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Finite element analysis of laminar and turbulent flowsusing LES and subgrid-scale models

T.L. Popiolek a,*, A.M. Awruch b, P.R.F. Teixeira a

a Department of Mathematics, Federal University Foundation of Rio Grande, Av. Italia km 8, 96200-000 Rio Grande,

RS, Brazilb Graduate Program in Civil Engineering, Federal University of Rio Grande do Sul,

Av. Osvaldo Aranha 99 -3 andar, 90035-190, Porto Alegre, RS, Brazil

Received 1 November 2003; accepted 18 March 2005Available online 17 May 2005

Abstract

Numerical simulations of laminar and turbulent flows in a lid driven cavity and over a backward-facing

step are presented in this work. The main objectives of this research are to know more about the structure ofturbulent flows, to identify their three-dimensional characteristic and to study physical effects due to heattransfer. The filtered NavierStokes equations are used to simulate large scales, however they are supple-mented by subgrid-scale (SGS) models to simulate the energy transfer from large scales toward subgrid-scales, where this energy will be dissipated by molecular viscosity. Two SGS models are applied: the classicalSmagorinskys model and the Dynamic model for large eddy simulation (LES). Both models are imple-mented in a three-dimensional finite element code using linear tetrahedral elements. Qualitative and quan-titative aspects of two and three-dimensional flows in a lid-driven cavity and over a backward-facing step,using LES, are analyzed comparing numerical and experimental results obtained by other authors. 2005 Elsevier Inc. All rights reserved.

Keywords: Laminar and turbulent flows; Large eddy simulation; Finite elements; Subgrid-scale model

0307-904X/$ - see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2005.03.019

* Corresponding author. Fax: +55 5323 15382.E-mail address: [email protected] (T.L. Popiolek).

www.elsevier.com/locate/apm

Applied Mathematical Modelling 30 (2006) 177199
mailto:[email protected]:[email protected]

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1. Introduction

Turbulent flows are of great practical interest in several engineering fields and may be defined as

a three-dimensional flow with highly disordered, intermittent and rotational fluid motion and withdiffusive and dissipative characteristics. Although formulation of mathematical models to simu-late numerically such complex flows is a challenging task, many researches have been developedand some reliable results have been obtained.

Flows with high Reynolds numbers, where the influence of the different turbulence scales mustbe taken into account, cannot be solved by direct numerical simulation (DNS) due to the largeamount of data and unknowns involved in the computational solution (and the correspondingrequirements in terms of CPU time and computer memory).

Turbulent flows may be simulated using the Reynolds Averaged NavierStokes (RANS) equa-tions. This approach is based in the separation of the instantaneous value of a specific flow vari-

able in its mean value and fluctuations with respect to this mean value. The well-known Reynoldsstress components are originated substituting mean values and fluctuations of the variables in theconservation equations. Details about this subject can be found in traditional texts such as Hinze[1], Schlichting [2] and Tennekes and Lumley [3], among others.

The RANS equations have more unknowns than equations, and for this reason it is necessaryto use closure models to define the Reynolds stress components. Several models have beenemployed by different authors in the last three decades, and most of these models are describedby the state of the art reviews presented by Launder and Spalding [4,5], Rodi [6] and Markatos[7], among others. In recent years several authors have implemented different modifications tothe original equations in order to get some improvements of the numerical models behaviour.

Alternatively, large eddy simulation (LES) may be used to analyze turbulent flows. This meth-

odology was initially proposed by Smagorinsky [8], and it consists in the separation of the largeeddies and subgrid-scales using a grid filter. Large eddies are associated to the low flow frequenciesand they are originated by the domain geometry and the boundaries. Subgrid-scales (SGS) areassociated to high frequencies and they have an isotropic and homogeneous behaviour, maintain-ing their independence with respect to the main stream. As in RANS equations, in LES is also nec-essary to use closure models, and due to the characteristics of the SGS (homogeneity, isotropy andno significant variations for different flows), they are more appropriated to be represented bymathematical models. Then, in LES the Large Eddies are simulated directly, whereas SGS are sim-ulated using closure models. Although RANS equations and LES seem to be similar, in the firstone the closure models simulate the momentum and energy transfer from the mean flow to the fluc-

tuating part, while in LES the closure models simulate the momentum and energy transfer from thelarge eddies to the small turbulence scales (or subgrid-scales). Comparisons, advances and trends ofturbulence models applied to bluff bodies were presented by Ferzinger [9], Leschziner [10] andMurakami [11].

In this work, studies to simulate turbulent flows using LES, with the classical Smagorinskysmodel and the dynamic subgrid-scale model are presented. The three-dimensional flow in alid-driven cavity is simulated, and statistical studies with respect to the velocity mean value,turbulence intensity and Reynolds stresses are performed. Two and three-dimensional flows ina backward-facing step are also analyzed in order to verify the behaviour of the two models com-paring results of this work with those obtained numerically and experimentally by other authors.

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2. Smagorinskys subgrid-scale model (SSGS)

Applying a grid filter to the continuity, momentum and energy equations, respectively, the fol-

lowing filtered expressions for a Newtonian slightly compressible fluid are obtained:

1

c2op

ot q

oui

oxi 0; 1

o

otqui

o

oxjquiuj

op

oxjdij

o

oxjqm

oui

oxjouj

oxi

k

ouk

oxkdij sij

! 0; 2

o

otq/

o

oxiqui/

o

oxiDij

o/

oxj si/

S/ 0; 3

where ui and pare the filtered velocity components and pressure, respectively,

/ is a filtered scalarfield (which may be the temperature, pollutant concentration, etc.), q is assumed to be constantand represent the specific mass, m is the molecular kinematic viscosity, k is the volumetric viscosity,Dij are the components of the molecular diffusion coefficient tensor,S/ is a filtered source, c is thesound speed and dij is the Kroenecker delta. sij and si/ are the components of the SGS stress ten-sor and of the SGS flux vector, respectively, representing the influence of the SGS in the flowstructure. They are given by

sij quiuj uiuj uiu0j uju0i u

0iu

0j qLij Cij u

0iu

0j ffi qu

0iu

0j; 4

si/ qui/ ui/ ui/0 u0i

/ u0i/0 qLi/ Ci/ u0i/

0 ffi qu0i/0. 5

In Eqs. (1)(5), the overbar indicates filtered quantities and represent components of the large tur-bulence scales, while () 0 in Eqs. (4) and (5) represents components of the small turbulence scales orsubgrid-scales. Crossed terms Cij uiu0j u

0iuj as well as Leonard terms Lij uiuj uiuj have not

much influence and they may be omitted.The eddy viscosity model is frequently used to represent the effects of SGS in LES. sij is

assumed as a non-linear function of the strain rate, and it may be written as follows:

sij 2qmtSij; 6

where the filtered strain rate component Sij and the eddy viscosity mt are given by

Sij 1

2

oui

oxj

ouj

oxi ; mt CsD

2

ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSijq CsD2jSj. 7

In Eq. (7) Cs is the Smagorinskys constant, which has values varying between 0.1 and 0.2 andD isthe grid filter width (representing a length scale). Usually, in three-dimensional flows D DxDyDz1=3, but in the finite element context D may be taken as the cubic root of the elementvolume.

si/ is also assumed as a non-linear functions of the strain rate, and may be written as follows:

si/ Dto/

oxiwith Dt

qmt

r; 8

where r is the Prandtl number for heat transfer problems.

T.L. Popiolek et al. / Applied Mathematical Modelling 30 (2006) 177199 179

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3. Dynamic subgrid-scale model (DSGS)

In Smagorinskys model Cs has a constant value over the whole domain, and this value remains

constant during the time marching process. Values ofCs depend on the users criterion and of thedifferent applications.

In the dynamic subgrid-scale model, formulated first by Germano et al. [12] and modified afterby Lilly [13], values ofCs have variations in space and time. These values may be calculated in asystematic way, without any interference of the user.

The dynamic subgrid-scale model is characterized by two filtering processes: In the first one,using the grid filter, the filtered expressions are given by Eqs. (1)(3), where the SGS Reynoldsstress and SGS flux vector components were included. In the second filtering process, a test filteris applied, and the corresponding equation are given by

1

c2o ph i

ot qohuii

oxi 0; 9

o

otqhuii

o

oxjqhuiihuji

ohpi

oxjdij

o

oxjqm

ohuii

oxjohuji

oxi

k

ohuki

oxkdij Tij

! 0; 10

o

otqh/i

o

oxiqhuiih/i

o

oxiDij

oh/i

oxj Ti/

hS/i 0; 11

where hi indicates the application of the second filtering process using a test filter, and Tij and Ti/are component of a stress tensor and a flux vector, with their components given by

Tij qhuiuji huiihuji hu0iu0ji; 12

Ti/ qhui/ji huiih/i hu0i/

0i. 13

Taking into account Eqs. (6) and (7), Tij may be expressed as follows:

Tij 2qChDi2jhSijhSiji; 14

where hDi 2D.Using Eqs. (4) and (12), it is obtained:

Lij Tij hsiji qhuiuji huiihuji. 15

This expression was also presented by Germano et al. [12].From Eqs. (6), (7), (14) and (15) the following system of equations is obtained:

Lij Tij hsiji 2CMij; 16

where

Mij qhDi2jhSijhSiji qhD

2jSjSiji. 17

Lilly [13] solved the system of equations given by Eq. (16), obtaining the following expressionfor C:


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C 1

2

LijMij

MijMij. 18

This model has some important characteristics: (a) The eddy viscosity is equal to zero in laminarflows. (b) The eddy viscosity may take negative values, simulating the energy transfer from smallto large scales (the backscatter phenomenon). (c) The model has an appropriated asymptoticbehaviour near the solid boundaries.

4. The model to simulate turbulent flows with LES

Taking into account the two models described in the previous sections, the system of equationsto simulate turbulent flows of slightly compressible fluids are given by

1

c2op

ot q

oui

oxi 0; 19

o

otqui

o

oxjquiuj

op

oxjdij

o

oxjqm mt

oui

oxjouj

oxi

k

ouk

oxkdij

! 0; 20

o

otq/

o

oxiqui/

o

oxiDij Dt

o/

oxj S/ 0; 21

where

mt CD2jSj and Dt

qmt

r. 22

With

jSj 2SijSij1=2

; Sij 1

2

oui

oxjouj

oxi

; D Element volume1

=3. 23

When classical Smagorinskys model (SSGS) is used, C= (Cs)2 remains constant, and commonly

values varying from 0.1 to 0.2 may be adopted.

If the dynamic subgrid-scale model (DSGS) is used, C has not a constant value, and it isobtained with Eq. (18), using Eqs. (15)(17). In the dynamics model, the field variables obtainedwith a first filtering process (using, for example, the box filter with its length scale defined by thefinite element mesh) are submitted to a second filtering process using a test filter. In this work, avariable resulting from the second filtering process, at a specific node i, was obtained calculatingthe mean value of the same variable with values computed in the first filtering process belonging toall nodes connected to node i.

Negative values of the total viscosity, obtained by the addition of the molecular and the eddyviscosities, were not allowed, because it was verified that in regions where the energy is transferredfrom small to large scales, numerical instabilities may occur.


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5. The finite element algorithm

The mass conservation for slightly compressible fluids, assuming constant entropy, may be

expressed by the following equation:

oq

ot

1

c2op

ot

oUi

oxi; 24

where c is the sound speed and Ui = qui.Expanding the momentum conservation equations in Taylor series the following expression is

obtained for the first time step (at time t + Dtn/2):

Un1=2i U

ni

Dt

2

oUniot

Uni Dt

2

ofnij

oxjosnij

oxjopn

oxi

1

2

oDp

oxi ; 25where fij = ujUi and p

(n+1)/2 = pn + Dp/2, with Dp = pn+1 pn. Using

eUn1=2i Uni Dt2 ofnij

oxjosnij

oxjopn

oxi

; 26

Eq. (25) is given by the following expression:

Un1=2i

eU

n1=2

i Dt

4

oDp

oxi. 27

In Eq. (26), sij represents viscous terms (including molecular and turbulence effects).Discretizing Eq. (24) in time and using Eq. (27), it is obtained:

Dq 1

c2Dp Dt

oUn1=2i

oxi Dt

o eUn1=2ioxi

Dt

4

o

oxi

oDp

oxi

" #. 28

The second time step is given by the following expression (at time t + Dtn):

Un1i Uni Dt

oUn1=2i

ot Uni Dt

ofn1=2ij

oxjos

n1=2ij

oxjopn1=2

oxi

. 29

Then the flow is analyzed, after space discretization, by the following algorithm: (1) DetermineeUn1=2i with Eq. (26). (2) Determine Dp with Eq. (28) and calculate pn+1 = pn + Dp. (3) DetermineU

n1=2i with Eq. (27). (4) Determine U

n1i with Eq. (29).

Applying the classical Galerkin method for space discretization, the following matrix expres-sions are obtained for Eqs. (26), (28), (27) and (29), respectively

XEeUn1=2jE ZX

N dX

bUni Dt2ZX

oN

oxjdX

fn

ij

ZX

oN

oxjdX

snij

ZX

oN

oxidX

pn

!;

30


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ZX

NT1

c2N dX

Dt2

4

ZX

oNT

oxi

oN

oxidX

Dp Dt

ZX

oNT

oxidX

eU

n1=2

iE

Z

CNTni dC eUn1=2iE !; 31

XEUn1=2iE XE

eUn1=2iE Dt4ZX

oN

oxidX

Dp; 32

ZX

NTN dX

U

n1

ZX

NTN dX

U

n

i Dt

ZX

oNT

oxjdX

f

n1=2

ijE

Dt

ZX

oNT

oxjN dX

snij Dt

ZX

oNT

oxjN dX

pn Dp=2

Dt ZC

NTnj dC fn1=2ijE Dt Z

C

NTNnj dC

snij

Dt

ZC

NTNni dC

p Dp=2. 33

In Eqs. (30)(33) N is a vector containing the shape functions, the index E indicates that the cor-responding variables are taken with a constant value over the element domainX, with boundariesC, NT is the transpose of vector N and n is a unit vector normal to the boundaryC. The symbol ^indicates vectors with nodal values of the corresponding variables. Eq. (31) is solved using theconjugate gradient method with diagonal pre-conditioning.

6. Numerical applications

6.1. Three-dimensional flow in a lid-driven cavity

The three-dimensional flow in a cavity, due to a prescribed velocity U0 applied to its top sur-face, going from the left to the right side, is analyzed in this section. The 12 edges have the samedimension (1.00 m) and non-slip boundary conditions were applied on the cavity walls.

The finite element mesh with 40,817 nodes and 184,320 linear tetrahedral elements is presented

in Fig. 1.Flows with Re = 1000 and 3200 were considered. In the first case, as the flow is essentially lami-nar, a steady state may be obtained. In the second case, for a three-dimensional flow, periodicoscillations occur [14] and the steady state cannot be attained, although according Zang et al.[15], for this Reynolds number, the flow is still a laminar flow.

For Re = 3200, a time interval, where oscillations of the velocity field occur, was identified andstatistical studies in this period were carried out.

The statistical analysis of the turbulence quantities (intensity and Reynolds stress components)were performed taking velocity field data corresponding to central horizontal and vertical linesbelonging to the plane of symmetry (z = 0).


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Dimensionless mean velocity components on the two lines defined previously and located at theplane of symmetry were calculated with the following expression:

Ui ui

U0i 1; 2; 34

where u1 and u2 are mean velocity components in x and y directions, respectively, and U0 is theprescribed velocity at the top surface.

Taking into account Reynolds hypothesis, the instantaneous velocity can be separated in itsmean value and a fluctuating part, and it may be written as follows:

ui ui u0i i 1; 2; 35

where u0i is the fluctuating part.The turbulence intensity may be defined by the following dimensionless expression:

Uirms 10u02i

1=2

U0i 1; 2. 36

The dimensionless Reynolds stress components are given by

U1U2 500u01u

02

U20. 37

X

Y

Z

Fig. 1. Finite element mesh.


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Constants 10 and 500, in Eqs. (36) and (37), respectively, were used to amplify values of turbu-lence intensity and Reynolds stress in order to get a suitable graphical representation.

The finite element algorithm described in the previous section and LES with the dynamic SGS

model were used.In Figs. 24, streamlines, pressure distribution and dimensionless mean velocity components at

the central horizontal and vertical lines belonging to the plane of symmetry, respectively, are pre-sented for Re = 1000 and good agreement have been obtained with respect to the result reportedby Tang et al. [16].

In Fig. 5 the path of a particle located initially near the opposite wall to the plane of symmetry(x = 0.63, y = 0.60, z = 0.46) is presented. The particle displacement takes place with a spiralmotion (characterized by a growing radius) toward the plane of symmetry, returning after tothe frontal wall, leaving the plane of symmetry region following a secondary vortex, located atthe left side of the cavity.

In Fig. 6 the particle motion from a position near the symmetry plane (x = 0.60, y = 0.50,z = 0.005) toward the frontal wall is present. In this case, the particle follows a secondary vortexlocated at the right side of the cavity.

Figs. 5 and 6 show the complex characteristics of a three-dimensional flow in a lid-driven cavity.A statistical study was carried out analyzing the three-dimensional flow in a cavity with

Re = 3200. In Fig. 7 the velocity field in a secondary vortex, located on the right side, at the bot-tom of the plane of symmetry, is presented. Instants with weak and strong oscillations are shown

x

y

0.00 0.25 0.50 0.75 1.00

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2. Streamlines for Re = 1000.


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x

y

0.00 0.25 0.50 0.75 1.00

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3. Pressure distribution for Re = 1000.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

X

U2

Present work

Tang et al. (1995)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1.0 -0.5 0.0 0.5 1.0

U1

Y

Present work

Tang et al. (1995)

(a) (b)

Fig. 4. Mean velocity components profiles at central lines belonging to the plane of symmetry: (a) horizontal line and(b) vertical line.


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and they are representative of a periodic phenomenon. In this work, velocity field data corre-sponding to a period of approximately 12 s or 40,000 time steps, where strong oscillations occur,were used to perform the statistical study.

Comparisons of numerical and experimental results for the mean velocity, turbulenceintensity and Reynolds stress components are presented in Figs. 810, respectively. Althoughsome differences can be observed, specially in the peak values, the same characteristics arepreserved.

6.2. Two- and three-dimensional flow over a backward-facing step

Two- and three-dimensional flows in a backward-facing step are analyzed in this section. Theyhave characteristics of a very complex flow with layers separation, reattachment and recirculation.

Fig. 5. Path of the particle starting from the point (0.63, 0.60, 0.46), close to the frontal wall, for Re = 1000.


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One of the more important characteristics is the relation between Reynolds number and the reat-tachment length, because it allows to compare results obtained in this work with numerical andexperimental results obtained previously by other authors.

The computational domain is depicted in Fig. 11, were w = 5.0 m, h = 1.0 m, s = 0.94 m,xe = 1.0 m and xt = 30.0 m. There are solid walls at the top, at the bottom and at the frontal faceof the channel, while in the other face there is a plane of symmetry.

Non-slip boundary conditions were applied at the solid walls, while in the plane of symmetrythe perpendicular velocity was taken equal to zero. At the channel exit only pressure wasprescribed (p = 0). At the channel entrance the velocity profile corresponds to a fully developedflow, perpendicular to the entrance plane. The inflow velocity profile is given by

Fig. 6. Path of the particle starting from the point (0.60, 0.50, 0.005), close to the plane of symmetry, for Re = 1000.


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Uy 2

3Umax 1

y h=2

h=2

2 ; 38

where Umax is the maximum velocity.For the thermal problem, the temperature at the entrance was taken equal to 1.0C and at the

solid boundaries the temperature remains constant and equal to zero. The value adopted for thethermal diffusion coefficient was 0.05 m2/s.

X

0.70 0.80 0.90 1.00

X

0.70 0.80 0.90 1.00

y

0

0.1

0.2

0.3

0.4

y

0

0.1

0.2

0.3

0.4

(a) (b)

Fig. 7. Velocity field in a secondary vortex: (a) instant with weak oscillations and (b) instant with strong oscillations.

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1.0 -0.5 0.0 0.5 1.0

U1

Y

Present work

Zang et al. (Exp.)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8

X

U2

Present work

Zang et al. (Exp.)

(a) (b)

Fig. 8. Mean velocity at central lines belonging to the plane of symmetry: (a) horizontal line and (b) vertical line.


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The Reynolds number for this problem is defined as [17]

Re UD

m

23Umax

2h

m

4

3

Umaxh

m. 39

For two-dimensional flows, a mesh with 7594 nodes and 18,255 tetrahedral elements wasadopted, using only one layer of elements in the perpendicular direction to the flow. Here, inthe two lateral surfaces the perpendicular velocity remains constant and equal to zero.

For the three-dimensional flow, a mesh with 64,549 nodes and 291,600 elements was adopted.This mesh is shown in Fig. 12.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

X

U2rms

Present work

Zang et al. (Exp.)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1.0 -0.5 0.0 0.5 1.0

U1rms

Y

Present work

Zang et al. (Exp.)

(a) (b)

Fig. 9. Turbulence intensity at central lines belonging to the plane of symmetry: (a) horizontal line and (b) vertical line.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

X

U1U2

Present work

Zang et al. (Exp.)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-1.0 -0.5 0.0 0.5 1.0

U1U2

Y

Present work

Zang et al. (Exp.)

(a) (b)

Fig. 10. Reynolds stress components at central lines belonging to the plane of symmetry: (a) horizontal line and(b) vertical line.


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In Fig. 13, experimental results obtained by Armaly et al. [17] and numerical results obtained byKim and Moin [18] for the reattachment length are compared with the results of the present work,simulating two-dimensional flows with several Reynolds numbers (until Re = 1000). Althoughnumerical results present good agreement, the differences with the experimental results grow asReynolds number increases.

In Fig. 14, the experimental work of Armaly et al. [17] and the numerical results given byWilliams and Baker [19] are compared to the results obtained in the present work for three-dimen-sional flows. Good agreement is observed between numerical and experimental simulations.Values of the reattachment lengths obtained with the classical Smagorinskys (SSGS) model is

y

xz

Plane of symmetry

Solid wall

Solid wall

Solid wall

Exit

Channel

entrancew

s

h

xrxext

Fig. 11. Computational domain.

X

Y

Z

Fig. 12. The finite element mesh for three-dimensional flows.


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smaller with respect to those obtained with the dynamic (DSGS) model, showing that SSGS

model is more dissipative than the DSGS model.In Fig. 15, results obtained for two- and three-dimensional flows in the present work, usingSSGS and DSGS models are shown, and three facts are confirmed (observing the correspondingresults for Re > 600): (a) Reattachment lengths with SSGS model are smaller with respect to thosesimulated by the DSGS models, which evidences the dissipative character of SSGS model. (b) Asnumerical and experimental results are coincident only for three-dimensional flows whenRe > 600, the three-dimensional characteristic of the turbulence phenomenon is also evident. (c)Reattachment lengths in three-dimensional flows are greater than those of two-dimensional flows.One of the reasons of this fact is the transversal flow generated by the non-slip boundaryconditions prescribed in one of the side walls (on the other side, the plane of symmetry is located).

1.0

3.0

5.0

7.0

9.0

11.0

13.0

15.0

17.0

0 200 400 600 800 1000

Re

Xr/S

Armaly et al. (Exp.)

present work (DSGS model)

present work (SSGS model)

Kim and Moin

Fig. 13. Reattachment length for two-dimensional flows.

1.0

3.0

5.0

7.0

9.0

11.0

13.0

15.0

17.0

0 200 400 600 800 1000Re

Xr/S

Armaly et al. (Exp.)

Williams and Baker

present work (DSGS model)

present work (SSGS model)

Fig. 14. Reattachment length for three-dimensional flows.


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In Fig. 16, streamlines corresponding to Reynolds numbers 400, 600, 800 and 1000, respec-tively, for two-dimensional flow are shown. A secondary vortex near to the top of the channelcan be observed for Reynolds numbers greater or equal to 600.

Dimensionless separation lengths of the secondary vortices obtained in the present work are5.52, 10.11 and 9.25 for Reynolds numbers 600, 800 and 1000, respectively. Armaly et al. [17]found (through an experimental work) for the same Reynolds numbers, separation lengths equalto 5.87, 9.17 and 10.40, respectively.

It can be observed that results of the present work for two-dimensional flow are similar to those

obtained experimentally by Armaly et al. [17]. However two aspects may be considered: (a)

1.0

3.0

5.0

7.0

9.0

11.0

13.0

15.0

17.0

0 200 400 600 800 1000Re

Xr/S

DSGS model. 3-D

SSGS model. 3-D

DSGS model. 2-D

SSGS model. 2-D

Fig. 15. Reattachment length for two- and three-dimensional flows.

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

Fig. 16. Streamlines for Re = 400, 600, 800 and 1000, respectively, for two-dimensional flow.


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Armaly et al. [17] affirmed that the separation length of the secondary vortex increases with Rey-nolds number until the transition regime, which begins when Re ffi 1200. During the interval cor-responding to the transition regime (Reynolds number between 1200 and 6600) the separation

length of the secondary vortices decreases. In the present work it seems that the transition regimebegins in the interval of Reynolds numbers between 800 and 1000. (b) Armaly et al. [17] reportedthat the secondary vortex is very thin (about 0.4 mm). In the two-dimensional simulations of thepresent work thickness greater than 0.4 mm were obtained. Other numerical results for two-dimensional such as those present by Kim and Moin [18], for Re = 600 and 800, and by Sohn[20], for Re = 800, have the same characteristics. It will be seen after that when three-dimensionalflows are simulated, the secondary vortex is very thin, and to capture this vortex it is necessary touse a refined mesh, with very small elements in this region.

The secondary vortex is a consequence of adverse pressure gradients. In Fig. 17 pressure con-tours for Reynolds numbers 400, 600, 800 and 1000, respectively, are presented.

In Fig. 18, temperature contours for Reynolds numbers 600, 800 and 1000, respectively, arepresent for a two-dimensional flow. For Re = 600, dominant heat transfer along the channel isdue to advection effect, but between the top and the bottom boundaries the effect of diffusionis more important. For Re = 800 the effect of advection between the top and bottom boundariesincreases, but it is still very weak. Finally, for Re = 1000, advection effects are dominant in bothdirections.

In Fig. 19, the streamlines on the plane of symmetry of a three-dimensional flow, for Reynoldsnumbers 600, 800 and 1000, respectively, are shown. It can be observed that the secondary vortexon the top of the channel, disappear. Chiang and Sheu[21] presented similar results for Re = 1000and Williams and Baker [19] expressed that the separation region is very thin for Re = 800 (theyreported a value equal to 1.0 mm). Secondary vortices was not captured in the present work

because the elements size in this region is approximately equal to 26.0 mm.

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

Fig. 17. Pressure distribution for Re = 400, 600, 800 and 1000, respectively.


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It is necessary to stand out that results of a three-dimensional flow is greatly influenced by thedimension adopted for the transversal direction (which is the width of the channel). If this dimen-sion is not large enough to weaken the transversal flow, the difference between a two- and three-dimensional flow will be more intense.

It is necessary to emphasize that results for Re = 800 and 1000 were plotted using time averagevalues to identify with accuracy the reattachment length, because, for these Reynolds numbers,vortex shedding, separation and small perturbations along the walls occur.

In order to study real turbulent flows, simulations with Re = 104 and 4 104 were performed.In Fig. 20, instantaneous values for t = 150 s and Re = 104 are shown. It can be observed that

separation bubbles are distributed near the walls at the top and the bottom of the channel. Astrong coherence may be verified between the velocity field and the other variables (streamlines,pressure and temperature). Advective effects are dominant in both directions, and the correspond-ing consequence is that the loss of temperature is faster in turbulent flows than in laminar flows,where advective effects are only important in the longitudinal direction.

0.11

0.11

0.22

0.22

0.33

0.33

0.450.56

0.670.780.89

x

y

0 5 10 15 20 25 30

0

1

2

0.11

0.22

0.22

0.33 0.45

0.78 0.670.56

0.45 0.330.89

x

y

0 5 10 15 20 25 30

0

1

2

0.09

0.18

0.18

0.27

0.270.36

0.550.91 0.82

0.73 0.640.64 0.18

0.46 0.180.36

0.27

x

y

0 5 10 15 20 25 30

0

1

2

Fig. 18. Temperature contours for Re = 600, 800 and 1000, respectively.

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

y

0 5 10 15 20 25 30

0

1

2

Fig. 19. Streamlines for three-dimensional flows with Re = 600, 800 and 1000, respectively, using DSGS model.


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In Fig. 21 time average values of the velocity and pressure fields as well as the correspondingstreamlines are shown. Comparing instantaneous velocity fields and streamlines, presented inFig.20 with the average values, presented in Fig. 21, it is possible to conclude that vortices locatednear the step oscillate, but in terms of mean values they remain always in the same positions. Thiscomparison shows also that vortices near the exit section oscillate and change continuously their

positions.In Figs. 22 and 23 instantaneous (at t = 150 s) and time average values of field variables for a

turbulent flow with Re = 4 104 are shown. Results are similar to those obtained for Re = 104,but comparing Figs. 20 and 22 where instantaneous values are presented, it is possible to observethe more intense oscillations and faster loss of temperature occur when Re = 4 104.

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

0.33 0.500.92 0.84 0.420

.08

0.080.08

0.250.17

0.170.59

x

y

0 5 10 15 20 25 30

0

1

2

Fig. 20. Turbulent flow for Re = 104 in t = 150 s. Velocity, streamlines, pressure and temperature fields, respectively.

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

Fig. 21. Turbulent flow for Re = 104. Time average values for velocity, streamlines and pressure fields, respectively.


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Using Figs. 21 and 23, separation length of the main vortices can be measured. For Re = 104 a

dimensionless value equal to 8.05 was obtained, which is very close to the value obtained in aexperimental work by Armaly et al. [17]. These authors reported a separation length equal to8.00. For Reynolds 4 104, the dimensionless separation length obtained in the present workwas 6.40, which is inside the interval 7.0 1.0 obtained experimentally by Kim et al. [22]. Thiswork was used as a reference by Sohn [20].

7. Conclusions

Smagorinskys classical model (SSGS model) and the dynamic model (DSGS model) were ana-lyzed to simulate laminar and turbulent flows in a lid-driven cavity and a backward-facing step.

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

0.08

0.15

0.62

0.620.930.85

0.77

0.150.08 0.080.23

x

y

0 5 10 15 20 25 30

0

1

2

Fig. 22. Turbulent flow for Re = 4 104 in t = 150 s. Velocity, streamlines, pressure and temperature fields, respectively.

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

x

y

0 5 10 15 20 25 30

0

1

2

Fig. 23. Turbulent flow for Re = 4 104. Time average values for velocity, streamlines and pressure fields, respectively.


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The three-dimensional flows in a lid-driven cavity for Re = 1000 and 3200 were analyzed. In thefirst case the results of this work were compared with those obtained numerically by Tang et al.[16] and good agreements were observed. In the second case, a statistical study were carried out

and results for the mean velocity, turbulence intensity and Reynolds stress components werepresented and compared with those obtained experimentally by Zang et al.[15] and, again, goodagreements were obtained.

With respect to the two- and three-dimensional flows in a backward-facing step, several studieswere accomplished such as the influence of Reynolds number, comparisons between the classicalSmagorinskys model and the Dynamic subgrid-scale model and differences between two- andthree-dimensional flows.

When Re < 600 good results for the separation length of the main vortex were obtained in two-and three-dimensional flows. For Reynolds number between 600 and 1000 and two-dimensionalflows, numerical results for the reattachment length are smaller than those obtained with experi-

mental works, but, for three-dimensional flows, good agreements are obtained between numericaland experimental results. For this interval of Reynolds numbers the SSGS model simulates smal-ler values of the separation length than those obtained with the DSGS model, and values of SSGSmodel decrease when Reynolds number increases.

From the previous commentaries, it was confirmed that the SSGS model is more dissipativethan the DSGS model and that turbulence is a three-dimensional phenomenon. In three-dimen-sional flows a transversal flow going from the lateral wall to the plane of symmetry occurs.

For two-dimensional simulations the separation length of the secondary vortex is similar tovalues given in the technical literature. However, in this work the transient regime (characterizedby a decrease of the separation length, among other reasons) occurs for a smaller Reynoldsnumber than that reported by the experimental work of Armaly et al. [17]. For three-dimensional

simulations it was not possible to capture the secondary vortex because it is very thin, and arefined mesh with very small elements would be necessary.

Finally, it was shown that for high Reynolds numbers (in the present workRe = 104 and 4 104

were simulated), vortices oscillations and loss of temperature due to dominant advective effect inall directions increase together with growing values of Reynolds numbers.

Future works will include applications to more practical engineering problems.

Acknowledgments

The authors wish to thank CNPq and CAPES due to their financial support and the Supercom-puting Center of the Federal University of Rio Grande do Sul.

References

[1] J.O. Hinze, Turbulence, an Introduction to Its Mechanism and Theory, McGraw-Hill Book Company, New York,1959.

[2] H. Schlichting, Boundary-layer Theory, sixth ed., McGraw-Hill Book Company, New York, 1968.[3] H. Tennekes, J.L. Lumley, A First Course in Turbulence, 15th ed., Cambridge, Massachusetts and London, 1972.[4] B.E. Launder, D.B. Spalding, Lectures in Mathematic Models in Turbulence, Academic Press, London, 1972.


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[5] B.E. Launder, D.B. Spalding, The numerical computation of turbulent flows, Comput. Meth. Appl. Mech. Eng. 3(1974) 269289.

[6] W. Rodi, Turbulence Models and Their Application in Hydraulicsa State of the Art Review, Institute

Association of Hydraulics Research (IAHR), Delft, Netherlands, 1984.[7] N.C. Markatos, The mathematical modelling of turbulent flows, Appl. Math. Model. 10 (1986) 190220.[8] J. Smagorinsky, General circulation experiments with the primitive equations: I. The basic experiment, Month.

Weather Rev. 91 (1963) 99164.[9] J.H. Ferzinger, Simulation of complex turbulent flows: recent advances and prospects in wind engineering, J. Wind

Eng. Ind. Aerodyn. 46&47 (1993) 195212.[10] M.A. Leschziner, Computational modelling of complex turbulent flowsexpectations, reality and prospects,

J. Wind Eng. Ind. Aerodyn. 4647 (1993) 3751.[11] S. Murakami, Current status and future trends in computational wind engineering, J. Wind Eng. Ind. Aerodyn.

67&68 (1997) 334.[12] M. Germano, U. Piomelli, P. Moin, W.H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A 3

(1991) 17601765.[13] D.K. Lilly, A proposed modification of the Germano subgrid-scale closure method, Phys. Fluids A 4 (1992) 633

635.[14] F.M. Denaro, Towards a new model-free simulation of high-Reynolds-flows: local average direct numerical

simulation, Int. J. Numer. Meth. Fluids 23 (1996) 125142.[15] Y. Zang, R.L. Street, J.R. Koseff, A dynamic mixed subgrid-scale model and its application to turbulent

recirculating flows, Phys. Fluids A 5 (1993) 31863196.[16] L.Q. Tang, T. Cheng, T.T.H. Tsang, Transient solutions for three-dimensional lid-driven cavity flows by a least-

squares finite element method, Int. J. Numer. Meth. Fluids 21 (1995) 413432.[17] B.F. Armaly, F. Durst, J.C.F. Pereira, B. Schonung, Experimental and theoretical investigation of backward-

facing step flow, J. Fluid Mech. 127 (1983) 473496.[18] J. Kim, P. Moin, Application of a fractional-step method to incompressible NavierStokes equations, J. Comput.

Phys. 59 (1985) 308323.[19] P.T. Williams, A.J. Baker, Numerical simulations of laminar flow over a 3D backward-facing step, Int. J. Numer.

Meth. Fluids 24 (1997) 11591183.[20] J.L. Sohn, Evaluation of FIDAP on some classical laminar and turbulent benchmarks, Int. J. Numer. Meth. Fluids8 (1988) 14691490.

[21] T.P. Chiang, T.W.H. Sheu, Time evolution of laminar flow over a three-dimensional backward-facing step, Int. J.Numer. Meth. Fluids 31 (1999) 721745.

[22] J. Kim, S.J. Kline, J.P. Johnston, Investigation of a reattachment turbulent shear layer: flow over a backward-facing step, J. Fluid Eng., ASME Trans. 102 (1980) 302308.


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