Reynolds Averaged Navier Stokes Computations of Jet Flows Emanating From Turbofan Exhausts Turbofan Egzoz Cikisi Jet Akisinin Reynolds Averaged Navier Stokes Ile Hesaplanmasi

REYNOLDS-AVERAGED NAVIER-STOKES COMPUTATIONS OF JET FLOWS EMANATING FROM TURBOFAN EXHAUSTS

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

SERPİL KAYA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

AEROSPACE ENGINEERING

SEPTEMBER 2008

ii

Approval of the Graduate School of Natural and Applied Sciences

Prof. Dr. Canan ÖZGEN

Director I certify that this thesis satisfies all the requirements as a thesis for the degree of Master of Science.

Prof. Dr. İ. Hakkı TUNCER Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. Prof. Dr. Yusuf ÖZYÖRÜK Supervisor

Examining Committee Members Prof. Dr. Yusuf Özyörük (METU, AEE) Prof. Dr. İ. Hakkı Tuncer (METU, AEE) Assist. Prof. Dr. Oğuz Uzol (METU, AEE) Dr. D. Funda Kurtuluş (METU, AEE) Assist. Prof. Dr. Nilay Sezer Uzol (TOBB ETU, ME)

iii

I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

Serpil KAYA

iv

ABSTRACT

REYNOLDS-AVERAGED NAVIER-STOKES COMPUTATIONS OF JET

FLOWS EMANATING FROM TURBOFAN EXHAUSTS

KAYA, Serpil

M.S., Department of Aerospace Engineering

Supervisor : Prof. Dr. Yusuf ÖZYÖRÜK

September 2008, 64 pages

This thesis presents the results of steady, Reynolds-averaged Navier-Stokes

(RANS) computations for jet flow emanating from a generic turbofan engine

exhaust. All computations were performed with commercial solver FLUENT

v6.2.16. Different turbulence models were evaluated. In addition to

turbulence modeling issues, a parametric study was considered. Different

modeling approaches for turbulent jet flows were explained in brief, with

specific attention given to the Reynolds-averaged Navier-Stokes (RANS)

method used for the calculations.

v

First, a 2D ejector problem was solved to find out the most appropriate

turbulence model and solver settings for the jet flow problem under

consideration. Results of one equation Spalart-Allmaras, two-equation

standart k-ε, realizable k-ε, k-ω and SST k-ω turbulence models were

compared with the experimental data provided and also with the results of

Yoder [21]. The results of SST k-ω and Spalart-Allmaras turbulence models

show the best agreement with the experimental data. Discrepancy with the

experimental data was observed at the initial growth region of the jet, but

further downstream calculated results were closer to the measurements.

Comparing the flow fields for these different turbulence models, it is seen

that close to the onset of mixing section, turbulence dissipation was high for

models other than SST k-ω and Spalart-Allmaras turbulence models. Higher

levels of turbulent kinetic energy were present in the SST k-ω and Spalart-

Allmaras turbulence models which yield better results compared to other

turbulence models. The results of 2D ejector problem showed that

turbulence model plays an important role to define the real physics of the

problem.

In the second study, analyses for a generic, subsonic, axisymmetric turbofan

engine exhaust were performed. A grid sensitivity study with three different

grid levels was done to determine grid dimensions of which solution does

not change for the parametric study. Another turbulence model sensitivity

study was performed for turbofan engine exhaust analysis to have a better

understanding. In order to evaluate the results of different turbulence models,

both turbulent and mean flow variables were compared. Even though

turbulence models produced much different results for turbulent quantities,

their effects on the mean flow field were not that much significant.

vi

For the parametric study, SST k-ω turbulence model was used. It is seen

that boundary layer thickness effect becomes important in the jet flow close

to the lips of the nozzles. At far downstream regions, it does not affect the

flow field. For different turbulent intensities, no significant change occurred in

both mean and turbulent flow fields.

vii

Keywords: Shear Layer, Axisymmetric Subsonic Jet, Reynolds-averaged

Navier-Stokes, Jet Noise, Computational Fluid Dynamics, Turbulence,

Ejector.

viii

ÖZ

TURBOFAN EGZOZ ÇIKIŞI JET AKIŞININ REYNOLDS-AVERAGED

NAVIER-STOKES İLE HESAPLANMASI

KAYA, Serpil

Yüksek Lisans, Havacılık ve Uzay Mühendisliği Bölümü

Tez Yöneticisi : Prof. Dr. Yusuf ÖZYÖRÜK

Eylül 2008, 64 sayfa

Bu tezde durağan, Reynolds-averaged Navier-Stokes denklemleri

kullanılarak tipik bir turbofan egzoz çıkışındaki jet akışının sayısal çözümleri

sunulmuştur. Tüm hesaplamalar ticari yazılım FLUENT v6.2.16 kullanılarak

yapılmıştır. Farklı türbülans modelleri kullanılarak yapılan değerlendirmeye

ek olarak parametrik bir çalışma da yapılmıştır. Türbülanslı jet akışlarını

modellemeye ilişkin yaklaşımlar kısaca açıklanmış, çalışmada kullanılan

RANS metodu detaylı olarak işlenmiştir.

İlk olarak, iki boyutlu bir ejektör geometrisi için çözüm yapılarak, jet akışını

modellemeye en uygun türbülans modelinin seçilmesi ve çözücü ayarlarının

ix

belirlenmesi istenilmiştir. Bir denklemli Spalart-Allmaras, iki denklemli

standart k-ε, realizable k-ε, k-ω ve SST k-ω türbülans modelleri ile Yoder’in

[21] çalışmasının sonuçları deneysel veri ile karşılaştırılmıştır. SST k-ω ve

Spalart-Allmaras modelleri deneysel veriye en yakın sonuçları vermiştir.

Deneysel veri ile uyuşmazlık özellikle jetin ilk oluşmaya başladığı bölgede

görülmüştür, jetin aşağı kesimlerinde sonuçlar deneysel veri ile

örtüşmektedir. Akış alanları karşılaştırıldığında, SST k-ω ve Spalart-

Allmaras haricindeki modeller, türbülanslı alanı daha erken yok etmiştir. SST

k-ω ve Spalart-Allmaras modelleri ile elde edilen çözümlerde ki türbülans

kinetik enerji seviyesinin daha yüksek olduğu görülmüştür. İki boyutlu ejektör

geometrisi için yapılan çalışmalar, türbülans modeli belirlemenin problemin

gerçek fiziğini yansıtması açısından çok önemli olduğunu göstermiştir. İkinci

bir çalışma olarak da gerçekçi boyutlarda, ses altı hızdaki, tipik eksenel

simetrik bir turbofan egzoz çıkışındaki akış için hesaplamalar yapılmıştır. Üç

farklı seviyede hazırlanan çözüm ağları ile yapılan çalışmaların sonuçları

değerlendirilerek, parametrik çalışma için uygun olan çözüm ağı

belirlenmiştir. Bu konfigürasyon için de, söz konusu akış şartlarında, farklı

türbülans modelleri kullanılarak çözümler alınmıştır. Elde edilen sonuçlar,

ortalama akış değerleri ve türbülans değişkenlerine bakılarak kıyaslanmıştır.

Ortalama akış değerleri büyük değişiklik göstermezken, türbülans

değişkenlerinde farklılıklar gözlemlenmiştir.

Parametrik çalışma için SST k-ω türbülans modeli kullanılmıştır. Bu

çalışmada sınır tabakası kalınlığının ve sınır bölgelerindeki türbülans

yoğunluğu değerlerinin akışa olan etkisi incelenmiştir. Sınır tabakası kalınlığı

özellikle lülenin uç kısımlarına yakın yerlerdeki jet akışını etkilemiştir. Aşağı

kesimlerde ise etkisini yitirdiği görülmüştür. Türbülans yoğunluğunun ise

hem ortalama hem de türbülans değerleri üzerindeki etkisinin önemsiz

olduğu gözlemlenmiştir.

x

Anahtar Kelimeler: Kayma Tabakası, Eksenel Simetrik Ses Altı Hızda Jet,

Reynolds-averaged Navier-Stokes, Jet Gürültüsü, Hesaplamalı Akışkanlar

Dinamiği, Türbülans, Ejektör.

xi

To the owner of everything, To My Parents,

To KAYA and GÜLSEREN families

xii

ACKNOWLEDGEMENTS The author wishes to express his gratitude to Prof. Dr. Yusuf ÖZYÖRÜK for guiding this work. The author would also like to thank Assist. Prof. Dr. Oğuz UZOL, Dr. D. Funda KURTULUŞ for their help during the thesis work and also to all committee members. Special thanks go to the colleagues and friends of the author, Büşra AKAY, Samih A. HAMID, N. Erkin OCER, Özgür DEMİR, Mustafa KAYA, Tahir TURGUT, Fatih KARADAL, Mehmet KARACA, Evrim DİZEMEN, Ali AKTÜRK. This work wouldn’t be done without the support of the author’s altruistic parents Şule, Hanifi KAYA, and dear sisters Serap, Seval and Sema. This study was supported for two terms by TÜBİTAK (Turkish Research and Science Council). The author would like to thank for the financial support to TÜBİTAK.

xiii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ............................................................................ xii

TABLE OF CONTENTS.............................................................................. xiii

LIST OF TABLES.........................................................................................xv

LIST OF FIGURES ..................................................................................... xvi

CHAPTER 1.................................................................................................. 1

INTRODUCTION .......................................................................................... 1

1.1 Motivation......................................................................................... 1

1.2 Literature Survey .............................................................................. 3

1.3 Objectives of the Present Research ................................................. 5

CHAPTER 2.................................................................................................. 7

THEORY OF TURBULENCE MODELING.................................................... 7

2.1 Turbulence Modeling and Simulation .................................................. 7

2.1.1 Direct Numerical Simulation (DNS)............................................... 8

2.1.2 Large Eddy Simulation (LES)........................................................ 8

2.1.3 Reynolds-averaged Navier-Stokes (RANS) .................................. 9

CHAPTER 3................................................................................................ 13

RANS CFD CALCULATIONS OF JET FLOWS .......................................... 13

3.1 RANS Computations for a 2D Ejector Nozzle ................................... 15

3.2.1 Problem Definition ...................................................................... 15

xiv

3.2.2 Computational Details................................................................. 17

3.2.2.1 Computational Grid.................................................................. 17

3.2.2.2 Boundary Conditions ............................................................... 18

3.2.3 Results and Discussion .............................................................. 19

3.3 Generic Turbofan Exhaust Analysis .................................................. 29

3.3.1 Problem Definition ...................................................................... 29

3.3.2 Computational Details................................................................. 30

3.3.2.1 Computational Grid.................................................................. 30

3.3.3.2 Boundary Conditions ............................................................... 33

3.4 Grid Sensitivity .................................................................................. 34

3.5 Turbulence Model Sensitivity............................................................. 37

3.6 Parametric Study............................................................................... 47

3.6.1 Effect of Boundary Layer Thickness on Turbulent Shear Layer.. 47

3.6.2 Effect of Turbulence Intensity ..................................................... 54

CONCLUSION............................................................................................ 59

REFERENCES ........................................................................................... 61

xv

LIST OF TABLES

Table 3.1 Flow conditions for ejector nozzle [21].

Table 3.2 Grid Dimensions for three separate zones.

Table 3.3 Flow conditions for generic turbofan dual stream engine nozzle.

Table 3.4 Grid dimensions of zones.

xvi

LIST OF FIGURES

Figure 1.1 Maximum perceived noise levels for different components [7].

Figure 2.1 Time averaging of velocity.

Figure 3.1 Schematic of the two-dimensional ejector nozzle [21].

Figure 3.3 Computational mesh for the 2D ejector.

Figure 3.4 Residuals with iterations for SST k-ω turbulence model.

Figure 3.5 Velocity profile for 2D ejector at x=0.0762 m.




Figure 3.9 Total temperature for 2D ejector profile at x=0.0762 m.


Figure 3.11 Axial velocity contours for the 2D ejector nozzle calculated with

three different turbulence models

Figure 3.12 Turbulent kinetic energy (m2/s2) contours for the 2D ejector

nozzle calculated with three different turbulence models.

Figure 3.13 Sketch of generic turbofan dual stream engine nozzle.

Figure 3.14 Three-block computational grid for axsymmetric generic turbofan

engine exhausts.

Figure 3.15 Details of the computational mesh.

Figure 3.16 Schematic of the boundary conditions.

xvii

Figure 3.17 Comparison of axial velocity distribution at x=1, 2, 4, 8 m for 3

levels of grids.

Figure 3.18 Comparison of turbulence intensity distribution at x=1, 2, 4, 8 m

for 3 levels of grids.

Figure 3.19 Schematic of the axial locations at which data are represented.

Figure 3.20 Axial velocity distribution in radial direction at x=-1.7, -1.5, -0.2, 0

m for different turbulence models.

Figure 3.21 Axial velocity distribution in radial direction at x=1, 2, 4, 8 m for

different turbulence models.

Figure 3.22 Axial velocity distribution in radial direction at x=25, 32 m for


Figure 3.23 Turbulent viscosity distribution in radial direction at x=-1.7, -1.5, -

0.2, 0 m for different turbulence models.


m for different turbulent intensities.



Figure 3.26 Turbulent kinetic energy (m2/s2) contours with three turbulent

models.


m for different nozzle wall lengths

Figure 3.28 Axial velocity distribution in radial direction at x=4, 8, 25, 32 m

for different nozzle wall lengths.

Figure 3.29 Turbulent kinetic energy distribution in radial direction at x=-1.7,

-1.5, -0.2, 0 m for different turbulence models.

xviii

Figure 3.30 Turbulent kinetic energy distribution in radial direction at x=4, 8,

25, 32 m for different nozzle wall lengths.

Figure 3.31 Turbulent kinetic energy (m2/s2) contours for different boundary

layer thickness configurations.


different turbulent intensities.



Figure 3.34 Turbulent kinetic energy distribution in radial direction at x=1, 2,

4, 8 m for different turbulent intensities.

Figure 3.35. Turbulent kinetic energy distribution in axial direction at x=25,

32 m for different turbulent intensities.

1

CHAPTER 1

INTRODUCTION

1.1 Motivation

The number of vehicles flying in air has been increasing very fast for both

civilian transport and military operation purposes. One of the important

things is to take many aspects into account while answering for this growing

need. For example, higher performance vehicles are always required which

might be possible with powerful engines, but it is important to keep in mind

that powerful engines mostly come together with higher level of noise

emission and some other atmospheric pollutants. Among these on noise

emission levels there are restrictions and regulations set by international

organizations which must be obeyed by aircraft manufacturers and airline

operators.

There are many research programs including experimental, theoretical and

computational investigations to reduce the level of noise and understand the

associated complicated physical phenomena [1-6]. One of the intense

research focuses in this area is to reduce the jet noise emanating from

exhausts of the engines. However, most of today’s large transport aircraft

are powered by high bypass ratio turbofan engines. This type of engines has

reduced jet speeds and consequently reduced jet noise levels. Nevertheless,

2

during especially take-off fan noise may be heard more than other noise

components due to increased fan loadings. Fan noise propagates forward

through the engine air intake and rearward through the bypass duct, and

then radiate to far-field. As can be seen in the Figure 1.1 below, a significant

amount of turbomachinery noise is radiated to the far-field through the core

and fan exhaust compared to other components [7]. Sound waves exiting

the bypass or core ducts radiate to far-field through the exhaust jets. These

waves are refracted by the jets and some amplification effects occur during

this process.

60 70 80 90 100 110

Turbine

Airframe

Combustion Core

Fan Inlet

Core Exhaust

Fan Exhaust

Total

Maximum perceived noise levels, dB

Figure 1.1 Maximum perceived noise levels for different components [7].

Considering the numerical studies, there are different approaches to predict

the far field noise. One of the methods is to use a two-step calculation. First

step is based on obtaining mean flow field and the second step corresponds

to the prediction of acoustic propagation by using the linearized Euler

equations [8-11]. This method brings the necessity of good representation of

3

the background turbulent jet flow field to predict the far-field noise accurately

because of the interactions between the acoustic waves and the mean flow.

These interactions may cause important refraction and amplification effects

[8]. In this approach of predicting the turbomachinery noise radiation from

engine exhaust, the required mean flow field is usually obtained by solving

the Reynolds-averaged Navier-Stokes equations (RANS). Motivated by this,

the present work studies several issues in computation of the mean

turbulent flow field for a typical subsonic axisymmetric generic turbofan

exhaust configuration, including the effects of turbulence modeling issues. In

addition to that, a parametric study is conducted to have a better

understanding of their influence on the turbulent jet flow field. One of the

parameters is the boundary layer thickness developed inside the core and

bypass ducts as well as on the external wall. The second parameter studied

is the turbulence intensity at the core and bypass inlet boundaries.

1.2 Literature Survey

As mentioned before, one of the approaches to predict the far field sound is

to solve the Euler equations linearized about the mean flow. The mean flow

variables interacting with the turbulent structures should be resolved with

enough accuracy in order to use as inputs for linearized Euler equations. In

the literature, there are many studies for developing models or

computational tools to represent the turbulent characteristics of jet flows. For

example, Koch et al. [12] investigated subsonic axisymmetric separate flow

jets with three flow solvers. Similar two-equation k-ε turbulence models were

implemented in WIND, NPARC, and the CRAFT Navier-Stokes solvers. The

results obtained from the three codes were generally in agreement with the

experimental data. Their results show that the mixing rate was lower than

that indicated by experimental results. The turbulent kinetic energy levels

4

were also lower, which corresponds to the slower mixing rate. Also, the

maximum turbulent kinetic energy locations were predicted further

downstream compared to the data. There are also many attempts to model

the turbulence field by using different turbulence models ranging from

algebraic to complex ones. However, there is not a universal model which

can be generalized to a wide range of problems. For most jet applications

linear two equation k-ε and k-ω models are typically used [13, 14]. Besides

these linear models, a significant amount of work has been done in the area

of advanced RANS models specifically for jets. Kenzakowski et al. [15]

made simulations for noise prediction by using advanced turbulence

modeling for a range of subsonic and supersonic 3D jets. They compared

the data obtained by using advanced explicit algebraic stress model

formulation (EASM/J) and an underlying k-ε turbulence model for a series of

nozzles. Both heated and cold jets were considered. Flow field results for

the cold jet cases indicate good agreement with available centerline data.

Hot jet comparisons were also reasonable. Engblom et al. [16] investigated

a series of hot and cold single flow subsonic nozzle flows including a

baseline round nozzle and several chevron nozzles. The spreading toward

the nozzle centerline was much lower than observed in experiments.

Georgiadis et al. [17] investigated a reference subsonic lobed nozzle flow

with linear two-equation and explicit algebraic stress turbulence models, and

found similar trends. Additionally, far downstream, the computed far field

mixing rate was predicted too high. This literature survey shows that, RANS

turbulence models are not adequate for accurate prediction of jet flow details.

In addition to the modeling issues of the background flow, there are many

studies to understand the physics of the jet flow and effects of some

parameters. For example, Gao et al. [18] made a numerical study of nozzle

5

boundary layer thickness on axisymmetric supersonic jet screech tones. The

axisymmetric Navier-Stokes equations and the two equation k-ε turbulence

model were solved in the generalized curvilinear coordinate system. They

chose a straight nozzle with several different initial boundary layer

thicknesses. Their results showed that the nozzle boundary layer thickness

has an important influence on the generation of supersonic jet screech tones.

They concluded that increasing boundary layer thickness would be an

effective way to reduce the supersonic jet screech tones. Viswanatham [19]

also carried out a detailed parametric study of the noise from dual-stream

jets. He evaluated the importance of the noise generated by primary and

secondary shear layers. He aimed to understand principal radiation

directions under different operating conditions. It is indicated that the

secondary-to-primary jet velocity ratio was an important parameter for

mixing noise while its effect is negligible on shock-associated noise. The

shock-associated noise was found to be dependent on the geometric details

of the nozzle.

1.3 Objectives of the Present Research

The primary goal of this research is to understand the influence of boundary

layer thickness and turbulence intensity on the jet flow development issuing

from the exhaust of generic turbofan engine geometry. The flow at the inlet

boundaries of the engine geometry is subsonic. The flows from the core and

the bypass ducts mix and a shear layer develops. Also a highly developed

shear layer occurs due to gradients through the flow from bypass duct and

co-external flow. Since turbulence plays an important role on the

propagation of acoustic waves in these mixing shear layers, it is crucial to

model the turbulent flow field accurately. The mean flow and turbulence

properties in the jet plume were determined from Computational Fluid

6

Dynamics (CFD) analysis using the Reynolds-averaged Navier-Stokes

(RANS) equations. Different turbulence models were used to close the

RANS equations and a sensitivity analysis was made.

The thesis is organized as follows. In Chapter 2, different approaches to

simulate such turbulent jet flows are explained. Special focus is given on the

Reynolds-averaged Navier-Stokes (RANS) modeling with brief explanations

on two other approaches; Direct Numerical Simulation (DNS) and Large

Eddy Simulation (LES). Chapter 3 is dedicated to the details of the RANS

CFD analysis for two different problems. First one is a 2D ejector problem

with experimental data [20, 21]. The aim to solve this problem is to asses the

flow solver and solver settings and compare with the experimental data. All

computational details are given. To have a better understanding, turbulence

model sensitivity study is also conducted for this case. The second problem

is a subsonic, axisymmetric generic turbofan engine nozzle with realistic

dimensions and operating conditions. The goal is to conduct a study to

understand the influence of some parameters on the mean flow field which

might cause different noise level predictions. The proposed parameters can

be listed as the effect of boundary layer thickness, and influence of

turbulence intensity at the inlet boundaries. Also, discussions for the grid

and turbulence model sensitivity are provided. The concluding remarks are

presented in the final section.

7

CHAPTER 2

THEORY OF TURBULENCE MODELING

2.1 Turbulence Modeling and Simulation

To simulate turbulent flows, many computational approaches and models

have been proposed and many are in use [22-28]. However, it is difficult to

develop an accurate theory or universal model for all types of turbulent flows.

In turbulent flows, the velocity field is three dimensional, time dependent,

and random. The largest turbulent motions are almost as large as the

characteristics width of the flow. Since the characteristic lengths change with

the boundary geometry, the largest turbulent motions are specific to problem

under consideration. As a result, there is a large range of time scales and

length scales. Relative to the largest scales, the smallest timescale

decreases as Re-1/2 and the smallest length scale as Re-3/4 [29]. Therefore,

resolving all the length and time scales of turbulent flows is a difficult task.

As mentioned above, there are different simulation approaches to model a

turbulent flow field. Direct numerical simulation (DNS), large eddy simulation

(LES), and Reynolds-averaged Navier-Stokes (RANS) are used to calculate

the turbulent flows. In this study, calculations were carried out by using only

8

RANS modeling, since the computational cost of DNS and LES is too high

for Reynolds number on the order of 1.5 million. For the sake of

completeness, brief explanations were also presented for DNS and LES

methods.

2.1.1 Direct Numerical Simulation (DNS)

In direct numerical simulation (DNS), the time dependent Navier-Stokes

equations are solved and all of the relevant length scales in the turbulent

flow are resolved. Since the time history of the entire unsteady turbulent flow

field is known, no turbulence modeling is required. It is conceptually the

simplest approach and provides the best accuracy. However, since all length

scales and time scales have to be resolved, DNS is computationally

expensive. The computational cost increases so rapidly with Reynolds

number, therefore this approach is currently limited to flows with low-to-

moderate Reynolds numbers, on the order of 3,000-4,000 for turbulent jets

[29]. As a result, this approach is not feasible for the high Reynolds number

application of modern jet engines.

2.1.2 Large Eddy Simulation (LES)

In large eddy simulation, the larger three-dimensional unsteady turbulent

motions are directly resolved, whereas effects of the smaller scale motions

are modeled. A spatial filter is applied to remove the small scales that are

not resolved by the grid [29]. Because the large scale unsteady motions are

represented explicitly, LES gives accurate and reliable results. However,

presently it is not practical to perform LES calculations for Reynolds

numbers that are consistent with modern jet engines, which can be on the

order of 10 million. When the internally mixed flow region is included, near

9

solid wall boundaries grid resolution becomes higher which increase the

computational cost significantly. Therefore, LES approach is also not

applicable for the real jet problems.

2.1.3 Reynolds-averaged Navier-Stokes (RANS)

The turbulent motion could be easily included into the Navier-Stokes

equations. Statistical methods are used to average the fluctuating properties

of flow in the turbulent case. To obtain the mean values, there are different

averaging techniques such as time, spatial and ensemble averaging. For

homogenous turbulence with uniform turbulent flow in all directions, spatial

averaging is used. For stationary turbulence on the average, does not vary

with time, time averaging is used. But ensemble averaging is the most

suitable averaging for flows decaying in time. For the flows that engineers

mostly deal with, time averaging is used. Time averaging yields an average

and a fluctuating part for a certain variable. These parts could be

represented as the part of the instantaneous parameter, for example,

velocity.

Here ( )txui , is expressed as the instantaneous velocity with, ( )xUi ; average

and ( )txui ,′ fluctuating part.

( ) ( ) ( )txuxUtxu iii ,, ′+=

(2.1 )

10

Figure 2.1 Time averaging of velocity.

If this instantaneous velocity given in Equation 2.1 is added into the Navier-

Stokes equations and then time averaging is applied, so called Reynolds Averaged Navier-Stokes (RANS) equations are obtained.

The quantity ''ij uu ⋅⋅− ρ is known as the Reynolds-stress tensor. In order to

determine the mean-flow properties of the turbulent flow, the value of the

Reynolds stress tensor should be found. The mean flow variables could be

solved (or computed) in the same manner as Navier-Stokes equations, but

the Reynolds stress tensor must be modeled.

Prandtl suggested a model for the eddy viscosity in which the eddy viscosity

is dependent on the kinetic energy of turbulent fluctuations, k. It was the first

introduction of one-equation turbulence model. Kinetic energy per unit mass

is described and related to the Reynolds stress tensor as,

( )''2 ijjijij

ij

i uuSxx

PxUU

tU

⋅⋅−⋅⋅∂∂

+∂∂

−=∂∂⋅⋅+

∂∂⋅ ρµρρ

( 2.2 )

11

Implementation of this relation into Reynolds stress tensor resulted in a

transport equation for the turbulent kinetic energy,

Each term has a different meaning regarding the turbulent flow. Production

term could be regarded as the mechanism of kinetic energy of the mean flow

turning into turbulent kinetic energy where as dissipation appears to be the

term describing the turbulent kinetic energy dissipated as thermal energy.

Last three diffusion terms could be explained as the turbulent energy

diffusion by fluid’s natural molecular transport, diffusion by turbulent

fluctuations and turbulent transport from pressure and velocity fluctuations in

the order of appearance.

The turbulence models do not involve a length scale. Since the turbulent

viscosity includes a velocity and a length scale (on dimensional grounds, the

kinematic viscosity υ appears to be in sm2

that is the product of a velocity

and a length scale), the models without involving a length scale are

regarded as incomplete.

⎟⎠⎞⎜

⎝⎛ ++⋅=

2'2'2'

21 wvuk

kuu iiii ⋅−=⋅−= 2''τ

( 2.3 )

{ ( ) ( )⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⋅⋅−⋅⋅⋅−∂∂

⋅⋅∂∂

+−∂∂⋅=

∂∂

⋅+∂∂

434214434421321434214434421DIFFUSIONPRESSURE

j

TRANSPORTTURBULENT

jii

DIFFUSIONMOLECULAR

jjNDISSIPATIO

PRODUCTION

j

iij

DERIVATIVELSUBSTANTIA

jj uPuuu

xk

xxU

xkU

tk ''''' 1

21

ρνετ

( 2.4 )

12

Kolmogorov introduced the first complete turbulence model by presenting a

time scale known as the rate of dissipation of energy in unit volume and time

and represented as “ω”. The absent length scale is provided by ω2

1k where

ω⋅k is analogue to the dissipation rate “ ε ”. This success of additional

equation showed up as the introduction of two-equation turbulence models.

Since the dissipation term in the “k” equation is evaluated by another

transport for “ω”, the model is complete. Second equation appears as,

Later Wilcox [29] had modified the Kolmogorov’s ω equation and formed

new correlations. Given the basic RANS formulation with one and two

equation formulas, Reference 29 should be referred as a source for detail

explanations.

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂⋅⋅⋅

∂∂

+⋅−=∂∂⋅+

∂∂

jT

jjj xxx

Ut

ωνσωβωω 2

( 2.5 )

13

CHAPTER 3

RANS CFD CALCULATIONS OF JET FLOWS

For many years, Reynolds-averaged Navier-Stokes (RANS) methods have

been used to calculate turbulent jet flow fields. The reason behind the

common usage of RANS methods is the low computational cost for industrial

flows compared to higher order methods like large eddy simulation (LES) or

direct numerical simulation (DNS). RANS methods replace all turbulent fluid

dynamics effects with a turbulence model. However, such turbulence models

have limitations. For jets with significant three-dimensionality, compressibility,

and high temperature streams, turbulence models are not capable to yield

accurate results [30, 31]. CFD analysis based on the RANS equations uses

models for the turbulence that employ many ad hoc assumptions and

empirically determined coefficients. Typically, these models cannot be

applied with confidence to a class of flow for which they have not been

validated.

In this chapter, comparisons of the RANS calculations with different

turbulence models, mostly used in literature for jet flows were made. First

part of the chapter includes the computational results for a 2D ejector nozzle

test case found from literature. It is aimed to set the proper solver settings

and once comparable results with the measurements were achieved, similar

boundary conditions and solver settings were applied to the generic turbofan

14

engine exhaust flow analyses in the second part. Before making a

parametric study on the generic turbofan engine exhaust flow, different

turbulence models have been evaluated for the sake of better understanding

of the sensitivity of the solution to turbulence modeling. Besides the

turbulence model variation, a grid study was performed to find out the grid

dimensions of which solution does not change. Choosing the turbulence

model and grid, a study was performed to understand the influence of some

parameters on the flow physics.

The parametric study includes two issues. First one aims to understand

effect of the different boundary layer thicknesses on the developing shear

layers from the core and bypass ducts. Second item is related with the effect

of turbulence intensity values at the bypass, core and free stream inlet

boundaries which are unknown prior to analyses in most of real applications.

It is proposed that the length of the core and bypass nozzle changing the

boundary layer thickness at the lip of the nozzles might have a significant

influence on the developed shear layer. For jet simulations finite domain is

used to decrease the computational cost. Therefore, influence of the

thickness of the boundary layer has been neglected. In a recent study, it was

shown that it has a significant effect on the screech tones of supersonic jet

[18]. In this study, main concern is to see the influence of boundary layer

thickness for subsonic jets. For this aim, length of the core and bypass duct

walls were doubled. Both the mean and the turbulent state variables were

compared for different boundary layer thicknesses.

15

A second parametric study was performed for different turbulent intensities.

The turbulent intensities at the inflow boundaries are usually not well known

and the values are estimated in analyses. In order to see the sensitivity of

the flow to the turbulence intensity, the results of the parametric study were

evaluated.

3.1 RANS Computations for a 2D Ejector Nozzle

In this section, it is aimed to determine if the RANS calculations can capture

the experimentally observed jet flow development for the 2D ejector

geometry. Different turbulent models were chosen for the computations and

the one which shows the best agreement with the measurements was used

to further examine details of the developing dual stream jets for the generic

turbofan engine exhaust problem.

3.2.1 Problem Definition

The turbulent flow through a two-dimensional ejector nozzle consisted of a

slot type primary nozzle and a two-dimensional mixing section was tested by

Gilbert and Hill [20] as shown schematically in Figure 3.1. The experimental

setup consists of a rectangular variable area channel formed by

symmetrically contoured upper and lower walls and the two flat sidewalls.

The widths of both the primary nozzle discharge slot and the mixing section

were 0.2032 m. Other dimensions of the system were shown in Figure 3.1.

Suction slots were placed in the corners of the mixing section to prevent flow

separation.

16

In this flow, the turbulent mixing occurs between the primary jet and

entrained secondary air. Another feature of the problem is the interaction

with the wall boundary layers. The experimental data to be used for

comparison purposes consists of velocity and temperature measurements at

several axial locations. The inflow conditions used in the numerical

computations were listed in the Table 3.1.

Table 3.1 Flow conditions for ejector nozzle [21].

Pt (Pa) Tt (K) Mach number

Secondary Inlet 101559.7 305.6 0.2

Primary Nozzle 246349.6 357.8 0.07

Downstream 92734.5(static) 305.6

Figure 3.1 Schematic of the two-dimensional ejector nozzle [21].

17

As shown in Table 3.1, the temperature gradient is not high between the

primary nozzle and secondary inlet whereas a high pressure difference

occurs. The flow in the primary nozzle is very slow compared to the

secondary inlet. Therefore, we expect a highly developed shear layer

between the two streams.

Comparisons were made between the FLUENT v6.2.16 realizable k-ε,

standard k-ε, Shear Stress Transport (SST) k-ω, Spalart-Allmaras

turbulence models and the experimental results. Also, the results found from

the literature obtained with the WIND solver were presented [21].

3.2.2 Computational Details

The fluid flow was simulated by solving steady, compressible, 2D, Reynolds-

averaged Navier-Stokes equations using an implicit, coupled, second order,

up-wind flux-difference splitting finite volume scheme and with variable

turbulence models. The computational grid and the boundary conditions

were explained in the following sections.

3.2.2.1 Computational Grid

Due to the symmetric nature of the ejector nozzle and mixing section, only

half of the ejector was modeled numerically. The 131x121 mesh of

Georgiadis, Chitsomboon, and Zhu [32] was used in these calculations. Only

format of the grid was modified in order to make it compatible with FLUENT

v6.2.16. The grid was packed to solid surfaces such that the first point of the

wall corresponded to a y+ between 0.1 and 1.6, depending on the flow

conditions as can be shown in the Figure 3.2. The viscous sublayer was

resolved well. The grid was composed from three blocks for the primary

18

inflow, secondary inflow and mixing regions. Relative size of each block was

shown in Table 3.2.

Table 3.2 Grid Dimensions for three separate zones.

Axial Direction Radial Direction

Primary Nozzle 31 41

Secondary Flow 31 71

Mixing Section 101 121

Figure 3.2 Wall y+ values for the nozzle and upper contoured walls.

3.2.2.2 Boundary Conditions

A schematic of the boundary conditions that are applied to each region of

the computational grid was shown in Figure 3.3. No slip condition was

19

applied at wall boundaries, and walls were assumed to be adiabatic.

Pressure inlet boundary condition was used for the primary nozzle and

secondary flow inlets; total pressures and total temperatures were specified.

Pressure outlet boundary condition was applied to the downstream

boundary, flow parameters were extrapolated from the interior. Symmetry

boundary condition, zero gradients were typically specified on the centerline.

Even though not shown in the figure, no slip boundary condition was applied

on the ejector wall.

Figure 3.3 Computational mesh for the 2D ejector.

3.2.3 Results and Discussion

As mentioned earlier, different turbulent models were used in the

calculations. The results of the standard k-ε, realizable k-ε, Spalart-Allmaras

and the SST k-ω turbulence models in FLUENT v6.2.16 code were

compared with the experimental data. The results carried out with WIND k-ε

turbulence model by Yoder [21] was also compared with the FLUENT

v6.2.16 results.

For the FLUENT v6.2.16 solutions, total 26000 iterations were done to have

a converged solution. Numerical stability problem appeared at the first

20

iterations due to developing shear layer between the primary and secondary

flows. Courant number (CFL) was gradually increased to have a better

convergence. First 8000 iterations were run with a CFL of 0.5, the next 4000

iterations with a CFL of 1, next 4000 with 2 and rest iterations were run by

setting CFL to 3. Continuity, axial velocity, momentum and energy residuals

for SST k-ω turbulence model are shown in the Figure 3.4, as representative.

The residuals of the variables dropped four to seven orders of magnitude,

which is enough to assume a converged solution.

Figure 3.4 Residuals with iterations for SST k-ω turbulence model.

Figure 3.5, Figure 3.6, Figure 3.7 and Figure 3.8 show the axial velocity

profiles obtained by using different turbulence models at x= 0.0762 m, 0.127

m, 0.1778 m and 0.2667 which were measured relative to the primary nozzle

exit plane and the vertical positions relative to the centerline. The vertical

positions were nondimensionalized by the local half-duct length.

21

In literature, for most of the jet flow problems k-ε model has been used in

RANS calculations. However, for the given problem as can be seen from

Figure 3.5 to Figure 3.8, FLUENT v6.2.16 SST k-ω and Spalart-Allmaras

turbulence models produce closer results to the experimental data. Except

at x=0.2667 m, all turbulence models underpredict the axial velocity

distribution. The velocity peak at the centerline was overpredicted for all

turbulence models under consideration.

As can be seen in Figure 3.5, at x=0.0762 m , at the first axial station, all

turbulence models of FLUENT v6.2.16 code and k-ε model with

compressibility correction of WIND code underpredict the axial velocity in the

mixing shear layer. The realizable k-ε turbulence model of FLUENT v6.2.16

code and k-ε model of the WIND code give comparable results. In the mixing

shear layer they underpredict the velocity distribution and give higher peak

velocities at the centerline.

At x=0.127 m, there is still discrepancy with the experimental data. The trend

for the axial velocity profile is similar at x=0.0762 m. All models underpredict

velocity in the mixing layer and higher peaks at the centerline. However,

agreement with the experimental data at this station is better compared to

x=0.0762 m which is closer to the mixing section. Here, again SST k-ω and

Spalart-Allmaras turbulence models yield the best results.

At farther downstream, at x=0.1778 m, the results are in good comparison

with the experimental data. There is still a discrepancy between the WIND

results and the experiment; however, SST k-ω turbulence model result

matches with the experimental data well. Spalart-Allmaras turbulence model

22

slightly overpredicts compared to the experimental data and the SST k-ω

turbulence model results.

Comparing the total temperature profiles at two axial locations; (1) close to

the lip of the nozzle and (2) at the downstream region, again there is a

mismatch with the experimental data. Figure 3.9 and Figure 3.10 show the

total temperature profiles at x=0.0762 m and x=0.266 meters, respectively. It

is interesting to note neither SST k-ω nor Spalart-Allmaras turbulence

models show no significant improvement compared to standard k-ε model as

in the case of axial velocity distribution.

Looking the contour plots for the x velocity and turbulent kinetic energy, the

difference in the solutions can be understood better. Figure 3.11 shows the x

axial velocity distribution for three different turbulence models. As can be

seen, the flow in the primary nozzle accelerates in the convergent sections

of the ejector such that it exceeds the speed at the secondary inlet. At the lip

of the ejector, two streams start to mix composing the shear layer. The

potential core lengths can be observed clearly in Figure 3.11. As shown,

realizable k-ε and standard k-ε turbulence models yield similar results

whereas the potential core length is shorter for SST k-ω turbulence model.

Similar to velocity distribution, turbulent kinetic energy profiles also shed

light on the reasons for the different solutions. As shown in Figure.3.12, the

turbulent kinetic energy diffuses in the downstream region more in the SST

k-ω turbulence model compared to realizable and standard k-ε turbulence

models. There is higher level of turbulent kinetic energy in close regions

where mixing starts for the SST k-ω turbulence model. In realizable and

standard k-ε turbulence models, this kinetic energy is dissipated.

23

Velocity Profile at x=0.0762 m

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350 400Velocity (m/s)

Hei

ght

Experiment FLUENT Realizable k-epsilon FLUENT SST k-omegaWIND k-epsilon [21] FLUENT Standart k-epsilon FLUENT Spalart-Allmaras



0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350

Velocity (m/s)

Hei

ght



24


0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300

Velocity (m/s)

Hei

ght




0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250Velocity (m/s)

Hei

ght



25

Total Temperature Profile at x=0.0762 m

0

0.2

0.4

0.6

0.8

1

1.2

300 305 310 315 320 325 330 335 340 345 350 355Total Temperature (K)

Hei

ght

Experiment FLUENT Realizable k-epsilon FLUENT SST k-omegaWIND k-epsilon [21] FLUENT Standart k-epsilon FLUENT Spallart-Allmaras


Total Temperature Profile at x=0.2667 m

0

0.2

0.4

0.6

0.8

1

1.2

300 305 310 315 320 325 330Total Temperature (K)

Hei

ght


Figure 3.10 Total temperature for 2D ejector profile at x=0.2667 m (10.5 in).

26

Figure 3.11 Axial velocity (m/s) contours for the 2D ejector nozzle calculated with

three different turbulence models.

27

Figure 3.12 Turbulent kinetic energy (m2/s2) contours for the 2D ejector nozzle

calculated with three different turbulence models.

28

For the simulations, no values at the boundaries were given for turbulent

variables in the experiment. In the simulations 2% of turbulence intensity

was assumed. As shown in SST k-ω results, higher level of turbulence at the

mixing region occurs, which results in better agreement with the

experimental measurements.

To conclude, there is a deficiency for accurate prediction of the initial jet

growth region. At further downstream regions, results get closer to the

experimental data. Comparing the different turbulence models, SST k-ω and

Spalart-Allmaras turbulence models are the most promising to yield better

results. For SST k-ω turbulence model, turbulence intensity can be defined

directly as boundary condition whereas it is not possible for Spalart-Allmaras

turbulence model. Since a parametric study concerning the turbulence

intensity will be conducted, SST k-ω turbulence model is chosen to be

proper for further calculations of the second turbofan dual stream jet flow

problem.

29

3.3 Generic Turbofan Exhaust Analysis

3.3.1 Problem Definition

Realistic geometry and engine conditions were chosen to enhance the

usefulness of the data obtained. The effects of boundary layer thickness and

the turbulence intensity on turbulent shear layers emanating from the

exhaust of the turbofan were evaluated in the presence of an external co-

flowing stream.

Figure 3.13 Sketch of generic turbofan dual stream engine nozzle.

The conditions at the inflow boundaries were given in Table 3.3, below. The

flow in the core is high temperature with a higher level of turbulence while

the flow in the fan is at higher pressure, lower temperature and at higher

speed. There are high velocity and pressure gradient between the fan

stream and the external co-flowing stream composing the shear layer.

Given the total values at the inflow boundaries, the static values were

calculated by using isentropic relations to apply as boundary conditions.

30

Table 3.3 Flow conditions for generic turbofan dual stream engine nozzle.

Core Fan Free stream

Pt (Pa) 140000 165000 106000

Tt (K) 720 300 287

Mach 0.68 0.85 0.22

Turbulence intensity 10% 6% 1%

3.3.2 Computational Details

For the background turbulent shear flow computations, as in the validation

case, FLUENT 6.2.16 was used. The computational grid was constructed by

using CFD-GEOM program. The details for the computational grid, boundary

conditions were given in the following parts. The fluid flow was simulated by

solving steady, compressible, axisymmetric, Reynolds-averaged Navier-

Stokes equations using an implicit, segregated, second order, up-wind flux-

difference splitting finite volume scheme and with various turbulence models.

3.3.2.1 Computational Grid

As mentioned above, the grid was created in CFD-GEOM program. This is a

multiblock grid, with 3 blocks as shown in Figure 3.14, below. The domain

extends 50 radius in axial direction and 30 radius in radial direction. The

extent in the axial direction is enough to have a developed shear layer and

far enough in the radial direction to apply far field boundary condition. Since

the problem is axisymmetric, only half of the model was meshed to save

computational time.

31

Table 3.4 Grid dimensions of zones.

Axial Direction Radial direction

Zone1 (Core flow) 747 101

Zone2 (Fan flow) 1013 101

Zone3 (Free stream) 981 211

Grid dimensions for different zones were shown in Table 3.4. As can be

seen, the grid is sufficiently fine in the axial direction to resolve the rapid

changes occurring in the shear layer. Especially, in the interior region of the

zones the grid lines were packed near the primary and secondary shear

layers. The grid was also clustered near the solid boundaries so that giving a

y+ of 1. This value corresponds to wall normal grid spacing of 10-6 m for the

flow conditions under consideration. This resolution is proper for all the

turbulence models applied here. The close view of the grid was shown in the

Figure 3.15.

32

Figure 3.14 Three-block computational grid for axsymmetric generic turbofan

engine exhausts.

Figure 3.15 Details of the computational mesh.

33

3.3.3.2 Boundary Conditions

A schematic of the boundary conditions that are applied to each region of

the computational grid was shown in Figure 3.16. No slip condition was

applied at wall boundaries, and walls were assumed to be adiabatic.

Pressure inlet boundary condition was used for the core, bypass stream

inlets and at free stream inlets; total pressures and total temperatures were

specified. Free stream boundary condition was used on the outer

boundaries to represent the simulated flight stream, and the inflow Mach

number was specified. Pressure outlet boundary condition was applied to

the downstream boundary, flow parameters were extrapolated from the

interior. Axis boundary condition, i.e., zero gradients was typically specified

on the centerline.

Figure 3.16 Schematic of the boundary conditions.

When the realizable k-ε, k-ω and SST k-ω turbulence models were selected,

the turbulence intensity and turbulent viscosity ratios were specified at the

inflow and free stream boundaries. For all of the turbulence models stated,

34

these values were specified using the following assumptions: (1) the core

and bypass streams had an inflow turbulence intensity of 6% and 10%,

respectively, (2) the free stream had an inflow turbulence intensity of 2%, (3)

the core and bypass streams had a turbulence viscosity ratio of 100, and

free stream had a turbulent viscosity ratio of 10. Those values correspond to

fully turbulent flows. These values are typical for the background turbulence

levels in jet flow experiments.

3.4 Grid Sensitivity

Three levels of grids were created to find out the grid independent solution.

They were labeled as coarse, medium and fine grids. The coarse grid was

built by decreasing one of every other two grid lines in the axial and radial

directions. Fine grid was made by increasing the grid dimensions in the axial

direction only. For all three levels, height of the first cell was kept constant to

make sure the viscous sub layer is well resolved near the wall regions for all

the grids. The medium grid for the generic turbofan engine nozzle contains

384,751 cells grid points. As shown in the Figure 3.17 below, at four different

axial locations velocity profiles were compared for three levels of grid. The

turbulence intensity profiles for all grid levels were shown in the Figure 3.18.

The velocity profiles essentially do not vary between the medium and fine

grid levels, whereas there is a slight difference with the coarse grid. When

we look at the turbulence intensity profiles, as can be seen there is a

discrepancy for all three level grids at the downstream regions x=4 m and

x=8 m where primary flow mixes with the secondary fan flow. Since mean

flow quantities do not vary and the aim is not to validate but to understand

the effects of some parameters, further grid sensitivity analyses were not

performed. The medium level great was used for further analyses.

35

Axial Velocity at x=1 m




Figure 3.17 Comparison of axial velocity distribution at x=1, 2, 4, 8 m for 3

levels of grids.

36

Turbulence Intensity at x=1 m




Figure 3.18 Comparison of turbulence intensity distribution at x=1, 2, 4, 8 m

for 3 levels of grids.

37

3.5 Turbulence Model Sensitivity

This section aims to compare four different turbulence models to get a better

understanding the influence of turbulence modeling on the complex shear

flow. Spalart-Allmaras, realizable k-ε, k-ω and Shear Stress Transport (SST)

k-ω turbulence models of FLUENT solver were chosen to assess the

modeling issue.

Spalart-Allmaras turbulence model solves the single transport equation for a

modified turbulent viscosity which is used to compute turbulent stresses and

so on. It is relatively simple compared to the other turbulence models. The

two-equation turbulence models were developed and calibrated for room

temperature, low Mach number, and plane mixing layer. For high

temperature jet flows, the standard turbulence models lack the ability to

predict the observed increase in the growth rate of mixing layer. Realizable

k-ε turbulence model solves two other transport equations for turbulent

kinetic energy and dissipation rates. It is mostly chosen for simulating shear

flows, jet flows. It has also some empirical constants derived from

experiments. Even though those constants are suitable for most of the fully

turbulent flows, calibration might be necessary. K-ω turbulence model

incorporates modifications for low-Reynolds-number effects, compressibility,

and shear flow spreading. It predicts free shear flow spreading rates that are

in close agreement with measurements for far wakes, mixing layers, and

plane, round, and radial jets, and is thus applicable to wall-bounded flows

and free shear flows. The shear-stress transport (SST) k-ω model was

developed to effectively bring together the robust and accurate formulation

of the k-ω model in the near-wall region with the free-stream independence

of the k-ε model in the far field. To achieve this, the k-ε model is converted

38

into a k-ω formulation. The SST k-ω model is similar to the standard k-ω

model, but includes some refinements [33]:

In order to evaluate the results, the axial velocity and turbulence viscosity

profiles for different turbulence models were plotted. The axial locations

where profiles were plotted were shown in the Figure 3.19, below.

Figure 3.19 Schematic of the axial locations at which data are represented.

As shown in the Figure 3.20 and Figure 3.21, axial velocity profiles at

upstream of x=4 m do not vary much with the turbulence model used. It is

interesting to note that SST k-ω and Spalart-Allmaras models produce

slightly different results at the very close mixing region of the bypass and

free stream flows where the gradients are higher compared to shear layer

developed from the core and bypass ducts. However, especially at locations

downstream of x=4 m turbulence models start to give different results. The

results of Spalart-Allmaras and SST k-ω models are comparable with each

other. Also, k-ε and k-ω models give similar results within each other.

However, as can be seen in Figure 3.22, at locations further downstream

(x=25 m and 32 m), k-ω turbulence model results become very different

compared to other three models. It produces higher values for axial velocity.

39

K-ω turbulence model differs from the SST k-ω model in terms of its closure

coefficients which are obtained by calibrating experimental data. Other than

that, a cross-diffusion derivative term is included to k-ω formulation to obtain

SST k-ω model [32]. These two refinements to k-ω model make the

difference between the results. Taking the results obtained from the 2D

ejector case into account, k-ω model closure coefficients should be re-

calibrated to get comparable results with other models. At further

downstream regions, turbulence modeling does not have a strong influence

on axial velocity distributions. As shown in the Figure 3.23 and Figure 3.24,

turbulent viscosity distribution variation for different turbulence models

cannot be ignored. The SST k-ω model yields higher values while the lowest

turbulent viscosities were obtained from k-ω turbulence model at the mixing

region of the bypass flow and free stream. Comparing the turbulent kinetic

energies sheds light on reason for difference. As shown in Figure 3.26, k-ω

turbulence model yields longer potential core length and lower turbulent

kinetic energy levels. SST k-ω turbulence model gives higher turbulent

kinetic energy compared to k-ε turbulence model especially close to onset of

free stream and bypass flow mixing region. Also, the potential core length is

longer for SST k-ω turbulence model than k-ε turbulence model.

In sum, it can be said that resolving the wall bounded regions are

comparable for each turbulence model. Variation between the results

obtained from different turbulent models occurs if the flow gradients are

strong. We see that the variation is pronounced in the mixing region of fan

and free stream flows at which high gradients are present. At further

downstream, the gradients become smaller and turbulence modeling does

not change the results much other than the k-ω turbulence model.

40

Axial Velocity at x=-1.7 m




Figure 3.20 Axial velocity distribution in radial direction at x=-1.7,-1.5, -0.2, 0


41







42





One should be careful while determining the turbulence model that will be

used and if necessary calibration should be made to represent the real

physics of the flow better.

43

Turbulent Viscosity at x=-1.7 m



Turbulent Viscosity at x=0 m



44





Figure 3.24 Turbulent viscosity distribution in radial direction at x=1, 2, 4, 8


45



Figure 3.25 Turbulent viscosity distribution in radial direction at x=25, 32 m

for different turbulence models.

46

Figure 3.26 Turbulent kinetic energy (m2/s2) contours with three turbulent

models.

47

3.6 Parametric Study

In this part, the flow physics has been explained before discussing the

influence of the parameters on the flow field. The feature of shear layer

composed by a dual stream nozzle depends on thermodynamic parameters;

pressure, temperature in the primary and secondary flows, velocity ratio of

secondary flow to the primary flow and also geometric parameters [19].

Other than the parameters listed above, it is proposed that turbulence

intensity, thickness of the boundary layer might have an impact to the mixing.

Change in the mean flow field also modifies the behavior of the sound

waves passing through the shear layer and therefore an evaluation to see

the effects of these parameters on the background turbulent flow field is

useful.

3.6.1 Effect of Boundary Layer Thickness on Turbulent Shear Layer

As pointed out before, propagated noise levels and propagation direction are

strongly influenced by the background turbulent flow field. In industrial

simulations, only representative values are used at the inlet locations.

Upstream nozzle geometry is not included in analyses which results in

uncertainties in specifying the mean and turbulent flow profiles at the

boundaries. It is very much important to include both the geometric and the

flow conditions details as realistic as possible for the sake of obtaining

accurate results. It is known that shear layer is associated with the flow field

inside the nozzle; therefore any change for the nozzle boundary layer can

significantly impact downstream mixing and change the direction and

amplification of the sound waves propagated through the turbulent jet.

48

To see the influence of the boundary layer thickness, two more cases were

evaluated. For the first case, only length of the core nozzle wall length was

doubled, in the second case bypass and core nozzle walls were extended

two times. The total values at the inlet boundaries were kept constant; in

other words, the loss due to boundary layer was assumed to be negligible

for longer walls. The static values at the inlet boundaries were calculated

from isentropic relations.

For the simulations SST k-ω turbulence model was used due to results

obtained for 2D ejector nozzle problem in the first part. Axial velocity and

turbulent viscosity profiles at different axial locations were plotted for

comparison.

Doubling the wall lengths result in roughly 1.8 times thicker boundary layer

at the lip of the nozzles. Doing so, as plotted in Figure 3.27 there is a slight

variation in the axial velocity distribution between the generic nozzle and the

longer nozzles. The impact of boundary layer thickness on the mean velocity

field is slightly more pronounced at downstream regions. The change in

turbulent field is more distinguishing compared to the mean flow. As shown

in the turbulent kinetic energy profile plots, change starts to occur in the wall

bounded regions of core duct flow with thicker boundary layer and it affects

the mixing regions. The influence of the boundary layer thickness loses its

importance at the further downstream regions, namely after x=32 m. It is

interesting to note that, increasing both the fan and core duct lengths does

not change the results; in other words, the influence of boundary layer

thickness in the core duct is eliminated due to effects of thicker boundary

layer thickness in the fan flow as can be seen from the plots below.

49






m for different nozzle wall lengths.

50



Axial velocity at x=25 m



different nozzle wall lengths.

51

Turbulent Kinetic Energy at x=-1.7 m



Turbulent Kinetic Energy at x=0 m

Figure 3.29 Turbulent kinetic energy distribution in radial direction at x=-1.7, -

1.5, -0.2, 0 m for different nozzle wall lengths.

52





Figure 3.30 Turbulent kinetic energy distribution in axial direction at x=4, 8,

25, 32 m for different nozzle wall lengths.

53

Figure 3.31 Turbulent kinetic energy (m2/s2) contours for different boundary

layer thickness configurations.

54

3.6.2 Effect of Turbulence Intensity

Another uncertainty in computing turbulent flows is the unknown value of

turbulence quantity at the boundaries. However, they are needed for

simulations as inputs. In this section effect of the turbulence intensity was

compared, again SST k-ω turbulence model was used in the computations.

The turbulence intensity both in the fan and core streams were doubled and

decreased to half. In other words, 3%, 6% and 12% turbulence intensity

values were used as parameters.

As shown in the Figure 32, different turbulent intensities do not affect the

axial velocity distribution; however, there is a slight influence is observed at

downstream locations as can be seen in the Figure 33. Similar trend occurs

for the turbulent kinetic energy distribution as shown in the Figure 34 and

Figure 35, below. According to the results, turbulent flow field is slightly

influenced only at downstream regions of x=25 m and further downstream

locations.

55







56





57





Figure 3.34 Turbulent kinetic energy distribution in radial direction at x=1, 2, 4,

8 m for different turbulent intensities.

58



Figure 3.35. Turbulent kinetic energy distribution in axial direction at x=25, 32

m for different turbulent intensities.

59

CONCLUSION

In this study, jet flow emanating from a turbofan engine exhaust was

computed by using Reynolds-averaged Navier-Stokes method with different

turbulence models of FLUENT solver. Turbulence modeling issues and a

parametric study were considered.

First, a 2D ejector problem was solved to find out the most appropriate

turbulence model. The results of 2D ejector problem showed that turbulence

model plays an important role to define the real physics of the problem. All

four turbulence models; Spalart-Allmaras, realizable k-ε, k-ω and SST k-ω

used here lack to predict accurately especially the initial jet growth region.

The models should be carefully evaluated and necessary modifications

should be done. For example, calibrating the constants used in the models

can be a choice to have better results.

For the parametric study, SST k-ω turbulence model was used by taking the

turbulence model sensitivity studies for both ejector and turbofan engine

exhaust analyses results into account. It is seen that boundary layer

thickness effect becomes important in the jet flow close to the lips of the

nozzles. At far downstream regions, it does not affect the flow field. For

different turbulent intensities, no significant change occurred in both mean

60

and turbulent flow fields. However, higher intensities might be possible for

real engine conditions which might impact the flow characteristics.

61

REFERENCES

[1] Viswanatham K., “Parametric Study of Noise from Dual-Stream Nozzles”,

J. Fluid Mechanics, Vol. 521, pp. 35-68, 2004.

[2] Bridges J., Wernet M., Brown C., “Control of Jet Noise Through Mixing

Enhancement”, NASA/TM-2003-212335.

[3] Bogey C., Bailly C., “A Numerical Study of the Acoustic Radiated by

Subsonic Jets Using Direct Computation of Aerodynamic Noise”

[4] Debonis J. R., “RANS Analyses of Turbofan Nozzles with Wedge

Deflectors for Noise Reduction”, AIAA paper, 46th AIAA Aerospace Sciences

Meeting, 7-10 January, 2008, Reno, NV, USA.

[5] Papamoschou D., “Parametric Study of Fan Flow Deflectors For Jet

Noise Supression”, AIAA Paper 2005-2890, 11th AIAA/CEAS Aeroacoustics

Conference, 23-25 May 2005, Monterey, California.

[6] http://www.x-noise.net/AssociatedProjects, “The X-NOISE Network

Homepage”, Aircraft Noise Reduction Efforts in Europe.

[7] NASA Facts Sheet, “Making Future Commercial Aircraft Quiter”, FS-

1999-07-003-GRC.

[8] Ozyoruk Y.,Dizemen E., Kaya S., Akturk A., “A Frequency Domain

Linearized Euler Solver For Turbomachinery Noise Propagation and

Radiation”, AIAA Paper 2007-3521, 13th Aeroacoustic Conference, May

2007, Rome, Italy.

[9] Bailly C., Juve D., “Numerical Solution of Acoustic Propagation Problems

Using Linearized Euler Equations”, AIAA Journal, Vol. 38, No. 1, January

2000.

62

[10] Bailly C., Bogey C., “Contributions of Computational Aeroacoustics to

Jet Noise Research and Prediction”, International Journal of Computational

Fluid Dynamics, August 2004, Vol. 18 (6), pp. 481-491.

[11] Bogey C., Bailly C., Juve D., “Computation of Flow Noise Using Source

Terms in Linearized Eulr’s Equation”, AIAA Journal Vol. 40, No. 2, February

2002.

[12] Koch L. D., Bridges J. E., Khavaran A., “Flowfield Comparisons from

Three Navier-Stokes Solvers for an Axisymmetric Separate Flow Jet”, AIAA

Paper 2002-0672,2002.

[13] Thies A. T., Tam C. K. W., “Computation of Turbulent Axisymmetric and

Nonaxisymmetric Jet Flows Using the k-epsilon Model”, AIAA Journal 1996;

34(2):309-16.

[14] Morgans R. C., Dally B. B., Nathan G. J., Lanspeary P. V., Fletcher D.

F., “Application of the Revised Wilcox k- Turbulence Model to a Jet in Co-

flow”, 6-8 December, 1999.

[15] Kenzakowski D. C., Kannepalli C., “Jet Simulation for Noise Prediction

Using Advanced Turbulence Modeling”, AIAA-2005-3086.

[16] Engblom W. A., Khavaran A., Bridges J.E., “Numerical Prediction of

Chevron Nozzle Noise Reduction Using Wind-MGBK Methodology”, AIAA

Paper 2004-2979, 2004.

[17] Georgiadias N. J., Rumsey C. L., Yoder D. A., Zaman KBMQ.,

“Turbulence Modeling Effects on Calculation of Lobed Nozzle Flowfields”, J.

of Propulsion and Power, 2006, 22(3):567-75.

[18] Gao J. H., Li X. D., “A Numerical Study of Nozzle Boundary Layer

Thickness on Axisymmetric Supersonic Jet Screech Tones”, AIAA-2006-

2675.

63

[19] Viswanatham K., “Parametric Study of Noise from Dual-Stream

Nozzles”, J. Fluid Mechanics, Vol. 521, pp. 35-68, 2004.

[20] Gilbert G. B., Hill P. G., “Analysis and Testing of two-Dimensional Slot

Nozzle Ejectors with Variable Area Mixing Sections”, NASA-CR-2251, May

1973.

[21] NPARC Alliance CFD Verification and Validation Website.

[22] Spalart, P. R., Jou, W.-H., Stretlets, M., and Allmaras, S. R. (1997),

“Comments on the Feasibility of LES for Wings and on the Hybrid

RANS/LES Approach”, Advances in DNS/LES, Proceedings of the First

AFOSR International Conference on DNS/LES.

[23] Launder, B. E., Reece, G. J. and Rodi, W. (1975), “Progress in the

Development of a Reynolds-Stress Turbulent Closure” Journal of Fluid

Mechanics, Vol. 68(3), pp. 537-566.

[24] Menter, F. R. (1993), “Zonal Two Equation k-ω Turbulence Models for

Aerodynamic Flows”, AIAA Paper 93-2906.

[25] Strelets, M. (2001), “Detached Eddy Simulation of Massively Separated

Flows”, AIAA 2001-0879.

[26] Wilcox, D.C. (1988), “Re-assessment of the scale-determining equation

for advanced turbulence models”, AIAA Journal, vol. 26, pp. 1414-1421.

[27] Germano, M., Piomelli, U., Moin, P. and Cabot, W. H. (1991), “A

dynamic sub-grid scale eddy viscosity model”, Physics of Fluids, A(3): pp.

1760-1765, 1991.

[28] Spalart, P. R. and Allmaras, S. R. (1992), “A One-Equation Turbulence

Model for Aerodynamic Flows”, AIAA Paper 92-0439.

[29] Pope S. B., “Turbulent Flows”.

64

[30] Lele S. K., “Compressibility Effects on Turbulence”, Annual Reviews in.

Fluid Mechanics, 1994, 26:211-54.

[31] Georgiadis N. J., Debonis J. R., “Navier-Stokes Analysis for Turbulent

Jet Flows with Application to Aircraft Exhaust Nozzles”, Prog. Aerospace Sci

(2007), doi: 10.1016/j.paerosci.2006.12.001.

[32] http://www.grc.nasa.gov/WWW/wind/valid/eject/eject.cgd

[33] Fluent User’s Manual.

65

Reynolds Averaged Navier Stokes Computations of Jet Flows Emanating From Turbofan Exhausts Turbofan Egzoz Cikisi Jet Akisinin Reynolds Averaged Navier Stokes Ile Hesaplanmasi

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