{"http:\/\/dx.doi.org\/10.14288\/1.0080886":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Applied Science, Faculty of","type":"literal","lang":"en"},{"value":"Mechanical Engineering, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Walsh, Paul","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2008-09-12T23:49:21Z","type":"literal","lang":"en"},{"value":"1991","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Applied Science - MASc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"The behaviour of a transient turbulent jet of natural gas as it is injected into a simulated combustion chamber of a diesel engine was investigated using numerical techniques. The TEACH code developed by A.D. Gosman of Imperial College, London, was used to investigate the influence of parameters such as injection angle, engine speed, and reservoir tank-to-chamber pressure ratio on the development of the jet.\r\nIt has been shown that the TEACH code is fully capable of predicting details of jet behaviour such as radial and axial velocity and concentration profiles when compared to known data. The code has been modified to use a compressing and expanding grid to simulate the effects of piston motion.\r\nA model of a fixed geometry combustion chamber revealed that the most influential parameter on jet behaviour is the injection angle. The jet had a tendency to adhere to either the top wall of the chamber near the injector tip or to the bottom wall directly opposite depending on the injector angle.\r\nThe compressing grid simulation showed that the presence of piston motion combined with other parameters such as injection angle and pressure ratio, produced jet characteristics that were dissimilar compared to the fixed boundary model. In general it was shown that the jet was less sensitive to injection angle and strongly influenced by increased pressure ratio as a result of the moving boundary.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/1925?expand=metadata","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/extent":[{"value":"5067901 bytes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/elements\/1.1\/format":[{"value":"application\/pdf","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"NUMERICAL ANALYSIS OF HIGH PRESSURE INJECTION OFNATURAL GAS INTO DIESEL ENGINE COMBUSTION CHAMBERSByPaul WalshB.A.Sc., The University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(MECHANICAL ENGINEERING DEPARTMENT)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember 1991\u00a9 Paul Walsh, 1991In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatuDepartment of if&C \/1.0171\/cg ( The University of British ColumbiaVancouver, CanadavtAevty^\/ 92_DateDE-6 (2\/88)ABSTRACTThe behaviour of a transient turbulent jet of natural gas as it is injected into a simulatedcombustion chamber of a diesel engine was investigated using numerical techniques. TheTEACH code developed by A.D. Gosman of Imperial College, London, was used toinvestigate the influence of parameters such as injection angle, engine speed, and reservoirtank-to-chamber pressure ratio on the development of the jet.It has been shown that the TEACH code is fully capable of predicting details of jetbehaviour such as radial and axial velocity and concentration profiles when compared toknown data. The code has been modified to use a compressing and expanding grid tosimulate the effects of piston motion.A model of a fixed geometry combustion chamber revealed that the most influentialparameter on jet behaviour is the injection angle. The jet had a tendency to adhere to eitherthe top wall of the chamber near the injector tip or to the bottom wall directly oppositedepending on the injector angle.The compressing grid simulation showed that the presence of piston motion combinedwith other parameters such as injection angle and pressure ratio, produced jet characteristicsthat were dissimilar compared to the fixed boundary model. In general it was shown that thejet was less sensitive to injection angle and strongly influenced by increased pressure ratioas a result of the moving boundary.iiTABLE OF CONTENTSABSTRACT^ iiTABLE OF CONTENTS^ iiiLIST OF FIGURES viACKNOWLEDGEMENTS^ xiNOMENCLATURE xiiCHAPTER 1: INTRODUCTION1.1 General^ 11.2 Gas Jets 31.3 Numerical Analysis^ 81.4 Thesis Objectives and Structure^ 13CHAPTER 2: MATHEMATICAL FORMULATION2.1 General^ 162.2 Governing Equations^ 162.3 k-e Turbulence model 192.4 Discretization of Governing Equations^ 222.5 The SIMPLE Algorithm^ 26CHAPTER 3: JET SIMULATION3.1 General^ 303.2 Configuration and Assumptions^ 303.3 Moving Boundary Modification 353.4 Density Change Due to Piston Motion and Gas Injection^ 403.5 Boundary Conditions^ 42iiiCHAPTER 4: ROUND FREE JET4.1 General^ 494.2 Steady-State Jet^ 504.3 Grid Size and Time Step Dependence^ 514.4 Comparison with Analytical and Experimental Results^ 534.5 Radial Profiles and Jet Penetration Data^ 57CHAPTER 5: FIXED PISTON RESULTS5.1 General^ 645.2 Comparison to Experiment^ 655.3 Injection Angle^ 745.4 Pressure Ratio 815.5 Combustible Mixture Region^ 875.6 Clearance Gap^ 91CHAPTER 6: MOVING PISTON MODEL6.1 General^ 986.2 Effect of Cylinder Pressure and Temperature Level^ 996.3 Pressure Ratio Effects^ 1006.4 Injection Angle Effects 1056.5 Engine Speed^ 1106.6 Combustible Mixture Region^ 115CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS^118REFERENCES^ 120ivAPPENDIX A: GAS JET ANALYSIS^ 123vLIST OF FIGURESFigure 2.4.1 Standard control volume configuration^ 21Figure 3.2.1 Conical gas jet in relation to piston and cylinder^ 30Figure 3.2.2 Details of the conical gas jet^ 31Figure 3.2.3 Computation grid inside combustion chamber^ 32Figure 3.2.4 Detailed computation grid^ 33Figure 3.3.1 Details of compressing and non-compressing grid regions.^35Figure 3.5.1 Injection speed^ 42Figure 3.5.2 Underexpanded jet details^ 43Figure 3.5.3 Boundary control volumes 44Figure 4.2.1 Axial velocity along the jet axis at various times; 10 m\/s initial jet speed 48Figure 4.2.2 Radial concentration profiles at various times; 0.46 m from jet orifice;10 m\/s initial jet speed^ 49Figure 4.2.3 Radial velocity profiles at various times; 0.46 m from jet orifice;10 m\/s initial jet speed 50Figure 4.3.1 Radial velocity profiles for various time step sizes; 0.435 m from jetorifice; 10 m\/s initial jet speed^51Figure 4.3.2 Radial concentration profiles at various time steps; 0.46 m from jetorifice; 10 m\/s initial jet speed; mass fraction in kg\/kg^52Figure 4.3.3 Radial concentration profiles at various grid sizes; 0.46 m from jetorifice; 10 m\/s initial jet speed; Cm is centerline concentration^53Figure 4.4.1 Comparison of numerical and analytical axial velocity profiles; U.is 10 m\/s initial jet speed; Um is the centerline velocity^54Figure 4.4.2 Corrected prediction of the axial velocity profile; U.is 10 rri\/s initial jet speed; Um is the centerline velocity^55Figure 4.4.3 Comparison of numerical and experimental axial concentration profilesfor a methane jet; 10 m\/s initial jet speed; C o is the jetoutlet concentration^ 56viFigure 4.5.1Figure 4.5.2Figure 4.5.3Comparison of velocity profiles of Tolhnien, Schlichting and the -numerical analysis; 10 m\/s initial jet speed; U. is the jetcenterline velocityComparison of numerical radial concentration profile with Tollmien'sdata ; 0.46 m from jet orifice; 10 m\/s initial jet speed; C. iscenterline concentrationComparison of a numerical model and Birch's data of a radial -concentration profile of a methane jet; 0.46 m from jet orifice; 10 m\/s initial jet speed; C. is centerline concentration575859Figure 4.5.4 Comparison of modelled jet penetration times with that of Kuo andBracco; 10 m\/s initial jet speed.^ 60Figure 5.1.1Figure 5.1.2Figure 5.1.3Figure 5.2.1Figure 5.2.2Figure 5.2.3Figure 5.2.4Concentration field for case 1 showing jet development with time,clearance gap 0.3125 inches, pressure ratio 2:1, jet angle 10 deg.contour interval 0.04 (kg\/kg), injector tip in top left corners.^63Concentration field for case 8 showing jet development with time,clearance gap 0.6250 inches, pressure ratio 5:1, jet angle 30 deg.contour interval 0.04 (kg\/kg), injector tip in top left corners.^64Velocity field for case 8 showing jet development with time,clearance gap 0.6250 inches, pressure ratio 5:1, jet angle 30 deg., injector tip in top left corners.^ 65Schlieren image compared to case 8; jet angle 30 deg; pressure ratio 5:1,clearance gap 0.6250 inches, concentration contours 0.08 (kg\/kg), injector tip in top left corners.^ 66Schlieren image compared to case 7; jet angle 10 deg; pressure ratio 5:1,clearance gap 0.6250 inches, concentration contours 0.08 (kg\/kg), injector tip in top left corners.^ 67Schlieren image compared to case 1; jet angle 10 deg; pressure ratio 2:1,clearance gap 0.3125 inches, concentration contours 0.08 (kg\/kg), injector tip in top left corners.^ 68Schlieren image compared to case 6; jet angle 30 deg; pressure ratio 5:1,clearance gap 0.3125 inches, concentration contours 0.08 (kg\/kg), injector tip in top left corners.^ 69Figure 5.3.1 Concentration fields for case 1 and 2 showing the jet angle effect,pressure ratio 2:1, clearance gap 0.3125 inches, contour interval0.04 (kg\/kg), injector tip in top left corners.^ 73viiFigure 5.3.2 Velocity fields for case 1 and 2 showing the jet angle effect,pressure ratio 2:1, clearance gap 0.3125 inches, injectortip in top left corners.Figure 5.3.3 Concentration fields for case 7 and 8 showing the jet angle effect,pressure ratio 5:1, time 3.59 ms, contour interval, 0.04 (kg\/kg),injector tip in top left corners.Figure 5.3.4 Velocity fields for case 7 and 8 showing the jet angle effect,pressure ratio 5:1, time 3.59 ms, clearance gap 0.6250 inches,injector tip in top left corners.Figure 5.3.5 Concentration fields showing jet angle sensitivity, pressure ratio5:1, time 2.13 ms, contour interval, 0.08 (kg\/kg),injector tip in top left corners.Figure 5.4.1 Concentration fields for case 1 and 5 showing pressure ratio effects,jet angle 10 deg., clearance gap 0.3125 inches, contour interval, 0.04(kg\/kg), injector tip in top left corners.Figure 5.4.2 Velocity fields for case 1 and 5 showing pressure ratio effects,jet angle 10 deg., clearance gap 0.3125 inches, injector tip in topleft corners.Figure 5.4.3 Concentration fields for case 8 and 3 showing pressure ratio effects,jet angle 30 deg.,time 2.13 ms, clearance gap 0.6250 inches, contourinterval 0.08 (kg\/kg), injector tip in top left corners.Figure 5.4.4 Velocity fields for case 8 and 3 showing pressure ratio effects,jet angle 30 deg., time 2.13 ms, clearance gap 0.6250 inches, injectortip in top left corners.Figure 5.4.5 Pressure fields for case 8 and 3 showing pressure ratio effects,jet angle 30 deg., time 2.13 ms, clearance gap 0.6250 inches, injectortip in top left corners.Figure 5.5.1 Combustible region development for case 7 ,pressure ratio 5:1, jet angle10 deg., clearance gap 0.6250 inches, contours given in mass fraction(kg\/kg), injector tip in top left corners.Figure 5.6.1 Concentration fields for case 6 and 8 showing the effect of clearancegap, pressure ratio 5:1, jet angle 30 deg.,time 2.13 ms, contourinterval 0.08 (kg\/kg), injector tip in top left corners.viiiFigure 5.6.2 Concentration fields for case 6 and 8 showing the effect of clearancegap, pressure ratio 5:1, jet angle 30 deg.,time 3.59 ms, contourinterval 0.08 (kg\/kg), injector tip in top left corners.^ 90Figure 5.6.3 Concentration fields for case 1 and 4 showing the effect of clearancegap, pressure ratio 2:1, jet angle 10 deg.,time 2.13 ms, contourinterval 0.08 (kg\/kg), injector tip in top left corners.^ 92Figure 5.6.4 Concentration fields for case 1 and 4 showing the effect of clearancegap, pressure ratio 2:1, jet angle 10 deg.,time 3.59 ms, contourinterval 0.08 (kg\/kg), injector tip in top left corners.^ 93Figure 5.6.5 Velocity fields for case 6 and 8 showing the effect of clearancegap, pressure ratio 5:1, jet angle 30 deg.,time 3.59 ms, injectortip in top left corners.^ 94Figure 6.2.1 Concentration fields for case 7 and 1 comparing actual engine and STPinitial conditions, pressure ratio 2:1, jet angle 30 deg.,engine speed 1200 RPM, contour interval 0.08 (kg\/kg), injector tipin top left corners.^ 98Figure 6.3.1 Concentration fields for case 1 and 2 showing pressure ratio effects,jet angle 30 deg., engine speed 1200 RPM, contour interval 0.08 (kg\/kg),injector tip in top left corners.^ 99Figure 6.3.2 Velocity fields for case 1 and 2 showing pressure ratio effects,jet angle 30 deg., engine speed 1200 RPM, injector tip in topleft corners.^ 100Figure 6.3.3 Concentration fields for case 8 and 10 showing pressure ratio effects,jet angle 10 deg., engine speed 1200 RPM, contour interval 0.08 (kg\/kg),injector tip in top left corners.^ 102Figure 6.4.1 Concentration fields for cases 8,1 and 3 showing injection angle effects,pressure ratio 2:1, engine speed 1200 RPM, crank angle 6.8 deg. BTDC,contour interval 0.08 (kg\/kg), injector tip in top left corners. 104Figure 6.4.2 Velocity fields for cases 8,1 and 3 showing injection angle effects,pressure ratio 2:1, engine speed 1200 RPM, crank angle 6.8 deg. BTDC,contour interval 0.08 (kg\/kg), injector tip in top left corners.^105Figure 6.5.1 Concentration fields for cases 2 (fixed boundary),6 and 1 showing effects ofengine speed, pressure ratio 2:1, jet angle 30 deg., time 5.3 ms afterinjection start, contour interval 0.08 (kg\/kg), injector tip in topleft corners.^ 107ixFigure 6.5.2 Velocity fields for cases 2 (fixed boundary),6 and 1 showing effects ofengine speed, pressure ratio 2:1, jet angle 30 deg., time 5.3 ms afterinjection start, injector tip in topleft corners.^ 107Figure 6.5.3 Concentration fields for cases 1 (fixed boundary),9 and 8 showing effectsof engine speed, pressure ratio 2:1, jet angle 10 deg., time 5.3 ms afterinjection start, contour interval 0.04 (kg\/kg), injector tip in topleft corners.^ 109Figure 6.5.4 Velocity fields for cases 1 (fixed boundary),9 and 8 showing effects ofengine speed, pressure ratio 2:1, jet angle 10 deg., time 5.3 ms afterinjection start, injector tip in top left corners.^ 110Figure 6.6.1 Combustible region development for case 1 ,pressure ratio 2:1, jet angle30 deg., engine speed 1200 RPM, contours given in mass fraction(kg\/kg), injector tip in top left corners.^ 113xACKNOWLEDGEMENTTh ,,, author would like to express his sincere appreciation to his supervisor Prof. P.G. hillfor all his expert advice and encouragement which made the completion of this thesis possiblein a timely and orderly fashion.I would also like to extent my gratitude to my co-supervisor Prof. M. Salcudean for hertechnical expertise and generous advice.I would also like to thank Mr. Patric Ouellette for the use of his experimental resultsrelated to this work.The author is also indebted to his family for their advice and support, and would like toexpress his appreciation. I would finally like to express my appreciation to Ms. DorothyMiyake for her assistance on matters related to computer software operation.xiNOMENCLATUREAe^- mi.,. of control volume faceA,ap.e\u201e - neighbouring coefficients of linearized equationsC^- concentrationd,D^- diameterde^- effective\/equivalent diameter (high pressure ratio)Fr^- body force termF,D^- convective,diffusive componentsG^- rate of strain termk^- turbulent kinetic energyPa,Po^- ambient,tank pressurePe^- Peclet number ( ratio of F over D)Red^- Reynolds number ( diametral)r,y^- radial coordinateS^- source termt,t*^- time, nondimensional timeU,u^- axial velocity ( mean,fluctuating)V,v^- radial velocity ( mean,fluctuating)y+^- nondimensional distance from wallz,x,x *^ - axial coordinate ( * - nondimensional )4^- piston face distance from injector tipK,a,d,ce^- constantsxiiGreek- turbulent energy dissipation\u2022 nondimensional axial coordinate ( z\/zh)r^- diffusivity (p\/a)- dependant variable ( U,V,k,e,C)p^- densityPeff - dynamic viscosity ( turbulent,laminar,effective)a^- Prandtl number- kinematic viscositytiW^- wall shear stressSuperscripts- future point in timeo - previous point in time- guessed value- averaged valueSubscriptso - jet inlet valuem^- jet centerline valuep,n,e,s,w^- neighbouring nodes1CHAPTER 1 : INTRODUCTION1.1 GENERALNatural gas has been used as an automotive fuel for many years. Only recently has it -come under close study as an alternative to conventional automotive fuels. As a newenvironmental awareness occupies public attention, the quest for automotive fuels that offerthe performance of standard fuels but with far fewer harmful emissions has expanded. Thepublic has already demanded that their commercial diesel carriers meet strict new guidelinesthrough legislation to be introduced in the next three years. The new laws will limit theamounts of nitrogen oxides (NOx) and particulate matter to levels considered to be restrictiveby the diesel engine manufacturing establishment. Particulate levels must be reduced to 1\/6their 1989 levels while NOx must be reduced by half. It may be that the only feasible mannerin which these new standards can be met is with the use of alternative fuels such as naturalgas.Unlike gasoline and standard diesel fuel, natural gas is a simple hydrocarbon resulting infewer combustion byproducts, giving it the potential to be a clean air fuel [34]. With carefulmixture and timing control, emissions of carbon monoxide, particulates, NOx and reactivehydrocarbons can be reduced to levels below those of standard spark ignition and dieselengines. Additionally, natural gas has a low photochemical reactivity meaning that unburnedfuel will not contribute significantly to smog problems plaguing urban centres. Clearly,natural gas has the potential to run cleaner than standard diesel fuel.Although natural gas has been used in spark ignition engines for some time, its use as adiesel fuel raises certain problems. The flame speed of natural gas is low, especially at high2compression ratios. This will lead to poor ignition and burning characteristics causing lowerpower output. Adding a small amount of 'pilot' diesel fuel to the natural gas to promoteignition is one possible remedy to this problem.Several diesel engine strategies are available related to the timing and to hardwarerequirements of dual fuel systems. Early cycle injection places the natural gas charge into thecylinder before the pilot diesel fuel, early in the compression stroke when cylinder pressuresare low, eliminating the need for a high pressure injection system. However, if the fuel iswell mixed, a detonation limit may be reached where the fuel pre-ignites once certainconditions are achieved. This will restrict compression ratios to values below which pre-ignition will not occur. This limit can hamper the achievement of the full operating potentialof the engine.The fumigation process injects the natural gas fuel into the intake air stream before thecompression stroke commences. The natural gas will displace approximately 10% of theintake air flow, reducing the maximum amount of intake air and performance compared tolate cycle injection systems. A detonation limit will restrict the operating range of an engineconfigured in such a manner. Such systems do not perform as well as late cycle directinjection engines.High pressure, late cycle, direct injection with electronic control of injection duration andtiming offers technical advantages. Since injection occurs late in the compression stroke, noneof the intake air is displaced by the natural gas. No fuel of any kind is present in the cylinderduring the compression stroke; removing any possibility of pre-ignition. Therefore, a latecycle injection can preserve the diesel cycle efficiency. Even though a late cycle high3pressure system appears to offer the best chance for a clean efficient system, the propertiming and mixing is still required to achieve peak performance.The Alternative Fuels Laboratory at the University of British Columbia is developing adual fuel, late cycle injection system with an innovative injector design. In contrast totraditional dual fuel systems which have one injector for each fuel, the U.B.C. projectcombines the two fuels in one injector. The aim is to introduce the pilot diesel directly intothe stream of natural gas as it is injected into the cylinder. It is thought that this process willresult in a uniform distribution of fine diesel droplets which should promote quicker ignitionand faster burning.In spite of the fact that late cycle direct injection systems can offer superior performancecompared to other systems, very few diesel engine researchers have attempted to implementit since it is technically the most challenging. Late cycle injection requires both a highpressure gas supply and fast acting injectors, which represents a considerable developmenteffort. Once constructed, the injector jet characteristics will need to be investigated to allowoptimization for peak engine performance. To meet such goals, both experimental andnumerical modelling techniques are called upon to investigate the jet performance.1.2 GAS JETSThe injector tip is designed to produce a continuous sheet of gas instead of several smallround jets as in the traditional liquid diesel fuel injector. Initially the jet is very thin,approximately one third of a millimetre in thickness. It emerges from the injector tip as ahigh speed axially symmetric conical sheet. The injector tip is centrally located in the enginecylinder giving axial symmetry to the jet-cylinder system. This provides the opportunity to4assume two dimensional flow for any model of this system.The pressure ratio between the fuel tanks and the engine cylinders may be as high as 5to 1. The jet may therefore be under-expanded, meaning that the pressure at its exit plane willbe higher than the cylinder pressure into which it emerges. The jet will expand to cylinderpressure as it leaves the injector, forming a region where compressibility effects will besignificant. Ewan and Moodie [11] have made a study of such jets. Their results show thatfor a fuel tank to cylinder pressure ratio of 5 to 1 the length of this expansion region will beapproximately 1.3 times as large as the exit diameter. Considering that the jet velocity dropsto one third its original velocity within approximately 18 exit diameters of this initialexpansion, any compressibility effects will therefore be limited to a very small region ( onetenth the cylinder diameter). It is not unreasonable then to assume that the flow field insidethe cylinder is mainly incompressible.The gas jet behaviour depends on the injection angle, the injection pressure, the chambergeometry and the engine speed. The effects of these parameters can best be assessed throughthe determination of the jet velocity and concentration fields. These fields can be determinedat least qualitatively by either analytical, experimental or numerical means. Simplified modelscan be solved analytically to predict flow fields and jet behaviour in an attempt to understandmixing and penetrating character. Experimental analysis uses physical representations eitherfull size or scaled models in order to simulate the actual in cylinder processes .Much analytical and experimental work has been conducted on the subject of freeincompressible steady-state turbulent jets. A review of some of these results will allow acomparison to be made to a steady-state version of the code used in this thesis. If the87c c ox (1+ _,,, ) 241 2\"3 K^1u -5numerical code can successfully predict turbulent jet behaviour, then this will be a positiveassessment of its analysis capabilities and will give confidence for its use in in-cylindertransient jet investigations.Schlichting [29] gives an analytical derivation created by Tollmien for the velocity profileof a circular turbulent jet emptying into free space. The governing equations are approximatedusing momentum integral techniques assuming that the centerline velocity is inverselyproportional to the axial distance. The solution gives an estimate of the axial velocity as afunction of the radial and axial coordinates.= 1 \\1 3 ,,rfe y11 -.I - i e . x where K= 2n f u2 ydy0Here, y and x are the radial and axial jet coordinates respectively, and K is the specificmomentum. The virtual kinematic viscosity for round free jets has been determinedexperimentally by Riechard [29] to bee\u00b0 = 0 . 0161 .1,8?This value is a constant for round free jets as a result of the centreline velocity , u m, beinginversely proportional to the axial distance x. The results computed using this analysis agreewell with experimental results presented by Schlichting. The universality of the velocityprofile for a turbulent jet is demonstrated by the reduction of the experimental data at variousaxial locations in the jet and of the predicted values onto one standard curve of iliu m vs. T.6Abramovich [2] also quotes Tollmien and gives tabulated results of velocity, concentrationand axial velocity decay for round free turbulent jets. Tollmien's axial velocity decay formulafor round jets begins with um = C\/x. Using momentum conservation it can be shown thaturn _ 0 . 96 ^u 0^ax1?\u201e,u. and R. are the initial jet axial velocity and radius respectively. The constant 0.96 is thedimensionless distance (ax\/Ro = 0.96) from the jet origin where x=0, to the transition crosssection where um\/u. = 1. This theory agrees well with data taken by Trupel and Gottingen[2], who found the constant a to be 0.066.A study of the concentration fields of steady-state turbulent round free jets of methanewas conducted by Birch et al [7]. They found that methane jets can be analyzed usingincompressible jet theory so long as a correction is made to the starting diameter to accountfor the density difference between the gas and the ambient air. The adjusted diameter wasreferred to as the effective diameter, and was computed from similarity conditions by Thringand Newby (1953). A free jet can be characterized by its initial diameter do, its totalmomentum M, and its total initial mass flow rate m. For an initially uniform velocity theseare:1 1^M = --4 ndo2 p o uo2^m = \u20144 n do2 pou0.From these Thring and Newby deduced that an incompressible jet of initial density p o ,differing from the ambient density p a, will behave as a jet having po equal to p a but having7a different starting diameter. For the same mass and momentum fluxes the effective diameter(dE ) will be:dE = d (p o\/p a ) 1 \/ 2The initial jet diameter is d, and the gas and air densities at ambient pressure and temperatureare respectively po and pa. Their experiments led them to conclude that the centerlineconcentration of a methane jet decayed with distance according to the following relation;C _ kideCo^(z +a)where C is concentration, C o is inlet concentration, k 1 and a are constants, and z is the axialcoordinate. Concentration profiles in the radial direction as a function of axial position werealso investigated and were found to obey the relationC = exP( -D(r\/ z) 2 ) .mHere, D is a constant, found experimentally to be 73.6. Cm is the centreline concentration,and r is the radial coordinate.Kuo and Bracco [21] investigated the transient nature of incompressible jets numerically.They compared their results produced by the implicit finite volume TEACH code, whichemploys the k-E turbulence model, to experiments of air injected into air to test theperformance of the code. Care was taken in the experiments to ensure a uniform velocityprofile at the nozzle outlet in order to justify such a velocity profile in the numerical8code. Time and axial position were non-dimensionalized for comparing cases.t * -t Llin^x* \u2014 ^XD (41\u00b053 ) D (4' \u00b053 )Here, t indicates time, x axial distance from the orifice, D the nozzle diameter, and D thekinematic viscosity.They observed the time required for a position on the centreline downstream to reach70.0% of its steady state velocity. The centreline velocity decay with axial position for thesteady jet was tested as well. The results of the numerical analysis appeared to agree fairlywell with the experimental data, validating the use of the TEACH code for transient turbulentjet studies. The penetration time of their transient jet investigation is given in the followingrelation.t*=0 .235x*2^x*k7With the experimental information on jet behaviour just described, the validity of usinga numerical code for its analysis can then be assessed.1.3 NUMERICAL ANALYSISThe purpose of the research described in this thesis is to obtain insight into jet-cylinderinteractions through numerical means. Numerical analysis attempts to investigate jet behaviourthrough an iterative solution of the governing equations of mass and momentum conservationby computer algorithm given a set of inlet and boundary conditions.9Numerical models of fluid dynamic phenomena, have become an increasingly popularresearch tool for a number of reasons. First, a more comprehensive insight of the phenomenaunder study can be obtained . The effects of numerous parameters such as injection pressureand spray angle can be investigated quickly and with minimal effort. Answers to importantquestions such as how piston motion affects the jet behaviour and the subsequent in-cylinderflow field may be obtained. Secondly, complete representations of concentration, velocity andpressure fields at any point in time are possible. Finally, new design ideas can be testedthoroughly over the complete range of engine operation before costly engine experiments areperformed.Numerical fluid flow models approximate the governing equations through various means.Numerous methods have been devised to reduce the governing equations to a series of linearalgebraic equations in a process known as discretization. The control volume method is onediscretization scheme used frequently in engine modelling. The region of interest issubdivided into a grid of nodes each surrounded by a control volume . The governingequations are integrated over each control volume to produce one linear equation perdependent variable per control volume. The values of dependent variables are calculated atthe node point within each control volume from the corresponding linear equation producedby the discretization process. Values at adjacent nodes are related through a number ofpossible weighting schemes, which will form the basis of the numerical code.The finite difference method which is also common in engine modelling derives thelinearized governing equations in a different manner. The terms appearing in the governingequations are approximated through a Taylor series expansion of the dependent variable.10The work described in this thesis is the development and application of a two-dimensionalfinite volume model for the study of transient injections of natural gas into a diesel enginecylinder. Codes such as this have only appeared in the last two decades with the introductionof high speed computation devices. The first true two-dimensional code to analyze fluidmotion in engines was written by Watkins in 1973 [32]. His code was a finite differencemodel that solved only mass and momentum conservation without combustion orcompressibility effects. Geometry was restricted to a rectangular enclosure without thecomplexities of inlet ports or piston bowls. Piston motion was simulated by a transformationof the axial coordinate to a non-dimensional time independent form, which allowed the gridto expand and contract. The flow field was further simplified by neglecting turbulence andconsidering only laminar viscous flow. The collapsing grid to simulate piston motion was themost significant contribution of his analysis.Soon afterwards Gosman et al. [13] developed a similar code called Reciprocating PistonMotion ( RPM) based on the same geometry as Watkins. Gosman's code included valvegeometry and used the k-E turbulence model. When compared to available engine data, theirresults were only qualitatively accurate. The most notable success was the code's ability topredict recirculating flow patterns seen in experiments when air was drawn through an openvalve. He also was able to compare his results to data taken in a motored engine. In somecases the discrepancy between the predicted and measured values of air motion wasconsiderable. The researchers felt that their biggest source of errors was the lack ofknowledge of the boundary conditions and the possible inability of the turbulence model toaccount for larger turbulent structures.11Later, Gosman developed the TEACH code which is the base program upon which theanalysis in this study is built. TEACH is a two dimensional finite volume code originallywritten to analyze steady-state fluid flow inside circular ducts. It uses the k-e turbulencemodel and assumes the fluid flow is incompressible.Later, Watkins [32] built on his previous work by incorporating into his piston motioncode the k-E turbulence model and a rudimentary compressibility model. His results showedthat flow behaviour was strongly dependent on initial conditions, such as at the valve inlet.A considerable portion of his work concerned the development of adequate boundaryconditions at valve inlets.In the past decade a considerable number of engine , simulation codes have appeared.Spalding [23] created the PHOENICS code as a fluid flow simulator, and it was one of thefirst codes to appear with the finite volume discretization technique. Its usefulness as anengine modelling code arises from its ability to analyze multiphase flow, three-dimensionalphenomena, and chemical kinetics.Cloutman [23] introduced the CONCHAS program in 1984. It was a finite difference codethat possessed a simple combustion model based on chemical kinetics and a simple Arrheniusrelation for flame speed. He simulated the combustion chamber of a spark ignition engine.His results showed that the numerical calculations could qualitatively predict the heat transferand nitrogen monoxide characteristics, but prediction of other variables was less accurate.This was only accomplished after the combustion model was adjusted by the variation ofseveral parameters. Cloutman found that the optimal parameter settings varied withequivalence ratio, volumetric efficiency, spark timing and even the choice of engine. He12showed a simple combustion model was incapable of the generality that was desired in anumerical combustion model.Some of the most advanced codes produced to analyze internal engine phenomena are theKIVA codes created by Amsden et al. [23]. The KIVA I code appeared in 1985 and wascapable of computing three dimensional flow fields with chemical reaction. It also possesseda liquid fuel spray break up and evaporation model for use with diesel engine configurations.The KIVA II code replaces the explicit nature of the original KIVA with an implicit finitevolume method. Liquid fuel jets are computed by a discrete particle technique, where anumber of droplets are represented by one computation particle. Fuel droplet size andcoalescence are determined statistically. The KIVA codes have gained popularity formodelling liquid diesel sprays in conventional diesel engines.Little work has been done on the numerical modelling of gaseous hydrocarbon injectionsinto engine combustion chambers. However, one study by Gaillard [12] is worth noting.Gaillard simulated the injection of four methane jets directed radially into a two dimensionalround combustion chamber using a semi-implicit fmite difference scheme. Initially, Gaillardcompared computed values of transient jet penetration with known results of Kuo and Bracco.He found that the jet penetration and concentration contours agreed well with experiment.Gaillard's intention was to study the effects that swirling air motion inside the cylinder wouldhave on jet development. His model showed the effects of swirl motion at various speeds ina two- dimensional plane. However, Gaillard did not study the effects of jet angle andpenetration in an axial direction.Many numerical codes exist that are capable of modelling internal combustion engines.13Virtually all codes that have been written model diesel engines have spent considerable effortanalyzing the break up of the liquid diesel fuel jet. Codes, such as KIVA II, have been usedin numerous studies on liquid fuels in diesel engines. Very little has been done on the studyof gaseous fuels injected in diesel engines. The work of Gaillard is the only study conductedin this area. Although his work is insightful on the effects of swirl on the transient jet, it doesnot consider the effects of jet angle, piston motion and pressure ratio. To answer thesequestions a two dimensional numerical model in radial and axial coordinates has beenconstructed and is presented in this thesis.1.4 THESIS OBJECTIVES AND STRUCTURENumerical analysis was used to investigate the injection of natural gas fuel into dieselengines for several reasons. Little experimental or numerical data exists on the behaviour ofsuch jets, making optimization of the dual fuel diesel injection system difficult. Apart frombeing less expensive and time consuming than experiment, numerical analysis may be ableto provide insight into areas where experiment cannot. Piston motion may have a stronginfluence on the jet behaviour, which is easily investigated numerically.The original TEACH I code was written to calculate steady-state incompressible pipe flow.It was selected for its simplicity and flexibility, and its reputation as a proven universal flowanalysis program. The code must be modified to perform transient computations through theintroduction of time-dependent source terms. Piston motion is simulated through atransformation of the axial coordinate to accommodate a compressing and expanding grid.The accumulation of mass inside the closed cylinder as gas is injected is accounted for14through a correction to the density field.Two models are used in this thesis: a fixed piston model will have a stationary lowerboundary that can be set at various positions; and a moving piston model, with a movinglower boundary that can be made to move at various speeds. In each model the jet angle andpressure ratio may be altered.The objectives of this thesis are stated now in the form of questions to be addressed.1). Can the TEACH code, with adjustment of the initial conditions, simulate a twodimensional transient jet ?2). Does the jet angle influence the jet penetration and development in both moving and fixedpiston models, and if so in what way ?3). Is there a change in jet penetration when a higher injector tip pressure ratio is used ?4). Does the chamber size influence the jet development ?5). How does engine speed influence the jet development ?6). Is there a difference in the jet character between the fixed and moving piston models ?The chapters in this thesis are outlined. Chapter two goes over the TEACH code, whichwas modified to perform this analysis. The governing equations and their discretization areall described.Chapter three describes the assumptions used in developing the models and theirconfiguration. A derivation of the general and local compressibility model as well as anintroduction to the moving boundary transformation is presented.Chapter four compares the free jet calculations to known analytical and experimental15results. The comparison will validate the use of the TEACH code for analyzing the turbulentjet under consideration.Chapter five gives the results for the analysis of the transient jet erupting into a fixed,closed environment. This study will identify the parameters that appear to affect the jetbehaviour the most. Jet angle, pressure ratio, and clearance will be considered.Chapter six gives the results of the moving boundary analysis. The effects of jet angle,pressure ratio and engine speed will be investigated.Chapter seven draws conclusions from the results, and recommends a course for furtherstudy.1 6CHAPTER 2 : MATHEMATICAL FORMULATION2.1 GENERALThe motion of fluid inside the combustion chambers of diesel engines can best bedescribed with the aid of the mass conservation and Navier-Stokes equations. The Navier-Stokes equations express the conservation throughout the computational domain of the fluidmomentum. The TEACH fluid flow analysis code , which was used in this investigation,solves the mass and momentum conservation relations.The second section ( 2.2 ) of this chapter reviews the basic governing equations used inthe TEACH code. The effect of turbulent fluctuations is introduced through a redefinition ofthe velocity field into mean and fluctuating components in a manner proposed by Reynolds( 1895 ). The development of the k-c turbulence model using the Reynolds form of theNavier-Stokes equations is outlined in the next section of the chapter ( 2.3 ). Reduction ofthe generalized governing equation through discretization to a linear algebraic form isreviewed in section 2.4, along with a description of the hybrid differencing scheme used inthe process. The SIMPLE algorithm used to solve the momentum equations is brieflyreviewed in section 2.5.2.2 GOVERNING EQUATIONSThe basic equations governing fluid motion are the mass conservation and the Navier-Stokes equations of momentum conservation, derived in detail in Schlichting [291. Theconservation of momentum provides one equation each for the radial and axial directions.The mass conservation equation provides a way to determine the pressure field, which willbe shown in a later section. The mass and momentum conservation equations in these17coordinates are shown below. The variables u and v are the axial and radial velocitycomponents respectively, z and r are the axial and radial coordinates respectively, t is time,p is pressure , p is density, and Fr and Fz are body force terms, in the radial and axialdirections per unit volume respectively.continuity: 4-Ea (Pry) + az (Pu) =\u00b0^(2.1)momentum conservation: radial direction:^ (2.2)^niav .4. ay^,,av\u2018^ap., 1 a ' r ay\\ y .321tlat \u2022 var \u2022 - az) - r^ar \u2022 7r ark- ' r 2 az2axial direction:^ (2.3)pi^pl au^au^u^F^ap^( 1 a (..au) -r a2 u)^at^ar^az) -^- az \u2022 _Lr ar ar^az2These equations imply constant viscosity p. For variable p and allowing the symbol 4) tostand for either the u or v velocity components, eqs. (2.2) and (2.3) can be shown to bespecial cases of the following general equation.18General Momentum Equation:^ (2.4)vti, _^(^) sata (4)p ) \u2014aaz ( Pu. ) +^a (rP^az 11 as\u2014E^r ar^arThe general momentum equation is valid for laminar and turbulent flows. The dependentvariables of the general momentum equation are instantaneous values, meaning they aredefined at every point in time and can fluctuate with turbulent motion. The dependentvariables can be separated into mean ( indicated by an overline) and fluctuating 4)'components,= (2.5)The mean values are determined through phase averaging, described by Reynolds (1980)[27]. The cycle-to-cycle variations in the cylinder of the moving piston model are removedby averaging over a great number of samples of a dependent variable taken at a specific pointinside the engine and in its cycle, shown in equation (2.6). The engine rotates with a periodof To .r, t)^Lim I 1 \\1\u00b0+1) (z r t 1-nTo )10 (2.6)If the fluctuating and mean components of the dependent variables are inserted into thegeneral momentum equation (2.4), the following equation is obtained by averaging. Theoverlines having been omitted on the mean values.19a^aat (4) + \u2014az (P14) \u2014r\u2014ar (rPvC2.7= 1( 11 1 -^4-1-(r11:4- rP7V) S\u2022The averaged product of the fluctuating components of the velocities (for example pu'v'),is the Reynolds stress term, which represents the diffusion of momentum by turbulentfluctuations.2.3 k - e TURBULENCE MODELThe presence of the Reynolds stresses complicates the solution of the general equation(2.7) by introducing further variables into the equations. With the Reynolds stress, thenumber of dependent variables will exceed the number of equations that can be used to solvethem. The Reynolds stress terms must them be modelled in terms of the other dependentvariables in order to invoke closure.Boussinesq, in 1877, developed the concept of an eddy viscosity to approximate theReynolds stress terms. He suggested that the turbulent shear stress ( Reynolds stress ) isequivalent to the product of the mean strain rate and the eddy viscosity ( PT ).-puivl = \u00b5T ay^(2.8)The turbulence model used in the TEACH code is the two equation k-E model. The eddyviscosity is derived from the turbulence kinetic energy k, and the turbulent energy dissipationrate e. Both turbulent kinetic energy and turbulent energy dissipation rate are determinedthrough their own transport equation. The governing equations for k and e and the derivationCo p k214T ^e20of the eddy viscosity from these parameters are given by Launder and Spalding (1972) [22].The experimentally determined constant C. is 0.09.(2.9)The effects of turbulence are introduced into the momentum equations through theeffective viscosity (peff). The eddy viscosity (pT) is added to the laminar viscosity (p i.) toproduce the effective viscosity.I 1 eff = 14 laminar + \u00b5 T^ (2.10)For variables other than the velocity components, a fluctuating component term similarto the Boussinesq approximation is used.-IT 7 ri au-pu Iv = i \u2014 .0 ay (2.11)The diffusivity term (F0) is determined from the eddy viscosity using the turbulent Prandtlnumber (a,) for the specific variable. The Prandtl number for diffusion of gaseous speciesthrough gaseous host media has been estimated by Launder and Spalding to be of order 1.r4, _ \"L eff_ a0 (2.12)With the equations for k and e defined by Launder and Spalding (1972), the completegeneralized governing equation for the TEACH code can be shown to be:21a ^v4, ) =^as\\ + ^irr ar)^c,(4)^(Pu4))^(rpat^ ar az ir az\/^--z.\" ar\u2018^arj^-4)variable^r4,^ So1 (mass)^0 0Neff^aaz (ileff :zu)^aan(riteffalzr)^apzV effaaz Neff UT:a\u201e^ay^,,,^apr ar^ar(r\"eff--) 4. \"eff\u2014r2 \u2014ark^II eft\/ ak^G - peNeff\/a t^ k (C 1G - C2pe)C (mass frac.)^peffia. 0au\u25a02 ( ar )^au +G = Per42RT;)^T:r1 J^arty 2( 7,1 2]The k-e model constants are:CP C1 C2 ak K0.09 1.44 1.92 1.0 0.4187where^a \u2014 K2(C2 - C1 )^\/ 2Standard Control Volume (c.v.)\u2022 Nh- AZ_ _Earr______,,\/A e7,2'1 _,\/__IAr88xw axe\u2022 r ,7--nSscalar c.v.222.4 Discretization of Governing EquationsThe discretization method used by the TEACH code is the control volume formulation.The region of interest is divided into a network of control volumes. Inside each controlvolume is a nodal point where the values of the dependent variables are to be evaluated. Thevelocities u and v are computed in control volumes that are staggered in relation to the scalarvariables control volumes (C,k,e, and p). The staggered grid prevents the possibility of anunrealistic solution to the pressure field ( Patankar 1980 ).Figure 2.4.1: standard control volume configurationA typical control volume arrangement is shown in figure 2.4.1. The scalar control volume isshown by the shaded region. The local scalar nodal point is indicated by P, while itsadjoining neighbours are indicated by N,S,E,W, referring to northern, southern, eastern, and23western locations . The faces of the scalar control volume are indicated by the lower caseletters; n,s,e,w. The velocity control volumes have nodal points at locations indicated by thesolid arrows in figure 2.4.1. The staggered grid allows the velocity nodal points to be locatedon the faces of the scalar control volumes, simplifying the calculation of convection throughthese faces.The integration of the governing equation over time and the control volume is shown inequation (2.13).t+At e 12 \u2014a 4)) + 1\u2014a^vit)f^f^(.13)^az^r ar (rp) r dr dz dtt w s.13)t+At e 11= ft f f (1(r.2)w s8\u2014 rr \u20141r ark^ar) +^rdrdzdtThe first term of the integrated equation can be evaluated as:t+At n ef f at (p4))) dz rdr dt rArAz( (p4)) 1 j, - (p4))%)t s w2.14The superscripts denote points in time; 1 indicates the present time, and o indicates thepreviously calculated point in time. The nodal point value is assumed to prevail over theentire control volume, allowing the integral to be evaluated.The other terms are not as easily integrated over time. Some assumption must be madeabout the variation of the integrand over time. A common practice is to assume that the resultof the integration is equivalent to various contributions of the integrand at the start and at theend of the time step, shown in equation 2.15:24t+Atf^dt = LEC + ( 1 - f)^A t^(2.15)where is any dependent variable (u,v,k, e,p,and C).The weighting factor f varies between0 and 1. If f is chosen to be 0 the discretization becomes fully explicit, and the linearequations can be solved directly from the values at previous time step. If f is chosen to be1, then the discretization is implicit, and the resulting equations must be solvedsimultaneously using an appropriate solver.Although the explicit scheme is computationally easier, it becomes unstable if the timestep exceeds the limits determined by the Courant condition [3]. For this reason the TEACHcode uses the implicit scheme. The remaining integrals can be computed as shown in equation2.16, again assuming that the integrand is constant over the control volume and omitting the1 superscripts for present time step.t+At e nf f _^ft (p u, - r^r a a^4 , zdtt w s= {(pu. - P\u2022 \u2014134') e - ( pub - Pe to- ) w irArAtaz aZ (2.16)The integration of the source term can be written in terms of a constant component S c , anda component linearly dependent on the variable, Op, shown in equation (2.17). The discretizedgoverning equations must be linear if they are to be solved easily. Therefore, the discretizedsource term must be no more complicated than a linear approximation.t+At e nf f f Sa rdrdzdt =^+ SpttpirArA tt w s (2.17)Once the governing equation is integrated, the results must be expressed in terms of the25known nodal values. The TEACH code uses the hybrid differencing scheme to express theconvective and diffusive fluxes in such terms. Equation 2.16 shows that the fluxes througha scalar control volume face can be written as a convective ( C ) and diffusive ( D )component. As an example the fluxes at the east face (Ce and De)are shown.Ce = (p eue.e ) e rAr,^De = (Iltz ) e rAr. .^(2.18)The hybrid scheme is a combination of the central and upwind differencing schemes.The central differencing scheme assumes that a linear variation of the dependent variablesexists between neighbouring nodes. The convective and diffusive fluxes can be written forthe eastern face of a control volume:(I) E ^4)p 4)p - 4)E) \u2022 rCe = (pu) 4 2 )rAr,^De = r4 8z. ra . (2.19)The central differencing scheme is accurate when diffusion is the important transportmechanism. In flows where convection is dominant and the Peclet number ( Pe = I puSxir I)is greater than 2, the central differencing scheme will become unstable.The upwind differencing scheme has been combined with the central differencing schemeto alleviate the high Peclet number instability of the central differencing scheme. In stronglyconvective flows the upwind scheme approximates the convective terms with values of 4)from the upwind location. An example is given in equation 2.20.Ce = p e ue4)prAr^for ue > 0C, = p e ue(bErAr^for u, < 0(2.20)26The hybrid differencing scheme uses the superior performance of central differencing forlow Peclet number flows and upwind differencing for high Peclet number flows. Diffusiveterms are always approximated with central differencing. Convective terms use centraldifferencing when IPe 15.2 and upwinding for IPeThe complete linearized governing equation for a dependent variable (I) is given inequation 2.21.apttp = adOE + awOw + aNON + asOs + b + Scz-ArAzap = as + a w + aN + as + ap\u00b0 - SprArAzrArAzp\u00b0 - ^ Pp0a Atb = ap041p(2.21)A typical coefficient of the hybrid scheme for the east face is given:aE = MAX (^, .D - ^.2 ^2(2.22)2.5 The SIMPLE AlgorithmSIMPLE stands for the Semi Implicit Method for Pressure Linked Equations, developedby Patankar and Spalding (1972). In brief, this is an algorithm to solve the momentumequations through an iterative procedure. Pressure and velocity correction equations aredeveloped from the continuity and momentum equations. Complete details of the algorithm27are given by Anderson et al [3] and Patankar [26].As an example of the derivation of the SIMPLE algorithm the axial momentum equationis considered:aeue = E abub + b + (Pp -PE)A.,^(2.23)where Ae is the area of the eastern face of a scalar control volume, subscript b is theneighbouring coefficients similar to equation 2.21, and b is the source term. The velocitieshave subscript e since they are evaluated in the velocity control volume on the east face ofa scalar control volume ( see figure 2.4.1 ).To solve the momentum conservation equations the pressure field must be known. If apressure field is guessed ( guess value denoted by P`) and the momentum equations solvedwith this incorrect pressure field, then the resulting velocities ( u* and v* ) will not satisfy thecontinuity equation. The momentum equation based on the guessed pressure field can bewritten as:aeu* = E abu* b + b + (P* p -P* E)Ae .^ (2.24)It is assumed that the velocities and pressures appearing in the discretized axial momentumequation (2.23) are each a sum of a guessed value (0 *), and a correction (4'), where 4) iseither a velocity component or the pressure ( = 4)* + 4' ). If a guessed velocity equation( 2.24) is subtracted from (2.23) then an equation of velocity and pressure correction termsis obtained (2.25).aeuie = E abu i b + ( Pip - PI E) A, .^(2.25)28The summation term, labdb , can be dropped from equation (2.25). The path taken toreach a solution is irrelevant so long as a solution is achieved that satisfies momentum andmass conservation. Neglecting this term in effect only alters the route taken in the iterativeprocess, and has no influence on the final solution. Equation (2.25) can then be manipulatedto give the velocity correction ue. If this is inserted into the original velocity definition,Ue + ue, then the velocity correction equation (2.26) is obtained.Ue =^+ cl,(P1 p - PC)^where de = a.^(2.26)A velocity correction equation for v can be obtained by similar methods.The velocity correction equations can be substituted into the discretized continuity equationshown in equation (2.27).(p p -pep ) rArAzAt^ +[(pu).- (pu),,,jrAr +[(pv)^(pv) sjAz = o 2.27With some rearrangement of terms the pressure correction equation (2.28) is obtained.apPi p = aEPI E + a wPI w + aNP1N + asPI s + bap = aE + aw + aNaE = p ederArb - (Pp\u00b0 - Pp) rArAz PPu*),,, - ( Pu s ) eirArA t+ [(pv*) .9 - ( pv*),21Az(2.28)The b term in this relation is the discretized continuity equation of 2.27 with guessed2 9velocity values.The SIMPLE algorithm iteratively guesses and corrects the velocity and pressure fieldsuntil convergence is satisfied. Initially, a pressure field is guessed, P* , which is then used tocompute the guessed velocity fields with the aid of equations such as 2.24. With the valuesfor guessed velocities, the pressure correction equation 2.28 is solved and the results areadded to the guessed pressures to obtain the pressure field ( P=P*+P' ). The velocity field isthen computed using equations for axial and radial velocity equivalent to (2.26). Once theupdated pressure and velocity fields are computed, relations governing other dependentvariables can then be solved. Finally, the newly computed pressure field is used as a guessfor the next iteration and the entire procedure is repeated.30CHAPTER 3: JET SIMULATION3.1 GeneralIn the previous chapter the governing equations were used in a derivation of fundamentalrelations use in the TEACH code. This chapter reviews the changes made to the TEACHcode and the boundary conditions necessary for in-cylinder engine analysis.The second section of this chapter ( 3.2 ), reviews the basic configuration of the gasinjector in the cylinder and how it is modelled. The assumptions upon which the models arebased are outlined. The third section ( 3.3 ), shows how the control volume grid is made tocontract and expand, simulating the piston motion. Section 3.4 shows how the mass injectedinto the closed cylinder is accounted for. The final section ( 3.5 ), looks at boundaryconditions used in the code.3.2 CONFIGURATION AND ASSUMPTIONSFour figures will be used in this section to show the location of the jet inside the engineand the manner in which the jet is modelled.Figure 3.2.1 gives a schematic drawing of the injector inside the piston-cylinderassembly of the engine. The injector is centrally located in the cylinder head between thetwo exhaust valves, and protrudes approximately 1.6 millimetres into the combustionchamber. The jet is an axially symmetric continuous conical sheet emanating from theinjector tip inside the cylinder. The moving piston is directly underneath the jet.A detailed view of the gas jet is seen in figure 3.2.2. This figure shows a cut view ofthe tip portion of the injector. The axial coordinate is on the centerline of the injector withthe radial coordinate perpendicular to it. The tip is a poppet valve which moves axially31Figure 3.2.1: Conical gas jet in relation to piston and cylindercreating a pathway for the gas. The gap created by the poppet is 0.3 millimetres. The jet isinclined to the radial axis at an angle determined by the injector tip. The dimensions used inthe fixed and moving piston models are the same as those in the actual engine combustionchamber.The combustion chamber can be divided into a series of two-dimensional control volumesor cells in the axial and radial plane. Figure 3.2.3 shows the computation grid with respectto the injector, piston and cylinder. The number of control volumes shown does notcorrespond to the number used in the models. The computation grid is densest near theinjector exit since at this point the gradients of all dependent variables are greatest. A higher32Figure 3.2.2: Details of the conical gas jet.grid density will better be able to resolve the high gradients at the injector tip. The symmetryaxis is at the extreme left edge of the grid at the centerline.A detailed diagram of a portion of the computation grid and its features is given in figure3.2.4. The computation grid includes 29 cells in the axial direction and 35 in the radialdirection, including the boundary and injector cells. Shaded control volumes along the outeredge, are boundary control volumes and do not take part in the analysis. Their purpose is toallow the imposition of boundary conditions which may change from side to side. In theupper left corner of the grid, a group of cells 5 by 9 is segregated by a heavy line. This grouprepresents the injector and has in each cell a value of 0 for all dependent variables. A solidobject inside a computational domain is often represented in this fashion. One control volumeGRIDDED COMBUSTION CHAMBERINJECTORCOMPUTATION GRID1 \u25a0 \u25a0\u25a0^\\ 1..\u25a0V\/ZI\/4111%\/-14IZA il 1 V.\/\/\u25a0 ',4\/\/\/:.r\/Adw\/p\/\/\/41\/\/\/\/4(N^2.125 in. )^1SYMMETRY AXIS33Figure 3.2.3: Computation grid inside combustion chamberin this region is used to represent the injector outlet. The values of all dependent variablesin this control volume are specified; in particular the velocities are set at specific values thatwill vary with time according to an injection velocity profile. Details of the injection velocityprofile will be given in a subsequent section.A simplifying assumption in these models is the axial symmetry. The axial symmetry ofthe jet allows the use of a two-dimensional grid in the axial-radial plane. The two-dimensional grid reduces the computational effort required to analyze the model relative toa three-dimensional grid.Another notable assumption is that of incompressibility. Compressibility effects are presentin high speed flows of gasses such as air and natural gas. Though the flow at the injector3 nodes a.9 nodesshaded regions arei____, 26 nodes boundary cells WA r .\/'^I. . A%^ .\/A24nodes34Computation Grid and Injector tip RegionFigure 3.2.4: Detailed computation gridexit will be sonic or supersonic, the velocity drop to \"low speed\" values occurs within adistance of roughly 10% of the cylinder radius ( as described in section 1.2 ). This means thatif a suitable equivalence is established for the injector flow the bulk of the fluid may betreated as though it were incompressible.One other significant assumption is that the initial velocities and turbulence quantities arezero. In reality this is not exactly true, at any time inside the cylinder there will be motion,a pressure field and some turbulence. However, the initial fields may be small compared tothe fields created by the jet. The effects of the initial fields on the subsequent results maythen be insignificant.353.3 Moving Boundary ModificationThe motion of the piston is simulated in the computation domain through the movementof the lower boundary of the computation grid shown in figure 3.3.1 ( see Watkins 1977) .This face is considered impermeable and is specified to move in a manner consistent withthat of a piston inside a cylinder. The computation grid will compress and expand uniformlyin the axial direction to simulate the piston motion.This is achieved through the transformation of the axial dimension z, into a non-dimensional counterpart , which is specified as:E = Z^ (3.1)ZHThe variable zH, is defined as the instantaneous piston location, or the distance from the loweraxial boundary to the tip of the injector. The advantage of this transformation is that thevalues of the new axial variable 4, defined at each node, do not change with piston position.For instance, if each control volume in the compressing region contracted a uniform amountin the axial direction, then the ratio of the new axial node location z to the axial pistonlocation zH will not change. The non-moving injector tip will have z=0 and 4=0, and thepiston face will have z=zH and 4=1. The original TEACH code with its non-moving gridformulation can then be used for a moving boundary model if 4 is used instead of z.The moving piston model is divided into compressing and non-compressing grid regionsas shown in figure 3.3.1. The two regions are divided by the z = 0 line. The non-compressinggrid region is used to prevent the injector tip dimensions from changing as the piston moves.To implement the transformation, the original governing equation must be transformed inNon-Compressing GridZ = 0=0Compressing Gridradial edge\u25a0boundary movingat piston speed:a z Hataxialedge36Figure 3.3.1 Details of compressing and non-compressing grid regions.terms of the new coordinate system. This is achieved by first realizing that the value of anydependent variable (0), must be the same in both systems.4)(z,r ,t) = 4) 1( ,r,t)^ (3.2)The prime represents the variable in the new coordinate system, t is the time, z and r areaxial and radial coordinates.The derivatives of these variables must also be equivalent; equation 3.2 is differentiated:(3.3)37Observing that the piston location zH is a function of time, the derivative of can beexpanded in the following manner:zH = za(t)^= (4zH) = (z,t)dt = at dz at,kaz^at(3.4)If this expression is introduced into the right side of 3.3 , then it becomes;ora4dr^-`1t) a4dtar^at az^at^atat' dr +^a dz + adv + atldtar^at az^at at^atThe derivative of the new axial variable can be shown to be,at =^at , ak dzH = _ dzHaz^zH ' at^azH dt^ZH ck \u2022Introducing equations 3.6 into 3.5 the derivative identities that will, allow the originalgoverning equation to be transformed are:4 aiv^att) = 1 4'^_ dzif 41 +ar = ar^az^=at^at^zH dt at^at \u2022The derivative identities shown in 3.7 can be substituted into the original governingequation shown in the previous chapter. The resulting general governing equation of alldependent variables with a moving boundary is:(3.5)(3.6)(3.7)38a^ o4))^a(4)p) - a^--(pu4)) 1 a (rPvCat^zH a ds^at^r ar(3.8)= 1 a (^\u20131 a rr \u20144) s .zH at zH at^r ar^3rThe new governing equation contains an additional term compared to its predecessor(equation 2.4). Discretization of this equation can be simplified by defining a new axialvelocity:a = u - E^.at(3.9)The new velocity is simply the speed of the fluid relative to the local grid (iI), which isequivalent to the absolute fluid velocity u, less the local speed of the grid ((az H\/at)). Thenew governing equation can be then written in the same form as the original governingequation (2.4):1 43 (p4)z,i)^1 8--(pu4))^1 a\u2013r \u2014ar(rPv.)zH at^zH 13E(3.10)= 1 a (Ar^+^(rr a-A) sat^r^\u00e4r^\u2022In a similar fashion the continuity equation can be written in the same format.11(pz,i)^a^a\u2014(po \u2013r\u2014ar (rpv) = 0zH at^zH(3.11)The new governing equation is discretized after having been multiplied by rzH. The result- Vz,e)fif r\u2014 (pzm )dtdrdt - A t rpArAt .ae n t+AtW s t(3.12)39will be a series of linearized equations similar to the linear equations obtained through thediscretization of the non-moving boundary governing equation. Terms such as Az will bereplaced by zHA4. The time derivative terms will contain zH at present and past time steps,shown in equations (3.12) and (3.13). For the continuity equation the time derivative termbecomes:Here rP is the cylinder radius at the local node p. The density p is the local node value shownin the next section. The superscripts indicated the present point in time (') and the previouspoint in time (\u00b0).For the governing equation the integrated time derivative is:e n t+At111 r\u2014(pzzA)dtdrelt - (Pizi1+1 Vzm\u00b041?) rpArAt .a A tW S t(3.13)To construct some of the coefficients of the linear equations resulting from thediscretization, the values of the velocity field with respect to the grid (II) will be required.The relative velocity, fi ,should be used for the axial velocity instead of the absolute velocityu. It would then be possible to compute fi directly. To obtain the governing equations forrelative velocity the following substitution is made,u = a + dz^ (3.14)dt40which is placed in the absolute velocity equation ( from eqn. 2.21 with \u2022:0 as u):Apu = EAnun + Ap\u00b0u\u00b0 + S ,^(3.15)to yield,dZH^AApti +^eTt. -^VinundZH+ An&--=') + APIs\u00b0 +^+ Sdt (3.16)p0AtZH\u00b0- ^rP ArA .In its final form the linearized axial momentum equation is then:A pt2 = E A nt2 + A p'14\u00b0 +s, s (EAn. _ Aptp)dzH Apclittidt(3.17)3.4 DENSITY CHANGE DUE TO PISTON MOTION AND GAS INJECTIONThe computational domain is a closed system with mass entering through the injector. Asmass enters and as the grid compresses and expands, the density and pressure of the gasinside the cylinder will change. The density and pressure fields can be corrected on a globallevel through an application of the mass conservation relation. A correction term for densityand pressure is determined which is then applied to all control volumes inside the domain.The density (p) is defined in terms of a guessed (p*) and correction value (p').41(3.18)The pressure (P) can be corrected by use of the correction term for the density.P =P' + p , apap(3.19)The density correction is derived from the summation of the mass conservation equationover each control volume in the computation domain; equation (3.20).(z\u201epp - zif\u00b0pZ)rpArAt - mi = 0a^Atmmwg.(3.20)The present time superscripts (') have been omitted. The term mo is the mass inflow throughthe injector and is considered to be constant over one time step. Inserting the redefineddensity (3.18) into (3.20) and rearranging terms the density correction is obtained.(zHpp - zi;pp)rpArAt -P^La. vol. At (3.21)zipArA\u201e\u201e,L,,i.^AtThis term is used to compute the pressure correction in equation (3.19). The pressuregradient term of equation (3.19) is evaluated assuming isentropic compression and expansion.The pressure gradient can be shown to be:ap = k P .^ (3.22a)aP^PThe variable k is the ratio of specific heats. The pressure and density corrections are addedto all control volumes during each iteration. These corrections drop to zero as the codeP \u2014 RTP (MWas,g)42reaches convergence.The density is adjusted for the presence of natural gas using the ideal gas law shown in(3.22b)equation (3.22b). The term MWe\u201eg is the average molecular weight of the gas at the localnode ( computed from the gas mass fraction), R is the universal gas constant ( 8.314 LkPa\/(K mol)), T is the temperature assuming isentropic compression.3.5 BOUNDARY CONDITIONSThe boundary conditions of this model are the jet inlet velocity and density, and theconditions at the symmetry axis and the solid walls.Figure 3.5.1 shows how the injection speed will initially vary with time. Initially the timestep size is 0.1 milliseconds, growing at a rate of 20% per time step. As the poppet valveopens the jet velocity will start at 0 and will increase rapidly until Mach 1 is attained. Mach1 is the speed of sound in natural gas at the jet orifice. After Mach 1 is reached the jet speedwill remain constant. It is assumed that the poppet valve lift is directly proportional to thespeed of the jet. This is done to provide a \"ramped\" starting velocity since impulse staringmay prevent convergence of the model.The pressure ratio between the upstream gas tank (Pe) and the engine cylinder (P e) willdictate the exit speed and density at the nozzle. The injector passageway behaves as aconverging nozzle, accelerating the gas. If the pressure ratio is above the critical value (above1.8), the jet will be underexpanded. The pressure at the exit plane P e, will then be higher than43Figure 3.5.1: Injection speed as a function of timethe cylinder pressure.As the underexpanded jet exits, its diameter increases as it expands to the engine cylinderpressure. Figure 3.5.2 shows the characteristic jet behaviour due to underexpansion.Experiments on underexpanded jets have been conducted by Ewan and Moodie 1986 [11],who show that the underexpanded jet is equivalent to a correctly expanded one emergingfrom a nozzle of different diameter D om. The equivalent diameter is defined as the diameterof the jet after it has expanded to cylinder pressure and acquired an assumed Mach numberof 1. Ewan and Moodie observed that the adjustment region was only a few jet diameters inlength so they neglected entrainment during the expansion process, and assumed constantmass flow rate in this rapidly expanding jet. Their experiments showed that with a pressure44Figure 3.5.2: Underexpanded jet details.ratio of 5 to 1 the length of the expansion region will only be about 1.3 times the nozzlediameter. With reference to figure 3.5.2, mass conservation implies:PAU. = pegi eqU,4^ (3.22c)Here, subscript e indicates the exit plane and eq indicates the end of the expansion region.Using the ideal gas law and assuming that the temperature at the exit plane is equal to thetemperature at the end of the expansion region and that M eg is 1, it can be shown that:pe lf,^P, (3.22d)p eqU 07 P.^A,45The equivalent diameter (Deq) is then:D =ay (3.23a)The term D 1, is the injector outlet diameter.The equivalent diameter calculated from eq. 3.23a was used as the initial diameter for thejet calculation. Given that the mathematical flow simulation employed for calculating the jetbehaviour implied incompressible flow, it could not exactly represent the effects of Machnumber on density and temperature. Hence it was decided to use a starting value of the jetdensity which was more representative of the downstream conditions than the conditions atthe end of the sudden expansion. With this in mind the starting value of the density was takento be:Pa MWgP R Ta(3.23b)where MWg is the molecular weight of the gas, and P a and T. are the cylinder pressure andtemperature.An approximation was required also for the starting value of the jet velocity. Consistentwith the idea ( shown to be reasonable by Ewan and Moodie ) that the jet Mach number justdownstream of the sudden expansion is nearly sonic, the steady initial value of the jetvelocity was taken to be:k R TaU \u2014 ^eg^MW(3.23c)in which the temperature representative of downstream conditions has been used to compute46. the sonic speed.In addition to exit conditions at the nozzle, conditions at the other boundaries must alsobe specified. There are two boundary conditions of interest on the edge of the computationdomain; solid wall and symmetry axis conditions. Figure 3.5.3 shows typical control volumesacross a wall and a symmetry boundary.Figure 3.5.3: Boundary control volumesThe symmetry axis is the simplest boundary in the model. Since values of all dependentvariables are equal on either side, no fluxes take place. Referring to figure 3.5.3, radialvelocity V. will be zero. The nodal index (I,J) is respectively the axial and radial positionsequentially in the grid. For example site (2,2) is the top left node inside the injector (referring to figure 3.2.4). To eliminate momentum transfer across the boundary, U. is set47equal to Ub. For variables such as gas concentration and turbulence parameters the linearizedequation coefficients for example a s at position (1,2), are set equal to zero.The wall boundary condition is more complex than the symmetry axis. The wall isconsidered impermeable, therefore velocities that penetrate the boundary such as U\u201e referringto figure 3.5.3, are set equal to zero. Velocities that are parallel to the wall such as V c, willbe influenced by the wall. A logarithmic velocity profile for wall boundary layers, describedby Launder and Spalding (1972) [22], is used to introduce wall effects.The logarithmic wall velocity (uw) profile for turbulent flow is defined as:uw = --11n(Ey .^ (3.25)The term u\u201e is the friction velocity and K and E are experimentally determined constantsdefined as 0.4187 and 9.783 respectively. The friction velocity is a function of the wall shearstress 'Ty,:1tw)iUT = (3.26)The non dimensional distance from the wall y+ , is a function of the friction velocity, thelaminar viscosity, and the normal distance from the wall y p .Y^P Ypu,^ (3.27)Wall effects are introduced into velocities such as V c in figure 3.5.3 through the wall shearstresses. The wall shear stress is computed using the logarithmic law of the wall.Coefficients such as aw, at position (2,J) are set to zero. The wall shear stress is then3 3c 4 k 2^U 2P. 11'e -^k = 7KYp Ic 2W(3.29)48^1 ^1pc 4 k 2 KuT w -^it^wln(Ey+)(3.28)introduced into the source term for the velocities such as Vc. The value of cp, is constant andcommonly accepted to be 0.09.Values near the wall for turbulent kinetic energy (k) and dissipation (e) are introduced ina manner similar to the velocities. The turbulence parameters near the walls are defined as:49CHAPTER 4: ROUND FREE JET4.1 GeneralA computer model of an injection of gas into a combustion chamber is of little value ifit is not known whether the phenomena is being simulated reasonably. The validity of a codecan be determined through a comparison of its results to experimental measurements of theflow being numerically simulated. However, it is not always feasible or practical to constructan experiment to obtain data to verify a numerical model.Data are available on the subject of the transient and steady-state incompressible roundfree jets. Even though the round free jet are not exactly the same as the confined jet understudy, data on such jets can be used to check the performance of the code.A numerical model of a transient jet inside a large closed chamber is used to compare theperformance of the TEACH code to known results ( see appendix A ). Briefly, the jet isformed by two control volumes in the wall of the chamber, giving a jet radius of 2 cm. Thewall opposite the jet is 1.2 m from the orifice and the adjacent wall is 0.8 m from thecenterline of the jet. Velocities in some areas of the chamber may indicate laminar flowconditions, invalidating the k-e turbulence model in these regions. Profiles are obtained inregions near the orifice of the jet where the flow is expected to be turbulent.Having the jet orifice formed by two control volumes limits the possible velocity profilesat the exit plane to uniform flow. More control volumes would need to be added to simulatea non-uniform velocity profile. However, this is not always possible since very small controlvolumes may cause numerical stability.5 0It must first be shown (section 4.2) that the walls which enclose the transient jet in thenumerical model are far enough from the origin so that the jet will behave, for a limited time,as though it were in a constant pressure environment. This is crucial if steady-state jet theoryand results are to be used. The next section (4.3), will show the behaviour of the code as thetime step size and grid size is changed. The final section ( 4.4 ), will compare both axial andradial profile data to results produced by the TEACH code. Details of the comparisons andof the grid configuration of these tests are given in appendix A.4.2 STEADY-STATE JETBefore a comparison can be made to steady-state data, it must be shown that the TEACHcode can produce such results. In spite of the fact that solid wall boundary conditions wereimposed to enclose the jet, the walls were made far enough from the jet so that theirinfluence was not immediate. Figure 4.2.1 shows the velocity on the jet axis as a functionof distance from the orifice. A uniform inlet velocity profile of 10 m\/s was imposed at timezero. A time step size of 0.02 seconds was used.It is clear that the jet does reach a steady-state; within 0.8 seconds the velocity profile alongthe jet axis is steady. As expected, the region nearest the jet inlet reaches the steady-statemost rapidly. In time the presence of the wall will induce a recirculating flow pattern andinvalidate the comparison to free jet data.Radial velocity and concentration profiles taken at a distance of 0.435 m and 0.46 mrespectively from the jet orifice are shown at various points in time in figures 4.2.2 and 4.2.3.Both these figures show very much the same behaviour. The profiles for velocity and51Figure 4.2.1: Axial velocity along jet axis at various times;10 m\/s initial jet speedconcentration reach a steady-state very rapidly after the jet starts. After roughly 0.2 secondsthe velocity and concentration fields have stabilized, again establishing the validity of thismodel for comparison to steady-state data.4.3 GRID SIZE AND TIME STEP DEPENDENCEIf the grid size and time-step size are chosen too large, the results of the analysis maycontain considerable error. In a finite-difference discretization it is possible to obtain anestimate of the order of magnitude of the error through truncation terms [3]. Commonly theerror would be found to be of the order of Ax or (Ax) 2, depending on the term, 'x' being anydependent variable. The control volume formulation will yield errors similar to that of theRADIAL CONCENTRATION PROFILES0.' 5 0.'^ 0.^O.^0. 5 0.'^0.105 0.12RADIAL DISTANCE FROM JET AXIS (M)Time\u2022 0.22 secNE 0.18 sec0 0.14 sec^ 0.10 secA 0.06 sec26P20.60.550.50.450.40.350.30.250.20.150.10.05052Figure 4.2.2: Radial concentration profiles at various times;0.46 m from jet orifice; 10 m\/s initial jet speed.finite-difference procedure. Clearly then a reduction of the time-step size grid spacing shouldreduce the error of the results. It is common practice in numerical analysis to run a modelwith various sized grids and time steps to determine the sensitivity of the results.Figures 4.3.1 and 4.3.2 show the effects of a reduction of the time step size on the radialprofiles of steady-state velocity and concentration. The velocity and concentration profilesappear very similar in nature and exhibit the same behaviour with changes in time step size.It is clear that only the largest time step size, 0.02 seconds, produces any deviation in velocityand concentration profiles. Other time step sizes appear to yield similar results as their curvestend to be equivalent. This suggests that the code tends to be insensitive to time step size aslong as the time steps are kept small.RADIAL VELOCITY PROFILES6-5.5 -5-4.5-4-3.5 -a2.521.5 -10.50--0.51 O oxiie 0.b3 0.d46 0.b6 0.075 ate 0.105 0.12RADIAL DISTANCE FROM JET AXIS (in)Time\u2022 0.22 secCIE 0.18 secO 0.14 sec^ 0.10 secA 048 sec53Figure 4.2.3: Radial velocity profiles at various times; 0.46m from jet orifice; 10 m\/s initial speed.Figure 4.3.3 shows steady-state radial concentration profiles with three different griddensities. The densest grids , 34 axially by 30 radially and 31 by 27, have relatively similarprofiles. This suggests that for these grid densities grid size independence has been attained.The concentration profiles of the least dense grid, 29 by 25, appears to deviate somewhatfrom the other two. This suggests that for an analysis of this nature the grid density shouldbe higher than 31 axial nodes by 27 radial nodes, to ensure step size independence.4.4 COMPARISON WITH ANALYTICAL AND EXPERIMENTAL RESULTSThe round free jet has been investigated by a number of researchers in the past who havedefined the jet behaviour both experimentally and analytically. One of the first to look at the54Figure 4.3.1: Radial velocity profiles for various time stepsizes; 0.435 m from jet orifice; 10 m\/s initial jet speed.round free incompressible jet was Tollmien [2]. From his experimental work he was able toconfirm the validity of a simple algebraic expression for the axial velocity profile;U\/U0=0.96\/(ax\/R.) where U is the axial velocity, R. is the initial jet radius, a is a constant (see appendix A ). Based on his analysis it is possible to compare his results to that predictedby the TEACH code.Figure 4.4.1 shows the axial velocity profile of Tollmien's analysis compared to that ofthe TEACH code. It is clear that there exists good agreement between the numerical resultsand that predicted by Tollmien. The shapes of the curves appear to match very closely. Thedifference between the two curves appears uniform over their length, suggesting that it is nota breakdown of the basic theory at fault but rather some assumption about the initialRADIAL CONCENTRATION PROFILES0.80.55TIME STEP SIZE^0.5^ \u2022 0.02^sac^0.46 A 0.013^sec0.4^ 0 0.01^sac0.35 0 0.0087 sec0.3^ X 0.004 mm0.'^a^0.1^0.120.250.20.150.10.050RADIAL DISTANCE (m)V55Figure 4.3.2: Radial concentration profiles at various timestep sizes; 0.46 m from jet orifice; 10 m\/s initial jetspeed; mass fraction in kg\/kg.conditions. To'Innen states that the virtual origin of the jet is 0.0878 m before the x = 0starting point, for a starting radius of 2 cm. The virtual origin is where the jet apparentlystarts from as a point source of infinite velocity but with finite momentum. If the virtualorigin location is now set to 0.125 m, then the curves tend to collapse to one, shown in figure4.4.2. The purpose in applying this new virtual origin location is to demonstrate the similarityin shape of the curves in figure 4.4.1.The sharp change in profile expected at the end of the core region of the jet is not wellproduced in the numerical results. A possible explanation is the tendency for many numericalcodes to overestimate diffusion, called false diffusion [26], which would account for the quick561 .21 .110.90.80.70.60.50.40.30.20.10Figure 4.3.3: Radial concentration profiles at various gridsizes; 0.46 m from jet orifice; 10 m\/s initial jet speed; C mis jet centerline concentration.initial drop off of the jet momentum on the axis as it quickly diffuses radially. As well, usingonly two cells for the jet outlet may lead to lack of resolution in the outlet region.Birch et al [7] have produced similar results for an incompressible jet of methane gas intoair. Figure 4.4.3 shows the axial concentration profile for a methane jet erupting into aircompared to results provided by the TEACH code for a similar situation. An offset of 0.02m is applied to the curve, similar to the previous figure. Again there is a reasonably goodagreement between the shapes of the two curves. The slow change in the profile at the endof the core region encountered in the previous case of air into air is also present in the caseof methane into air, indicating that the problem may be dependent on the numericalAXIAL VELOCITY PROFILE 1 .21 .11M90.80.70.60.50.40.30.20.10D+ TOLLMIENNUMERICAL0.2^0.4^0.6^0.8^1^1.2\u2022AXIAL DISTANCE (m)57Figure 4.4.1: Comparison of numerical and analytical axialvelocity profiles; Ucl is 10 m\/s the initial jet speed; U m isjet centerline velocity.methodology and not specific to the phenomena.4.5 RADIAL PROFILES AND JET PENETRATION DATATollmien has investigated the radial profiles of concentration and velocity fields. He wasable to develop tables of non-dimensionalized variables to describe their profiles at anyreasonable distance from the jet orifice. His results showed that the radial profiles could becollapsed onto one universal curve [2]. Selecting the velocity profile at 0.435 m away fromthe jet orifice in the numerical results, a comparison can be made to Tollmien's data.Schlichting [29] also has made an analysis of the round free turbulent jet and developed ananalytical model of its steady-state structure. Figure 4.5.1 shows a comparison of radial58Figure 4.4.2: Corrected prediction of the axial velocityprofile; U0 is 10 m\/s the initial jet speed; U m is the jetcenterline velocity.velocities between Tollmien , Schlichting and the numerical analysis.The radial distance has been non-dimensionalized by dividing by the radial distance r c,which is where the velocity has dropped to half its centerline velocity U.. The profiles arereasonably similar in nature and deviate only slightly near the jet edges. This would suggestthat there is a slight discrepancy between the numerical and experimental results on thelocation of the jet boundary. However, in general the overall agreement is reasonably good.Tollmien has also investigated the profiles of concentration in turbulent jets. The resultsobtained by Tollmien and those for a numerical simulation are compared in figure 4.5.2.Again the results are plotted in a non-dimensional form. The agreement between the0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.559Figure 4.4.3: Comparison of numerical and experimental axialconcentration profiles for a methane jet; 10 m\/s initial jetspeed; Co is the jet outlet concentration.experimental and numerical analysis in this case is quite good. Very little deviation betweenthe two curves occurs, and in general the profiles are virtually indistinguishable.The concentration profile in a methane jet has been studied by Birch et al. [7]. Anumerical model of the round free transient jet of methane into air has been constructed tocompare results with Birch's data. The results are shown in figure 4.5.3, where the radialcoordinate is non-dimensionalized in this case by division with the axial distance from thejet (0.46 m). The agreement here is not as good as in the previous case. Although the profilesare similar, there is some deviation near the centre of the profiles. It appears as though thenumerical code slightly overestimates the diffusion of the methane in the radial direction.RADIAL VELOCITY PROFILESRESULTS\u2022 TINIAVENX NUMERICAL0 SCHLICTINGDIMENSIONLESS RADIAL DISTANCE r\/rC0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.1.21.110.9a 0.80.70.60.50.40.3020.10-0.1-0.260Figure 4.5.1: Comparison of velocity profiles of Tollmien,Schlichting and the numerical analysis; 10 m\/s initial jetspeed; Um is the jet centerline velocity.This behaviour would be consistent with the high momentum diffusion in the case of theaxial jet. The methane and air jets possess different profiles as a result of the densitydifferences between the two.Transient behaviour in free jets has been studied by Kuo and Bracco [21]. Their interestwas in the penetration times of a transient round jet erupting into a quiescent medium. Theymeasured the time required for the velocity at a given point on the jet axis to reach a certainpercentage of its steady-state velocity. In figure 4.5.4, the non-dimensionalized penetrationcurve for a jet to reach 99.9% of its steady-state velocity is presented. The time and distancecoordinates are non-dimensionalized by the Reynolds number at the orifice and the initial jetCONCENTRATION PROFILESa1.21.110.90.80.70.60.50.40.30.20.10.25^0.5^0.^1^125^1.50DIMENSIONLESS RADIAL DISTANCE r\/reRESULTS\u2022 TOLLMIEN'SCIE NUMERICAL61Figure 4.5.2: Comparison of numerical radial concentrationprofile with Tollmien's data; 0.46 m from jet orifice; 10 m\/sinitial jet speed; Cm is the jet centerline concentration.diameter, details of which are given in appendix A.Clearly, the curves are very similar in behaviour in their rates of penetration. The overallshape of the plots appears to be in good agreement. It is only when the penetration times ofthe two sets of data are compared is there some difference. The TEACH code appears tooverestimate the penetration time of the free jet compared to the results of Kuo and Bracco.The variation at the far ends of the curves are most likely the effects of the wall. Thedifference between the two curves is consistent for the most part meaning that on aqualitative basis the TEACH code is justified in such an analysis.In general, from the results presented it is clear that the TEACH code can predict the axialRADIAL CONCENTRATION PROFILE\u2022 NUMERICALA( BIRCHa^0.1^0.15^02^0.25^0.31.21.110.9a 0.8Li)^0.7L) 0.60.50.40.30.20.10DIMENSIONLESS RADIAL DISTANCE (r\/z)62Figure 4.5.3: Comparison of a numerical model and Birch'sdata of a radial concentration profile of a methane jet; 10m\/s initial jet speed; 0.46 m from jet orifice; Cm is the jetcenterline concentration.and radial concentration and velocity profiles of round free turbulent jets. In some instancessuch as the axial profile and some radial profiles of air into air, the agreement between theexperimental and analytical results to that of the model is very good. The grid sizing and time-step size necessary to achieve grid and time-step size independence appears to be reasonable.Clearly, the comparisons made in this chapter give a good indication of the validity of usingthe TEACH code for the applications discussed in this thesis.X starT star1401301201101009090708050403020100JET PENETRATION+ NUMERICALKUO & BRACCO11 13 15 17 19 21 23 25 2763Figure 4.5.4: Comparison of modelled jet penetration timeswith that of Kuo and Bracco; 10 m\/s initial jet speed.64CHAPTER 5: FIXED PISTON RESULTS5.1 GENERALBefore proceeding to the moving piston model, the fixed piston model is studied toidentify the variables which have the greatest influence on the jet development. The enginecombustion chamber outlined in chapter 3 is modelled with a fixed piston height. Parameterssuch as the injection angle, the pressure ratio between the fuel tank and the combustionchamber, and the distance between the piston face and the cylinder head ( clearance gap ),are varied. For each of these three parameters there are only two values selected; theinjection angle will either be 10 or 30 degrees from the cylinder head, the pressure ratio iseither 5:1 or 2:1, and the clearance gap is either 0.3125 inches or two times this value. Thegap 0.3125 inches is used since this will be the actual clearance in an engine with a flatpiston face, 17 to 1 compression ratio, and a 5 inch piston stroke. A table is shown belowwhich summarizes the various cases.Clearance Gap Jet Angle Pressure RatioCase 1 0.3125\" 10 2: 1Case 2 0.3125\" 30 2: 1Case 3 0.625\" 30 2: 1Case 4 0.625\" 10 2: 1Case 5 0.3125\" 10 5 : 1Case 6 0.3125\" 30 5 : 1Case 7 0.625\" 10 5 : 1Case 8 0.625\" 30 5: 165As an introduction, the development with time of the jet sheet of case 1 is shown in figure5.1.1 in terms of the mass fraction concentration of the natural gas. Each contour representsa line of equal gas concentration in kilograms of gas per kilogram of air\/gas mixture. The jetplume penetrates outward from the injector tip in the top left corner of each plot, along thetop wall of the chamber. By 3.59 milliseconds after the start of injection, the jet hasencountered the far wall and begins to fill the chamber with natural gas. Figures 5.1.2 and5.1.3 give concentration and velocity plots for the 30 degree case, showing its development.The thirty degree jet propagates toward the bottom wall and not the top wall as in theprevious case. Again the jet encounters the far wall and turns to fill the remaining chamber.The velocity fields show zones of recirculation under the injector, which appear to correspondto zones of high gas concentration. Figures such as the ones just shown are used throughoutthe next two chapters to demonstrate the jet character.This chapter is divided into six sections. The second section of this chapter (5.2),compares experimental results with numerical results under similar initial and boundaryconditions. The third section ( 5.3 ) looks at how a change in jet angle influences jet mixingand penetration. The effects of an increase in pressure ratio are described in section 5.4. Ahigher pressure ratio injects a greater amount of fluid momentum into the chamber and itseffects are shown. Section 5.5 shows the development of a region combustible mixture as thejet penetrates into the chamber. The final section (5.6) investigates the effects of the clearancegap on the jet behaviour.5.2 COMPARISON TO EXPERIMENTOne of the best methods by which the performance of a numerical model can be assessedradial distance (m) time = 1.14 ms0Oo0OOOOOcotime --a 2.13 mso.time EN 3.59 ms160.04time nu 5.72 ms0.4o n4^nfut Figure 5.1.1 Concentration field for case 1 showing jet development with time, clearance gap0.3125 inches, pressure ratio 2:1, jet angle 10 deg., contour interval 0.04 (kg\/kg), injector tipin top left corners.6600O010100O00 0cocn0OO11- 1111111^111111111O 0OOO IIIIIIIIITIIIIIIIIIO021000co00time = 3.59 mstime = 5.72 ms0 0 0-P\u20220 00 0 0 0 000cri^co0 0 0 0co^ coradial distance (m) time = 2.13 ms67Figure 5.1.2 Concentration field for case 8 showing jet development with time, clearance gap0.625 inches, pressure ratio 5:1, jet angle 30 deg., contour interval 0.08 (kg\/kg).\u2022 40441,4444. 0 4 44 0 4 11 4* AMAMI. Pt IR \u2022 Ot IP It a aRt ~MR Itkg,' R'RMMIRRIR IR IR01,1 ,1C, I^4MR R a^4.44 4Qtr 44144444.444.4 i p R a a a a.. a 4 4...1 4416 Malan kit 6 6 44444444- 4. ^a a^466* 64416**^^464* 46444666^444 46064~^444 444444461464 V. i * .6+16446444464***4 44 \u2022 at 44 a64+4 446644444 * * i ...4441444MN.144 4. 16 It 116 i 14644 4 4444444***** * * 4. *a)U)O0C-)C\/)C)OQ.)U)0O444 4 .164444444644 #^ ... 4 + P I6\u00b544444441i144444446.4^ . 44 4} t D I444 4, 4.1444, 4 4 4 41,1. ^..444 4 4444444466 4 \u2022^^ . + . .\u2022t I4 \u201e : * P I T:, ,,ii,,, ri , ii. -. . . p 1 T-a .., 7. 7 i I414 4 441444444 4 4 4 4 4 * 4 a111141111111114 4 4 4 4 4 - -4 --. '. za 2\u2022 r11114444 14 a A 4 .,..,--\"- *:%* 4141 ^61 --1 ' ''' ,^\/m- --- --_-_ -_-4)4- --,e --if,.\"r4-----4 -4--0I.4 r ea , v^tiles - ^V\"44**4444 4^4 4-4-4*-64.---kb* 4114661460*- 46-4 4- 4- e- 4- 4- 4 f \u2666 r 46461446444644414.16^\u2022 a r 4 A t I I\u201c1.1.414,14. ^ ... +I.??4441\u00b5444444444t^ at t464 4 466444^....^\u2022 *414+ 41414441 \u2022 4 is414+ 1144444 ^r P4444 4\u00b5444444M4 tat 44 4 4 4 4 1- r---...,rf)Q.)^ 6 4 t t I Ic..) +44 4 4444444+^E0^. . t t Ic:- 7:......,,,\u201e4U14\":^. 4 4.\u201e,TI4414 4144444444 4 4 4 4 4 4^' ' 7 I I4111i 1411111114414 4 4 4 * 4 4 4 4 A '''' -' 7 t I--1 Au ^ii:ii: 7_ .,:1) 41,4444kit i. 44 . -- `,Z * 41:01 4- ,,,- :-!-:-.,----441\u20221144444\u20224 ^* 4 %,,a1 , : .3:1,1:6-\" - 4 - \" a 'a.,,,,,: 4.--4 _+_, 0^,` ),a ),,t--\u25a0 :, : ,..-- :_.\u2014:444 4446446664 \u2022 4. 4. + a 4464 +44444444 ^4\u00b51444464444i1^414 4111444444 4^444 4 4444466-44 .6 1 + 4 4 44 4 * a * '''r 1......-.4...> 17.7 C...)Lit^4,-444 44444444441114^+ 4 4 469time \u2014 2.13 mstime = 1.14 msFigure 5.2.1 Schlieren image compared to case 8, jet angle 30 deg., pressure ratio 5:1,clearance gap 0.625 inches, concentration contours 0.08 (kg\/kg), injector tip in top leftcorners.70Figure 5.2.2 Schlieren image comparison to case 7, jet angle 10 deg., pressure ratio 5:1,clearance gap 0.625 inches, concentration contour interval 0.08 (kg\/kg), injector tip in the topleft corners.71time = 1.14 mstime = 2.13 msoghtFigure 5.2.3 Schlieren image comparison to case 1, jet angle 10 deg., pressure ratio 2:1,clearance gap 0.3125 inches, concentration contour interval 0.08 (kg\/kg), injector tip in topleft corners.time = 1.14 mstime = 2.13 msFigure 5.2.4 Schlieren image compared to case 6, jet angle 30 deg., pressure ratio 5:1,clearance gap 0.3125 inches, concentration contour interval 0.08 (kg\/kg), injector tip in topleft corners.7273is by comparison to experiments. Work on an experimental model of similar shape and initialconditions as the work done in this chapter was performed concurrently by Patric Ouellette[25]. Some of the results he has obtained are presented.Four cases were selected for the comparison, two 10 degree and two 30 degree injectionangle cases. Figures 5.2.1 through 5.2.4 show schlieren photographs of a jet of natural gascorresponding to the cases considered. In each figure there are two schlieren images; thetopmost is at 1 millisecond after the start of injection and the bottommost is at 2milliseconds. The photographs are of the complete chamber while the concentration plots areof half the cylinder with the centerline on the left edge.Figure 5.2.1 shows the experimental result for the 30 degree case, with a larger chamber.The image in the photograph clearly shows the shape and the extent to which the jet haspenetrated. The jet appears roughly bell shaped which is closely parallelled by the numericalresults shown below. The qualitative agreement between the photograph and the numericalmodel suggests that the assumptions made in the model were reasonable.Figure 5.2.2 shows the comparison to the 10 degree, large chamber case. The jet is clearlypenetrating into the chamber along the top wall which was predicted in the numerical model.The jet penetration appears to be well predicted. The 2 ms images both have penetrations onthe order of 4.5 cm. The numerical model predicts a jet held tight against the top wall, whilethe photograph shows a much looser jet which has penetrated axially into the chamber. Itshould be considered that the results of the numerical analysis are averaged values, while thephotograph gives an image that is one instant in time, which may suggest one reason for thedifference.74Figure 5.2.3 shows the comparison to the 10 degree, small chamber case. Thephotographic images are not very well defined in this case, but it is possible to ascertain theedge of the jet. The 1 ms photograph clearly shows the jet clinging to the top wall which isalso seen in the numerical model. In the 2 ms photograph it is again apparent that the actualjet has penetrated axially more and radially less than in the numerical model results. Thisindicates that the 10 degree results may not be as accurate as the 30 degree cases.Figure 5.2.4 shows the comparison to the 30 degree, small chamber case. The image inthe photograph does not offer the detail of previous images. However, it is possible to definethe edge of the jet. What can be seen is a jet that has reached the bottom wall at 1 ms in thefirst photograph, unlike the numerical model. The second photograph suggests that the modeland the experiment give the same jet penetration.In general, the photographs compare well with the numerical results. In most cases thenumerical results were qualitatively similar to the experiment. The resolution was notsufficiently detailed to make a quantitative comparison. However, the results obtained by theexperiment indicate that the numerical code simulates the real flow reasonably well.5.3 INJECTION ANGLEThis section investigates the influence of jet angle on the gas jet inside a fixed pistonchamber to ascertain if it is of significance. Two comparisons are made; a 10 degree and a30 degree jet each with similar pressure ratios and chamber sizes.The character of the jet appears radically altered by the initial injection angle. Theconcentration plots for 2:1 pressure ratio and small clearance gap, shown in figure 5.3.1,demonstrate the magnitude of the change brought about by a 20 degree change in injection75angle. The jet in the 30 degree case starts off at 30 degrees but as it penetrates into thechamber, shifts direction until the jet is nearly vertical. The jet then attaches itself to thebottom wall and proceeds to fill the chamber from this position. The velocity plot , shownin figure 5.3.2, confirms the jet path shown in the previous figure. It is clear that a highdegree of recirculatory flow is developed just under the injector. A zone of high gasconcentration exists under the injector suggesting gas is drawn into the region of recirculation.The 10 degree case behaves in a manner that is in complete contrast to the thirty degreecase. The jet now clings to the top wall as it fills the chamber. Figure 5.3.2, showing thevelocity field for this case gives no indication of the recirculation zone under the injector thatwas apparent in the thirty degree case. In general the magnitudes of the velocities in the 10degree case appear less than in the 30 degree case, suggesting a cause for the difference inthe concentration fields.Figure 5.3.3 shows concentration fields for cases with larger clearance gaps and pressureratios of 5 to 1. It is quite clear that the jet patterns seen in the previous cases are againrepeated in these cases. The thirty degree jet still moves towards the bottom wall and the tendegree jet is still fixed to the top wall, in spite of the change of pressure ratio and clearancegap. The velocity fields for these two cases in figure 5.3.4 show the same pattern as seen incases 1 and 2. Clearly the injection angle has a considerable influence on the jet characterregardless of the clearance gap or pressure ratio.It has been established that the jet is sensitive to injection angles, but the range andcharacter of this sensitivity has not been explored. It has yet to be shown if the jet graduallyswitches between the top and bottom walls or if this change is abrupt.time 2.13 msjet angle 10 deg.jet angle N- 30 deg.time = 3.59 msjet angle = 10 deg.jet angle = 30 deg.Figure 5.3.1 Concentration fields for case 1 and 2 showing the jet angle effect, pressure ratio2:1, clearance gap 0.3125 inches, contour intervals 0.04 (kg\/kg), injector tip in top rightcorners.76time am 2.13 msjet angle 10 deg.77time = 3.59 msVELOCITY SCALE.^100 m\/secjet angle 30 deg.jet angle an 10 deg.VE OCI A E .^100 m\/secVELOCITY SCALE.^100 m\/secU( Tjet angle = 30 deg.Figure 5.3.2 Velocity fields for cases 1 and 2, showing jet angle effects, pressure ratio 2:1,clearance gap 0.3125 inches, injector tip in top left corners.jet angle 10 deg.jet angle = 30 deg.78Figure 5.3.3 Concentration fields for cases 7 and 8 showing jet angle effect, pressure ratio5:1, time 3.59 ms, contour interval 0.04 (kg\/kg), injector tip in top left corners.\u25ba TTh s Vi4 \u2022 \u2022Trimi^414411.1t t+ }44}1444144}4h1UqCDCD1,1 CA\u25a01) 4.)CDPDCr4 CLC'lC)-tNul. Ci)cbofa.,CD 000 4.o.0. t-a 05' (IQFr 500 nCD CD\"P(#3CDCDq1=t.0^'---- ^T-4 1*-4- \u2022+ -1-4\u202244\u00f7-V\u2022er+^+ ::\u2022\u2022\u2022\u2022 \u2022\u25ba\u25a0\u2022\u2022\u2022^+ \u2022 \u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u25a0\u2022 ^+A+A A A ++a424 a a a ++++ ^\u2022 A +++++ A A II 4 A A A A4 ,14\u2022 4 1\u2022\u2022\u2022\u2022\u2022\u2022\u25a0\u2022\u25a0 A A AxAmANA A %%AAA+^AIMNA A^ AAAAAAAAAN4} 4 A 4 AAAAAAAAAAAA.A A A A A A A A A A A AAAA44,f 'A A %%AAA% AA\u2022 4 S A 9. A A \u2022 a A An\u2022%maaaa444,1\u2022 A \u2022 +++++ 1114,4.111f A 4 4***AAA +++++ .4444441+++++ +44+1.4411+1, 444+1.91.1.1.1144 AAR^4 5^++ ^^4 ^++^AA 491 ,9141\u25a0444-44449+44 4444 +++ ^a a 4 r f+ 4 ,4 4++44444+944-44411a \u2022 \u2022 + f \u2022 \u2022 + f \u2022 414+4414444+++ 4 4 4 ARAN^AA^ ++ 4444730,CDjet angle 18.75 deg.jet angle = 19.0 deg.jet angle 19.375 deg.80Figure 5.3.5 Concentration fields showing jet angle sensitivity, pressure ratio 2:1, time 2.13ms, contour interval 0.08 (kg\/kg), injector in top left corners.81Figure 5.3.5 gives a clear indication of the very abrupt nature of the switch. Theconcentration contours for the 19 degree case looks similar to that of the 30 degree plots. The18.75 degree field looks like the 10 degree case. The difference in jet patterns between the19.0 and the 18.75 degree cases is clear evidence of the jet's extreme sensitivity to the jetangle. As a confirmation of these results, an experimental investigation of the 20 degree jetperformed by Patric Ouellette [25] revealed that the switch over occurred abruptly near 20degrees jet angle.In general the path of the gas jets between stationary walls has been shown to be highlydependent on the angle at which they are injected into the chamber. The presence of a wallwill prevent the jet from entraining surrounding air as would be the case in a \"free\" jet. Alow pressure zone develops between the wall and the jet, which is shown in the next section.The low pressure zone may bend the jet towards the adjacent wall. The abrupt change in jetbehaviour is an indication of the relative intensity of the low pressure regions on either sideof the jet. In the next chapter the effect of jet angle with the moving boundary (pistonmotion) is described.5.4 PRESSURE RATIOHigh speed gas jet injection requires a high pressure ratio between the injector and thecombustion chamber. It is possible that this pressure ratio will be great enough to make thejet underexpanded. In this case a greater amount of fluid momentum will be injected into thechamber through a higher inlet density than a lower pressure ratio case.Figure 5.4.1 shows the effect of the pressure ratio on concentration fields for two 10degree jets in the small chamber driven by different pressure ratios. In both cases the gas jet82clings to the top wall and induces a clockwise recirculation inside the entire chamber seenin the velocity plot of figure 5.4.2. The jet shapes and penetration distances are much thesame, with the higher pressure ratio case having slightly greater penetration . The similarityin concentration and velocity fields together suggest that for the 10 degree cases the pressureratio is not a strongly influencing factor.Figure 5.4.3 shows the pressure ratio effect on concentration fields for two 30 degree jetsin the large chamber driven by different pressure ratios. The higher pressure ratio shows asignificantly different concentration and velocity field than the lower pressure ratio case.Here, the jet has a tendency to move out into the chamber before moving towards the bottomwall. The lower pressure ratio jet appears to immediately move underneath the injector andalong the symmetry axis. The differences in the two are stressed by the velocity fields infigure 5.4.4 , which shows the dissimilar paths taken initially.It appears that the extra momentum of the high pressure ratio case causes the jet tocontinue in the path it was initially directed for a longer period of time than the low pressureratio case. This is evident in the velocity fields where the high momentum jet strikes thebottom wall further radially than the low momentum jet. An investigation into the relativepressure fields produced by these cases may explain the variation in behaviour.Figure 5.4.5 shows the pressure fields near the injector tip 2.13 milliseconds afterinjection start for the 5:1 and 2:1 pressure ratios respectively. Clearly, the two pressure fieldsare completely different. As has been mentioned, the jet is a thin sheet. Therefore, it may besensitive to low pressure regions, which was stated in the last section.The low pressure ratio case has one zone of high pressure in the corner opposite the0.04pressure ratio 2:10.\u2018time= 2.13 mspressure ratio 5:1pressure ratio 2:1time 3.59 mspressure ratio 5:1Figure 5.4.1 Concentration fields for cases 1 and 5 showing pressure ratio effects, jet angle10 deg., clearance gap 0.3125 inches, contour intervals 0.04 (kg\/kg), injector tip in the topleft corners.83CITY,\"!\u00a7C^100 m\/sec0111KOM11111 (i (V EL84time = 2.13 mspressure ratio 5:1pressure ratio 2:1time = 3.59 ms\\TEL CITA;^100 m\/sec^pressure ratio 5:1MI I WIN= I I' 1 I pressure ratio 2:1Figure 5.4.2 Velocity fields of cases 1 and 5 showing pressure ratio effects, jet angle 10 deg.,clearance gap 0.3125 inches, injector tip in top left corner.pressure ratio 5:1pressure ratio 2:1Figure 5.4.3 Concentration fields for cases 8 and 3 showing pressure ratio effects, jet angle30 deg., time 2.13 ms, clearance gap 0.625 inches, contour interval 0.08 (kg\/kg), injector tipin the top left corners.85Mm 8 ,48880444.44418^4400S^ PPPPPPPPP 4 4 444,80441\/81.88, If V 8 P r, IP S 4 4 4 4^1144 484848^ WWWWW R \/8 R R 4 4 44- +11444.1* 441-4 Off of .48 st. 4 4li.\u201c % MINNA a a r 4- Er Or 4 + 4. 4 - 1 4 4 4 w 8 4 4a.)C.) W*0 \u2022 ao,66444 WW^A4.4. 444~60 WW.141.444444444 f 4 f 4 4 16 4 f C 4 f 4^TO 44+ 4 44444444.4 \u2022 4 W^ WIM+441414++4+464 1 4 4 4 4 4 8 I t444 4444444^ W 4 f* 4 4 W \u2022 1' Tr^4 14,14444444 4 4 4^ WWWCri4414-11.44444iii*** t I v*. WL.)\u2022 1,444.4,6444.(ii...^+ -4 A r l \/cf)^444,\/,,a\u201eii^\"14\u2022 saW.e \/40C-ICA'.4.\u25aa 187injector tip and one zone of low pressure directly underneath it. The values of pressure aregiven relative to the zero datum. The lowest pressure in this case is approximately -6400 Pa,underneath the injector. Observing the corresponding velocity field of figure 5.4.4, the lowpressure zone under the injector is located in a region of intensely recirculating flow. Thehigh pressure region is associated with an area where the velocity field is rapidly changingdirection or is stagnating, as in the corner of the chamber. The presence of high pressurezones indicate the existence of stagnation points. The presence of low pressure regions tendto indicate the existence of recirculating flow patterns.The high pressure ratio case has three zones of high pressure (stagnation), one underneaththe injector, one in the corner opposite the injector and one located radially one centimetrefrom the corner high pressure region. Two low pressure regions exist; one directly adjacentto the injector tip radially outward, and one beneath it centred in the middle of the highpressure regions. The very low pressure zone opposite the injector ( with the majority of thecontours removed for clarity ), is associated again with a region of recirculatory flowbehaviour as seen in the corresponding velocity plot of figure 5.4.4. It appears as though alow pressure region underneath the injector, that may divert the jet, never develops in thehigh pressure ratio case.It is evident that in the 10 degree jet angle cases the effect of pressure ratio on the jetbehaviour is minimal. Only in the case of large injection angle and clearance gap does thepressure ratio have a significant effect.5.5 COMBUSTIBLE MIXTURE REGIONInformation on the mass fraction of natural gas present in the chamber at any point in time^Pp^p^PPP^P^PPS)^\u00a7og^\u00a74\u00a7^2^t2\/^Ei^2^2 ipressure ratio 5:1 contour interval 250 Papressure ratio 2:1 contour interval 1000 Pa88Figure 5.4.5 Pressure fields for cases 8 and 3 showing pressure ratio effect, jet angle 30 deg.,clearance gap 0.625 inches, injector tip in top left corners.89provides an opportunity to investigate the development of a combustible mixture region. Inthe actual engine combustion will commence soon after a combustible mixture is produced.This will eventually distort the flow field and concentration field to configurations other thanwhat is predicted here. Nevertheless, this section will give an insight into the speed and shapeof a combustible region as it begins to develop. For natural gas the combustible range canarbitrarily be defined as the volume lying between the concentration contours of 0.035 to0.07. These numbers represent the flammability limit at standard temperature and pressure,and correspond to a relative air-fuel ratio of 1.6 and 0.8. The model is run assuming standardtemperature and pressure conditions at all times. The results shown have only a qualitativesignificance since the high temperatures of the actual engine will alter the combustible limits.The higher pressure of the actual engine has little effect on these limits.The 10 degree jet in the large chamber is selected to demonstrate the growth of thecombustible region. Figure 5.5.1 shows the region of combustible mixture at various pointsin time. As is expected, the combustible region moves and expands as the jet does. For mostof the jet, the combustible zone is limited to a band of very small thickness. Only when thejet hits the far wall and turns to fill the cavity does the zone attain appreciable thickness. Itis clear that the jet entrains air as it expands outward into the cavity. The combustible regionremains thin near the jet outlet where considerable entrainment of air is expected. Only whenthe jet has circulated into a region of relatively slow moving fluid does the combustible zoneenlarge.time = 1.14 mstime \u2014 2.13 ms o .070.035time =-- 5.72 ms0 .02,..............^.... ,.......,IFigure 5.5.1 Combustible region development for case 7, pressure ratio 5:1, jet angle 10 deg.,clearance gap 0.625 inches, contours given in mass fraction (kg\/kg), injector tip in top leftcorners.90915.6 CLEARANCE GAPAn investigation of clearance gap effects will be a useful exercise in gaining a morecomprehensive understanding of the jet behaviour. The effect of clearance gap is mostapparent when parameters such as the injection angle and pressure ratio are held fixed andthe clearance gap is allowed to vary.Figures 5.6.1 and 5.6.2 compare the effects of clearance on the concentration fields fortwo jets of 30 degrees angle each of a different sized chamber. Clearly, the concentrationfields of both cases are similar. The jet, once it has emerged from the injector, attaches itselfto the piston face. Both jets appear to have roughly the same penetrations, with the smallerchamber jet having propagated slightly further. In both cases a zone of high gas concentrationexists just below the injector tip. The one noticeable difference in behaviour between thetwo cases is the mixing occurring in the chambers. The smaller chamber is nearly completelyoccupied by gas of some concentration at 3.59 milliseconds, while this is not the case for thelarger chamber. The gas appears to spread upwards, proportionally, at a faster rate in thesmaller chamber, referring specifically to figure 5.6.2. If this behaviour were repeated in thecomparison of 10 degree jets then it can be inferred that the smaller chamber generally yieldsslightly higher mixing rates than the larger.Figures 5.6.3 and 5.6.4 show two cases with a jet angle of 10 degrees and a pressure ratioof 2:1. It is clear that the behaviour of the jet in these cases is similar to that of the previoustwo. The results suggest that the clearance gap has only a slight influence on the overall jetbehaviour. In both sets of cases just compared, the jets do not change significantly withclearance gap. However, the proportionally faster spreading seen previously in the smallerclearance gap = 0.3125 inchesclearance gap 0.625 inchesradial distance (m)Figure 5.6.1 Concentration fields for cases 6 and 8 showing the effect of clearance gap,pressure ratio 5:1, jet angle 30 deg., time 2.13 ms, concentration contours 0.08 (kg\/kg),injector tip in the top left corner.92clearance gap = 0.3125 inchesclearance gap = 0.625 inchesFigure 5.6.2 Concentration fields for cases 6 and 8 showing clearance gap effect, pressureratio 5:1, jet angle 30 deg., time 3.59 ms, concentration contour interval 0.08 (kg\/kg), injectortip in the top left corners.9394chamber is again present in figure 5.6.4.It is clear from the results just shown that the clearance gap has only a small effect on thecomplete jet character, independent of pressure ratio or jet angle.sz pA,\u2014..,.......\u25a0,..clearance gap a- 0.3125 inchesclearance gap = 0.625 inchesFigure 5.6.3 Concentration fields for cases 1 and 4 showing clearance gap effect, pressureratio 2:1, jet angle 10 deg., time 2.13 ms, concentration contour interval 0.08 (kg\/kg), injectoris in the top left corners.95\u2014illiWjRr=------- IT;:rr;- -----\u00b0------J\"\u20140.04.,^ .-----o.04clearance gap 03125 inchesclearance gap 0.625 inchesFigure 5.6.4 Concentration fields for cases 1 and 4 showing clearance gap effect, pressureratio 2:1, jet angle 10 deg., time 3.59 ms, concentration contour interval 0.04 (kg\/kg), injectorin the top left corners.96\u2022\u202249clearance gap 0.3125 inchesVELOCITY SCALE:^100 m\/sec\u2022*----7- EFF_FFFFFF_F^F_^f^f\u20224,44*.IP V1; \" . e10.V4\u2018ss;^\\ ,AN\u2022fl^tV 1.411clearance gap =1 0.625 inches97VELOCITY SCALE. , 100 m\/secI+kr.:-^f_r_f_Efff t i^i^f^f^,._ \u2022^t^i^i i^[tiff!!! 1 1 1 i^I^1d\u2022\u20221^.i...).4._ I.laa^I^1i ; \\I . ,^t 4 \u2018). ),^a \u2022^4-^40,4 \u2022 4- + +^4-^4.'40\u2018')),^4 ,, 4,- 4- +^+^+^4-,^, 4.^4^4.^4-+ + \u20224- I- \u2022\u2022^4-^\u2666*\u2022Iiirc7!..r\u2022kk4,_*-- .,^\\ \\ ,'s\\g, \u2018\u201e, 4- i',^.': y 9 -9^\u2022----\u2022 --\u2022 --\u2022 --\u25a0 \u2014\u2022^--.-....\u201e^--.^-\u2022^-\u2022\u2022r4-F4*^4-4-44.\u2022F4.4-+A-4^4.4-*4.4*4-+a\u2022\u2022444\u2022 \u20224-4-+*1--4-0^4-4-44.4-4-4-0-\u2022\u20140^44*4-+4\u2022^44-\u2022 4\u2022-4+^\u2022\u20224-4-.-4-4-444-04^4-+\u2022+4-4-4\u2022\u2022\u2022-0\u20224\u20224-4\u2022\u2022\u2022\u2666\u2022V4\u2022\u2022\u2022\u20224\u20220,0.I.\u2022\u20224\\ : , ,,24 \\\\; :ie^.,,,^*^4-^4.^4.,\u2018 :,., \\, \\, \\^4 4. 4.^4.1^1 ,,^\u25a0^, .. .\\^) .^\u2022^4-**..144+41-444.4.44.4_,___^,6 .....,4' ---'4\u2014_. \u2014_, \u20144 \u20144 --4 \u20144 ____. __4^\u20149^-4^\u20140^\u20144^\u2014\u2022^-4Figure 5.6.5 Velocity fields for cases 6 and 8 showing clearance gap effect, pressure ratio 5:1,jet angle 30 deg., time 3.59 ms, injector in the top left corner.98CHAPTER 6: MOVING PISTON MODEL6.1 GENERALThe moving piston model investigates the effects of piston motion through a movingboundary. The flat bottom wall of the chamber is made to approach the stationary top wallusing the coordinate transformation of the axial spatial variable outlined in chapter 3. A bowlshape could have been imposed on the piston, but was not done in this analysis. The movingboundary characteristically moves in a manner consistent with the piston of a 71 seriesDetroit Diesel engine. The motion of the piston is obtained through an analysis similar to thatof Heywood [15], given the true engine dimensions. Injection of the gas in the engine occurs,at the earliest, 35 degrees crank angle rotation before the top most piston location ( 35degrees before top dead centre; BTDC ). The injection in the model will be started from thiscrank position with a velocity profile equivalent to the fixed piston model. In the actualEngine Speed Jet Angle Pressure RatioCase 1 1200 RPM 30 degrees 2 : 1Case 2 1200 RPM 30 degrees 5 : 1Case 3 1200 RPM 60 degrees 2 : 1Case 4 1200 RPM 60 degrees 5 : 1Case 5 600 RPM 60 degrees 2 : 1Case 6 600 RPM 30 degrees 2 : 1Case 7 1200 RPM 30 degrees 2 : 1Case 8 1200 RPM 10 degrees 2 : 1Case 9 600 RPM 10 degrees 2 : 1Case 10 1200 RPM 10 degrees 5 : 1engine the injection starting angle and its duration will depend on engine load and speed. The99earliest injection crank angle is selected to give an opportunity to observe all possible pistonmotion effects. Ten cases are considered, tabulated above.Three parameters are selected to observe their effects on the development of the jet;engine speed , jet angle and pressure. Case 7 is similar to case 1, only the initial conditionsof the air and gas in the cylinder are consistent with that of the actual engine at 35 deg.BTDC. This comparison is made, in section 6.2, to demonstrate the effect of variation in gastemperature and pressure. Section 6.3 focuses on the effect of pressure ratio on the jetbehaviour, and will compare results to the fixed boundary. The effect of injection angle onthe jet behaviour is then investigated in section 6.4. In section 6.5 the engine speed isconsidered to study its influence on the jet. What is also shown is that the results of themoving piston model are fundamentally different than the fixed piston analysis through acomparison of both models with similar conditions. In the final section ( 6.6 ) thedevelopment of a combustible mixture region under the influence of a moving piston isconsidered.6.2 EFFECT OF CYLINDER PRESSURE AND TEMPERATURE LEVELThe compression ratio of the engine is approximately 17:1, giving a high temperature andpressure of the host air when the jet commences. In all cases the jet commences at 35 degreesBTDC. At this point the compression ratio is at 6 to 1, giving a host pressure and temperatureof 1279 kPa and 590 K. Ideally, the initial temperature and pressure of the moving pistonmodel should be consistent with the engine. However, to make a valid comparison to thefixed piston model the initial conditions must be similar. Therefore, it would be expected thatthe same standard temperature and pressure used in the fixed piston case will be used in the100moving piston model. To justify this approach it is necessary to show that the dynamics ofthe phenomena do not depend appreciably on the absolute values of the pressures anddensities of the air and gas but on their ratios.Figure 6.2.1 shows the concentration fields of a 30 degree jet with both standardtemperature and pressure and engine initial conditions at 6.8 and 16.3 degrees BTDC. Theresults appear almost identical, which suggests that the values of the pressure and densityfields are not the controlling factors of the jet dynamics. In fact it appears that the ratio ofthe densities of the injected gas and the host gas is the important factor. The small increasein the laminar viscosity of the fluids with the increased temperature (proportional to T 314) ofthe actual engine configuration appears to be insignificant. This seems logical when it isrealized that for most of the flow field the turbulent viscosity may be much larger than thelaminar viscosity. It can then be concluded that for velocity and concentration fields it matterslittle if the actual engine conditions or that of standard temperature and pressure are used asinitial conditions.6.3 PRESSURE RATIO EFFECTSAs in the fixed piston model, the influence of pressure ratio on the jet behaviour isinvestigated. The higher pressure ratios are expected to impart a higher momentum to thejet, which coupled with the effects of the piston motion may yield results that aresignificantly different than the fixed boundary model. Previously it was shown that thepressure ratio had only a very limited effect on the jet behaviour. Only in the case of a 30degree jet with enlarged chamber was there a significant effect.Two values of pressure ratio between the fuel tanks and the combustion chamber are used.C.A. Ins 16.3 deg. BTDCEngine initial conditionsSTP initial conditionsC.A. = 6.8 deg. BTDCEngine initial conditionsSTP initial conditionsFigure 6.2.1 Concentration fields for cases 7 and 1 comparing actual engine and STP initialconditions, pressure ratio 2:1, jet angle 30 deg., engine speed 1200 RPM, contour intervals0.08 (kg\/kg), injector tip in top left corners.101102The 2 to 1 ratio is considered since it is near this value that the flow through the injector willbecome choked ( Mach 1 ). The higher pressure ratio will be choked as well, but will alsobe underexpanded. In the engine, the pressure ratio will change as the cylinder pressureincreases with compression. A constant pressure ratio is used in the moving piston model.The grid spacing would need to be adjusted at every time step otherwise to account for thepressure ratio change, based on the effective diameterarguments.Figures 6.3.1 and 6.3.2 show the concentration and velocity field for a 30 degree jet withpressure ratios of 2 and 5 to 1. Clearly the difference between the two cases is considerable.The higher pressure ratio case has a velocity and concentration field that attaches itself to thetop wall and moves along this boundary. At no time does this jet attach itself to the lowerwall as in the fixed piston model. The lower pressure ratio jet appears to initially movetowards the top wall and then as top dead centre is approached, switches to the bottom wall.It is evident from these results that the pressure ratio in conjunction with the effects ofpiston motion can have a significant effect on the gas propagation in the chamber. In thefixed piston model the 30 degree jet always moved towards the bottom wall with only aminor change resulting from an increase in pressure ratio. There, the higher pressure ratiolarge chamber case had the 30 degree jet hitting the lower wall further radially than the lowerpressure ratio case. In the moving piston model the jets behave quite differently in severalways. First, the jet does not immediately attach itself to the bottom wall. The lower pressureratio jet eventually attaches itself to the lower wall, but the higher pressure ratio case neverdoes. Secondly, the jets develop in a completely different manner. The high pressure ratio jetis firm against the top wall as it entrains air to fill the far end of the chamber. The lowC.A. =. 16.3 deg. BTDCPres. Ratio am 5:1Pres. Ratio ma 2:1C.A.= 3.2 deg. ATDCPres. Ratio = 5:1Pres. Ratio in 2:1103Figure 6.3.1 Concentration fields for cases 1 and 2 showing pressure ratio effects, jet angle30 deg., engine speed 1200 RPM, contour interval 0.08 (kg\/kg),injector tip in top left corners..4, 0440=44\u20224164164\/66, \u2022 66 \u2022 \u2022 \u2022 4 44414011114144164444 \u2022 4 4 4 4^+41044141114444444444 ^\u2022 4 \u2022 + 44H84L4.4M?f4,444 4 4 4 \u2022 44444414144414444444 66 +^+ 4444444 4 4 4\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u25a0\u2022\u2022\u2022\u2022\u2022\u2022\u2022 I. \u2022 a \u2022 a \u2022 \u202244\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 a A 4 \u2022 44444441 11114411440\u20221444 4 446644441111111144*44144 i 61 t M 66 444444\u202211MP\/ 4444*48. ^\u20224444 *40 44.\"\"\"\" 11 \u2022 *1--IO_JLLJ4444-^44- 4-^Iv^4Z4 4 4 t4rounItft.44.4\u00b1\u00b1 t_t 4 \u2022 *\u2022\u2022Ad 4 4 144 4 416 4 4 4If Id 8 t 4ark V 6 II *41- le 16 Vrrt t 16 stIr^1141t48 8\u2022\u20221tlill1411114844444.4 \u2022 4 4 4CDn INk.,....\u201e,,,3434 *o 111 INkuirve******* 4 ainIll lillharterwa *v.* \u2022 *'If^4444 4 + 444 4 4 *\u20184^444 4 \u2022 \u2022!Ir .44.I I '^161648 4 4 44^8164 4 418\u20228844 4tan 16446 4\".^84 4^ 16. 4'4 4 8 4 44 6-4 4 8 444.4 4 444 4 11 44444 4+4 4 4 li611444 t\u2022 4 4-.426U\u2022\u2022I nI0I I 1 1 ^16 a \u2022 \u2022 16 \u2022 16 4 + 4O III1 11140111\u202214.41616 Wu 4 16 4 4 4 4III\u2018M41010441644 4 416 4 4 4111.19444414\u2022\u2022414:14 4 4 4 st 4 4 411,, :,,,,..,....,k^ar16416 16 It % 4 * 4,.,161616 4 * la 4 46 4 44481414116 4 8 4 4\u25a0i ,,.:17tick 8 8 8 16 4 4 4tlahr 4 4 1681616 4 *11\u2022466 \u2022 66 16 16 4 446446- 4 4 4 4 16 16 4It* 6-k \u2022 4 4 8 4 4444 4 4 4 k 16 16 4444 4 8 4 4 8 44444 4 4 4 4 8 44- 4 \u2022 4 4 \u2022 4 44 4 4 4- 4 4 \u2022 484-4 4 \u2022 \u20224. 4- 44 4_4_14 46_4_, 4844U105pressure ratio jet gradually attaches itself to the bottom wall, initially filling the area nearestthe injector tip. Finally, the higher pressure ratio adds stability to the jet behaviour. The highpressure ratio jet does not change its path away from the top wall. The lower pressure ratiojet switches movement from the top wall to the bottom wall as the piston approaches.The pressure ratio effects mentioned have only become apparent in the presence of themoving piston. Other cases may not show the same variation in behaviour from pressure ratioeffects as is seen in these cases.Figure 6.3.3 shows concentration plots for a 10 degree jet, at 2 and 5 to 1 pressure ratio.In this instance the concentration fields appear to have the same general structure,independent of pressure ratio. The high pressure ratio jet has a slightly greater penetrationdistance than does the lower pressure ratio case. The similarity in the concentration profilessuggests that the pressure ratio effects seen in the previous case may be influenced by theinjection angle. The change in jet behaviour with pressure ratio seen previously is not presentin the 10 degree jet.In general it is quite clear that the combination of pressure ratio and moving boundaryeffects have a profound influence on the jet behaviour, except at 10 degrees injection angle.At 30 degrees angle the higher pressure ratio cases tend to have the jets cling to the top wallat any point in time while the lower pressure cases have the jet near the bottom wall.6.4 INJECTION ANGLE EFFECTSAs has already been shown in the previous section, injection angle is a parameter that caninfluence the development of the gas jet. In the previous chapter it was shown that the pistonmotion had a strong influence on the effects of pressure ratio changes. The purpose of thisC.A. as 16.3 deg. BTDCPres. Ratio \u2022N 5:10.04Pres. Ratio \u2022\u2022 2:10.04 ^0.04C.A. 32 deg. ATDCPres. Ratio - 5:1Pres. Ratio 2:1.04 0.04Figure 6.3.3 Concentration fields for cases 8 and 10 showing pressure ratio effects, jet angle10 deg., engine speed 1200 RPM, contour interval 0.08 (kg\/kg), injector tip in top leftcorners.106107section is to determine if a change in the injection angle is also influenced by the movingpiston.Figures 6.4.1 and 6.4.2 show concentration and velocity fields for jets of 10, 30 and 60degrees angle. Similar to the fixed piston model, the moving piston model also exhibitsvariability in jet behaviour with changes in injection angle. In these cases however thechanges due to jet angle are less abrupt than the fixed piston model. The jet behaviour is lesswell defined as indicated by the 30 degree plot shown in figure 6.4.1. In this case the jetspends a great deal of time between the two boundaries not approaching either wall until wellafter injection has started . The 60 degree jet firmly adheres to the bottom wall while the 10degree injection stays on the top wall. It is worth noting that in the fixed piston model, the30 degree injection always moved immediately towards the bottom wall. In the movingboundary model this is true of the 60 degree case but not necessarily in the 30 degreeinjection.The effects of the piston motion on the injection angle characteristics of the jet are clear.In contrast to the fixed piston model the immediate attachment of the jet to either the top wallor the bottom wall is not seen in the moving piston case. The 30 degree jet, shown in aprevious section, initially moves towards the top wall and switches to the bottom wall wellafter injection has started. It appears as if the moving piston has some moderating influenceon the effects of jet angle. The moving piston effect manifests itself as a net migration offluid towards the top wall. This obviously causes the jet to be less sensitive to injection angleas compared to the fixed piston case. One final parameter that must be explored in contextwith the moving piston model is the rate of piston movement on the jet.\u25a0.......-....0.040.04Injection Angle - 10 deg.Injection Angle = 30 deg.Injection Angle = 60 deg.Figure 6.4.1 Concentration fields for cases 8,1,and 3 showing injection angle effects, pressureratio 2:1, engine speed 1200 RPM, crank angle 6.8 deg. BTDC, contour interval 0.08 (kg\/kg),injector tip in top left corners.108VEL SCALE (100 m\/sec) =CITrOtarR R 11 It R R IL 11*^ PC R R R R IL 4.*ff.** \u2022 4.4, 4.^# \u20221.844014-44.4- \u2022 4. 4- 4. 4- 4- 4- +#^* 4.^\u2022\u2022 4-^4-^\u2022^\u2666 \u2022 4 4-^\u2022\u2022 \u2666 \u2022^4-^4.\u2022 4-^4-^+r r\u2022+ 4-+ +e4- 4 4.4- \u2022 + ( 1Injection Angle = 10 deg.109Injection Angle = 30 deg.VELOCIF1 SCALE (100 misec) =Injection Angle = 60 deg.VELOCIF1 SCALE (100 m\/ seal =rr 1r rrr^Y4,\" r-41, k' ^ 4,^*tr- *r L114_4,--ft --ft --ft --ft --t\u2022 --4 -- -P^\u2014- 4 Figure 6.4.2 Velocity fields for cases 8,1, and 3 showing jet angle effects, pressure ratio 2:1,engine speed 1200 RPM, crank angle 6.8 deg. BTDC, injector tip in top left corners.1106.5 ENGINE SPEEDAs has been demonstrated in the previous sections, the piston movement has aconsiderable influence on the jet dynamics in conjunction with jet angle and pressure ratio.This section looks at the effects of the rate of piston motion alone by comparing cases thatdiffer in their engine speeds only. The rate of piston motion will determine the size of someof the source terms in the governing equation for the axial momentum. It is the effect ofthese terms that will be considered in this section.Figures 6.5.1 and 6.5.2 shows the concentration and velocity fields for a 30 degree jetwith a piston moving at 0, 600 and 1200 revolutions per minute (RPM). Figures 6.5.3 and6.5.4 show similar plots for a 10 degree jet. The figures appear noticeably different in nature.Comparing the moving boundary plots with the fixed boundary plots of both figures, it isclear that the motion of the piston has a substantial effect on jet development.Several points are clear about the effect of the piston speed. First, the behaviour of the 30degree jet changes with piston speed. The plots for each speed appear different in figure6.5.1. Secondly, the jet propagation seems inhibited by the motion of the piston. The fixedpiston models (0 RPM) have jets that have penetrated into almost all of their chamberscompared to the moving piston models. Finally, the 10 degree jet appears sensitive to themoving piston, but not its speed. The difference in the 600 RPM and 1200 RPM plots of the10 degree jet are insignificant. The fixed piston model shows substantial jet propagation overthe two moving piston cases.One possible explanation for the inhibited propagation of the gas jets may be the differentamounts of compression between air and natural gas under a similar change in pressure. It40.28 a28,,,o\u00b0'^......\/ -Fr's\".' ----\u25a0 \u2014___0 RPM111600 RPM1200 RPMFigure 6.5.1 Concentration fields for cases 2 (fixed boundary), 6 and 1 showing the effectsof engine speed, pressure ratio 2:1, jet angle 30 deg., time 5.3 ms after injection start, contourintervals 0.08 (kg\/kg), injector tip in top left corners.VELOCITY SCALE:^100 m\/sec4 -4^1H 1 I 1_ m \u2022 f^ Ve^I^I^I^r^4 ...\u2022 S^!^4^4^et^ee^f a-^\u2022 et * \u2022 kJa^ 4 \u2022 et et f^4- el. \u2022 * * \u20224^\u2022 :::.. 0 2.^\u00b0 ^r^it^\u2022^r e-^r^\u2022 +^to1 +^4,' * \u2022 \u2022^\u2022 ,,' ---\u2022 \u20144 JO ja^II^4^\u2022^..-\u2022^\u2022^A^\u2022^\u2022^etet^ff^4f^\u2022 \u2022\u2022^k\u2022\u2022^Itkg^*ai-\u2022 A^x^\u2022^Or^tx^f 4- . 4 *k 114.^1.\u2022\u2022 \u2022\u2022^,T, ..ekelf- ,i\u2014 kl 4.'1,4 ! ..,.. '\u00b0 -* \u20140 4,^4^\u2022^se^sr \u2022^4. 4-^4^\u2022^r^rpr-ttvtorrr\u2018r \u2022,,, .... .r.,,4 \\,,, y, ,, , ,... ---. --\u20221,\u2022\u2022\u2022\u2022\u2022\u2022\u20224444.4.4-4_4--4\u2014 4-- 4- 4, 4 --\u2022 -\u2022 -10^4^4^4^V^4^4^4^4 41120 RPM600 RPMVELOCIT r SCALE Ho rn\/secltr1200 RPMv'ELOCI^SCALE (100 misec) =Figure 6.5.2 Velocity fields for cases 2 (fixed boundary), 6 and 1 showing the effects ofengine speed, pressure ratio 2:1, jet angle 30 deg., time 5.3 ms after injection start, injectortip in top left corners.^....\u2022\u25a0\u2022\u25a0\u2022.^m..16 ...^ 0.04...0.1812^ 0.120.040 RPM600 RPM1200 RPMFigure 6.5.3 Concentration fields for cases 1 (fixed boundary), 9 and 8 showing the effectsof engine speed, pressure ratio 2:1, jet angle 10 deg., time 5.3 ms after injection start, contourinterval 0.04 (kg\/kg), injector tip in top left corners.113VE OCI^A E\u2022^100 m\/sec\u2022 71c..((rr^ \u2666(1 1 14 1 1VELl I^SC LE (100 m\/sec) =Allir 'IgiriririCir01. R It R It R OLt R Or ot Ot it St *14441. KKKKKKKK1`4.114.00ra la ft Cleat Pr g F It 0 \u2022=4.4.44t K^ K^II II F\u2022 \u2022IL IL f\u202241*^ F 0. \u2022 \u2022 \u2022 \u2022ItRIE00\u2022\u2022\u2022\u2022\u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022\u2022\u2022 \u2022 \u2022\u2022 \u2022\u2022 \u20224-4-VEL I CIT , SC LE (100 m\/sed =^R . i^al=^4'404* 4-4-4-4-4-4- 44-4f+4.1.*^ 4-4-4-4-4-4-4-4. 4.44:100.0\u2022\u2022\u2022\u2022\u20221.4. \u2022 \u2022 11. \u2022 \u2022 \u2022 \u2022 \u2022 \u2022^ \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022\u2022\u2022\u20224-0-^\u2022^4.^0-^*-\u2022 \u2022 \u2022 \u2022 \u2022\u2022 \u2022 \u2022 \u2022 \u2022\u2022 \u2022 \u2022 4-^\u2022\u20224-0 RPM600 RPM1200 RPMFigure 6.5.4 Velocity fields for cases 1 (fixed boundary), 9 and 8 showing the effects ofengine speed, pressure ratio 2:1, jet angle 10 deg., time 5.3 ms after injection start, injectortip in top left corners.114115can be shown that the amount by which a region of gas compresses under a specific changein pressure is proportional to its molecular weight. The relation shown (6.5.1), based onisentropic behaviour, indicates that the ratio of the change of specific volume (u), isproportional to the molecular weight (M) of each species. Since air has a higher molecularweight (28.9 g\/mol) than natural gas (17.0 g\/mol),81) CH4 _ Y AIR C (6.5.1)8 V AIR ( Y CH44) -^MAIR)(1.4)(1-3^MCH4the change in specific volume of the gas will be greater than that for air. The natural gas willtherefore be inhibited from expanding outward as the pressure in the chambers increase. Thiseffect is clearly seen in figure 6.5.3 where the moving piston cases have not propagatednearly as far as the stationary piston case.In general it is clear that the piston motion has a significant effect on jet development,especially at higher injection angles.6.6 COMBUSTIBLE MIXTURE REGIONIn the fixed piston model the development of a combustible mixture region was computedfor the 30 degree jet with a pressure of 2 to 1 and an engine speed of 1200 RPM. The resultsare shown in figure 6.6.1. Again it must be stated that combustion in an engine environmentwould distort the images presented in this section. These plots are intended to give anindication of the development of a combustible zone preceding combustion. The combustiblelimits are taken to be between 0.07 and 0.035 kg\/kg mass fraction, which corresponds to arelative air\/fuel ratio range of 0.8 to 1.6 respectively. These limits will change with highertemperatures in a real engine.116The combustible region development shown in figure 6.6.1 shows several details. First,the combustible zone shows the same inhibited growth as the jet in the previous section. Thissuggests that the combustible mixture zone exists imbedded in the larger jet concentrationfield. Secondly, the combustible zone is always a thin region in the moving piston model. Theprevious chapter showed that in the fixed piston model the combustible region grewconsiderable after the jet had developed. The compressible effects mentioned in the lastsection may inhibit the expansion of the combustible zone.C.A. = 23.1 deg. BTDCC.A. = 16.3 deg. BTDCC.A. = 6.8 deg. BTDC0.07C.A. - 3.2 deg. ATDC0\u20221Ais ..\u201e._117Figure 6.6.1 Combustible region development for case 1, pressure ratio 2:1, jet angle 30 deg.,engine speed 1200 RPM, injector tip in top left corners.118CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONSThe purpose of this study was to investigate the behaviour of a transient injection ofnatural gas into a simulated combustion chamber. The effects of a number of differentparameters including injection angle, engine speed and storage-tank-to-chamber pressure ratiowere investigated. A numerical model based on the conservation equations of mass,momentum and species concentration was employed to facilitate this investigation.The TEACH code developed by A.D. Gosman of Imperial College, London was used tomodel the injection of gas into a fixed piston and moving piston combustion chamber. Thenumerical code was found to be capable of predicting axial and radial velocity andconcentration profiles with acceptable accuracy.The conclusions drawn from the investigation are:1). The concentration contours calculated from the fixed piston numerical model are inapproximate concordance with the photographed shape and penetration of the jet at varioustimes. The 30 degree high pressure ratio jet showed a close correlation in jet shape andpenetration to the experimental photograph.2). The jet angle is the most influential parameter in determining the jet shape. The jet of thefixed piston model is highly sensitive to changes in jet angle. The jet switchesattachment from the bottom wall to the top wall near 19 degrees angle. Low pressurezones develop adjacent to the jet that can direct it from the bottom to the top wall. Themoving piston model has less sensitivity to jet angle.3). A higher pressure ratio has significant effects only at high jet angle and in a movingpiston chamber. The jet of the fixed piston model showed a small variation with pressure119ratio increase in the 30 degree jet and no change with the 10 degree jet. The movingpiston model jet showed significant change at 30 degrees jet angle.4). There is a significant difference in the jet shape and penetration between the moving andfixed piston models. Both the 10 and 30 degree jets showed significant change whenpiston motion was introduced. The engine speed has a significant effect on the jet shape,only at higher jet angles. The 30 degree jet of the moving piston model showedappreciable change over all engine speeds, while the 10 degree jet showed little variation.5). The size of the fixed piston chamber had no significant effect on the shape of the jet orits tendency to cling to the top or the bottom wall, for the clearance gaps considered.The recommendations for further research are:1). A proven compressibility model should be implemented in the code to account forcompressibility effects, allowing piston bowl shapes to be studied.2). Initial air motion in the cylinder created by the scavenging process should be studiedthrough the addition of air inlet ports and exhaust valves. The air motion inside thecylinder was assumed to be non existent at the start of computation.3). It is recommended that similar calculations be performed on a code with superioraccuracy than the. TEACH code, such as KIVA, to validate the results of this study. Thepresent results obtained are only expected to be qualitatively accurate.120REFERENCES[1]. Abramovich, S. and Solan, A., \"The Initial Development of a Submerged Laminar RoundJet\", J. Fluid Mech., Vol. 59, part iv, pp. 791-800, 1973.[2]. Abramovich, G.N., The Theory of Turbulent Jets. MIT press, Cambridge, mass, 1973.[3]. Anderson, D.A., Tannehill, J.C., Pletcher, R.H., Computational Fluid Mechanics and HeatTransfer, Hemisphere Pub. co., New York, 1984.[4]. Bassoli, C. et al., \"Two Dimensional Combustion Chamber Analysis of a Direct InjectionDiesel\". S.A.E. Transactions paper no. 840228, 1984.[5]. Beck, N.J., \"Electronic Fuel Injection for Dual Fuel Methane\", S.A.E. Transactions paperno. 891652, 1989.[6]. Birch, A.D. et al., \"The Structure and Concentration Decay of High Pressure Jets ofNatural Gas\", Comb. Sci. and Tech. vol 36, pp. 249-261, 1984.[7]. Birch, A.D., et al., \"The Turbulent Concentration Field of a Methane Jet\", J. Fluid Mech.,vol. 88, pp. 431-449, 1978.[8]. Butler, T.D., et al., \"Multidimensional Numerical Simulation of Reactive Flow in InternalCombustion Engines\", Prog. Energy Comb. Sci., vol. 7, pp. 293-315, 1981.[9]. Canadian Resourcecon Ltd., \"Market Assessment: Intensifier Injector for DDECControlled Engines\" unpublished report, Sept. 1989.[10]. Diwakar, R., \"Assessment of the Ability of a Multidimensional Computer Code toModel Combustion in a Homogeneous Charge Engine\", S.A.E. Transactions paper no.840230, 1984.[11]. Ewan, B.C.R., and Moodie, K., \"Structure and Velocity Measurements inUnderexpanded Jets\", Comb. Sci. and Tech., vol. 45, pp.275-288, 1986.[12]. Gaillard, P., \"Multidimensional Numerical Study of the Mixing of Unsteady GaseousFuel Jets with Air in Free and Confined Situations\", S.A.E. Transactions paper no.840225, 1984.[13]. Gosman, A.D., and Tsui, Y.Y., and Watkins, A.P., \"Calculation of the ThreeDimensional Air Motion in Model Engines\", S.A.E. Transactions paper no. 1989.121[14]. Gosman, A.D., Johns R.J.R., Watkins A.P.,\"Development of Prediction Methods forin_cylinder Processes in Reciprocating Engines\" in Combustion Modelling in Reciprocating Engines,Plenum Press, New York, 1980.[15]. Heywood, J.B., Internal Combustion Engine Fundamentals, McGraw Hill co., NewYork, 1988.[16]. Henriot, S., and LeCoz, J.F., and Pinchon, P., \"Three Dimensional Modelling of Flowand Turbulence in a Four Valve Spark Ignition Engine - Comparison with LDVMeasurements\", S.A.E. Transactions paper no. 890843, 1989.[17]. Hiroyasu, H., and Arai, M., \"Structure of Fuel Sprays in Diesel Engines\", S.A.E.Transactions paper no 900475, 1990.[18]. Hutchinson, B.R., and Galpin, P.F., and Raithby, G.D., \"Application of AdditiveCorrection Multigrid to the Coupled Fluid Flow Equations\", Num. Heat Transfer, pp 133-147, 1988.[19]. Kollmann, W., Computational Fluid Dynamics, Hemisphere Pub. co., Washington D.C.,1979.[20]. Kundu, P.K., Fluid Mechanics, Academic Press Inc., San Diego Ca., 1990.[21]. Kuo, T.W., and Bracco, F.V., \"On the Scaling of Transient Laminar, and TurbulentSpray Jets\", S.A.E. Transactions paper no. 820038, 1982.[22]. Launder, B.E. and Spalding, D.B., Mathematical Models of Turbulence, Academic Pressco, London, 1972.[23]. Markatos, N.C., Computer Simulation of Mass Transfer and Combustion in Reciprocating Engines, Hemisphere Pub. co., New York, 1989.[24]. Needham, J.R., et al., \"Technology for 1994\", S.A.E. Transactions paper no. 891949,1989.[25]. Ouellette, P., \"High Pressure Injection of Natural Gas in Diesel Engines\", M.A.Sc.Thesis, Univ. of British Columbia, 1992.[26]. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere Pub. co., NewYork, 1980.[27]. Reynolds W.C., \"Modelling of Fluid Motions in engines - an Introductory Overview\"in Combustion Modelling in Reciprocating Engines,Plenum^Press, New York,1980.122[28]. Ricou, F.P., and Spalding, D.B., \"Measurements of Entrainment by AxisymetricalTurbulent Jets\", J. Fluid Mech., vol. 11, pp. 21-32, 1961.[29]. Schlichting, H., Boundary Layer Theory 7th. ed., McGraw Hill co., New York, 1979.[30]. Tahry, S.H., \"k-c Equation for Compressible Reciprocating Engine Flows\", J. Energy,vol. 7, pp. 345-353, 1983.[31]. Wakenell, J.F., et al., \"High Pressure Late Cycle Direct Injection of Natural Gas in aRail Medium Speed Diesel Engine\", S.A.E. Transactions paper no. 872041, 1987.[32]. Watkins, A.P., \"Flow and Heat Transfer in Piston Cylinder Assemblies\", Ph.D. Thesis,Univ. of London, London, U.K., 1977.[33]. Weaver, C., \"Cost Effectiveness of Alternative Fuels and Conservational Technologiesfor Reducing Transit Bus Emissions in Santiago Chile\", S.A.E. Transactions paper no.891100, 1989.[34]. Weaver, C., \"Natural Gas Vehicles - A Review of the State of the Art\", S.A.E.Technical Report no. 892133, 1989.COMPUTATION GRIDnot to scalejet orifice, 0.02 m radius25nodes0.8 m\/ tjetorifice29 nodes1.2 m ^123APPENDIX A: GAS JET ANALYSISA schematic drawing of the computational grid used to compare the numerical results toanalytical and experimental results is given below. Two computation nodes at the bottom leftcorner have their axial velocities fixed to simulate the jet orifice. The jet inlet velocities havea uniform profile, starting at time zero and maintaining a constant speed of 10 m\/s. Thestandard grid used has 29 control volumes in the axial direction and 25 in the radial direction.Three free slip walls and one symmetry axis form the boundaries of the grid. Free slip wallshave viscous effects removed by setting the velocity inside the wall equivalent to the velocityadjacent to it. The drawing is not to scale nor does it contain the indicated number of nodes,it is merely given to provide a reference frame.The control volume size is set at 1 cm growing at a rate of 10% per control volume from the124origin.Tollrnien's and Schlichting's data The following table lists the plotted data obtained from Tollmien's and Schlichting'sanalysis. The original data tables of Tollmien given in Abramovich [2] have been treated toa regression analysis using the spreadsheet Lotus 123. The resulting values and thosedetermined by Schlichting and those determined numerically are given.Y\/Yc Tollmien Numerical Schlichting0.1 0.983092 1.012758 0.9886990.2 0.93502 0.966224 0.9325140.3 0.885116 0.916393 0.8768660.4 0.833718 0.863756 0.8218790.5 0.781166 0.808803 0.7676780.6 0.727798 0.752024 0.7143870.7 0.673955 0.693911 0.6621320.8 0.619975 0.634954 0.6110370.9 0.566197 0.575644 0.5612261 0.512962 0.516471 0.5128261.1 0.460607 0.457926 0.4659591.2 0.409472 0.400499 0.4207511.3 0.359897 0.344682 0.3773281.4 0.312221 0.290965 0.3358121.5 0.266783 0.239838 0.296331.6 0.223922 0.191793 0.2590061.7 0.183977 0.147319 0.2239641.8 0.147288 0.106907 0.191331.9^0.114194^0.071049^0.1612271252 I 0.085033 I^0.040234^0.133782 HThe equations use by Schlichting to obtain the radial velocity profile are listed below.3 K^187c e_x^1Q^(1 + -7-n 2 )4_ 14Tay- .71^\u20ac0 xK = 2 n f u 2ydyExperimentally it has been determined that;e\u00b0 = 0.0161Here, y and x are the radial and axial coordinates respectively. Using the data for an axialdistance of x = 0.4357 m, the previous data table was produced.Kuo and Bracco iet penetration relations The transient incompressible jet penetration relations which are given, have been usedto obtain the following table. The time taken for the jet to reach 70.0% of its steady statespeed at a specified stationary point is used in the following scaled parameters.t* -t Uin126 (R0.053 Dx* - ^D ( RL\"3Here, t is the time, D the jet starting diameter (0.04 m), Ufr, is the jet starting velocity ( 10mis), and Red is the Reynolds number based on the diameter D. Experimentally, the followingrelation was found to govern jet penetration.t* = 0.235x`2 for x* Z 7numerical^predictedt star x star t star17.5752 7.0706 11.748622.5273 8.2673 16.061727.5079 9.6434 21.853836.3473 11.2259 29.615146.7265 13.0459 39.995661.1586 15.1388 53.857779.9567 17.5456 72.3443100.5042 20.3135 96.9697119.5145 23.4965 129.7403132.3801 27.1570 173.3135127Birch's Radial and Axial Concentration Profiles Birch's axial concentration profile for a steady state methane turbulent jet is given in thefollowing expression.C _Ca^(z+a) de = d (Rs-) 112The constant k1 is taken to be 5.05, and the constant 'a' is taken to be 0.02 m, whichdiffers from the value specified by Birch in his experiments. The reason for this may be theinability of the numerical code to adequately model the jet outlet conditions. The densitiesof the gas ( p0) and the air (p.) are 0.677 and 1.21 kg\/m^3 respectively. Here, dE is theeffective diameter, d the initial jet diameter, z the axial coordinate, C the concentration, andCo the initial jet concentration. Data tables obtained are given.axial distance Numerical Predicted0.026 0.999762 10.0381 0.998982 10.05141 0.996656 10.066051 0.990969 10.082156 0.97912 10.099872 0.957569 10.119359 0.922892 10.140795 0.873327 0.9396820.164374 0.810078 0.8195070.190312 0.737445 0.718438......._ .... ___ . \/1^.\/.106 \u25a0\u2022\u2022\u2022\u2022 \u2022\u25a0\u20221280.250227 0.588474 0.5591440.28475 0.522306 0.4958030.322725 0.464827 0.4408670.364497 0.415726 0.392970.410447 0.372608 0.3510210.460992 0.329336 0.3141340.516591 0.272753 0.2815850.57775 0.185615 0.2527750.645025 0.077745 0.2272030.719028 0.011601 0.2044520.800431 0.000343 0.184167Birch's radial concentration profile for a steady state methane jet is given in the followingexpression.= exp ( -D ( ) 2 )The constant 'D' was found experimentally to be 73.6 and is used to produce the followingdata table at an axial distance of z = 0.46 m. Here, r and z are the radial and axialcoordinates, C the concentration, and C. is the concentration on the centerline of the jet.r\/z numer. C\/Cm exper. C\/Cm0.010846 1 0.9913790.032539 0.93474 0.9250340.0564 0.814196 0.7912670.082648^0.655518^0.6048721290.11152 0.474357 0.4003770.14328 0.280157 0.22070.178216 0.109119 0.0965590.216645 0.029441 0.0316050.258918 0.005616 0.0071980.305417 0.000726 0.0010430.356566 0.000063 0.000086Tollmien's Axial Velocity Profile The axial velocity profile for a steady state turbulent jet developed by Tollmien andquoted by Abramovich is given as follows;U _ 0.96U0^axRoThe constant 'a' is given by Tollmien to be 0.066 for jets with a uniform initial velocity.The axial coordinate 'x' is adjusted to compensate for the undeveloped region of the startingjet. Here, U is the axial velocity and U 0 is the initial axial velocity. The adjustment isspecified by Tollmien as:axaxo^ax+^where^\u00b0 - 0. 29R,^Ro R,For an initial radius Ro of 0.02 m, this leads to xo = 0.0878 m, confirmed experimentally.For purposes of comparison this value of xo and another at 0.125 m is plotted. The 0.125 m13 0value compares best to the numerical tests, and again most like indicates the numerical codeslack of sophistication in jet boundary conditions.Predicted Numerical Axial dist.1.0000 1.0000 - 0.00501.0000 0.9982 0.01501.0000 0.9980 0.02651.0000 0.9987 0.03971.0000 0.9995 0.05491.0000 0.9992 0.07241.0000 0.9965 0.09251.0000 0.9884 0.11571.0000 0.9704 0.14231.0000 0.9360 0.17290.8735 0.8798 0.20800.7789 0.8021 0.24850.6926 0.7113 0.29500.6144 0.6195 0.34850.5437 0.5360 0.41000.4802 0.4642 0.48080.4233 0.4038 0.56220.3726 0.3527 0.65580.3275 0.3082 0.76340.2874 0.2661 0.88710.2520 0.2202 1.02940.2207 0.1635 1.1931A schematic drawing of the starting region of the steady state jet showing the initial offset131xo is given.Location of Virtual Origin withrespect to the jet orifice at x 0.0start speed 10 m\/sRo = 0.02 m","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"1992-05","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0080886","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Mechanical Engineering","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"For non-commercial purposes only, such as research, private study and education. 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