction to Turbulence and Measurement, Perg- amon Press, Oxford.

Bosch, G., and Rodi, W., 1996, Simulation of Vortex Shedding Past a Square Cylinder Near a Wall, Int. J. Heat and Flow, Vol. 17, pp 267-275.

Bosch, G., and Rodi, W., 1998, Simulation of Vortex Shedding Past a Square Cylinder with Different Turbulence Models, Int. J. for Numeric Methods in Fluids, Vol. 28, pp 601-616.

Choudhury, D., 1993, Introduction to the Renormalization Group Method and Turbulence Modeling, Fluent. Inc. TM 107.

Deb, P.. Biswas, G., and Mitra, N. K., 1995, Heat Transfer and Flow Structure in Laminar and Turbulent Flows in a Rectangular Channel with Longitudinal Vortices, Int. J. Heat Mass Transfer, Vol. 38, pp 2427-2444.

Durbin, P. A., 1995, Separated Flow Computations with k-e-v2 Model, AIAAJ., Vol. 33, pp. 659-664.

Dutta, S., and A chary a, S., 1993, Heat Transfer and Flow Past a Backstep with Nonlinear k-ϵ Turbulence Model and the Modified k--ϵ Turbulence Model, Numerical Heat Transfer, Part-A, Vol. 23. pp 281-301.

Gibson, M. M., and Launder, B. E., 1978, Ground Effects on Pressure Fluctuations in the Atmospheric Boundary Layer, J. Fluid Mech., Vol. 86, pp 491-511.

Gupta, A. K., Laufer, J. and Kaplan, R. E., 1971, Spatial Structure in the Viscous Sublayer, J. Fluid Mech., Vol. 50, pp. 493-512.

Harlow, F. H., and Welch, J. E., 1965, Numerical Calculation of Time-dependent Viscous Incompressible Flow of Fluid with Free Surface, The Phys Fluids, Vol. 8, pp 2182-2188.

Hadid, A. H., Sindir, M.M., and Issa, R. I., 1992, Numerical Study of Two- Dimensioned Vortex Shedding from Rectangular Cylinders, Compt. Fluid Dyn. Vol J-2, pp 207-214.

Hirt, C. W., Nichols, 13. D., and Komero, is. U., iy/o, 5ULA - a ixumericai ouiu- tion Algorithm for Transient Fluid Flows, Los Alamos Scientific Laboratory Report, LA - 5652.

Hoffman, G. and Bcnocci, C., 1994, Numerical Simulation of Spatially-Developing Planer Jets, AGARD, CP-551, pp.26.1-26.6.

Hussain, A. K. M. F., 1983, Coherent Structures-Reality and Myth, Phys. Fluids, Vol. 26, pp 2816-2850.

Jayatilleke, C.L.V., 1969, The Influence of Prandtl number and Surface Roughness of the Resistance

of the Laminar Sub-Layer of Momentum and Heat TVansfer, in Progress in Heat and Mass Transfer, ed by U. Grigull and E. Hahne, Pregamon Press, vol 1, pp 193-329.

Jones, W. P., and Launder, B. E., 1972, The Prediction of Laminarization with a Two Equation Model of Turbulence, Int. J. Heat Mass Transfer, Vol. 15, pp 301-314.

Kato, M. and Launder, B. E., 1993, The modelling of turbulent flow around stationary and vibrating square cylinders, Proc. 9th Symposium on Turbulent Shear Flows, Kyoto, Japan, 10-4-1 to 10-4-6.

Kawamura, T., Takami, H., and Kuwahara, K., 1986, Computation of High Reynolds Number Flow Around a Circular Cylinder with Surface Roughness, Fluid Dynamics Research, Vol. 1, pp 145-162.

Launder, B. E., and Spalding, D. B., 1974, The Numerical Computation of Turbulent Flows, Comput. Meth. Appl. Mech. Engg, Vol. 3, pp 269-289.

Lee, S. H., Ryou, H. S., and Choi, Y. K., 1999, Heat Transfer in a Three- Dimensional Turbulent Layer with Longitudinal Vortices, Int. J. Heat Mass Transfer, Vol. 42, pp 1521-1534.

Leonard, B. P., 1979, A Stable and Accurate Convective Modeling Procedure Based on Quadratic Upstream Interpolation, Comput. Meth. Appl. Mech. Engg., Vol. 19, pp 59-98.

Lyn, D. A., Einav, S., Rodi, W. and Park J. H., 1995, A Laser-Doppler Ve- locimetry Study of Ensemble-Averaged Characteristics of Turbulent Near Wake of a Square Cylinder, J. Fluid Mech., Vol. 304, pp. 285-319.

Myong, H. K. and Kasagi, N., 1990, Prediction of Anisotropy of the Near- Wall Turbulence with an Anisotropic Low-Reynolds Number k-e Turbulence Model, Journal of Fluids Engg (ASME) Vol. 112, pp 521-524.

Nallasamy, M., 1987, Turbulence Models and Their Applications to Prediction of Interna] Flows: A Review, Computers and Fluids, Vol. 15, pp 151-194.

Orlanski, I., 1976, A Simple Boundary Condition for Unbounded Flows, J. Compt. Phys., Vol. 21, pp. 251 - 269.

Pauley, W. R., and Eaton, J. K., 1988 (a), The Fluid Dynamics and Heat Transfer Effects of Streamwise Vortices Embedded in a Turbulent Boundary Layer, Rept MD-51, Thermosciences Division, Department of Mechanical Engineering, Stanford University, USA.

Pauley, W. R., and Eaton, J. K., 1988 (b), Experimental Study of the Development of Longitudinal Vortex Pairs Embedded in a Turbulent Boundary Layer, AlAA Vol. 26 (7), pp 816-823.

Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington D.C.

Rai, M. M., and Moin, P., 1991, Direct Simulations of Turbulent Flow Using Finite Difference Schemes, J. Comp. Phys., Vol. 96, pp 15-53.

Saha, A. K., Biswas, G-, and Muralidhar, K., 1999, Numerical Study of Turbulent Unsteady Wake Behind a Partially Enclosed Square Cylinder Using RANS, Comput., Meth. Appl. Mech. Engg., Vol. 178, pp 323-341.

Shih, T. H., Zhu, J., and Lumley, 1993, A Realizable Reynolds-Stress Algebraic Equation Model, NASA Tm 105993.

Speziale, C. G., 1987, On Nonlinear k-l and k-e Models of Turbulence, J. Fluid Mech., Vol. 178, pp 450-475.

Yakhot, V., and Orszag, S. A., 1986, Renormalization Group Analysis of Turbulence, I. Basic Theory, Journal of Scientific Computing, Vol. 1, pp 3-51.

Yakhot, V., Orszag, S. A., Thangam, S., Gatski, T. B. and Speziale, C. G., 1992, Development of

Turbulence Models for Shear Flows by a Double Expansion Technique, Phys. Fluids A, Vol. 4 pp. 1510-1520.

Zhu, J. X., Fiebig, M., and Mitra, N. K., 1993, Comparison of Numerical and Experimental Results for a Turbulent Flow Field with a Longitudinal Vortex Pair, J. Fluids Engg. (ASMEJ, Vol. 115, pp 270-274.