Experimental study of a flow into an engine cylinder using PIV

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Experimental Study of a Flow into an Engine Cylinder

Using PIV

Alessandro Davario Vincenzo Di Lella

June 2017

Master of Science Thesis Royal Institute of Technology

KTH Mechanics

SE-100 44 Stockholm, Sweden

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Experimental Study of a Flow into an Engine Cylinder

Using PIV

Master Thesis in Fluid Mechanics

School of Industrial Engineering and Management

Authors:

Alessandro Davario Vincenzo Di Lella

Academic supervisors:

Prof. P. Henrik Alfredsson Prof. Alessandro Talamelli Industrial supervisor : Dr.Eng. Björn Lindgren

June 2017 KTH Mechanics SE-100 44 Stockholm, Sweden

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Preface

The present master thesis work was performed as part of a Dual Master programme between the University of Bologna and KTH, Stockholm, between January and June 2017. The work was conducted as a collaboration between the KTH Mechanics department and the Engine Development, Gas Exchange Performance (NMGP) division at Scania CV AB, situated in Södertälje, Swe- den.

June 2017, Stockholm Alessandro Davario Vincenzo Di Lella

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Abstract

This degree project is focused on the study of the tumble and the swirl motions, which develop during the intake stroke, inside a cylinder of a Diesel engine.

Nowadays, the reduction of fuel consumption and emissions is a primary aspect for automotive companies, including Scania. Then, an efficient combustion process is required, and a fundamental role is played by tumble and swirl motion.

These motions have been studied by means of the Particle Image Ve- locimetry (PIV) technique. In particular, two different cylinder head designs have been investigated, focusing on the structures present in the flow and their evolutions inside the cylinder. Finally, these results have been compared with LES results, in order to validate the latest.

Analysing the swirl motion, it has been possible to identify three main regions, along the cylinder, characterized by different vortex structures. In addition, the velocity field into the entire cylinder volume has been extracted by means of a three-dimensional three-component reconstruction.

Keywords: Swirl motion, Tumble motion, PIV, intake stroke, Diesel engine.

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List of Figures

2.1 p − V diagram of the engine cycle . . . . 5

2.2 Tumble and swirl motion visualization . . . . 7

2.3 Different swirl generation solutions . . . . 8

2.4 The energy cascade process . . . 15

3.1 Illustration of a PIV setup in a wind tunnel . . . 23

3.2 From interrogation volume to interrogation area . . . 25

3.3 Intensity field I . . . 26

3.4 Peaks decomposition in cross-correlation function . . . 28

3.5 Determination of an optimal ∆t . . . 28

3.6 Third velocity component reconstruction . . . 29

3.7 Mono-configuration setup . . . 32

3.8 Stereo-configuration setup . . . 33

3.9 Schematic rapresentation of the stereo-configuration setup . . 34

3.10 Schematic stereo-plane position . . . 35

3.11 Two level calibration plate for PIV calibration. . . 37

3.12 Schematic representation of the Scheimpflug rule . . . 38

3.13 Scheimpflug adapter component . . . 39

3.14 Flow chart of stereo PIV vector field computation . . . 40

3.15 Misalignment between laser sheet and calibration planes . . . 42

3.16 Non-reflective plastic ring. . . 44

4.1 Simple illustration of the 3D3C reconstruction process. . . 48

4.2 Overlap region due to redundant information . . . 48

4.3 Instantaneous swirl centre calculation . . . 50

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5.1 Visualization of uncertainty analysis . . . 56 5.2 Instantaneous u velocity component values . . . 57 5.3 Valve position for mono PIV configuration measurements . . . 58 5.4 Swirl plane main results for CH1, valve lift 15 mm . . . 59 5.5 Swirl plane main results for CH1, valve lift 10 mm . . . 60 5.6 Instantaneous swirl centre position for two different snapshots 61 5.7 Swirl centre distribution for different cases . . . 62 5.8 Analysis of radial, tangential velocity profile . . . 63 5.9 Analysis of radial, tangential velocity profile: point evaluated . 63 5.10 Mode energy variation for CH1 . . . 64 5.11 POD analysis for the 350 mm measurement plane with 15 mm

valve lift for CH1 . . . 66 5.12 POD analysis for the 80 mm measurement plane with 15 mm

valve lift for CH1 . . . 67 5.13 Kinetic energy content variation along the cylinder axis, asso-

ciated to Mode 1, for CH1 . . . 68 5.14 Total kinetic energy variation along the cylinder axis, for CH1 68 5.15 Positive direction of rotation of the measurement plane in

stereo configuration . . . 69 5.16 Tumble plane 1 main results for CH1, valve lift 15 mm . . . . 70 5.17 Streamtubes obtained from the 3D3C reconstruction for CH1 . 71 5.18 3D3C flow visualization for CH1 . . . 72 5.19 Swirl plane main results for CH2, valve lift 15 mm . . . 74 5.20 Swirl plane main results for CH2, valve lift 10 mm . . . 75 5.21 POD analysis for the 350 mm measurement plane with 15 mm

valve lift for CH2 . . . 77 5.22 POD analysis for the 350 mm measurement plane with 15 mm

valve lift for CH2 . . . 78 5.23 Energy content variation along the cylinder axis, associated to

the Mode 1, for CH2 . . . 79 5.24 Total kinetic energy variation along the cylinder axis, for CH2 79 5.25 Tumble plane 1 main results for CH2, valve lift 15 mm . . . . 80 5.26 Streamtubes obtained from the 3D3C reconstruction for CH2 . 81

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LIST OF FIGURES

5.27 3D3C flow visualization for CH2 . . . 82 5.28 Visualization of swirl motion obtained using a 3D3C recon-

struction for CH2 . . . 84 5.29 Visualization of swirl motion obtained using a 3D3C recon-

struction for CH1 . . . 85 5.30 Comparison between mono PIV results and LES simulations (1) 87 5.31 Comparison between mono PIV results and LES simulations (2) 88 A.1 Swirl plane main results for CH1: 42 mm and 60 mm, 15 mm

valve lift . . . 93 A.2 Swirl plane main results for CH1: 80 mm, 100 mm, 120 mm

and 140 mm, 15 mm valve lift . . . 94 A.3 Swirl plane main results for CH1: 228 mm, 300 mm and 350

mm, 15 mm valve lift . . . 95 A.4 Swirl plane main results for CH1: 42 mm, 60 mm, 80 mm and

100 mm, 10 mm valve lift . . . 96 A.5 Swirl plane main results for CH1: 120 mm, 140 mm, 170 mm

and 228 mm, 10 mm valve lift . . . 97 A.6 Swirl plane main results for CH1: 300 mm and 350 mm, 10

mm valve lift . . . 98 B.1 Tumble plane 1 main results for CH1: 0^◦ and 10^◦ . . . 99 B.2 Tumble plane 1 main results for CH1: 20^◦, 30^◦, 40^◦ and 50^◦ . . 100 B.3 Tumble plane 1 main results for CH1: 60^◦, 70^◦, 80^◦ and 90^◦ . . 101 B.4 Tumble plane 1 main results for CH1: 100^◦, 110^◦, 120^◦ and 130^◦102 B.5 Tumble plane 1 main results for CH1: 140^◦, 150^◦, 160^◦ and 170^◦103 B.6 Tumble plane 1 main results for CH1: 180^◦, 190^◦, 200^◦ and 210^◦104 B.7 Tumble plane 1 main results for CH1: 220^◦, 230^◦, 240^◦ and 250^◦105 B.8 Tumble plane 1 main results for CH1: 260^◦, 270^◦, 280^◦ and 290^◦106 B.9 Tumble plane 1 main results for CH1: 300^◦, 310^◦, 320^◦ and 330^◦107 B.10 Tumble plane 1 main results for CH1: 340^◦ and 350^◦ . . . 108 C.1 Swirl plane main results for CH2: 42 mm and 60 mm, 15 mm

valve lift . . . 109

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C.2 Swirl plane main results for CH2: 80 mm, 100 mm, 120 mm and 140 mm, 15 mm valve lift . . . 110 C.3 Swirl plane main results for CH2: 170 mm, 200 mm, 230 mm

and 300 mm, 15 mm valve lift . . . 111 C.4 Swirl plane main results for CH2:350 mm, 15 mm valve lift . . 112 C.5 Swirl plane main results for CH2: 42 mm, 60 mm, 80 mm and

100 mm, 10 mm valve lift . . . 113 C.6 Swirl plane main results for CH2: 120 mm, 140 mm, 170 mm

and 200 mm, 10 mm valve lift . . . 114 C.7 Swirl plane main results for CH2: 300 mm and 350 mm, 10

mm valve lift . . . 115 D.1 Tumble plane 1 main results for CH2: 0^◦, 20^◦, 40^◦ and 60^◦ . . 117 D.2 Tumble plane 1 main results for CH2: 40^◦, 60^◦, 80^◦ and 100^◦ . 118 D.3 Tumble plane 1 main results for CH2: 120^◦, 140^◦, 160^◦ and 180^◦119 D.4 Tumble plane 1 main results for CH2: 200^◦, 220^◦, 240^◦ and 260^◦120 D.5 Tumble plane 1 main results for CH2: 280^◦, 300^◦, 320^◦ and 340^◦121

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List of Tables

3.1 Distance of each measurement plane from the cylinder head, in mono-configuration. . . 31 3.2 Positions of the stereo-planes. . . 34 3.3 Experimental parameters for different PIV measurement cases. 36 5.1 Region limits for both valve lifts for CH1 . . . 57 5.2 Region limits for both valve lifts for CH2 . . . 73

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Acknowledgements

We would like to thank our KTH supervisor, Prof. Henrik Alfredsson, and our Scania supervisor, Dr. Eng. Björn Lindgren, for their availability and help during this thesis project.

A huge thank to Manuel and Timo for helping us during the tests performed, and to the NMGP division for welcoming us in the office.

We would like to express our gratitude to Prof. Alessandro Talamelli for this exchange opportunity.

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1 Introduction

Nowadays, fuel consumption and emissions are a main concern for car and truck industries. A reduction of both is required, due to several aspects.

In particular, the constant rise of fuel price and the increasingly stringent regulations lead to a continuous research of new solutions to satisfy these objectives.

The emission standards are defined by a series of European Union di- rectives. These concern most part of the vehicles: cars, trucks, locomotives and similar machinery. Several emissions are of main interest for the environmental safeguard: nitrogen oxides (NO_X), non-methane hydrocarbons (NMHC), total hydrocarbon (THC), carbon monoxide (CO) and particulate matter (PM). With regards to Diesel engines, NO_X and PM are of main concern. In an industry with an increasing environmental awareness, Scania developed a complete package for reducing fuel consumptions and emissions, called Ecolution. The main objective of this programme is the decrease of fuel consumption by more than 10%.

To reach all these requirements, it is important to have the best combustion process. Then, understanding air and fuel motion inside a diesel engine is a key aspect. A fundamental role, during the combustion process, is played by tumble and swirl motions. Inducing these two motions, a better mixing of the fuel and air, inside the combustion chamber, can be achieved. For this reason, important care is taken in the design of the cylinder head.

In this degree project, tumble and swirl motions are investigated by means of the Particle Image Velocimetry (PIV) technique. For this purpose, two different cylinder heads are tested. In particular, the work is focused on the

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identification of these main structures of swirl and tumble, induced during the intake stroke of Diesel engine. Strong emphasis is posed on the study of their evolution. In addition, the entire velocity field inside the cylinder volume is reconstructed. Finally, the results obtained are compared with LES simulations.

In this report, a brief overview of the engine working principle and of the flow inside the cylinder is presented in Theoretical Background chapter.

In the Experimental Method and Setup chapter, first the PIV technique is explained, including the calibration procedure and the related issues. In addition, the instrumentation used and experimental configurations adopted are described in detail.

The data obtained have been processed using different methods, presented in the Post-processing techniques chapter.

Finally, the results and the related conclusions are presented in the Results and Discussion chapter.

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2 Theoretical Background

In this chapter, a brief description of all the fundamental theoretical aspects regarding this degree project are treated. In particular, the working principle of a Diesel engine and the flow characteristics inside its cylinder are described, including the equations which govern the fluid motion.

The main characteristics of a complex process, such as the flow motion inside the cylinder, can be captured by means of the Proper Orthogonal Decomposition (POD), whose mathematical background is treated.

2.1 Engine background

By definition, an engine is a machine which transforms one type of energy into mechanical energy. Electric motors convert electrical energy into mechanical motion, heat engines produce heat burning fuel, pneumatic motors use compressed air. For our purpose, heat engines will be analysed in the following sections.

Heat engines are usually divided into:

• Internal combustion engine: the combustion of a fuel occurs with an oxidizer, which is usually air, inside a combustion chamber.

• External combustion engine: the internal working fluid is separated from the fuel-air mixture. Through its combustion and by means of the engine wall or a heat exchanger, the working fluid is heated and mechanical energy is produced, in a thermodynamic cycle.

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The internal combustion engines, which are of our interest, can be classi- fied, according to the type of ignition, as follows:

• Compression-ignition engine: also known as Diesel engine or CI engine, is an engine in which the combustion of the fuel injected inside the combustion chamber is ignited by means of the high temperature reached by the compressed air.

• Spark-ignition engine: generally a petrol engine, where the combustion of the air-fuel mixture is caused by a spark from a spark plug.

In the following analysis, only the Diesel engine has been considered, in particular the four-stroke Diesel engine, since it is the main interest in the Scania engine production.

2.1.1 The four-stroke Diesel engine

The four-stroke engine is an engine characterized by internal combustion. To complete an operating cycle, it requires four distinct piston strokes. With the term stroke, we are referring to a complete movement of the piston inside the cylinder in either direction. In this section, each of them will be discussed, posing particular attention on how the flow develops in each of the different phases. In Figure 2.1, a pressure-volume diagram of an engine cycle is shown.

The first stroke is usually called Intake Stroke and it corresponds to the suction of the air into the cylinder. During this phase, the inlet valves stay in an open position, while the exhaust valves remain closed. Moreover, the piston moves from the highest position (TDC: top dead centre), to the lowest position (BDC: bottom dead centre). Regarding the flow during this phase, two important aspects need to be mentioned. First, due to the particular geometry of the inlet ports and valves, two different types of organized motion are induced inside the cylinder: the first one is known as swirl, directed around the cylinder axis; the second one is addressed as tumble and it is orthogonal to the cylinder axis. In the following section a deeper understanding of these motions will be provided. The second aspect is related to the jet itself, which is turbulent, and part of the non-turbulent energy contained in

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2.1. Engine background

Figure 2.1: p−V diagram of the engine cycle. The marked points correspond to: (1) inlet valve closing; (2) start of combustion; (3) maximum pressure;

(4) exhaust valve opening. Reproduced from Ref [1].

the jet is transformed into turbulence. This will lead to a high turbulence level during this first phase.

The second stroke is called Compression Stroke. During this phase, all the valves are closed and the piston moves from the BDC to the TDC. This phase is necessary in order to increase the pressure. Moreover, also the temperature will rise significantly, a necessary condition to allow combustion of the fluid during the next phase. An important parameter characteristic of this phase is the compression ratio, which links the volume of the cylinder when the piston is at the BDC to the volume of the combustion chamber. It is fundamental to point out that the temperature of the compressed flow is a function of the compression ratio. From this fact follows the necessity of having high compression ratio, typical of a Diesel engine.

The third stroke is called Combustion Stroke, also referred as power stroke. During this phase the inlet valves and exhaust valves remain closed and the piston is in the end position of the compression stroke (TDC). In Diesel engine, the fuel is sprayed into the combustion chamber by means of

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high pressure injectors. Due to the presence of air at high temperature, the fuel droplets start to evaporate and a mixture of air and fuel is reached. Fi- nally, since the temperature and pressure are above the fuel’s ignition point, this mixture spontaneously ignites and it will result in a movement of the piston until the BDC position. This phase is fundamental in order to produce mechanical work to turn the crankshaft.

The last stroke is called Exhaust Stroke. During this phase, the piston will move again from the BDC to the TDC. In addition, the exhaust valves are open, while the intake valves remain closed. While the piston moves, the spent air-fuel mixture is expelled into the atmosphere.

2.1.2 Fluid motion inside cylinder: tumble and swirl

Due to the specific geometry of valves and ports, two different types of organized motion are induced inside the cylinder. As mentioned before, these motions have a specific direction. However, since the mean velocities are usually smaller than turbulent velocities, these organized motions, usually referred in the literature as coherent motion, are "hidden" inside the disor- ganized turbulence. Therefore, in order to find the organized part and thus to have a clear visualization of it, it is necessary to have a number of measurements enough to allow a statistical analysis and to find the average flow pattern. In Figure 2.2, a visualization of both tumble and swirl motion is reported. These two kinds of motion are related, since as swirl is generated, tumble motion is also induced. There are several reasons about why it is so important to induce swirl and tumble inside the cylinder. The most important reason is to provide high turbulence levels at ignition. Indeed, this fact will result in higher flame speeds, thus leading to a more efficient combustion process. In addition, another fundamental aspect lies on the energy content.

Indeed, the free turbulence is more dissipative than an organized motion, which will result in a rapid decay of the energy in the inlet jet. However, the basic idea of such motions is to try to retain some of the momentum of the inlet jet, in order to have high energetic levels for a longer period.

Another important aspect to underline is an interesting difference between

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2.1. Engine background

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Figure 2.2: (a) visualization of tumble motion; (b) visualization of swirl motion. Reproduced from Ref [2].

tumble and swirl. Indeed, as the piston reaches the TDC, the tumble always breaks down to turbulence, since there is not enough space to contain a vortex of such dimensions. The same do not happens for the swirl, in case of particular combustion chamber shapes, as reported in [2]. Therefore, swirl do not break up into turbulence and this will result in a good way to control the combustion process and the flame propagation.

Normally, for both swirl and tumble, a swirl ratio and a tumble ratio are defined. They are determined as following:

R_S = ω_S

2πN (2.1)

R_T = ω_T

2πN (2.2)

where R_S and R_T are the swirl ratio and the tumble ratio, ω_S is the angular velocity of the solid body having the same angular momentum as the real swirl flow, ω_T is the angular velocity of the solid body having the same angular momentum as the real tumble flow, N is the angular velocity of the

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(a) (b)

Figure 2.3: (a) Different types of swirl-generating inlet ports: (a.a) deflector wall; (a.b) directed; (a.c) shallow-ramp helical; (a.d) steep-ramp helical. (b) Shrouded inlet valve and masked cylinder head approaches. Reproduced from Ref [2].

crankshaft (revolutions/sec).

2.1.3 Swirl Generation

There are several ways to generate swirl motion inside the cylinder.

The first strategy when dealing with a two inlet valve head consists in keeping one of the inlet valves closed, or almost closed. In this way, high swirl motion is obtained and since the air flow is very small, usually this configuration is used to have a fuel-rich cloud close to the cylinder head, which will result in a stratified charge.

Moreover, swirl can be induced by the geometry of the inlet port, or by shrouding or masking the valve, as can be seen in Figure 2.3. By means of deflector wall port and directed port, the flow is brought in the desired tangential direction. However, they result in lower discharge coefficients compared with the ones reached by the helical ramp port. Moreover, it is possible to shroud the valve, option which is mainly realized on experimental engines, not on production engines, because in this configuration the valve has to be prevented from turning. To avoid this problem, masking has been introduced and included in a production engine.

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2.2. Flow inside the cylinder

2.2 Flow inside the cylinder

Starting from physical laws or principles, it is possible to express the equations of fluid motion in mathematical terms. In this way, we are capable to understand and, at the same time, to provide a representation of the dynamics of fluids.

It is important first to point out that in this derivation, the validity of the continuum hypothesis is assumed. In this sense, our description provides good predictions, at least whenever the smallest volume taken in consideration contains enough molecules that statistical averages are reasonable.

In particular, we can define a central concept of this hypothesis: the fluid particle, as expressed in Ref [3]:

"A fluid particle is a volume of fluid that is sufficiently small to be considered infinitesimal with respect to the spatial variations of any macroscopic quantity, but large enough to contain a number of molecules

that is sufficiently high to allow the average value of each quantity to be statistically stationary".

We will start deriving the equations of motion of a fluid for any deformable continuum. Then, further relations will be introduced, in order to specify the physical behaviour of the continua treated. Finally, our description will be specialized to a specific class of fluid, the so called Newtonian fluids.

2.2.1 From the general conservation of Mass, Momen- tum and Energy to the Navier-Stokes equations

As stated before, we derive the equations of motion expressing in mathematical form the following fundamental balances of the mechanics of a continuum:

• the balance (or conservation) of mass;

• the balance of momentum;

• the balance of total energy.

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This procedure leads to the following set of equations. Further details related to the derivation are reported in [3].

∂ρ

∂t + ∂

∂xi

(ρu_i) = 0 (2.3)

∂ρui

∂t + ∂

∂x_j(ρu_iu_j) = ∂

∂x_j(pδ_ij + τ_ij) (2.4)

∂ρe₀

∂t +∂ρu_ie₀

∂xi

= ∂

∂xi

(−u_ip + u_jτ_ij − q_i) (2.5) Here, ρ is the local density of the fluid, u_i is the component of velocity vector in the i direction (i = 1, 2, 3 and summation is assumed over j = 1, 2, 3), p is the mechanical pressure, which is equal to minus the mean normal stress, τ is the viscous stress tensor, e0 is the total energy per unit mass, and it is equal to e₀ = e + V²/2, where e is the internal energy, V is the modulus of the velocity and q_i is the heat flux through the surface in the i-direction.

The contribution given by the body (or volume) forces has been neglected, compared to the other contributions.

At this point, it is important to identify the unknown functions that are present in the equations above. The density ρ, the three components of the velocity vector ~u, the six independent components of the symmetric stress tensor τ_ij , the internal energy e and the three components of the heat flux vector ~q. Thus, we have 14 unknown scalar functions and only 5 scalar equations. We need to introduce appropriate constitutive equations, in order to close the mathematical problem.

The first equation that we introduce is a relation, which links density, internal energy and temperature:

e = e(ρ, T ). (2.6)

It is fundamental to point out that this relation does not reduce the number of unknown functions, since it introduces a new quantity, the absolute temperature T . However, this quantity is usually introduced due to its di- rect measurability and the presence of other quantities dependent on it. In

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particular, by means of the Fourier law, it is possible to link the heat flux vector and the temperature as following:

q_i = −k∂T

∂xi

(2.7) where k is the thermal conductivity. It holds that:

k = k(T ) (2.8)

Then, we introduce also the following relation, which links pressure, density and temperature:

p = p(ρ, T ) (2.9)

Finally, we can introduce the expression for the viscous stress tensor in case of a particular class of fluids, the so called Newtonian fluids. Considering also the Stoke’s hypothesis given by the relation λ = −²₃µ, where λ is the second viscosity coefficient and µ is the dynamic viscosity coefficient, we obtain the following relation:

τ_ij = −2 3µ∂u_i

∂x_iδ_ij + µ ∂u_i

∂x_j +∂u_j

∂x_i

!

(2.10)

For the dynamic viscosity, the following relation holds:

µ = µ(T ) (2.11)

2.2.2 Incompressible Navier-Stokes equations and Vor- ticity Dynamics Equation

The set of equations introduced in the previous section has an important role in the physics. However, in practice it is quite complex to analyse or simulate. Therefore, when studying flows usually some simplifications are assumed, considering the state of the fluid involved. For our purpose, since only the intake stroke is taken into consideration, it is reasonable to assume an incompressible flow (Mach number M<0.3), in which variations of den-

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sity due to its motion may be neglected. In particular, we are assuming that temperature fluctuations in our test rig are negligible. This important assumption leads to the so called Incompressible Navier-Stokes equations:

∂u_i

∂x_i = 0 (2.12)

ρ ∂u_i

∂t + u_j∂u_i

∂x_j

!

= −∂p

∂x_i + µ ∂²u_i

∂x_j∂x_j (2.13)

It is important to highlight that now the density is not an unknown, since it does not change during the motion. On the other side, the pressure p is an unknown.

Here it is of interest to recall also another quantity, which plays a fundamental role in the following analysis. The concept of vorticity tells us the tendency of a fluid particle to rotate at a particular point. From a mathematical point of view, it is defined as the curl of velocity, as stated below:

~

ω = ~∇ × ~u (2.14)

Finally, reshaping the momentum balance equation for homogeneous incompressible flow, and taking the curl, it is possible to obtain the following vorticity dynamics equation:

∂~ω

∂t + (~u · ∇)~ω = (~ω · ∇)~u + ν∇²~ω (2.15)

It is easy to understand that the terms on the left-hand side refer to the material derivative of vorticity; this means that the terms on the right-hand side represent the possible sources that change the vorticity of a moving par- ticle. In particular, the first term is referred in literature as Vortex Stretching and it derives from the conservation of angular momentum, while the second term represents the variation of vorticity due to diffusion by viscous effect.

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2.2.3 Reynolds-Averaged Navier-Stokes equations

The Reynolds-Averaged Navier-Stokes equations, usually referred in literature with the acronym RANS, are time-averaged equations of motion for fluid flow. Principally, they are used to describe turbulent flows. Indeed, turbulence is a particular flow regime characterized by some interesting features that make its study a challenge for scientists. Turbulence is mainly characterized by irregularity, since the flow consists of structures with different scales, aspect ratio, and orientation. In addition, it exhibits high diffusiv- ity, which increases the exchange of momentum. Moreover, it shows also three-dimensionality, unsteadiness and dissipation.

In flows characterized by turbulence, the field properties become ran- dom functions of space and time. For this reason, in order to derive the RANS equations, it is necessary to introduce the Reynolds decomposition.

By means of this technique, since we are considering a time independent flow, it is possible to decompose an instantaneous feature of interest into its time-averaged and fluctuating parts. Thus, the field variables u_i and p can be expressed as the sum of mean and fluctuating part as following:

u_i = U_i+ u⁰_i (2.16)

p = P + p⁰ (2.17)

where the mean and fluctuating parts satisfy:

< u_i >= U_i, < u⁰_i >= 0 (2.18)

< p >= P, < p⁰ >= 0 (2.19) where the expression < · > denotes the time average. For our purpose, it is reasonable to not make any distinctions between time average and statistical average, due to the assumption of steady flow. In this way, it is possible to define the above quantities by means of a statistical average based on N

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samples as following:

U_i =< u_i >= 1 N

N

X

k=1

u^k_i (2.20)

P =< p >= 1 N

N

X

k=1

p^k (2.21)

Finally, it is possible to derive the Reynolds-Averaged Navier-Stokes equations for a stationary, incompressible Newtonian fluid, in the following Ein- stein notation:

∂U_i

∂x_i = 0 (2.22)

∂U_i

∂t + ∂

∂x_j(U_iU_j) = −1 ρ

∂P

∂x_i + ν ∂²U_i

∂x_jx_j − ∂

∂x_j < u⁰_iu⁰_j > (2.23)

The last term on the right-hand side, −ρ < u⁰_iu⁰_j >, is referred to in the literature as the Reynolds stress tensor. It is symmetric and it represents the correlation between fluctuating velocities. This non-linear tensor requires additional modelling to close the system. Indeed, the system derived above is not closed: we have 10 unknown terms but only 4 equations.

By means of the decomposition and equation defined above, it is possible to express some quantities, which will be used in the following analysis.

To begin with, we can determine the root mean square velocities RM S(u_i), using the following equation:

RM S(u_i) =

v u u t

1 N − 1

N

X

k=1

(u^k_i − U_i)² (2.24)

U_{RM S} =

s1

3(< u⁰²> + < v⁰² > + < w⁰²>) (2.25) In addition, we can define also the turbulent kinetic energy as follows:

K = 1

2 < u⁰_iu⁰_i >= 3

2U_{RM S}² (2.26)

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Figure 2.4: A description of the energy cascade. Reproduced from Ref [5].

2.2.4 Turbulent energy cascade and Kolmogorov mi- croscales

As pointed out in the previous section, turbulent flows contain a wide range of scales; in other words the velocity field contains eddies that are large enough to fill the entire space available (in our case the engine cylinder), but also eddies of around a millimeter or less in size. It is important to highlight an important feature related with the eddy size: the kinetic energy. Indeed, this property changes according to the size: the largest eddies contain a relatively small amount of energy. As the size decreases, the energy reaches a peak, where the most energetic scales are located [4].

Another interesting aspect about turbulence consists in dissipation. The process through which mean flow energy is converted to large scale turbulence, which breaks down into smaller scales, until the smallest scale is reached, is generally referred to as an energy cascade. For this reason, the dissipation, designated by is defined as the kinetic energy per unit mass of the flow that is transformed into heat.

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The Kolmogorov microscales are the smallest scales in turbulent flow.

The small eddies characterizing these scales are where the mechanical energy is converted into heat. They are characterized by the following length, time and velocity scales:

η = ν³

!1/4

(2.27)

τ_η = ν

!1/2

(2.28)

u_η = (ν)^1/4 (2.29)

where is the dissipation and ν is the kinematic viscosity of the fluid. In most cases, the Kolmogorov microscale η is of the order of few tenths of millimiter.

In order to provide a relation between the Kolmogorov scales and the largest scales (integral scales) it is necessary to estimate the dissipation rate, representing it by means of the large scale properties. Indeed, considering that the energy production at large scales has to be equal to the energy dissipation, it is possible to give an approximation to , as follows:

 = u⁰³

L (2.30)

where u⁰ is a turbulent velocity scale (e.g. the root mean square fluctuating velocity) and L is the size of the most energetic eddies. It is clear that Equation (2.30) comes from the fact that u⁰² is proportional to the kinetic energy of the flow and L/u⁰ is related to the time scale of the large eddies.

Finally we have that:

L

η = u⁰L ν

!3/4

= Re^3/4_L (2.31)

T

τ_η = u⁰L ν

!1/2

= Re^1/2_L (2.32)

where T = L/u⁰ is usually referred in literature as "large eddy turnover" time

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2.3. Proper Orthogonal Decomposition: POD

and Re_L is the Reynolds number of the turbulence, based on the large scale properties.

2.3 Proper Orthogonal Decomposition: POD

Proper orthogonal decomposition is a useful method to reduce high dimensional processes representing them with few basis modes, which are able to capture the dominant components of the process considered, without losing too much information [11].

2.3.1 POD Approximation

Given a function U (x) in a domain Σ, we can approximate it as a linear combination of basis function ϕⁱ(x):

U (x) '

M

X

i=1

a_i· ϕⁱ(x) (2.33)

Chosen the basis, the best approximation is given by amplitudes’ value which minimizes the error in a least square sense:

U (x) −

M

X

i=1

a_i· ϕⁱ(x)

→ min (2.34)

Obviously, there is not a unique choice for the basis functions, and it is difficult to evaluate if the ones chosen are the best choice. Using the POD approximation the basis are chosen in such way that

Z

Ω

ϕ_k1(x)ϕ_k2(x)dx =







1 k₁ = k₂ 0 k₁ 6= k₂

(2.35)

this imply

a_i =

Z

Ω

U (x) · ϕⁱ(x)dΩ (2.36)

which means that a_idepends only on the function ϕⁱ(x). With non-orthogonal

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basis function we should solve a set of linear equation to determine the amplitudes.

Moreover, satisfying Equation (2.34) we can find the first i orthonormal functions such that they give the best possible i-term approximation. These are called proper orthogonal modes for the function U (x) and Equation (2.33) is called the Proper Orthogonal Decomposition of U (x).

2.3.2 Discrete POD

In our case a discrete version of POD analysis is used, since we are dealing with a finite number of data. The name given in literature to discrete POD is Principal Component Analysis (PCA) [11].

In PCA the goal is to reduce the dimension of a set of correlated variables, without losing so much information regarding variation in the original data set. So, we find a new uncorrelated set of variables, called Principal Compo- nents (PCs), ordered in such a way that the first keep most of the variation presented in the original data set. Then, keeping only the first few PCs the original space of variables can be represented, reducing its dimension, without loosing too much accuracy.

The aim is to find orthogonal directions in N-dimensional space, ordered in such a way that the first i orthogonal directions give the best possible i-component approximation. Satisfying Equation (2.34), they guarantee that there cannot be another basis with a better approximation.

POD Basis: Singular Value Decomposition approach To construct the POD basis different approaches exist:

• PCA derivation minimizing the error of approximation;

• PCA derivation based on the correlation matrix;

• singular value decomposition (SVD) approach.

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2.3. Proper Orthogonal Decomposition: POD

In our work the last method has been used. Description for the other methods can be found in [11]. SVD, differently from the other two methods, can be applied also to non-square matrices.

Given any real N × M matrix U, there exist orthogonal matrices, N × N V¹ and M × M V², such that

(V¹)^TUV² = S (2.37)

where S is a N × M matrix with all element equal to zero, except for the main diagonal, in which these are equal to

s_ii = σ_i, σ₁ > σ₂ > ... > σ_r, r = min(M, N ) (2.38) where σ_i are the singular values of the matrix U, columns of V¹ are the left singular vectors, and columns of V² are the right singular vectors. A non negative σ is called singular value of the matrix U if and only if there exists a unit-length N -dimensional vector v¹, and a unit-length M -dimensional vector v², such that

Uv² = σv¹ and U^Tv¹ = σv² (2.39) The number of non-zero singular values gives the rank of the matrix U.

We can rewrite Equation (2.37) as

U = V¹S(V²)^T (2.40)

In order to obtain a matrix with a lower rank to approximate U, we can take only the first K singular values into a square K × K diagonal matrix SK, obtaining

U ' VK1SK(VK2)^T (2.41) where in V_K¹ and V_K² are collected respectively the first K columns of matrices V¹ and V². With this decomposition, no other matrix with a rank equal to K can better approximate the matrix U, in a last square sense.

Pre-multiplying Equation (2.37) by U^T from the left, it can be shown [11]

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that, defining D = U^TU, and S² = S^TS,

DV² = V²S² (2.42)

where the columns of V² are eigenvectors of matrix D (modified correlation matrix), and the square roots of eigenvalues of matrix D are singular values of matrix U.

Similarly, pre-multiplying Equation (2.37) by U^T from the right, it can be shown that the columns of V¹ are eigenvectors of matrix C = UU^T.

Based on the theoretical background explained, a MATLAB code [12] has been used to perform a POD decomposition.

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3 Experimental Method and Setup

The Particle Image Velocimetry (PIV) technique is used to study the flow motion inside the cylinder. In the following sections, the main characteristics, with the associated mathematical background, are presented. In addition, the instrumentation used and the configurations adopted are described in detail. Moreover, the calibration required and the main issues that can occur performing tests are included.

3.1 PIV technique

The Particle Image Velocimetry (PIV) technique is based on a very simple idea: to track particle movement in a flow, in order to determine the velocity field. A simple way to understand the technique is to think about snowflakes during a snowstorm: observing their trajectories we can have a qualitative understanding of the wind characteristics.

PIV technique is not a new concept, the earliest observations of this kind were conducted by Leonardo Da Vinci [6] [10]. However, modern techniques have been developed during the last thirty years, due to important improve- ments in the technological field.

PIV technique is a very broad topic, and despite that it has existed since many years, it is still improving and is in development. Then, in the following sections the authors aim is not to provide an extended description of PIV, but to focus on the basic working principles of it, and its advantages. A more

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complete and depth description can be found in [6] and [7].

3.1.1 PIV characteristics

The PIV technique consists of several different subsystems, which should work harmonically to obtain good measurement results. In Figure 3.1 an illustration of a PIV setup in a wind tunnel is shown . The main components are:

Laser and laser sheet optics To illuminate the zone of interest Mirror There can be more than one to direct the laser light Particles Tracking them the flow characteristics can be studied

Imaging optics-camera To capture particles movement in the lightened zone

Depending on the type of camera, particle shifts are recorded in a single frame or in a double frame. The pictures are divided into interrogation areas, in which the local displacement vector of the particles of the first and second illumination is evaluated through statistical methods (e.g: cross-correlation, auto-correlation etc...). Assuming that inside the interrogation area particles have moved homogeneously, and velocity vector is calculated knowing the time delay of the two frames.

The main characteristics of PIV measurement are:

Non-intrusive measurement PIV is a non-intrusive optical technique. This allows to take measurements also in difficult situations, such as super- sonic flow, boundary layer, without disturbing the flow.

Whole field technique PIV allows to analyse large region of the flow field.

This is not possible with other techniques, such as hot-wires or Pitot tubes, which give only punctual informations. However, due to the current technological restriction – frame per second – with PIV we cannot have the same temporal resolution of the classical techniques.

Experimental study of a flow into an engine cylinder using PIV

Experimental Study of a Flow into an Engine Cylinder

Using PIV

Alessandro Davario Vincenzo Di Lella

Experimental Study of a Flow into an Engine Cylinder

Using PIV

Preface

Abstract

Contents

List of Figures

List of Tables

Acknowledgements

1 Introduction

2 Theoretical Background

2.1 Engine background

2.2 Flow inside the cylinder

2.3 Proper Orthogonal Decomposition: POD

3 Experimental Method and Setup

3.1 PIV technique