Direct and Large Eddy Simulations Using the SIMPLE Algorithm

This article was downloaded by: [Monash University Library]On: 02 August 2013, At: 11:51Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Numerical Heat Transfer, Part A:Applications: An International Journal ofComputation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unht20

Direct and Large Eddy Simulations Usingthe SIMPLE AlgorithmHanif Montazeri aa Department of Mechanical and Industrial Engineering, University ofToronto, Toronto, CanadaPublished online: 05 Dec 2011.

To cite this article: Hanif Montazeri (2011) Direct and Large Eddy Simulations Using the SIMPLEAlgorithm, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation andMethodology, 60:10, 827-847, DOI: 10.1080/10407782.2011.627799

To link to this article: http://dx.doi.org/10.1080/10407782.2011.627799

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/unht20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/10407782.2011.627799

http://dx.doi.org/10.1080/10407782.2011.627799

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

DIRECT AND LARGE EDDY SIMULATIONS USING THESIMPLE ALGORITHM

Hanif MontazeriDepartment of Mechanical and Industrial Engineering, University of Toronto,Toronto, Canada

The standard SIMPLE algorithm is examined for simulation of turbulent channel flow. The

algorithm is currently underutilized for turbulent simulations. It is explained and demon-

strated why its implicit time integration scheme should be considered for practical and generic

turbulent simulations. Taking advantage of the implicit time integration feature of SIMPLE

algorithm, a systematic error reduction method is presented, by which highly accurate LES

solutions can be obtained while computational cost is substantially reduced. In addition to the

effect of time steps, the effects of two popular spatial discretization schemes for convective

terms are examined for fully developed channel flow. Smagorinsky and dynamic Smagor-

insky subgrid models are also implemented and tested for different spatial resolutions. It is

demonstrated that these subgrid models do not necessarily improve turbulent solutions.

1. INTRODUCTION

Direct numerical simulation (DNS) and large eddy simulation (LES) of turbu-lent flows have been developed for a number of applications using different methodsand techniques [1–12]. These methods are characterized with their spatial and timediscretization methods, numerical schemes and flow algorithms. Due to the sensitivenature of turbulent simulations, researchers have been studying the effects of differ-ent techniques in variety of frameworks such as spectral method, finite difference,finite element and finite volume.

Despite the higher accuracy of spectral methods, finite volume and finite differ-ence methods are becoming more popular for simulations of direct and large eddysimulations. Their primary advantage lies in the ease with which they can be appliedto complex geometries. Midpoint-rule approximation of surface integral in finite vol-ume methods defaults these methods to second-order accuracy, and therefore makethem less promising for high-order accurate simulations. Finite volume techniques,however, benefit from its conservative properties which can be considerably impor-tant both for large and direct simulation of turbulent flows. To the best of the

Received 4 March 2011; accepted 12 September 2011.

Author would like to sincerely thank professors Javad Mostaghimi and Cyrus K. Madnia for their

valuable comments that improved the manuscript. This work could not be finalized without the invaluable

support from professor Mostaghimi.

Address correspondence to Hanif Montazeri, Department of Mechanical and Industrial

Engineering, University of Toronto, 5 Kings College Road, Toronto, Ontario M5S 368, Canada.

E-mail: [email protected]

Numerical Heat Transfer, Part A, 60: 827–847, 2011

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2011.627799

827

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

author’s knowledge, finite difference methods have received more attention thanfinite volume methods, mainly because implementing higher order numericalschemes is easier for these methods.

Rai and Moin [13] extensively studied the reliability of finite difference methodsfor direct simulation of turbulent flows. They compared a kinetic-energy-conservingtype of central difference with a high-order-accurate upwind scheme. Despite the dis-sipative nature of upwind schemes, they recommended high-order-accurate upwindscheme for direct simulations of turbulent flows when finite-difference schemes wereused. On the other hand, similar studies for large eddy simulations revealed differentperspectives [14–16]. Tafti [16] suggested conservative central difference discretizationas a suitable scheme for LES simulation because it maintains the integrity of theresolved spectrum better than high-order upwind methods. Whereas, fifth order non-conservative upwinding discretization of convective terms and second order central dif-ference discretization for pressure equation were recommended for DNS. Furthermore,Tafti found little evidence to justify the computational cost of high-order treatment ofgradient operators. The article also provides evidences that using high-order conserva-tive treatment of convection terms, in spite of the second order finite volume operatorexhibits high order accuracy. Najjar and Tafti [15] also found that the dynamic pro-cedure in implementing dynamic turbulent models senses the dissipative nature ofthe non-conservative upwind biased approximation and returns a lower turbulent vis-cosity than with a second order central difference approximation. They indicated thatthe inclusion of the dynamic model or any subgrid scale model with finite-differenceapproximations will not always result in better predictions of the flow field.

As mentioned earlier, finite volume simulations of turbulent flows have receivedless attention than finite difference techniques, some of these works can be found in[17–23]. Smagorinsky-based models were evaluated in the framework of finite-volumeby Majander and Siikonen [18]. They found that none of the tested models improvedthe results of low Reynolds number simulation for any grid level compared to the cal-culations with no model; however, at the higher Reynolds number the subgrid-scalemodels stabilize the computation. They analyzed the accuracy of a resolved field andrealized that the discretization error overwhelms the subgrid term at Reb¼ 2800 in the

NOMENCLATURE

L filtering operator

ui flow velocities

p pressure

t time

D filter-width

vs turbulent viscosity

CS smagorinsky coefficient

C dynamic smagorinsky coefficient

sij turbulent stress tensor

v kinematic viscosity

us wall-shear velocity

d channel half-width

X horizontal direction

Y vertical direction

Z lateral direction

yþ non-dimensional Y direction

Uþ non-dimensional root-mean-square

velocity in X direction

uþ non-dimensional root-mean-square

velocity fluctuation in X direction

vþ non-dimensional root-mean-square

velocity fluctuation in Y direction

wþ non-dimensional root-mean-square

velocity fluctuation in Z direction

uvþ non-dimensional root-mean-square

reynolds shear stress

828 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

most part of the computational domain. Error-assessment of central finite-volumediscretizations in large-eddy simulation was studied in reference [19]. They examinedsecond and fourth order central difference discretization for convective and viscousterms (including subgrid-scale models), denoted as 2 and 4, respectively. They useddifferent combinations for convective and viscous terms represented as 2-2, 2-4,4-4, and 4-2. They observed that the use of equal-order finite-volume methods (2–2and 4–4) leads to better results compared to mixed-order combinations (2-4 and4-2). A 2–2 scheme generally obtains the smallest errors at coarser resolutions, closelyfollowed by the 4–4 scheme. They concluded that, at coarse resolutions, the asymp-totic error behaviour as expressed by the order of the spatial discretization is not asuitable indicator for the total error in large-eddy simulations. Xu et al. [24] proposeda finite volume method with compact fourth order accuracy scheme for large eddysimulation (LES), showing that the higher accurate finite volume method for LESis proved to be a promising numerical method. In reference [23], the feasibility ofan optimal finite-volume large-eddy simulation model for isotropic turbulence wasassessed. They used a stochastic estimate of the fluxes in a finite-volume frameworkto approximate the ideal LES. Furthermore, the variational multiscale (VMS)approach is applied to the Smagorinsky model and developed within the context ofunstructured finite-volume solver in reference [20].

Explicit numerical algorithms have been extensively used both for direct andlarge eddy simulations. Perhaps the most common algorithm used for this groupof simulations is the fractional step method of Kim and Moin [25]. Explicit algo-rithms are justified both for DNS and LES, since small time steps are required toaccurately resolve flow details. However, it was shown in reference [26] that time steprestriction of explicit methods is quite stringent for such simulations. Kim et al. [27]

used 0:0676n=u2s as the computational time step for their explicit algorithm of simu-lating channel flow, where n is kinematic viscosity and us is wall-shear velocity.Whereas Choi and Moin [26] obtained successful simulations even when they used

1:2n=u2s as the time step for their fully implicit algorithm. They showed that this timestep is only half of the Kolmogorov time scale in their simulation, while a compara-ble time scale to Kolmogorov scale was expected. Although they found successfulsolutions for large time steps, they lost accuracy because of using larger time steps.No remedy was presented in the article to eliminate or reduce the error caused by theimplicit time integration. Furthermore, Hou and Mahesh [28] overcome the acoustictime-scale limit in their implicit algorithm for direct and large eddy simulation ofcompressible turbulent flows. Park [29] studied the effects of time-integrationmethod using PISO Algorithm. The standard PISO algorithm is basically a modifi-cation of SIMPLE-type algorithm [30, 31] for unsteady simulation of a flow field anduses EULER implicit scheme for time integration. Seeking second order accuracy intime, Park [29] modified the algorithm and compared its simulation results with thefractional step method. It was demonstrated that the modified PISO algorithm suc-cessfully predicts turbulent flows while using time steps considerably higher thantime step restriction of explicit methods. It was shown if the first order implicit

EULER scheme is used, the time step has to be smaller than 0:2n=u2s . The aboveresults and reports seem to justify implicit algorithms for simulation of direct andlarge eddy simulations.

DIRECT AND LARGE EDDY SIMULATIONS USING SIMPLE ALGORITHM 829

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

Although both the PISO and fractional-step methods are designed forunsteady simulations, due to the splitting of the convection-diffusion and projection

steps, they have an error proportional to square root of time step Dt2 [30]. Therefore,they are best suitable for explicit time integration rather than implicit time inte-gration. This splitting operator has also brought concerns regarding first-orderaccuracy in time for pressure [32–35]. Moreover, these types of methods en-counter difficulties when the boundary velocity is discontinuous, as demonstratedin reference [34].

In summary, within the framework of finite difference or finite volume meth-ods, the above review and discussion suggests the following.

. Numerical time step restriction in explicit time integration methods might sub-stantially be smaller than the smallest existing time scale in a turbulent flow(e.g., for the test case presented in reference [26], explicit time step is 35 timessmaller than the Kolmogorov time scale).

. Implicit time integration methods allow using a comparable time step toKolmogorov time scale for both DNS and LES simulations [26, 29].

. High-order unpwinding discretization of convective terms and low order centraldifference discretization of other operations are optimized schemes for DNS simu-lations [13, 16].

. Equal-order central difference schemes 2-2 and 4-4 are recommended for LESsimulations, with little ambition for higher order schemes [16, 19].

. This is evidence that the second order finite volume operator does not preventobtaining higher order accuracy for turbulent flows [16].

Considering all of these, one might look for an alternative robust algorithm suit-able for implicit time integration expandable for higher order accuracy. Perhaps,SIMPLE-type algorithms [36] are suitable candidates satisfying all these criteria.SIMPLE-type algorithms, unlike fractional step methods, do not produce the split-ting error when large time steps are used and also higher order accuracy in time forpressure can be readily achieved. Besides, in these types of algorithms there are noconcerns regarding suitable boundary conditions for intermediate velocity field aswell as frustrations due to the discontinuity of velocity field. Hence, they seem tobe a promising approach for DNS and particularly for LES simulation of generic flowfields with complex geometries. Nevertheless, the standard simplification done in theiterative procedure for finding pressures in the standard SIMPLE [30, 31] might be ofsome concern for these sensitive simulations, especially for low spatial resolutions.

Over the years, SIMPLE-type algorithms have been extensively used anddeveloped for different types of flow simulation [37–46], but received considerablyless attention for direct and large eddy simulations [14, 47, 48]. In addition to multi-grid techniques [49, 50], recently new techniques have been proposed to acceleratethe convergence and increase the accuracy of these types of methods for unsteadysimulations [47, 51]. An interested reader can find an extensive review of these typesof methods in reference [52].

Considering all of these, in this article we extensively evaluate the performanceof the standard SIMPLE algorithm, as we found no careful study in the literature

830 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

testing these types of solvers for such simulations. To solve the accuracy problemcaused by using large time steps in implicit algorithm as shown in reference [26],we propose a method by which highly accurate solutions can be obtained. It isdemonstrated how the method greatly reduces the numerical cost while maintainingthe accuracy of solutions. As the above study suggests, we first use second order cen-tral difference discretization for all operators and then test third order QUICKscheme [53] for convection terms. Different spatial resolutions are used to test thereliability and performance of SIMPLE algorithm for turbulent simulations. Wealso evaluate the effect of two different turbulent models, i.e., Smagorinsky anddynamic Smagorinsky models. Moreover, the effects of time steps on the simula-tions are presented for both LES and DNS simulations. We take advantage of con-servative properties of finite volume methods to discretize flow equations. Also, wetest a number of sparse linear solvers to compare their performance for solving thesetypes of flows.

2. GOVERNING EQUATIONS

2.1. Standard Filtered Flow Equations

We start with a space-time convolution filter operation L and assumean unbounded domain. The filter L is linear and its operation is defined by[6, 54, 55]:

f x; tð Þ ¼ Lðf Þðx; tÞ ¼Z þ1

�1dt

0Z þ1

�1G x�x

0; t�t

0� �

f x0; t

0� �

dx0¼ G�f ð1Þ

where Z þ1

�1dt

0Z þ1

�1G x�x

0; t�t

0� �

f x0; t

0� �

dx0¼ 1 ð2Þ

The aim of this filter is to decompose flow variables into large and small scales.Therefore, to obtain the governing equations for large scales of flow, we applythis filter to flow equations. Hence, the filtered Navier-Stokes and continuityequations yield,

qtui þ qj uiuj� �

þ qip�1

Reqjjui þ qjsij ¼ 0 ð3Þ

qiui ¼ 0 ð4Þ

where

sij ¼ uiuj � uiuj ¼ L P ui; uj� ��

�P L uið Þ;L uj� ��

¼ L;P½ � ui; uj� �

ð5Þ

In which the commutator operator f ; g½ � is defined

f ; g½ �U ¼ f � g Uð Þ � g � f Uð Þ ð6Þ


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

Turbulent stress tensor sij or its dynamic consequence qjsij should be modeledbased on the resolved velocity field. These models, so called subgrid models, pri-marily involve structures smaller than the filter-width D. In most present-daysimulation studies, D is directly related to grid spacing h, e.g., D¼ 2 h. However,for most common turbulent models, D appears as a result of considering it as alength scale representing the unresolved structures. In some models, however,such as similarity models [6], this length scale is not explicitly used yet its actualeffect appears in an explicit filtering procedure.

2.2. Different Models

Employing the eddy-viscosity concept, the Smagorinsky model is considered asone of the very first modeling procedures for LES. Based on molecular viscosity, thismodel adds additional viscosity to the flow equation. Assuming D as a turbulentlength scale and DjSj as a velocity scale, turbulent viscosity reads as

ns � uuð Þ ll� �

� D Sj jð Þ Dð Þ ) ns ¼ CsDð Þ2 Sj j ð7Þ

Sij ¼ qiuj þ qjui S ¼ :5SijSij ð8Þ

and

qjsij ¼ nsqjjui ð9Þ

Cs is called the Smagorinsky constant. Over the years, various values for Cs havebeen suggested. For homogeneous isotropic, for instance, turbulence Cs is suggestedas 0.17 [6], yet in most applications 0.1 is preferred [6, 54]. This constant has a sig-nificant effect on flow simulation. Overpredicted value damps all turbulent fluctua-tions, resulting in a laminar solution for the flow field. To find a more accurateestimation for this constant, a dynamic Smagorinsky model was proposed [4]. In thismodel, an additional filter, the test filter, is used to find a dynamic value for the Sma-gorinsky coefficient [4, 6, 55, 56]:

c ¼ �hRijbiji2hbijbiji

ð10Þ

where

Rij ¼ duiujuiuj � buiui bujuj ð11Þ

bij ¼ bDD2 bSS�� bSSij � D2 dS�� SijS�� Sij

� �ð12Þ

and

sij �dij3skk ¼ �2cD

2S�� Sij ð13Þ

Here, :ð Þ denotes LES-filter with width D and :ð Þ is the test filter with widthc:ð Þ:ð Þ (see

Figure 1). The width of the combined filterc:ð Þ:ð Þ can be defined exactly for two

832 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

Gaussian filters as bDD2

¼ D2 þ bDD2

; this relation can be optimal for other filters such as

the top hot filter. In this type of modeling, only the ratio bDD= D needs to be adjustedwhich is commonly set to 2. To implement the test filter, the following numericalstructure was employed with the trapezoidal rule [57].

W�i;j;k ¼ :25 Wiþ1;j;k þ 2Wi;j;k þ Wi�1;j;k

� �ð14Þ

W��i;j;k ¼ :25 W

�i;jþ1;k þ 2W

�i;j;k þ W

�i;j�1;k

� �ð15Þ

eWWi;j;k ¼ :25 W��i;j;kþ1 þ 2W

��i;j;k þ W

��i;j;k�1

� �ð16Þ

3. CHANNEL FLOW SIMULATION

3.1. Simulation Results

A symmetric channel flow was simulated using periodic boundary condition[58, 59] in two homogenous directions X and Z. Initial velocity profile of the periodicchannel flow was randomly perturbed using cosine function. Then flow equationswere integrated in time by the implicit Euler method. Time averaging was performedafter the statistically steady state was reached. Turbulent Reynolds number is about180 based on the turbulent wall-shear velocity and the channel half-width d. Thechannel size is chosen as p� 0:289p in streamwise and spanwise directions [26].The half-width is chosen as 1 and computational nodes are distributed using tangenthyperbolic in the normal wall direction [7].

yj ¼1

2tanh �1þ 2 j � 1ð Þ

NJ � 1

� �tanh�1a

� j ¼ 1; 2;NJ ð17Þ

The transformation parameter a can vary between 0 and 1. The higher a the morenodes close to walls will be distributed. For simulations in this article we used

Figure 1. Test and grid filters for 2-D space.


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

a¼ 0.98346. As mentioned earlier, flow equations were colocatedly discretized withthe finite volume method and solved in the framework of the standard SIMPLEalgorithm.

We tested a series of DNS and LES solutions using different time steps, spatialresolutions, and two different schemes for convection terms. In Table 1, different testcases are listed and detailed. For each test case, time averaging began after flow fieldreached a stationary steady state. For this purpose, plane-averaged wall-shear ratesqu=qyjwd=U and root-mean-square turbulent statistics were monitored (some of themare shown in Figures 2 and 3). U is centerline velocity. The exact nondimensional

Figure 2. Nondimensional velocity gradient at the wall of channel flow versus nondimensional time is

shown. Second order accurate central difference scheme is used for convection terms. Time step was set

as 10�4, and 129 nodes were used in the wall-normal direction (color figure available online).

Table 1. Different test case conventions and their numerical specifications

Test case

convention

Type of

modeling

Numerical

resolution Dt � Dtu2sv Discretization scheme

Other

Descriptions

DNS — — — 4th order Cent. Diff. Ref. [63]

DNS 1 — 16� 129� 32 10-4 0.004 2nd order Cent. Diff.

DNS 2 — 16� 129� 32 10-3 0.04 2nd order Cent. Diff.

DNS 3 — 24� 129� 32 0.009 0.3 2nd order Cent. Diff.

LES 1 No Model 16� 65� 32 10-4 0.004 2nd order Cent. Diff.

LES 2 No Model 16� 65� 32 0.003 0.1 2nd order Cent. Diff.


LES 4 No Model 16� 65� 32 0.012 0.4 2nd order Cent. Diff. Failed




LES 8 No Model 13� 65� 16 0.006 0.2 2nd order Cent. Diff. Failed

LES 9 Smag. 16� 65� 32 10-3 0.04 2nd order Cent. Diff.

LES 10 Dyn. Smag. 16� 65� 32 10-3 0.04 2nd order Cent. Diff.

LES 11 Smag. 13� 65� 16 10-3 0.04 2nd order Cent. Diff.

LES 12 Dyn. Smag. 13� 65� 16 10-3 0.04 2nd order Cent. Diff.

LES 13 No Model 16� 65� 32 10-4 0.004 3rd order QUICK Failed

LES 14 No Model 24� 65� 16 10-3 0.04 3rd order QUICK

LES 15 Smag. 24� 65� 16 10-3 0.04 3rd order QUICK Failed


834 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

wall-shear rate qu=qyjwd=U for a laminar channel flow is 2. If a numerical algorithm

fails to maintain the turbulent instability modes, its averaged wall-shear rateconverges to 2.

To study the effect of time steps on simulations, five test cases are considered.LES 1–4 and LES 8 use four different time steps to illustrate the effect of time inte-gration on turbulent statistics. Figure 4 shows that the smaller the time steps are, themore accurate the results are. However, little difference can be seen for the meanvelocity profile and Reynolds shear stresses. Similar results are shown in Figure 5for different DNS solutions. Two factors contribute to the less accurate resultswhen larger time steps are used, i.e., larger truncation errors and disregarding flowstructures with smaller time scales. Truncation errors can be considerably reduced

Figure 3. Effect of four different time steps on nondimensional velocity gradient at the wall of the channel

flow versus nondimensional time is shown. In these test cases (LES 1, LES 2, LES 3, and LES 4), second

order accurate central difference scheme is used for convection terms, and 65 nodes were used in the wall–

normal direction. Due to a large time step, LES 4 converges to the laminar flow solution (color figure

available online).


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

by using more accurate time integration methods [60]. However, to resolve the smal-ler structures, one must use smaller time steps. The balance between the size of timesteps and the accuracy of time integration methods is similar to the balance betweenspatial spacing and the accuracy of spatial schemes. A suitable balance can beestimated using Fourier analysis of different schemes. An interested reader is referredto study the concept of modified wave number in references [30, 61].

Figure 4. Effect of time steps on root-mean-square of different turbulent statistics is illustrated. In these

test cases (LES 1, LES 2, and LES 3), a second order accurate central difference scheme is used for con-

vection terms, and 65 nodes were used in the wall-normal direction (color figure available online).

836 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

LES 4 shows that a standard SIMPLE algorithm with Euler time integrationconverges to a laminar solution if the computational time step in wall units

Dtn=u2s exceeds 0.2. This is the same time step found in reference [29] as the upperlimit when first-order Euler method is used. However, DNS 3 disproves this con-clusion. In DNS 3, successful turbulent solution is obtained even when a nondimen-sional time step is set to 0.3, showing the importance of the spatial resolution on thelaminarization. LES 8 confirms this conclusion, as even earlier laminarization isobserved when a courser mesh is used. Therefore, not only truncation errorsinvolved in time integration are important, but also spatial resolution contributesin an early laminarization.

The time step used in DNS 3 is 4.5 times higher than what Kim et al. [27] used(0.067), in their explicit simulation. Nevertheless, in our SIMPLE algorithm thelaminarization starts earlier than the implicit algorithm of Choi and Moin [26], as

their algorithm laminarization starts after 1:2n=u2s . This might answer the questionthey raised. Having used second order accurate Crank-Nicolson scheme in theirarticle [26], they noticed that the laminarization begins earlier than using the

Figure 5. Effect of time steps on the root-mean-square of different turbulent statistics is shown. In these

test cases (DNS 1, DNS 2), a second order accurate central difference scheme is used for convection terms,

and 129 nodes were used in the wall-normal direction (color figure available online).


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

Kolmogorov time scale. The Kolmogorov time scale for this channel flow is about2.4 wall units, while the laminarization occurs for any time step larger than 1.2. Theywere suspicious of the amplification factor of the Crank-Nicolson scheme, yet theyruled out this possibility. Considering even earlier laminarization when we used afirst order accurate Euler scheme, the laminarization might be attributed to lowaccuracy of numerical schemes used in both methods. Perhaps a third or fourthorder time advancement scheme allow implicit algorithms to use time steps closeto Kolmogorov time scales. However, the numerical cost for these higher accurateschemes must be carefully studied.

To test the performance of SIMPLE algorithm for a courser spatial resolution,

the uniform spacings used in the streamwise and spanwise directions of Dxþ � 35

Figure 6. Effect of spatial resolution on LES simulations on the root-mean-square of different turbulent

statistics is shown. In these test cases (LES 1, LES 5), a second order accurate central difference scheme is

used for the convection terms (color figure available online).

838 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

and Dzþ � 5ð16� 32Þ increased to Dxþ � 43 and Dzþ � 10ð13� 16Þ. The results areshown in Figure 6. Less accurate yet acceptable turbulent statics are obtained.Results of DNS 1, LES 1, and LES 5 ensure us the SIMPLE algorithm is able tomaintain turbulent fluctuations even on a course mesh. This suggests no concernregarding the simplification in the iterative formula of the pressure-correctionequation [30–31].

The performance of the SIMPLE algorithm is furthered investigated by intro-ducing subgird models for two different spatial resolutions (LES 9–12). Smagorinskyand dynamic Smagorinsky models are used to simulate the channel flow. The resultsin Figure 7 demonstrate low performance of both models. The results also show thatspatial resolution affects the dynamic Smagorinsky more than Smagorinsky model.For this test case, Smagorinsky model obtains more accurate solutions when spatialresolution is decreased. Generally, the added turbulent viscosity by both models doesnot improve the result of the simulation; nonetheless dynamic Smagorinsky modelconverges to DNS solution better than Smagorinsky model. These results are alignedwith the results of other references such as [18].

Figure 7. Root-mean-square of different turbulent statistics is compared using Smagorinsky and dynamic

Smagorinsky models for different test cases: LES 6, LES 9, LES 10, LES 11, and LES 12. Their results are

also compared with DNS (color figure available online).


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

LES 13–16 are designed to study the effect of two different schemes forconvective terms. LES 13 uses a third-order accurate Quadratic Upwind Interp-olation for Convective Kinematics or QUICK [53]. The dissipative nature of thisupwinding scheme damps all the turbulent fluctuations and therefore a laminarsolution is obtained. However, we noted if the number of nodes in streamwisedirection is increased, a turbulent solution can be achieved. Hence, the resultsobtained using QUICK in LES 14 are compared with the results of LES 16 inwhich a second order central difference is used. Superior results of a second ordercentral difference does not justify using the more expensive QUICK scheme forthese types of simulations (Figure 8). Worth noting, as shown in reference [7] moresatisfactory results are expected for higher order upwinding schemes, yet thenumerical cost must be justified. Finally, LES 15 shows adding turbulent viscosityusing a Smagoronisky model removes all turbulent modes and therefore a laminarsolution is found.

3.2. Higher Accuracy, Lower Computational Cost

It was shown in the previous section that an implicit algorithm allows us to uselarger time steps to simulate turbulent flows. However, it was shown both in Figures 4and 5 that less accurate solutions are obtained when larger time steps are used. In thissection, we devise a method by which more accurate solutions can be obtained. We

Figure 8. Effect of different numerical schemes for the convection terms is shown. LES 14 and LES 16 use

a third order QUICK and second order central difference schemes, respectively (color figure available

online).

840 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

follow a similar methodology as in Richardson’s extrapolations technique [60], yet it iscustomized for this particular application. Assume we have estimated a turbulentquantity h using a particular time discretization method. Therefore,

h|{z}ExactSolution

¼ hðDtÞ|fflffl{zfflffl}NumericalApproximation

þEðDtÞ|fflffl{zfflffl}Error

ð18Þ

Figure 9. More accurate solutions using previous solutions. Combined solutions are obtained using LES 2

and LES 3 (color figure available online).


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

If two different time steps are used to estimate h then,

h ¼ h1 Dt1ð Þ þ E1 Dt1ð Þ ð19Þ

h ¼ h2 Dt2ð Þ þ E2 Dt2ð Þ ð20Þ

Having an estimate for the time discretizationmethod, one can approximate error andimprove the solution. For instance, in a standard SIMPLE algorithm numerical timeintegration method has an error of

E Dtð Þ / O Dtð Þ ) E1 Dt1ð ÞE2 Dt2ð Þ ffi

Dt1Dt2

ð21Þ

If we assume Dt2 < Dt1, then a better estimation for h can be found using Eq. (21):

h ffi h2 Dt2ð Þ þ 1Dt1Dt2

� 1h2 Dt2ð Þ � h1 Dt1ð Þ½ � ð22Þ

We call this more accurate estimation a combined solution. To illustrate the effect ofthis formulation, we use the solutions of LES 2 (h2) and LES 3 (h1). Using Eq. (22),their combined solution is shown in Figure 9. The combined solution closely followsthe solutions of LES 1 which uses a considerably smaller time step. To compare thecost of finding a solution for LES 1 and the combined solution, assume we need tointegrate flow equations for 100 nondimensional time steps. Therefore, for LES 1,LES 2, and LES 3 we need 25,000, 1,000, and 500 realizations, respectively. Thismeans the combined solution only needs 1,500 realizations which is 16.6 times lessthan the number of realizations for LES 1. Taking advantage of larger time steps inDNS simulations, we can further reduce the computational cost. Assume combiningthe solution of h2 Dt2 ¼ 0:2ð Þ and h1 Dt1 ¼ 0:4ð Þ then for 100 nondimensional timesteps, we only need 500 and 250 realizations, respectively; comparing it to the numeri-cal cost of DNS 1, there is around 33 times reductions in the numerical cost. Not onlywas an accurate solution obtained, but the cost of time integration was considerablyreduced. Because of the flexibility of using large time steps, this combination techniqueis particularly attractive for implicit algorithms, whereas it becomes insignificant forexplicit algorithms.

4. DIFFERENT LINEAR SOLVERS

To compare the cost effectiveness of different sparse linear solvers for thesetypes of simulations, we tested seven common different solvers, which are listed inTable 2. Except for the TDMA method, all other methods were imported fromthe CXML library [62]. The overall error was set to 10�6 for all equations. To mea-sure CPU times, we started a time integration using a fully turbulent stationary sol-ution of DNS 2. The relative CPU time shown in the table clearly shows the TDMAmethod is the fastest method among these methods. Comparing different CPU timessuggests an extensive investigation for varieties of linear solvers by which implicittime integration of turbulent flows can be reduced.

842 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

5. CONCLUSION

The standard SIMPLE algorithm is tested for the simulation of turbulent chan-nel flow. The algorithm, which is discretized using the control volume method, resultsin satisfactory results both for direct and large eddy simulations when the secondorder central difference scheme is used for convective terms. However, when a thirdorder accurate QUICK scheme was used, early laminarazation was observed in com-parison to when the second order central difference was used; hence to obtain a tur-bulent solution, one has to use higher spatial resolution to avoid the laminarazation.Not only does the dissipative nature of QUICK scheme prohibit large eddy simula-tion to maintain unstable modes for low spatial resolution, but it also results in lessaccurate results in comparison with the second order central difference scheme.

We took advantage of an implicit time integration scheme of SIMPLE algor-ithm to increase the size of the time steps. Even though the first order Euler methodis used, a turbulent solution is obtained with time steps which are 4.5 times largerthan explicit methods. Compared with the results of other references [26, 29],laminarization is considerably affected by both the accuracy of the time integrationmethod and spatial resolution. Although our results confirm the time step restrictionreported in reference [29], we found the laminirazation can be delayed or promptedby spatial resolution. The higher the spatial resolutions are, the higher time steps thatcan be used. It was discussed that first and second order time advancement methodsdo not allow using a time step as large as the Kolmogorov time scale. Therefore, itwould be an interesting test using third or higher order implicit algorithm to studywhether a turbulent simulation can be obtained using time steps comparable tothe Kolmogorov time scale.

We found utilizing the Smagorinsky or dynamic Smagorinsky models has anegative effect on the simulation of turbulent channel flow when the SIMPLE algor-ithm is used. Yet, dynamic Smagorinsky model works considerably better than theSmagorinsky model when a higher spatial resolution is used. This trend, however,changes when a course resolution is used. The Smagorinsky model seems to be lessdependent to spatial resolution and therefore results in more accurate solutions forcourser resolutions.

Different spatial resolutions were used to test whether the SIMPLE algorithmresults in satisfactory solutions when a coarse spatial resolution is used. We foundthese types of algorithm are able to produce similar results, as in the populartwo-step projection methods.

Table 2. Comparing CPU time for different sparse linear solvers

Solver Relative CPU time Description

TDMA 1 Single sweeping in each direction

PCGS 7.1 100 internal iterations, error tolerance¼ 10�6

PBCG 9.6 100 internal iterations, error tolerance¼ 10�6

PGMRES 11.1 100 internal iterations, error tolerance¼ 10�6

PTFQMR 11.9 100 internal iterations, error tolerance¼ 10�6

PLSCG >>20 100 internal iterations, error tolerance¼ 10�6

PCG >>20 100 internal Iterations, Error Tolerance¼ 10�6


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

Several popular sparse solvers were tested to solve the implicit linear equations.Among them, TDMA remarkably converges faster than others. Nevertheless, moreresearch needs to be done to find optimal solvers for turbulent solutions.

In general, due to their discretization method (avoiding the splitting operatorerrors), SIMPLE-type algorithms are promising algorithms for general purposesimulations. Their implicit time integration scheme allows more stability, especiallywhen fine mesh is used. Considering the stingy time step restriction of explicit meth-ods, larger time steps comparable to the Kolmogorov time scale can be used whichmight facilitate simulations considerably. Benefiting from larger time steps inimplicit methods, we showed how accurate solutions (combined solutions) can beobtained while the numerical cost is considerably reduced. Although we estimated16 and 33 times reduction in the numerical cost of the LES and DNS simulationsof channel flow, this varies depending on the time steps and the required accuracyof an engineering application.

REFERENCES

1. E. Balaras, C. Benocci, and U. Piomelli, Finite-Difference Computations of High Rey-nolds Number Flows Using the Dynamic Subgrid-Scale Model, Theor. Comp. FluidDyn., vol. 7, no. 3, pp. 207–216, 1995.

2. J. Bardina, J. H. Ferziger, and W. C. Reynolds, Improved Turbulence Models Based onLES of Homogeneous Incompressible Turbulent Flows, Department of MechanicalEngineering, Report No. TF-19, Stanford University, Stanford, CA, 1984.

3. T. Brandt, A Priori Tests on Numerical Errors in Large Eddy Simulation Using FiniteDifferences and Explicit Filtering, Int. J. Numer. Meth. Fl., vol. 51, no. 6, pp. 635–657,2006.

4. M. Germano, U. Piomelli, P. Moin, and W. H. Cabot, A Dynamic Subgrid-Scale EddyViscosity Model, Phys. Fluids A, vol. 3, no. 7, pp. 1760–1765, 1991.

5. O. Desjardina, G. Blanquart, G. Balarac, and H. Pitsch, High Order Conservative FiniteDifference Scheme for Variable Density Low Mach Number Turbulent Flows, J. Comput.Phys., vol. 227, no. 15, pp. 7125–7159, 2008.

6. B. J. Geurts, Elements of Direct and Large- Eddy Simulation, R. T. Edward, 2004.

7. P. Moin and J. Kim, Numerical Investigation of Turbulent Channel Flow, J. Fluid Mech.,vol. 118, pp. 341–377, 1982.

8. B. Song, G. R. Liu, and R. S. Amano, Applications of a Higher-Order BoundedNumerical Scheme to Turbulent Flows, Int. J. Numer. Meth. Fl., vol. 35, pp. 371–394,2001.

9. A. V. Trofimova, A. E. Tejada-Martinez, K. E. Jansen, and R. T. Lahey Jr., DirectNumerical Simulation of Turbulent Channel Flows Using a Stabilized Finite ElementMethod, Comput. Fluids, vol. 38, no. 4, pp. 924–938, 2009.

10. L. Y. Hu, L. X. Zhou, and Y. H. Luo, Large-Eddy Simulation of the Sydney SwirlingNonPremixed Flame and Validation of Several Subgrid-Scale Models, Numer. HeatTransfer, B, vol. 53, no. 1, 2008.

11. S. L. Rani, C. E. Smith, and A. C. Nix, Boundary-Layer Equation-Based Wall Model forLarge-Eddy Simulation of Turbulent Flows with Wall Heat Transfer, Numer. Heat Trans-fer, B, vol. 55, no. 2, 2009.

12. G. E. Lau, G. H. Yeoh, V. Timchenko, and J. A. Reizes, Large-Eddy Simulation of Tur-bulent Natural Convection in Vertical Parallel-Plate Channels, Numer. Heat Transfer, B,vol. 59, no. 4, 2011.

844 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

13. M. M. Rai and P. Moin, Direct Simulations of Turbulent Flow Using Finite-DifferenceSchemes, J. Comput. Phys., vol. 96, no. 1, pp. 15–53, 1991.

14. F. M. Najjar and D. K. Tafti, Evaluation of the Dynamic Subgrid-Scale Stress Model inFinite Differenced LES: Effects of Grid Resolution and Inhomogeneous Test Filtering,Symposium on LES and DNS of Turbulent Flow, 242 FED, pp. 229–236, 1996.

15. F. M. Najjar and D. K. Tafti, Study of Discrete Test Filters and Finite DifferenceApproximations for the Dynamic Subgrid-Scale Stress Model, Phys. Fluids, vol. 8, no.4, pp. 1076–1088, 1996.

16. D. Tafti, Comparison of Some Upwind-Biased High-Order Formulations with aSecond-Order Central-Difference Scheme for Time Integration of the IncompressibleNavier-Stokes Equations, Comput. Fluids, vol. 25, no. 7, pp. 647–665, 1996.

17. J. G. M. Eggels, F. Unger, M. H. Weiss, J. Westerweel, R. J. Adrian, R. Friedrich, and F.T. M. Nieuwstadt, Fully Developed Turbulent Pipe Flow: A Comparison Between DirectNumerical Simulation and Experiment, J. Fluid Mech., vol. 268, pp. 175–209, 1994.

18. P. Majander and T. Siikonen, Evaluation of Smagorinsky-Based Subgrid-Scale Models ina Finite- Volume Computation, Int. J. Numer. Meth. Fl., vol. 40, pp. 735–774, 2002.

19. J. Meyers, B. J. Geurts, and P. Sagaut, A Computational Error-Assessment of CentralFinite-Volume Discretizations in Large-Eddy Simulation Using a Smagorinsky Model,J. Comput. Phys., vol. 227, pp. 156–173, 2007.

20. P. Sagaut, and M. Ciardi, A Finite-Volume Variational Multiscale Method Coupled witha Discrete Interpolation Filter for Large-Eddy Simulation of Isotropic Turbulence andFully Developed Channel Flow, Phys. Fluids, vol. 18, 115101, 2006.

21. P. K. Subbareddy, and G. V. Candler, A Fully Discrete, Kinetic Energy ConsistentFinite-Volume Scheme for Compressible Flows, J. Comput. Phys., vol. 228, no. 5, pp.1347–1364, 2009.

22. X. Xu, J. S. Lee, and R. H. Pletcher, A Compressible Finite Volume Formulation forLarge Eddy Simulation of Turbulent Pipe Flows at Low Mach Number in CartesianCoordinates, J. Comput. Phys., vol. 203, no. 1, pp. 22–48, 2005.

23. P. S. Zandonade, J. A. Langford, and R. D. Moser, Finite-Volume Optimal Large-EddySimulation of Isotropic Turbulence, Phys. Fluids, vol. 16, no. 7, pp. 2255–2271, 2004.

24. L. Xu, G. Cui, C. Xu, Z. Wang, Z. Zhang, and N. Chen, High Accurate Finite VolumeMethod for Large Eddy Simulation of Complex Turbulent Flows, Int. J. Turbo. JetEng., vol. 23, no. 3, pp. 191–210, 2006.

25. J. Kim and P. Moin, Application of a Fractional-Step Method to IncompressibleNavier-Stokes Equations, J. Comput. Phys., vol. 59, no. 2, pp. 308–323, 1985.

26. H. Choi and P. Moin, Effects of Computational Time Step on Numerical Solutions ofTurbulent Flow, J. Comput. Phys., vol. 113, pp. 1–4, 1994.

27. J. Kim, P. Moin, and R. Moser, Turbulence Statistics in Fully Developed Channel Flowat Low Reynolds Number, J. Fluid Mech., vol. 177, pp. 133–166, 1987.

28. Y. Hou and K. Mahesh, A Robust Colocated Implicit Algorithm for Direct NumericalSimulation of Compressible Turbulent Flows, J. Comput. Phys., vol. 205, pp. 205–221,2005.

29. T. Park, Effects of Time-Integration Method in a Large-Eddy Simulation Using thePISO Algorithm: Part I- Flow Field, Numer. Heat Transfer A., vol. 50, no. 3, pp. 229–245, 2006.

30. J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, Springer-Verlag,New York, 1999.

31. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corp.,New York, 1980.

32. J. B. Perot, Comments on the Fractional Step Method, J. Comput. Phys., vol. 121, no. 1,pp. 190–191, 1995.


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

33. J. B. Perot, An Analysis of the Fractional Step Method, J. Comput. Phys., vol. 108, no. 1,pp. 51–58, 1993.

34. C. Pozrikidis, On the Relationship Between the Pressure and the Projection Function inthe Numerical Computation of Viscous Incompressible Flow, Eur. J. Mech. B-Fluid.,vol. 22, no. 2, pp. 105–121, 2003.

35. Y. Wang and Y. Ren, Accuracy Analysis of Approximate Projection Methods for SolvingIncompressible Flows, Qinghua Daxue Xuebao=J. of Tsinghua U., vol. 48, no. 2, pp. 280–284, 2008.

36. S. V. Patankar, and D. B. Spalding, A Calculation Procedure for Heat, Mass and Momen-tum Transfer in Three-Dimensional Parabolic Flows, Int. J. Heat Mass Transfer, vol. 15,pp. 1767–1806, 1972.

37. B. R. Baliga and S. V. Patankar, A Control-Volume Finite Element Method for Two-Dimensional Incompressible Fluid Flow and Heat Transfer, Numer. Heat Transfer, PartB, vol. 6, pp. 245–261, 1983.

38. I. Demirdzic, and S. Muzaferija, Numerical Method for Coupled Fluid Flow, Heat Trans-fer and Stress Analysis Using Unstructured Moving Meshes with Cells of Arbitrary Top-ology, Comput. Methods Appl. Mech. Eng., vol. 125, pp. 235–255, 1995.

39. C. F. Hsu, A Curvilinear-Coordinate Method for Momentum, Heat and Mass Transfer inDomains of Irregular Geometry, Ph.D. thesis, University of Minnesota, Minneapolis,MN, 1981.

40. K. C. Karki and S. V. Patankar, Calculation Procedure for Viscous IncompressibleFlows in Complex Geometries, Numer. Heat Transfer, Part B, vol. 14, pp. 295–307,1986.

41. F. S. Lien, A Pressure-Based Unstructured Grid Method for All-Speed Flows, Int. J.Numer. Methods Fluids, vol. 33, pp. 355–374, 2000.

42. C. R. Maliska and G. D. Raithby, A Method for Computing Three-Dimensional FlowsUsing Non-Orthogonal Boundary-Fitted Coordinates, Int. J. Numer. Methods Fluids,vol. 4, pp. 519–537, 1984.

43. S. R. Mathur and J. Y. Murthy, A Pressure-Based Method for Unstructured Meshes,Numer. Heat Transfer A, vol. 31, pp. 195–216, 1997.

44. S. V. Patankar, Calculation Procedure for Two-Dimensional Elliptic Situations, Numer.Heat Transfer, Part B, vol. 4, no. 4, pp. 409–425, 1981.

45. C. Prakash and S. V. Patankar, A Control-Volume-Based Finite-Element Method forSolving the Navier-Stokes Equations Using Equal-Order Velocity-Pressure Interpolation,Numer. Heat Transfer, Part A, vol. 8, pp. 259–280, 1985.

46. W. Shyy, S. S. Tong, and S. M. Correa, Numerical Recirculating Flow CalculationUsing a Body-Fitted Coordinate System, Numer. Heat Transfer, Part B, vol. 8, pp.99–113, 1985.

47. M.-J. Ni and M. A. Abdou, A Bridge Between Projection Methods and SIMPLE TypeMethods for Incompressible Navier–Stokes Equations, Int. J. Numer. Meth. Eng., vol.72, pp. 1490–1512, 2007.

48. H. Suzuki and H. Usui, Direct Numerical Simulation of a Turbulent Duct Flow byHigh-Ordered Scheme for SIMPLE, The 13th Japanese Symposium of Computational FluidDynamics (JSCFD), pp. B04–B03. Tokyo, Japan, 1999.

49. C. F. Lange, M. Schafer, and F. Durst, Local Block Refinement with a Multigrid FlowSolver, Int. J. Numer. Methods Fluids, vol. 38, pp. 21–41, 2002.

50. F. Moukalled and M. Darwish, A High-Resolution Pressure Based Algorithm for FluidFlow at All Speeds, J. Comput. Phys., vol. 168, pp. 101–133, 2001.

51. M. Ijaz and N. K. Anand, Simulation of Unsteady Incompressible Viscous Flow UsingHigher-Order Implicit Runge-Kutta Methods-Staggered Grid, Numer. Heat Transfer B,vol. 52, pp. 471–488, 2007.

846 H. MONTAZERI

Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

52. S. Acharya, B. R. Baliga, K. Karki, J. Y. Murthy, C. Prakash, and S. P. Vanka,Pressure-Based Finite-Volume Methods in Computational Fluid Dynamics, J. HeatTransfer, vol. 129, pp. 407–424, 2007.

53. B. P. Leonard, A Stable and Accurate Convective Modeling Procedure Based on Quad-ratic Upstream Interpolation, Comput. Method. Appl. M., vol. 19, pp. 59–98, 1979.

54. M. Lesieur, Turbulence in Fluids, 4th ed., Springer, London, 2008.55. P. Sagaut, Large Eddy Simulation for Incompressible Flows, 3rd ed., Springer-Verlag, Ber-

lin, 2006.

56. U. Piomelli, High Reynolds Number Calculations Using the Dynamic Subgrid-ScaleStress Model, Phys. Fluids A, vol. 5, no. 6, 1993.

57. Y. Zang, R. L. Street, and J. R. Koseff, A Dynamic Mixed Subgrid Model and Its Appli-cation to Turbulent Recirculation Flows, Physics Fluids A, vol. 5, no. 12, 1993.

58. S. V. Patankar, C. Liu, and E. Sparrow, Fully Developed Flow and Heat Transfer inDucts Having Streamwise-Periodic Variations of Cross-Sectional Area, ASME J. HeatTransfer, vol. 99, pp. 180–186, 1977.

59. H. Montazeri, A Consistent Numerical Method for Simulating Interfacial TurbulentFlows, Ph.D. thesis, the University of Toronto, Toronto, Canada, 2010.

60. S. C. Chapra and R. P. Canale, Numerical Methods for Engineering, 6th ed., McGrawHill,

New York, 2010.61. J. M. C. Pereira and J. C. F. Pereira, Fourier Analysis of Several Finite Difference

Schemes for the One-Dimensional Unsteady Convection–Diffusion Equation, Int. J.Numer. Meth. Fluids, vol. 36, pp. 417–439, 2001.

62. Compaq Extended Math Library Reference Guide, Compaq Computer CorporationHouston, Texas, Jan. 2001.

63. H. Abe, H. Kawamura, and Y. Matsuo, Direct Numerical Simulation of a FullyDeveloped Turbulent Channel Flow with Respect to Reynolds Number Dependence,ASME J. Fluids Eng., vol. 123, pp. 382–393, 2001.


Dow

nloa

ded

by [

Mon

ash

Uni

vers

ity L

ibra

ry]

at 1

1:51

02

Aug

ust 2

013

Date post:	09-Dec-2016
Category:	Documents
Upload:	hanif
View:	214 times
Download:	1 times

Direct and Large Eddy Simulations Using the SIMPLE Algorithm

Documents