FELIPE AUGUSTO FERREIRA GOMES
Numerical Simulation of Unsteady Flow
Inside Single-Cylinder Engine’s Intake and
Exhaust Manifolds
Simulação Numérica de Escoamento
Transiente Dentro dos Coletores de Admissão
e Exaustão de Motor Monocilíndrico
CAMPINAS 2019
Numerical Simulation of Unsteady Flow
Inside Single-Cylinder Engine’s Intake and
Exhaust Manifolds
Simulação Numérica de Escoamento
Transiente Dentro dos Coletores de Admissão
e Exaustão de Motor Monocilíndrico
Dissertation presented to the School of Mechanical Engineering of the University of Campinas in par-tial fulfillment of the requirements for the degree of Master in Mechanical Engineering, in the area of Thermo- Fluids.
Dissertação apresentada à Faculdade de Engenharia Mecânica da Universidade Estadual de Campinas como parte dos requisitos exigidos para obtenção do título de Mestre em Engenharia Mecânica, na Área de Térmica e Fluidos
Orientador: Prof. Dr. Waldyr Luiz Ribeiro Gallo ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA DISSERTAÇÃO DEFENDIDA PELO ALUNO FELIPE AUGUSTO FERREIRA GOMES, E ORIENTADO PELO PROF. DR. WALDYR LUIZ RIBEIRO GALLO.
CAMPINAS 2019
Rose Meire da Silva - CRB 8/5974
Gomes, Felipe Augusto Ferreira,
G585n GomNumerical simulation of unsteady flow inside single-cylinder engine's intake and exhaust manifolds / Felipe Augusto Ferreira Gomes. – Campinas, SP : [s.n.], 2019.
GomOrientador: Waldyr Luiz Ribeiro Gallo.
GomDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica.
Gom1. Dinâmica dos gases. 2. Motores de combustão interna. 3. Métodos numéricos. I. Gallo, Waldyr Luiz Ribeiro, 1954-. II. Universidade Estadual de Campinas. Faculdade de Engenharia Mecânica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Simulação numérica de escoamento transiente dentro dos
coletores de admissão e exaustão de motor monocilíndrico
Palavras-chave em inglês:
Gas dynamics
Internal combustion engines Numerical methods
Área de concentração: Térmica e Fluídos Titulação: Mestre em Engenharia Mecânica Banca examinadora:
Waldyr Luiz Ribeiro Gallo [Orientador] Rogério Gonçalves dos Santos
Sílvio Carlos Aníbal de Almeida
Data de defesa: 08-03-2019
Programa de Pós-Graduação: Engenharia Mecânica
Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0002-8964-8035
- Currículo Lattes do autor: http://lattes.cnpq.br/6716407458302681
COMISSÃO DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA
DEPARTAMENTO DE ENERGIA
DISSERTAÇÃO DE MESTRADO ACADÊMICO
Numerical Simulation of Unsteady Flow
Inside Single-Cylinder Engine’s Intake and
Exhaust Manifolds
Simulação Numérica de Escoamento
Transiente Dentro dos Coletores de Admissão
e Exaustão de Motor Monocilíndrico
Autor: Felipe Augusto Ferreira Gomes
Orientador: Prof. Dr. Waldyr Luiz Ribeiro Gallo
A Banca Examinadora composta pelos membros abaixo aprovou esta Dissertação: Prof. Dr. Waldyr Luiz Ribeiro Gallo
DE/FEM/Unicamp
Prof. Dr. Rogério Gonçalves dos Santos DE/FEM/Unicamp
Prof. Dr. Sílvio Carlos Aníbal de Almeida DEM/UFRJ
A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vida acadêmica do aluno.
Agradeço em primeiro lugar a minha família, todos os primos (as) e tias (os) por todo o apoio e confiança que me foi conferido. A minha mãe e ao meu irmão Cezar que sempre me observaram mesmo distantes, ao meu irmão Fernando que foi uma das razões para eu entrar no mestrado. Vocês foram a base para o meu crescimento.
Agradeço também a minha namorada, Denise Emy, por todo o suporte emocional, me escutando sempre que precisei, me motivando e até mesmo me ajudando nesse trabalho. Mesmo com a distância sempre me senti próximo a ela.
Não poderia deixar de expressar minha extrema gratidão pelo meu orientador, Waldyr Gallo, por todo o ensino e apoio em todas etapas da realização deste projeto.
Agradeço a todos os meus amigos, cujo apoio foi essencial para que esse mestrado fosse con-cluído, e especialmente à minha amiga Giovanna Piaulino, por sua contribuição nas imagens desse trabalho.
Agradeço também aos professores e funcionários da Unicamp, que por meio do seu suporte técnico contribuíram para o meu desenvolvimento acadêmico.
O presente trabalho foi realizado com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Código de Financiamento 001.
As características do escoamento dentro dos coletores de admissão e escape influenciam diretamente o desempenho do motor e um modelo capaz de estimar seu comportamento se torna essencial. Vários modelos numéricos foram usados, de baseados em dados experimentais até modelos tridimensionais de alta complexidade. Porém, a abordagem não-viscosa unidi-mensional (1D) se mostrou como uma boa aproximação para o problema. Existem diversos métodos numéricos de diferentes graus de acurácia para aproximar um escoamento 1D tran-siente e compressível. Entre esses, o método das características (MC), que possui acurácia de primeira ordem, foi o primeiro método a ser aplicado em escoamento dentro dos coletores com resultados confiáveis. Outro método muito usado é dois passos de Lax-Wendroff (LW2), que possui segunda ordem de precisão nas regiões suaves do escoamento e soluções espúrias na presença de ondas de choque e superfícies de contato. A literatura indica que ambos os méto-dos apresentam resultaméto-dos similares em relação ao desempenho do motor. Por essa razão, duas abordagens diferentes do MC; a abordagem clássica (MC-Winterbone) e a modificação feita por Payri (MC-Payri), e LW2 foram implementados e comparados entre si em simulações de tubos com área variável. No intuito de representar os coletores adequadamente, as condições de con-torno mais comumente encontradas nos sistemas de admissão e escape foram implementadas. Essas são: extremo livre, conexões de tubos, válvula e válvula borboleta. Os modelos foram adicionados ao Simulador de Motores desenvolvido pelo Laboratório de Motores de Biocom-bustíveis (LMB) da Universidade Estadual de Campinas e comparados a trabalhos encontrados na literatura e com programas bem estabelecidos; OpenWAM e GT-Power. Os resultados no tubo de área variável mostraram que MC-Payri melhorou a conservação de massa e a qualidade das propriedades estimadas quando comparado ao MC-Winterbone, enquanto LW2 apresentou resultados consistentes contra OpenWAM. Todas as condições de contorno foram testadas sep-aradamente para isolar seus efeitos no escoamento. Os modelos das conexões de tubos e da válvula apresentaram resultados praticamente idênticos aos do OpenWAM enquanto o modelo da válvula borboleta subestimou a queda de pressão comparado a dados experimentais para uma onda de pressão chegando na condição de contorno, apesar de ter capturado razoavelmente bem o seu comportamento. O LMB Simulador de Motores foi testado em uma simulação de um motor real e os resultados mostraram uma boa concordância qualitativa para potência e torque, mesmo com várias simplificações feitas na geometria dos coletores.
Palavras-chave: Dinâmica dos gases 1D, coletores de admissão e exaustão, método das carac-terísticas, dois passos de Lax-Wendroff
The flow characteristics inside intake and exhaust manifolds direct influence engine per-formance and a model capable of estimating its behavior becomes essential. Many numerical models have been used, from experimental based models to highly complex three dimensional models. However, the inviscid one-dimensional (1D) approach has shown to be a suitable ap-proximation for the problem. There are many numerical methods of different levels of accuracy to approximate 1D unsteady compressible flow. Among them, the method of characteristics (MC), which has first order accuracy, was the first method to be applied to gas motion inside engine manifolds with reliable results. Another widely used method is the two step of Lax-Wendroff (LW2), which has second order accuracy on smooth regions of the flow and spurious solution in the presence of shock waves and contact surfaces. The literature indicates that both methods presented similar results regarding engine performance. For that reason, two different approaches for MC; the classical method (MC-Winterbone) and the modification made by Payri (MC-Payri), and LW2 were implemented and compared with each other in a tapered pipe sim-ulation. In order to represent the manifolds properly, the most common boundary conditions encountered in gas exchange systems were implemented. These are open end, pipe junctions, poppet valve and throttle valve. The models were added to the Engine Simulator developed by the Laboratory for Biofuel Engines (LMB) in the University of Campinas and compared to works found in the literature and well-established software, OpenWAM and GT-Power. The results in a tapered pipe showed that MC-Payri improved the mass conservation and the quality of the estimated properties when compared to MC-Winterbone while LW2 presented consistent results against OpenWAM’s. All the boundaries were tested separately to isolate its effect on the flow. Pipe junction and poppet valve models presented results almost identical to OpenWAM’s while the throttle valve model under-predicted the pressure drop compared to experimental data for an incident pressure wave arriving at the boundary, although it captured fairly well its behav-ior. The LMB Engine Simulator was tested in a real engine simulation and the results showed a good qualitative agreement for power and torque, even though multiple simplifications were made to the geometry of the manifolds.
Keywords: 1D gas dynamics, intake/exhaust manifold, method of characteristics, two step Lax-Wendroff
2.1 General manifold systems (Blair, 1999) . . . 22
2.2 p-V diagram of engine cycle . . . 23
2.3 Control volume in a general pipe . . . 25
2.4 Characteristics mesh grid for subsonic flow . . . 41
2.5 General computational cell for one-dimensional conservation equations (Win-terbone and Pearson, 2000) . . . 43
2.6 Computational stencil Two-step Lax-Wendroff scheme (Winterbone and Pear-son, 2000) . . . 51
2.7 General information propagating in a mesh grid for a single time step . . . 52
3.1 General boundary condition, showing the incident characteristic and pathline (Winterbone and Pearson, 2000) . . . 55
3.2 Valve layout and its physical interpretation . . . 60
3.3 Physical behavior of steady flow through a sudden enlargement (Winterbone and Pearson, 2000) . . . 66
3.4 Schematic for a general N-pipe junction . . . 71
3.5 (a) Schematic adiabatic device inside a pipe (b) Pressure behavior for a wave interacting with a adiabatic device . . . 72
5.1 Tapered pipe connected to a constant pressure and temperature chamber . . . . 87
5.2 Pressure profile inside the tapered pipe after 2𝑚𝑠 . . . 88
5.3 Pressure variation at the entry location, 𝑥 = 0 . . . 89
5.4 Pressure variation at the exit location, 𝑥 = 1𝑚 . . . 90
5.5 Temperature variation at the exit location, 𝑥 = 1𝑚 . . . 91
5.6 Mass flow rate at the exit location, 𝑥 = 1𝑚 . . . 91
5.7 Mass flow profile in steady state, after 1𝑠 . . . 93
5.8 Van Hove tapered pipe scheme . . . 94
5.9 Mass flow profile in steady state of divergent pipe, after 1𝑠 . . . 95
5.10 Sudden enlargement simulation scheme . . . 96
5.11 Pressure variation at both pipe ends for sudden expansion test . . . 97
5.12 Steady state for a pipe sudden enlargement, 𝑡 = 1𝑠 . . . 97
5.16 Velocity variation for three pipe junction . . . 101
5.17 Steady state for three pipe junction, 𝑡 = 1𝑠 . . . 102
5.18 Shock tube test bench schematic (Chalet and Chesse, 2010) . . . 102
5.19 Simulated pressure profile using MC-Payri at 𝑡 = 4.01𝑚𝑠 for 𝑝𝑖𝑛𝑖 = 0.7 . . . . 104
5.20 GT-power schematic model . . . 105
5.21 Intake and Exhaust valve lift . . . 106
5.22 Intake and Exhaust coefficient of discharge . . . 107
5.23 Pressure variation with crank angle degree inside intake and exhaust port . . . . 107
5.24 Mass flow variation with crank angle degree inside intake and exhaust port . . . 108
5.25 G50 Manifold scheme (Blair, 1999) . . . 109
5.26 Approximation for the 240𝑚𝑚 pipe in the exhaust system . . . 110
5.27 Simulated and experimental results for G50 engine . . . 111
5.28 Simulated volumetric efficiency and IMEP values for G50 engine . . . 112
5.29 Simulated static pressure at intake valve for G50 engine . . . 113
5.30 Simulated mass flow rate through intake valve for G50 engine . . . 114
A.1 Velocity variation at the entry location, 𝑥 = 0 . . . 126
A.2 Temperature variation at the entry location, 𝑥 = 0 . . . 127
A.3 Mass flow variation at the entry location, 𝑥 = 0 . . . 127
A.4 Velocity variation at the entry location, 𝑥 = 1 . . . 128
A.5 Pressure variation for divergent pipe simulation . . . 129
A.6 Velocity variation for divergent pipe simulation . . . 129
A.7 Temperature variation for divergent pipe simulation . . . 130
B.1 Pressure variation at both pipe ends for sudden expansion test . . . 131
B.2 Temperature variation at both pipe ends for sudden expansion test . . . 131
B.3 Pressure variation at both pipe ends for sudden contraction test . . . 132
B.4 Temperature variation at both pipe ends for sudden contraction test . . . 132
C.1 Pressure variation for three pipe junction . . . 133
C.2 Temperature variation for three pipe junction . . . 133
D.1 Cylinder pressure variation with crank angle degree . . . 134
D.2 Cylinder temperature variation with crank angle degree . . . 134
E.1 Simulation with isentropic flow through valves and experimental results for G50 engine . . . 135
E.2 Simulated static pressure at exhaust valve for G50 engine . . . 135
3.1 𝐾𝑡ℎvalues for a throttle valve (Benson, 1982) . . . 73
5.1 Tapered pipe: simulation condition . . . 87
5.2 Tapered pipe: simulation condition . . . 94
5.3 Shock tube main characteristics (Chalet and Chesse, 2010) . . . 103
5.4 Simulated pressure drop across a throttle valve in the shock tube . . . 104
5.5 Engine and manifolds characteristics . . . 106
Coefficients 𝜓 - Area ratio of throat to pipe
𝐶𝐷 - Coefficient of discharge
𝑘𝑡ℎ - Resistance coefficient
Engine related parameters 𝜂𝑉 - Volumetric efficiency
𝜆𝑐𝑜𝑚𝑏 - Equivalent ratio
AFR - Air-fuel ratio BDC - Bottom dead center e.v.c - Exhaust valve closing e.v.o - Exhaust valve opening
IMEP - Indicated mean effective pressure i.v.c - Intake valve closing
i.v.o - Intake valve opening TDC - Top dead center 𝑇𝑖 - Indicated torque
𝑉𝑠𝑣 - Swept volume
𝑊𝑖 - Indicated work
˙
𝑊𝑖 - Indicated power
Dimensions and physical quantities
𝜌 - Density
𝜖 - Pipe roughness
𝐷 - Pipe diameter
𝑒 - Specific internal energy
𝑒0 - Specific stagnation internal energy
𝐸 - Internal energy
𝐹 - Cross section area
𝐹𝑙 - Wall surface area per unit of length
ℎ0 - Specific stagnation enthalpy 𝐻 - Stagnation enthalpy 𝐿 - Pipe length 𝑚 - Mass 𝑝 - Pressure 𝑝0 - Stagnation pressure
𝑞 - Heat transfer rate per unit of mass ˙
𝑄 - Heat transfer rate
𝑠 - Specific entropy 𝑡 - Time 𝑇 - Temperature 𝑇0 - Stagnation temperature 𝑉 - Volume ˙ 𝑊 - Work rate 𝑥 - Length 𝑋 - Non-dimensional length 𝑍 - Non-dimensional time
Gas and flow parameters
𝜇 - Fluid viscosity
𝜏𝑤 - Sheer force
𝜆 - Left-travel Riemann variable
𝜆𝑖𝑛 - Approaching the boundary incident characteristic
𝜆𝑜𝑢𝑡 - Leaving the boundary incident characteristic
𝛽 - Right-travel Riemann variable
𝑎 - Speed of sound
𝐴 - Non-dimensional speed of sound 𝑎0 - Stagnation speed of sound
𝐴0 - Non-dimensional stagnation speed of sound
𝑎𝐴 - Entropy Level
𝐴𝐴 - Non-dimensional entropy Level
𝑐 - Wave speed
𝐶𝑓 - Fanning friction factor
𝐶𝑝 - Heat capacity at constant pressure
𝑓𝑐 - Pressure loss coefficient
ℎ𝑐 - Convection heat transfer coefficient
𝑘 - Heat capacity ratio
𝑘𝑐 - Thermal conductivity of a fluid
˙
𝑚 - Mass flow rate
𝑀 - Mach number Nu - Nusselt number Pr - Prandtl number 𝑄 - Volumetric flow 𝑅 - Gas constant Re - Reynolds number 𝑢 - Flow velocity
𝑈 - Non-dimensional flow velocity
Matrices and Vectors C(U) - Source term vector
F(U) - Vector of fluxes
U - Vector of conserved variables
Other notations
LW - Lax-Wendroff schemes
LW2 - Two step Lax-Wendroff schemes MC - Method of characteristics
List of Figures 8
List of Tables 10
List of Abbreviations and Acronyms 11
CONTENTS 14 1 INTRODUCTION 17 1.1 Aim . . . 19 1.2 Objectives . . . 19 1.3 Dissertation outline . . . 19 2 LITERATURE REVIEW 21 2.1 Engine manifolds and four-stroke engines performance . . . 21
2.2 Brief history of the methods applied to engine manifolds . . . 24
2.3 Governing Equations . . . 25
2.3.1 Continuity equation . . . 25
2.3.2 Momentum equation . . . 26
2.3.3 Energy equation . . . 27
2.3.4 Equations in non-conservation law form . . . 28
2.3.5 Friction and Heat Transfer terms . . . 30
2.4 Method of Characteristics . . . 32
2.4.1 General solution . . . 33
2.4.2 Governing equations solutions . . . 34
2.4.3 Entropy level variable . . . 36
2.4.4 Non-dimensional characteristic equations . . . 38
2.4.5 Riemann variables . . . 40
2.4.6 Numerical solution . . . 41
2.5 Conservative Discretization . . . 42
2.6.3 Lax-Wendroff scheme . . . 48
2.6.4 Richtmyer scheme . . . 50
2.6.5 The Courant-Friedrichs-Lewy stability criterion . . . 52
3 BOUNDARY CONDITIONS 54 3.1 Incident characteristics . . . 54 3.2 Flow direction . . . 56 3.3 Open End . . . 58 3.3.1 Outflow . . . 58 3.3.2 Inflow . . . 58 3.4 Poppet Valve . . . 60 3.4.1 Outflow . . . 61 3.4.2 Inflow . . . 62 3.4.3 Closed valve . . . 65
3.5 Single Pipe Junction . . . 65
3.5.1 Sudden Enlargement . . . 65
3.5.2 Sudden Contraction . . . 68
3.6 Multi pipe junction . . . 70
3.7 Throttle valve . . . 72
4 NUMERICAL PROCEDURE 76 4.1 Mesh Method of Characteristics . . . 76
4.1.1 Winterbone and Pearson method (Benson method) . . . 77
4.1.2 Payri modification . . . 79
4.2 Two step Lax-Wendroff . . . 81
4.2.1 Source Term discretization . . . 82
4.3 Friction and heat transfer terms . . . 83
4.4 Laboratory for Biofuel Engines - Engine Simulator . . . 84
4.5 Final remarks . . . 85 5 RESULTS 86 5.1 Tapered pipe . . . 86 5.1.1 Unsteady Analysis . . . 88 5.1.2 Steady Analysis . . . 92 5.1.3 Divergent pipe . . . 93
5.3 Multi Pipe Junction . . . 100 5.4 Throttle Valve . . . 102 5.5 Motoring simulation . . . 104 5.6 Engine Simulation . . . 108 6 CONCLUSIONS 115 6.1 Future works . . . 116 REFERENCES 118 APPENDICES 126 A – Tapered pipe complementary results 126 A.1 Results at the pipe entry . . . 126
A.2 Results at the pipe exit . . . 128
A.3 Results for Divergent pipe . . . 129
B – Simple pipe junction complementary results 131
C – Multi pipe junction complementary results 133
D – Motoring simulation complementary results 134
E – Engine simulation complementary results 135
F – Numerical Simulation of Unsteady flow in Engine Intake and Exhaust
1
INTRODUCTION
Simulations started to become more important around the middle of the twentieth century and since then it has been a major feature of engineering design. They allow accurate insight of the physical phenomenon in multiple situations that are not easily, or even impossible, to measure. They also enabled a fast paced growth of new technologies because it is less time consuming and cheaper financially. Racing teams have extensively used simulations to improve the design of their cars in competitions and many of their developed technology have become standard in daily vehicles.
Multiple research groups around the world have developed their own engine simulator in order to have full control of the models and methods applied to the simulations, some exam-ples are Engine Simulation Program (ESP) from Stanford, GASDYN from The Politecnico di Milano, OpenWAM from The Polytechnic University of Valencia and FKFS working with the University of Stuttgart. The Laboratory for Biofuel Engines group from the University of Campinas is currently developing its own Engine Simulator and this work is a complementary part of this development. It is responsible to estimate the flow filed inside the manifolds of an engine.
The gas exchange systems of an engine heavily contributes to its performance. It affects the flow motion inside the manifolds which in turn influences volumetric efficiency. Mostly nowadays engines run in four-stroke cycles with poppet valves in which the gas exchange is divided in intake and exhaust processes.
The intake process is responsible for the fresh charge of air-fuel that allows the combustion to occur. This process starts when the intake valve opens (i.v.o), a few crank angles before the top dead center (TDC) while the cylinder pressure is higher than the manifold’s. This causes a portion of the combustion gases to flow into the intake pipe, being called backflow. As the piston moves toward the bottom dead center (BDC) and the cylinder pressure falls below intake pressure, an air-fuel mixture starts to fill the cylinder. The intake valve closing (i.v.c) occurs after the piston reaches BDC in order to utilize the flow inertia to help filling the cylinder, which increases the volumetric efficiency. The unsteady nature of the process is characterized by its pulsating flow with traveling waves inside the pipes. These waves interact with other parts of
the manifold such as pipe junctions, throttle body and other equipment and they also are able to facilitate or hinder the air-fuel charge into the cylinder (Heywood, 1988).
In the exhaust process the burned gases are removed from the cylinder. The exhaust valve opening (e.v.o) happens around 50∘before BDC causing the burned gases to leave the cylinder
due to pressure difference. After BDC, the gases are pushed out by the piston as it moves towards the TDC. A portion of the burned gases remain in the cylinder after exhaust valve closing (e.v.c), which are called residual gases. Similar to intake, the propagating waves inside the exhaust pipes influence the cylinder clearance and, therefore, the volumetric efficiency. The pressure difference between the cylinder and exhaust manifold at e.v.o is usually high enough to cause the flow to chock while passing through the valve (Heywood, 1988; Gupta, 2006).
The influence of intake and exhaust systems on engine performance have been accounted by basically three approaches: quasi-steady, filling and emptying and wave action (Benson, 1982). Quasi-steady models assume constant gas properties and rely on empirical data. Filling and emptying approach corresponds to zero dimensional models applied to a finite volume. It takes into account the unsteady behavior of the flow but considers the gas properties homogeneous throughout the volume. Additionally, it is still vastly used to estimate the operation of engine systems such as the cylinder and plenum.
Wave action approach consists of dimensional models of the unsteady compressible flow inside the intake and exhaust systems. Although 3D models give theoretically more accurate results about the flow field, they are very computationally expensive and time demanding. Their use is more common analyzing complex junctions and subparts of the system, such as valves. One-dimensional (1D) models have presented the most success despite of its simplifications. They have shown good agreement with experimental data and relative low computational cost (Winterbone and Pearson, 2000). However, these models encounter limitations in the presence of more complex geometries and apparatus such as valves, throttle bodies, multi pipe junctions and sudden area change in which multidimensional effects cannot be ignored. Correlations and approximated methods have been developed for the 1D models taking into account those effects, however, many of them require experimental data (Nikita et al., 2015; Bassett et al., 2003; Winterbone and Pearson, 2000).
The present work consists of developing a one-dimensional simulator capable of estimating the characteristics of the unsteady compressible flow inside engine manifolds systems and its
interactions with frequently found components. This flow simulator is then integrated to the main software developed by the University of Campinas.
1.1 Aim
• Develop a one-dimensional simulator for unsteady compressible flow with boundary con-ditions commonly found in engine systems.
1.2 Objectives
• Implement and compare multiple wave capture schemes. • Integrate and test the boundary models.
• Include and test area variation on the source term.
• Integrate and validate the Engine Simulator against works found in the literature and well established software.
1.3 Dissertation outline
• Chapter 2: A literature review about some commonly found methods applied to engine systems and the basic equations utilized by them.
• Chapter 3: A methodology to treat some of the most encountered boundary conditions. • Chapter 4: The numerical procedure employed to estimate the flow field.
• Chapter 5: Results about the accuracy and validity of the employed models compared to well-established software and literature works.
2
LITERATURE REVIEW
2.1 Engine manifolds and four-stroke engines performance
This section aims on given a brief overview about some characteristics of engine manifolds and some relevant parameters of performance. All the content present in this section was derived from (Heywood, 1988; Blair, 1999; Winterbone and Pearson, 2000) in which a more detailed discussion can be find.
The intake and exhaust manifolds are composed by tubes and components in which the gases travel within the systems before or after entering the cylinder. Fig. 2.1 illustrates some of the main components and features presented in almost every modern engine; 1 is bellmouth connected to atmospheric conditions, 2 is a plenum to diminish the pressure waves amplitude, 3 is an air filter, 4 is throttle body controlling the overall intake mass flow, 5 and 6 are pipe connections of some sort, 7 and 8 are intake and exhaust valves, respectively, 9 are bends in the pipes, 10 is an expansion or contraction depending on the flow direction, 11 is a tapered pipe, 12 is catalyst, and 13 is a silencer element. It is important to point out that in many commercial engines the throttle body and the filter may be located before the plenum.
The flow inside both manifolds vary greatly depending on the engine geometry and operat-ing conditions. For naturally aspirated engine, the average intake and exhaust pressure are close to atmosphere; intake pressure distribution may vary from 0.6 to under 2.0 bar while exhaust’s can range from 0.3 to under 3.0 bar depending on the tuning of the engine.
Regarding engine design and performance analysis, some the most relevant parameters are going to be introduced. Volumetric efficiency, defined in Eq. 2.1, compares the amount of mass supplied during the intake period with a reference mass which represents the mass necessary to fill the swept volume (𝑉𝑠𝑣) under a reference condition of pressure and temperature, in this
case, atmospheric condition.
𝜂𝑉 = 𝑚𝑖𝑛𝑡 𝑚𝑟𝑒𝑓 (2.1) 𝑚𝑟𝑒𝑓 = 𝑝𝑎𝑡𝑚𝑉𝑠𝑣 𝑅𝑇𝑎𝑡𝑚 (2.2)
Figure 2.1: General manifold systems (Blair, 1999)
The concept of air-to-fuel ratio (AFR) represents the mass ratio of air and fuel in the com-bustion process. Another useful parameter is the relative air/fuel ratio, 𝜆𝑐𝑜𝑚𝑏, which is defined
in Eq. 2.3. It indicates in a proportional manner whether an air-fuel mixture is lean or rich in relation to its stoichiometric value.
𝜆𝑐𝑜𝑚𝑏 =
𝐴𝐹 𝑅 𝐴𝐹 𝑅𝑠𝑡𝑜𝑖𝑐ℎ
(2.3) The work transferred from the gas to the piston during the whole cycle is defined in Eq. 2.4 and it is referred to the net indicated work per cycle because it also accounts for exhaust and intake strokes.
𝑊𝑖 =
∮︁
𝑝𝑑𝑉 (2.4)
Another very useful performance variable is the mean effective pressure (MEP). It is better expressed in Fig. 2.2 where the area of the 𝑝 − 𝑉 diagram (indicated work) is equivalent to the area of the rectangle IMEP × 𝑉𝑠𝑣, then the indicated mean pressure can be defined as
IMEP = 𝑊𝑖
𝑉𝑠𝑣
Figure 2.2: p-V diagram of engine cycle
From the net indicated work, it is just natural to arrive at the definition of power. However, the indicated power represents only the work done on the piston during the compression and expansion strokes, the power cycle. Thus, for the engine speed in revolutions per minute (RPM) the indicated power output is given by Eq. 2.6 while the indicated torque can be expressed as function of power in Eq. 2.7.
˙ 𝑊𝑖 = 𝑊𝑖× EngSpeed 120 (2.6) 𝑇𝑖 = ˙𝑊𝑖 30 𝜋× EngSpeed (2.7)
In engine testing, the experimental values for torque are acquired in a test bed and, then, power can be estimated analogously to the equations above. All the measured quantities are often referred with the adjective ’brake’; brake power, brake MEP (BMEP), brake thermal effi-ciency and so on. In addition, brake power is supposed to be a fraction of the indicated power because it accounts for mechanical losses such as friction.
Many other useful terms such as thermal efficiency, mechanical efficiency, trapping effi-ciency, scavenging effieffi-ciency, specific fuel consumption, and others, are explained in details in (Heywood, 1988; Blair, 1999).
2.2 Brief history of the methods applied to engine manifolds
Benson et al. (1964) was one of the first to apply computational one-dimensional models to engine manifold systems. They developed a mesh method of characteristics, which was based on the graphical technique of the method. This method became the most used for geometrical studies of intake and exhaust systems until the decade of 1990. It is still relevant to this day because of its applicability on boundary conditions. The numerical method of characteristics as well as the graphical solution are well developed in Benson (1982).
In the mid 80’s finite difference and finite volume schemes started to be extensively used. The two step Lax-Wendroff and MacCormack schemes were the most employed on engine performance and even noise prediction (Dwyer et al., 1974; Azuma et al., 1983; Poloni et al., 1987; Payri et al., 1996). Even though both schemes demonstrated to be faster than the mesh method of characteristics, they presented spurious oscillations when faced with discontinuities such as shock waves and contact surfaces.
Due to the presence of non-physical overshooting, flux limiting schemes rapidly became of standard use (Bulaty and Niessner, 1985; Chen et al., 1992; Liu et al., 1996; Onorati et al., 1997; Pearson and Winterbone, 1997). They usually consists of flux corrector techniques (FCT) or total variation diminishing (TVD) schemes, which were introduced by Boris and Book (1973) and Harten (1983), respectively. The concepts behind those techniques are well described in Toro (1999) and LeVeque (2002).
More recently Montenegro et al. (2007) presented the 1D-3D model, which couples a one-dimensional solver with a three-one-dimensional one. This model has shown to overcome the limits of 1D models on simulating complex geometries. It has been accurately use to pre-dict noise while solving engine performance parameters (Montenegro et al., 2011b,a, 2016). This model has been included in many commercial softwares but it is still being improved Della Torre et al. (2017).
2.3 Governing Equations
The fundamental equations for one-dimensional unsteady compressible flow are derived from conservation of mass, momentum and energy applied to an infinitesimal control volume with variable cross-section area, Fig. 2.3. The fluid is assumed as an ideal gas and both specific heat capacity, at constant pressure and constant volume, are considered uniform inside the con-trol volume. 𝑢 is velocity in 𝑥 direction, 𝜌 is density, 𝑝 is pressure, 𝐹 is cross-section area, 𝜏𝑤is
the shear force, 𝑑𝑥 is control volume length. Since area change is take into account, this system is often call quasi-one dimensional. When the area variation is gradual, the properties can be considered uniform across any cross-section more accurately.
Figure 2.3: Control volume in a general pipe
2.3.1 Continuity equation
The continuity equation represents that the rate of change of mass inside the control volume is equal to the net mass flow rate that crosses its boundaries. This balance is expressed by Eq. 2.8. Ignoring the terms multiplied by 𝑑𝑥2, Eq. 2.9 can be obtained.
𝜕 𝜕𝑡(𝜌𝐹 𝑑𝑥)− 𝜌𝑢𝐹 + (𝜌 + 𝜕𝜌 𝜕𝑥𝑑𝑥)(𝑢 + 𝜕𝑢 𝜕𝑥𝑑𝑥)(𝐹 + 𝜕𝐹 𝜕𝑥𝑑𝑥) = 0 (2.8)
𝜕 (𝜌𝐹 )
𝜕𝑡 +
𝜕 (𝜌𝑢𝐹 )
𝜕𝑥 = 0 (2.9)
2.3.2 Momentum equation
The momentum conservation expresses that the rate of change of momentum within the control volume plus net flux of momentum must be equal to the sum of shear and pressure forces acting on the surfaces of the control volume, which is shown in Eq. 2.10.
𝜕 (𝜌𝑢𝐹 𝑑𝑥) 𝜕𝑡 − 𝜌𝑢 2𝐹 + (︂ 𝜌 + 𝜕𝜌 𝜕𝑥𝑑𝑥 )︂ (︂ 𝑢 +𝜕𝑢 𝜕𝑥𝑑𝑥 )︂2(︂ 𝐹 + 𝜕𝐹 𝜕𝑥𝑑𝑥 )︂ = 𝑝𝐹 − (︂ 𝑝 + 𝜕𝑝 𝜕𝑥𝑑𝑥 )︂ (︂ 𝐹 + 𝜕𝐹 𝜕𝑥𝑑𝑥 )︂ + 𝑝𝜕𝐹 𝜕𝑥𝑑𝑥 + 𝜏𝑤𝐹𝑠 (2.10)
Where 𝐹𝑠 is the superficial area of the element. The second last term of Eq. 2.10 is
de-termined by geometric analysis of the control volume. The shear force can be related with the Fanning friction factor in Eq. 2.11
𝐶𝑓 =
𝜏𝑤 1 2𝜌𝑢2
(2.11) The shear force has to be negative since it always acts in the opposite direction of the flow. Substituting Eq. 2.11 into Eq. 2.10 and simplifying, the momentum equation can be written as Eq. 2.12. Here, 𝐹𝑙represents to the wall surface area per unit of length.
𝜕 (𝜌𝑢𝐹 ) 𝜕𝑡 + 𝜕 (𝜌𝑢2𝐹 ) 𝜕𝑥 =− 𝜕 (𝑝𝐹 ) 𝜕𝑥 + 𝑝 𝑑𝐹 𝑑𝑥 − 1 2𝐶𝑓𝜌𝑢 2𝐹 𝑙 (2.12)
2.3.3 Energy equation
The energy balance is given by the first law of thermodynamics Eq. 2.13, where 𝐸 is the total stagnation energy inside the control volume, ˙𝑄 is the heat transfer rate, ˙𝑊 is the work done by or on the control volume, 𝐻 is the stagnation enthalpy.
𝜕𝐸
𝜕𝑡 = ˙𝑄− ˙𝑊 − 𝜕𝐻
𝜕𝑥𝑑𝑥 (2.13)
The internal energy is given by Eq. 2.14, where 𝑒 is the specific internal energy. ˙𝑄can be written as heat transfer rate per unit of mass 𝑞, Eq. 2.16, and the total enthalpy is given by Eq. 2.15. The control volume does not face any external or internal work then ˙𝑊 = 0.
𝐸 = 𝜌𝐹 (︂ 𝑒 + 𝑢 2 2 )︂ 𝑑𝑥 (2.14) 𝐻 = 𝜌𝑢𝐹 (︂ 𝑒 + 𝑢 2 2 + 𝑝 𝜌 )︂ (2.15) ˙ 𝑄 = 𝑞𝜌𝐹 𝑑𝑥 (2.16)
Substituting these equations into Eq. 2.13 lead to the differential equation of energy. Defin-ing 𝑒0 = 𝑒 + 𝑢2/2as stagnation internal energy per unit of mass and ℎ0 = 𝑒 + 𝑢2/2 + 𝑃/𝜌as
stagnation enthalpy per unit of mass. The governing equations for one-dimensional unsteady compressible flow with variable area change considering wall friction and wall heat transfer are summarized in Eq. 2.17 to Eq. 2.19, which are continuity, momentum and energy, respectively. The term 𝑢|𝑢| replaced 𝑢2 to ensure that the wall friction is always in the opposite direction of
the flow. 𝜕 (𝜌𝐹 ) 𝜕𝑡 + 𝜕 (𝜌𝑢𝐹 ) 𝜕𝑥 = 0 (2.17) 𝜕 (𝜌𝑢𝐹 ) 𝜕𝑡 + 𝜕 ((𝜌𝑢2+ 𝑝)𝐹 ) 𝜕𝑥 − 𝑝 𝑑𝐹 𝑑𝑥 + 1 2𝐶𝑓𝜌𝑢|𝑢|𝐹𝑙 = 0 (2.18) 𝜕 (𝜌𝑒0𝐹 ) 𝜕𝑡 + 𝜕 (𝜌𝑢ℎ0𝐹 ) 𝜕𝑥 − 𝑞𝜌𝐹 = 0 (2.19)
This set of equations are characterized as non-linear hyperbolic partial differential equations. The governing equations are commonly expressed in vector form as Eq. 2.20 or in a simpler manner as Eq. 2.21.U is called vector of conserved variables, F(U) is the vector of fluxes which is a function of the components inU and C(U) is the source term which is also a function of U (Toro, 1999). 𝜕 𝜕𝑡 ⎡ ⎢ ⎣ 𝜌𝐹 𝜌𝑢𝐹 𝜌𝑒0𝐹 ⎤ ⎥ ⎦ + 𝜕𝑥𝜕 ⎡ ⎢ ⎣ 𝜌𝑢𝐹 (𝜌𝑢2+ 𝑝) 𝐹 𝜌𝑢ℎ0𝐹 ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ 0 −𝑝𝑑𝐹 𝑑𝑥 + 𝑔𝜌𝐹𝑙 −𝑞𝜌𝐹 ⎤ ⎥ ⎦ = 0 (2.20) 𝜕U 𝜕𝑡 + 𝜕F(U) 𝜕𝑥 +C(U) = 0 (2.21) where, 𝑔 = 1 2𝐶𝑓𝑢|𝑢|
2.3.4 Equations in non-conservation law form
The governing equations written in non-conservation law form is mostly used by the meth-ods of characteristics. This can be achieved by expanding continuity, momentum and energy equations. It is important to mention that in order to obtain the final momentum equation (Eq. 2.23), the non-conservation form of continuity (Eq. 2.22) is required while for energy (Eq. 2.24), both non-conservation equation of continuity and momentum are needed.
𝜕𝜌 𝜕𝑡 + 𝑢 𝜕𝜌 𝜕𝑥 + 𝜌 𝜕𝑢 𝜕𝑥 + 𝜌𝑢 𝐹 𝜕𝐹 𝜕𝑥 = 0 (2.22) 𝜕𝑢 𝜕𝑡 + 𝑢 𝜕𝑢 𝜕𝑥 + 1 𝜌 𝜕𝑝 𝜕𝑥 + 𝑔 𝐹𝑙 𝐹 = 0 (2.23) 𝜕𝑒 𝜕𝑡 + 𝑢 𝜕𝑒 𝜕𝑥 − 𝑝 𝜌2 [︂ 𝜕𝜌 𝜕𝑡 + 𝑢 𝜕𝜌 𝜕𝑥 ]︂ − (︂ 𝑞 + 𝑢𝑔𝐹𝑙 𝐹 )︂ = 0 (2.24)
Both systems of the basic equations contain three equations and four unknowns, 𝜌, 𝑢, 𝑝 and 𝑒. Therefore, it is needed more relations to find the solution of the systems. Those are the ideal
gas state equation Eq. 2.25 and some thermodynamic relations. 𝑞 and 𝐶𝑓 are considered known
variables and they are going to be related to the properties of the flow in Sec. 2.3.5. 𝑝
𝜌 = 𝑅𝑇 (2.25)
Since the fluid can be considered a perfect gas, internal energy and enthalpy can be ex-pressed as functions of temperature alone. By the definition, the specific heat capacity at con-stant volume is 𝐶𝑣 = (︂ 𝑑𝑒 𝑑𝑇 )︂ 𝑉 ≈ (︂ 𝜕𝑞 𝜕𝑇 )︂ 𝑉 (2.26) For thermodynamic processes with constant specific heat and constant composition, a change from a 0 state of reference, at 𝑇0 = 0𝐾, gives internal energy as
𝑒 = 𝑒0+ 𝐶𝑣𝑇 (2.27)
Considering the same reference state for all processes, the equation may be expressed from now on as Eq. 2.28. For a perfect gas, it is convenient to express the specific heat capacities in terms of specific heats ratio, 𝑘 = 𝐶𝑝/𝐶𝑣. This is possible to reveal by using the relation
𝐶𝑝 = 𝐶𝑣 + 𝑅. The result for 𝐶𝑣 is shown in Eq. 2.30, where 𝑅 is the specific gas constant.
Although this procedure was only demonstrated for 𝐶𝑣, and its relation to internal energy, the
same may also be done for enthalpy by analyzing specific heat at constant pressure, 𝐶𝑝, which
is expressed in Eq. 2.29 and Eq. 2.31.
𝑒 = 𝐶𝑣𝑇 (2.28) ℎ = 𝐶𝑝𝑇 (2.29) 𝐶𝑣 = 𝑅 𝑘− 1 (2.30) 𝐶𝑝 = 𝑘𝑅 𝑘− 1 (2.31)
The system of equations may be reduced to four equations and four unknown, 𝜌, 𝑢, 𝑝 and 𝑒, where the internal energy is given by Eq. 2.32 ans 𝑎 is the speed of sound. The other three equations can be expressed in conservation or non-conservation law form. Solutions for both set of the governing equations are demonstrated in further sections.
𝑒 = 𝑝
𝜌 (𝑘− 1) = 𝑎2
𝑘(𝑘− 1) (2.32)
2.3.5 Friction and Heat Transfer terms
Despite the flow inside the manifolds having a pulsating behavior, the classical relations for friction factor used in incompressible flow are vastly applied (Winterbone and Pearson, 2000). For laminar flow in rounded tubes, which is valid for Re < 2000, the exact solution for Newtonian fluid can be employed, Eq. 2.33, where Re is Reynolds number, defined in Eq. 2.34, and 𝜇 is the flow viscosity.
𝐶𝑓 =
16
Re (2.33)
Re = 𝜌𝑢𝐷
𝜇 (2.34)
There are many correlations for estimating friction factor of turbulent flow, where most of them are given as Darcy friction factor, 𝑓. However, the governing equations were developed using Fanning friction factor, 𝐶𝑓. Then, the expression 𝑓 = 4𝐶𝑓 may be used to relate both
factors. For turbulent flow, Blasius equation, Eq. 2.35, is a simple relation of easy estimation with accurate range of applicability of 4000 < Re < 105.
𝑓 = 0.316
Re1/4 (2.35)
Relations accounting for pipe roughness 𝜖 are the most used in internal turbulent flows. The Colebrook-White equation, Eq. 2.36, have shown to present very good agreement with experimental measurements. Since this equation is implicit, an iterative solution is necessary.
1 √ 𝑓 =−2 log10 (︂ 𝜖/𝐷 3.7 + 2.51 Re√𝑓 )︂ (2.36) An explicit approximation of the Colebrook-White equation is the correlation developed by Swamee-Jain, Eq. 2.37.
𝑓 = 0.25
log10(︀3.7𝐷𝜖 +Re5.740.9
)︀ (2.37)
Another approximation of the Colebrook-White equation is the Serghides equation, Eq. 2.38, which is divided in three different variables as functions of each other. Although it is necessary to solve three extra relations, the Serghides equation is explicit and have better results than Eq. 2.37. 𝐴 =−2 log10 (︂ 𝜖/𝐷 3.7 + 12 Re )︂ 𝐵 =−2 log10 (︂ 𝜖/𝐷 3.7 + 2.51𝐴 Re )︂ 𝐶 =−2 log10 (︂ 𝜖/𝐷 3.7 + 2.51𝐵 Re )︂ 1 √ 𝑓 = 𝐴− (𝐵− 𝐴)2 𝐶− 2𝐵 + 𝐴 (2.38)
A model for approximating viscosity is required and Winterbone and Pearson (2000) sug-gests two models, which are function of flow temperature. These are the Sutherland’s equation for air, Eq. 2.39, and a relations given by Blair which is valid for both intake and exhaust, Eq. 2.40.
𝜇 = 1.458× 10
−6𝑇3/2
110.4 + 𝑇 (2.39)
𝜇 = 7.457× 10−6+ 4.1547× 10−8𝑇 − 7.4793 × 10−12𝑇2 (2.40) These are a few of the most used correlations on flow simulation. Some well established softwares such as OpenWAM and GT-Power use these correlations in their models.
Regarding the heat transfer factor, the most common approximation is to only considers convective heat transfer and assume that the analogies between momentum and heat transfer
derived for steady state are also valid for non-steady flow. From the Newton’s law of cooling, the heat transfer per unit of mass is
𝑞 = ℎ𝑐
𝐹𝑙
𝐹 (𝑇𝑤− 𝑇𝑔) (2.41)
Where ℎ𝑐is convective heat transfer coefficient, 𝑇𝑤 and 𝑇𝑔 are temperatures of the wall and
gas, respectively. The Colburn analogy relates the friction factor to the convective parameters as
𝐶𝑓
2 =
Nu RePr
The Nusselt number is Nu = ℎ𝑐𝐷/𝑘𝑐, where 𝑘𝑐 the thermal conductivity of the fluid, and
Prandtl number is Pr = 𝐶𝑝𝜇/𝑘𝑐. Rearranging them, it is possible to estimate the convective heat
transfer coefficient explicitly in Eq. 2.42. ℎ𝑐=
1
2𝐶𝑓𝜌𝑢𝐶𝑝𝑃 𝑟
−2/3 (2.42)
The drawback of this equation is the need to estimate 𝑘𝑐 using another correlation. The
Reynolds analogy, which assumes Pr = 1, is the simplest way to avoid such complication, although it is less precise.
2.4 Method of Characteristics
Regarding applications on engine manifolds, the method of characteristics was first used as a graphical technique in the early 1950’s. A numerical model based on the graphical approach was developed by Benson et al. (1964), naming it mesh method of characteristics (MC). This method have been applied to a wide variety of intake and exhaust systems. Overall, the simu-lated results have shown good agreement when compared to experimental data, a few works are (Benson and Baruah, 1965; Benson, 1971; Benson et al., 1975; Low et al., 1981; Winterbone et al., 1985).
Payri et al. (1986) suggested some modifications of the classical approach. These removed nonphysical pressure discontinuities produced by the classical method and successfully im-proved the results of the flow field with the drawback of higher computational expense. Another modification was proposed by Van Hove and Sierens (1991) in order to improve the mass flow conservation of the classical method when applied to tapered pipes.
MC became the dominant numerical technique used to simulate the wave effects encoun-tered inside engine manifolds until middle of 1980’s. Even though this method has only first order accuracy, it is based on the exact solution. Therefore, it is expected to present good results when discontinuities in the properties are encountered (Smith, 1985).
The solution of hyperbolic quasi-linear equation by method of characteristics consists of reducing a partial differential equation (PDE) to a ordinary differential equation (ODE) with the solution along a specific curve called characteristic (Smith, 1985). A general example with one equation is going to be presented first to help explaining the process of finding a solution and then the solution for the governing equations in Sec. 2.3 is described later.
2.4.1 General solution
For a single hyperbolic equation of the type 𝑎𝜕𝑢
𝜕𝑡 + 𝑏 𝜕𝑢
𝜕𝑥 = 𝑐 (2.43)
where 𝑎, 𝑏 and 𝑐 are functions of 𝑢, 𝑥 and 𝑡 but not of 𝜕𝑢/𝜕𝑡 and 𝜕𝑢/𝜕𝑥. This characterize the equation as quasi-linear. The main goal is to find a curve 𝐶 where the integral of Eq. 2.43 becomes the integral of an ODE. This means that on this curve the equation to be integrated is independent of the partial derivatives, 𝜕𝑢/𝜕𝑡 and 𝜕𝑢/𝜕𝑥. Assuming that the partial derivatives exist in such a curve, this indicates that the total variation of 𝑢 must exist on 𝐶 in the form of Eq.2.44.
𝑑𝑢 = 𝜕𝑢 𝜕𝑡𝑑𝑡 +
𝜕𝑢
𝜕𝑥𝑑𝑥 (2.44)
𝜕𝑢
𝜕𝑥(𝑎𝑑𝑥− 𝑏𝑑𝑡) + (𝑐𝑑𝑡 − 𝑎𝑑𝑢) = 0 (2.45)
as mentioned before Eq. 2.45 must be independent of the partial derivatives. This leads to so-called characteristic curve, Eq. 2.46, and to the solution for 𝑢 along this particular curve, Eq. 2.47. 𝑑𝑥 𝑑𝑡 = 𝑏 𝑎 (2.46) 𝑑𝑢 𝑑𝑡 = 𝑐 𝑎 (2.47)
Notice that a solution could be obtained by eliminating 𝜕𝑢/𝜕𝑥 instead, which would lead for a solution of the form 𝑑𝑢/𝑑𝑥 = 𝑐/𝑏. Since the PDE was reduced to an ODE, the more common methods for solving ODE are applicable. However, its solution is only acceptable along the characteristic curve Eq. 2.46.
2.4.2 Governing equations solutions
In order to find the ODEs for the governing equations, Eq 2.22 to Eq. 2.24 must be reduced to only three partial derivatives. This can be done by substituting Eq. 2.32 into non-conservation energy equation, Eq. 2.24, and rearranging the terms leads to Eq. 2.48
𝜕𝑝 𝜕𝑡 + 𝑢 𝜕𝑝 𝜕𝑥 + 𝑘𝑝 𝜌 [︂ 𝜌𝜕𝑢 𝜕𝑥 + 𝜌𝑢 𝐹 𝜕𝐹 𝜕𝑥 ]︂ − (𝑘 − 1) 𝜌 (︂ 𝑞 + 𝑢𝑔𝐹𝑙 𝐹 )︂ = 0 (2.48)
The procedure described for the general solution is applied for the system of quasi-linear hyperbolic equations represented by Eq. 2.22, Eq. 2.23 and Eq. 2.48. Then, adding to the total
differential of 𝜌, 𝑢 and 𝑝 the system of six equations is represented by Eq. 2.49 to Eq. 2.54 𝜌𝑡+ 𝑢𝜌𝑥+ 𝜌𝑢𝑥+ 𝑓1 = 0 (2.49) 𝑢𝑡+ 𝑢𝑢𝑥+ 1 𝜌𝑝𝑥+ 𝑓2 = 0 (2.50) 𝑝𝑡+ 𝑘𝑝𝑢𝑥+ 𝑢𝑝𝑥+ 𝑓3 = 0 (2.51) 𝜌𝑡𝑑𝑡 + 𝜌𝑥𝑑𝑥 = 𝑑𝜌 (2.52) 𝑢𝑡𝑑𝑡 + 𝑢𝑥𝑑𝑥 = 𝑑𝑢 (2.53) 𝑝𝑡𝑑𝑡 + 𝑝𝑥𝑑𝑥 = 𝑑𝑝 (2.54)
where the subscript 𝑡 and 𝑥 represents the partial derivative of the variable with respect to the subscript and the 𝑓s are given by Eq. 2.55 to Eq. 2.57.
𝑓1 = 𝜌𝑢 𝐹 𝑑𝐹 𝑑𝑥 (2.55) 𝑓2 = 𝑔 𝐹𝑙 𝐹 (2.56) 𝑓3 = 𝑘𝑝𝑢 𝐹 𝑑𝐹 𝑑𝑥 − (𝑘 − 1) 𝜌 (︂ 𝑞 + 𝑢𝑔𝐹𝑙 𝐹 )︂ (2.57) The system of equation can be reduced for only one equation by multiple substitutions; first, eliminating all time derivatives and then reducing the other equations, leading up to Eq. 2.58. Since the fluid is a perfect gas, a relation for speed of sound 𝑎, given by Eq. 2.59, was used in order to simplify Eq. 2.58.
𝜌𝑥 (︂ 𝑢− 𝑑𝑥 𝑑𝑡 )︂ [︃(︂ 𝑢− 𝑑𝑥 𝑑𝑡 )︂2 − 𝑎2 ]︃ + (︂ 𝑑𝜌 𝑑𝑡 + 𝑓1 )︂ [︃(︂ 𝑢− 𝑑𝑥 𝑑𝑡 )︂2 − 𝑎2 ]︃ +𝑑𝑝 𝑑𝑡 − 𝜌 (︂ 𝑢− 𝑑𝑥 𝑑𝑡 )︂ (︂ 𝑑𝑢 𝑑𝑡 + 𝑓2 )︂ + 𝑓3 = 0 (2.58) 𝑎2 = 𝑘𝑝 𝜌 = 𝑘𝑅𝑇 (2.59)
As for the general solution, Eq. 2.58 must be independent of partial derivatives, thus, the first term of the equation must be zero. This leads to three possible results, which are the
characteristic curves, Eq. 2.60, Eq. 2.61 and Eq. 2.62 𝑑𝑥 𝑑𝑡 = 𝑢 (2.60) 𝑑𝑥 𝑑𝑡 = 𝑢 + 𝑎 (2.61) 𝑑𝑥 𝑑𝑡 = 𝑢− 𝑎 (2.62)
Substituting each of the characteristic curves into Eq. 2.58 results in the respective solution along that characteristic. Therefore, the solution along Eq. 2.60, Eq. 2.61 and Eq. 2.62 are Eq. 2.63, Eq. 2.64 and Eq. 2.65, respectively. They are called the compatibility equations of the characteristic curves. 𝑑𝑝 𝑑𝑡 − 𝑎 2𝑑𝜌 𝑑𝑡 − (𝑘 − 1) 𝜌 (︂ 𝑞 + 𝑢𝑔𝐹𝑙 𝐹 )︂ = 0 (2.63) 𝑑𝑝 𝑑𝑡 + 𝜌𝑎 𝑑𝑢 𝑑𝑡 + 𝜌𝑎𝑔 𝐹𝑙 𝐹 − (𝑘 − 1) 𝜌 (︂ 𝑞 + 𝑢𝑔𝐹𝑙 𝐹 )︂ + 𝜌𝑢𝑎 2 𝐹 𝑑𝐹 𝑑𝑥 = 0 (2.64) 𝑑𝑝 𝑑𝑡 − 𝜌𝑎 𝑑𝑢 𝑑𝑡 − 𝜌𝑎𝑔 𝐹𝑙 𝐹 − (𝑘 − 1) 𝜌 (︂ 𝑞 + 𝑢𝑔𝐹𝑙 𝐹 )︂ +𝜌𝑢𝑎 2 𝐹 𝑑𝐹 𝑑𝑥 = 0 (2.65)
The characteristics curves represent the speed from which information, disturbances, prop-agates through the fluid. For instance, Eq. 2.61 and Eq. 2.62, which are often call Mach waves, propagate information relative to the local flow speed, 𝑢 ± 𝑎, while Eq. 2.60, which is called pathline characteristic, propagates at the local flow speed by advection. The Mach waves pro-voke changes in pressure, temperature, velocity and density, while the pathline carries energy level and composition (Winterbone and Pearson, 2000).
2.4.3 Entropy level variable
The compatibility equations, Eq. 2.63 to Eq. 2.65, have 𝜌, 𝑢, 𝑝 and 𝑎 which are functions of 𝑥 and 𝑡. It is possible to replace density and pressure by only one variables called entropy level, 𝑎𝐴. Benson (1982) describes 𝑎𝐴 as "the speed of sound at the reference pressure due to
an isentropic change of state from the pressure 𝑝". It can be demonstrated that the entropy level is direct related to entropy by using Gibbs relation, Eq. 2.66.
𝑇 𝑑𝑠 = 𝑑𝑒 + 𝑝𝑑 (︂ 1 𝜌 )︂ = 𝑑𝑒− 𝑝 𝜌 𝑑𝜌 𝜌 (2.66)
For an ideal gas with constant specific heats, 𝑒 can be expressed as Eq. 2.32, and from the definition of entropy level, an isentropic change can be expressed by Eq. 2.67 or Eq. 2.68
𝑝 𝑝𝑟𝑒𝑓 = (︂ 𝜌 𝜌𝐴 )︂𝑘 (2.67) 𝑝 𝑝𝑟𝑒𝑓 = (︂ 𝑎 𝑎𝐴 )︂2𝑘/(𝑘−1) (2.68) 𝑎2𝐴 = 𝑘𝑝𝑟𝑒𝑓 𝜌𝐴 (2.69) Where the subscript 𝐴 represents the final state of the process. Eq. 2.69 indicates the sound speed at final state. The derivative of the equations 2.32, 2.67, 2.68 and 2.69 are
𝑑𝑒 = 2𝑎 𝑘 (𝑘− 1)𝑑𝑎 (2.70) 𝑑𝑝 𝑝 = 2𝑘 𝑘− 1 (︂ 𝑑𝑎 𝑎 − 𝑑𝑎𝐴 𝑎𝐴 )︂ (2.71) 𝑑𝜌 𝜌 = 𝑑𝜌𝐴 𝜌𝐴 + 1 𝑘 𝑑𝑝 𝑝 (2.72) 𝑑𝜌𝐴 𝜌𝐴 =−2𝑑𝑎𝐴 𝑎𝐴 (2.73) Combining these derivatives with Eq. 2.66 lead to a direct relation of entropy and 𝑎𝐴, Eq
2.74. 𝑑𝑠 = 2𝑅 (︂ 𝑘 𝑘− 1 )︂ 𝑑𝑎𝐴 𝑎𝐴 (2.74) Looking at the compatibility equations, it can be noticed that finding relations for 𝑑𝑝/𝜌 and (𝑑𝑝− 𝑎2𝑑𝜌)/𝜌 as functions of 𝑎 and 𝑎
𝐴 are enough to reduce the number of variables of the
system. It is possible to find the first relation by dividing Eq. 2.71 by 𝜌 and isolating 𝑑𝑝, which gives Eq. 2.75. 𝑑𝑝 𝜌 = 2𝑎 𝑘− 1 (︂ 𝑑𝑎− 𝑎𝑑𝑎𝐴 𝑎𝐴 )︂ (2.75)
The second relation may be find by substituting Eq. 2.73 into Eq. 2.72 and manipulation of the terms, leadings to Eq. 2.76.
𝑑𝑝− 𝑎2𝑑𝜌
𝜌 = 2𝜌𝑎
2𝑑𝑎𝐴
𝑎𝐴
(2.76) Having both relations and substituting them into Eq. 2.63, Eq. 2.64 and Eq. 2.65 allows the compatibility equations to be reduced to a function of 𝑢, 𝑎 and 𝑎𝐴.
𝑑𝑎 𝑑𝑡 ± 𝑘− 1 2 𝑑𝑢 𝑑𝑡 = 𝑎 𝑎𝐴 𝑑𝑎𝐴 𝑑𝑡 − 𝑘− 1 2 𝑔 𝐹𝑙 𝐹 (︁ ±1 − (𝑘 − 1)𝑢 𝑎 )︁ + (𝑘− 1) 2 2 𝑞 𝑎 − 𝑘− 1 2 𝑢𝑎 𝐹 𝑑𝐹 𝑑𝑥 (2.77) 𝑑𝑎𝐴 𝑑𝑡 = 𝑘− 1 2 𝑎𝐴 𝑎2 (︂ 𝑞 + 𝑢𝑔𝐹𝑙 𝐹 )︂ (2.78)
2.4.4 Non-dimensional characteristic equations
The non-dimensionalizing is very useful in numerical simulations due to round-off errors, when the quantities of the variables are expected to be very small. Although the magnitude of the variables are not close to usual round-off error of modern computers, the equations for the method of characteristics are going to be written in a non-dimensional manner to maintain consistency with the literature.
Reference values for length, pressure and temperature are required. The reference speed, 𝑎𝑟𝑒𝑓, is the speed of sound at the reference temperature, and the reference entropy, 𝑠𝑟𝑒𝑓, is also
determined from the reference temperature. Therefore, the non-dimensional variables are 𝑍 = 𝑡𝑎𝑟𝑒𝑓 𝐿𝑟𝑒𝑓 𝑋 = 𝑥 𝐿𝑟𝑒𝑓 𝑈 = 𝑢 𝑎𝑟𝑒𝑓 𝐴 = 𝑎 𝑎𝑟𝑒𝑓 𝐴𝐴 = 𝑎𝐴 𝑎𝑟𝑒𝑓
Applying these to the characteristic curves and its compatibility equations leads to • Mach waves 𝑑𝑋 𝑑𝑍 = 𝑈± 𝐴 (2.79) · Compatibility equations 𝑑𝐴 𝑑𝑍 ± 𝑘− 1 2 𝑑𝑈 𝑑𝑍 = 𝐴 𝐴𝐴 𝑑𝐴𝐴 𝑑𝑍 − 𝑘− 1 2 (︂ 1 2𝐶𝑓𝑈|𝑈| )︂ 𝐹𝑙 𝐹 [︂ ±1 − (𝑘 − 1)𝑈 𝐴 ]︂ 𝐿𝑟𝑒𝑓 +(𝑘− 1) 2 2 𝑞 𝐴 𝐿𝑟𝑒𝑓 𝑎𝑟𝑒𝑓 − 𝑘− 1 2 𝑈 𝐴 𝐹 𝑑𝐹 𝑑𝑋 (2.80) • Pathline 𝑑𝑋 𝑑𝑍 = 𝑈 (2.81) · Compatibility equation 𝑑𝐴𝐴 𝑑𝑍 = 𝑘− 1 2 𝐴𝐴 𝐴2 [︃ 𝑞𝐿𝑟𝑒𝑓 𝑎3 𝑟𝑒𝑓 + 𝑈 (︂ 1 2𝐶𝑓𝑈|𝑈| )︂ 𝐹𝑙 𝐹 𝐿𝑟𝑒𝑓 ]︃ (2.82)
2.4.5 Riemann variables
The way the compatibility equations associated with the Mach waves are posed, indicates that both ODEs need to be solved simultaneously. It is possible to decouple them by defining two new variables named Riemann variables, Eq 2.83 and Eq. 2.84.
𝜆 = 𝐴 + 𝑘− 1
2 𝑈 (2.83)
𝛽 = 𝐴−𝑘− 1
2 𝑈 (2.84)
differentiating these equations
𝑑𝜆 = 𝑑𝐴 +𝑘− 1
2 𝑑𝑈 (2.85)
𝑑𝛽 = 𝑑𝐴− 𝑘− 1
2 𝑑𝑈 (2.86)
The derivative of the Riemann variables can be replaced in Eq. 2.80. In addition, all the properties and characteristic curves can be written as function of 𝜆 and 𝛽.
𝐴 = 𝜆 + 𝛽 2 (2.87) 𝑈 = 𝜆− 𝛽 𝑘− 1 (2.88) 𝑝 = 𝑝𝑟𝑒𝑓 (︂ 𝐴 𝐴𝐴 )︂2𝑘/(𝑘−1) (2.89) 𝑇 = 𝑇𝑟𝑒𝑓 (︂ 𝜆 + 𝛽 2 )︂ (2.90) This way, all the characteristic and compatibility equations may be written as functions of 𝜆, 𝛽 and 𝐴𝐴.
2.4.6 Numerical solution
As demonstrated before Eq. 2.80 and Eq. 2.82 are solution for the original set on partial derivative equations only along their characteristic curves. This means that if the properties of a specific point in time and space have to be estimated, it is necessary to identify the three characteristics that passes through such point. The physical meaning of the solution can be better acknowledged by analyzing the waves in the 𝑥 − 𝑡 plane for a subsonic flow. Figure 2.4 illustrates this flow, with 𝑢 > 0 from left to right, on a regular mesh grid.
Figure 2.4: Characteristics mesh grid for subsonic flow
It can be noticed that for mesh point 𝑖 at time 𝑛 + 1 the three characteristics originate from different locations at time 𝑛, as expected. The starting position depends on the slope of the characteristics. For instance, in a subsonic flow (|𝑢| < 𝑎) the wave 𝜆 always travels rightward at 𝑢+𝑎speed. The compatibility equation represents the 𝜆 ratio of change along the characteristic. The same analysis is also valid for 𝛽, which propagates on the opposite direction at 𝑢 − 𝑎, and the entropy level, which travels at the particle speed. In the case of supersonic flow (|𝑢| > 𝑎) the slope of the of the characteristics associated with the Riemann variable may change sign depending on the flow direction. This condition will not be demonstrated in this work due to supersonic flow being highly unlikely inside engine manifolds.
For a mesh grid with constant ∆𝑥, the CFL condition, which is going to be explained in details in Sec. 2.6.5, determines that all characteristic curves passes through some point at time level 𝑛 within 𝑖 − 1 and 𝑖 + 1. Once the properties of the flow are known at 𝐿, 𝐾 and 𝑅, it is possible to calculate the properties in 𝑖 after a time step ∆𝑡. Multiple numerical methods for solving a system of ODEs exist. In order to maintain consistency with the solutions found in the literature, the method of Euler was used. Even though it has a first order accuracy, it is the simplest to implement. Therefore, the Riemann variable and the entropy level can be estimated as: 𝜆𝑛+1𝑖 = 𝜆𝑛𝐿+ 𝑑𝜆 𝑑𝑍 ⃒ ⃒ ⃒ ⃒ 𝑛 𝐿 ∆𝑍 (2.91) 𝛽𝑖𝑛+1= 𝛽𝑅𝑛+ 𝑑𝛽 𝑑𝑍 ⃒ ⃒ ⃒ ⃒ 𝑛 𝑅 ∆𝑍 (2.92) 𝐴𝑛+1𝐴 = 𝐴𝑛𝐴 𝐾 + 𝑑𝐴𝐴 𝑑𝑍 ⃒ ⃒ ⃒ ⃒ 𝑛 𝐾 ∆𝑍 (2.93)
The derivatives are calculated by the compatibility equations at their respective points at time level 𝑛. Once the values of 𝜆𝑛+1, 𝛽𝑛+1 and 𝐴𝑛+1
𝐴 are obtained, the fluid properties may
be estimated by Eq. 2.87 to Eq. 2.90. The two mesh methods of characteristics, presented in Chapter 4, differ from each other on how to find the values of the properties at 𝐿, 𝑅 and 𝐾.
2.5 Conservative Discretization
The integral form of the conservation law presents the advantage over the differential form because it allows discontinuities inside the control volume. The differential form generates in-finite gradients when encounter with discontinuities, such as shock waves and contact surfaces, since it assumes the properties of being differentiable (Anderson, 1995). Knowing that contact surfaces are encountered in engine manifolds in every cycle in the exhaust process and in even-tual back flow, the integral form of the governing equations are going to be used. In addition, finite volume methods are derived from the integral form of the conservation law, thus, this is the approach used in this work.
Finite volumes are very closely related to finite difference methods. In fact, finite vol-umes often generate the same discrete equations as finite differences method and, thus, it may present the same limitations. This is actually the case for the two-step Lax-Wendroff that will be demonstrated in further sections.
Figure 2.5: General computational cell for one-dimensional conservation equations (Winter-bone and Pearson, 2000)
A finite volume method is characterized by dividing the spacial domain into grid cells, the finite volumes, and integrating the conserved quantities over each of these volumes (LeVeque, 2002). Fig. 2.5 illustrates a general grid cell of a one-dimensional flow, showing in 𝑥 − 𝑡 space. In the 𝑥 axis, 𝑖 is located in the middle of the cell and the distance between cells is ∆𝑥. However, the subscript 𝑖 also denote the cell itself. Thus,U𝑖 represents the conserved variables
in cell 𝑖. A similar analogy is made for time, where 𝑛 is an arbitrary time and 𝑛+1 is equivalent to time in 𝑛 + ∆𝑡. Applying the concept of finite volume to Eq. 2.21 results in the integral form of the governing equations, Eq. 2.94.
∫︁ 𝑖+1/2 𝑖−1/2 (︂ 𝜕U 𝜕𝑡 + 𝜕F(U) 𝜕𝑥 +C(U) )︂ 𝑑𝑥 = 0 (2.94)
In order to simplify the system and to be consistent with the literature, the source term is going to be ignored for now, it is properly treated in section 4.2.1. This results in the one-dimensional Euler equations, in which after the spacial integration become Eq. 2.95
∫︁ 𝑖+1/2
𝑖−1/2
𝜕U
𝜕𝑡𝑑𝑥 +F𝑖+1/2− F𝑖−1/2 = 0 (2.95)
Integrating Eq. 2.95 in time from 𝑛 to 𝑛 + 1 and dividing by ∆𝑥 is possible to create an explicit equation capable of estimating the cell average ofU at time 𝑛+1, as shown in Eq. 2.96.
1 ∆𝑥 ∫︁ 𝑖+1/2 𝑖−1/2 U𝑛+1 𝑖 𝑑𝑥 = 1 ∆𝑥 ∫︁ 𝑖+1/2 𝑖−1/2 U𝑛 𝑖𝑑𝑥− 1 ∆𝑥 ∫︁ 𝑛+1 𝑛 (︀ F𝑖+1/2− F𝑖−1/2 )︀ 𝑑𝑡 = 0 (2.96) This equation allows the calculation of the conserved variables after one time step by know-ing the average of U at time 𝑛 and the time integral of the fluxes on the edge of the cells. Nevertheless, such integral cannot be evaluated since the exact solution is not known. In order to avoid this the mean values with time of the fluxes might be used instead (LeVeque, 2002). Then, the basic equation for explicit finite volume methods is expressed as Eq. 2.97.
U𝑛+1 𝑖 =U 𝑛 𝑖 − ∆𝑡 ∆𝑥 (︀ F𝑛 𝑖+1/2− F 𝑛 𝑖−1/2 )︀ (2.97) In this equation,U expresses the spatial average values of the conserved variable inside the cells, Eq. 2.98, andF the time average for the fluxes on the edges of the cell, Eq. 2.99. This nomenclature is going to used from now on.
U𝑛 𝑖 = 1 ∆𝑥 ∫︁ 1+1/2 𝑖−1/2 U𝑛𝑑𝑥 (2.98) F𝑛 𝑖±1/2 = 1 ∆𝑡 ∫︁ 𝑛+1 𝑛 F𝑖±1/2𝑑𝑡 (2.99)
When the governing equations are written in integral form; mass, momentum and energy are explicitly conserved. This can also be shown by summing Eq. 2.97 from all cells, resulting in Eq. 2.100. 𝑖∑︁𝑚𝑎𝑥 𝑖𝑚𝑖𝑛 U𝑛+1 𝑖 − 𝑖𝑚𝑎𝑥 ∑︁ 𝑖𝑚𝑖𝑛 U𝑛 𝑖 =− ∆𝑡 ∆𝑥 (︀ F𝑛 𝑖𝑚𝑎𝑥+1/2− F 𝑛 𝑖𝑚𝑖𝑛−1/2 )︀ (2.100) In this equation, all the internal fluxes canceled out except for the those at the end of the pipe. Moreover, this equation indicates that the time variation of the conserved variables across the
whole domain is only dependent of the fluxes at the boundaries. This guaranties the conservative properties of the governing equations.
The finite volume methods basically differ on how the integral form of the conservation equations are discretized. Some methods are going to be shown in the next section.
2.6 Two Step Lax-Wendroff
The two step Lax-Wendroff (LW2) was firstly proposed by Richtmyer (1962). It consists of numerically solve the governing equations in two-steps. The first step is the Lax-Friedrichs method (Lax, 1954) and the second is the Leapfrog method, which is based on central differ-ences discretization in time and space.
Regarding its application on engine manifold systems, Azuma et al. (1983) was one of the first to use the two step Lax-Wendroff method on manifolds, applying to a multi-cylinder tur-bocharged marine diesel engine. They compared the simulations with mesh method of charac-teristics and experimental results. The two step Lax-Wendroff presented equal or better results than the method of characteristics regarding mass flow and also showed a reduction in compu-tational time by 30 to 60%.
Poloni et al. (1987) also compared the two-step Lax-Wendroff to the mesh method of char-acteristics. The simulations were employed to a constant cross section area pipe considering friction and heat transfer, which is connected to a constant pressure and temperature chamber ("De Haller" problem). Similarly to Azuma et al. (1983), they concluded that both methods presented overall similar results and the computational expense was reduced to almost half. In addition, a steady state analysis showed that the mass conservation of the two step Lax-Wendroff was two order of magnitude smaller, when compared the mass flow rate on both ends.
Sung and Song (1996) studied the pressure waves and its effects on volumetric efficiency for different intake geometries and engine operating conditions employing the LW2 scheme. The method predict fairly well the pressure waves for most of the geometries. Regarding the volumetric efficiency, the results were overestimated up to 20% but it showed similar tendencies. The authors relate this difference to the idealized boundary conditions used in the flow model.
Although having a second order accuracy on time and space on the smooth regions of the flow, the LW2 has the drawback of presenting overshooting when encountered with high gra-dients. Flux correction techniques (FCT) and total variation diminishing (TVD) schemes have been developed in order to diminish this spurious solution and some of these adjustments have been adapted to the two step Lax-Wendroff.
Bulaty and Niessner (1985) used a one step Lax-Wendroff with a flux corrector of the naive form applied to a two cylinder engine system in order to diminish the overshooting. Although successful, the mass conservation was violated with errors up to 17%. It also showed an increase of around 30% on the computational time.
Liu et al. (1996) did a comparison study on several schemes; the classical MC, LW2 and two different LW2 with FCT (being one modified by the authors). The tests were applied in tapered pipes on steady and unsteady conditions. The MC and one of the LW2-FCT showed mass conservation errors above 10% in tapered pipes, while the unchanged LW2 and the LW2-FCT modified by the authors presented error below 4%.
Payri et al. (2004) compared multiple methods on tapered pipes, including high resolution schemes with flux limiters and total variable diminishing (TVD). They observed that the LW2 have about the same mass conservation of the high resolution schemes while being the fastest among them. Pearson and Winterbone (1997) drew similar conclusions regarding mass conser-vation and computational expense. They compared multiple schemes on four different engines in a wide range of engine speeds. In addition, they also illustrated that the pressure distribution on the exhaust system is fairly close to the high resolution schemes and to experimental data.
The LW2 can be adapted to support a variety of high resolution schemes, which makes it a very flexible scheme. Stockar et al. (2013) mention how the LW2 with flux limiters or TVD can be seen as a provider of benchmark results. The open source software developed by the University Politècnica de València uses the LW2 associated with and without limiters to simulate engine systems (CMT-Motores Térmicos, 2014).
Overall, LW2 has shown to be suited to estimate the behavior of the unsteady flow inside engine manifolds. Its advantages are good mass conservation, low computational effort and easy implementation. However, it has the drawback of presenting non-physical overshooting when shocks or contact surfaces are encountered. Several works on engine simulations demonstrated that this problem does not have great impact on engine performance.
2.6.1 Lax-Friedrichs scheme
Regarding Eq. 2.97, the easier way to approximate the fluxes at the edge of the cells is to get the mean value of the fluxes inside adjacent cells at time 𝑛. This leads to Eq. 2.101, which is an unstable scheme. U𝑛+1 𝑖 =U 𝑛 𝑖 − ∆𝑡 2∆𝑥 (︀ F𝑛 𝑖+1− F 𝑛 𝑖−1 )︀ (2.101) The Lax-Friedrichs scheme, also called the Lax method (Lax, 1954), overcomes this prob-lem by averaging the conserved variables with the surrounding points, Eq. 2.102. It represents the first step of the Richtmyer modification of Lax-Wendroff method.
U𝑛+1 𝑖 = 1 2 (︀ U𝑛 𝑖+1+U 𝑛 𝑖−1 )︀ − ∆𝑡 2∆𝑥 (︀ F𝑛 𝑖+1− F 𝑛 𝑖−1 )︀ (2.102) Equation 2.102 was not derived from the finite volume approach in Eq. 2.97. However, it can be written as such when the fluxes in Eq. 2.97 are replaced by Eq. 2.103 form.
F𝑛 𝑖+1/2 = 1 2 (︀ F𝑛 𝑖+1/2+F 𝑛 𝑖+1/2 )︀ − ∆𝑥 ∆𝑡 (︀ U𝑛 𝑖+1− U 𝑛 𝑖 )︀ (2.103) Notice that Eq. 2.103 is the sum of the unstable part in 2.101 plus a second term. LeVeque (2002) explains that this term is alike an approximation to the flux of a diffusion equation, which has the form of Eq. 2.104 with 𝛽 = 1
2(∆𝑥) 2/∆𝑡. F𝑖−1/2 =−𝛽𝑖−1/2 𝜕U 𝜕𝑥 ⃒ ⃒ ⃒ ⃒ 𝑖−1/2 =−𝛽𝑖−1/2(︂U 𝑖− U𝑖−1 ∆𝑥 )︂ (2.104) This diffusion term is the reason for the stability of the method because it dissipates the oscilla-tions generated by Eq. 2.101. The extra term may also be called numerical diffusion. Using the flux in Eq. 2.103 means that an advection-diffusion equation is being modeled. However, as the mesh is being refined the value of 𝛽 tends go to zero, thus, being consistent with the original advection equation.
The Lax-Friedrichs scheme has first order precision and it is also known for being very dissipative (Toro, 1999), adding more diffusion than necessary.
