stress modeling to study turbulent ow in non-circularducts such as Gessner and Emmery (1981), Nakayamaand Koyama (1986), Speziale (1984), and Abdel Gawadet al. (1997). Few experimental investigations of this oware available. Comparisons are made with the previousexperimental and numerical results wherever possible.

2. GOVERNING EQUATIONS

2.1 Large Eddy Simulation

In LES, the resolved (large-scale) ow �eld is de�nedby a smoothing or �ltering operation which eliminatessmall scales that cannot be represented on a speci�edmesh. Whereas, scales smaller than the grid size are mod-eled. Applying the �ltering operation to the incompress-ible Navier-Stokes and continuity equations to obtain thefollowing �ltered (resolved) equations of motion:

@ui

@t+@ui uj

@xj= �

1

�

@P

@xi

+�@2ui

@xj@xj�@�ij

@xj(1)

@ui

@xi= 0 (2)

where �ij = uiuj � ui uj , is the subgrid-scale stress ten-sor. This term is modeled by the dynamic eddy viscositymodel (Germano et al. (1991)) which assumes an eddyviscosity representation for small scales:

�ij �1

3�kk �ij = �2 �T Sij

= �2C(x; t)�2jSj Sij (3)

where �ij is Kronecker's delta, and � is the �lter width

(�3 = �x �y �z), jSj = [2 Sij Sij ]1

2 is the magnitude ofthe large-scale strain rate tensor which is de�ned as:

Sij =1

2(@ui

@xj+@uj

@xi) (4)

The coe�cient C is determined by the least-squares min-imization procdure proposed by Lilly (1992). Thus, thecoe�cient is given by:

C �2 = �1

2

< Lij Mij >

< Mij Mij >(5)

where

Lij = duiuj � bui buj (6)

Mij =c�2

�2jbSj cSij �

d

jSjSij (7)

where (b) denotes a �ltered quantity (with �lter widthb� = 2 �) and (< :: >) means averaging over homoge-neous directions (along the duct in the present case).

2.1. Nonlinear K-� Model

A two-equation model uses the concept of eddy viscos-ity �t so as to describe the magnitude of turbulence inten-sity and its spatial extent. In the K-�model, the referencevelocity of turbulence is represented by K0:5 determinedfrom the turbulent kinetic energy (K). The characteristiclength scale is given by the eddy length scale L� = K0:5=�

. The complete model can be stated to form from:- Continuity equation:

@U=@x+ @V=@y + @W=@z = 0 (8)

- Momentum equations:

@U2=@x+ @UV=@y + @UW=@z =

�(1=�)(@P=@x) + [@=@x(�eff@U=@x) +

@=@y(�eff@U=@y)]� [@=@x(�t@U=@x) +

@=@y(�t@V=@x)] (9)

@UV=@x+ @V 2=@y + @V W=@z =

�(1=�)(@P=@y) + [@=@x(�eff@V=@x) +

@=@y(�eff@V=@y)]� [@=@x(�t@U=@y)

+@=@y(�t@V=@y)] (10)

@UW=@x+ @V W=@y + @W 2=@z =

�(1=�)(@P=@z) + [@=@x(�eff@W=@x) +

@=@y(�eff@W=@y)] (11)

- K-equation:

Ui@k=@xi = �t(@Ui=@xj + @Uj=@xi)@Ui=@xj �

@=@xi((�t=�k) @k=@xi)� � (12)

- �-equation:

Ui@�=@xi = @=@xi(�t=�� @�=@xi) +

(�=k)[C�1 �t @Ui=@xj(@Ui=@xj + @Uj=@xi)� C�2�] (13)

- Turbulent Viscosity Equation:

�t = C� K2=� (14)

2