CFD full cycle analysis of a spark ignition engine at high part-load conditions

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POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master of Science in Mechanical Engineering

CFD FULL CYCLE ANALYSIS OF A SPARK

IGNITION ENGINE AT HIGH PART-LOAD

CONDITIONS

Author:

Davide Arnaboldi

ID

: 905104

Supervisor:

Prof. Gianluca D’Errico

Co-supervisor:

Ing. Stefano Fantoni

Ing. Donato Colangelo

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Acknowledgements

With this work that lasted more than 6 months ends a chapter of my life, spent mostly in the lively Milano but also in the erudite Bologna. Even if it is impossible to thank all the people who helped me during the university years, with these few lines I will try to do my best.

First of all, I want to thank Engineers Stefano Fantoni and Donato Colangelo who gave me the outstanding opportunity to spend 6 months of work at Ducati Motor Holding, aimed at the drafting of this Master Thesis. I would again like to thank Donato Colangelo, who helped me in the daily routine and was my mentor during this period. I would also thank the other guys at the Engine Simulation Office, who welcomed me in their ”family”.

Then i want to thank my Supervisor, Professor Gianluca D’Errico, who introduced me to the Internal Combustion Engine technical world with his course and then has been available to support me when i decided to develop this thesis at Ducati.

I wish to thank my family: my Father Alfredo who passed on his passion to me for the world of motors, my Mother Anna who, together with Alfredo, always helped and supported me and my Brother Matteo who is a silent but strong presence at my side. I will never be able to thank you all enough for the support, motivations and possibilities you have offered me. I would like to thank also all the other components of my family, who always supported me. Without all of you, I would have never made it.

Finally, a big thank goes to my girlfriend and all my friends: thank you for your support in the difficult moments and the laughs together in the good ones.

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Abstract

In the present thesis 3D computational fluid dynamics (CFD) technique has been used to analyse the fluid dynamic behaviour of a high specific power naturally aspirated engine for motorbike application, at strong part load conditions and low engine speed. The work covers a full cycle both with and without including the injection simulation; the combustion and the expansion phases were not part of the investigation. The study has been performed using the commercial CFD software AVL FIRE.

Today the major challenges in the design of engines for high power motorbike appli-cations are to reduce pollutant emissions while minimizing consumption without affecting performance. For these reasons has become essential to investigate also engine’s operational points that were not studied in such detail before. To meet the mentioned requirements is fundamental to understand how turbulence is generated and determine the quality of the mixture in such critical conditions, featured by part load.

Hence, main targets of the investigation are mixture formation, tumble motion and tur-bulence generation. Moreover, the dynamic analysis has been carried out following two different approaches: the first assumes as working fluid an already homogeneous mixture while the latter considers ambient air and the port fuel injection simulation. The simula-tion which include the injecsimula-tion lend itself to a statistical analysis of the mixture during compression, with the aim to evaluate the quality of the air-fuel mixing process. Finally, a comparison between the two methodologies is accomplished.

Key words: CFD, engine, full cycle, turbulence, tumble, injection, mixture formation, part load, AVL FIRE

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Sommario

Nella presente tesi `e stato utilizzato il metodo della fluidodinamica computazionale 3D (CFD) per analizzare il comportamento fluidodinamico di un motore aspirato per appli-cazione motociclistica, caratterizzato da alta potenza specifica, in condizioni di forte parzial-izzazione e basso regime di rotazione. L’analisi, la quale riguarda un ciclo motore completo, viene svolta sia includendo che non includendo la simulazione dell’iniezione; le simulazioni di combustione ed espansione non sono oggetto dello studio. L’indagine `e stata svolta uti-lizzando il software CFD commerciale AVL FIRE.

Al giorno d’oggi le maggiori sfide nella progettazione di motori di alta potenza per applicazioni motociclistiche sono di ridurre le emissioni di inquinanti e i consumi senza intaccare le prestazioni. Per questi motivi è diventato essenziale indagare anche punti motore che non erano soggetti a tale dettaglio di studio in precedenza. Per raggiungere i sopra citati obiettivi è di fondamentale importanza capire, nelle particolari condizioni di carico parzializzato, come viene generata turbolenza e determinare la qualità della miscela.

Di conseguenza i principali obiettivi dello studio sono l’indagine della formazione della miscela, il moto di tumble e la generazione di turbolenza. L’analisi dinamica `e stata svolta considerando due diversi approcci: il primo assume come fluido di lavoro una miscela omoge-nea, mentre il secondo considera aria in condizioni ambiente e la simulazione dell’iniezizone (PFI). La simulazione che include l’iniezione si presta ad un’analisi statistica della miscela durante la fase di compressione, con l’obiettivo di valuatare la qualit`a del processo di missce-lamento tra aria e benzina. Infine, viene svolto un confronto tra le due metodologie adottate.

Parole chiave: CFD, motore, full-cycle, turbolenza, tumble, iniezione, formazione mis-cela, parzializzato, AVL FIRE

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Introduction

Motivation of the study

Numerical investigations, thanks to the rapid development of computational capabilities of computers, are becoming essential engineering tools. In the industrial and scientific world these kind of studies are nowadays integral part of the design process since, in addition to expand the design horizons, they allow to fast obtain results without excessive need of expensive classic experimental tests.

The numerical branch that deals with the study of fluids and therefore with the well known Navier-Stokes equations is Computational Fluid Dynamics. In many industrial fields, like automotive, aeronautics, aeroacoustics, environmental engineering and biomechanics this tool has become indispensable and has led to a revolution in the way to elaborate and approach problems. This branch is very wide and ranges from simple and fast tools to very reliable and complex methods.

As mentioned above, the automotive industry is increasingly exploiting Computational Fluid Dynamics, especially in the internal combustion engines design. These are complex machines in which CFD is a very powerful tool, allowing to greatly enhance the design capabilities which previously mainly relied on experimental procedures.

Until not long ago the engine design, particularly in the high power motorbike field, focused mainly on high load conditions with the purpose of optimize the maximum per-formances. Nowadays, due to the continuous demand in reducing pollutant emissions and fuel consumption, is becoming more and more necessary to investigate and optimize en-gine operational points characterized by low load and enen-gine speed, since they are the most frequent engine operating conditions and therefore critical for the type approval test. Part load conditions are very critical in naturally aspirated engines because of the issues in proper air-fuel mixing and turbulence generation, which could lead to a poor quality combustion process with subsequent higher fuel consumption and pollutant emissions.

In the current study has been exploited the potential of the CFD technique, to perform an analysis with the aim to study the fluid dynamic performances, in terms of mixture for-mation and turbulence generation, of a four stroke naturally aspirated spark ignition engine

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at high part-load conditions and low engine speed. The engine is devoted to motorcycle applications and is characterized by high specific power. Other fundamental purpose of the study is the comparison between two approaches to the dynamic analysis, which consider an incremental degree of accuracy: the first assumes as working fluid a homogeneous mixture while the latter considers ambient air and the port fuel injection simulation.

Thesis content

The first and most simple approach during the design of an Internal Combustion Engine is the steady analysis, carried out both numerically and experimentally. This study is impor-tant to have a first fluid dynamic characterization of the intake system of the engine and to perform comparison among different design solutions, but is not enough to fully understand the fluid dynamic behaviour of the engine. The first part of this thesis considers a three-dimensional CFD steady analysis of the engine under study in wide open throttle condition.

Internal Combustion Engines are cyclic machines, in which dynamic effects play a major role and therefore dynamic CFD analyses, both mono-dimensional and three-dimensional, are the most accurate methods to investigate the their fluid dynamic performances. The multi-dimensional analysis is the most accurate but also the most onerous in terms of com-putational effort, hence is often coupled with the mono-dimensional one in order to optimize the methodology. This kind of analyses, in order to be as accurate as possible, often rely on experimental data which provide a fundamental benchmark.

The second part of the thesis is the most relevant one and concern the multi-dimensional CFD dynamic analyses of the engine, carried out at part load condition and low engine speed. This particular operational point is important due to its relevance for the type approval test. Two different approaches are carried out and compared, in order to investigate the fluid dynamic performances in this particular condition and to understand how the degree of accuracy of the simulation affects the results. The two methods are:

• Case HOM : The first method is based on the assumption that at the inlet boundary, which is placed just upstream the throttle valve, the engine aspires an already homo-geneous mixture. This simplification lead to a simpler and faster engine simulation, with the drawback that nothing can be said on the quality of mixture formation • Case INJ : The second method add the complexity of the injection simulation, with

related phenomena. This approach is more complex and requires more computational effort than the first one, but allows to understand how the mixture is formed

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CONTENTS 7 A short recap of the thesis content, is shown below:

1. Chapter 1 : A brief theoretical background on the relevant subjects covered in this thesis is proposed.

At first is treated the gas exchange process in ICE, describing the most important features and parameters. Volumetric efficiency, flow through the valves, influence of fuel supply and charge motions, with particular emphasis on tumble, are discussed. The second section deals with the fundamentals of liquid spray theory and the asso-ciated phenomena are presented, such as primary and secondary break-up, spray-wall interaction, collision, coalescence and evaporation. Moreover, the main parameters used in spray characterization are described.

The third section contains a brief recall of fluid dynamics theoretical bases together with the governing equations used to describe the flow behaviour of a generic system domain. A discussion on turbulence modelling is also available. The section is com-pleted with a short summary on numerical methods including discretisation techniques, grid arrangements and the most common solution strategies.

The last section treats the spray and wallfilm modelling for CFD. The models of sec-ondary break-up, spray-wall interaction, evaporation and wallfilm are briefly discussed. 2. Chapter 2 : A description of the steady numerical analysis and discussion of the results

is presented.

At first the description of the analysis and the meshing strategy are discussed.

Then are reported the results in terms of flow coefficient and tumble number. A com-parison between numerical and experimental flow coefficient results is also presented. Finally, the flow field inside the cylinder is investigated.

3. Chapter 3 : Is the most important chapter, where the description of the dynamic analyses, discussion of the results and comparison between the two methods adopted is presented.

At first the simulation method is described explaining the meshing strategy, the nu-merical set-up and highlighting the difference between Case HOM and Case INJ. The second section contains the discussion of the results coming from the Case HOM dynamic analysis. A comparison with the CFD mono-dimensional model is proposed for all the part of the simulation. The main focus is on the compression phase, where the tumble motion and turbulence generation are studied.

The third section contains the discussion of the results coming from the Case INJ dynamic analysis. A comparison with the CFD mono-dimensional model is proposed for all the part of the simulation. The main focus is on the compression phase, where

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the tumble motion and turbulence generation are studied together with a statistical analysis of the air excess coefficient to determine the quality of the mixture.

The last section includes the comparison between Case HOM and Case INJ. At first the exhaust and intake simulation are investigated. The analysis focuses on the com-pression phase. The comparison between tumble motion and turbulence generation is presented.

4. Chapter 4 : Presentation of the thesis conclusions

Working environment

The present thesis has been drawn up at the end of a stage held at the Engine Simulation Office of Ducati Motor Holding, which is a worldwide leading company in the motorcycle business. Ducati is best known for high performance motorcycles characterized by large-capacity four stroke engines, equipped with a desmodromic valve control system. The above mentioned office is part of the R&D department and its task is, together with the Engine Design Office, to design all the engines of the motorbike selections of Ducati.

Due to confidentiality reasons, all the quantitative results of the work have been either hidden, reported as non dimensional or scaled.

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Chapter 1

Theoretical background

In this chapter is briefly summarized the theory behind each topic touched in this thesis. This introduction would be only an insight into the theory of these topics and the reader is invited to follow the references for further details.

First of all, is reported an overview of the gas exchange process in internal combustion engines [1], since the main target of the investigation is the evaluation of the fluid dynamics properties of the engine under study. One important feature of the work is the presence of the injection simulation, therefore the fundamentals of liquid sprays which include the basics of linked physical phenomena and spray characterization methods, are discussed [10]. Afterwards, being the work a numerical analysis based on Computational Fluid Dynamics, the principles and basic theory of this technique are also reported [4]. In CFD simulations the analysis of fuel spray evolution is usually performed by means of a lagrangian approach and phenomenological models [10] [11], in order to avoid direct calculations of too small scale processes. A brief recall of the models used in this work is presented.

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1.1 Gas exchange process in ICE

The gas exchange process in a four stroke, naturally aspirated (NA) spark ignition (SI) engine is fundamental, since the work obtained is mainly limited by:

• the amount of air aspired • the speed of air-fuel mixing • the speed of combustion • pollutant emissions

Furthermore, must be considered that Internal Combustion Engines (ICE) are cyclic machines in which each cylinder aspires air and expels exhaust gases cyclically, generating relevant dynamic effects such as accelerations of fluid columns and generation of pressure waves. Therefore, can be said that the air supply of an ICE is a complex and critical process, which has strong influence on power, efficiency and emissions.

The most common and synthetic parameter used to perform evaluation on the air supply system, the volumetric efficiency, is discussed in the first of the following sections. Con-cerning four stroke engines the most significant flow restrictions, along all the intake and exhaust systems, are usually located in the valves and ports. Therefore this chapter is also devoted to the analysis of the main features of the complex flows through these important components. Must be noted that the intake flow is also affected by the fuel supply when dealing with a Port Fuel Injection (PFI) system.

The last section of this chapter treats an other fundamental phenomenon, which has a great impact on air-fuel mixing, turbulence generation and therefore on the combustion process: the charge motion inside the cylinder. Main focus is on the tumble motion, since its the most relevant in the class of engines under study.

1.1.1 Volumetric efficiency

To measure the effectiveness of the global intake process the reference parameter is the volumetric efficiency (or filling coefficient) ηv, defined as:

ηv =

m_a ma,th

where ma is the actual aspirated air mass present inside the cylinder at the end of the

intake process of the single cycle, while ma,th is the ideal air mass that could fill the swept

volume, considering reference fluid conditions, which can be: inside of the intake manifold or ambient conditions. In the first case ηv measures the performance of the intake valve and

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1.1. GAS EXCHANGE PROCESS IN ICE 11 The amount of air mass actually aspired, differs from the ideal one due to the following reasons:

• burned gases which fills the combustion chamber at the end of the exhaust process have a pressure higher than the atmospheric one, thus at the beginning of the intake they expand occupying part of the displaced volume

• the pressure inside the cylinder at the end of the induction stroke is lower than ambient pressure, because energy must be spent to overcome frictional losses along the induction system and to accelerate the fluid through the inlet ports, while the acquired kinetic energy is mostly dissipated

• heat exchange between the hot engine walls and the entering fresh fluid during the intake process, which reduces the air density

• temperature reduction of the entering air due to fuel evaporation, if the fuel injection occurs simultaneously with the intake process

• dynamic effects, which can improve or hinder the cylinder filling, depending on whether the geometry of the induction system is tuned or not with available times (related to the engine rotational speed n)

• in presence of fixed angular valve timing, times available for the gas exchange process, can be optimized only for a limited number of n. To avoid this limitation some engines are equipped with a Variable Valve Timing (VVT) system

1.1.2 Valve flow coefficient

The flow through a valve is mainly influenced by the available area and pressure head across cylinder and pipe, which are both varying with the crank-angle. According to the different valve lift is possible to express the effective available area and in general is possible to distinguish between:

• medium and low lifts: the available area depend on the actual lift and is given by the lateral surface of a truncated cone. In common practice is often considered a cylindrical instead of a conical area, named curtain area

• high lifts: the available area is the maximum possible and is given by the minimum transversal section of the valve seat, reduced by the valve stem

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(a) Main geometrical parame-ters

(b) Passage areas for small (1) and high (2) valve lifts

Figure 1.1: Valve geometry

A first method to characterize the permeability of the intake system and the charge motion in the chamber, is the flow test in steady conditions. From this test many useful information can be derived, despite the presence of several assumptions, such as:

• constant pressure difference across the valve • absence of the piston

• the test loses the information related to the valve motion, since different behaviour can occur during valve opening or closing at same valve lift

Through this technique can be computed the flow coefficient Ceand other charge motion

parameters. The flow coefficient is calculated for different valve lifts, as the following ratio: C_e= m˙

˙ mid

Where ˙m is the measured mass flow rate through the valve and ˙mid is ideal mass flow

rate that would pass through a reference area under the following hypothesis: • ideal flow (isentropic + ideal fluid (constant density and negligible viscosity)) • steady and mono-dimensional flow

• adiabatic flow

Being the reference area arbitrary, different choices are possible such as valve seat area (A_v), cylinder area and curtain area. Hereafter, as done in most cases, the valve seat area is considered: ˙ mid= Arefρ01a01Φf p2 p01 (1.1.1)

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1.1. GAS EXCHANGE PROCESS IN ICE 13 where Φf is the compressible flow function, quantities labelled with subscript ”2” refers to

the cylinder and have been introduced the total ambient quantities, marked by the subscript ”01”, with total sound speed defined as a01=

√ kRT01 and k = _ccp_v T01= T1 1 +k − 1 2 · u2 kRT1 (1.1.2) p01 = p1 1 +k − 1 2 · u2 kRT1 _k−1k = p1 T01 T1 _k−1k (1.1.3) ρ01= ρ1 T01 T1 _k−1k (1.1.4) Φf p2 p01 = v u u t 2 k − 1 " p2 p01 2_k − p2 p01 _k−1k # (1.1.5)

Can be also adopted a simplified equation to evaluate the ˙mid, with ρ density in the

cylinder, ρ0 ambient density:

˙ mid= Avρ r 2∆p ρm (1.1.6) ρ = ρ0 p0− ∆p p₀ _k1 (1.1.7) ρm = 1 2(ρ0+ ρm) (1.1.8)

The choice of valve seat area as reference provides the advantage that the flow coeffi-cient includes not only the typical losses of an actual flow (friction, heat exchanges, energy dissipation, etc.) but also the effect of continuous change in geometric flow area.

In case of turbulent flows Ce is independent from the Reynolds number, hence on the

pressure drop across the valve; however, it varies continuously depending on the valve lift and available area, which depend on the crank angle. It’s possible to define a mean flow coefficient: ¯ Ce = 1 (θIV C − θIV O) Z θIV O θIV C Ce(θ)dθ (1.1.9)

To highlight the characteristics of the flow, is convenient to consider an alternative defi-nition of the flow coefficient, indicated with Cf, which adopt a variable reference area such

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Is useful to refer at the following exemplary values of flow coefficient, plotted against _dh v, which consider a typical seat-valve group with sharp edges:

Figure 1.2: Typical C_f - h/d_v plot

Can be recognized that the Cf curve is divided into three main branches, which can be

related to three flow conditions illustrated in the figure. At small valve lifts, after separation which occur due to the presence of sharp edges, the flow is able to reattach. At high valve lifts the flow is detached and present a conical shape, while at medium lifts typically the flow can reattach only on one side.

1.1.3 Influence of fuel supply

The fuel supply in PFI engines is a critical process which influences the performance, the efficiency and the pollutant emissions. The two main relevant effects are: the charge cooling due to fuel evaporation and the formation of a liquid film on the walls.

The charge cooling is a positive effect, because of the air temperature decrease correspond a density increase, so there is an improvement in the cylinder filling.

The formation of the wallfilm is instead a negative effect. Since part of the injected mass remains attached to the intake port walls in liquid phase, the air-fuel ratio inside the cylinder does not match the expected one, causing efficiency reduction and increase in pollution emission. Moreover, if liquid fuel is present inside the cylinder, emissions of unburned hydrocarbons increase.

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1.1. GAS EXCHANGE PROCESS IN ICE 15

1.1.4 Charge motions

In addition to the filling process, is critical to optimize also the gas motion within the engine cylinder. Considering an SI engine, these phenomena have a great influence on combustion in terms of: mixture formation, lighting of the first mixture kernel close to the spark plug and flame front propagation. Efficiency and completeness of the combustion will vary according to the previously mentioned characteristics. The main purpose is to have turbulent motions, which breaking down into smaller and smaller vortices, increase the air-fuel mixing and accelerate the combustion process.

The three main charge motions are:

• Swirl: is a structured rotational flow on a tangential plane, about the cylinder axis, which is formed during the intake process. It’s typical of diesel engines

• Tumble: is a structured rotational flow on a cylinder axial plane, generated during the intake process, but maintained and increased by the following compression

• Squish: is a structured rotational flow on a cylinder axial plane, generated towards the end of the compression stroke. The fluid motion producing squish is due to the non-uniform decrease in time of the volume of different zones of the combustion chamber The fluid-dynamic characterization of these organised gas motion is a difficult and com-plex task. It is therefore convenient the use of global indexes, obtained by experimental measurements or three-dimensional CFD predictions. In general, can be said that whenever energy is spent to produce organised charge motions, the cylinder filling is penalized.

Tumble

Tumble is the main structured charge motion in SI engines. It is obtained by directed intake ports, which orient the entering air toward the area underneath the exhaust valves (extrados side of curtain area). Here, interacting with the cylinder walls and piston head, the inducted flow undertakes a sort of ‘tumble’, which reverses its movement direction and organizes the flow on an axial plane. Then, equally important to the turbulence generation, is the action on rotational flow, done by the piston during the compression stroke, which reduces the vortex size as it moves towards TDC. Neglecting the effect of fluid friction, the angular momentum of the isolated system is conserved. Thus, as the moment of inertia of the vortex is reduced, its angular velocity must increase.

Experience shows that it’s better to generate one single macro-vortex to achieve higher turbulence and reduced cyclic variability. However, must be noted that an additional counter rotating structure (called reverse tumble) could be generated under the intake valve by the air flow through the curtain area, directed towards the cylinder and not under the exhaust valve (intrados side of curtain area). The intensity of the tumble motion depends on the

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geometry of the intake duct and on the stroke/bore (S/B) ratio. Moreover, it’s difficult to generate one single macro-vortex in presence of S/B lower than 0.7-0.6 (over-square engines).

Figure 1.3: Example of different tumble generation

The main reasons which lead to the choice of generating a tumble motion are:

• it’s the most effective way to generate useful turbulence during combustion. Turbu-lence is first generated during the intake process due to the flow separation and fluid interaction with the cylinder walls. Then the tumble, thanks to the deformation caused by the piston motion, generates high velocity gradients which determine the dissipa-tion of kinetic energy into turbulence generadissipa-tion. The achieved effect is combusdissipa-tion duration reduction and reduction in cyclic variability

• it’s fundamental to generate a stratified charge in lean burn engines, to decrease the tolerated equivalence ratio

• it’s a way to increase the tolerated EGR fraction The drawbacks carried by tumble generation are:

• increased fluid-dynamic losses due to higher fluid speed determined by the acceleration of the flow towards the side of the curtain area facing the exhaust valve

• reduction of the volumetric efficiency due to partial exploitation of the curtain area Only exploiting CFD unsteady simulations is possible to make tumble motion character-ization also in real engine operating conditions.

However, the characterization of tumble motion is usually done at first through flow tests and computer simulations, both in steady conditions. Starting from a model of the port-valve system is possible to determine the angular momentum with a system analogous to the one used to determine the flow coefficient. The test is based on a fixed pressure difference

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1.1. GAS EXCHANGE PROCESS IN ICE 17 and intake valves lifts. In this case the test rig needs the installation of an additional measure device and a tumble-swirl converter. The converter is necessary considering that the tumble motion is normal to the cylinder axis, while the measure device can be made of a paddle wheel or an impulse torque meter constituted by a honeycomb matrix. The global indexes obtained by these two methods are:

• Tumble Coefficient: Ct = ω_vt_isD

Where ωt is the paddle wheel angular speed, D the bore diameter and vis the axial

velocity of the flow due to an ideal isentropic expansion under the given pressure difference

• Tumble Number: Nt = 8_m_˙_aM_Dvt_is

Obtained comparing the total angular momentum of the tumble vortex (Mt) with the

angular momentum of the axial flow

The steady evaluation of the tumble presents the same drawbacks previously mentioned when talking about the Ce. Besides, the absence of the shielding effect of the piston is even

more critical, because the surface of the piston and the relative position between piston and cylinder head are fundamental for the formation of tumble motion.

Hence, the designer is interested in having an estimation of the angular momentum of the mass of fluid entrapped in the cylinder at the end of the intake process. The index usually used with this goal is the Tumble Ratio:

Tr =

ntb

nmot

Where ntb is the angular speed of an equivalent rigid body rotating with the same

angular momentum of the fluid, and nmot is the engine speed. This index expresses a global

information in which are included the effects of eventual counter rotating structures, without specifying their number, dimension or location. In order to determine ntbthere are two ways:

• Exploit a three-dimensional CFD simulation of the intake process and evaluate the an-gular momentum of the charge, with respect to an axis normal to the tumble symmetry plane, which passes through the centre of mass of the charge itself. This evaluation, un-less inaccuracies related to the solution of the velocity field, is correct since is obtained at IVC from the integration of each discretized fluid mass contribution

• Estimate ntb starting from data which come from steady flow tests, integrating the

tumble coefficients obtained at different valve lifts and weighting them with respect to the flow coefficients

Is possible to use this coefficient also when dealing with steady simulations even if the engine speed can’t be specified directly. To overcome this problem can be considered a

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fictious engine speed which can be calculated assuming that the mean piston speed is equal to the mean axial velocity of the steady state flow:

nmot=

30¯uax

S =

30 ˙m ρAcylS

To distinguish this artificial case with respect to the real one, the Tumble Ratio takes the name of Tumble Number, Nt.

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1.2. FUNDAMENTALS OF LIQUID SPRAY 19

1.2 Fundamentals of liquid spray

In this section are discussed the general spray theory, mechanisms and characterization parameters in order to have the means to understand the engine’s injection behaviour.

In SI internal combustion engines, injection of fuel can be realized in a specific point of the intake duct (Port Fuel Injection) or directly into the combustion chamber (Direct Injection). This task is performed by means of injectors, which are specific nozzles. Thanks to a high pressure, the liquid fuel that exits the injector nozzle is vaporized to properly mix it with air in order to enhance evaporation, which is essential for good combustion achieve-ment. For this reason fuel-air mixing in engines is a process which heavily affects ignition behaviour, heat release, fuel consumption, pollutant formation and emissions, such as un-burned hydrocarbons (HC) and nitrogen oxides (N Ox). In PFI engines spray phenomena

are very complex because the liquid fuel droplets interact in multiple ways with each other, with the turbulent gas phase and with the intake port and intake valve walls.

Numerical simulations including injection represent a useful tool that can provide in-formation on the characteristics and quality of the air-fuel mixing process. An alternative strategy to completely model all the injection phenomena, is to perform specific experimen-tal tests to determine some characterization parameters, later introduced, which can be used to tune the numerical model of the spray.

1.2.1 Primary break-up

When the liquid fuel is forced by a large pressure to flow through the injector holes with high speed, the liquid jet (cone shaped spray) breaks up in small droplets because of its high velocity, relative to the surrounding air, the turbulence in the jet itself and in the air, and also cavitation which happens inside the nozzle. Cavitation occur when the fuel pressure falls below the fuel vapour pressure. This is the primary break-up.

At the nozzle outlet, the turbulent movements in the liquid stream and the implosion of the cavitation bubbles generate in the liquid jet unstable surface waves. Then the interaction between the jet and the air turbulence produces a rapid and selective grow of the surface waves. The surface waves amplification brings the liquid column leaving the nozzle to break up in ligaments and droplets. Primary break-up is mainly controlled by Reynolds nozzle number: Re,n = uDρ_µ l

l .

The movement of the liquid inside the nozzle depends on:

• Pressure difference between cylinder (Pcyl) and inside the nozzle (Pinj)

• Physical properties of the liquid

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Figure 1.4: Scheme of a liquid spray

1.2.2 Secondary break-up

Far away from the nozzle, the air mass entrained within the spray increases and generates turbulent vortexes in which the drops, whose shape is continuously changing due to aerody-namic interaction, undergo a further secondary break-up. This further division is favoured by high values of inertia forces (ρu2_r) due to the relative velocity between the air and single droplet (ur) and contrasted by the surface tension forces (_Dσl

d). The process is controlled by the Weber number, which compares the relative importance of previous forces:

W_e = ρu

2 rDd

σl

The break-up mechanisms vary according to We number:

M echanism We V ibrational ≈ 12 Bag < 20 Bag − jet/streamer < 50 Stripping < 100 Catastrophic > 100

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1.2. FUNDAMENTALS OF LIQUID SPRAY 21

Figure 1.5: Break-up mechanisms according to Wierzba [12]

1.2.3 Spray-wall interaction

Spray-wall interaction occur if a spray penetrating into a gaseous atmosphere impacts a wall, which can be the backside of the intake valve and the wall of the induction system in case of port fuel injection, or the combustion chamber wall in case of a direct injection engine. Two main physical processes can be involved: wall-spray development and wallfilm evolution. The interaction between wall and spray droplet mainly depends on the physical properties of the droplet, their velocity and direction during the impact; in particular the different impingement regimes are described in section 1.4.3. Both processes mentioned above may strongly influence combustion efficiency and the formation of pollutants. Whether wall impingement occurs or not depends on the penetration length of the spray and on the distance between injection nozzle and wall. High injection pressures as well as low gas densities and temperatures increase penetration and the possibility of wall impact.

1.2.4 Collision and coalescence

Droplet collision is an important effect in the dense spray region near the injection nozzle, where the number of droplets per unit volume is large and the probability of collision is high. The result of a collision event depends on the impact energy, the ratio of droplet sizes, and ambient conditions like gas density, gas viscosity, and the fuel-air ratio of the gas surrounding the droplets during impact. The possible outcomes of a collision event can be divided into five regimes: bouncing, coalescence, reflexive separation, stretching separation, and shattering.

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Figure 1.6: Collision and coalescence mechanisms

1.2.5 Evaporation

In addition to the break-up of the spray and the mixing processes of air and fuel droplets, the evaporation of liquid droplets also has a significant influence on ignition, combustion, and formation of pollutants. The formation of fuel vapour due to evaporation is a prerequi-site for the subsequent chemical reactions. The evaporation process determines the spatial distribution of the equivalence ratio, and thus strongly affects the timing and location of ignition. The energy for evaporation is transferred from the combustion chamber gas to the colder droplet due to conductive, convective, and radiative heat transfer, resulting in diffusive and convective mass transfer of fuel vapour from the boundary layer at the drop surface into the gas. This again affects temperature, velocity, and vapour concentration in the gas phase. Hence, there is a strong linking of evaporation rate and gas conditions, and, for this reason, there must always be a combined calculation of heat and mass transfer processes.

1.2.6 Spray characterization

A spray is characterized by three main physical properties:

• Atomization: describes disintegration of the liquid jet and it’s evaluated through proper droplet diameters

• Penetration: describes how fast and far from the nozzle the droplets are able to travel • Diffusion: describes the spreading of the spray and it’s evaluated through proper angles Granulometry

To describe the atomization process and to investigate the granulometry of a spray, it is necessary to use statistical quantities, such as a mean droplet diameter and a proper distribution around its mean value. If f (x) is the Probability Density Function (PDF) of

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1.2. FUNDAMENTALS OF LIQUID SPRAY 23 the casual variable droplet diameter x, the following mean values can be defined according to their physical meaning:

• Surface mean diameter, is given by the condition that the global surface of the real spray droplets equals the one of droplet set of mean diameter (xm is the min real

diameter and xM is the max real diameter):

dsur =

s Z xM

xm

x2_{f (x)dx}

• Volume mean diameter, is given by the condition of equal volumes:

dvol= 3

s Z xM

xm

x3_{f (x)dx}

• Sauter Mean Diameter (SMD), is the diameter of a model drop whose volume to surface area ratio is equal to the ratio of the sum of all droplet volumes in the spray to the sum of all droplet surface areas:

SM D = RxM xm x 3_{f (x)dx} RxM xm x 2_{f (x)dx}

Considering that the time required by the droplet vaporization is proportional to the ratio between its volume (∼ mass) and its surface (∼ heat transfer), the SMD is widely used. Can be also adopted a reference diameter which comes from an arithmetic average. Penetration

Consists in the evaluation of the distance travelled by the spray with respect to time. The distance is evaluated considering as reference a fraction of the mass of the spray. The outlet velocity of the liquid from the nozzle is the most relevant parameter affecting penetration and is dependent on the pressure difference and losses across the injector.

Diffusion

In the characterization of a spray three main angles are considered, which are evaluated considering a reference mass fraction contained inside the geometry:

• Split angle (αsplit)

• Cone angle (βcone)

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1.3 Computational Fluid Dynamics

Computational Fluid Dynamics (CFD) is the analysis of systems involving fluid flow, heat transfer and associated phenomena such as chemical reactions by means of computer-based simulations, which include the numerical solution of the Navier-Stokes equations. This methodology was born in the aeronautical field during the sixties and nowadays, thanks to the improvement in computer performances, is a fundamental tool in many engineering applications.

The most important limit of CFD is the available computational power, while the main advantages are:

• possibility to study complex systems which are difficult or even impossible to be in-vestigated experimentally

• flexibility in the analysis of several geometries and operating conditions

• possibility to monitor all the desired physical quantities simultaneously and without perturb the system

• time reduction in the evaluation of different design solutions • cost reduction

From the above mentioned benefits of this technique, is easy to understand its wide application in the engines’ world. Because of the complexity of these machines, to match the increasing demands of emissions reduction and better fuel consumption, together with high performances, is only possible with a combination of experimental and simulation methods.

In this section is proposed a brief recall of the theory at the basis of CFD, mainly focusing on the solutions adopted in the current work. Specifically are introduced the governing equations of fluid dynamics, the modelling of turbulence and numerical methods.

1.3.1 Governing equations

Fluid dynamic problems are described by a set of fully coupled nonlinear partial differen-tial equations (PDE), called Navier-Stokes equations. These governing equations represent mathematical statements of the conservation laws of physics:

• conservation of mass: continuity equation

• conservation of momentum: momentum equation • conservation of energy: energy equation

Due to the complexity of the Navier-Stokes equations, analytical solutions are available only for simple problems. However, experience shows that these equations faithfully describe

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1.3. COMPUTATIONAL FLUID DYNAMICS 25 the flow for a Newtonian fluid, therefore for all cases in which is not available an analytical solution is possible to rely on numerical resolution. Must be noted that in computational fluid dynamics the fluid is considered as a continuum.

In internal combustion engines (ICE) two different fluid phases are present: the gas, like the air flow inside the induction system, is easily describable by means of an eulerian ap-proach, and the liquid fuel injected, which is better represented by a lagrangian description.

Material derivative

There are two ways to describe fluid motion: the Lagrangian and the Eulerian description. The first approach is based on the motion of fluid particles, which are followed as they move through a flow field. It is the direct extension of single particle kinematics to a whole field of fluid particles that are labelled by their location, r0, at a reference time, t = t0. The

subsequent position r of each fluid particle as a function of time, r(t; r0, t0), specifies the

flow field. Here, r0 and t0 are boundary or initial condition parameters that label fluid

particles and are not independent variables. Thus, the current velocity u and acceleration a of the fluid particle that was located at r0 at time to are obtained from the first and

second temporal derivatives of particle position r(t; r0, t0). The Lagrangian description of

fluid motion is used in some simulations of combustion and multiphase flows. The second approach focuses on flow field properties at locations or in regions of interest, and involves four independent variables: the three spatial coordinates represented by the position vector x, and time t. Thus, in this field-based Eulerian description of fluid motion, a flow-field property F depends directly on x and t : F = F (x, t). Even though this description complicates the calculation of the acceleration, because individual fluid particles are not followed, it is the favoured description of fluid motion. Kinematic relationship between the two description can be determined by requiring equality of flow filed properties when r and x define the same point in space, both are resolved in the same coordinate system and a common clock is used to determine the time t. Under these assumptions the following equation can be derived:

d dtF [r(t; r0, t0), t] = D DtF (x, t) = (∇F ) · u + ∂F ∂t (1.3.1)

The final equality defines _DtD as the total time derivative in the Eulerian description of fluid motion. It is the equivalent of the total time derivative _dtd in the Lagrangian description and is known as the material derivative.

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General formulation of conservation equations

The integral general form of the conservation equations is the following: ∂ ∂t Z V ρφ dx + I A ρφV dA = I A Γ∇φ dA Z V Sφdx (1.3.2)

Is possible to identify four terms, from left to right, all referred to the generic quantity φ:

• the unsteadiness term related to the time variation of φ inside the control volume V • the convective term related to the net flux of φ due to convection through the surface

of the control volume

• the diffusive term related to the net flux of φ due to diffusion through the surface of the control volume

• the source term related to the generation of φ inside the control volume

Due to the complexity of the Navier-Stokes equations, analytical solutions are available only for simple problems. However, experience shows that these equations faithfully describe the flow for a Newtonian fluid, therefore for all cases in which is not available an analytical solution is possible to rely on numerical resolution.

Continuity equation

The conservation of mass states that, in absence of a source term, mass is neither created nor destroyed, and result in the following equation:

∂ρ

∂t + ∇ · (ρ~v) = S

l

m (1.3.3)

Sl

m is the source term which expresses the variation of gas mass in the domain, by the

effect of phenomenons of evaporation and condensation of the liquid phase.

Momentum equation

The conservation of momentum is derived directly from the second law of dynamic of New-ton, which states that the variation in time of the momentum is equal to the sum of all the forces acting on the fluid:

dq dt =

d(mu)

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1.3. COMPUTATIONAL FLUID DYNAMICS 27 The forces acting on a fluid basically are of two kind:

• body forces commonly arise from gravitational or electromagnetic force fields. In addition, in accelerating or rotating frames of reference, fictitious body forces arise from the frame non-inertial motion. By definition body forces are distributed through the fluid and are proportional to the mass (or electric charge, electric current, etc.). Usually these forces are specified per unit mass and carry the units of acceleration • surface forces, act on fluid elements through direct contact with the surface of the

element. They are proportional to the contact area and carry units of stress (force per unit area). Surface forces are commonly resolved into components normal and tangential to the contact area

The momentum equation can be written in the following way: ∂(ρu)

∂t + ∇ · (ρuu) = −∇p + ∇ · (τij) + ρg + S

l

u (1.3.5)

Where τij is the viscous shear stress tensor for Newtonian fluids, which is defined as a

function of the fluid viscosity and local deformation rate or strain rate:

τ_ij = 2µ(S_ij− Siiδij

3 ) (1.3.6)

δij is the tensor delta di Kronecker while Sii and Sij are respectively the components

normal and tangential of the viscous stress tensor. Sl

u represents the exchange of momentum among the two fluid phases.

Energy equation

The equation for energy conservation comes from the first principle of thermodynamic, which states that the rate of change of energy of a fluid particle is equal to the rate of heat addition to the fluid particle plus the rate of work done on the particle. Defining the total energy per unit mass e = i +u₂2 as the sum of the internal energy i and the kinetic energy, the energy equation is given by:

∂(ρe)

∂t + ∇ · (ρeu) = Qvol+ ∇ · Qs (1.3.7)

Qsrepresents the surface source term, inside which there are the following contributions:

• the work of pressure stresses ∇ · (−pu) and shear stresses ∇ · (τ · u)

• the heat flux ˙Q = −∇ · q ; here q is defined through the Fourier law q = −k∇T with k coefficient of heat transmission

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Qvol instead is the term that includes contributions relating to generation/dissipation

of energy that take place inside the control volume, among them the most important is the work done by the gravitational force. Substituting these terms inside Qs and Qvol the

expression of the balance of total energy is obtained:

∂(ρe)

∂t + ∇ · (ρeu) = −∇ · (pu) + ∇ · (τ · u) + ∇ · (k∇T ) + ρg · u + S

l

e+ ˙Qch (1.3.8)

in which, again it has been introduced a source term Sl

eto take into account the exchange

between liquid and gas, whereas ˙Q_ch represents the heat power caused by potential chemical reaction which can occur inside the domain.

In presence of compressible flows, the energy equation is often expressed as a function of the specific enthalpy h = i +p_ρ where the specific total enthalpy is h0 = h +1₂(u2+ v2+ w2).

∂(ρh) ∂t + ∇ · (ρhu) = Dp Dt + ∇ · (τ · u) + ∇ · (k∇T ) + ρg · u + S l h+ ˙Qch (1.3.9) Equation of state

Based on what has been said so far, the study of the behaviour of a generic fluid dynamic system can be attributed to the solution of five partial differential equations. However, it is underlined how the system of equation is not complete, since the unknown quantities are six (p ; ρ ; e or h ; ux; uy; uz). Therefore, to close the system an additional relation is required:

the equation of state. This relation provides the link between the momentum and continuity equation with the energy equation. Under the assumption of perfect gas:

p

ρ = RT (1.3.10)

When dealing with incompressible flows, the density is considered constant, therefore the velocity field can be solved without the energy equation and the link provided by the equation of state is not necessary. The present study considers the assumption of ideal gas.

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1.3. COMPUTATIONAL FLUID DYNAMICS 29

1.3.2 Turbulence modelling

Within the set of Navier-Stokes equations, the main non-linearity is provided by the convec-tion term that is responsible for the physical phenomenon of turbulence and shock waves, which can be generated with compressible flows. Turbulence is a three-dimensional, un-steady, rotational fluid motion with broad-banded fluctuations of flow quantities (velocity, pressure, temperature, etc.) occurring in both time and space. Its onset is triggered by fluid dynamic instabilities that, for Reynolds number beyond a certain limit (depending on the problem), broke the regular laminar profiles leading to the formation of unsteady vortices. Turbulence is a dissipative phenomenon characterized by the presence of eddies of different sizes, which range from the characteristic size of the geometry to the molecular level of the Kolmogorov scale, where dissipative viscous phenomena occur. Moreover, turbulent flows present enhanced mixing via transport, resulting in mass, momentum and energy diffusion rates much larger than molecular ones.

There are three main numerical approaches to deal with the presence of turbulence in CFD:

• DNS (Direct Numerical Simulation): simulates the whole range of the turbulent sta-tistical fluctuations at all relevant physical scales, and this needs a huge computational effort

• LES (Large Eddy Simulation): simulates directly the largest scales and models the small-scale vortexes. The computational effort is still high, but lower than that of DNS

• RANS (Reynolds Averaged Navier-Stokes): include the highest level of approximation and the lower computational effort. Is based on the solution of time averaged equations and rely on the modelling of turbulence

The RANS approach is by far the most adopted one, especially in the industrial field, due to its good trade-off between computational effort and quality of the solution. RANS method is based on the Reynolds decomposition, which considers each flow quantity to be written as the summation of a mean and a fluctuating component:

¯ ϕ(x) = 1 ∆T Z t+∆T₂ t−∆T 2 ϕ(x, t)dt (1.3.11) ϕ(x) = ¯ϕ(x) + ϕ0 (1.3.12)

Applying the temporal average to the Navier-Stokes equations lead to the Reynolds Av-eraged Navier-Stokes equations:

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                   ∂ρ ∂t + ∇ · (ρ~¯V ) = S l m ∂(ρ~V )¯ ∂t + ∇ · (ρ~¯V ~¯V ) = −∇¯p + ∇ · (¯τij) + ρ~g + ∇ · (ρ~v~v) + S l u ∂ρc ¯T ∂t + ∇ · (ρc ¯T ~¯V ) = k∇ 2_{T − ∇ · (ρcT}¯ 0_~_{v) + S}l e

As can be seen the averaging procedure lead to the appearance of additional terms. In the momentum equation the term ∇ · (ρ~v~v) acts like a stress added to the viscous molecular one. It’s called Reynolds stress and it is a symmetric tensor combining velocity fluctuations. In the energy equation, the non-linear convective term causes the presence of a combined fluctuating term analogous to the Reynolds stress, which is called turbulent heat flux. The equations do not provide any direct expression for these terms; therefore, a closure problem arises, and modelling is needed to achieve a solution.

As Boussinesq proposed, under the assumption of Newtonian fluid, Reynolds turbulent stresses can be modelled in analogy with viscous shear stresses. In this way the Reynolds stresses are reduced to be proportional to the mean strain rate, through a scalar coefficient, called turbulent or eddy viscosity µt. A similar shortcoming was proposed for the turbulent

heat flux, introducing the eddy diffusivity kt. To link this two quantities a turbulent Prandtl

number is introduced (pr,t = cµ_kt

t).

k-ε model

The model is the most widely used turbulence model, particularly for industrial computa-tions and has been implemented into most CFD codes. It is numerically robust and has been tested in a broad variety of flows, including heat transfer, combustion, free surface and two-phase flows. It is known that this model presents few shortcomings, since the need of wall functions lead to inaccuracies in the turbulence solution at the wall in presence of pressure gradients, due to curvature, and compressibility effects. However, it is generally accepted that the k − ε model usually yields reasonably realistic predictions of major mean-flow fea-tures in most situations. It is particularly recommended for a quick preliminary estimation of the flow field, or in situations where modelling other physical phenomena, such as chem-ical reactions, combustion, radiation, multi-phase interactions, brings in uncertainties that outweigh those inherent in the k − ε turbulence model.

The model solves modelled equations for the Turbulence Kinetic Energy, k, and the dissipation rate ε : ρ∂k ∂t + ρUj ∂k ∂xj = P + G − ε + ∂ ∂xj µ + µt σk ∂k ∂xj (1.3.13)

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1.3. COMPUTATIONAL FLUID DYNAMICS 31 ρDε Dt = Cε1P + Cε3G + Cε4k ∂U_k ∂xk − Cε2ε ε k + ∂ ∂xj µt σε ∂ε ∂xj (1.3.14) where: P = −2µtS : S − 2 3[µt(tr(S) + k)]tr(S) G = − µt ρσp ∇ρ µt= Cµρ k2 ε

The constants can be determined through experimental test or DNS simulations:

Cµ Cε1 Cε2 Cε3 Cε4 σk σε σρ

0.09 1.44 1.92 0.8 0.33 1 1.3 0.9 Table 1.2: Constants of k − turbulence model

Standard Wall Function

The common way to treat wall boundaries is to avoid to solve the molecular layer and buffer zones adjacent to a wall and to bridge the solutions at the first control cell (assumed to be fully turbulent) with the wall properties. This is achieved by using the “wall functions”, the set of semi-empirical functions, which have been derived from experimental evidence and similarity arguments. For mean momentum and energy equation, the wall functions are based on the assumed logarithmic velocity and temperature distribution, scaled with the “inner-wall” scales: the friction velocity Uτ = τ ω_ρ and viscosity µ for mean velocity, and

wall heat flux and temperature diffusivity for mean temperature. The standard logarithmic “laws of the wall” are modified as to relax conditions where τ ω → 0. For mean velocity the following wall functions are used:

U+ _{= y}+ _y+ _{< 11.63} U+ ₌ 1 kV ln(Ey +_{) y}+ _{> 11.63} where: U+= C 1 4 µ k 1 2 P U2 τUP ; y +_{= C}14 µ ρk 1 2 P µ yP

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with kV = 0.41, Von Karman constant. T+= σ_T 1 kV ln(Ey+) + Y where: T+= C 1 4 µk 1 2 P ρcp(TP − Tw) ˙ qw Y = 9.24 " Pr σT 0.75 − 1 # 1 + 0.28 exp −0.007Pr σT

with Pr molecular Prandtl number, σT turbulent Prandtl/Schmidt number and w

iden-tifing the wall values.

Han-Reitz Model [6]

Han and Reitz developed another formulation in the context of reciprocating engines which includes the variation of gas density:

T+ = 2.1 ln(y+) + 2.5 where: T+= C 1 4 µk 1 2 P ρcpTPln(T_TP_w) ˙ qw k-ς-f model [7]

This model has been developed by Hanjalic, Popovac and Hadziabdic (2004). The authors propose a version of eddy viscosity model based on Durbin’s elliptic relaxation concept (1991). The aim is to improve numerical stability of the original v2_{− f model by solving a}

transport equation for the velocity scale ratio ς = v_k2 instead of velocity scale v2_.

The eddy viscosity µt is obtained in the same way as in the k − ε model, while the other

variables are obtained from the following equations:

ρDε Dt = ρ(Pk− ε) + ∂ ∂xj µ + µt σk ∂k ∂xj (1.3.15) ρDk Dt = ρ (C_ε1∗ Pk− Cε2)ε T + ∂ ∂x_j µ + µt σ_k ∂ε ∂x_j (1.3.16) ρDς Dt = ρf − ρ ς kPk+ ∂ ∂xj µ + µt σk ∂ς ∂xj (1.3.17)

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1.3. COMPUTATIONAL FLUID DYNAMICS 33 Additional modifications to the ε equation is that the constant Cε1 is dampened close to

the wall thus:

C_ε1∗ = Cε1 1 + 0.0045

r 1 ς

!

Hybrid Wall Treatment [8]

This wall treatment should ensure a gradual change between viscous sublayer formulations and the wall functions. Popovac and Hanjalic (2005) extended Kader’s (1981) proposal for the description of temperature profile in the wall boundary layer also on all turbulence properties, thus: U+ = y+e−Γ+ 1 kV ln(Ey+)e−1Γ µw = µ y_P+ U_P+ Γ = 0.01(Pry +₎4 1 + 5P3 ry+

This wall treatment provides the standard wall function for the large values of y+ as well as the integration of equations up to the wall for the very small values of y+.

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1.3.3 Numerical methods

To solve the system of PDE Navier-Stokes, the conservation equations must be discretized. The critical point is to convert the partial differential equations that govern a physical phenomenon into a numerically manageable algebraic system of equations. To achieve this goal, the most widespread methodology is the Finite Volume Method. Some basic notions on this approach, grid arrangements and related solution strategies are here presented. Finite Volume Method (FVM)

FVM is the most widely used method in CFD thanks to its generality, its conceptual sim-plicity and its ease of implementation for arbitrary grids, structured as well as unstructured. The FVM discretize the integral form of the conservation equations directly in the physical space. For instance, the discretized equation of the general conservation equation (1.3.2) is the following: ∂(ρϕ) ∂t + nf X f =1 ρfufϕf · Af = nf X f =1 Γ∇ϕf · Af + SV (1.3.18)

The computational domain is subdivided into a finite number of contiguous control volumes, which form the mesh, and for each one of those an algebraic equation can be obtained. The variable values are calculated at the centroid of the control volumes, while the values at cell faces are computed from discretization techniques.

Discretization techniques

These techniques allow to evaluate variable values and gradients at cell faces. Must be noted that any discretization will generate errors, and this is a direct consequence of the replacement of the continuum model by its discrete representation (truncation error).

For simplicity it is analysed a mono dimensional grid, as can be seen in the figure below, where P is the cell centre, W and E are the neighbouring centre points respectively at distance δ_{xW P} and δ_xEP from P; w and e are the face centre points respectively at distance δ_xwP and δ_xeP from P and ∆x is the cell dimension.

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1.3. COMPUTATIONAL FLUID DYNAMICS 35 • Upwind, this scheme considers the direction of the flow and impose at the face the

value of ϕ, according to:

Φ_e=    ΦP uE ≥ 0 ΦE uE < 0 Φw =    ΦP uW ≥ 0 ΦE uW < 0 ∂Φ ∂x w = ΦP−ΦW ∆x ∂Φ ∂x e = ΦP−ΦE ∆x

• Central differencing, this scheme assumes a linear trend of φ between two adjacent cells: Φw = ΦW+Φ₂ P Φe = ΦP+Φ₂ E ∂Φ ∂x w = ΦP−ΦW ∆x ∂Φ ∂x e = ΦP−ΦE ∆x

• MINMOD, is a TVD (Total Variation Diminishing) scheme which adopts a flux lim-iter function, f (r). In CFD, TVD schemes are employed to capture sharper shock predictions without any misleading oscillations, when variation of field variable Φ is discontinuous. TVD schemes enables sharper shock predictions on coarse grids saving computation time and, as the scheme preserves monotonicity, there are no spurious oscillations in the solution.

Φe = ΦP +

1

2f (r)(ΦE − ΦP)

r = ΦP − ΦW ΦE − ΦP

f (r) = max[0, min(1, r)] ; lim

r→∞f (r) = 1

Limiter functions are designed such that they pass through a certain region of the solution, known as the TVD region, in order to guarantee stability of the scheme. Second-order, TVD limiters satisfy at least the following criteria:

 



f (r) < 2r 0 < r < 1 f (r) ≤ 2 r ≥ 1

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The admissible limiter region for second-order TVD schemes is shown in the Sweby Diagram below (shaded area), and the blue line shows the MINMOD limiter function.

Figure 1.8: MINMOD limiter fuction

Grid arrangements

The two most popular grid arrangements are the staggered and the co-located. The aim of having a staggered grid arrangement for CFD computations is to evaluate the velocity components at the control volume faces while the rest of the variables governing the flow field, such as the pressure, temperature, and turbulent quantities, are stored at the central node of the control volumes. The evaluation of the velocities at the control volume faces allows a straightforward evaluation of the mass fluxes that are used in the pressure correction equation, which is described in the following paragraph. This arrangement therefore provides a strong coupling between the velocities and pressure, which helps to avoid some types of convergence problems and oscillations in the pressure and velocity fields.

(a) Staggered (b) Co-located

Figure 1.9: Staggered and co-located arrangements of velocity components on a finite-volume grid

However, when the use of nonorthogonal grids became commonplace because of the need to handle complex geometries, alternative grid arrangements had to be explored because of some inherent difficulties in the staggered approach. In particular, if the staggered approach

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1.3. COMPUTATIONAL FLUID DYNAMICS 37 is used in generalized coordinates, curvature terms are required to be introduced into the equations that are usually difficult to treat numerically, and may create nonconservative errors when the grid is not smooth. Nowadays, the alternative grid arrangement that is frequently adopted in many commercial CFD codes is the co-located grid arrangement. Here, all the flow-field variables including the velocities are stored at the same set of nodal points. For the finite-volume grid as shown in figure 1.9, they are stored at the central node of the control volumes (open symbols). The collocated arrangement offers significant advantages in complicated domains, especially the capability of accommodating slope discontinuities or boundary conditions that may be discontinuous. Furthermore, if multi-grid methods are used, the collocated arrangement allows the ease of transfer of information between various grid levels for all the variables. The widespread use of the co-located grid arrangement is also due to developments of the pressure-velocity coupling algorithms such as the Rhie and Chow (1983) interpolation scheme. This scheme, which has provided physically sensible solutions on structured collocated meshes, generated much interest for unstructured meshing applications.

SIMPLE algorithm

The acronym SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations and this scheme is essentially a guess-and-correct procedure for the calculation of pres-sure through the solution of a prespres-sure correction equation. The main application of this algorithm is for incompressible flows, but there are also applications for compressible cases. In incompressible flows the decoupling dependence of density on pressure, also result in the decoupling of the energy equation from the rest of the system. As seen in section 1.3.1, the pressure term appear in all momentum equations but is not available an explicit equation for the pressure field. Furthermore the velocity field, which depend on the presssure one, has to satisfy the continuity equation. However the system is closed since there are four unknowns and four equations. The algorithm can be schematized as follows:

1. Perform an initial guess for the flow variables 2. Solve the dicretized momentum equations 3. Solve the pressure correction equation 4. Correct pressure and velocities

5. Solve all others discretized transport equations 6. Repeat until convergence

SIMPLE algorithm without under-relaxation is unstable, and will probably diverge. Di-vergence is due to the fact that pressure correction contains both the pressure as a physical

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variable and a component which forces the discrete fluxes to become conservative. After each iteration, at each cell, a new value for variable φ is then updated using following equation:

φnew,used_i = φold_i + αur(φnew,predicted_i − φoldi ) (1.3.19)

where αur is the under-relaxation factor. Under-relaxation may slow down speed of

convergence, but increases the stability of the computation, i.e. it decreases the possibility of divergence or oscillations in the solution. The choice αur = 1 corresponds to no

under-relaxation, while the choice αur < 1 is under-relaxation. This

PISO algorithm

The PISO algorithm, which stands for Pressure Implicit with Splitting of Operators, is a pressure–velocity calculation procedure developed originally for non-iterative computation of unsteady compressible flows.

PISO algorithm may be seen as an extension of SIMPLE, used to solve Navier-Stokes equations applied to unsteady problems. The idea of PISO can be summarized as follows:

• the pressure-velocity system contains to complex coupling terms: the non-linear con-vection term and the linear pressure-velocity coupling

• under the assumption of low Courant number (small time step), the pressure velocity coupling is much stronger than the non linear coupling

It is therefore possible to repeat a number of pressure correctors without updating the momentum equation discretization. The first pressure corrector will create a conservative velocity field, while the second and following will establish the pressure distribution. It is no longer necessary to under-relax the pressure.

The algorithm can be schematized in the following way:

1. Use the available pressure field from previous corrector or time step 2. Discretise the momentum equation with the available flux field 3. Solve the momentum equation using the guessed pressure 4. Calculate the new pressure based on the velocity field

5. Based on the pressure solution, assemble conservative face flux

6. Explicitly update cell centred velocity field with the assembled momentum coefficients 7. Return to pressure correction step

CFD full cycle analysis of a spark ignition engine at high part-load conditions

POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master of Science in Mechanical Engineering

CFD FULL CYCLE ANALYSIS OF A SPARK

IGNITION ENGINE AT HIGH PART-LOAD

CONDITIONS

Author:

Davide Arnaboldi

ID

: 905104

Supervisor:

Prof. Gianluca D’Errico

Co-supervisor:

Ing. Stefano Fantoni

Ing. Donato Colangelo

Contents

Acknowledgements

Abstract

Sommario

Introduction

Chapter 1

Theoretical background

1.1

Gas exchange process in ICE

1.1.1

Volumetric efficiency

1.1.2

Valve flow coefficient

1.1.3

Influence of fuel supply

1.1.4

Charge motions

1.2

Fundamentals of liquid spray

1.2.1

Primary break-up

1.2.2

Secondary break-up

1.2.3

Spray-wall interaction

1.2.4

Collision and coalescence

1.2.5

Evaporation

1.2.6

Spray characterization

1.3

Computational Fluid Dynamics

1.3.1

Governing equations

1.3.2

Turbulence modelling

1.3.3

Numerical methods