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

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) ii

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

iii

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.

iv

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 SpalartAllmaras 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.

v

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.

vi

Keywords: Shear Layer, Axisymmetric Subsonic Jet, Reynolds-averaged Navier-Stokes, Jet Noise, Computational Fluid Dynamics, Turbulence, Ejector.

vii

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 viii

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 SpalartAllmaras 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. ix

Anahtar Kelimeler: Kayma Tabakası, Eksenel Simetrik Ses Altı Hızda Jet, Reynolds-averaged Navier-Stokes, Jet Gürültüsü, Hesaplamalı Akışkanlar Dinamiği, Türbülans, Ejektör.

x

To the owner of everything, To My Parents, To KAYA and GÜLSEREN families

xi

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.

xii

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 xiii

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

xiv

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.

xv

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 3.6 Velocity profile for 2D ejector at x=0.127 m. Figure 3.7 Velocity profile for 2D ejector at x=0.1778 m. Figure 3.8 Velocity profile for 2D ejector at x=0.2667 m. Figure 3.9 Total temperature for 2D ejector profile at x=0.0762 m. Figure 3.10 Total temperature for 2D ejector profile at x=0.2667 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. xvi

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 different turbulence models. Figure 3.23 Turbulent viscosity distribution in radial direction at x=-1.7, -1.5, 0.2, 0 m for different turbulence models. Figure 3.24 Axial velocity distribution in radial direction at x=-1.7, -1.5, -0.2, 0 m for different turbulent intensities. Figure 3.25 Turbulent viscosity distribution in radial direction at x=-1.7, -1.5, 0.2, 0 m for different turbulence models. Figure 3.26 Turbulent kinetic energy (m2/s2) contours with three turbulent models. Figure 3.27 Axial velocity distribution in radial direction at x=-1.7, -1.5, -0.2, 0 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.

xvii

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. Figure 3.32 Axial velocity distribution in radial direction at x=1, 2, 4, 8 m for different turbulent intensities. Figure 3.33 Axial velocity distribution in radial direction at x=25, 32 m for 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.

xviii

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, 1

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.

Total Fan Exhaust Core Exhaust Fan Inlet Combustion Core Airframe Turbine 60

70

80

90

100

110

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 2

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 3

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 4

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 5

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.

6

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 7

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-tomoderate 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 8

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.

ui ( x, t ) = Ui ( x) + ui′( x, t )

(2.1 )

Here ui ( x, t ) is expressed as the instantaneous velocity with, U i ( x ) ; average and ui′ ( x, t ) fluctuating part.

9

Figure 2.1 Time averaging of velocity.

If this instantaneous velocity given in Equation 2.1 is added into the NavierStokes equations and then time averaging is applied, so called Reynolds Averaged Navier-Stokes (RANS) equations are obtained.

ρ⋅

(

∂U i ∂U i ∂P ∂ ' ' + ρ ⋅U j ⋅ =− + 2 ⋅ µ ⋅ S ji − ρ ⋅ u j ⋅ ui ∂t ∂x j ∂xi ∂x j

)

( 2.2 )

The quantity − ρ ⋅ u j ⋅ ui 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,

10

k=

2 2 1 ⎛ '2 ⋅ ⎜ u + v ' + w ' ⎞⎟ ⎠ 2 ⎝

( 2.3 )

τ ii = −ui ' ⋅ ui ' = −2 ⋅ k Implementation of this relation into Reynolds stress tensor resulted in a transport equation for the turbulent kinetic energy, ⎛ ⎞ ⎜ ⎟ ⎟ ∂U i ∂k ∂k ∂ ⎜ ∂k 1 1 ' ' ' ' +U j ⋅ = τ ij ⋅ − + ⋅⎜ ν ⋅ − ⋅ ui ⋅ ui ⋅ u j − ⋅ P ' ⋅ u j ⎟ ( 2.4 ) ε{ ∂t ∂x j ∂x j ∂x j ⎜ ∂x ρ 4243 ⎟ 2 44244 1 3 1 14 4244 3 1 424 3 DISSIPATION 123j TURBULENT ⎜ ⎟⎟ PRESSURE SUBSTANTIAL PRODUCTION ⎜ MOLECULAR TRANSPORT DIFFUSION DERIVATIVE ⎝ DIFFUSION ⎠

(

) (

)

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 2 kinematic viscosity υ appears to be in m

s

that is the product of a velocity

and a length scale), the models without involving a length scale are regarded as incomplete.

11

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 k

1

2

ω where

k ⋅ ω 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,

∂ω ∂ω ∂ +U j ⋅ = −β ⋅ ω 2 + ∂t ∂x j ∂x j

⎛ ∂ω ⎞⎟ ⋅ ⎜ σ ⋅ν T ⋅ ⎜ ∂x j ⎟⎠ ⎝

( 2.5 )

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.

12

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 13

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.

14

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.

15

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].

16

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, Reynoldsaveraged 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 17

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 18

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 19

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.

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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

21

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. 22

1.2

Velocity Profile at x=0.0762 m 1

Height

0.8

0.6

0.4

0.2

0 0

50

100

150

200

250

300

350

400

Velocity (m/s) Experiment WIND k-epsilon [21]

FLUENT Realizable k-epsilon FLUENT Standart k-epsilon

FLUENT SST k-omega FLUENT Spalart-Allmaras

Figure 3.5 Velocity profile for 2D ejector at x=0.0762 m. 1.2

Velocity Profile at x=0.127 m 1

Height

0.8

0.6

0.4

0.2

0 0

50

100

150

200

250

300

Velocity (m/s) Experiment WIND k-epsilon [21]

FLUENT Realizable k-epsilon FLUENT Standart k-epsilon

FLUENT SST k-omega FLUENT Spalart-Allmaras

Figure 3.6 Velocity profile for 2D ejector at x=0.127 m.

23

350

1.2

Velocity Profile at x=0.1778 m 1

Height

0.8

0.6

0.4

0.2

0 0

50

100

150

200

250

300

Velocity (m/s) Experiment WIND k-epsilon [21]

FLUENT Realizable k-epsilon FLUENT Standart k-epsilon

FLUENT SST k-omega FLUENT Spalart-Allmaras

Figure 3.7 Velocity profile for 2D ejector at x=0.1778 m.

1.2

Velocity Profile at x=0.2667 m 1

Height

0.8

0.6

0.4

0.2

0 0

50

Experiment WIND k-epsilon [21]

100

Velocity (m/s)

150

FLUENT Realizable k-epsilon FLUENT Standart k-epsilon

200

FLUENT SST k-omega FLUENT Spalart-Allmaras

Figure 3.8 Velocity profile for 2D ejector at x=0.2667 m.

24

250

1.2

Total Temperature Profile at x=0.0762 m 1

Height

0.8

0.6

0.4

0.2

0 300

305

310

315

320

Experiment WIND k-epsilon [21]

325

330

Total Temperature (K)

335

340

FLUENT Realizable k-epsilon FLUENT Standart k-epsilon

345

350

355

FLUENT SST k-omega FLUENT Spallart-Allmaras

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

1.2

Total Temperature Profile at x=0.2667 m 1

Height

0.8

0.6

0.4

0.2

0 300

305

Experiment WIND k-epsilon [21]

310

315

Total Temperature (K)

320

FLUENT Realizable k-epsilon FLUENT Standart k-epsilon

325

330

FLUENT SST k-omega FLUENT Spalart-Allmaras

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

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Figure 3.11 Axial velocity (m/s) contours for the 2D ejector nozzle calculated with three different turbulence models. 26

Figure 3.12 Turbulent kinetic energy (m2/s2) contours for the 2D ejector nozzle calculated with three different turbulence models.

27

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.

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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 coflowing 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.

29

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 NavierStokes equations using an implicit, segregated, second order, up-wind fluxdifference 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.

30

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.

31

Figure 3.14 Three-block computational grid for axsymmetric generic turbofan engine exhausts.

Figure 3.15 Details of the computational mesh.

32

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, 33

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. 34

Axial Velocity at x=1 m

Axial Velocity at x=2 m

Axial Velocity at x=4 m

Axial Velocity at x=8 m

Figure 3.17 Comparison of axial velocity distribution at x=1, 2, 4, 8 m for 3 levels of grids. 35

Turbulence Intensity at x=1 m

Turbulence Intensity at x=2 m

Turbulence Intensity at x=4 m

Turbulence Intensity at x=8 m

Figure 3.18 Comparison of turbulence intensity distribution at x=1, 2, 4, 8 m for 3 levels of grids. 36

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

37

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. 38

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 recalibrated 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.

39

Axial Velocity at x=-1.7 m

Axial Velocity at x=-1.5 m

Axial Velocity at x=-0.2 m

Axial Velocity at x=0 m

Figure 3.20 Axial velocity distribution in radial direction at x=-1.7,-1.5, -0.2, 0 m for different turbulence models. 40

Axial Velocity at x=1 m

Axial Velocity at x=2 m

Axial Velocity at x=4 m

Axial Velocity at x=8 m

Figure 3.21 Axial velocity distribution in radial direction at x=1, 2, 4, 8 m for different turbulence models. 41

Axial Velocity at x=25 m

Axial Velocity at x=32 m

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

42

Turbulent Viscosity at x=-1.7 m

Turbulent Viscosity at x=-1.5 m

Turbulent Viscosity at x=-0.2 m

Turbulent Viscosity at x=0 m

Figure 3.23 Turbulent viscosity distribution in radial direction at x=-1.7, -1.5, 0.2, 0 m for different turbulence models. 43

Turbulent Viscosity at x=1 m

Turbulent Viscosity at x=2 m

Turbulent Viscosity at x=4 m

Turbulent Viscosity at x=8 m

Figure 3.24 Turbulent viscosity distribution in radial direction at x=1, 2, 4, 8 m for different turbulence models. 44

Turbulent Viscosity at x=25 m

Turbulent Viscosity at x=32 m

Figure 3.25 Turbulent viscosity distribution in radial direction at x=25, 32 m for different turbulence models.

45

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

46

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.

47

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.

48

Axial Velocity at x=-1.7 m

Axial Velocity at x=-1.5 m

Axial Velocity at x=-0.2 m

Axial Velocity at x=0 m

Figure 3.27 Axial velocity distribution in radial direction at x=-1.7, -1.5, -0.2, 0 m for different nozzle wall lengths. 49

Axial Velocity at x=4 m

Axial Velocity at x=8 m

Axial velocity at x=25 m

Axial velocity at x=32 m

Figure 3.28 Axial velocity distribution in radial direction at x=4, 8, 25, 32 m for different nozzle wall lengths. 50

Turbulent Kinetic Energy at x=-1.7 m

Turbulent Kinetic Energy at x=-1.5 m

Turbulent Kinetic Energy at x=-0.2 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. 51

Turbulent Kinetic Energy at x=4 m

Turbulent Kinetic Energy at x=8 m

Turbulent Kinetic Energy at x=25 m

Turbulent Kinetic Energy at x=32 m

Figure 3.30 Turbulent kinetic energy distribution in axial direction at x=4, 8, 25, 32 m for different nozzle wall lengths. 52

Figure 3.31 Turbulent kinetic energy (m2/s2) contours for different boundary layer thickness configurations.

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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.

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Axial Velocity at x=1 m

Axial Velocity at x=2 m

Axial Velocity at x=4 m

Axial Velocity at x=8 m

Figure 3.32 Axial velocity distribution in radial direction at x=1, 2, 4, 8 m for different turbulent intensities. 55

Axial velocity at x=25 m

Axial velocity at x=32 m

Figure 3.33 Axial velocity distribution in radial direction at x=25, 32 m for different turbulent intensities.

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Turbulent Kinetic Energy at x=1 m

Turbulent Kinetic Energy at x=2 m

Turbulent Kinetic Energy at x=4 m

Turbulent Kinetic Energy at x=8 m

Figure 3.34 Turbulent kinetic energy distribution in radial direction at x=1, 2, 4, 8 m for different turbulent intensities. 57

Turbulent Kinetic Energy at x=25 m

Turbulent Kinetic Energy at x=32 m

Figure 3.35. Turbulent kinetic energy distribution in axial direction at x=25, 32 m for different turbulent intensities.

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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 59

and turbulent flow fields. However, higher intensities might be possible for real engine conditions which might impact the flow characteristics.

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