{"Affiliation":[{"label":"Affiliation","value":"Applied Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Mechanical Engineering, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Tse, Daniel C.M.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2009-04-09T00:00:00","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1994","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Doctor of Philosophy - PhD","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"The problem of multiple turbulent jet interactions is investigated with special attention\r\nto applications in kraft recovery boilers. The phenomena due to turbulence are simulated\r\nwith the k-\u03b5 turbulence model, and a multigrid numerical technique is applied to solve\r\nthe time-averaged Navier-Stokes equations governing the flows. Investigations carried out\r\ninclude a study on the simulation of primary level jets and on the characteristics of a row\r\nof jets discharging into a confined crossflow. For the primary level jets, the interaction\r\nand merging of the jets are investigated. The jets merge rapidly and a suitable open\r\nslot representation gives an adequate description of the velocity field. For jets in a\r\nrow interacting with a confined crossflow, the effects of varying the jet spacing on flow\r\ncharacteristics are investigated. At moderate spacing, the penetration decreases as the\r\nspacing is reduced. It is also observed that the vorticity structures of a jet within the row\r\ncan be substantially different from those of an isolated jet. The penetration of rectangular\r\njets from orifices having different aspect ratios is then studied. A quantitative analysis\r\nis carried out to examine the extent of mixing between the jets and the crossflow. The\r\napplicability of a correlation by Holdeman and his co-workers is extended to rectangular\r\njets. The correlation yields information on the penetration at various values of jet spacing,\r\nconfinement size, and jet-to-crossflow momentum ratio. Holdeman\u2019s correlation is also\r\nfound to be applicable to a crossflow having a peaked non-uniformity in the velocity\r\nprofile. The use of Holdeman\u2019s correlation indicates that, for a given mass flow from\r\nthe jets, large jets at a low momentum can penetrate as far as smaller jets at a higher\r\nmomentum. Furthermore, because of their low momentum, these large jets introduce a\r\nlower degree of flow non-uniformity in the mainstream.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/7006?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"Extent":[{"label":"Extent","value":"3457855 bytes","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/extent","classmap":"dpla:SourceResource","property":"dcterms:extent"},"iri":"http:\/\/purl.org\/dc\/terms\/extent","explain":"A Dublin Core Terms Property; The size or duration of the resource."}],"FileFormat":[{"label":"FileFormat","value":"application\/pdf","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/elements\/1.1\/format","classmap":"edm:WebResource","property":"dc:format"},"iri":"http:\/\/purl.org\/dc\/elements\/1.1\/format","explain":"A Dublin Core Elements Property; The file format, physical medium, or dimensions of the resource.; Examples of dimensions include size and duration. Recommended best practice is to use a controlled vocabulary such as the list of Internet Media Types [MIME]."}],"FullText":[{"label":"FullText","value":"MULTIPLE JET INTERACTIONS WITH SPECIALRELEVANCE TO RECOVERY BOILERSByDaniel C.M. TseB. Sc. (Mathematics and Physics) University of British Columbia, 1987M. Math. (Applied Mathematics) University of Waterloo, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAFebruary 1994\u00a9 Daniel C.M. Tse, 1994In 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.(Signature)__________________________Department of kcx ccJThe University of British ColumbiaVancouver, CanadaDate fr&cDE-6 (2\/88)AbstractThe problem of multiple turbulent jet interactions is investigated with special attentionto applications in kraft recovery boilers. The phenomena due to turbulence are simulatedwith the k \u2014 e turbulence model, and a multigrid numerical technique is applied to solvethe time-averaged Navier- Stokes equations governing the flows. Investigations carried outinclude a study on the simulation of primary level jets and on the characteristics of a rowof jets discharging into a confined crossflow. For the primary level jets, the interactionand merging of the jets are investigated. The jets merge rapidly and a suitable openslot representation gives an adequate description of the velocity field. For jets in arow interacting with a confined crossflow, the effects of varying the jet spacing on flowcharacteristics are investigated. At moderate spacing, the penetration decreases as thespacing is reduced. It is also observed that the vorticity structures of a jet within the rowcan be substantially different from those of an isolated jet. The penetration of rectangularjets from orifices having different aspect ratios is then studied. A quantitative analysisis carried out to examine the extent of mixing between the jets and the crossflow. Theapplicability of a correlation by Holdeman and his co-workers is extended to rectangularjets. The correlation yields information on the penetration at various values of jet spacing,confinement size, and jet-to-crossflow momentum ratio. Holdeman\u2019s correlation is alsofound to be applicable to a crossflow having a peaked non-uniformity in the velocityprofile. The use of Holdeman\u2019s correlation indicates that, for a given mass flow fromthe jets, large jets at a low momentum can penetrate as far as smaller jets at a highermomentum. Furthermore, because of their low momentum, these large jets introduce alower degree of flow non-uniformity in the mainstream.11Table of ContentsAbstract iiTable of Contents iiiList of Tables viiiList of Figures ixNomenclature xvAcknowledgement xix1 Introduction : Simulation of Kraft Recovery Boilers 11.1 A Kraft Recovery Boiler 11.2 Recovery Boiler Air System 21.2.1 Primary Air 41.2.2 Secondary Air 41.2.3 Tertiary Air 51.2.4 Issues Related to the Air System 51.3 Simulation of the Recovery Boiler Air System 61.4 Merits of Simplified Studies 91.5 Objectives and Contributions of the Present Study 102 Dynamics of Turbulent Jets : A Literature Review 132.1 Mathematical Description of Jet Flows 132112.2 Jets in Quiescent Environments . 193 Numerical Solution of the Navier-Stokes System3.1 Discretization of Differential Equations2.2.1 Axisymmetric Jets 202.2.2 Jets from Rectangular Nozzles 212.2.3 Twin Jet Flow 222.2.4 Multiple Interfering Jets 232.2.5 Primary Jets in Recovery Boilers 252.3 Single Jet in a Crossfiow 262.3.1 General Shape of a Single Jet in a Crossfiow . . . 262.3.2 Jet Trajectory and Penetration 282.3.3 Jet Width and Thickness 302.3.4 Effects of Confinement 302.3.5 Results from Detailed Flow Visualization 312.3.6 The Counter-Rotating Vortex Pair 322.3.7 Turbulence Characteristics 342.4 Multiple Jets in a Crossfiow 352.4.1 Effects of Varying Jet Spacing and Momentum Ratio 362.4.2 More Comprehensive Parametric Variations 392.5 Prediction Methods for Turbulent Jets 412.5.1 Empirical Models 422.5.2 Analytical Models 422.5.3 Numerical Models 442.6 Application of the k \u2014 c Model to Jet Flows 452.7 Chapter Summary . . 464848iv3.2 Boundary Conditions 543.3 Numerical Solution Procedure 553.3.1 Sequential Solution Methods 553.3.2 Coupled Solution Methods 563.3.3 Convergence Difficulties 573.4 Multigrid Procedure 583.4.1 Full Multigrid Strategy 613.4.2 Design of a Multigrid Algorithm 623.5 Multigrid Technique as Applied to the Navier-Stokes Equations 643.5.1 Stability of Discretization 643.5.2 Choice of Smoother 653.5.3 Inter-Grid Transfer of Information. . . 663.5.4 Construction of Coarse Grid Equation 703.6 Extension of the Multigrid Technique to Turbulent Flow Problems . . 703.6.1 Difficulties with Implementation 713.6.2 Experience with Applying Multigrid to the k \u2014 c Model 723.7 Multigrid Code MGFD for Solving Complex Turbulent Flows 753.7.1 Features of MGFD 753.7.2 Single Jet in a Crossflow : Validation of MGFD 783.8 Chapter Summary 864 Simulation of Primary Level Jets 884.1 Interaction of Corner Jets 894.2 Full Discrete Jet Simulation 944.3 Slot Representations of Primary Jets . . . 974.3.1 Open Slots . 97V1004.4 . 1014.5 . 1064.6 . . 1184.7 . . 1265 Characteristics of a Row of Jets in a Confined Crossflow5.1 Problem Description and Boundary Conditions5.2 Convergence Performance5.3 Jet Penetration5.4 Development of the Cross-section of a Jet5.5 Vorticity Dynamics for a Row of Square Jets in a Crossflow .5.5.1 Magnitudes of the Streamwise Vorticity5.5.2 Distribution of the Streamwise Vorticity5.6 Concluding Remarks6 Multiple Jet Interactions with a Crossflow6.1 Jet Penetration Study : Effects of Parametric Variations . .6.1.1 A Non-dimensional Relationship6.1.2 A Row of Rectangular Jets in a Crossflow6.1.3 Results for Uniform Crossflow6.1.4 Experimental Investigation on Jet Penetration6.2 Mixing and Vorticity Characteristics6.2.1 Quantitative Study of Jet Mixing6.2.2 Vorticity Considerations6.3 Consequences of Holdeman\u2019s Relationship6.4 Implications for Tertiary Level Jets in a Recovery Boiler4.3.2 Porous SlotsComparison of Mean Up-flow VelocityComparison of Turbulence QuantitiesVarying the Turbulence Input Parameters in Slot Simulations .Concluding Remarks127128130133135140145147155157157158160161167172172177\u2022 . . 180181vi6.5 Non-uniform Crossflow 1846.6 Interlaced Jets on Both Sides of the Chamber 1876.7 Concluding Remarks 1907 Conclusions and Recommendations 1927.1 Results of the Present Study 1927.2 Recommendations for Further Studies 194Appendices 198A Flow Visualization Experiment 198B A Quantitative Description of Jet Mixing 201Bibliography 205viiList of Tables2.1 Parameters in the k \u2014 c model 182.2 Terms representing the time-averaged Navier-Stokes equations with thek \u2014 c turbulence model and a transport equation for an inert scalar. . . . 193.1 Discretization of the source terms 536.1 Parameters chosen for simulations of square jets 1616.2 Parameters chosen for simulations of rectangular jets with AR = 3 and 6 1646.3 Optimal values of C for orifices having different aspect ratios 1676.4 Values of W, J and V\u20ac employed in the simulation of tertiary-type jets 183viiiList of Figures1.1 A schematic representation of a kraft recovery boiler 31.2 Domain of a recovery boiler used in numerical simulations 82.1 Orifices for (a) axisymmetric and (b) rectangular jets 202.2 Development of a row of circular jets 242.3 A schematic representation of a jet in a crossifow 272.4 A schematic view describing a row of jets discharging into a confined cross-flow 352.5 Effect of jet spacing on jet trajectory, J = 100, Ivanov (1959) 362.6 Effect of jet spacing on jet trajectory, H\/D = 24, J = 8 and 72, Kamotaniand Greber (1974) 383.1 Variable arrangements in a staggered grid in two-dimensions 503.2 FAS (h,H) two-grid method 593.3 The full multigrid (FMG) algorithm 623.4 Staggered mesh arrangement showing storage of variables in the fine andcoarse grids 673.5 Interface of two segments having different grid densities 773.6 A schematic representation of the single square jet in a crossflow studiedby Simitovi\u00e9 (1977) 793.7 Grid distribution for the calculation of a single jet in a crossflow 803.8 Mass error reduction histories for TEMA and MGFD 82ix3.9 Measured and predicted streamwise velocity at y\/W 1.0, (a) x\/W = 3.5,(b) x\/W = 8.253.10 Measured and predicted streamwisevelocity at z\/W 0.0, (a) x\/W = 3.5,(b) x\/W = 8.253.11 Measured and predicted streamwise velocity at z\/W = 0.5, (a) a\/W 3.5,(b) x\/W = 8.253.12 Measured and predicted profiles of jet tracer concentration, (a) z\/W =0.0,x\/W = 1.75, (b) z\/W = 0.5,z\/W = 1.75, (c) z\/W = 0.0,x\/W = 4.5,(d) z\/W = 0.5, x\/W = 4.5A schematic illustration of the Plymouth water modelDomain for simulating corner jet interactionConvergence history for the simulation of corner jet interaction4.4 Computation mesh and the velocity field at the jet entrance elevation(x = 4.24 cm) 93A three-dimensional view of the computation domainComputation mesh used in the primary discrete jet simulationConvergence history for the full primary discrete jet simulationRepresentations of primary level jets: discrete jets, open slot, and porousslotComputation mesh used in the slot jet simulationConvergence history for the open slot simulationDistribution of the vertical velocity component at x = 4.24 cm.Distribution of the vertical velocity component at x = 7.62 cm.Distribution of the vertical velocity component at x 17.78 cm.Distribution of turbulent kinetic energy at x 4.24 cm4.14.24.3838484859091924.54.64.74.84.94.104.114.124.134.14959696979999103\u2022 . . 104\u2022 . . . 105107x4.15 Distribution of dissipation rate of turbulent kinetic energy at x = 4.24 cm 1084.16 Magnified views of the distributions of turbulent energy and its dissipationrate for discrete jets 1094.17 Distribution of turbulent kinetic energy at x = 7.62 cm 1114.18 Distribution of turbulent kinetic energy at x = 17.78 cm 1124.19 Distribution of dissipation rate of turbulent kinetic energy at x 7.62 cm 1134.20 Distribution of dissipation rate of turbulent kinetic energy at x 17.78 cm.1144.21 Distribution of turbulent viscosity at x = 4.24 cm 1154.22 Distribution of turbulent viscosity at x = 7.62 cm 1164.23 Distribution of turbulent viscosity at x = 17.78 cm 1174.24 Distributions of turbulent energy and dissipation at a = 4.24 cm for (a)k\u20ac = 0.05 x and (b) kjet = 0.5 X 1204.25 Distributions of turbulent viscosity and mean vertical velocity at x = 4.24cm for (a) kjet 0.05 x and (b) k3\u20ac = 0.5 x T\/2e 1214.26 Distributions of turbulent energy and dissipation at x 7.62 cm for (a)0.05 x and (b) kjet 0.5 X 1224.27 Distributions of turbulent viscosity and mean vertical velocity at x = 7.62cm for (a) kjet = 0.05 x and (b) k,et = 0.5 X 1234.28 Distributions of turbulent energy and dissipation at x = 17.78 cm for (a)kjet = 0.05 x and (b) k\u20ac = 0.5 X 1244.29 Distributions of turbulent viscosity and mean vertical velocity at x = 17.78cm for (a) kjet = 0.05 x and (b) kjet = 0.5 X 1255.1 Schematic representation of jet injection from a row of square orifices. . 1295.2 Convergence histories for different cases of S\/W and J 1315.3 Convergence history for S\/W 2 at J = 72; time dependence calculation 134xi5.4 Velocity profiles for S\/W = 2, 4, 8, and 16 at J 8 along the jet center-plane 1365.5 Jet tracer concentration profiles for S\/W = 2, 4, 8, and 16 at J = 8 alongthe jet center-plane 1375.6 Velocity profiles for S\/W = 2, 4, 8, and 16 at J = 72 along the jet center-plane 1385.7 Jet tracer concentration profiles for S\/W = 2, 4, 8, and 16 at J = 72 alongthe jet center-plane 1395.8 Jet tracer concentration profiles for S\/W 2, 4, and 8 in the cross-streamplane x\/W = 5 at J = 8 1415.9 Jet tracer concentration profiles for S\/W = 2, 4, and 8 in the cross-streamplane x\/W = 10 at J = 8 1425.10 Jet tracer concentration profiles for S\/W 2, 4, and 8 in the cross-streamplane x\/W = 5 at J = 72 1435.11 Jet tracer concentration profiles for S\/W = 2, 4, and 8 in the cross-streamplane x\/W = 10 at J = 72 1445.12 Variations in the maximum streamwise vorticity with downstream distanceforS\/W=2,4,and8 1465.13 Distribution of the streamwise vorticity in the cross-stream plane x\/W =lOforS\/W=2,4,and8atJ=8 1485.14 Secondary flow vectors in the cross-stream plane x\/W = 10 for S\/W 2,4,and8atJ=8 1495.15 Distribution of the streamwise vorticity in the cross-stream plane x\/W =10forS\/W=2,4,and8atJ72 1505.16 Secondary flow vectors in the cross-stream plane x\/W 10 for S\/W = 2,4, and 8 at J = 72 151xii5.17 Distributions of the production and diffusion terms in the cross-streamplane x\/W = 10 for S\/W = 4 at J = 72 1545.18 Distribution of the streamwise vorticity in the cross-stream plane x\/W2OforS\/W=4atJ=72 1546.1 A schematic description of the domain for one-side injection 1606.2 Distribution of jet tracer concentration in the center-plane (z = 0) forsquare jets at J = 6 and M = 0.15 1636.3 Distribution of jet tracer concentration in the center-plane (z = 0) forsquare jets at J = 25 and M 0.31 1636.4 Distribution of jet tracer concentration in the center-plane (z = 0) forAR=3jetsat J=6andM=0.15 1656.5 Distribution of jet tracer concentration in the center-plane (z = 0) forAR=6jetsat J=6andM=0.15 1656.6 Jet tracer concentration in the cross-sectional plane x\/H = 0.5 at thespacing S\/H = 0.5 1666.7 Jet tracer concentration in the cross-sectional plane x\/H = 1.0 at thespacing S\/H = 0.5 1666.8 Photographs of the Kamloops water model showing the overall model andthe orifices for tertiary jets 1696.9 The two templates used in the experiment: square orifices and rectangularA? = 4 orifices 1706.10 Photographs displaying jet penetration. Top: crossfiow. Middle: squarejet injection. Bottom: rectangular jet injection. Jets enter from the rightside only 1716.11 Profiles of f() for A? = 6 and S\/H = 0.25 at various downstream locations.174xiii6.12 Variation of & with x\/H for jet-crossflow mixing for various cases ofA? and S\/H 1766.13 Variation of the maximum streamwise vorticity with x\/H for jets fromorifices that have different aspect ratios and at various lateral spacings 1786.14 Distributions of streamwise vorticity, production, and diffusion for the caseA? = 3 and S\/H = 0.5 in the cross-stream plane x\/H = 1.0 1796.15 Jet tracer concentration in the center-plane for uniform crossflow 1826.16 Schematic representation of opposing jets in a crossflow with penetrationreduced to avoid collision between jets 1846.17 Velocity profile of the peaked crossflow 1856.18 Jet tracer concentration in the center-plane for non-uniform crossflow. 1866.19 Plan view of the interlaced jet injection scheme 1876.20 Jet tracer concentration in the cross-stream plane x\/H = 1.0 due to mixingwith interlaced jets 1886.21 Standard deviation of the streamwise velocity component at various downstream positions for the three sizes of interlaced jets 188A.1 Schematic of the flow pattern in the east-west plane within the modelgenerated by primary jets 199A.2 Schematic of the experimental set-up for flow visualization 200B.1 A schematic drawing representing the mixing between a jet and a crossflow.202xivNomenclatureSI units for all physical quantitiescoefficients of finite difference equationsA? aspect ratio of jet orifice (= L\/W)C1,C2,C1. turbulence model constantsCe convective flux through the east cell faced defect or residual quantityD diameter of circular jet orificeDe diffusive flux through the east cell faceD,D(12) diffusion term in the vorticity equationfGf) probability density function off(e) f() computed at the streamwise location xC generation term of turbulent kinetic energyG grid used in the discretization of domainH dimension of the confinement facing a jetI prolongation operator forIf restriction operator for dIf restriction operator forJ jet-to-crossflow momentum ratiolv turbulent kinetic energyL length of jet orificedimensions of the domain in the single jet in crossflow simulation\u00a3 non-linear differential operatordiscrete approximations to \u00a3 on grids Oh and C, respectivelyxvdissipation length scaleM jet-to-crossflow mass flow ratiop pressureP modified pressure (= p + pk)P production term in the vorticity equationQ i volume flow rate of the cross-streamQ 2 volume flow rate of the jetQ solution vectorQh discrete approximation to Qq approximation to Qhsmoothed version of qcorrection toS spacing between neighboring jet orificesS, source term in the transport equation for \u20181S ,S coefficients of linearized source termt timeU velocity vectorUi,U2,U3 Cartesian velocity components, also written as U,V,W in Eq.(3.1)u2 fluctuating velocity componentsUc crossflow velocity far upstream of a jetjet velocityW width of jet orificex,y,z Cartesian coordinatesY2d distance along y at which discrete jets have been merged to becomeeffectively two-dimensionalxviGreek Symbolsnormalized standard deviation measuring the degree of mixing8x2 grid spacingC dissipation rate of turbulent kinetic energyI\u2019 diffusion coefficient for \u2018I\u2019von Karman constantdynamic mass diffusivity for 4(= product of density [kg m3] and kinematic mass diffusivity [m2 s\u2019])dynamic viscosityut turbulent viscosityI\u2019eff effective viscosity (= + )v kinematic viscosityvorticity vectorx-component of the vorticity vectorilr general variabletime-averaged component of \u2018I\u2019fluctuating component of \u2018Ppassive scalar, also denotes its fractional concentration valuefluctuating component ofau fully mixed concentration value of 4p densityparameters in k, c, and equationsxviivariance of f(E)maximum variance computed before mixingdeviation of 4 from 4av ( 4) 4)av)porosity factorSuperscriptsc coarse gridj generic index representing the stage of iterationf fine gridSubscriptsC value associated with the crossfiowci\u2019 for critical spacing ratioh value associated with the fine gridH value associated with the coarse gridjet value associated with the jetn,s,e,w,b,t cell facesP,N,S,E,W,B,T grid pointsxviiiAcknowledgementI would like to express my sincere gratitude to the many individuals who have helpedme in different ways during the course of my research. This study was carried out underthe supervision and guidance of Professors Martha Salcudean and Ian Gartshore whosepatience, sensitivity, encouragement, and contributions have been most helpful duringthe many trying moments. I would also like to acknowledge Dr. Paul Nowak and Dr.Zia Abdullah for teaching and sharing their insights on problems with numerical simulations. The code MGFD employed in this study was developed by Dr. Paul Nowak.The assistance of Mr. Mike Savage with the experiments and graphics, together withhis sincere friendship, are whole-heartedly appreciated. The many stimulating conversations with Dr. Yacov Barnea have been enjoyable and often brought brightness into anotherwise monotonous day. The communications with Dr. Pen Sabhapathy, Mr. PaulMatys, Ms. Fariba Aghdasi, Mr. Steve Ketler, and Mr. Jeff Quick have been vaiuable inthe understanding of flow phenomena in recovery furnaces. Finally, I wish to rememberthe fellowship I have been deeply enjoying with many friends at CCC whose support andcare have helped me to journey through this period of my life.The financial support for this study has been provided partly by NSERC, BC ScienceCouncil, and Weyerhaeuser Company Limited.xixChapter 1Introduction : Simulation of Kraft Recovery BoilersThe objective of the present study is to investigate characteristics of jet interactionsrelevant to the design and operation of kraft recovery boilers. This type of boiler iscentral to a pulp mill because of its role in the kraft pulping process. Spent cooking fluidleft after the pulping process is called black liquor; it contains inorganic chemicals suchas phosphates and sulphates, as well as organic lignin. The purpose of a recovery boileris to provide a medium both for the combustion of organic matter to produce heat energyand for reduction reactions to take place to recover valuable chemicals such as sulphides.Most of the energy consumed by a pulp mill is generated by the recovery boiler. This,together with its role in recovering useful chemicals, makes it a very critical componentin the kraft pulp mill. The operational efficiency of the mill is very dependent on theproper functioning and efficiency of the boiler. Thus, it is necessary to ensure thatthe combustion process within the boiler is efficiently carried out, with due regard toenvironmental requirements.1.1 A Kraft Recovery BoilerA detailed description of the physical and chemical processes that take place in a kraftrecovery boiler can be found in the monograph by Adams and Frederick (1988). Additional design considerations can be found in Barsin (1989). The recovery boiler is avery large piece of equipment. The furnace section alone can have dimensions of 10 mwidth by 10 m length by 40 m height. A schematic representation of a typical boiler is1Chapter 1. Introduction : Simulation of Kraft Recovery Boilers 2shown in Figure 1.1. The current design is the result of numerous modifications basedon operational experience over the past several decades.Two main sections constitute a kraft recovery boiler: a furnace and a convectiveheat transfer section. At the base of the furnace is the char bed which is formed by theaccumulation of partially burned black liquor deposits. It is a design goal to maintain thebed at a high temperature for chemical reduction to take place. Another design goal isthat, higher up in the furnace, mixing of black liquor with air and subsequent combustionshould be completed. In addition, a substantial amount (about 40%) of the heat transferfrom the combustion gas should also be completed in the furnace. Heat transfer to theboiler water which forms high pressure steam is then completed in the convective heattransfer section. The bullnose shown in Figure 1.1 shields the heat transfer surface fromextreme conditions in the furnace, and its location is the general demarcation betweenthe two main sections.The furnace walls consist of numerous vertical tubes set in columns. Water is usedas the heat absorbing agent. It is made to run in the tubes and receives heat fromradiation due to the char bed and from gases in the furnace. Ports must be provided forintroduction of air and liquor guns. These ports are formed by bending one or severalwall tubes to the side. It is common practice to use ports in the shape of vertical slots sothat a given port area can be obtained with the minimum number of bends in the tubes.1.2 Recovery Boiler Air SystemThe requirement that a recovery boiler has both high chemical and thermal recoveryefficiency leads to difficulties in designing a good air delivery system. The reason isthat these two tasks can only be optimized in diametrically opposing environments. Tomaximize the recovery of inorganic chemicals, the smelting area of the furnace must beChapter 1. Introduction : Simulation of Kraft Recovery Boilers 3Waterwal IsLiquorGunsSteam Flowto MillSecondary AirSmelt toDissolving TankSuperheater BoilerBank EconomizerToElectrostaticPrecipitatorBuilnoseFurnaceTertiary AirUuuuBuDu10 metersPrimary AirFigure 1.1: A schematic representation of a kraft recovery boiler.Chapter 1. Introduction : Simulation of Kraft Recovery Boilers 4in an oxygen deficient reducing atmosphere so that the inorganic chemicals recovered willbe in a useful form for the pulping process \u2014 in the form of sodium suiphide instead ofsodium sulphate. In contrast, to optimize the thermal efficiency of the unit, the organicmaterial must be exposed to an oxygen rich oxidizing atmosphere where the combustiblematerial can be burnt to completion and thereby liberates the required heat to producesteam.As a result of these divergent process demands, the recovery boiler is divided intodistinct zones and the introduction of air usually takes place at three different levels, asoutlined in the following sections.1.2.1 Primary AirThe primary air ports provide 30% to 50% of all air supplied. This maintains the reducingcondition in the bed. Their sizes and locations are important to obtain a high reductionefficiency. Generally, these air ports are located on four walls approximately 1 m abovethe furnace floor. They supply air to maintain sufficient distribution of oxygen aroundthe full furnace periphery. The air contributes oxygen for the carbon burnout whichproduces the heat for the reduction reaction to proceed.1.2.2 Secondary AirThe secondary air ports are located above the char bed, which has a maximum heightof about 1 m to 3 m above the primary air level. They provide up to 50% of the totalair required and their role is to control the top of the char bed, since their momentum issufficient to provide scouring across the top. The full penetration of these secondary airjets into the bulk furnace gas is critical for completing burnout low in the furnace, forbreaking up the combustion gas cone, and for assuring mixing and combustion of the airwith the volatile gases rising from the char bed.Chapter 1. Introduction Simulation of Kraft Recovery Boilers 51.2.3 Tertiary AirThe tertiary air ports are located 1.5 m to 5 m above the black liquor spray nozzles,which themselves are located 3 m to 7.5 m above the furnace floor. These air portssupply up to 30% of the total air required. They are usually located on only two walls.The momentum of jets developed at this level is critical: to complete combustion ofthe partially burned fuel components flowing up from the lower furnace; to completethe break up of the flame cone, and to insure oxygen availability throughout the upperfurnace which eliminates unwanted sulphide gases. It is also the role of the tertiary airjets to aid in the establishment of a uniform flow profile and uniform heat distributionin the upper furnace tube banks.1.2.4 Issues Related to the Air SystemConcerns for the design of the air distribution system stem from the multitude of tasks thesystem is required to fulfill. In a recovery boiler, unlike many other combustion devices,the fuel, in the form of black liquor, and air are introduced separately. Good jet mixing ismandatory in these types of designs to obtain complete combustion. The size and numberof nozzles at each level are mainly dictated by the amount of combustion air required.Other considerations do apply; for instance, the practical range of the quantity of airbelow the liquor gun is from 80% to 95% of the stoichiometric requirement for the liquor.This amount of air is used to ensure bed stability and to maximize bed temperaturein order to improve reduction efficiency and SO2 control. However, increased air in thelower furnace also increases the potential for entrainment of the char, which contributesto unwanted \u2018carryover\u2019 of char particles.Considerations such as those mentioned above have created a need to predict theflow characteristics in a boiler unit. Although numerous studies have been carried outChapter 1. Introduction : Simulation of Kraft Recovery Boilers 6to investigate the physics of fluid jets issuing into quiescent, co-flowing, or cross-flowingsurroundings, most of the established observations for single jet and related correlationsare not directly applicable to recovery boiler applications. In the monograph by Adamsand Frederick (1988, p.158), the authors remarked that the use of single jet relationshipsleads to over-prediction of jet penetration due to the neglect of multiple jet interactions.Moreover, the non-uniformity of the crossflow could create additional complications in theprediction process. Thus, there is this need to investigate the flow field using experimentaland numerical methods.1.3 Simulation of the Recovery Boiler Air SystemMany investigators have carried out experimental and numerical simulations to predictthe effects of design changes on the aerodynamic characteristics inside a boiler. Experimental studies in scaled models running air or water have been carried out by Lefebvreand Burelle (1988), Chapman and Jones (1990), and Ketler et al. (1992,1993). Theseexperimental investigations reveal valuable information regarding the flow patterns observed in boilers. For example, the results of the water model experiment by Ketler etal. (1993) show that the flow is often asymmetric and contains large regions of low frequency unsteadiness. However, measurement problems do cause difficulties in assessingthe validity of the observation and complicated thermal-chemical processes are difficultor impossible to simulate in an experimental facility. Thus, in view of the rapid development of computational fluid dynamics (CFD), a numerical technique is an essential toolfor predicting flows in recovery boilers.It is necessary to apply numerical methods to compute the flow in a furnace becausethere is a lack of analytical techniques available for the prediction of complex jet interactions. A number of investigators have attempted to perform parametric studies using fullChapter 1. Introduction : Simulation of Kraft Recovery Boilers 7scale numerical models for the flow fields in kraft recovery boilers. The simulations byUppstu et al. (1989), Grace et al. (1990), and Chapman and Jones (1990) used commercially available codes. In these simulations, certain simplifying assumptions were made inthe representation of an actual boiler. For the purpose of fluid flow calculation, the shapeof the domain may be simplified to that shown in Figure 1.2, where many complicatedstructures in the top exit portion of the boiler have been ignored. Much valuable insighthas been gained through those simulations. For example, the results of Grace et al. andChapman and Jones show that a simple isothermal model is adequate to capture theessential flow characteristics of a boiler.Many difficulties have been experienced with these numerical simulations. It wasfound that even when all complex thermal and chemical processes were neglected, thefull scale calculations for boilers were difficult to converge. A reason for this difficulty wasthe need to use a very large number of grid cells to represent the domain of the interiorof the boiler. This problem arises due to large variations in dimensions that are found ina typical boiler. Recall that the furnace portion of a typical boiler has the dimensions of10 m wide by 10 m long by 40 m high, while a secondary air port has dimensions of 5 cmby 25 cm in size. Thus, to resolve each of these ports within such a large domain, a veryfine grid relative to the domain is needed. This requirement causes problems in bothcomputational time and memory allocation. Moreover, the use of a fine grid also strainsthe performance of most common numerical solution techniques: slow convergence is tobe expected and very often stalling in error reduction occurs.Because of this lack of efficiency in standard numerical techniques for the study oflarge flow simulations of this type, our research group at UBC has been developingan algorithm that will deal with this type of problem more effectively. The algorithmuses a multigrid solution acceleration procedure together with segmentation capabilityto partition the domain for efficient calculation. Some details of the algorithm will beChapter 1. Introduction Simulation of Kraft Recovery Boilers 8FRONTWALL \\\u2014 BULLNOSEBACK WALL\u2014 SECONDARYPRIMARYFigure 1.2: Domain of a recovery boiler used in numerical simulations.Chapter 1. Introduction : Simulation of Kraft Recovery Boilers 9described in Chapter 3. Preliminary results have been reported in Salcudean et al. (1992)and show fast convergence and promising robustness. Presently, the code has been usedfor isothermal flow simulations. A modified version that takes into account heat transferand chemical reactions is under development.Even with this improvement in numerical methodology, comprehensive full scale numerical simulation is still plagued with problems. At this stage of the development of thealgorithm, a complete simulation of multiple jet interactions in a boiler still requires substantial computer memory and CPU time. This problem can certainly be improved in thefuture with better computer technology. However, a major short-coming with full scalesimulation is that it can be difficult to decipher the effects of different flow parameterswhen they are subjected to variations. This difficulty is compounded by the long computation time involved for each simulation such that systematic parametric studies couldbecome prohibitive. In view of this limitation of full scale numerical modelling, it is ofgreat value to perform simplified flow simulations to study the basic flow characteristicsof multiple jet interactions.1.4 Merits of Simplified StudiesAn obvious advantage with performing simplified jet flow studies is the large saving incomputing resources so that more parametric cases can be investigated in a given amouutof time. This advantage is utilized in the study of simulations for primary jets and inthe systematic parametric investigation into the details and phenomena of multiple jetinteractions in the presence of a crossflow. The results obtained can be related to thefull scale problem if those simplifying assumptions made do not deviate too far fromthe phenomena in the full scale model. For example, in the study of multiple jets in acrossflow, parameters such as the jet spacing, jet size, jet momentum, and jet shape areChapter 1. Introduction : Simulation of Kraft Recovery Boilers 10designed to vary systematically for the investigation of the effect on those parametersof jet penetration and mixing. These studies are carried out to identify trends thatcan be useful both for the understanding of observed phenomena and for future designconsiderations. With proper verification using the full scale simulation, such trends willbe useful in providing guidelines for design. In addition, this parametric study canreveal aspects of jet dynamics such as the vorticity characteristics that often have beenoverlooked without such a detailed investigation.Another important by-product from these simplified studies is that these comparatively small calculations provide good examples for the testing and debugging of ouralgorithm. This is particularly necessary since the application of the multigrid techniqueto equations representing turbulent flows has rarely been tested. The simulations involving jets in a crossflow provide flow situations with small enough domains so that only amoderate number of grid points is needed and yet the flow characteristics are complexenough to test the efficiency and robustness of the numerical algorithm. It has beenfound that the results of these computations provide experience that is necessary for thedevelopment of the algorithm and the calculation of more complex cases of full boilersimulations.1.5 Objectives and Contributions of the Present StudyMost systematic studies on multiple jet interactions have been carried out on circularjets with application to the design of small combustion chambers. The specific needsof kraft recovery boilers, such as the use of rectangular jets and the presence of nonuniformity of the ambient flow, have seldom been addressed. In the present investigation,studies are carried out on the interaction of turbulent jets from rectangular orifices insimple geometrical domains. These lead to a better understanding of the complexities inChapter 1. Introduction : Simulation of Kraft Recovery Boilers 11numerical modelling and in physical phenomena for kraft recovery boilers. The objectivesof the present study are as follows: to investigate the interaction of multiple jets in thepresence and in the absence of a crossflow using numerical methods; to investigate theinteraction of jets similar to those found in a recovery boiler; to make recommendationsfor full boiler modelling; to carry out parametric studies on multiple jet interactions, andto identify trends for design considerations. The following results are obtained:1. The application of the multigrid algorithm to obtain solutions of turbulent flowproblems has been extensively tested. The performance of the algorithm has beenexamined for a variety of flows with turbulent jets. Some difficult cases have beenidentified and experience has been gained to improve the robustness of the algorithm. When the algorithm is implemented with precaution, it can return a fastconvergence performance for the type of problems under study.2. With the aid of the multigrid solution technique, the simulation of primary level jetsis successful. Because these jets are closely spaced, they merge rapidly, which allowsfor their effective representation by using slot equivalences. The use of open slotsand porous slots has been examined and it is observed that both representationsyield good approximations to the velocity distribution.3. The high efficiency of this solution technique has also allowed for a detailed parametric investigation on the characteristics of multiple jets placed in a row injectinginto a crossifow. The variation of jet penetration with orifice spacing has been examined. Our mathematical model predicts that the vorticity characteristics of eachjet in this row configuration will undergo unexpected changes as the jet momentumincreases. This finding is of interest in clarifying and contrasting the dynamics ofmultiple jets versus a single jet in a crossflow.Chapter 1. Introduction : Simulation of Kraft Recovery Boilers 124. A quantitative study of jet mixing has been carried out to clarify the effectivenessof jet mixing according to the extent of jet penetration into the cross-stream. Itis also found that as the geometric parameters are varied, the changes in the jetmixing effectiveness are consistent with the corresponding decay rate of the stream-wise vorticity of the jet. This observation confirms the important role of vorticitydynamics on jet mixing.5. The parametric study of a row of rectangular jets in a crossflow has confirmed theuse of a semi-empirical relationship relating geometric and operational parametersfor effective jet penetration and mixing between the jet fluid and the cross-streamfluid.6. An application of this semi-empirical relationship has led to the realization that at agiven mass flow rate, larger, slower jets can penetrate as deep into the cross-streamas smaller, faster jets. In addition, it is found that for interlaced jets interactingwith a non-uniform crossflow having a peaked velocity profile, these larger jetsoperating at a lower momentum can satisfy the mass flow requirement and willcreate less flow non-uniformity downstream.In the following chapters, attention will be focussed on a series of studies of multiplejet interactions, both with and without the presence of a cross-stream. First, a literaturereview on the dynamics of turbulent jets that are relevant to problems in kraft recoveryboilers is presented. Secondly, the numerical solution algorithm employed in this studyis described. Results and discussion of the simulations, together with the difficultiesencountered, will be addressed in subsequent chapters.Chapter 2Dynamics of Turbulent Jets : A Literature ReviewThere have been many studies on the properties of jets in quiescent environments andin crossflows. The monographs by Rajaratnam (1976) and by Schetz (1980) provide adetailed description of the characteristics of many types of jet mixing phenomena. Thischapter presents an overview of those aspects of jet dynamics that are relevant to theoperation of kraft recovery boilers. Attention is limited to incompressible turbulent jets.For jets in quiescent environments, also known as free jets, the focus is on the flowcharacteristics which are affected by the orifice aspect ratio, the merging of multiple freejets, and flow stability. For jets in crossflows, the focus is on the flow properties thatare characterized by jet penetration, the extent of mixing, and the extent of interactionbetween neighboring jets. These two common types of jets are found in kraft recoveryboilers. Their respective physical characteristics are described in the sections to follow.2.1 Mathematical Description of Jet FlowsThe equations describing the flow of incompressible turbulent jets are the Navier-Stokesequations, which have the following vector form:VU=0 (2.1)=\u2014Vp+V.(VU) (2.2)In Cartesian tensor notation, the above equations become(2.3)13Chapter 2. Dynamics of Turbulent Jets: A Literature Review 14andOU, 8 op 8 1 loU, 8U\u2019\\p--+-a_(UiUj)=t-JI (2.4)These equations represent the conservation of mass and momentum within the fluid.In the above mathematical model, it is assumed that the fluid is Newtonian, has constantdensity p and molecular dynamic viscosity ,u. Also, the effects of body forces have beenneglected. These are valid assumptions for gas flow.The equation governing the transport of a passive inert quantity by the fluid isa a I la+ p\u2014(U3)= jA (2.5)where 4 is the scalar concentration value and A is the molecular diffusivity of the inertquantity. Examples of such quantities are propane and ethylene gases, which serve asmarkers in many flow experiments using air. In the above equation, it is assumed that theflux of the scalar is related to its spatial gradient through a Fourier-type law. Additionalequations describing the energy balance may also be coupled with the above equations,although they are not considered in the present study.To complete the description of fluid motion, we state the principle of conservation ofangular momentum. This principle is expressed through the vorticity vector , definedby=VxU (2.6)The vorticity vector is transported by, and evolved in, the flow field according to thevorticity equation, which is obtained by taking the curl of the momentum equation (2.2).The result is+ (U. V)= ( . V)U + vV2 (2.7)where ii is the molecular kinematic viscosity. The left hand side of the above equationdescribes the convective transport of the vorticity vector. The first term on the rightChapter 2. Dynamics of Turbulent Jets: A Literature Review 15hand side is customarily labelled the cproduction term for the vorticity, even though itonly refers to the strengthening or weakening of the vorticity in a fluid element throughthe respective action of stretching or compressing that element. The last term in theequation refers to the diffusion of vorticity through the action of viscosity.Time-AveragingAt high Reynolds numbers, a flow can be said to be steady only on an average basis,since small scale high frequency fluctuations are always present. In the precise numericalsimulation of this type of flow, it is necessary to solve the full three-dimensional time-dependent form of the equations of motion with a very fine scale of resolution in order tocapture small turbulent eddies. This is still beyond the capability of present technologyfor most practical problems.A standard method of by-passing this stringent computational requirement is to introduce time-averaging to obtain equations for average quantities. The time-averagingprocess consists of decomposing a general variable \u2018I\u2019 into its mean component and itsfluctuating component b as(2.8)The time-averaged value, I1, is defined as\u2014 1 to+ (2.9)where t, is a reference point in time, and the average time At is greater than the longesttime scales of the turbulent motion.For non-reacting incompressible turbulent flows, it is justifiable to neglect fluctuationsof fluid viscosity and density. After introducing the decomposition in Eq.(2.8) for thedependent variables in Eqs.(2.3-2.5) and time-averaging, the following set of equationsChapter 2. Dynamics of Turbulent Jets : A Literature Review 16are obtained for a statistically steady flow:aUi0 (2.10)6 6p 8 1 lou2 OU\u2019\\p\u2014(U2U3)= \u2014\u2014 + i.t\u2018\\-a\u2014 + \u2014 (2.11)p-(U7) j_ {.) (!) \u2014 (2.12)where the overbars for the mean variables have been omitted for clarity.These equations are similar to their instantaneous counterparts, if the instantaneousquantities are replaced by mean values, with the exception of extra terms \u2014p\u00fc and\u2014p appearing respectively in Eqs.(2.11) and (2.12). These terms, known respectivelyas Reynolds stresses and turbulent scalar fluxes, consist of mean products of fluctuatingcomponents and arise from the averaging of the nonlinear convective terms in Eqs.(2.4-2.5). Physically, these terms represent diffusion of momentum or scalar quantities byturbulent motion.For the vorticity equation (2.7), the procedure of time-averaging leads to the followingequation form given by Sykes et al. (1986):(U. V) = (. V)U + D() (2.13)where all the variables listed above are the time-averaged quantities. The diffusion term,labelled D(), refers to the diffusive action on the vorticity due to the turbulent motion.It can be computed by subtracting the first term on the right hand side of Eq.(2.13) fromthe convective term on the left hand side.Turbulence ModellingTo close the system of equations (2.10-2.12), the mean products of fluctuating quantitiesneed to be related either to existing mean quantities or derived by solving additionalChapter 2. Dynamics of Turbulent Jets : A Literature Review 17equations. A standard model which is widely used in applied research is the \u2018k \u2014 &turbulence model, where k stands for the turbulent kinetic energy, and e denotes thedissipation rate of k.The k \u2014 e model is based on the eddy viscosity concept and its derivation was given byLaunder and Spalding (1974). The model assumes the following relation for the Reynoldsstresses:\/8U. 8U,N 2\u2014 = ut H-\u2014) \u2014 pkSj (2.14)where k is defined ask=(u+u+u) (2.15)The term involving the Kronecker delta on the right hand side of Eq.(2.14) ensures thatthe sum of the normal stresses is equal to 2k.The eddy or turbulent viscosity is related to definable quantities through the mixinglength concept and dimensional reasoning. In this model, the expression for ,t\u2019t is= C pk2\/ (2.16)where C is an empirical constant of proportionality.The turbulent stress values can be estimated if values for Ic and e are known. Theequations governing the transport of these quantities are given by Launder and Spalding(1974) and written as8k 8 (Ueff Ok\u2019\\ (OU 8U2 8UpU3 = I J + ILeff I + I \u2014 (2.17)OX3 OX3 \\ k OX3 \\ OX3 UX J OX3and86 8 (Ieff 86\u2019\\ \u20ac (8ti O(J\u2019\\ 0U2 (.8)OX3 OX3 \\ U OX3J \\ OX3 OX2 J OX3The effective viscosity neff is the sum of the molecular and eddy viscosities:ILeff = II + I-\u2019t (2.19)Chapter 2. Dynamics of Turbulent Jets: A Literature Review 18The eddy viscosity I\u2019t also appears in the equation relating the turbulent scalar flux\u2014pq5u, with the mean scalar gradient:\u2014I_Lto,\u2014 pqu3 = (2.20)J UxiThe empiricism of the above model lies in parameters C, C1, C2, k, o and o.Launder et al. (1972) made extensive examinations of free turbulent flows to estimatevalues for these parameters. The values employed in this study are listed below:C,., C1 C2 k0.09 1.44 1.92 1.0 1\/2 0.4187 1.0(C2C1) gTable 2.1: Parameters in the k \u2014 e model.To summarize, with the use of the k \u2014 e turbulence model, the governing equationscan be written in the following general form:\u2014(pU\u2019I\u2019) = \u2014 (i) + (2.21)Equations for continuity, momentum, species concentration, turbulent kinetic energy,and dissipation rate of turbulent kinetic energy are presented in Table 2.2, in terms ofa general dependent variable \u2018, a diffusion coefficient 1\u2019,, and a source term S. InTable 2.2 the diffusion coefficient for scalar is written as \/teff\/0q, by assuming that thevalues of both the laminar Schmidt number representing the ratio t\/) and its turbulentcounterpart, u, have values near unity, and that kUt is usually much greater than ii inmany flow situations.This turbulence model is attractive because the equations for laminar and turbulentflows have the same form; only the diffusivity coefficients 1\u2019 are calculated differently.Also, the model has been tested in a variety of flow problems and was found to be robustChapter 2. Dynamics of Turbulent Jets: A Literature Review 19s,r1 0 0U, \u2018i = 1,2,3 Peff \u20148P\/8x + \u00f6\/\u00f6Xj[(iUeff(OUj\/\u00f6Xi)]I\u2019eff\/\u00b0k 0k fLeff\/JJg G\u2014pe6 \/\u2018eff\/\u00b0c (c\/k)(C1G\u2014 C2pe)P =p+ pkG =\/.t6ff[(t9U\/Xi)+ (aU\/ax)](\u00f4U2\/th)Table 2.2: Terms representing the time-averaged Navier-Stokes equations with the k \u2014 6turbulence model and a transport equation for an inert scalar.in many cases. Despite the empiricism, the model gives, at least qualitatively, reasonableresults in the majority of cases.Having discussed a mathematical model for turbulent jets, attention is now turnedto the properties of these jets based on experimental observations. The experience withthis turbulence model applied to the flow field simulation of these turbulent jets will bedescribed later in this chapter.2.2 Jets in Quiescent EnvironmentsJets issuing into a relatively quiescent environment are found in many engineering applications. Some examples are jets used in thrust augmenting ejectors for VTOL aircraftand jets associated with air conditioning devices. Primary air system jets in kraft recovery boilers are free jets because there is little crossflow in the lower part of the furnace.We will first describe the characteristics of an isolated axisymmetric and rectangular jet.This will be followed by a description of multiple-interfering free jets.Chapter 2. Dynamics of Turbulent Jets: A Literature Review 20(a)2.2.1 Axisymmetric Jets(b)Axisymmetric jets refer to jets issuing from circular orifices. They are the simplest typeof free jets and their characteristics have been investigated in detail by Wygnanski andFiedler (1969). The flow field characteristics depend upon the inlet geometry leading tothe jet exit and the magnitude of the turbulence intensity at the jet exit plane. If theReynolds number of the jet is high enough for fully turbulent flow to exist a short distancedownstream from the jet exit, then the flow field is not dependent on the Reynoldsnumber. The orifice for such a jet is shown in Figure 2.1(a).Near the jet exit is the potential core region where the velocity is uniform since themixing initiated at the boundaries has not yet permeated into the jet core. Downstreamof the potential core, the jet is characterized by the momentum transported per unittime over each cross-section. By applying the principle of conservation of momentumand assuming similarity in the velocity distribution in the fully developed region of thejet flow, dimensional analysis shows that, at zero pressure gradient, the width of the jetis proportional to y and the center-line velocity decays as y1.Figure 2.1: Orifices for (a) axisymmetric and (b) rectangular jets.Chapter 2. Dynamics of Turbulent Jets: A Literature Review 212.2.2 Jets from Rectangular NozzlesThe investigations by Sfeir (1979), Krothapalli et al. (1980), Marsters (1981), and Quinn(1991) reveal a wealth of information about the flow fields associated with rectangularorifices. A diagram of such an orifice is shown in Figure 2.1(b). The flow fields are morecomplex because of their three-dimensional nature. Nozzle shape and aspect ratio playa major role in the development of rectangular jets. When the decay of the axial meanvelocity is used to describe the flow field, the flow is characterized by the presence ofthree distinct regions. These regions are the potential core region, a characteristic decayregion, and an axisymmetric region. The characteristic decay region is influenced directlyby the shape and aspect ratio of a slot or nozzle, and originates when the shear layersin the plane containing the short dimension of the nozzle meet. Correspondingly, theaxisymmetric region originates approximately where the two shear layers in the planecontaining the long dimension of the nozzle meet. A more detailed description of theflow field, together with graphical illustrations, can be found in the works of Sfeir andKrothapalli et al.A prominent observation for this type of jet is the rapid spreading in the directionof the small dimension of the jet orifice. Sfeir describes this straining of the jet in thex \u2014 z plane as due to the presence of vortex rings of elliptical shape surrounding thejet. The velocities induced by these vortex rings have also been suggested as the causeof the csaddleback shape in the streamwise (y) velocity distribution along the spanwise(x) direction.The effects of orifice aspect ratio on the flow and mixing characteristics for turbulentfree jets issuing from sharp-edged rectangular orifices were examined by Quinn (1991).This is of interest to recovery boiler applications because slender orifices of differentaspect ratios are often used. Quinn examined jets from sharp edged orifices with aspectChapter 2. Dynamics of Turbulent Jets: A Literature Review 22ratios 2, 5, 10 and 20. The results show that the saddle-shaped mean streamwise velocityprofiles are present in many cases, especially when the aspect ratio is large. The saddle-shaped profiles are characterized by steep gradients which give rise to a high productionof turbulence and thus facilitates effective mixing. The highest shear-layer values ofthe turbulent kinetic energy are found for the aspect ratio 20 jet. The hypothesis thatmixing is more effective for high aspect ratio jets is supported by the far-field meanvelocity decay rates, which are observed to increase with increases in slot aspect ratio. Itis also supported by observing the lengths of the potential cores in various cases, whichbecome shorter with increases in slot aspect ratio. Furthermore, the spreading rate of thejet, which is determined by the half-velocity widths in both central planes of symmetry,increases with increasing slot aspect ratio. This latter observation implies better far-fieldmixing and is consistent with the higher velocity decay rates.2.2.3 Twin Jet FlowThe above description for a single axisymmetric or rectangular turbulent free jet revealsmany interesting flow characteristics. However, in many applications, jets are issuedin a multiple jet configuration where the interaction between neighboring jets can besignificant. A basic unit of this multiple jet configuration is a pair of jets located side-by-side. A study of this flow can lead to some understanding of the nature of multiplejet interactions. A careful experimental investigation of the flow field associated withsuch twin jet flow was performed by Miller and Comings (1960). They studied subsonicturbulent slot jets at = 22 rn\/s which corresponded to Re = 17800 based on slotwidth. The slot spacing was S\/W = 6, where W denotes the width of a slot, and Sdenotes the spacing between the slots. Under these conditions, the experimental resultsshow that the flow is highly symmetrical about the symmetry plane between the pairof jets. This observation suggests that the lateral interaction of the jets does not causeChapter 2. Dynamics of Turbulent Jets A Literature Review 23instability in the flow field. Another observation is that the entrainment in the confinedregion between the jets produces pronounce negative pressures in this region. Hence, thejet streamlines are deflected inwards as the jets are forced towards one another. Anotherconsequence of the negative pressure in this confined region is that the momentum ofthe merged flow is less than the combined momentum of the flow emerging from the twoslots. Downstream, the merged jets lose their individual identities and behave as a singletwo-dimensional jet with subsequent conservation of the remaining momentum.The deflection of jets towards one another is significant only for slot jets. The investigation by Alexander et al. (1953) reveals that for a pair of axisymmetric jets, because theflow around the jets is vented to the surroundings, the jets are only slightly deflected towards one another and the downstream momentum is very nearly equal to the combinedmomenta of the jets.2.2.4 Multiple Interfering JetsKnystautas (1964) performed a comprehensive investigation on the interaction of multipleround jets in a quiescent environment. Figure 2.2 shows the configuration under study.The combination of such a row of jets was labelled a quasi-two-dimensional jet, andthree values of jet spacings were examined: S\/D = 1.5, 2, and 3, where jet diameterD = 0.0127 m (0.5 inches). Based on D, the Reynolds number tested range from 6090 to52800. These values are large enough so that the flow field was insensitive to variationin the Reynolds number.An important objective of the investigation was to determine the downstream distancefrom the jet exit plane where the amplitude of the undulating mean velocity profile hadfirst decreased to the point where it could be considered, effectively, two-dimensional.Let this distance be denoted as Y2d, and as the following data from the investigationindicate, ma was found to increase with increasing jet separation.Chapter 2. Dynamics of Turbulent Jets: A Literature Review 24y2d\/D = 20 when S\/D = 1.5y2d\/D = 28 when S\/D = 2.0y2d\/D = 37 when S\/D = 3.0In the two-dimensional region of the quasi-two-dimensional jet, the streamwise velocity profiles are self-similar. In addition, the inverse square of the mean center-line velocitydecays in proportion to the downstream distance, in accordance with the properties of aplane jet.The turbulence characteristics of jets from an array of rectangular lobes were investigated by Krothapalli et al. (1980). The aspect ratio L\/W of each lobe was 16.7 andthe spacing S\/W = 8, where W = 3 mm was the small dimension of a lobe. The meanjet velocity was 60 rn\/s corresponding to Re = 12000 based on W. For such a configuration, neighboring jets do not attract one another, each jet mixes with ambient air quiteindependently, and the jets merge completely for y\/W 60. In the downstream regiony = y2Figure 2.2: Development of a row of circular jets.Chapter 2. Dynamics of Turbulent Jets : A Literature Review 25where the jets are merging, the interaction between the jets results in a lower turbulencelevel compared to a single jet at corresponding locations.2.2.5 Primary Jets in Recovery BoilersThere are differences between primary jets in recovery boilers and the conditions studiedin the above mentioned investigations. For instance, the primary jets are located on allfour walls in a boiler so jets from adjacent walls may interact. Also, the presence of thechar bed could affect the flow field by limiting jet penetration.A study of multiple rectangular free jets was carried out by Sutinen and Karvinen(1992) with applications to kraft recovery boilers in mind. A non-isothermal calculationwas performed to simulate the flow field for a row of jets in situations similar to thosefound for the primary level jets. The goal was to estimate where the two-dimensionalassumption for the primary jet flow would be valid for their char bed simulation program.The jet velocity was 40 rn\/s at 400 K and the furnace temperature was 1200 K, while thebed surface temperature was 1500 K. The distance between adjacent nozzles was set at0.5 rn and the nozzle dimension was 0.10 m by 0.25 m. It was found that the primary jetsmerge almost completely at about 1.5 m from the injection plane. Sutinen and Karvinenthen concluded that the gas flow above the char bed is effectively two-dimensional.All the studies discussed have illustrated the fact that closely spaced jets will mergequickly not far from the exit plane. Therefore, it is feasible to numerically model primaryjets by using suitable slot equivalences. The study of primary jet simulation using slotswill be the subject of Chapter 4.Chapter 2. Dynamics of Turbulent Jets: A Literature Review 262.3 Single Jet in a CrossflowUnlike primary jets, the higher level secondary and tertiary jets operate in an environmentwhere there is a crossflow about the jets. This situation also occurs in many other types ofengineering applications that include effluent dispersement, jets in combustion chambers,and jets for turbine blade film cooling. In this study, the crossflow is also labelled thecross-stream, mainstream, or main flow. The degree of crossflow uniformity and theimportance of confinement varies for different cases. As well, the flow Reynolds numberis different for different applications, and for subsonic flows this can range from slowlaminar jets to higher speed turbulent jets. The Reynolds numbers of secondary andtertiary jets in a recovery boiler are high so that they are in the turbulent regime.The basic features of a single jet discharged into a crossflow have been thoroughlystudied and form the basis for the interpretation of multiple jets dynamics. Detailedreview work can be found in Simitovi\u00e9 (1977), Sherif (1985), and Blackwell (1990). Inthis section, we describe the phenomena associated with a single jet in a crossflow. Numerous experimental, analytical, and theoretical investigations have been performed byresearchers to study the dependence of gross characteristics such as jet trajectory, penetration, and spreading on injection parameters. Detailed investigations have also beencarried out on the nature of turbulent entrainment and mixing, as well as the characteristics of vortex structures associated with the jet. These results are outlined in thefollowing subsections.2.3.1 General Shape of a Single Jet in a CrossflowThe general appearance of a jet discharging into a crossflow is shown in Figure 2.3. Asthe jet is discharged into a crossflow, the jet path is deflected towards the directionof the main flow. This arises because the jet creates a blockage in the crossflow andChapter 2. Dynamics of Turbulent Jets: A Literature Review 27kidney-shaped jetwith internalvortex pairucupflow combustiongasesFigure 2.3: A schematic representation of a jet in a crossifow.consequently, the flow immediately ahead of the jet decelerates, causing an increase inpressure. Downstream, a low pressure wake region occurs and this, combined with theincreased upstream pressure, provides a force that deforms and bends the jet. The jetacts like an obstruction to the crossflow but the boundaries of the jet are compliant andentraining. The rapid entrainment of the cross-stream fluid into the jet also helps to pushthe jet over in the cross-stream direction because of the addition of momentum broughtin by the entrained fluid.It is well known that downstream from the orifice in the x-direction, the cross-sectionof an isolated jet assumes a kidney shape, as illustrated in Figure 2.3. The developmentof the kidney shape is a consequence of the intensive mixing between the jet fluid andthe cross-stream fluid. The shear experienced by the jet, due to the crossflow, causes theformation of a turbulent shear layer. Peripheral particles of the jet, having less velocitythan the particles of the core, are more forcefully bent by the deflecting flow away fromportopeningChapter 2. Dynamics of Turbulent Jets : A Literature Review 28the initial direction and are moved along more curved trajectories. This movement ofthe peripheral fluid particles leads to the development of a kidney shape in the jet\u2019scross-section.2.3.2 Jet Trajectory and PenetrationNumerous experiments have been performed by researchers to study the penetration ofa jet in a crossflow. The extent of penetration is usually inferred from the jet trajectory.An important parameter that affects the jet penetration is the ratio of the jet momentumto the cross-stream momentum, defined as= PjetYt (2.22)PC UcThe investigation by Andreopoulos and Rodi (1984) reveals that for jets with J 4, thejet path is only mildly affected at the exit and penetrates into the crossfiow stream beforeit is bent over. The variation of the trajectory with injection parameters has been thesubject of many studies. The details of experimental methods and results are found inSimitovi\u00e9 (1977) and Blackwell (1990). A point to note is that the determination of jettrajectory can be different for different experimental methods employed. For experimentsthat use temperature measurements to locate the jet fluid, the trajectory is usuallydefined as the locus of the points of maximum jet temperature in the plane of symmetrydefined by the jet orifice axis and the direction of the crossfiow. When the velocity ismeasured, the jet trajectory is usually defined as the locus of points of maximum resultantvelocity in the same plane as previously defined. With flow visualization experiments,the trajectory may be defined as the median line between the visible boundaries of thejet fluid.The expression for the jet trajectory is usually correlated with the momentum fluxratio J. These various definitions for the jet trajectory lead to differences in the resultsChapter 2. Dynamics of Turbulent Jets: A Literature Review 29for the trajectory. For circular jets, in the case of normal jet injection into a relativelyunconfined crossfiow, the trajectories are found to be correlated with the following expression:y = aDbxcJdI (2.23)where D is the orifice diameter, a 1, b , c , and < d < . Dimensionalanalysis requires that b + c 1, a constraint on the equation. If the injection angle isnot 90\u00b0, then additional dependence on the angle has to be expressed.Effects of Orifice ShapeThe effect of orifice shape on jet penetration has been investigated by Hawthorne etal. (1944), Ruggeri et al. (1950), and Reilly (1968). These authors found that as theaspect ratio of the jet orifice increases, with the long side lying in the direction of themain flow, the penetration of the jet increases in many instances. This is due to thefact that as the aspect ratio increases, the rearward part of the jet is deflected less thanthe forward part. In addition, the area of the jet presented to the crossflow decreasesand this can lead to a reduction in the drag force exerted by the crossflow on the jet.However, for a given value of J, there appears to be an optimum orifice aspect ratio whichmaximizes penetration. Exceeding this optimum value, the increase in shear forces dueto the increased jet circumference nullifies the decrease in pressure forces due to thestreamlining of the jet and as a result the penetration is reduced.Effects of Inlet Velocity Profile and Turbulence IntensitySimitovi\u00e9 (1977) reported a study by Kamotani and Greber (1974) on the effects ofturbulence intensity and velocity distribution in the incoming jet flow on the temperature and velocity trajectories. They compared two basic types of jet inlet structures,Chapter 2. Dynamics of Turbulent Jets: A Literature Review 30one with a uniform velocity profile and 0.3% turbulence intensity and the other witha fully-developed pipe flow profile and a level of turbulence of 2.4% on the axis. Theyfound that the penetration of the jets for the two cases differed by about 10%, which maybe regarded as insignificant, but no details were given for the nature of the difference. Itwas not clear whether this effect was caused by the change of turbulence structures, orthe inlet velocity profiles, or both.2.3.3 Jet Width and ThicknessThe width and thickness of a jet can be determined from flow visualization measurements,and together these two properties characterize the extent of the mainstream that isaffected by the jet. The observations by Rajaratnam and Gangadharaiah (1981,1982)reveal that for the velocity ratio, V\u20ac\/Uc, in the range from 2.7 to 23.4, the widthand the effective thickness of the jet increase in proportion to the distance along thetrajectory from the nozzle. Also, it appears that the kidney shape of the jet becomesless pronounced as the velocity ratio increases and the jet becomes more rounded fardownstream. A number of empirical correlations describing the width, thickness, andthe profile of the jet are presented in the review work by Blackwell.2.3.4 Effects of ConfinementConfinement in either the lateral (z direction, Figure 2.3) or transverse (y) direction willlimit the jet spreading and affect the characteristics of jets in crossflows. Impingementonto the wall facing the jet occurs if either the jet velocity is too high or the opposingwall is too close to the jet entrance. If there is no impingement, then it has been observedthat the jet trajectory has little dependence on the extent of transverse confinement, andthe correlation shown in Eq.(2.23) can be used to estimate the trajectory. The lateralconfinement will modify the development of the cross-sectional shape of the jet.Chapter 2. Dynamics of Turbulent Jets: A Literature Review 312.3.5 Results from Detailed Flow VisualizationUp to this point, the discussion has been on the more easily observed properties concerning jets in crossflows. Attention is now shifted to the more detailed flow structures of ajet to obtain a better understanding of the mixing process.The detailed flow visualization experiments performed by Fric and Roshko (1989) andSmith et al. (1993) reveal the nature of the intensive mixing between the crossflow andjet fluid. This intensive mixing, in most practical situations, is due to the flow beinghighly turbulent. Turbulence arises, in part, from the instability of the laminar shearlayers at high Reynolds numbers which leads to a rapid formation of a turbulent shearlayer around the periphery of the jet. Smith et al. (1993) showed many photographs displaying the intense instantaneous mixing caused by the pronounced large-scale intrusionof mainstream fluid around the jet periphery.Downstream of the injection plane, the flow field is observed to be dominated by several vorticity structures. These vortex systems are believed to affect the entrainment andmixing characteristics between the jet and the cross-stream. Fric and Roshko (1989) provided detailed visualizations of the near-field and identified four main vorticity structureswhich comprise the flow. They are as follows:(a) Distorted shear layer ring vortices at the circumference of the bending jet.(b) The inception of the counter-rotating pair of vortices which eventually dominatesthe far field jet structure.(c) A system of horse-shoe or collar vortices at the crossflow wall.(d) A system of wake vortices nearly aligned with the initial jet direction.It is commonly believed that the vorticity structures (a) and (c) have only a small influence on the shape or global behavior of the jet. The shape of the jet is influenced mainlyChapter 2. Dynamics of Turbulent Jets: A Literature Review 32by the system (b). The mixing of jet fluid and cross-stream fluid can be influenced byboth systems (b) and (d).A key observation made by Fric and Roshko (1989) is that most of the source of thewake vorticity is in the crossflow boundary layer. The crossflow boundary layer separatesat the downstream side of the jet to subsequently feed the wake. The fact that the wakevorticity comes from the crossflow boundary layer fluid and not from the jet fluid is ofsignificance when considering the mixing in this flow. This observation suggests that,although the wake region is highly turbulent, it does not add substantially to the mixingof crossflow fluid with jet fluid; the wake contains essentially no jet fluid. This was quitecontrary to some long held hypotheses about the wake vortices being shed from the jet,and this observation had been subsequently confirmed by Smith et al. (1993). The latterinvestigators applied a planar laser-induced fluorescence visualization technique whichallowed for a very detailed examination of the instantaneous structure and mixing of ajet in a crossflow. They observed that only a very small amount of jet fluid could enterthe wake through wake vortices in the region of the jet that was undergoing curvature.The above results elevate the importance of the role of the counter-rotating vortexpair in contributing to the mixing between the two fluid streams. In addition, Kulisa etal. (1992) suggest that this vortex pair is responsible for the particular jet cross-sectionalshape and the vortex motion serves to transport fluid around the perimeter of the jetand hence facilitates mixing. The discussion in the next section is focussed on this vortexpair because of the significance of its dynamics.2.3.6 The Counter-Rotating Vortex PairThe counter-rotating vortex pair is a remarkable characteristic of a jet in a crossifow andhas been examined by many investigators. Its origin was discussed by Andreopoulos andRodi (1984): the vortex pair evolves from the shear layer vorticity of the jet; that is, itsChapter 2. Dynamics of Turbulent Jets: A Literature Review 33source is in the vorticity issuing from the nozzle. As most of the vorticity issuing fromthe pipe is reoriented and stretched by the flow, it bundles up into a pair of vortex tubesbound to the lee side of the jet and accentuates the kidney shape of the jet.The vorticity generated in the jet is propagated downstream by the jet and intensified by the interfacial shear of the initially orthogonal jet and the crossflow stream.This vorticity is also diffused by the turbulent transport process as shown in the vorticity equation (2.13). The characteristics of this vortex pair have been examined bymany researchers. Fearn and Weston (1974) correlated the vortex strength with different injection rates and positions along the jet path. Their empirical model was used tobuild a description of the velocity field in a cross-sectional plane. The observations byRathgeber and Becker (1983) reveal that the maximum concentration in the jet is at thecenters of the two vortices and can be 30% to 75% higher than the concentration on thejet axis. By assuming that the centers of the two low-pressure cells correspond to thecenters of the two vortices of the bound vortex system, Rajaratnam and Gangadharaiah(1983) developed correlations for the maximum pressure defect, the separation distance,and the radial distance of the vortices from the axis of the deflected jet. Recently, thedetailed visualization results by Smith et al. (1993) reveal that the transient developmentof the counter-rotating vortex pair can be non-symmetric with significant undulations inthe streamwise direction. The variation in the vortex strengths between the vortex pairis proportional to the velocity ratio. The two ends of the kidney profile oscillate at awavelength comparable to the local jet diameter.Results for the vortex pair have also been obtained by numerical methods. Using aversion of the Reynolds stress closure model, Sykes et al. (1986) showed that the evolutionof the vortex pair is from the original vorticity in the sides of the jet. This result isobtained by noting the Lagrangian distortion of vortex lines through the calculation oftheir trajectories in the three-dimensional vector vorticity field. For impinging jets in aChapter 2. Dynamics of Turbulent Jets: A Literature Review 34crossflow, the numerical results by Catalano et al. (1989) using the k \u2014 model show thepersistence of the vortex structure even after impingement.2.3.7 Turbulence CharacteristicsThe turbulence characteristics of a jet in a crossflow have been investigated by manyresearchers. This type of flow field is an example of a complex free turbulent shear flow.The primary goal of these investigations is to examine the structure and characteristicsof velocity fluctuations. The data could also be used to guide the development, improvement, and evaluation of better prediction methods and turbulence models for this typeof flows.In relation to mathematical modelling of turbulence characteristics, the experimentalresults have provided insight into the applicability of turbulence viscosity models suchas the one indicated by Eq.(2.14). The results of Andreopoulos and Rodi (1984) showthat not all components of the Reynolds stress tensor can be described realistically bya scalar turbulence viscosity model. Specifically, the component which influences thelateral spreading of the jet may not be described well by the simple effective viscosityconcept. More favorable results were reported by Pietrzyk et al. (1988) for jets inclinedto the direction of the cross-stream. They found coincident peaks in the mean velocitygradient and the turbulent quantities. This points to the possible use of a turbulenceviscosity model for this type of flow field. The turbulence field for an impinging jet ina crossflow has been studied by Catalano et al. (1989), who found that the turbulencefield was highly anisotropic in the initial region, although there were tendencies towardsisotropy further downstream. This latter result suggests that a turbulence model whichtakes anisotropy into account is needed for the accurate prediction of the flow field in theinitial region.Chapter 2. Dynamics of Turbulent Jets: A Literature Review 35_D jIiIUc\u2014\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/-uc\u2014\u2019..\/\/\/\/\/\/\/,\u2018777A V\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/Figure 2.4: A schematic view describing a row of jets discharging into a confined crossfiow.2.4 Multiple Jets in a CrossflowExperimental evidence shows that for multiple jets in a crossflow, each jet behaves individually until just before merging. The configuration is depicted in Figure 2.4. Theunderstanding acquired for a single jet in a crossfiow cannot be applied directly to multi-pie jet configurations, due to the complexity introduced by jet interaction. Nevertheless,the characteristics of the single jet case may still be useful to interpret some of the observations for multiple jets. This section describes some of the relevant background materialin the study of a row of jets discharging into a confined crossfiow.HChapter 2. Dynamics of Turbulent Jets: A Literature Review 3635xID302520151050J = 100Figure 2.5: Effect of jet spacing on jet trajectory, J = 100, Ivanov (1959).2.4.1 Effects of Varying Jet Spacing and Momentum RatioAn early investigation into the penetration of a row of jets, when the spacing betweenjets varies, was made by Ivanov (1959) and his results are quoted by Niessen (1978). Thetrajectories of jets at J = 100 were measured for jet spacings of S\/D = 4, 8, and 16,and the results are illustrated in Figure 2.5. No details were given for the dimension ofconfinement H. A notable decrease in the jet trajectory is observed as S\/D is reducedfrom 16 to 8. The interpretation by Niessen suggests that as the jet spacing is reduced, thejets merge into a curtain, like that produced by a plane jet. It appears that the blockageeffect of the curtain impedes the flow of the cross-stream around the jets and increasesthe effective deflecting force due to the crossflow. There is only a minor reduction inthe trajectory as the jet spacing is reduced from 8 to 4. This observation is due tothe apparent rapid merging of jets close to the injection orifices already occurring atS\/D=8.25 30ylDChapter 2. Dynamics of Turbulent Jets: A Literature Review 37NASA sponsored a series of systematic investigation into the characteristics of multiple jets in a crossflow. The applications were aimed at the design of air mixing systemsfor combustion chambers but, because the experimental conditions were turbulent, theresults should also be applicable to other systems of turbulent flows having differentphysical dimensions.Kamotani and Greber (1974) studied the dynamics of multiple jets in a linear arraywhen the spacing S between jet nozzles and the confinement height H were varied. Thearrangement of this jet configuration is shown in Figure 2.4. The values of J rangedfrom 8 to 72 and jet spacings were 2, 4, 6 and 10 nozzle diameters. In their work,they defined the center-plane jet trajectory as the locus of maximum speed at each cross-stream plane intersecting the jet center-plane. They evaluated the penetration in differentcases by comparing those jet trajectories. Results that are relevant to jet penetration aresummarized as follows:(1) At very large jet spacings, S\/D > 10, each jet behaves independently and the effectof neighboring jets is minimal. Also, the jet penetration is not sensitive to theextent of the confinement.(2) At moderate jet spacings, 10 > S\/D > 2, the jet penetration is only mildly affectedby the presence of a confinement.(3) At the close jet spacing S\/D = 2, the jets merge rapidly and the resulting flow fieldresembles that of a slot jet. The penetration depends strongly on H.(4) There is a monotonic decrease in penetration as S\/D is reduced until a critical value(S\/D)cr is reached.Their results on jet penetration are displayed in Figure 2.6, and they are consistentwith those of Ivanov (1959). The results indicate that there is increasing jet deflectionChapter 2. Dynamics of Turbulent Jets: A Literature Review 38(a) J=8 (b) J72xID20 xID1510500Figure 2.6: Effect of jet spacing on jet trajectory, H\/D = 24, J = 8 and 72, Kamotaniand Greber (1974).by the crossfiow as the orifice spacing decreases until a certain critical value is reached.Kamotani and Greber offered an explanation based on the interaction of vorticity ofneighboring jets. As discussed earlier, in the far field, a jet in crossflow may be representedby a pair of counter-rotating vortices. The interaction of neighboring vortices from twoneighboring jets has two effects. First, the vortex strength of each jet will be reduceddue to cross diffusion of vorticity of opposite signs. Secondly, the velocity field inducedby each vortex will drive its neighboring vortex down towards the injection wall. Thisresults in less penetration of each jet. Thus, the trajectories of a row of jets will becomemore deflected as the spacing ratio decreases. This hypothesis was numerically verifiedby Huang (1989), who took the velocity field induced by the counter-rotating vortexsystem into account in his calculations.Kamotani and Greber suggested that when the jet spacing is below the critical value(S\/D)cr, the jets start to interfere with the entrainment of cross-stream fluid of theirneighboring jets. This interference results in a lack of entrainment by each jet andconsequently a slower decay of the initial momentum flux in the y direction of the jet.105 10yID510 15ylDChapter 2. Dynamics of Turbulent Jets: A Literature Review 39This slower decay, together with the increase in the momentum flux per length of thearray of jets as the spacing becomes smaller, cause the jets to be more resistant to thedeflection by the crossflow. The value of the critical spacing ratio (S\/D). was found toincrease with J. They estimated that (S\/D),. 2 for J 8 and (S\/D),. 4 for J = 72.2.4.2 More Comprehensive Parametric VariationsA comprehensive study of the effects of geometric and operational parameters on thejet penetration and mixing of multiple cold air jets into a ducted subsonic heated mainstream flow was performed by Walker and Kors (1973) and Holdeman and Walker (1977).The application was for air jet mixing in combustion chambers. In their experiments,the momentum flux ratio J ranged from 6 to 60. A variety of jet configurations and flowconditions were tested, which include the variation of orifice size, orifice spacing, extentof transverse confinement (H), and different levels of turbulence in the cross-stream. Intheir experiments, the spacing ratio S\/D ranged from 2 to 6 while the confinement ratioH\/D ranged from 4 to 16. The mixing effectiveness for different cases was assessed byexamining the uniformity and skewness of temperature distributions at positions downstream of the injection orifices. The main results are summarized below:(1) Single jet correlations do not adequately describe multiple jet results.(2) The jet to mainstream momentum ratio is the single most important operatingvariable influencing jet penetration and mixing.(3) The absolute momentum flux level does not influence jet penetration or mixingsignificantly.(4) The effect of the turbulence level on jet penetration and mixing was insignificantwithin the range of turbulence levels examined.Chapter 2. Dynamics of Turbulent Jets : A Literature Review 40(5) The spacing between orifices has a significant effect on lateral spreading of thejets, jet penetration, and jet mixing. Closely spaced orifices (S\/D = 2) inhibit jetmixing.(6) For both low and high momentum flux ratios, temperature center-line penetrationdepth does not increase significantly with increasing x\/H beyond a short downstream distance. Instead, a flattening of the temperature profile occurs with increasing x\/H for both low and high momentum flux ratios.(7) The temperature profile does not change shape if S\/H and J are held constant andthe orifice diameter is varied. Only the magnitude of the temperature distributionchanges.Item (6) is important for the establishment of the following mixing criterion for combustion chamber application: effective mixing between the jet and the mainstream isachieved when the jet penetrates to approximately half way across the chamber as rapidlyas possible. This objective serves to establish a symmetrical temperature distributionprofile across the width of the chamber. The action of turbulence mixing flattens thetemperature profile so that a uniform distribution is found downstream. The observationstated in item (7) is significant. It suggests that for a given momentum flux ratio, thereexists a value of S\/H such that nearly even temperature distribution across the chambercan be achieved, with the distribution not skewing too much to one side of the chamberor to the other. The hole size may then be chosen based on the desired jet mass-flow-rate.The fact that a consistent mixing profile can be obtained when S\/H and J are heldconstant suggests that a correlation involving these quantities can be derived to leadto a desirable temperature profile or mixing profile based on the criterion explained inthe last paragraph. Such a correlation was developed by Holdeman et al. (1984), byexamining the experimental results from various cases. They observed that at a givenChapter 2. Dynamics of Turbulent Jets A Literature Review 41jet-to-mainstream mass flow ratio, similarity in the temperature profiles in the x\u2014yplane can be obtained, independent of orifice diameter, when the spacing to confinementratio S\/H is inversely proportional to the square root of the momentum flux ratio. Inother words, a consistent extent of jet penetration is obtained when the parameters 5,H and J are related byS CH=(2.24)For their results with cooling jets in a hot crossflow, they found that a value of C = 2.5leads to a jet fluid distribution that is approximately centered across the channel height.A value of . \/J that is a factor of two greater or smaller than this optimal valuewill result in over-penetration or under-penetration, respectively. The observation thatthe spacing and momentum flux ratio can be coupled in such a simple way for optimalpenetration is useful. It allows for a simple estimation of the penetration and mixingperformance. Equation (2.24) also illustrates that it is more appropriate to use largewidely spaced jets at low momentum flux ratios, while small closely spaced jets are moreappropriate at high momentum ratios.Extension of the above jet mixing study to two-sided injection was carried out byHoldeman et al. (1984). The results of their experiments support the hypothesis that aconfiguration that mixes well with one-side injection performs even better when everyother orifice is moved to the opposite wall. That is, two-sided injection with jets staggeredrelative to those in the opposing row could give rise to very rapid and effective mixingwith the cross-stream.2.5 Prediction Methods for Turbulent JetsThe complexity of the flow fields associated with turbulent jets causes difficulties withpredictions. This complexity increases when multiple jets are involved. Many of theChapter 2. Dynamics of Turbulent Jets: A Literature Review 42available prediction methods are summarized and presented in Rajaratnam (1976) andDemuren (1986). These methods may be divided into three broad classes, as empirical,analytical, and numerical, in ascending order of computational complexity. Each of thesemethods will be briefly discussed.2.5.1 Empirical ModelsThe development of empirical models depends largely on the correlation of experimentaldata for properties such as jet penetration and spreading. The accuracy of the model isvalid within the range of the database used for the correlation. Empirical models havebeen used often for the determination of jet trajectory for a single jet in an unconfinedcrossflow. The trajectory correlation given in Eq.(2.23) is an example of this type ofmodel. Besides jet trajectory, characteristics such as the jet width, mean velocity profile,and mean concentration profile have all been empirically correlated for a single free jetand a single jet in a crossflow. These models offer simple methods to obtain first-orderestimates and a qualitative picture of the evolution of the jet structure as it leaves theorifice.2.5.2 Analytical ModelsAnalytical models refer to the group of models where physical phenomena associated withthe flow are modelled with mathematical relations which are empirical to varying degrees.These relations are then used to simplify the governing differential equation system suchas in Eqs.(2.1O-2.12). A loose criterion for a model to be under this classification is thatthere should not be too much computing effort required to solve the simplified equationsystem.An example of analytical models can be found in the theoretical predictions for theshape of the mean velocity profile of an axisymmetric turbulent free jet. The classicalChapter 2. Dynamics of Turbulent Jets : A Literature Review 43results by Toilmien (1926) and Goertler (1942) were obtained by assuming characteristicsof the turbulence. Tollmien applied the mixing-length hypothesis for the determinationof the Reynolds stress while Goertler assumed that the eddy viscosity relating theReynolds stress to the mean velocity gradient to be proportional to the product of thecenter-line velocity and the half width of the jet. Their results compare well with experimental data.Demuren (1986) presented examples of analytical models used in the prediction of jettrajectory for a jet in a crossfiow. In these examples, integral equations are derived eitherby considering a balance of forces acting over an elemental control volume of the jet or byintegrating in two spatial directions of the three-dimensional partial differential equationsgoverning the turbulent jet flow. The resulting set of ordinary differential equations canthen be solved analytically or numerically. Empirical relations are needed for physicalphenomena such as pressure drag, entrainment of cross-stream fluid, and spreading rates.Because of the use of integral equations, these models presented by Demuren are calledintegral models. With ingenuity and experience, good comparison between predictionsand experimental results can be obtained as well as insight into the nature of the flow.Another variant which may be considered an analytical model is the inviscid three-dimensional vortex sheet model for the problem of a jet in crossflows. This predictionmethod is favored by some applied mathematicians where certain characteristics of theflow field are assumed to simplify the equation set. In this vortex sheet model, the flowswithin and without the jet are assumed to be potential and the boundary between themis assumed to be a vortex sheet. The aim is to explain the mechanism responsible forthe deflection of the jet and the evolution of vortices. Usually, the perturbation methodis used to analyze the equation set where the perturbation parameter is the ratio ofcrossflow speed to jet speed, and this parameter is assumed to be small. Applicationsof the technique are found in Needham et al. (1988) and Coelho and Hunt (1989). TheChapter 2. Dynamics of Turbulent Jets : A Literature Review 44latter group used an empirical formula to include the effects of turbulent entrainment tostudy the dynamics of the near field of strong jets in a crossflow. Their analytical andexperimental results show that turbulent entrainment and the transport of the transversecomponent of vorticity largely control the dynamics of the jet and its bounding shearlayer in the near field of strong jets in a crossflow. In particular, they found that thediffusion of vorticity into the wake is weak and therefore the jet does not act on thecrossflow like a bluff body. The dominating mechanism for jet deflection is entrainmentrather than the pressure-drag effect. Such insight would be difficult to obtain using othermethods.2.5.3 Numerical ModelsA more realistic modelling of a complex three-dimensional jet flow is through the use ofnumerical models. Prediction methods based on numerical models involve the solutionto the partial differential equations governing the turbulent transport of mass or speciesconcentration. Usually, the time-averaged form of the equations is solved using the finitedifference technique. Turbulence models are required for closure of the equations andthese models usually follow from the eddy-viscosity concept. Other turbulence modelsare available. Full Reynolds stress modelling and large eddy simulations are frequentlyemployed in turbulence research. The next chapter is devoted to a discussion of a numerical methodology that is used in the present study.The attractiveness of numerical models stems from the fact that no assumptionsare required for the evolution of the jet within the flow domain, but this evolution isobtained as a result of the computations. The complexity of interactions among multiplejets makes numerical models the only suitable means for prediction. The accuracy ofthe results are dependent upon the quality of the mathematical model employed in therepresentation of the physical phenomena, the discretization scheme chosen, and theChapter 2. Dynamics of Turbulent Jets : A Literature Review 45treatment of boundary conditions. A popular model used in the practical numericalsimulation of turbulent jets is the k \u2014 model described in section 2.1. The followingsection discusses the applicability of such a model.2.6 Application of the k \u2014 c Model to Jet FlowsAs mentioned earlier in section 2.1, the two-equation k \u2014 c model was derived based onsimple turbulent flows such as boundary-layer type shear flows. The model may needmodifications before it can be used to analyze free jets or jets in crossflows.For free jets, McGuirk and Rodi (1977) applied a modified form of the k \u2014 6 modelin the parametric study of the flow field due to a single rectangular jet emitted fromdifferent orifices having the same size but with different aspect ratios. A modificationwas made to the empirical parameter C1 (see Table 2.1) in the 6-equation to reflect theobservation that the velocity decay rate affects the scale of turbulence eddies of the jet.The prediction of the velocity decay rate was good, but some finer flow features such asthe saddle-back velocity profile were not captured. Unfortunately, the extension of thismodification to the configuration of multiple jets is not simple. Some investigators likeSutinen and Karvinen (1992) used the standard k \u2014 e model for the prediction of the flowfield due to multiple jet interactions.There have been other studies on the applicability of the standard k \u2014 6 model to theproblem of jets in crossflows. An example is the study by Jones and McGuirk (1979), whocomputed a round turbulent jet discharging into a confined crossflow. It was found thatthere was good agreement for the gross features of the flow, such as the area of mixedfluid, the rate of jet dilution, and the jet trajectory. Results from other investigationsprior to 1984 were summarized by Demuren (1986) who showed many good qualitativeagreements between numerical predictions and experimental data. In addition, specificChapter 2. Dynamics of Turbulent Jets: A Literature Review 46verification studies were carried out by Claus (1985), Barata et al. (1988), and Claus andVanka (1990) to study the errors due to numerical discretization and the k \u2014 turbulencemodelling. The general conclusion is the same, namely that the two-equation model isgenerally adequate in predicting the gross features of the flow field such as jet penetration.However, the study by Claus and Vanka also pointed out that the model increasesthe effective viscosity of the fluid in the region near the jet entrance. This increasedviscosity works to damp-out any small scale structures such as horse-shoe vortices thatmay form. Moreover, the turbulence levels were generally underpredicted in comparisonswith data. Nevertheless, as stated by Claus and Vanka, the results of their comparisonsfor the turbulence intensity levels do not invalidate the use of a two-equation model forthis jet-in-crossflow geometry, even though the effect of counter-gradient transport asreported by Andreopoulos and Rodi (1984) was not accounted for by the model.2.7 Chapter SummaryWe have presented a brief description of a number of established results that are relevantto the study of air jet dynamics in a recovery boiler. Concerning the interaction ofmultiple free jets in an array, the results show that the jets merge relatively quickly andthat the lateral attraction caused by a decrease in pressure between each pair of jets isnot likely to be significant enough to cause flow instability. These results are valuable inour study of the proper modelling of primary level jets in a boiler.For the complex problem of multiple jet interactions with a crossflow, numerical techniques appear to be the only viable means for the prediction of the flow field. Numericalsolution methods have become more feasible with improvements in algorithms and computer technology. The use of the k \u2014 e model provides reasonable results for phenomenaassociated with turbulent jets and is adequate for engineering analysis.Chapter 2. Dynamics of Turbulent Jets: A Literature Review 47Regarding experimental results, the observations made by Holdeman and his coworkers reveal trends that are useful in providing guidance for establishing a schemewhere good jet mixing with the crossflow can be achieved. Specifically, they have deriveda correlation relating design and operation parameters that are useful in steering the jettrajectory down the middle of a channel. This correlation will be examined in Chapter6 for jet flows from rectangular orifices in conditions that are typical of those found ina recovery boiler, where the crossflow may exhibit a peaked non-uniformity, and wherethe momentum ratio J is high.Chapter 3Numerical Solution of the Navier-Stokes SystemIn this chapter, we discuss some numerical solution methods for solving problems withincompressible turbulent jet flow. Solutions are obtained by solving the time-averagedNavier-Stokes equations coupled with the k\u2014c turbulence model. The multigrid techniqueis needed to accelerate the convergence of iterative algorithms employed for solving thesystem of equations. We will discuss the care that is needed with the implementation ofthe multigrid procedure to solve fluid flow problems. Finally, we will apply this multigridalgorithm to simulate the flow field of a single square jet in a crossflow. The latter studyis carried out to validate the implementation of the algorithm and to examine the qualityof the mathematical model describing the flow.3.1 Discretization of Differential EquationsTo study the complex phenomena associated with multiple jet interactions, numericalsolution technique appears to be the only practical prediction method available. Thesystem of equations to be solved was described in Chapter 2. The form of the equation,as shown in Eq.(2.21), together with Table 2.2 are rewritten here for reference. The mathematical formulae describe the conservation of mass, momentum, species concentration,turbulent kinetic energy, and the dissipation rate of turbulent kinetic energy.= t- (r4-!-) + Sw (2.21)48Chapter 3. Numerical Solution of the Navier-Stokes System 49w1 0 0U1, i = 1,2,3 neff \u2014OP\/Ox1+ O\/OXj[(Peff(UUJ\/Oxi)]I1eff\/0qS 0lv \/leff\/Jk 0\u2014peC [\u2018eff\/Uc (c\/k)(C1G\u2014 C2pe)F =p+ pkG = ,Ueff[(OUI\/OXj) + (Ou\/ox1)](ocr\/oxTable 2.2: Terms representing the time-averaged Navier-Stokes equations with the lv \u2014 cturbulence model and a transport equation for an inert scalar.The above system of differential equations is cast in the strong conservation form,which is convenient for numerical integration. In this form, all terms arising from thedivergence operator are under differential operators, and when the differential equationsare integrated over a finite number of control volumes and properly discretized, theresulting fluxes from the strong conservation form would cancel in pairs at all interiorcontrol volume (cell) faces when summed, so that only boundary fluxes remain. Thisguarantees overall conservation of the transported quantity.A large system of difference equations will result upon the discretization of the equations. These difference equations must mirror the properties of the differential equations.That is, they must preserve the properties of conservation, boundedness, and transitivity. In addition, the numerical scheme for differencing should exhibit good accuracy andstability characteristics. Detailed discussions of the above topics can be found in theworks of Patankar (1980) and Syed et al. (1985).The above requirements on the differencing scheme have dictated that much care isneeded in the construction of the finite difference equations. The stability requirement,Chapter 3. Numerical Solution of the Navier-Stokes System 504-.\u2014 U velocities1 t V velocitiesScalar variablesIw p E\u2022 w \u2022 e \u2022SS. . .Figure 3.1: Variable arrangements in a staggered grid in two-dimensions.which will be discussed in section 3.5.1, has led to the use of a staggered grid arrangementfor the scalar and velocity variables. A two-dimensional example of the grid arrangementis shown in Figure 3.1. Also shown in the figure are two sets of labels. The labels {e,w,n,s}refer respectively to the east, west, north, and south faces of the scalar control volume.The labels {E,W,N,S,P} refer respectively to the East, West, North, and South scalarnodes relative to the scalar node P. Velocity variables follow similar labelling schemes.Extension to three-dimensions is straight-forward.The finite volume integration procedure is the first step in constructing differenceequations. Volume integration is performed for Eq.(2.21) around a control volume suchas the one shown in Figure 3.1 which represents a two-dimensional projection. Theintegration is carried out asJJJ{_(U) + -(pV) + dVdV + JJJSdv (3.1).N.nChapter 3. Numerical Solution of the Navier-Stokes System 51The volume integrals can be transformed to surface integrals using the divergencetheorem, yielding the following formula:Fe\u2014Fw+Fn\u2014F8+Ft Fb=JJJSdV (3.2)where {Fe,Fw,Fn,Fs,Ft,Fb} denote the sums of convective and diffusive fluxes throughthe {e,w,n,s,t,b} faces, respectively, of a three-dimensional control volume cell with t andb referring to the top and bottom cell faces. The expression for F isFe= Jf (pU_P) dydz (3.3)= Ce + Dewith Xe being the location of the cell\u2019s east face, andCe= ff (pU dydz (3.5)is the convective flux, while.De= JJ (_) dydz (3.6)is the diffusive flux. Similar expressions can also be derived for fluxes through otherfaces.In Cartesian coordinates, the diffusive flux is usually approximated by central differencing; in other words,Dc We(P_E) AYAzk (3.7)where Ip and \u201cE refer to the values of \u20181\u2019 at grid points P and B, respectively. Thequantity ax2 is the distance between P and B, while L\\y and \/zk are the dimensions ofthe east cell face. The expression for De is formally second order accurate in Sx.To approximate the convective flux, Eq.(3.5) is first written asCe PeUeeYjk (3.8)Chapter 3. Numerical Solution of the Navier-Stokes System 52where the subscript e for p, U, and I1 denotes that the values of those quantities are tobe taken at the east face of the cell. How \u2018e is approximated affects the characteristicsof the resulting finite difference equations. It is well known that the central difference approximation is numerically unstable if the cell Peclet number Fe Ce\/De is greater than2. The use of upwind differencing will lead to better numerical stability characteristics,but at the expense of accuracy due to numerical diffusion. More accurate differencingschemes are available. For example, the hybrid scheme combines both central differencingand upwind differencing. In addition, upwind weighted schemes such as the power-lawscheme and the exponential scheme have been derived to attempt better approximationsto the convective and diffusive fluxes. Detailed descriptions of these various schemes canbe found in the monograph by Patankar.Other schemes for the approximation of the convective flux have also been derivedaiming to reduce the effect of numerical diffusion in instances when the flow is skewedrelative to the grid. Examples include the QUICK scheme and the skew differencingscheme. The application of these schemes requires a larger differencing molecule, andmore care is needed to ensure numerical stability. Details of these aspects are presentedin the work of Syed et al.The discretization of the source term starts with its linearization into the followingform:fffs dV S\u2019Pp + S (3.9)The goal is to construct expressions for S and S so that the approximation is accurateand the resulting difference equations have good stability characteristics. The numericalstability of difference equations is contingent upon the condition that S must be nonpositive. In addition, for always-positive variables such as turbulent energy and itsdissipation rate, the source term must be discretized so that the variable will neverChapter 3. Numerical Solution of the Navier-Stokes System 53become negative during the iterative solution process. This goal will be achieved if S isalways positive. Patankar (1980) offers a more detailed discussion on the discretizationstrategy. For the k \u2014 e turbulence model, the discretization of the source terms for variousequations is presented in Table 3.1.After inserting expressions presented in Eq.(3.2) and Eq.(3.9) into Eq.(3.1), we obtainthe following general finite volume equation which is cast into the following quasi-linearform:with(ap\u2014 = aN\u2019I\u2019N + as\u2019I\u2019s + aE\u2019E + aww + aB\u2019I\u2019B + aT[\u2019T + Sap=a (i=N,S,E,W,B,T)(3.10)Expressions for the coefficients {a2} for various differencing strategies can be found inthe works of Patankar and Syed et al.Table 3.1: Discretization of the source terms.(3.11)Chapter 3. Numerical Solution of the Navier-Stokes System 543.2 Boundary ConditionsThe proper setting of boundary conditions is necessary for the successful simulation ofa flow phenomenon. The imposition of boundary conditions reflects either the needto represent a physical boundary for the flow such as a wall, or the need to simplifythe representation of the domain. The latter type of boundary conditions includes thesymmetry conditions and the inflow and outflow conditions. The introduction of theseboundary conditions usually affects the convective or diffusive fluxes entering the flowdomain. For example, at a symmetry plane, there is no convective or diffusive fluxacross the plane. At an outflow boundary, the diffusive flux is neglected since the flow isassumed to exhibit a local parabolic behavior. Because the values of the coefficients {a2}appearing in Eq.(3.1O) depend on the convective and diffusive fluxes entering the cell,boundary conditions are usually introduced through modifications to the coefficients forcells adjacent to the boundary.For simulations of turbulent flows using the k \u2014 model, additional complications ariseat a solid wall boundary. The reason is that the model assumes high Reynolds numberflow, a condition that is violated near a wall where the fluid has to be decelerated becauseof a no-slip condition at the wall surface. To overcome this deficiency in the model, asemi-empirical wall function treatment was introduced by Launder and Spalding (1974)to bridge the gap between the wall and the outer flow. The logarithmic law-of-the-wall isapplied to account for the effect of the wall on the shear stress distribution. Consequently,the diffusive fluxes for cells adjacent to the wall boundary are modified. Details of theimplementation of this wall function, together with other types of boundary conditions,can be found in the thesis by Djilali (1987).Chapter 3. Numerical Solution of the Navier-Stokes System 553.3 Numerical Solution ProcedureFor problems with large and complex domains, the discretization procedure describedabove leads to a large system of algebraic equations. This, coupled with the nonlinearnature of the equations, make it mandatory for the application of iterative or relaxationmethods to seek a solution to the system of equations. For the incompressible NavierStokes equations, there are primarily two types of iterative methods used by codes basedon the primitive-variable formulation: (i) a sequential or decoupled procedure such asthe SIMPLE algorithm described by Patankar (1980), and (ii) a coupled method suchas the symmetrically coupled Gauss-Seidel (SCGS) scheme described by Vanka (1986).These variations arise due to the fact that for incompressible flows, the pressure fieldis determined by the entire hydrodynamic equations system and not just by one of theequations. For the determination of the pressure field we need to resolve the couplingbetween momentum and continuity. Rodi et al. (1989) considered the use of these twosolution strategies to represent compromises between memory storage requirements andcomplexity of programming. These two solution methods are discussed in the next twosections.3.3.1 Sequential Solution MethodsThe algorithm for a sequential or decoupled procedure is usually of the predictor-correctortype. In the predictor step, a new velocity is computed by approximately solving themomentum equations by performing a few standard relaxation sweeps. In the correctorstep, both velocity and pressure are updated. The aim is to compute these changes suchthat the continuity equation is satisfied.A pressure-correction based scheme such as SIMPLE is one such example. In thisalgorithm, velocity correction is defined in terms of pressure changes. The use of theChapter 3. Numerical Solution of the Navier-Stokes System 56continuity equation leads to an equation for the pressure correction. At each iterationstep, the computed pressure and velocity fields are updated through the solution tothe pressure correction equation. The solutions to equations for scalar quantities andthe turbulence model are carried out by solving those equations sequentially after thehydrodynamic part of the system has been treated. An important advantage of this classof algorithm is the ease with which additional equations can be incorporated into thesolution scheme. Another advantage of this procedure is the low memory requirementas the coefficients constructed for an equation can be over-written once the relaxation ofthat equation is finished.Besides SIMPLE, there are a number of other sequential type solution schemes. A wellknown one is the PISO algorithm, which is a two-step procedure of pressure correctiontype. Other more refined algorithms, derived from SIMPLE, are the SIMPLER andSIMPLE-C schemes. The efficiency of these schemes for laminar and turbulent flowproblems have been assessed by Latimer and Pollard (1985), Jang et al. (1986), andWanik and Schnell (1989). The general observation is that for laminar problems wherepressure-velocity coupling is significant, PISO, SIMPLER and SIMPLE-C can all performbetter than SIMPLE, but the improvement to a system of equations coupled with aturbulence model is modest.3.3.2 Coupled Solution MethodsIn a coupled solution scheme, the hydrodynamic variables associated with a control voiume cell are updated simultaneously. For the staggered grid arrangement, this is carriedout by solving the momentum equations and the continuity equation for the velocitycomponents along the edges of the cell and the pressure in the center. A Gauss-Seideltype relaxation method can be used in this process, and the variables can be updated onecell at a time until all cells are considered. This method is known as the symmetricallyChapter 3. Numerical Solution of the Navier-Stokes System 57coupled Gauss-Seidel (SCGS) relaxation scheme. Alternatively, the equations can be setup in such a way that a line-relaxation technique can be employed to obtain the nodalpressure and the face velocities, simultaneously, along a complete line of grid nodes.Any additional equations such as the k and e equations are solved in a sequentialmanner after the flow variables are computed by the coupled method. This part of thesolution procedure can be similar to that used in the sequential solution strategy.An advantage of this type of algorithm is the close coupling of hydrodynamic equations that is maintained by the algorithm since a whole set of unknowns in velocity andpressure is updated simultaneously. This has been shown to be advantageous when agood efficiency is desired in reducing simultaneously the errors present in the results.3.3.3 Convergence DifficultiesIt has been commonly observed that the convergence rate of an iterative solution schemedeteriorates as the grid is made finer. Moreover, most basic schemes suffer convergencestalling problems as the iteration progresses. In the terminology of numerical analysis,the reduction of the residues of a set of equations by an iterative solution procedure iscalled error smoothing. Thus, the above observation amounts to the fact that most basiciteration schemes exhibit good error smoothing properties only during the initial stageof the solution process.This observation can be explained by a discrete Fourier analysis on the error or residual functions. Upon decomposing a residual function into its frequency components ormodes, it was found that an iterative scheme tended to be effective in reducing the amplitudes of high frequency error components but ineffective in reducing the low frequencycomponents. The realization of this inherent deficiency of most iterative schemes can beused to our advantage with the recognition that on coarser grids, these low frequencymodes would appear as having high frequencies and the iterative method would againChapter 3. Numerical Solution of the Navier-Stokes System 58be effective. A systematic exploitation of this principle of performing relaxation on asequence of grids of different coarseness has led to the development of the multigrid technique. This technique can return very rapid smoothing rates by effectively smoothingerrors of all wavelengths by working on grids with different coarseness. The basic ideasand implementation of this technique are the focus of the following section.3.4 Multigrid ProcedureThe practical utility of the multigrid technique was developed in the acclaimed work byBrandt (1977). One of its original applications was to accelerate the convergence rate ofan iterative procedure when applied to solve a discretized elliptic boundary value problem,such as the Poisson equation with Dirichiet-type boundary conditions. It is well knownthat any stable discretization of such an elliptic operator will lead to a system of algebraicequations which is in turn elliptic, in the sense defined in Stflben and Trottenberg (1982).This property is critical for the availability of a relaxation process with sufficient errorsmoothing properties that are independent of the grid size. Because of this, the Poissonequation is an ideal test problem for the multigrid method to showcase its performance.Excellent convergence performance was reported by Brandt (1977).The multigrid Coarse Grid Correction (CGC) scheme has been developed for linearproblems. For non-linear problems such as the Navier-Stokes equation, the Full Approximation Storage (FAS) scheme should be used. Detailed descriptions of this method canbe found in Brandt (1977) and Stilben and Trottenberg (1982). The salient features ofthe multigrid method will be described in the following paragraphs. This description isessential for our later discussion in section 3.6 on the difficulties encountered when themultigrid method is applied to problems describing turbulent flows.Chapter 3. Numerical Solution of the Navier-Stokes System 59f-li________\u2014 j+1\u2022\u20181h +Sh qhV1 relax v2 relaxd LH(H+H\u2019)- LHH=dHFigure 3.2: FAS (h,H) two-grid method.Let a system of nonlinear differential equations such as Eq.(2.21) be written as= 0 (3.12)where \u00a3 is some nonlinear differential operator and Q is the solution vector. Let thediscretized system on a grid Gh be denotedLhQh = 0 (3.13)where h denotes the characteristic grid size, Lh is a linear difference operator approximation to \u00a3 linearized about some values of Q, and Qh is a discrete approximation toQ.The Full Approximation Storage (FAS) algorithm can be described on the basis of anFAS (h,H) two-grid method. This is shown in Figure 3.2 illustrating the computation ofq\u2019 from q, where qj, is an approximation, to the solution Qh, with the superscript jdenoting the iteration index. The indices h and H denote respectively the fine grid andcoarse grid used by the method. The complete FAS algorithm is obtained by applyingthis two-grid method recursively to the equation in the coarser grid.In the diagram, is the result of v1 relaxation steps starting with q as first approximation. The quantity is the residual or defect quantity in the multigrid terminology.It is smoothed by the relaxation process and is a measure of the accuracy of the computedsolution. The quantities Ff and 4 are corresponding quantities in the coarser grid GH\u2022HChapter 3. Numerical Solution of the Navier-Stokes System 60The operators If, If, and I are intergrid operators for transferring values from Gh toG11, and vice versa. When the transfer is from Gh to C11, it is called restriction; andit is called prolongation when it is from GH to Gh. The reason for having two differentrestriction operators, If and If, is that higher accuracy is sometimes sought for thevalue of the solution represented in the coarser grid; but very often it is sufficient touse the same restriction operator as for the residue. Finally, the quantity F1 and its finegrid counterpart . are corrections to be added to the approximate solutions h! and ,respectively, so that Eq.(3.13) will be approximately satisfied on both grids.In the algorithm mentioned above, the coarse grid is only visited once from the finegrid. This algorithm is also known as the FAS V-cycle, and is the most basic form of amultigrid algorithm. The following components of the algorithm can be identified:(i) Pre-Smoothing: Smoothing is carried out v1 times to reduce the amplitude of thehigh frequency error components present in 4.(ii) Restriction of defect: By using an appropriate weighting scheme, the smoothedresidue from Gh is transferred to a coarser grid GH. There, the low frequency errorwould appear as a high frequency mode.(iii) Coarse grid smoothing: The GH version of the operator \u00a3 is constructed and anappropriate smoothing procedure is employed to smooth out the high frequencymode in the correction .-.(iv) Prolongation of correction: An appropriate interpolation procedure I is chosen totransfer the correction 4 to the finer grid to become .(v) Post-Smoothing: Smoothing is performed v2 times to smooth out any high frequencynoise introduced during the interpolation process.Chapter 3. Numerical Solution of the Navier-Stokes System 61A unique feature of this FAS method is the way in which the coarse grid defectequation is set up and solved. To illustrate, let s be the exact correction to ; that is,+ s = Qh (3.14)Then Eq.(3.13), together with the definition of 4 indicated in Figure 3.2, imply thefollowing relationship between s and 4 on the grid Gh:Lh( + 4) \u2014 = (3.15)This equation is then approximated on GH by\u2014j ,\u2018j \u2014j \u20143LH(q + SH) \u2014 LHq = dH (3.16)which is to be solved for the full approximation + ., other terms in the equationbeing known.It is important to note that it is the correction Ff that is transferred back to the finegrid and not the full approximation itself. The reason is that it is the correction and defectquantities that are smoothed by relaxation process and can therefore be approximatedwell on coarser grids.3.4.1 Full Multigrid StrategyIn the multigrid algorithm described, the calculation is first carried out on the finestgrid, while the coarser grids are used to speed up the smoothing of residual quantities.However, the algorithm can be implemented more efficiently through a strategy knownas the full multigrid (FMG) algorithm. The procedure is illustrated in Figure 3.3.In this algorithm, smoothing is first done on the coarsest grid. Upon reaching areasonable level of convergence, interpolation is carried out to transfer the solution to afiner level where the multigrid cycle will be carried out. This is continued until the finest,Chapter 3. Numerical Solution of the Navier-Stokes System 62restrictionprolongationprolongation ofI converged solutionV- Cyclesconvergedm solutionapproximatem solutionFigure 3.3: The full multigrid (FMG) algorithm.pre-determined level is reached. This method offers an efficient means to generate anaccurate solution field to serve as an initial solution estimate for a finer grid level. Indeed,in many applications of the multigrid method for non-linear boundary value problems,the preferred solution procedure is a combination of this FMG strategy together withthe FAS algorithm.3.4.2 Design of a Multigrid AlgorithmWhen the multigrid algorithm is implemented correctly, it can return a very fast convergence rate to the iterative procedure. Given a specific problem, unfortunately, thereis no general rule for choosing individual components of the algorithm that will lead toan optimal scheme. Nevertheless, some general guidelines can be provided. The knowledge of these guidelines is obtained through experience and local Fourier analysis for testproblems, and has been discussed by Brandt (1982). Several essential points are outlinedbelow.Chapter 3. Numerical Solution of the Navier-Stokes System 63(1) The number of smoothings on a particular grid should not be too large to avoid there-introduction of high frequency error modes due to interaction with boundaryconditions.(2) The restriction operator should be chosen so that the weighting of residues wouldnot introduce high frequency errors. This concern arises when the equations to besolved are nonlinear or have rapidly varying coefficients. In such cases, nontrivialweighting should be used.(3) The interpolation procedure for prolongation should be chosen so that the orderof interpolation is of the same order as the differential equation. This is done tominimize the generation of spurious frequency modes through the interpolationprocedure.(4) The efficiency of the technique also depends on the choice of the coarse grid operator.The important consideration is that the coarse grid defect equation (3.16) should bea faithful representation of its fine grid counterpart. One way to evaluate the coefficients defining the coarse grid operator is simply to average the corresponding finegrid coefficients. Another way would be to compute new coefficients directly fromthe restricted variables. The former strategy may save some computations, but forhighly nonlinear equations with large source terms it may not lead to convergencesince inconsistencies can be magnified during the solution procedure.It is important to note that the guidelines mentioned above were gained through theapplication of the multigrid method to relatively simple linear and nonlinear boundaryvalue problems. To extend the method to a system of nonlinear equations such as theNavier-Stokes equations presents many challenges, and the above guidelines may needmodifications. It is the objective of the following section to outline some of the attemptsChapter 3. Numerical Solution of the Navier-St okes System 64made in extending the procednre to solve fluid flow problems.3.5 Multigrid Technique as Applied to the Navier-Stokes EquationsThe multigrid technique can be applied in many ways to solve the Navier-Stokes equations. For example, the multigrid solver can be implemented to accelerate the convergenceof the momentum equations or the pressure correction equation within the SIMPLE algorithm. However, this approach is inefficient because only a time-consuming componentin an iterative solution algorithm is replaced by a multigrid solver. The convergence ofthe global iteration remains slow and any increase in efficiency is very limited.A more effective approach would be to construct a multigrid procedure for the NavierStokes equations as a whole. In this section, we discuss the special attention that is neededin the construction of such a procedure.3.5.1 Stability of DiscretizationBrandt and Dinar (1979), Brandt (1980), and Linden et al. (1988) discussed aspects aboutstability characteristics when the differential equations are discretized. They emphasizedthe importance of constructing stable and high order finite-difference approximations togeneral elliptic partial differential systems. This is important because the constructionof a robust and efficient smoother is possible only if the finite difference system itself canbe shown to be stable when subjected to an iterative solution procedure. The ellipticitymeasure of the finite difference equations is one means of showing this condition.For viscous, incompressible flow problems, there are two sources of numerical instability. First, at high Reynolds numbers, the ellipticity measure of central differencing forthe convection-diffusion part of the momentum equations decreases. To counter this, anChapter 3. Numerical Solution of the Navier-Stokes System 65additional measure of ellipticity has to be introduced to keep the discrete equations stable. This can be achieved, for example, by using upwind differencing. The second sourceof instability is due to the fact that pressure appears in the differential equations only interms of its first derivatives. Consequently, a simple central differencing for the pressuregradient term can lead to a system of difference equations which admits a solution that isperiodic in the pressure variation across the grid. Such a system of difference equationsis unstable to such periodic perturbations. For flow problems having simple rectilineardomains, the simplest remedy for this instability is to use a staggered grid, as illustratedearlier in Figure 3.1.3.5.2 Choice of SmootherTo construct an efficient and robust multigrid solution aigorithm for Navier- Stokes problems, it is important to use an equation solver that has good error smoothing propertiesand robustness when the flow Reynolds number changes. Researchers have examined numerous smoothers for their suitability for the above purpose. Examples of solvers testedare the distributive Gauss-Seidel (DGS) scheme and the pressure gradient averaging(PGA) scheme, both belong to the class of coupled solvers, and their descriptions can befound in the works by Brandt (1979) and Fuchs (1984). Brandt made the recommendation that a locally strongly coupled block of unknowns should be relaxed simultaneously;Fuchs observed that the PGA scheme is more robust than the DGS scheme at highReynolds numbers.Arakawa et al. (1987) and Linden et al. (1988) compared the use of sequential smoothing versus coupled smoothing for multigrid purposes for laminar flow problems. Bothgroups of investigators found that coupled smoothing is preferred to sequential smoothing, especially at higher Reynolds numbers, for the following reasons: first, sequentialsmoothing schemes are conceptually more complicated and, moreover, are sensitive toChapter 3. Numerical Solution of the Navier-Stokes System 66the details of implementation. Secondly, the application of the multigrid techniques inconnection with the sequential procedure can be problematic because, after smoothingthe momentum equations, errors of higher frequencies can be introduced through thevelocity corrections. Thirdly, numerical experiments show that coupled smoothers aremore robust, particularly at high Reynolds numbers.The SCGS smoother was used by Vanka (1986), who found that it was very robustfor Reynolds number up to 5000 for the driven cavity problem. It was also employed byThompson and Ferziger (1989), who accounted for its good performance by its attempt tosatisfy continuity for each cell at each iteration step. They found that the DGS approachhad difficulty when the cell face fluxes did not match well.3.5.3 Inter-Grid Transfer of InformationIt is customary to partition a coarse grid volume into eight fine grid volumes, or, in twodimensions, a course grid cell into four fine cells. The use of the staggered grid, however,makes inter-grid transfer of variables difficult. The reason is that different formulae areneeded for velocities, which are stored at scalar cell faces, and for scalar quantities, whichare stored at the scalar cell centers. To illustrate a typical grid transfer strategy, it issufficient to consider the situation shown in Figure 3.4, which depicts a two-dimensionalcell with uniform grid spacing. Extension to three dimensions and non-uniform gridspacing is straight-forward.RestrictionRestriction of grid functions can be made by averaging nearby values. For problemswith fluid flows, mass conservation needs to be ensured during grid transfer, and thisChapter 3. Numerical Solution of the Navier-Stokes SystemFine grid velocitiesCoarse grid velocitiesFine grid scalar variablesCoarse grid scalar variablesFine gridCoarse gridI I4 \u2022t\u2022 \u2022I ih---- I I67(3.17)(3.18)(3.19)(3.20)Figure 3.4: Staggered mesh arrangement showing storage of variables in the fine andcoarse grids.requirement leads to the following expression:(pUA)coarse = (pUA)finewhere the summation is over all fine-grid control volume faces coinciding with the coarse-grid face. Thus, the coarse-grid velocity is given by\u2014 >Zfjne(P(L4)Ucoarse \u2014 (pA)coarseIn studies of incompressible flows discretized with a uniform grid, the above requirement leads to the following expressions in two dimensions: let the superscripts c and fdenote coarse and fine grid values. Let (ic,jc) and (if,jf) denote coarse and fine meshindices, respectively. Also, let U,.112,3 be referred to as U and U_112, be referred toas and similar expressions for the other velocity component V. Then we haveif=2(ic)\u20141,jf=2(jc)\u20141,andUc(ic,jc) [U(if,jf) +U1(if,jf - 1)]Vc(ic,jc)= [V(if,jf) + V1(if\u20141,jf)]For scalar variables such as the inert tracer 4, restriction is achieved by simple linearChapter 3. Numerical Solution of the Navier-Stokes System 68interpolation among fine-grid values. The expression isc(jCjC)= [(if,jf) + (if - 1,jf) + (if,jf -1) + (if - 1,jf \u2014 1)] (3.21)This formula is applicable to other solution variables stored at the scalar node, such asthe pressure. Similarly, the restriction of solution residues is carried out in an analogousmanner.ProlongationProlongation relations are derived by bilinear interpolation. For each coarse grid node,four fine grid values are derived. For the U-velocity, they areU1(if,jf) = (3Uf+U) (3.22)U1(if,jf+i) = (3U+U) (3.23)U(if + 1,jf) = (3Uf + U + 3U + U) (3.24)U(if+1,jf+l) = (3U+Uf+3U+U) (3.25)whereU Uc(ic,jc)U = Uc(ic,jc+1)U = UC(ic1,jc)U = Uc(ic+1,jc+1)The V-velocity is prolongated by equivalent relations obtained by rotating the coordinatesby ninety degrees. The scalar prolongations have different weights because of their cellcentered locations. The relations are](if,jf) = (9 + 3 + 3 + ) (3.26)Chapter 3. Numerical Solution of the Navier-Stokes System 69(if,jf + 1) = + 3 + 3 + ) (3.27)](if + 1,jf) = (9 + 3 + 3 + ) (3.28)(if + 1,jf + 1) = + 3 + 3 + ) (3.29)with= c(jjC)= 4c(ic+1,jc)= c(jCjC+ 1)4 \u2014 c(jC+1,jC+1)The use of a collocated variable arrangement where all variables are stored in thecenter of a cell simplifies the intergrid transfer algorithm, since the same set of formulaecan be used. To deal with the stability problem, a special interpolation practice, calledthe momentum interpolation by Majumdar (1988), has been introduced. This interpolation procedure is designed for the determination of cell face velocities representing theconvective mass fluxes in order to avoid oscillatory solutions which can cause numericalinstability.In examining the effect of altering the order of the interpolation, Thompson andFerziger (1989) found that the use of more accurate higher order interpolation schemesactually degraded the convergence rate. Their test results show that it is especially badto use high-order interpolation for prolongation. This may have been caused by theintroduction of high-frequency noise which the fine grid iterations need to remove later.Moreover, cubic or higher order polynomial fits can be inconvenient to use in practicalflows with complex geometrical shapes and blocked regions. Hence, for most practicalproblems, the use of simple linear interpolation formulae seems to be adequate.Chapter 3. Numerical Solution of the Navier-Stokes System 703.5.4 Construction of Coarse Grid EquationAs shown in Figure 3.2, the construction of the coarse grid equation(3.30)requires the restriction of the defect or residual quantity and the solution vector fromthe fine grid Gh to the coarse grid 0H, together with the setting-up of the coarse gridoperator LH. The restriction of the solution vector can be carried out with formulaesimilar to those given in the previous section. For the residual quantities, the coarse-cell residues are obtained from an appropriate summation of residues of the fine cellsconstituting the coarse cell.A simple approach in constructing the coarse grid operator LH for the Navier-Stokessystem is to restrict the coefficient fields defining the fine grid operator Lh onto the coarsegrid directly. As discussed in section 3.4.2, this strategy reduces the computational cost,but is not robust because it does not maintain the consistency between the sources andthe associated variable fields. It is preferable to compute the coefficients and the sourcesby using the coarse grid solution . This preference becomes mandatory when thosequantities are dependent on the associated variables.Thus far, the discussion on applying the multigrid method to fluid flow problems hasbeen based mainly on the laminar Navier-Stokes system. To extend the method to solveturbulent flow problems requires the consideration of a larger set of equations with morecomplicated source terms. A discussion is presented in the following section on extendingthe solution procedure to this class of problems.3.6 Extension of the Multigrid Technique to Turbulent Flow ProblemsThe extension of the multigrid method to turbulent flow problems faces many additionaldifficulties, which arise due to the nature of the turbulence model equations. For theChapter 3. Numerical Solution of the Navier-Stokes System 71ic \u2014 c model, which is the most widely tested model used in multigrid applications, ithas been observed that there are a number of difficulties when the implementation of themultigrid method is sought.3.6.1 Difficulties with ImplementationSeveral major difficulties with the implementation of the method are listed below:(1) The best way of incorporating the k and e model equations into the smoothingscheme is not clear when a coupled solver such as SCGS is used.(2) The use of the wall function to link the flow in the near wall region to the mainflow can be problematic for multigrid applications. It is difficult to consistentlyassign values to grid nodes closest to the wall since those nodes are moved duringrestriction. Specifically, the fixing of the value of c at the near-wall node impliesthat every time restriction is carried out, the logarithmic region will extend fartherand farther from the wall.(3) The nonlinearity and the dominance of the source terms in both the k and c equationscan adversely affect the performance of the multigrid method because the successof the method relies on the smoothness of the error and correction quantities. Anydisturbances introduced into the residues through inaccurate handling of the sourceterms can lead to serious degradation or complete failure of the algorithm. The eequation has a strong nonlinear (1\/k) coupling with the k equation through theproduction and dissipation terms. This coupling is observed to be much moredominant than the convective and diffusive transport, and unless it is resolvedaccurately, the solution of these equations can have serious errors. Under-relaxationof these source terms is needed to ensure stability during the solution process.Chapter 3. Numerical Solution of the Navier-Stokes System 72(4) The constraint of the physical realizability imposed by the turbulence model callsfor concerns. Specifically, the values of k and 6 must always be positive and theturbulent viscosity I\u2019t defined in Eq.(2.16) should not be unrealistically larger thanthe molecular viscosity, say iO times larger. Failure to observe these constraintsduring smoothing or prolongation could lead to numerical instability which couldcause the iterative solution procedure to diverge.Special attention is required to deal with the above concerns.3.6.2 Experience with Applying Multigrid to the k \u2014 e ModelThere have been only a limited number of reports on the application of the multigridmethod to equations coupled with the k \u2014 6 turbulence model. One of the earliestreports is by Phillips et al. (1985) who applied the FAS multigrid method to simulatea two-dimensional axisymmetric turbulent flow in an expanding duct using the k \u2014model with wall function. Both the Stone\u2019s method and the Gauss-Seidel method wereemployed as smoothers. The authors did not discuss any difficulties with their solutionprocess. A comparison of the convergence histories between a single grid calculation anda 4-level multigrid calculation shows that the multigrid procedure requires 76% of theCPU time for the single grid calculation. This inefficiency shows that there might beproblems with their implementation of the algorithm.Vanka (1987) extended his coupled solution multigrid scheme to solve for the turbulentflow problem of an axisymmetric sudden expansion. Again, the k \u2014 6 model with wallfunction was used. The continuity and momentum equations were solved simultaneouslyby the point SCGS solver. Vanka noted the complexities due to the dominance of thesource terms and their nonlinearity in the k and 6 equations. He also noted the difficultywith the use of the wall function. In his preliminary trials, Vanka had difficulties obtainingChapter 3. Numerical Solution of the Navier-Stokes System 73convergence when the k and e equations were solved in a coupled manner with themomentum and continuity equations. He circumvented this problem by solving the kand e equations by a single grid procedure on the finest grid only. He observed thathis multigrid solution required 20 times less CPU time than a corresponding SIMPLEsolution for the same problem.Despite the recommendations by Arakawa et al. (1987) and Linden et al. (1988), manyinvestigators prefer to apply SIMPLE-type algorithms as smoothers in their multigridroutines to solve flow problems described by the k \u2014 e model. One reason could be theobvious ease with which the turbulence model equations are to be implemented intothe solver. Peric et al. (1989) applied a finite volume multigrid method to computethe turbulent flow over a backward-facing step using the SIMPLE algorithm for errorsmoothing. They paid special attention to the computation of fluxes on different grids.For instance, the cell-face convective fluxes on the coarse grid were obtained by summingthe corresponding fine grid convective fluxes, while the diffusive fluxes were recalculated;the source terms were likewise recalculated for consistency reasons. On a very fine grid,the computational work for multigrid was found to be a hundred times less than thatfor the standard single-grid procedure. They also found that increasing grid refinementwould reduce the range of useful under-relaxation factors.Lien and Leschziner (1991) also made many interesting observations regarding the useof the multigrid method for calculating complex recirculating turbulent flows. They studied axisymmetric flow in a circular pipe with a sinusoidal constriction. The numericaltechnique employed was a three-dimensional, non-orthogonal, collocated finite-volumeprocedure together with the SIMPLE smoothing algorithm. Compared to single gridresults, the speedup ratios for the high Reynolds number k \u2014 e model generally rangedfrom 1.2 to 5.9, with the ratio increasing as the number of control volumes increased.This speedup ratio was significantly less than their laminar results. They observed thatChapter 3. Numerical Solution of the Navier-Stokes System 74the convergence characteristics were strongly dependent on flow type and geometry, andthat the skewness and amount of stretching of the grid affected the performance of thesmoother and consequently the multigrid procedure. In addition, the authors presentedstrategies such as under-relaxing the coarse-grid solutions and conditioning the prolongation process that would help to maintain the realizability constraints k > 0 and e> 0.Rubini et al. (1992) examined the application of the multigrid method to turbulent,variable density flow over a three-dimensional, backward-facing step, using a staggeredgrid discretization. The standard FMG-FAS technique was used for convergence acceleration while a SIMPLE-type algorithm was used as the smoother. The authors stressedthe importance of maintaining mass conservation during restriction and prolongation,and noted that the highly nonlinear nature of the k \u2014 equations required careful treatment within a multigrid procedure to achieve optimal convergence rates. Additionallinearization of the source terms was found necessary to prevent negative values of k ore, which could arise during the prolongation operation in regions of large gradients. Theproblem was circumvented by simply not updating any locations that would result in anegative value. Their multigrid technique yielded a 25 fold reduction in computer time,but they could not obtain the expected linear increase of computing time with respect tothe number of grid nodes. The authors attributed this problem to the use of a low-orderprolongation scheme.Shyy et al. (1993) presented a number of valuable insights into the extension of themultigrid technique to turbulent flows. They discussed three special treatments that wererequired for the successful implementation of the algorithm. First, the corrections for kand e needed to be checked for negative values after the prolongation procedure. If therequirement was violated, then the same strategy used by Rubini et al. was employed:prolongation would not be carried out for the solution correction at that particular iterative step. Secondly, an upper bound for the turbulent viscosity I-Lt was imposed to ensureChapter 3. Numerical Solution of the Navier-Stokes System 75that the values of k and remained realistic during the course of the multigrid procedure. They set the bound to be iO times the laminar value. Thirdly, to alleviate theproblem with consistency due to the use of the wall function in different grid levels, thegrid line next to the solid boundary was retained during the grid restriction procedure.The convergence rate was found to be improved with this treatment.So far, several previous studies on applying the multigrid method to turbulent flowproblems have been outlined. Our research group has developed a versatile code that cansimulate turbulent flows in complicated geometry, with a special focus on applications toproblems related to kraft recovery boilers. The problem of a single jet in a crossflow willbe used to validate the code and is described in the following section.3.7 Multigrid Code MGFD for Solving Complex Turbulent FlowsThe code developed in our research group is called MGFD and it utilizes the FMGFAS multigrid algorithm to simulate turbulent flows in complicated geometries. In thissection, some salient features of the code are described and the code is validated throughthe simulation of a single jet in a crossflow. In addition, the efficiencies of several solutionmethods are examined. A detailed description of the code is given in the report by Nowak(1992).3.7.1 Features of MGFDThe multigrid scheme used in MGFD is a version of the proven FMG-FAS scheme, whichwas developed by Nowak (1984). The unique feature of the present version is that thecoarse grid solution is taken from the results obtained in the previous level ratherthan restricted from the fine grid solution. In some trial runs, this strategy performs aswell as the standard FAS scheme, but with less computational cost.Chapter 3. Numerical Solution of the Navier-Stokes System 76The choice of smoother for the code is based on the need for robustness. The flowpatterns in a recovery boiler are very complex. There can be numerous recirculatingzones and regions of high velocity gradients due to intense jet interaction. Based onthe investigations by Vanka (1986) and Linden et al. (1988), the coupled type smootherline-SCGS scheme was chosen, partly because of its demonstrated reliability and partlybecause of its conceptual simplicity.The smoothing of the hydrodynamic equations is carried out as one unit. The relaxation of the k, c, and any other scalar equations is carried out separately, sequentially,and on the finest grid only. The basic Gauess-Seidel scheme is employed with smoothingdone along lines traversing in turn to cover all three coordinate directions. In this version,a fixed number of iterations is performed for the equations at each grid level.The restriction of residual or defect quantities is based on the area-weighted summation of fine grid quantities. Similarly, prolongation of grid variables is carried out usingbilinear interpolation. The same prolongation routine is used for both solution transferand correction transfer. The results have been found to be satisfactory.Other unique features of the code include its capability to accept a domain subdividedinto segments and to perform smoothing sequentially for blocks of cells within a segment.These characteristics are described next.Segmentation CapabilityEven with the multigrid technique, the algorithm may not be optimal for certain types ofproblems. An important feature of MGFD is that it allows the domain to be subdividedinto segments, which are rectangular blocks of cells, for efficient grid distribution. Asmentioned earlier in Chapter 1, in a recovery boiler simulation, there is a large variationin dimensions within the domain. The overall dimension of the boiler is much largerthan that of an air port. The segmental capability of our code allows the domain to beChapter 3. Numerical Solution of the Navier-Stokes System 77Segment with Segment withfine grid Segmental coarse gridInterfaceFigure 3.5: Interface of two segments having different grid densities.covered by a union of segments, each having a different grid density that characterizesthe flow phenomena that need to be resolved in different parts of the domain.For this treatment to be successful in which a domain is divided into segments, it isimportant to have a conservative scheme to govern the passage of fluxes across segmentalboundaries. Working with a similar concept, Thomson and Ferziger (1989) emphasizedthat the convergence of the method could depend critically on how well this mass conservation is maintained.The passage of the information of mass fluxes across segmental boundaries is achievedin the following way. Referring to Figure 3.5, at the interface of two segments havingdifferent grid densities, the mass flux is computed on the segment that has a finer gridbecause of the expected higher accuracy in the calculation with this grid. This mass fluxis computed through full momentum balance as governed by the Navier-Stokes equations.For each coarse grid cell on the other segment where this mass flux information is to bereceived, its flux value is obtained by a continuity preserving interpolation process. Inthis way, global mass conservation is satisfied. The details of the implementation arepresented in the report by Nowak (1992).Chapter 3. Numerical Solution of the Navier-Stokes System 78Smoothing Procedure by Blocks within Each SegmentThe iterative relaxation process is performed segment by segment sequentially. Moreover, each segment at each multigrid level is divided into smoothing blocks, each of whichconsists of a cube of cells. The smoothing within a segment takes place by smoothingwithin each block sequentially in a \u2018black-red\u2019 manner, thus ensuring efficient coverageof all cells in a minimal number of sweeps. An advantage with this usage of blocks isthat the smoothing for cells in different blocks can be carried out with parallel computerprocessing. Within each block, a line-SCGS process is performed to solve for the variables. The resulting equation is written as a tn-diagonal system, which is solved by theThomas algorithm. Each coordinate direction is traversed in turn. For most problems,under-relaxation is necessary for velocity, pressure, scalar quantities, turbulent energy,dissipation, and the eddy viscosity. A demonstration of the solution procedure can befound in Salcudean et al. (1992). The validation of the code is the subject of the nextsection.3.7.2 Single Jet in a Crossflow : Validation of MGFDThe experimental and numerical results of Simitovi\u00e9 (1977) for a single square jet in aconfined crossflow is chosen for comparison because of its resemblance to many of ourflow situations to be simulated in later chapters. A description of the problem is providedin Figure 3.6. The following values are specified for the problem: W = 0.0254 m, Uc = 6m\/s, L\/W = 2, and L\/W = 4. These give the following value for the Reynolds numberRe = pUcDh = 2.7 x (3.31)for properties of air at room temperature, where Dh=is the hydraulic diameterof the tunnel. Different values of jet-to-crossflow velocity ratio Yjet\/U were investigatedChapter 3. Numerical Solution of the Navier-Stokes System 79LtbX____HtHw H fr\u2014wVJETFigure 3.6: A schematic representation of the single square jet in a crossflow studied bySimitovi\u00e9 (1977).by Simitovi\u00e9, but for the validation study to be performed here, we only study the casewhere \u2018et = Uc.In the experiment, the mean velocity was measured by a pitot-static pressure probewhile the jet fluid concentration was measured by a flame ionization detector, whereethylene was used as the tracer gas. Simitovi\u00e9 estimated that the errors in the jet tracerconcentration measurements were about 2 \u2014 6%.For the numerical prediction, Simitovi\u00e9 applied a 3D-TEACH code to solve the governing equations with the k \u2014 e turbulence model. The domain was extended 9W upstream and lOW downstream, and the symmetry condition was utilized so that only halfof the jet was simulated. The discretization was done on a non-uniform 26 x 26 x 26grid. Uniform velocity profiles were prescribed at both the main and the jet flows inlet.The zero gradient condition was imposed at the flow outlet. Near the solid boundary,the shear stress was calculated using the wall function.In our investigation, the TEMA code developed by Lai and Salcudean (1985) is applied to solve the flow problem. The code is similar to the 3D-TEACH. Then multigridsimulations are performed using MGFD. A 36 x 28 x 28 grid is used on the finest gridlevel. The grid distribution is shown in Figure 3.7.Chapter 3. Numerical Solution of the Navier-Stokes System 80(a) x-y plane (b) y-z plane2 2yJW_________________________________________yIW1_ _10-8 -6 -4 -2 0 2 4 6 8 00x\/W z\/WFigure 3.7: Grid distribution for the calculation of a single jet in a crossfiow.Assessment of Multigrid EfficiencyThe efficiency of the multigrid procedure is now examined. First, a comparison is madebetween the basic SCGS relaxation scheme used in MGFD and the sequential SIMPLE-Cscheme used in TEMA. Secondly, a comparison is made for the use of different multigridlevels in the calculation. The results of these comparisons validate the correctness, basedon performance, of the multigrid implementation.The convergence of the solution process is monitored by observing the reductionin mass error, which is the residue of the continuity equation. The total mass error iscalculated by summing the absolute values of the mass residue of all cells, and is obtainedby the following expression:total mass error= > r (3.32)all cellswhere r is the mass residue of a typical cell labelled with the subscript i.The mass error reduction histories are displayed in Figure 3.8 for the following cases:relaxation with SIMPLE-C on a single grid; relaxation with SCGS on a single grid;relaxation with multigrid V-cycle on two grid levels, and relaxation with multigrid V-cycleon three grid levels. The abscissa represents fine grid work units, defined as computertime required for one iteration on the finest grid using the corresponding single-gridscheme. The ordinate represents the total mass error expressed in Eq.(3.32), normalized1 2Chapter 3. Numerical Solution of the Navier-Stokes System 81by the total mass flow from the mainstream and the jet. In these calculations, thechoices for under-relaxation factors are 0.6 for velocity components, 1.0 for pressure, 0.5for turbulence quantities, and 1.0 for tracer concentration. No particular difficulties inconvergence are encountered. The results of the convergence performance are discussednext.The SIMPLE-C algorithm produces the usual \u2018ripples\u2019 in the residual reduction history that is typical of sequential pressure-correction type algorithms. The overall convergence rate is slow, but it is steady and shows no sign of stalling. The single gridcalculation using SCGS shows good reduction of residues during the first 100 iterations.After that, however, the reduction rate slows down significantly. Together these tworesults illustrate the slowness in attaining convergence by relaxation methods on a singlegrid. For a smoother to be effective within the multigrid framework, it needs to havegood error smoothing properties for high frequency error components. The SCGS schemeshows very rapid reduction of residues in the early stage of the iteration process. Thissuggests that for the problem considered, it is more effective in reducing high frequencyerrors than the SIMPLE-type method. This characteristic makes the SCGS scheme moredesirable for use with the multigrid algorithm.Compared to the one-level results, the convergence rate of the two-level calculation ismuch more rapid. The three-level calculation shows even faster convergence performance.The reason for the better performance is that when using three different grids, a widerrange of frequency components of the error function can be smoothed effectively. Withproper adjustments made to multigrid parameters such as adjusting the number of iterations to be carried out for the solution correction, it is believed that the \u2018humps\u2019 shownon the error reduction curve for the three-level multigrid calculation (Figure 3.8(d)) canbe removed.Chapter 3. Numerical Solution of the Navier-Stokes System 82(a) TEMA: SIMPLE-C0InU)U)COtci)NCOB0z0 1000 2000 3000 4000Work Units(c) MGFD : 2 Levels2InU)U)CO00)NCOB0z750(b) MGFD: 1 Level0I-InU)U)CO-oci)NCOB0z0 500 1000 1500 2000Work Units(d) MGFD: 3 Levels2InU)U)COtNCOB0z1(0 250 500Work Units Work UnitsFigure 3.8: Mass error reduction histories for TEMA and MGFD.Chapter 3. Numerical Solution of the Navier-Stokes System 83(a) (b)ul\/uc Ul\/uc1.0 1.0Figure 3.9: Measured and predicted streamwise velocity at y\/W = 1.0, (a) x\/W = 3.5,(b) x\/W = 8.25.Comparison of ResultsThe velocity and jet tracer concentration profiles obtained from both the experimentaland numerical results are now compared at selected positions. Specifically, the streamwisecomponent of the velocity is compared at the locations a\/W = 3.5 and 8.25 at y\/W = 1.0,and x\/W = 3.5 and 8.25 at z\/W = 0.0 and 0.5. The tracer concentration is compared atx\/W = 1.75 and 4.5 at z\/W 0.0 and 0.5. These locations are typical of those chosenby Simitovi\u00e9 in his study. The results are displayed in Figures 3.9-3.12.The results show that the agreement between the predictions from the two codes iswithin a few percents. The results obtained with MGFD exhibit slightly steeper gradientsthan those obtained with TEMA. This difference is mainly due to the use of differentdifferencing methods in the two codes: hybrid scheme in TEMA and power-law schemein MGFD. Our predictions using TEMA are visually very similar to those reported bySimitovi\u00e9.The numerical results exhibit fair agreement with experimental measurements inmany regions of the flow field, and is often within +10%, which is probably within1.50.5-1.0 0.0 1.0 2.0z\/W zJWChapter 3. Numerical Solution of the Navier-Stokes System1.5ul\/Uc(a) (b)Figure 3.10: Measured and predicted streamwise velocity at(b) x\/W = 8.25.z\/W = 0.0, (a) x\/W = 3.5,0.5 1.0 1.5 2.0Figure 3.11: Measured and predicted streamwise velocity at z\/W 0.5, (a) x\/W = 3.5,(b) x\/W = 8.25.841.5ul\/uc1.00.5yIWMGFDTEMA\u2022 DATA0.5 1.0 1.5yAW2.0U1(a) (b)1.5ul\/UG1.00.5MGFDTEMA\u2022 DATAyAW yAWChapter 3. Numerical Solution of the Navier-Stokes System 85IJ(a) (b)(d)Figure 3.12: Measured and predicted profiles of jet tracer concentration, (a)z\/W = 0.0,x\/W = 1.75, (b) z\/W = 0.5,x\/W 1.75, (c) z\/W = 0.0,x\/W = 4.5,(d) z\/W = 0.5, x\/W = 4.5.yIW y\/W(c)1.000.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0yJW y\/WChapter 3. Numerical Solution of the Navier-Stokes System 86the range of experimental inaccuracy. The largest discrepancy is found in the velocitydistribution in the wake of the jet, which is shown in Figure 3.10(a) and Figure 3.11(a).In the latter figure, the numerical simulation predicts a significant degree of velocitydefect; however, this is not supported by the data. Simitovi\u00e9 attributed the discrepancymainly to the measuring problem with the pitot-static pressure probe. The reliability ofthis type of measurement technique can be affected by strong flow deflection and highturbulence levels. These conditions are very significant in the wake region of a jet in acrossflow. Other causes of discrepancy include the inadequacy of the mathematical modelused in the prediction of this type of jet flow. The assumption employed by the k \u2014 6model does not allow for the anisotropy of the turbulent diffusion processes. Moreover,the set of empirical parameters used in the k \u2014 e model has not been tuned for complexflows with recirculation. In addition, the uncertainties about the conditions in the jet asit enters the main duct could cause errors. A more detailed discussion can be found inSimitovi\u00e9 (1977).3.8 Chapter SummaryIn this chapter, methods are presented for solving the time-averaged form of the NavierStokes equations coupled with the k \u2014 e model and a scalar transport equation. Aniterative solution procedure is required, for which the multigrid algorithm can be appliedto accelerate the convergence. The feasibility of applying the multigrid technique to theNavier-Stokes system is examined. When applying the technique, careful attention isneeded for the multigrid procedure to work as theoretically expected. Important issuesare the efficiency and robustness of the equation solver, the conservation of fluxes duringgrid transfer, and the consistency between the coarse grid problem and the fine gridproblem. When the multigrid algorithm is implemented correctly, it can return very fastChapter 3. Numerical Solution of the Navier-St okes System 87convergence rates.Using the FMG-FAS multigrid algorithm, our research group has developed a code forthe prediction of complex turbulent flows. The code is validated through comparison withresults obtained with the TEMA code and with experimental data for the problem of asquare jet in a crossflow. This problem has relevance for other turbulent jet simulationsto be performed later in this study. It is demonstrated that the mathematical modelprovides a generally good qualitative agreement with the experimental results, and thisobservation adds confidence to our study of multiple jet interactions to be carried out inthe following chapters.Chapter 4Simulation of Primary Level JetsThe numerical simulations of primary level jets are presented in this chapter. One ofthe difficulties with recovery boiler simulation is to represent the large number of smallprimary air ports that are located around the lower perimeter of the boiler. In a typicaldesign, there can be over a hundred of these primary air ports. A straight-forward modelling effort that includes the representation of every primary air port would require agrid so fine that the modelling effort would become prohibitively expensive and impractical. Moreover, the solution of such a large system of finite difference equations can bevery difficult to obtain. Thus, from a numerical modelling perspective, it is necessary tomake simplifying assumptions to represent primary jets.A common practice employed by Jones et al. (1990) in the modelling of these primaryports is to lump several of them together, according to the design of the grid. A simplerapproach, used by Salcudean et al. (1992), is to numerically construct a continuous slotspanning across the space that is originally occupied by a row of ports. This practice hasbeen encouraged by the experimental findings in the rapid merging of closely spaced jetsby Knystautas (1964), as described in Chapter 2. It is the purpose of this investigationto perform an examination of the quality of continuous slot simulation for primary jetsin a kraft recovery boiler.There are concerns with the turbulent flow simulation of slot jets representing discretejets. First, flow parameters such as the velocity and mass flow rate need to be selectedto reflect similarities between the flows from slots and from discrete ports. Secondly, in88Chapter 4. Simulation of Primary Level Jets 89the slot representation the interaction between jets from neighboring ports is neglected,and this can have an effect on the turbulence characteristics of the flow field, and can inturn affect the transport of momentum and scalar quantities. These concerns must beaddressed.This chapter begins with an investigation of the extent of interaction of primary jets.This enables us to judge whether it is appropriate to use the steady state solution algorithm for this application. Then the interaction of primary jets is studied by modellingindividual jet openings, and this is referred to as the prototype case. Next, the rowsof primary jets are modelled by two types of slot equivalences: open slots and porousslots. Simulations with these slots are performed and the convergence behavior for eachsimulation is discussed. In each of these simulations, the domain is set up to reflect thebottom portion of the Plymouth water model in the UBC laboratory. Consequently, thechar bed is not implemented at this stage in the numerical models. Results of the slotsimulations are compared with those of the prototype case. The primary basis of comparison is the distribution of the vertical component of the mean velocity. The distributionsof turbulence quantities are investigated to observe any changes in the turbulence characteristics when slot jets are used in place of discrete jets. The effects of different choices ofturbulence boundary conditions for the open slot simulation are then addressed. Finally,conclusions and recommendations are made.4.1 Interaction of Corner JetsA concern with the prediction of jet flows in a kraft recovery boiler is that the flow fieldmay be unsteady due to the intense interaction of the jets. This has been experimentallyobserved by Ketler et al. (1992) and numerically simulated by Abdullah et al. (1993). Aconsequence of this observation is that the prediction of the flow field using an algorithmChapter 4. Simulation of Primary Level Jets 90FRONTWALLfor steady flow may not converge. In the simulation of primary jets, even though theirvelocities are low, there is a possibility that the interaction of jets near the corner of twoadjacent walls may lead to flow instability. To examine the suitability of using a solverfor steady flow for primary jet simulation, the interaction of several corner jets operatingunder typical primary jet conditions is studied. A numerical calculation could reveal ifunsteadiness exists in the flow field.The operational characteristics of these corner jets are taken from those of the Plymouth recovery boiler model used in the laboratory at UBC. This is a 1: 28 scale modelconstructed primarily of plexiglass. Water is introduced into the model to simulate flowphenomena in a furnace. A schematic drawing of the model is shown in Figure 4.1. Adetailed description of the model can be found in Ketler (1993).Figure 4.1: A schematic illustration of the Plymouth water model.Chapter 4. Simulation of Primary Level Jets 91Symmetry planesx=88.9x=4.24x= 0All dimensions in centimeters (Not to scale)Figure 4.2: Domain for simulating corner jet interaction.In the simulation, the dimensions of primary ports and the jet velocity are adjustedto reflect the conditions of an actual experiment. The closest seven jets to a corner of themodel are simulated: four on one wall and three on the adjacent wall. Each jet orificehas dimensions 0.159 cm (W) by 0.610 cm (L) (0.0625 inches by 0.24 inches). A sketchof the domain is shown in Figure 4.2.The boundary conditions imposed are as follows: plug-flow condition for the jet yelocity; no-slip condition at solid walls; symmetry condition at the two orthogonal planesenclosing the domain, and zero-gradient condition at the top exit plane. The wall function is used to compute the shear stress near the no-slip walls. Symmetry planes are usedto limit the size of the domain. These are placed at 0.127 m (5 inches) from the wallcorner so they do not affect the interaction of the jets. The top exit plane is located at0.889 m (35 inches) above the floor. This distance is sufficiently far above the jets suchthat it is reasonable to assume that the flow is uni-directional at the plane.At a jet entrance, the magnitude of the jet velocity is set at et = 2 m\/s, with thezj7yChapter 4. Simulation of Primary Level Jets 92100101ct 10-4. io\u00b0Figure 4.3: Convergence history for the simulation of corner jet interaction.jet directed slightly downward at an angle of depression of 100. The turbulent kineticenergy kjet is set at kjet = 0.005 x and the dissipation length scale is chosen to be= 0.5 x W for the determination of dissipation rate through Ejet =k2\/Ld3.Thesechoices are made following the usual practice used in jet-in-crossflow calculations.The domain is segregated into three segments for efficient distribution of grid nodes.A two-level multigrid procedure is carried out to compute the flow field. When thenumber of iterations performed in each multigrid step is selected properly, convergenceis readily obtainable using the steady state algorithm. The convergence history showingthe reduction in the total mass error as defined in Chapter 3 is displayed in Figure 4.3.The normalization factor is the total mass inflow from the jets. It is significant to notethat the convergence history with the steady state solution algorithm shows no sign ofstalling. This observation suggests that a steady state solution does exist and can beobtained by our solution procedure.This simulation with corner jets provides more challenges for our numerical algorithmthan the previous single jet in a crossflow case. Specifically, the number of iterations to becarried out before restriction needs to be increased in order to achieve convergence. For500 1000 1500 2000 2500 3000Work UnitsChapter 4. Simulation of Primary Level Jets 93= 1 mIs0.125 I 0.125 -S----- \u2014 \u2014 \u2014y(m) :________y(m)0100EIII II N N0.025 IHIHINNii I0.000 .1 .____________________1,.!?,:0.000 0.025 0.050 0.075 0.100 0.125 5 0.100 0.125x(m) x(m)Figure 4.4: Computation mesh and the velocity field at the jet entrance elevation(x = 4.24 cm).the jet-in-crossfiow calculation, a good convergence rate is obtained with the number ofiterations in this pre-smoothing step set at 10. For calculation of the corner jets, the samenumber needs to be increased to 50. No attempt is made to optimize the convergencerate by adjusting the number of pre-smoothing iterations.A velocity vector plot at the elevation of the jet entrances, together with the computational mesh used, is shown in Figure 4.4. The graph shows the collision of the corner jetsand the occurrence of recirculating regions in the flow field. The degree of jet interactionis significant, but apparently these primary jets are so weak that their collision is notintense enough to induce any flow unsteadiness. Hence, any flow instability observed inthe recovery boiler water model is likely due to the interaction of stronger jets at the twohigher levels. This suggestion has been verified by the experimental evidence of Ketler(1993), who observed that the flow field due to primary jets alone was quite steady.Although the convergence rate is steady, it is slower than the case where a crossflowis present. Physically, the presence of a crossflow can have a stabilizing effect on theflow field by reducing the extent of jet collision. It is apparent that a high degree of0.0L. H ii0.000 0.025Chapter 4. Simulation of Primary Level Jets 94stability in the physical system will manifest itself numerically in improved convergencecharacteristics.4.2 Full Discrete Jet SimulationHaving established the suitability of applying the steady state solution algorithm forthe computation of primary jet flows, we now apply the algorithm to simulate a morecomplete set of primary jets. The results will be used as our prototype case, to whichthe results of the slot simulations will be compared.In the Plymouth model, there are 156 primary ports arranged almost symmetricallyaround a square cross-section in close proximity to the floor. Since only the dynamicsof primary jet interaction is desired, the representation of the domain is simplified byneglecting the complex physical structures that are presented in the upper portion of thePlymouth model. Due to computer memory limitations, only a quarter of this bottomregion can be simulated with all the primary ports included. The domain of calculationis similar to that depicted in Figure 4.2, but there are now 17 jets along the y-directionand 22 jets along the z-direction. A three-dimensional sketch of the domain is presentedin Figure 4.5.The domain is again divided into three segments. The grid used is shown in Figure4.6. The simulation requires 153984 cells, which is much too numerous to be incorporatedinto a full scale boiler simulation. Boundary conditions are prescribed in much the sameway as in our earlier calculation for corner jets, except that the symmetry planes arenow located at 0.194 m (7.65 inches) from the corner, which is the midpoint locationon one side of the Plymouth model. A two-level multigrid cycling process is used forthe calculation and the convergence history is displayed in Figure 4.7. The results ofthe flow field will be shown later when compared with the slot representations. TheChapter 4. Simulation of Primary Level Jets 95Symmetry planesx = 88.9(Fj6ioH -O.159x=4.24xx=OzyAll dimensions in centimeters (Not to scale)Figure 4.5: A three-dimensional view of the computation domain.computational requirement for this full discrete jet simulation is about 43 megabytes ofRAM.The convergence rate is similar to the corner jet simulation. However, compared tothe previous case, a much longer computational time is required for this calculation sincethe domain is comprised of many more cells. It is important to remark that in a trial calculation where smoothing is performed only on a single grid level, the convergence stallsafter approximately a thousand iterations. This observation illustrates the significance ofthe multigrid method: not only does it provide faster convergence, in some particularlydifficult cases it gives convergence where a standard iterative method fails.Having established the prototype case for comparison, the two types of slot representations for primary jets are now considered.Chapter 4. Simulation of Primary Level Jets 96yIHFigure 4.6: Computation mesh used in the0InU)Cl)asVa)NCUE0zprimary discrete jet simulation.xIH4.0x\/H4.00.0 z\/H 1.0Work UnitsFigure 4.7: Convergence history for the full primary discrete jet simulation.Chapter 4. Simulation of Primary Level Jets 97DiscreteJets_________________________Open SlotPorous SlotFigure 4.8: Representations of primary level jets: discrete jets, open slot, and porousslot.4.3 Slot Representations of Primary JetsThe two types of slot representations, open slots and porous slots, are examined. Thecriteria for the simulation are to maintain the same mass flow rate and the jet velocity.The slots span across the same space that is occupied by the discrete primary ports.A schematic illustration contrasting the three simulations is shown in Figure 4.8. Themodelling of these slot jets are discussed next.4.3.1 Open SlotsAn advantage of simulating primary jets with open slots is the simplicity of the representation. These slot jets are simulated simply by allocating cells on the sides of the domainto constitute the opening of the slots. This is done by assigning the velocity values atthose cells in accordance with the desired jet velocity, which is chosen to be the sameas that of the original primary jets. To maintain the same mass flow rate, the height ofeach slot must be less than the height of an individual air port so that the total areaChapter 4. Simulation of Primary Level Jets 98is the same. For the simulation, the following dimensions are used: in the y-direction,the slot spans from 4.89 cm to 19.4 cm (1.909 inches to 7.65 inches). In the z-direction,the slot spans from 1.04 cm to 19.4 cm (0.409 inches to 7.65 inches). A height of 0.115cm (0.0451 inches) is selected to give the slots a combined total area of 3.77 cm2 (0.585square inches), the same total area of the discrete jet orifices for which the simulation isintended.Boundary conditions are treated the same way as in the prototype case: plug-flowcondition for velocity; no-slip condition at solid walls; symmetry condition at the twoorthogonal planes bounding the domain; zero-gradient condition at the top exit plane,and the wall function for cells near the no-slip walls. The main concern is with the choiceof the turbulence parameters. The objective is to choose boundary values for k and 6such that the simulated flow field is similar to the prototype case.The value for kjet is chosen as before, that is, kjet 0.005 x T\u2019. However, the choicefor 6jet is not as obvious. There is no clear choice for the dissipation length scale Ld23.In the simulation with discrete jets, L,33 is chosen to be half of the smaller dimension(width) of an orifice, which is a common practice. For the open slot simulation, the slotis only a mathematical abstraction of the row of jets it represents, and hence the propervalue of Ld58 is not clear. For simplicity reasons, Ld53 is chosen to be the same as that ofthe earlier case, Ld283 = 0.5 x W where W is the width of an original primary jet orifice.Then 6jet is computed through jet =k2\/Ldj55,which yields a value the same as thatused in the prototype case.A two-level multigrid scheme is used for the calculation. This grid is much coarserin the y and z directions than in the prototype case. Figure 4.9 shows the mesh usedin this simulation. This simulation requires 36432 cells, about 24% of that used in thefull discrete jet simulation. The convergence history is displayed in Figure 4.10. Theconvergence rate is similar to that of the previous cases, but the computational time is100101Lj io(0io-io-10io-z1 081 o-IIIE E3.(2.C1.0\u2019NIIChapter 4. Simulation of Primary Level Jets 99I--Figure 4.9: Computation mesh used in the slot jet simulation.0.00.0 yIH 1.00.00.0 z\/H 1.0Work UnitsFigure 4.10: Convergence history for the open slot simulation.Chapter 4. Simulation of Primary Level Jets 100much shorter due to the coarser grid. The results are shown later in section 4.4 wherecomparisons are made.4.3.2 Porous SlotsIn the simulation with porous slots, slots of the same span are again introduced torepresent the rows of primary jets, but this time the height of each slot is set to bethe same as that of an individual primary port. This physical parameter is preservedbecause it is thought that the resulting prediction for the velocity field may be more likethe prototype case, particularly in regions near the slots.The velocity of the fluid entering through these enlarged slots is set to be the sameas that used for open slots. To keep the same amount of mass flux entering throughthe slots, the increase in the entrance area must be countered by the introduction of aporosity factor. For incompressible flow, because the density is constant, this porosityfactor can be defined simply as the ratio of the total area of the original ports to thearea of the porous slot representing those ports. That is,\u2014Area of Discrete Jets 4 1\u2014 Area of SlotThis parameter is introduced into the code to adjust the mass flux entering throughboundary cell faces where a porous slot is located. Specifically, at a boundary cell facedenoted as (Ax)(Ly), The mass flux entering the face is set to beMass Flux = C X [Pjetet(h\u2019)(1Y)lenIarged porous slot (4.2)This value of the mass flux is used in the computation of the coefficients in the finitedifference equations. To summarize, the product of this effort is to have jets entering thedomain at the same magnitude of velocity but through larger slots; the total amount ofthe mass flux entering is maintained as in the previous cases with the use of this porosityChapter 4. Simulation of Primary Level Jets 101factor. This strategy is sometimes used in computer simulations by researchers in theboiler industry.Before the set-up in our simulation is discussed, it is necessary to point out a concernfor numerical difficulties with this porosity approach. In this approach, the effective areais different from the total area on the side of the cell where the slot is located. Fromcontinuity of mass there will be changes of velocity or inflow\/outflow in the first cell,leading to possible changes in momentum flux on the opposite side. The pressure changenecessary to effect this change in momentum flux is an artifact of the porosity assumptionand is not physically correct. The momentum flux leaving the cell determines in partthe characteristics of the jet and this causes a new and physically incorrect boundarycondition for the jet entry. The effect of this change of boundary conditions and thedegree to which grid refinement may affect the overall result are problem specific andneed to be examined further. Present comparisons are interesting but must be treatedwith some reservation.In our simulation, the following values of the porosity factor are used: 0.185 forthe slot spanning along the y-direction and = 0.190 for the slot spanning along thez-direction. The mesh used for this simulation is identical to that used in the previousopen slot case. The boundary conditions are also the same. Specifically, the same choicesof kjet and Ejet are made. The convergence rate is similar to the open slot simulation.The quality of this simulation will be considered in the next section, where comparisonswill be made for all three simulations discussed.4.4 Comparison of Mean Up-flow VelocityA criterion for assessing the quality of each slot representation is the distribution of thevertical component of the mean velocity at various elevations. This criterion is chosenChapter 4. Simulation of Primary Level Jets 102because the regions of up-flow and down-flow are represented by the distribution of thisvelocity component. The extent of these regions of up-flow and down-flow affect thenature of flow interactions in the upper region of the boiler. Results of the distributionof the mean velocity component at the elevations x = 4.24 cm (1.67 inches), 7.62 cm (3inches), and 17.78 cm (7 inches) are shown in Figures 4.11-4.13 for the three simulations.The lowest elevation corresponds to the jet entrance level.An examination of Figures 4.11-4.13 reveals that the use of either open slots or porousslots gives results that are in reasonably good agreement with the prototype results. Thisis already evident at the jet entrance level, and more so at higher elevations. At the jetentrance level, shown in Figure 4.11, the extent of up-flow and down-flow is similaramong the three cases, with the exception of the region near the lower right corner aty\/H = 1.0 and z\/H = 0.0. This discrepancy is caused by the use of continuous slotsacross y\/H = 1.0 in both the open slot and porous slot simulations. By examining themagnitude of the velocity peaks located at the center of each graph in Figure 4.11, it isseen that the porous slot simulation gives results that show slightly more resemblanceto the prototype case than do the results of the open slot simulation. The porous slotspreserve the height of the jet opening, and give a better representation of the flow field atelevations close to the jet entrances. However, further up at x = 7.62 cm and 17.78 cm,the results shown in Figures 4.12-4.13 reveal that both slot representations give similarvelocity distribution to the prototype case, and that the advantage of the porous slotsimulation diminishes.To summarize, our results verify that it is reasonable to represent rows of closelyspaced jets such as the primary level jets with slots. In the simulation of slot jets usingthe k \u2014 e model, we obtain results of the velocity field that resemble the prototypecase when we choose values for the turbulent energy and its dissipation rate at the jetentrances to be the same as those used in the prototype case. The results with porousChapter 4. Simulation of Primary Level Jets 103DiscreteJets1.0zIHOpenSlotsPorousSlotszIHaiII 1\/tN0.50.00.0 0.5 y\/H 1.0Levels UNjet9 0.168 0.127 0.086 0.045 0.004 -0.043 -0.082 -0.121 -0.16Figure 4.11: Distribution of the vertical velocity component at x = 4.24 cm.Chapter 4. Simulation of Primary Level Jets 104DiscreteJetsPorousSlots1.0zIHzIH OpenSlots0.00.0 0.5 y\/H 1.01.0z\/HLevels UNjet9 0.168 0.127 0.086 0.045 0.004 -0.043 -0.082 -0.121 -0.16Figure 4.12: Distribution of the vertical velocity component at x 7.62 cm.Chapter 4. Simulation of Primary Level Jets 105DiscreteJets1.0z\/H0.51.0z\/H0.50.00.01.0z\/H0.50.00.0OpenSlotsPorousSlotsy\/H 1.0 Levels UNjet9 0.168 0.127 0.086 0.045 0.004 -0.043 -0.082 -0.121 -0.160.5 y\/H 1.0Figure 4.13: Distribution of the vertical velocity component at x = 17.78 cm.Chapter 4. Simulation of Primary Level Jets 106slots are only marginally better than those with open slots, and the benefit is limited toelevations close to the jet entrances.4.5 Comparison of Turbulence QuantitiesA concern with these slot representations is that a lower level of turbulent energy may bepredicted compared to the prototype results due to the neglect of turbulence generatedby shear in regions between neighboring jet orifices. A series of jets in-line will developintense turbulence intensity near the jet outlets due to the shear layers surrounding eachjet. A slot will lack this intensity in the near field and it is therefore important to checkthe turbulence characteristics in the simulation of closely spaced jets by slots.The distributions of the turbulent kinetic energy and its dissipation rate are examinedat the jet entrance elevation for the three simulations. The results are shown in Figures4.14-4.15, which illustrate the steep gradients in both k and e near the jet outlets. Magnified views for the turbulent energy and its dissipation rate for the prototype simulationwith discrete jets are shown in Figure 4.16 for clarity.The results shown in Figure 4.16 reveal that there is substantial generation of turbulent energy in regions between each pair of neighboring jets, but the regions of highlevels of k also correspond to those of high levels of c. Thus, the turbulent energy generated is dissipated rapidly over a short distance, and the distinction in the turbulencecharacteristics due to the simulation of discrete jets is very localized, as can be seen inFigures 4.14 and 4.15.Results of turbulent energy distribution at higher elevations are shown in Figures4.17-4.18. The results illustrate that the prototype (discrete jets) case gives values ofhi that are larger than the slot representations give. The corresponding results of thedissipation rate e are shown in Figures 4.19-4.20. A comparison of these latter graphs toChapter 4. Simulation of Primary Level Jets 107DiscreteJetsOpenSlots1.0z\/Hz\/H1.0zIH0.50.00.0PorousSlotsLevels k\/Vjet29 0.1008 0.0507 0.0206 0.0155 0.0104 0.0073 0.0052 0.0021 0.0010.5 y\/H 1.0Figure 4.14: Distribution of turbulent kinetic energy at x 4.24 cm.Chapter 4. Simulation of Primary Level Jets 108DiscreteJets1.0z\/H\u2022 41.0z\/H0.50.00.01.0z\/H0.50.00.0OpenSlotsPorousSlots0.5 y\/H 1.0 Levels E \/ (Vjet3\/H)9 5.0008 1.5007 0.5006 0.1005 0.0504 0.0103 0.0052 0.0021 0.0010.5 y\/H 1.0Figure 4.15: Distribution of dissipation rate of turbulent kinetic energy at x = 4.24 cm.Chapter 4. Simulation of Primary Level Jets 1090.24 0.26 0.28 0.30 0.32_____ ______y\/HFigure 4.16: Magnified views of the distributions of turbulent energy and its dissipationrate for discrete jets.TurbulenceEnergyLevels k I V29 0.0988 0.0867 0.0746 0.0625 0.0504 0.0383 0.0262 0.0141 0.0020.10z\/H0.080.060.040.020.000.220.10z\/H0.080.060.040.020.000.22 0.24 0.26 0.28 0.30 0.32y\/HEnergyDissipationLevels 8 I (V1e3\/H)9 2.458 2.157 1.856 1.555 1.254 0.953 0.652 0.351 0.05Chapter 4. Simulation of Primary Level Jets 110Figures 4.12 and 4.13 yields the observation that the regions of low values of c correspondto the regions of down-flow. The steep gradients of c near the wall are the result of theadoption of the wall function formulation in the simulations.Of more interest to the phenomenon of fluid mixing is the distribution of the turbulentviscosity p = C,2pk\/c. This quantity affects the extent of diffusive transport of scalartracer and momentum. An examination of the turbulent viscosity distribution at thejet entrance level, displayed in Figure 4.21, shows that the detailed distribution of theturbulent viscosity resulting from the various simulations are not alike, particularly nearthe jet entrances. In the prototype case, much higher values of are predicted near thejet entrances compared to the slot cases.The distribution of p at higher elevations is shown in Figures 4.22-4.23. The shapesof the distribution obtained from various simulations attain similarity in overall shapesas the elevation increases. However, the prototype case gives values of lit that are consistently around one and a half times greater than those produced by slot representations.The fact that higher values of 1ut are simulated in the prototype case is of interest.Because of the role of ,ut in the diffusive transport of scalar tracer and momentum, alower value predicted by slot jets may imply that the transport of fluid can be appreciablyaffected. This observation points to the precaution one must take when attempting torepresent a row of discrete jets by their slot equivalences in boiler simulations. Theseemingly good agreement in the velocity distribution shown in the last section throughthe use of either open slots or porous slots may not be a sufficient assurance for thevalidity of slot representations.Chapter 4. Simulation of Primary Level Jets 111DiscreteJetsPorousSlotsz\/Hz\/H OpenSlotszIHLevels k \/ Vjet29 0.00278 0.00247 0.00216 0.00185 0.00154 0.00123 0.00092 0.00061 0.0003Figure 4.17: Distribution of turbulent kinetic energy at x = 7.62 cm.Chapter 4. Simulation of Primary Level Jets 112OpenSlotsDiscreteJets1.00.56OCQ\u2022\u00e7 0.5 yIH \u00b0z\/H1.0z\/H PorousSlots0.50.00.0Levels k \/ Vjet29 0.00188 0.00167 0.00146 0.00125 0.00104 0.00083 0.00062 0.00041 0.00020.5 y\/H 1.0Figure 4.18: Distribution of turbulent kinetic energy at x = 17.78 cm.Chapter 4. Simulation of Primary Level Jets 113DiscreteJetsPorousSlots1.0z\/Hz\/H OpenSlotsz\/HLevels 6 I (Vjet3\/H)9 0.001458 0.001 307 0.001156 0.001005 0.000854 0.000703 0.000552 0.000401 0.00025Figure 4.19: Distribution of dissipation rate of turbulent kinetic energy at x 7.62 cm.Chapter 4. Simulation of Primary Level Jets 1141.0z\/H DiscreteJetsPorousSlots0.50.00.0 0.5 yIH 1.0zIH OpenSlotsz\/HLevels E I (Vjet3\/H)9 0.000368 0.000327 0.000286 0.000245 0.000204 0.000163 0.000122 0.000081 0.00004Figure 4.20: Distribution of dissipation rate of turbulent kinetic energy at x = 17.78 cm.Chapter 4. Simulation of Primary Level Jets 115DiscreteJetsPorousSlotsz\/H0.00.0 0.5 y\/H 1.01.0z\/H OpenSlotsz\/HLevels llt\/1LL9 270.08 240.07 210.06 180.05 150.04 120.03 90.02 60.01 30.0Figure 4.21: Distribution of turbulent viscosity at x = 4.24 cm.Chapter 4. Simulation of Primary Level Jets 116N \u2014-\u20180.5 yIH 1.0zIH1.00.50.0zIHDiscreteJetsOpenSlotsPorousSlotsz\/HLevels LtIJL9 530.08 470.07 410.06 350.05 290.04 230.03 170.02 110.01 50.0Figure 4.22: Distribution of turbulent viscosity at x 7.62 cm.Chapter 4. Simulation of Primary Level Jets 117DiscreteJetszIH1.0z\/H0.50.00.5 yIH 1 .00.0OpenSlotsPorousSlotsz\/HLevels \u20181\/119 630.08 560.07 490.06 420.05 350.04 280.03 210.02 140.01 70.0Figure 4.23: Distribution of turbulent viscosity at x = 17.78 cm.Chapter 4. Simulation of Primary Level Jets 1184.6 Varying the Turbulence Input Parameters in Slot SimulationsIn the simulations using slots, lower values of j are produced in the flow field as aconsequence of the omission of the extra turbulence intensity near the outlets of discretejets. Thus, it is important to examine the effects of increasing the turbulence intensityused in slot simulations. Because of similarities in the results produced by open slots andporous slots, the attention here will be focussed only on simulations with open slots.Our aim is to observe if there are any changes to the flow field when the value of thejet turbulent energy k\u20ac is varied. The choice for Lj83 = 0.5 x W is maintained suchthat the value of Ejet increases as k\u20ac increases. The following two cases are considered:kjet = 0.05 x Vj and kjet = 0.5 x which correspond to an increase in the jet turbulentenergy by factors of 10 and 100, respectively.Calculations are carried out with the same multigrid procedure as before. The convergence rates of these latter simulations are similar to the previous open slot simulation.The distributions of turbulent energy and its dissipation rate are examined at the jetentrance elevation and are shown in Figure 4.24. The graphs there illustrate the resultsthat there are steep gradients in k and e near the jet entrances. A comparison of Figure4.24(a) with 4.24(b) and with the open slot case in Figure 4.14 shows that there is anincrease in the turbulent energy near the jet entrances as the value of kjet increases.However, the corresponding energy dissipation rate e also attains higher values near thejet entrances.In order to examine how the mean velocity may be affected, the distributions of theturbulent viscosity and the vertical component of the mean velocity are compared at thesame jet entrance elevation, and the results are shown in Figure 4.25. A comparison ofthe graphs in this figure to those in Figures 4.11 and 4.21 shows that similar distributionsare obtained for both the turbulent viscosity and velocity fields for the different choicesChapter 4. Simulation of Primary Level Jets 119of kjet.The results of our simulations reveal the following characteristic in the calculation ofprimary jet flows with the k \u2014 turbulence model: when the input turbulent energy isincreased while the dissipation length scale is maintained the same, the model will beself-adjusting so that the extra turbulent energy will be consumed. Consequently, similarlevels of turbulent viscosity are predicted. Moreover, within the bulk of the flow field,the turbulence intensity appears to be governed primarily by the shearing action withinthe flow field and is not very sensitive to the incoming turbulence intensity level.Additional graphs displaying the distributions of k, , , and the vertical componentof velocity at higher elevations are shown in Figures 4.26-4.29. These results are comparedwith the corresponding earlier results of open slots shown in Figures 4.12-4.13, 4.17-4.20,and 4.22-4.23. The point to note is that for the cases examined, there is a high degreeof similarity in the results. It is apparent that the extra turbulent viscosity simulatedin the prototype case cannot be reproduced easily by increasing the turbulence intensityprescribed for the slot jets.In passing, we mention that we have also tried varying the dissipation length scaleLd28 by increasing it first to Ld58 = 5 x W, and then to 50 x W, while maintainingkjet 0.005 x i\u2019;2. These two cases correspond to decreasing jet by factors of 10 and100, respectively. The convergence characteristics of these simulations are similar to theoriginal open slot simulation and the results of the mean vertical velocity componentand turbulence quantities are also similar. This observation further confirms that in thesimulations with slots, the turbulence and mean flow characteristics are influenced morestrongly by the intense jet interactions than by the prescribed boundary data of theturbulence quantities.Chapter 4. Simulation of Primary Level Jets 120(a)(b)TurbulentEnergyLevels k I V29 0.1008 0.0507 0.0206 0.0155 0.0104 0.0073 0.0052 0.0021 0.001EnergyDissipationLevels \u00a3 \/ (Vjet3\/H)9 5.0008 1.5007 0.5006 0.1005 0.0504 0.0103 0.0052 0.0021 0.001Figure 4.24: Distributions of turbulent energy and dissipation at x = 4.24 cm for (a)kjet = 0.05 x and (b) k,\u20ac = 0.5 Xz\/HzIH z\/Hy\/H y\/H 1.0Chapter 4. Simulation of Primary Level Jets 121_____ ______Upflow__________andDownflowFigure 4.25: Distributions of turbulent viscosity and mean vertical velocity at r = 4.24cm for (a) kjet = 0.05 x and (b) kjet 0.5 X1.zIH z\/H(a)(b)zIH z\/H00.yIHTurbulentViscosity1.0yIHLevels Pt1lt9 270.08 240.07 210.06 180.05 150.04 120.03 90.02 60.01 30.0Levels U I V9 0.168 0.127 0.086 0.045 0.004 -0.043 -0.082 -0.121 -0.16Chapter 4. Simulation of Primary Level Jets 122TurbulentEnergyFigure 4.26: Distributions of turbulent energy and dissipation at1 \u2014 fl fl T72 fi...\\ 1 \u2014 rr2jet \u2014 U.UU X Vjet anu IU) \u2018\u2018jet \u2014 U.U X Vjet\u2022= 7.62 cm for (a)zIH(a)(b)zfHy\/H 1.0 y\/H 1.0EnergyDissipationLevels k \/Vjet29 0.00278 0.00247 0.00216 0.00185 0.00154 0.00123 0.00092 0.00061 0.0003Levels e \/ (Vjet3\/H)9 0.001458 0.001307 0.001156 0.001 005 0.000854 0.000703 0.000552 0.000401 0.00025Chapter 4. Simulation of Primary Level JetsTurbulentViscosityLevels9 530.08 470.07 410.06 350.05 290.04 230.03 170.02 110.01 50.01.cz\/HUpflowandDownflowLevels U\/Vjet9 0.168 0.127 0.086 0.045 0.004 -0.043 -0.082 -0.121 -0.16Figure 4.27: Distributions of turbulent viscosity and mean vertical velocity at x = 7.62cm for (a) kjet = 0.05 x and (b) kjet = 0.5 Xz\/H123(a) 0.51.cz\/H0.5 y\/H 1.0(b) 0.5Th 2\\1-. 10.0 0.5 yIH 1.0 y\/H0.G \u20140.0 0.5 y\/H 1.0TurbulentEnergyLevels k I Vjet29 0.00188 0.00167 0.00146 0.00125 0.00104 0.00083 0.00062 0.00041 0.0002EnergyDissipationLevels 2 \/ (Vjet3\/H)9 0.000368 0.000327 0.000286 0.000245 0.000204 0.000163 0.000122 0.000081 0.00004Figure 4.28: Distributions of turbulent energy and dissipation at x = 17.78 cm for (a)1 \u2014 n n 172 1 (1.. 1 \u2014 rr2\u2018\u201cjet \u2014 V.UcJ X Vjet anu Y) \u2018jet \u2014 V.J Xz\/HChapter 4. Simulation of Primary Level Jets 1241.0k )0.50.c0.0 0.5(a)(b)zIHy\/H 1.01.cz\/H0.5* [4Jy\/HFigure 4.29: Distributions of turbulent viscosity and mean vertical velocity at x 17.78cm for (a) kjet = 0.05 x and (b) kjet = 0.5 X V.z\/HChapter 4. Simulation of Primary Level Jets 125(a)zIH1.cz\/H(b) 0.5y\/H 1.0\u2018J .\u2018_,0.0 0.5TurbulentViscosityy\/H 1.0UpflowandDownflowLevels l\u2019\u201dl-9 630.08 560.07 490.06 420.05 350.04 280.03 210.02 140.01 70.0Levels \/ jt9 0.168 0.127 0.086 0.045 0.004 -0.043 -0.082 -0.121 -0.16Chapter 4. Simulation of Primary Level Jets 1264.7 Concluding RemarksWhen jets are closely spaced in a row, they may be represented numerically by slots.Our computational experience shows that the use of slot representation for primary jetsis essential for numerical modelling. If individual primary jets were to be modelled, thesystem of equations could become so large that convergence would be difficult to achieve.In such a case, the multigrid method may be difficult to apply unless the computer systemhas sufficient memory capability.In the simulations with slot jets, the important flow parameters that need to bemaintained are the jet speed and the total mass flow rate. Our results show that thevelocity profiles are captured fairly well by the use of either open slots or porous slots. Incomparison to open slot simulations, modelling with porous slots yields a slightly betteragreement with the prototype case in the regions near the jet entrances. However, athigher elevations, there is no advantage in adopting porous slots.It is important to note that the values of turbulent viscosity are under-predictedby slot simulations. This raises the concern that properties such as the extent of fluidmixing may not be reproduced well by slot simulations without further refinement to themodelling methodology. Moreover, note that only a single level of jets is simulated. In anactual boiler simulation with several levels of jets, one may encounter surprising resultsnot illustrated with the present simple configuration.Chapter 5Characteristics of a Row of Jets in a Confined CrossfiowA study is carried out on the characteristics due to a row of jets injecting into a confinedcrossfiow. Such a configuration will also be called a row-jet injection scheme. This studyis relevant to improving the understanding of the dynamics of jet interaction in a kraftrecovery boiler because the secondary and tertiary level jets operate in an environmentwhere a crossfiow is present. The results of our investigation provide information thatcan be used to interpret the observations arising from more comprehensive modellingstudies, such as the effects on jet penetration when both orifice spacing and dimensionsare varied.In this chapter, attention is restricted to the interaction of a single row of jets with aconfined uniform crossfiow. This simple geometric configuration provides a suitable basisfor the study of jet dynamics that should precede the consideration of more complexconfigurations. However, even with this simplification, there are still many geometric andoperational parameters that make a systematic study difficult. Examples of geometricparameters are the shape and size of jet orifices, their separation, and the extent ofconfinement. Examples of operational parameters are the jet velocity, the mainstreamvelocity, and the ratio of jet momentum to mainstream momentum.Among the many possible parametric variations, two studies which deserve attentionare the effects of varying jet spacing while maintaining jet momentum constant, andthe effects of varying jet momentum while holding spacing constant. One purpose ofthe present investigation is to verify the observations by Kamotani and Greber (1974)127Chapter 5. Characteristics of a Row of Jets in a Confined Crossfiow 128concerning the changes in jet penetration when the jet spacing is varied. Another purposeis to reveal detailed features such as the vorticity dynamics of the flow field due to theinteraction between a row of jets and a crossflow. This study provides results whichcontrast the flow characteristics of a row of jets with those of an isolated jet in a crossflow.First, a description is presented for the jet-in-crossflow problem under consideration.Then the solution method and the convergence characteristics are discussed. Next, thejet penetration and the development of the cross-section of a jet is examined for differentvalues of jet spacing and momentum ratio. Finally, the vorticity dynamics is investigated.5.1 Problem Description and Boundary ConditionsThe geometry of the injection scheme is displayed in Figure 5.1. The following geometricconditions are selected: jet width W = 0.00635 m (0.25 inches), dimension of channelH = 24 x W, and spacing between a pair of jets S\/W = 2, 4, 8, and 16.Two values of the jet-to-crossflow momentum ratio are considered: J = 8 and 72.The speed of the crossflow is set at Uc = 6 rn\/s. Consequently, the speeds of the jetsare = 17 and 51 rn\/s, respectively. These conditions are chosen to emulate theexperimental work by Karnotani and Greber (1974), where round jets having a diameterof 0.00635 m were used.In the numerical model, the domain of the flow field is represented as a rectangularbox, much like that in the previous computation for a single jet. Due to flow symmetry,only half of a jet needs to be computed. Symmetry conditions are imposed on the twolateral planes bounding the space between the jet center-line and the distance mid-waybetween two adjacent jets. The separation between these two symmetry planes is variedto permit simulation of different values of S\/W.The wall function is used to evaluate the shear stress near solid boundaries. The choiceChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 129x = streamwise coordinatey = cross-stream coordinatez = lateral coordinateIII II IxIt.0-s IlpiFIsZDji VJET4ywIHI I tttttuc U0Figure 5.1: Schematic representation of jet injection from a row of square orifices.for the location of the inlet plane for the crossflow is influenced by two considerations:first, the location needs to be sufficiently far from the jet so that the presence of thejet will not lead to an unrealistic velocity distribution near the plane. Secondly, theplane should not be too far upstream to avoid unnecessarily thick boundary layer atthe jet entrance. Based on these two considerations, it is decided that the inlet planeto be located at about 10 jet widths upstream of the jet, where a uniform velocity Ucis assigned in the plane. The inlet turbulent energy is specified by icc = 0.005 x Ur,while the dissipation length scale is chosen to be Ld88 = 0.1 x H and the dissipationrate at the inlet is set at Ec = kj2\/LdI33. At the jet entrance, due to the strength ofthe jet, uniform values for the jet velocity, turbulence quantities, and the jet tracer arespecified. The turbulent kinetic energy is set at k3 = 0.005 x T\u2019t, and the dissipationlength scale is set at Ld135 = 0.5 x W so that the dissipation rate is jet = k;2\/(o.s x W).Based on past experience with the calculation of a single jet in a crossfiow, the turbulenceChapter 5. Characteristics of a Row of Jets in a Confined CrossBow 130characteristics for this type of flow field should be governed mainly by the shearing actionof the flow and not by the prescribed boundary data for k and c. A unit value of aninert tracer is prescribed at the jet entrance to mark the dispersion of the jet fluid. Forthe exit condition, the exit plane is placed at up to 100 jet widths downstream of the jetentrance. The zero-gradient condition for the dependent variables is imposed.5.2 Convergence PerformanceFor the type of flow field considered in this chapter, our multigrid method generallyprovides very fast convergence. However, the convergence rate deteriorates as the jetmomentum is increased and as the jet spacing is reduced. To illustrate this, the convergence histories for the cases S\/W = 4 at J = 8 and S\/W = 2 and 4 at J = 72 arepresented in Figure 5.2. The mass residues have been normalized by the total incomingmass flow from the cross-stream and the jet.The deterioration of the performance of the multigrid solver is apparent from theabove results. Moreover, it needs to be noted that for the difficult case S\/W = 2 atJ = 72, the number of iterations carried out before the restriction cannot be too small;otherwise, stalling or divergence can occur. This observation is similar to that for theprimary jet simulations carried out in the previous chapter. This is a little surprisingbecause in past experience with applying the multigrid method to nonlinear equations asoutlined in section 3.4.2, if too many smoothing iterations are carried out, high frequencyerrors may be re-introduced into the solution due to interactions with boundary conditions. However, our results suggest that for difficult turbulent flow simulations causedby close jet spacing and high jet-to-crossflow momentum ratio, a significant amount ofsmoothing is needed to eliminate the high frequency error components. If this is notdone before the solution is transferred to the coarser grid, there can be too much \u2018noise\u2019Chapter 5. Characteristics of a Row of Jets in a Confined Crossflow(a)10.2io2 io4L io10.6a310.8- io10\u00b0Z io10.14io(b)10.2iO: iow 1010.8iO- 10.8. iO-C 10\u00b0100Z io10.13(c)0LiiCi)COci)NE0zS\/W=4 J=81.75E4131Fine Grid Work UnitsSIW=4 J=72FineGridWorkUnitsSIW=2 J=72Fine Grid Work UnitsFigure 5.2: Convergence histories for different cases of S\/W and J.Chapter 5. Characteristics of a Row of Jets in a Confined Crossfiow 132in the auxiliary problem in the coarse grid and convergence becomes difficult to achieve.It has been commonly observed in many flow situations that convergence becomesdifficult when the flow Reynolds number is increased. Examples include the laminardriven cavity problem and the flow over a backward-facing step. Vanka (1986) suggestedthat the problem is due to an increase in the nonlinearity measure of the equations asthe Reynolds number is increased. Thomson and Ferziger (1989) also remarked that asthe Reynolds number is increased, the flow may develop more fine-scale structures. Ifthe grid is too coarse to resolve these structures, the solution may be slightly wigg1y\u2019and convergence may be hindered.One reason for the deterioration in the convergence performance is that when thejets are closely spaced, they merge to form a curtain that impedes the crossflow frompassing between them. As a result, a recirculating bubble is formed behind each jet, andconsequently a much longer domain is needed to place the exit plane. Thus, certain gridcells may be highly stretched, which can cause numerical anisotropy and may adverselyaffect the rate of convergence. This latter effect is caused by the slowness in the transferof information along the direction parallel to the long side of the grid cells. The useof a line solver is inefficient for problems with a long domain since the solver updatesinformation in a relatively local manner. In this regard, the use of a global iterative solvermay improve convergence. An example of such a global solver is the lower-upper (LU)factorization algorithm discussed in many standard textbooks such as Maron (1987).The convergence rate for this case of high momentum value and close jet spacing canbe improved if the system of equations possesses additional numerical stability characteristics. These characteristics may be introduced by treating the problem as a transientone, and the steady state solution, if it exists, can be obtained by stepping through time.The transient equations of motion are obtained by including the time rate of changeChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 133terms into Eqs.(2.11-2.12). The equations become8 up a I IOU2 OU\u2019\\ Si (5.1)ut OX3 OX2 OX3 \\ OX3 OX2 J Ja a I Ia \u2014Si+ p\u2014(4)U3)= \u2014 \\ u\u2014) \u2014 pq,u (5.2)These modified equations describe the gross unsteadiness in the mean flow field, if suchan unsteadiness exists. For Eqs.(5.1-5.2) to be physically meaningful, the time scale ofthe unsteadiness has to be much larger than the longest time scales of the turbulentmotion.The inclusion of the time dependent terms improves the numerical stability characteristics because it increases the diagonal dominance of the resulting system of finitedifference equations. In the calculation, a time step value of 0.5 seconds is chosen. Starting at time t = 0, we march forward in time. The convergence history is displayed inFigure 5.3, which is to be compared to that presented in Figure 5.2c. The results of theflow field obtained here are visually indistinguishable to those obtained earlier using thesteady state solver. It is evident that a faster convergence has been achieved and thatthe overall residue reduction rate is rather constant, even though it is still significantlyslower than in the cases where the jet spacings are larger.5.3 Jet PenetrationOur study on jet penetration complements the experimental investigations by Ivanov(1959) and by Kamotani and Greber (1974) discussed earlier in Chapter 2. The focus ofthis section is on the effect of jet spacing on the penetration for moderately spaced jets.The results of the numerical simulations are depicted in Figures 5.4-5.7. In Figure5.4, the center-plane velocity profiles are displayed for S\/W = 2, 4, 8, and 16 at J = 8.It is apparent that at this low momentum ratio, the blockage effect caused by the jetChapter 5. Characteristics of a Row of Jets in a Confined Crossfiow 134S\/W=2 J=72 Time Dependence CalculationFigure 5.3: Convergence history for S\/W = 2 at J = 72; time dependence calculation.on the crossflow at low values of S\/W is not significant, and this is revealed by theabsence of a recirculating bubble behind the jet root for the case S\/W = 2. The jettracer concentration profiles along the same longitudinal plane are shown in Figure 5.5for the same cases. These profiles are used to estimate the jet penetration in eachcase. Also shown in the graphs in Figure 5.5 are streamlines representing paths of fluidparticles leaving the center of the jet orifice. These streamlines are representative of thejet trajectories in this center-plane. In passing, we note and recall that the definitions ofjet trajectories are varied in different investigations. Kamotani and Greber defined thecenter-plane jet trajectory to be the locus of maximum speed at each cross-stream planeintersecting the jet center-plane.Similar results of the velocity profiles and jet tracer concentration for the high momentum case J = 72 are presented in Figures 5.6 and 5.7, respectively. At this highermomentum, each jet spreads farther and the blockage effect on the crossflow is moreprominent. When S\/W = 2, a large reverse flow region is formed behind each jet.Consequently, the exit plane has to be placed very far downstream at = 100 x W, acondition that may adversely affect solution convergence as discussed in section 5.2._ io0io3b40N io21067.50E3 1 .00E4Fine Grid Work UnitsChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 135Our numerical results also indicate that for both J = 8 and 72, there is a monotonicreduction in jet penetration as S\/W is reduced. The reduction is more significant whenS\/W decreases from 16 to 8 and less so from 8 to 4. Comparing the results in Figures5.5 and 5.7 to those shown in Figures 2.5-2.6, our results are similar to those reportedby Ivanov and slightly different from those by Kamotani and Greber. At J = 72, ourresults predict a rise in jet penetration as reported by Kamotani and Greber when S\/Wis reduced from 4 to 2, but only in the region close to the jet entrance (0 < x\/W < 30).The discrepancy may be attributed to factors such as different definitions used for the jettrajectory, experimental uncertainties, and deficiencies in the mathematical model usedfor the flow prediction.The observed decrease in jet penetration with reduction in orifice spacing is usefulin providing a physical basis to explain Holdeman\u2019s correlation. Equation (2.24) is usedfor obtaining consistent jet penetration, and will be discussed in the next chapter. Besides penetration, other characteristics that are of interest for the problem of jet mixinginclude the spreading and vorticity dynamics associated with each jet in this row-jetconfiguration. In addition to its role on jet mixing, the dynamics of the pair of counter-rotating vortices has been suggested by Kamotani and Greber (1974) to be instrumentalin reducing the penetration of the jet as the jet spacing decreases. These characteristicsare examined in the next section.5.4 Development of the Cross-section of a JetThe effects of jet spacing or lateral confinement on the development of the cross-sectionalshape of a jet are now discussed. The development of the cross-section as the jet ispropagating downstream is described in Figures 5.8-5.11. The figures show the evolutionChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 136(a) S\/W=2 (b) SIW=4(c) S\/W=8 (d) S\/W= 16jig [11111111 itJIttillltiilllllil1 ti\/lit 111111111iIlilitti 1111111.111\/li ii III iitill!! ii 11111till\/tIll Ill IItill\/fl I Ill IIi\/I\/iIl\/\/ili\/\/I\u2018If50x\/W403020100-10lilt I Ii lii11111 II Ill.1 1 tIllu,t1\u2018I0 10 20 y\/WI 111111111 I I I I1111 IlIl1ll1ll1l1111111111 lIltIttIltItItIltlIIL!ifllt I I I Illhull I I Illti\/ft 1 1 1 110 10 20 yIW= 20 mIs= 20 mIs50xJW403020100-1050x\/W403020100-10=20 mIs=20 mIsI I 111111111111Ill 11111111111Ill II1111 11I liltII II111111 Iit\/li! IiiFill\/it Illill\/if illtilli 11till! Itill! IIll\/Iil\/I!1\/!\/1J150x\/W403020100-100 10 20 yIW 0 10 20 y\/WFigure 5.4: Velocity profiles for S\/W = 2, 4, 8, and 16 at J = 8 along the jet center-plane.Chapter 5. Characteristics of a Row of Jets in a Confined CrossflowTracerConc.0.350.250.150.05TracerConc.0.350.250.150.05137Figure 5.5: Jet tracer concentration profiles for S\/W = 2, 4, 8, and 16 at J = 8 alongS\/W=2 SIW=4TracerConc.0.350.250.150.05Levels432TracerConc.0.350.250.150.05Levels43210 20 yIW(a)50xIW403020100-100(c) S\/W=850xIW403020100-10(b)50x\/W403020100-100 10 20 yIW(d) S\/W=1650x\/W403020100-10Levels432Levels4320 10 20 y\/W 0 10 20 y\/Wthe jet center-plane.Chapter 5. Characteristics of a Row of Jets in a Confined Crossflow 138(a) SIW=2 (b) SIW=4(c) S\/W=8!iiiIIIlIiHiIIItiiiiiiitiillttI40 ritl\/\/iiitiilIIttttiiiiititt1i1l30 F1II!\/\/\/\/IIIH 1111!\/\/\/\/\/\/I!tI 11IH\/\/\/\/\/\/\/tlI 1120 III\/\/\/\/\/\/1ttt\/\/\/\/\/\/ it I\/\/\/\/\/i! Li,\/\/,-2\/\/ IL i\/\/--\u20184ji10 \/(d) S\/W= 16Figure 5.6: Velocity profiles for S\/W = 2, 4, 8, and 16 at J 72 along the jet cen100x\/W9080706050403020100-10iiiiIIlIIIIIlITl\u2022I I I I I I I I,\\HHHItH 11 III II. , \u2014I:fJ111111111tIlt II Iittttt111,11! tL1\/ui!11,,,!1\/\/ui1\/u, \/I0 10 20 y\/W0 10 20 y\/W50x\/W4030+ 20=20 m\/s100-1050x\/W4030+ 20=20 mIs100-1050x\/W=20mIs= 20 mIsIIII1iI\/\/\/\/\/\/\/\/,lilt!lIlt!1I\/\/\/\/\/\/\/I!lIlillI\/\/\/\/\/\/\/\/!111111 \/ \/\/\/\/\/\/\/1 IlltIt\/\/\/\/\/\/\/\/\/ttI 1tI\/\/\/\/\/\/\/\/IttIt!! 1\/\/\/Z\/\/\/1tl11 \/\/\/\/z-\u2019\/\/-100 10 20 y\/W 0 10 20 y\/Wter-plane.Chapter 5. Characteristics of a Row of Jets in a Confined Crossflow 139Figure 5.7:the jet center-plane.TracerConc.0.350.250.150.05(b) S\/W=4TracerConc.0.350.250.150.05Levels432Levels432(a) S\/W=250x\/W403020100-100 10 20 yIW(C) S\/W=850x\/W403020100-100 10 20 yIW50__x\/W__403020100-10(d) S\/W=1650Tracer___Conc. x\/W___ ____0.35 400.250.150.05 3020100-100 10 20 y\/WJet tracer concentration profiles for S\/W = 2, 4, 8, and 16 at J = 72 alongLevels432TracerConc.0.350.250.150.05Levels4320 10 20 y\/WChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 140in the jet fluid distribution from x\/W = 5 to 10 and for the cases S\/W = 2, 4, and8 at J = 8 and 72. The following observations are made: first, the jets spread outfaster laterally as J increases, and as the jet spacing is reduced the jets merge rapidly.Secondly, the jet fluid distribution bifurcates into two cores about the symmetry planez = 0 only when the jet spacing is sufficiently large (S\/W 4); otherwise, the maximumconcentration of the jet fluid lies along the symmetry plane. The latter phenomenon iscaused by the severe lateral confinement which prevents the jet fluid from bifurcatinginto two plumes.As mentioned in Chapter 2, the pair of counter-rotating vortices associated with eachjet has a significant role in the following areas: the mixing of the jet fluid; the cross-sectional shape of the jet, and the velocity distribution in the cross-sectional plane. Aquantitative study of the vortices is difficult because a large amount of velocity datais required to construct the vorticity field. Thus, numerical simulations offer a viablealternative for the study provided the mathematical model can yield correct informationabout the flow field. Some results concerning the vorticity dynamics of a single jet in acrossflow have already been discussed in Chapter 2. The vorticity dynamics for multiplejets, however, has rarely been discussed. A study of this vorticity dynamics is the focusof the following section.5.5 Vorticity Dynamics for a Row of Square Jets in a CrossflowIn the setting of multiple jets, the mutual interaction between neighboring jets bringsadditional complications to the vorticity phenomena. There are two conflicting andcompeting mechanisms that affect the vorticity characteristics of multiple jets. First,any blockage caused by a row of jets can lead to an acceleration of the crossflow aroundeach jet. Thus, the streamwise vorticity is intensified by this extensional strain rate.Chapter 5. Characteristics of a Row of Jets in a Confined CrossflowFigure 5.8: Jetplane x\/W = 5tracer concentration profiles for S\/W 2, 4, and 8 in the cross-streamatJ=8.141Levels765432TracerConc.0.4600.3940.3280.2630.1970.1310.066Levels765432TracerConc.0.3440.2950.2460.1970.1480.0980.049(b) SIW=4y\/W20 -15 -10 -00.0 1.0 2.0zIW(a) S\/W=2y\/W201510(C)yIW201510500.0S\/W=8Levels765432TracerConc.0.3240.2770.2310.1850.1390.0920.0461.0 2.0 3.0z\/W 4.0Chapter 5. Characteristics of a Row of Jets in a Confined CrossflowFigure 5.9: Jet tracer concentration profilesplane x\/W=1O atJ=8.for S\/W = 2, 4, and 8 in the cross-stream(a) S\/W=2 (b) SIW=4y\/W y\/W142Levels765432TracerConc.0.3560.3050.2540.2030.1530.1020.051Levels76543220 20 -15 15 -10 - 10 -!:6---- -0 -00 1.0z\/W\u2022z\/WTracerConc.0.2420.2070.1730.1380.1040.0690.03500.0S\/W=81.0Levels7654322.0TracerConc.0.2470.2120.1760.1410.1060.0710.035(c)yIW201510500.0 1.0 2.0 3.0z,\u2019W4.0Chapter 5. Characteristics of a Row of Jets in a Confined Crossflow(a) S\/W=2 (b) SIW=4143Figure 5.10: Jet tracerplanex\/W=5at J=72.concentration profiles for S\/W = 2, 4, and 8 in the cross-streamyIWLevels765432Levels765432TracerConc.0.2660.2280.1900.1520.1140.0760.03820 -15 -TracerConc.0.4130.3540.2950.2360.1770.1180.059S\/W=8yIW201510500.0 1.0zIW(c)y\/W -201510500.01.0 2.0z\/WLevels765432TracerConc.0.2200.1890.1570.1260.0940.0630.0311.0 2.0 3.0 4.0z\/WChapter 5. Characteristics of a Row of Jets in a Confined Cross11owFigure 5.11: Jet tracer concentration profiles for S\/W = 2, 4, and 8 in the cross-streamplanex\/W=lOatJ=72.144Levels765432Levels765432TracerConc.0.2200.1890.1570.1260.0940.0630.031(b) S\/W=4yIW20 -155TracerConc.0.3660.3140.2610.2090.1570.1050.052S\/W=8(a) S\/W=2y\/w2015500.0 1.0z\/W(C)y\/W201510500.0Levels765432TracerConc.0.1620.1390.1160.0930.0690.0460.0231.0 2.0 3.0z,\u2019W4.0Chapter 5. Characteristics of a Row of Jets in a Confined Crossflow 145Secondly, the adjacent vortices from two neighboring jets have opposite signs, and theirinteraction leads to a reduction in their strength. Hence, in a multiple jet configuration,it is of interest to note the distributions of the jet fluid and vorticity as they could be quitedifferent from those of a single jet. It has been suggested by Stevens and Carrotte (1988)that the decay rates of the vorticity present in the counter-rotating vortex pair affectthe jet mixing with the main flow, but the interpretation of the effect of jet spacing onvorticity is not trivial. The following study is aimed at elucidating some characteristicsof the vorticity dynamics in this multiple jet context.5.5.1 Magnitudes of the Streamwise VorticityIt was suggested by Stevens and Carrotte that the blockage caused by confined, laterallyspaced jets would result in a significant increase in the magnitude of the vorticity. Ourresults show that the suggestion needs to be stated more carefully.At a cross-sectional plane sufficiently far downstream of the jet, where most of the vorticityis in the streamwise (x) direction, we compute the maximum value of the streamwisecomponent of the vorticity of half of the jet that is being simulated. This streamwisevorticity is defined as8U3 8U253(.)The variation of the maximum streamwise vorticity with the downstream position isplotted in Figure 5.12 for various values of S\/W and for J = 8 and 72. The values of thevorticity are made non-dimensional through multiplication by W\/Uc.When the jets are very closely spaced at S\/W = 2, the maximum vorticity is at thelowest compared to other spacings for both J = 8 and 72. For J = 72, the maximumvorticity is still lower for S\/W = 4 than for S\/W = 8. This result is due to the rapidmerging of jets at close spacings, especially when the momentum ratio is high. When theChapter 5. Characteristics of a Row of Jets in a Confined Crossflow(a)(x)ma________(UIW)(b)(x)max(U0\/W)Figure 5.12: Variations in the maximum streamwise vorticity with downstream distanceforS\/W=2,4,and8.146J=8S\/W=2A SIW=40 S\/W=81.000.750.500.250.002.001.501.000.500.00) 5 10 15 20 25xJWJ = 721 S\/W=2A S\/W=40 S\/W=85 10 15 20 25x\/WChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 147jets merge, the crossflow is prevented from passing through the space between neighboringjets. The decrease in the lateral shear, together with an increase in the vorticity diffusionacross jets, contribute to a reduction in the vorticity. Hence, for J = 72, there is adecrease in the vorticity as the jet spacing is reduced for the cases shown. For the lowermomentum case J = 8, the results displayed in Figure 5.12(a) show that there is anincrease in the vorticity when the spacing is reduced from S\/W 8 to 4. Nevertheless,the increase is slight and the trend reverses a short distance downstream.To summarize, the results of our simulations show that the increase in blockage causedby a reduction in the spacing between neighboring jets does not always lead to an increasein the maximum vorticity of each jet. The competing effects of vorticity production dueto lateral shear and vorticity diffusion need to be considered in the estimation of thevorticity maximum.5.5.2 Distribution of the Streamwise VorticityAn examination of the streamwise vorticity distribution reveals the vorticity developmentfor various cases. At the location x\/W 10, the cross-sectional distribution of thestreamwise component of vorticity, together with the secondary flow vectors, are plottedin Figures 5.13-5.16 for the cases S\/W = 2, 4, and 8 at J = 8 and 72. In each graph,the distribution of the streamwise vorticity is shown for half of the jet between the twolongitudinal symmetry planes. The vorticity distribution of the other half of the jet isthe mirror image of the distribution shown but with the opposite sign.A number of observations are noted. First, by comparing the vorticity distributionwith the corresponding vector field of the secondary flow in the same cross-sectionalplane, it is seen that in many cases, the vorticity cores correspond to the centers of fluidrotation. However, for the case S\/W = 2 at J = 72 shown in Figure 5.15(a) and Figure5.16(a), the vorticity core is well defined even though there is no apparent recirculatingChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 148(a) S\/W=2 (b) SIW=4Levels 2I(UIW)7 0.4806 0.4055 0.3304 0.2553 0.1802 0.1051 0.031Figure 5.13: Distribution of the streamwise vorticity in the cross-stream plane x\/W 10for S\/W=2,4,and8atJ=8.y\/W yIWLevels765432Levels c2\/(U\/W)7 0.5036 0.4295 0.3554 0.2823 0.2082 0.1341 0.061QI(U1\u2019I1)0.2660.2270.1890.1500.1110.0720.033S\/W=820151000.01.02.00.0 1.0z\/W(c)yIW201510500.0 1.0 2.0 3.0z,\u2019W 4.0Chapter 5. Characteristics of a Row of Jets in a Confined Crossflow 149(a) S\/W=2 (b) S\/W=4yIW___________yi\u2019W__________>_ i >=1m\/s20 2015 H 15I I I10 10S\/\/I III \/ ,_-______._.5 5o\u2019\u2019_ 00.0 1.0 0.0 1.0 2.0z\/W z\/W(c) S\/W=8yIW20 > =1m\/s15 -10-:\/ \u2014\u2014\u2014\u2014\u2014\u2014----- \u2014 ------\u2014\u2014-\u2014--- .- \u2014 .. S\/ \u2014 \u2014-\u2014\u2014\u2014-----\u2014----\u2014-------- \u2014 \u2014-. .54- I s.. \\4jjS\/ \/ \\ \\I\u2014 \u2014 \/ \/44 \u2014- \u2014\u2014-\u2014\u2014-\u2014- \u2014\u2014\u2014 -00.0 1.0 2.0 3.0z\/W4.0Figure 5.14: Secondary flow vectors in the cross-stream plane 4W = 10 for S\/W = 2,4, and 8 at J = 8.Chapter 5. Characteristics of a Row of Jets in a Confined CrossflowyIW20151050-0.0 1.0 2.0 3.0z,\u2019W4.0Levels cI(U\/W)7 0.8026 0.6805 0.5584 0.4363 0.3142 0.1911 0.069Figure 5.15: Distribution of the streamwise vorticity in the cross-stream plane x\/W = 10for S\/W=2,4,and8at J=72.150Levels QX\/(UDIW)SIW=47654321(a) S\/W=2y\/W201510500.0(b)yIW201510500.0Levels c\/(U\/W)7 0.6356 0.5435 0.4514 0.3593 0.2682 0.1761 0.0840.4730.4030.3320.2620.1920.1220.051S\/W=81.0z\/W1.0 2.0z\/W(C)Chapter 5. Characteristics of a Row of Jets in a Confined Crossfiow 151(b)>=2m\/sFigure 5.16:4, and 8 at J = 72.Secondary flow vectors in the cross-stream plane x\/W = 10 for S\/W 2,SIW=4(a) S\/W=2y\/W201510Hy\/W201510>=1m\/sIIII illIllIlI IllIllI\/Il,,,, Ill,1,I,I \/ \/> =2m\/sS\/W=8II liii0 -0.0 1.0z\/W500.0 1.0 2.0z\/Wt \/ \/ \/ \u2014 \u2014 \u2014 \u2014 - - -\/ \/ \/ \/______ \u2014 \u2014 \u2014 \u2014- \u2014 - -(C)y\/W201510500.0\u2014 \u2014 ,\\ \u2018.-\u2014\u2014-_.\u2018.-_.._..\u2014-. - .\u2014\u2014\u2014 \u2014 \u2014 \u20141.0 2.0 3.0z,\u2019W 4.0Chapter 5. Characteristics of a Row of Jets in a Confined Crossflow 152pattern in the vector field. For this case, at the downstream position x\/W 10, the jetis still moving primarily upward as shown earlier in Figure 5.6(a). Thus, the secondaryflow field lacks any recirculating appearance. Nevertheless, the vortex core has alreadybeen developed at that downstream position.Secondly, in this multiple jet configuration, the distribution of vorticity and the scalartracer may not be alike in certain cases. An example is seen at the close jet spacingS\/W = 2 at J = 8 shown in Figures 5.9(a) and 5.13(a), displaying the jet tracer concentration and vorticity distribution at x\/W = 10. As discussed in Chapter 2, for asingle jet, it has been well accepted that the maximum concentration of the jet fluid isin the vortex cores, with a concentration that is 30-70% higher than that on the jet axis.For row-jet injection where the jets are closely spaced, the lateral confinement forcesthe jet fluid concentration profile to stop bifurcating into two plumes. The streamwisecomponent of the vorticity, however, must be identically equal to zero in the symmetryplane. Therefore, the development of individual vortex cores in the two halves of thecross-section of the jet is guaranteed. The contrast between Figure 5.9(a) and Figure5.13(a) illustrates the above discussion.At larger jet spacings, however, the vorticity distribution is similar to the jet fluiddistribution. For S\/W 4 and 8 at J = 8, the regions of high jet fluid concentration lieclose to the maximum vorticity locations. This observation is similar to that for a singlejet made by Rathgeber and Becker (1983) and by Sykes et al. (1986) at high values ofjet-to-crossflow velocity ratio.The most intriguing observation, however, occurs at J = 72. In Figure 5.15(b), itis seen that at the spacing S\/W = 4, instead of one vortex core in the half-plane foreach jet, two cores are observed. Contrasting Figure 5.15(b) with Figure 5.11(b), thecross-sectional profile of the jet fluid distribution displays the usual \u2018kidney\u2019 shape, whilethe locations of the twin vortex cores do not correspond to the position of maximum jetChapter 5. Characteristics of a Row of Jets in a Confined Crossilow 153fluid concentration. Farther downstream, the streamwise vorticity is dominated by theupper vortex core, which will be demonstrated later. These twin vortex cores are notfound for other integral values of S\/W at the same momentum ratio.The formation and intensification of the upper vortex core can be understood byexamining the individual terms in the vorticity equation (2.13) in the form proposed bySykes et al. (1986), which is recast below:(U.V)[1=((Z.V)U+D(12)The terms representing vorticity production and diffusion are examined. The streamwisecomponents of the production and diffusion terms are F = (1Z. \\7)U1 and D = D((Z),respectively. As mentioned in Chapter 2, the production term represents vortex stretchingwhile the diffusion term is due to the turbulence transport process. Their distributionsin the plane x\/W = 10 are shown in Figure 5.17. The values are made non-dimensionalthrough multiplication by (W\/U)2. Note that the distribution of the diffusion termexhibits two negative \u2018peaks\u2019 while the production term has two positive peaks. Thesepeaks are located near the vortex cores. The upper peak of the production term representsthe shearing action of the crossfiow passing over the upper edge of the merging jet. Thisshearing action is more significant at this spacing than at larger spacings since a smallerarea is available for the crossfiow to pass between neighboring jets. The shearing causesfluid elements to be stretched and thus intensifies the upper vortex core, while fartherdownstream, the lower vortex core has its significance reduced due to the less intenseshear experienced on the lower side of the jet. This dominance of the upper vortex coreover the lower one is supported by comparing Figure 5.15(b) with Figure 5.18, whichdisplays the vorticity distribution at a later downstream position at x\/W = 20.Finally, the results in Figure 5.17 show that the magnitude of the lowest negativeChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 154Figure 5.18: Distribution of the streamwise vorticity in the cross-stream plane x\/W = 20for S\/W = 4 at J 72.(a) Vorticity Production[levelsyIW II 920 87615I54312lOb 1,\u2014 5\u2014400.0 1.0 2.0zJWPROD0.02470.01580.0068-0.0021-0.0111-0.0200-0.0290-0.0379-0.0469(b) Vorticity Diffusiony\/W2015107\u2019\u00b00.01 2.0Levels98765432DIFF0.03860.0169-0.0049-0.0267-0.0484-0.0702-0.0920-0.1137-0.1355y\/WFigure 5.17: Distributions of the production and diffusion terms in the cross-stream planex\/W=10 for S\/W=4atJ=72.S\/W =4Levels c\/(UcJW)7 0.2416 0.2065 0.1714 0.1363 0.1012 0.0661 0.0311.0 2.0z\/WChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 155vorticity diffusion is several times the highest positive vorticity production. Thus, turbulence diffusion is actively reducing the vorticity whereas the effect of production is lesssignificant. The same trend is observed for other values of jet spacing ratio S\/W at bothJ = 8 and 72. This observation illustrates the dominance of diffusion over stretching onthe magnitude of maximum vorticity for this type of flow in the far field.5.6 Concluding RemarksThe numerical simulation for row-jet injection into a crossflow can be difficult to perform.Indeed, our attempt to compute the flow field for the case of high jet-to-crossflow momentum ratio at small lateral jet spacing reveals the slowness in convergence. Many iterationsteps are needed in the pre-smoothing stage of the multigrid algorithm to achieve convergence. In addition, our results show that the use of a transient solution method has apositive stabilizing effect on the overall iterative procedure and improves the convergencerate. These observations are useful in understanding and improving the robustness ofour present numerical procedure, which in turn is valuable for the prediction of morecomplex flows such as those found in full recovery boiler simulations.In this study, a number of characteristics concerning the flow fields of multiple jets ina confined crossflow are identified. First, using the jet fluid distribution as the criterion,the jet penetration is observed to decrease as the jet spacing is reduced. This observationagrees in principle with the experimental observations of Ivanov (1959) and Kamotaniand Greber (1974). The result is also useful in understanding the Holdeman correlationformula for jet penetration which will be studied in the next chapter.The results of the vorticity dynamics in this row-jet configuration serve to quantifysome basic characteristics regarding this type of flow field. When the spacing betweena pair of jets is small, the jets merge rapidly and the diffusion of vorticity across theChapter 5. Characteristics of a Row of Jets in a Confined Crossflow 156jets weakens the vorticity of each individual jet. It is observed that an increase in lateralshear on the jet due to a reduction in jet spacing does not increase the maximum vorticitysignificantly. In fact, at the high value of J = 72, diffusion dominates production at closejet spacing and the maximum value of vorticity is reduced.The results of the jet fluid distribution in the cross-stream planes and the corresponding streamwise vorticity distribution reveal that the usual understanding of the dynamicsfor a single jet in a crossflow needs to be modified substantially when applied to multiplejets. This is demonstrated in the observation that the locations of the maximum jet fluidconcentration do not always correspond to the centers of streamwise vortices. Indeed,at J 72 and S\/W = 4, multiple cores of the streamwise vorticity are numericallysimulated.Chapter 6Multiple Jet Interactions with a CrossifowIn the previous chapter, the characteristics of a row of square jets in a confined crossflowwere investigated. These characteristics are useful in providing a basis for the understanding and interpretation of the phenomena observed in situations where multiple jetsare used, such as changes in jet penetration when an array of jets are under variousgeometric and operating conditions.For applications to air systems in kraft recovery boilers, it is essential to study thedynamics of jets issuing from rectangular orifices. The aim of this chapter is to studythe penetration and mixing of multiple rectangular jets in a confined crossfiow.6.1 Jet Penetration Study: Effects of Parametric VariationsAs discussed in the last chapter, the major difficulty in the study of a row of jets interacting with a crossflow is the presence of many geometric and operational parametersthat are associated with the description of the flow problem. Fortunately, the resultsof Walker and Kors (1973) indicate conditions on geometric and operational parametersthat influence jet penetration. These conditions were discussed in Chapter 2 and theresults of Walker and Kors were correlated by Holdeman and his co-workers. The extension of Holdeman\u2019s relationship to rectangular jets of different aspect ratios is the focusof the following sections.157Chapter 6. Multiple Jet Interactions with a Crossflow 1586.1.1 A Non-dimensional RelationshipThe investigations by Holdeman and Walker (1977) and Holdeman et al. (1984) identified the jet-to-mainstream momentum flux ratio J as the most important parameter ingoverning jet penetration. Also, their results were grouped to define \u2018desirable\u2019 mixingin which the jet trajectory was bent by the crossflow so that it travelled down the centerof the confining chamber. This strategy seems to be reasonable for applications relatedto combustion, where it is desirable to have air jets penetrating into the interior of thechamber without significant over-penetration or under-penetration. In any specific application of course the desirable mixing is problem dependent, but similar non-dimensionalrelationships are likely to exist in most cases.Upon examining their results, Holdeman and his co-workers identified that when thegeometric and operational parameters are related through the formulaS C (6.1)then the jet penetration will be desirable, in the sense that it becomes central in theconfining chamber. In the above expression, which was also displayed as Eq.(2.24), Sis the center-line spacing of the jets in a long array, H is the cross dimension of theconfining chamber (see Figure 6.1), and C is a dimensionless parameter whose value forround cold jets entering a hot crossflow was found to be 2.5. The above expression impliesthat more widely spaced jets (S\/H large) require a lower momentum to penetrate to thecenter-line of the chamber, or conversely that closely spaced jets (S\/H small) requiremore momentum to penetrate to the chamber center-line. It can also be viewed as acorrelation of jet penetration: for . = 2.5, the jets penetrate to the center ofthe chamber; for . > 2.5, the jets penetrate more deeply than the center; for< 2.5, the jets do not penetrate as far as the center-line. In the present study, weconsider penetration from one side to the chamber center-line, following Holdeman et al.Chapter 6. Multiple Jet Interactions with a Crossfiow 159(1984) in considering this a \u2018good\u2019 mixing arrangement.The derivation of Eq.(6.1) by Holdeman and his co-workers was based entirely onexperimental observations. However, it can be put on a firmer basis by showing that italso reflects the results of Kamotani and Greber (1974). First, the latter experimentalresults reveal that at a certain jet spacing that is not too small (S\/W > 3), increasing theconfinement dimension H does not affect the jet trajectory too much. This is reflectedin Eq.(6.1) through the recognition that if S is kept constant, then to maintain the sameextent of penetration when H is increased, \/J needs to be increased proportionally. Inother words, the jet velocity V6 needs to be increased in proportional to any increasein H to maintain suitable penetration. Another observation by Kamotani and Greberconcerns the decrease in jet penetration as spacing S is reduced for moderately spacedjets (3 < S\/W < 10) examined in the last chapter. If H is held fixed in Eq.(6.1), thenfor the same extent of penetration, 1\/i\/i needs to be changed in proportion to S. Inother words, a reduction of S requires an increase of V116t to maintain the penetration forfixed H. Hence, once again there is consistency between the physical trend observed andthe empirical correlation.The results by Kamotani and Greber also reveal the limitation of the applicability ofEq.(6.1). The correlation is useful only when the spacing S divided by jet width W isneither too small (very closely spaced jets) nor too large (essentially isolated jets). Closelyspaced jets coalesce and the resulting lack of three dimensional entrainment increases thepenetration so that Eq.(6.1) is no longer appropriate. The validity of the correlation fora range of jet sizes and shapes is investigated here using numerical simulations. Thepossible implications of the correlation for recovery furnaces will then be considered.Chapter 6. Multiple Jet Interactions with a Crossflow 160I II Vii -i\u2014r\u2019i ri JETLU zi4ii tI.I [11 \/I I I LII [1 iuuiLiuI IwI I Httttt tttttuc ucFigure 6.1: A schematic description of the domain for one-side injection.6.1.2 A Row of Rectangular Jets in a CrossflowThe configuration for one-sided injection is illustrated in Figure 6.1. Again, the flow fieldis obtained by solving the time-averaged form of the Navier-Stokes equations coupledwith the k \u2014 turbulence model. Boundary conditions are handled the same way as inthe last chapter. Symmetry conditions are applied on lateral sides of a jet to minimize thesize of the domain. Hence, in the study, only half of a jet is simulated. Other boundaryconditions applied include imposed values for the velocity and scalar at the flow inletand jet entrance, the zero-gradient exit condition, and the wall function formulation atno-slip walls. A unit of a passive tracer is injected with the jet to indicate the dispersionof the jet fluid. Multigrid calculations are performed for square jets and rectangularjets of various aspect ratios. With proper attention to the placement of the exit planeand the choice of the number of iterations carried out within the multigrid procedure,convergence is achieved readily for the problems studied in this chapter.To extend Holdeman\u2019s well-mixing criterion to the case of rectangular jets of differentChapter 6. Multiple Jet Interactions with a Crossflow 161aspect ratios, numerous simulations are performed with different values of S\/H and Jto examine the appropriate choice of C for each case. The control parameter is the jet-to-mainstream mass flow ratio M, which for constant density flow is evaluated from theformulaM_._W.\/i 62HS U HSThis quantity is significant in combustion applications, where its value is fixed by thestoichiornetric requirement of air. In our simulations, at a given value of J, M is heldfixed when the extent of jet penetration is examined in various cases.6.1.3 Results for Uniform CrossflowSquare JetsTurbulent flow simulations are carried out to emulate the experiments of Holdeman andWalker (1977). The results should not be affected significantly by changes in actual sizeor Reynolds number, provided other dimensionless ratios are kept constant. Thus, thenon-dimensional correlations derived are useful in describing flows in larger devices.H = 0.1016 rn (4 inches)Uc = 15 rn\/sJ M S\/H W\/H6 0.15 0.25 0.12500.50 0.17680.75 0.21650.31 0.10 0.03940.20 0.05570.30 0.0682Table 6.1: Parameters chosen for simulations of square jets.Chapter 6. Multiple Jet Interactions with a Crossilow 162The values of the geometric and operational parameters considered are listed in Table6.1. Two values of J are chosen to represent the cases where the jets have momentumthat is low or moderate. For each value of J, three values of S\/H are chosen to studythe effects of varying the jet spacing. A value of M is then chosen and the jet width Wcan be determined from Eq.(6.2). A larger value of M is chosen for J = 25 so that thecorresponding value of W will not be too small for each choice of S\/H.Following Holdeman and Walker, the jet fluid distribution in the symmetry plane z =o is examined and the results are shown in Figures 6.2-6.3. By examining these profiles,and applying the requirement that jet penetration be such as to bring the concentrationmaxima to the approximate center of the chamber, we deduce that for J 6, thejet penetration is optimal when S\/H is between 0.5 and 0.75, and for J = 25, it isS\/H = 0.3. Based on these observations, a value of C 1.5 correlates the results foroptimal jet penetration for square jets in constant density flow. This value is about onehalf that recommended by Holdeman and his co-workers. Our lower value for C may bedue to the use of isothermal flow in the present study.Chapter 6. Multiple Jet Interactions with a CrossflowLevels765432TracerConc.0.950.800.650.500.350.200.05163Figure 6.2:jets at J =Distribution of jet tracer concentration in the center-plane (z = 0) for square6 and M = 0.15.(a) S\/H =0.1 (b) S\/H =0.2 (c) S\/H =0.3Levels765432TracerConc.0.950.800.650.500.350.200.05(a) S\/H = 0.25 (b) S\/H = 0.50 (c) S\/H = 0.75x\/H0.5 1y\/H0.0 0.5 1.0y\/H0.0 0.5 1.0yIHx\/H1.51.00.5\u2022 ,.0y\/HFigure 6.3: Distribution of jet tracer concentration in the center-plane (z 0) for squarejets at J=25andM=0.31.y\/H y\/HChapter 6. Multiple Jet Interactions with a CrossBow 164Rectangular Jets of Aspect Ratio 3 and 6Similar calculations are carried out for orifices having aspect ratio AR = L\/W of 3 and6. Values of J = 6 and M = 0.15 are used as before. These cases are listed in Table 6.2.H = 0.1016 m (4 inches)Uc = 15 m\/sJ = 6, M = 0.15AR S\/H W\/H3 0.25 0.07200.50 0.10210.75 0.12506 0.25 0.05100.50 0.07220.75 0.0884Table 6.2: Parameters chosen for simulations of rectangular jets with AR = 3 and 6.Again the jet tracer concentration contours are plotted in the center-plane z = 0.The results are shown in Figures 6.4-6.5 for AR = 3 and 6, respectively. In addition,the cross-sectional profiles of the jet tracer concentration for jets at S\/H = 0.5 but withdifferent values of AR are shown in Figures 6.6-6.7 for comparison. The results show thedifferences in jet characteristics caused by changes in the shape of the jet orifice.The following results are revealed. First, higher aspect ratio jets penetrate deeper intothe crossflow when conditions such as the orifice area, jet spacing, and jet momentum areequal. Secondly, the regions of high jet fluid concentration in the jet\u2019s cross-section splitabout the center-plane z = 0 for jets with AR 3 and 6. In these cases, the locations ofhigh jet fluid concentration lie below the jet concentration trajectory in the center-plane.These observations influence the choice of optimal penetration depth.For jets with Al? = 3, optimal penetration occurs at a spacing close to S\/H = 0.5,which is less than that for square jets. For jets with AR = 6, the optimal spacing shouldChapter 6. Multiple Jet Interactions with a CrossflowLevels765432TracerConc.0.950.800.650.500.350.200.05165Figure 6.4: Distribution of jet tracer concentration in the center-plane (z = 0) for AR = 3jets at J=6andM=0.15.(a) S\/H = 0.25 (b) S\/H =0.50 (c) S\/H =0.75Levels765432TracerConc.0.950.800.650.500.350.200.05(a) S\/H = 0.25 (b) S\/H = 0.50 (c) S\/H = 0.75x\/H0.0 0.5 1.0 0.0 0.5 1.0y\/H y\/H0.0 0.5 1.0y\/HxlH3.02.0o.cb.i.oy\/HFigure 6.5: Distribution of jet tracer concentration in the center-plane (z = 0) for AR = 6jets at J=6andM=0.15.Chapter 6. Multiple Jet Interactions with a Crossflow 166(a) AR=1 (b) AR=3 (c) AR=6TracerLevels Conc.7 0.516 0.445 0.364 0.293 0.222 0.141 0.07Figure 6.6: Jet tracer concentration in the cross-sectional plane x\/H = 0.5 at the spacingS\/H = 0.5.(C) AR=60.25TracerLevels Conc.7 0.406 0.345 0.284 0.223 0.172 0.111 0.05Figure 6.7: Jet tracer concentration in the cross-sectional plane x\/H = 1.0 at the spacingS\/H = 0.5.(a) AR=1 (b) AR=30.25zIHChapter 6. Multiple Jet Interactions with a Crossflow 167be slightly greater than S\/H = 0.25. In making these choices for rectangular jets, thejet trajectory in the center-plane is preferred to be slightly beyond the middle of thechamber, so that the jet fluid will not be too close to the injection wall. Thus, thefollowing values for C are estimated for proper jet penetration: C 1.0 for A? = 3 andC 0.65 for AR = 6. These estimates are listed in Table 6.3. For jets whose aspect ratiois between 1 and 6, the optimal value of C may be obtained by interpolation.AR Optimal CF 1.53 1.06 0.65Table 6.3: Optimal values of C for orifices having different aspect ratios.6.1.4 Experimental Investigation on Jet PenetrationAn experimental investigation is carried out to verify the difference in jet penetrationbetween wide jets and slender jets that is observed in the numerical simulation. Theexperiment is performed in the 1 : 30 scaled Kamloops boiler model shown in Figure 6.8in which the flow of water is used to simulate the fluid motion within a recovery boilerunit. There are three levels of jets in the model, and in the experiment the lowest primarylevel jets are all turned on to let in water to generate a cross-flow for the jets located athigher elevations. The total water flow rate entering these primary orifices is 7.7 x iOm3\/s (122 US gal\/mm). A flow visualization experiment is carried out to observe the jetpenetration. The details of the experimental set-up are provided in Appendix A.An important reason for performing the experiment in the Kamloops model ratherthan in a wind tunnel or water tunnel is to observe the behavior of jet penetration whenthe conditions of the cross-stream deviate from the ideal. In the numerical simulation,Chapter 6. Multiple Jet Interactions with a Crossflow 168jet penetration is examined when the incoming conditions for the crossflow are perfectlyuniform. However, in many practical situations as in a recovery boiler, the cross-streamcan exhibit great non-uniformity. Thus, the objectives of the experiment are to comparejet penetration when the shape of the jet orifice changes and to observe the effect ofcross-stream non-uniformity on jet penetration.In the experiment, the jets at the secondary level are turned off and only one side ofthe tertiary injection system is used. The two templates shown in Figure 6.9 act as thejet inlet holes. The area of each orifice opening is 4 cm2 and the rectangular orifices haveaspect ratio AR = 4. The flow rate for each jet is set constant at 2.5 x i0 m3\/s (4 USgal\/mm), which corresponds to an average jet velocity of 0.63 rn\/s. The focus of the flowvisualization study is on the middle jet, where the cross-stream is moving upward mostprominently in the vicinity of the jet entrance. A sheet of laser light is used to shine alongthe center-plane cutting longitudinally across the jet orifice. An S-VHS camera is used torecord approximately five minutes of the flow field in each case of square jet injection andrectangular jet injection. Typical results are shown in Figure 6.10, which are photographstaken from a video display. The exposure time is 0.125 seconds to allow for the showingof streak lines on the photographs. These streak lines indicate the movement of smallpolystyrene particles carried by the flow and the lines are good indicators of the flowpattern appearing in the boiler model.The flow structures displayed in the photographs in Figure 6.10 show that the penetration of rectangular jets is deeper than square jets. This observation is consistentwith our numerical predictions. During the time of filming, it is observed that there aremoments of flow unsteadiness where there are changes to the crossflow pattern. Suchchanges involve fluid flowing laterally instead of upwards, and this can affect jet penetration by allowing it to be deeper or not as deep. Nevertheless, it is generally observedthat, in the situation considered, rectangular jets penetrate deeper than square jets.Chapter 6. Multiple Jet Interaction with a Crossflow 169Figure 6.8: Photographs of the Kamloops water model showing the overall model andthe orifices for tertiary jets.IICDC .CD CD 0 CD CD Co Co CD CD CD CD CD r.n 3 CD 0 CD Co CD n 2ci 3 CD U) 0 U) 3 3 CD CDI.I. i171Figure 6.10: Photographs displaying jet penetration. Top: crossflow. Middle: square jetinjection. Bottom: rectangular jet injection. Jets enter from the right side only.Chapter 6. Multiple Jet Interaction with a CrossflowChapter 6. Multiple Jet Interactions with a Crossflow 1726.2 Mixing and Vorticity CharacteristicsThe previous discussion on optimal jet penetration is based on visual inspection of thejet trajectory in the center-plane. The goal is to have the jet penetrating to the center ofthe chamber. A quantitative study is now carried out to examine the degree of mixing invarious cases to verify whether jets penetrating to the center of a chamber is indeed anoptimal strategy. In addition, the characteristics of the streamwise vorticity are studiedfor jets that have different aspect ratios and at different jet spacings. An objective ofthis study is to investigate the relationship between the extent of jet mixing and vorticityreduction for various cases.6.2.1 Quantitative Study of Jet MixingThe methodology employed here has application to combustion in furnaces. Specifically,attention is focussed on the volume of fluid per unit time passing through a cross-sectionalplane having concentration of a scalar tracer in a certain range. The implication forcombustion is that it is important to know the concentration of combustibles withincertain critical ranges where burning occurs.The above objective is met by considering the probability density function f() derived in Appendix B. At a cross-sectional plane with elemental area dS located at stream-wise coordinate x, this density function is denoted as=1 J (U . dS) having concentration in range ( )Q1+Q2where =\u2014with av being the average or fully mixed concentration value, andtie being some small increment of.In addition, Qi and Q2 are the volume flow ratesof the cross-stream and the jet, respectively. It is shown in Appendix B that the meanof f(C) is zero and that for perfectly mixed fluid, the variance of f(C) is also zero.Chapter 6. Multiple Jet Interactions with a Crossflow 173At the same cross-sectional plane, the variance o of f() can be computed as= (6.4)It is defined in Appendix B that the quantity, S, is a normalized measure for mixing. Itis defined as= (6.5)Oowhere a0 is the maximum variance computed before any mixing has occurred. The valuesof S vary from 1 (no mixing) to 0 (fully mixed), and is a good quantitative measure ofjet-crossflow mixing.The values of f() and subsequently S are computed for the cases with J = 6 listedin Tables 6.1 and 6.2. The evaluation is carried out in cross-stream planes in the range0.5 x\/H 3.0. The graphs of f) for A?= 6 and S\/H = 0.25 at x\/H = 1.0, 2.0, and3.0 are shown in Figure 6.11 to demonstrate the change in the shape of the profile as thedownstream position is varied. It is seen that the spread of the distribution reduces as thedownstream position increases, indicating that more mixing has taken place, resulting ina more uniform distribution of the tracer flux.The graphs for the variation of S are shown in Figure 6.12. Comparison is first carriedout for jets that have the same aspect ratio but at different spacings. For square jets(A? = 1) at the spacings S\/H = 0.5 and 0.75, the values of 6 experience sharper decayspast x\/H = 1 than at S\/H = 0.25. This sharper reduction in S occurring at S\/H = 0.5and 0.75 is consistent with our earlier objective to have the jets penetrate into the middleof the chamber, as discussed in section 6.1.3. Figure 6.2 illustrates that the jet penetratesinto the middle of the chamber at S\/H = 0.5 and slightly over-penetrates at S\/H = 0.75.Thus, the results have quantified the observation by Walker and Kors (1973) that overpenetration (at S\/H = 0.75) is preferred to under-penetration (at S\/H = 0.25) for thepurpose of jet-crossflow mixing. Similar observations are also apparent for rectangularChapter 6. Multiple Jet Interactions with a Crossilow(a) x\/H=1.O50f ()403020100-0.155040302010174-0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25=- avxIH=2.O(b)f ()(c)I..,.-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.=-xIH = 3.0=- \u2018)avFigure 6.11: Profiles of f(cf) for A? = 6 and S\/H = 0.25 at various downstream locations.Chapter 6. Multiple Jet Interactions with a Crossilow 175jets. For jets with AR = 3, the fact that S assumes lower values at S\/H = 0.5 than atthe other two spacings (S\/H = 0.25 and 0.75) is again consistent with the penetrationprofiles in the center-plane shown in Figure 6.4. The figure shows that the jet propagatesdown the chamber too close to the injection wall at S\/H = 0.25 and too close to theopposing wall at S\/H = 0.75. Finally, for jets with AR = 6, the values of 5 are thehighest at S\/H = 0.75. These high values correspond to the results shown in Figure 6.5that the jet over-penetrates significantly at this spacing. The near-impingement of thejet with the opposing wall reduces the amount of jet volume available for mixing withthe mainstream. To conclude, the above observations confirm that the strategy to havethe jet propagates down the middle of a chamber is indeed desirable for good mixing.Another observation from Figure 6.12 is that at the close jet spacings of S\/H = 0.25and 0.5, better mixing is achieved with higher aspect ratio jets. This is related to theobservation of the cross-sectional shapes of the jets displayed in Figures 6.6-6.7, whichshow that jets from slender orifices have more elongated cross-sectional shapes. Thesejets propagate downstream for a longer time before they merge with the neighboring jets.Since the merging of jets reduces the amount of mainstream volume that will be affectedby each jet, the postponement in merging results in better mixing by high aspect ratiojets at close lateral spacings. On the other hand, when the spacing is large (S\/H = 0.75),jets with AR = 1 and 3 both have ample amount of time to develop independently beforeany merging will occur. Consequently, the degree of mixing is similar for the two kinds ofjets, as shown in Figure 6.12(c). Jets with AR = 6, however, mix less effectively at largedownstream locations because of the significant over-penetration as discussed in the lastparagraph.Figure 6.12: Variation of 8a with x\/H for jet-crossflow mixing for various cases of 4d? andS\/H.176D AR=1.. AR=3p AR=60.5 1.0 1.5 2.0 2.5Downstream Positions x\/HS\/H =0.503.0 3.5Chapter 6. Multiple Jet Interactions with a Crossflow(a) S\/H =0.250.60 \u20140.50cci>ci)0.40VCsV0.30Cl)0.200.0(b)C0cci>ci)VcciVCcciCl)(c)xC0_______cci>ci)-DCeCCeU)0.200.0 0.5 1.0 1.5 2.0 2.5 3.0Downstream Positions x\/HS\/H=0.753.50.600.500.400.300.200.0E AR=1.. AR=3p AR=60.5 1.0 1.5 2.0 2.5Downstream Positions x\/H3.0 3.5Chapter 6. Multiple Jet Interactions with a Crossflow 1776.2.2 Vorticity ConsiderationsAs shown in the last chapter, the counter-rotating vortex cores of each jet dominate thefar field jet structure. These vortices provide a mechanism for mixing between the cross-stream fluid and the deflected jet. As in the previous study, the streamwise component ofthe vorticity, defined by Eq.(5.3), is computed for the cases of J = 6 listed in Tables 6.1and 6.2 at various downstream positions x\/H. Far enough downstream, this streamwisecomponent of vorticity represents almost all of the vorticity the jet possesses.The variations of the maximum values of the streamwise component of vorticity areplotted in Figure 6.13 for the three jet spacings. The values of these vorticity maximaare affected by vortex stretching and turbulence diffusion. To evaluate the relative importance of these two effects, the vorticity production and diffusion terms in the vorticityequation are examined following the practice in section 5.5.2. In almost every case thelocation of maximum vorticity also corresponds to the locations of lowest negative vorticity diffusion and highest positive vorticity production. These results are similar to thosestated in section 5.5.2. Typical results are displayed in Figure 6.14. The figure showsthe distributions of the streamwise vorticity, its production, and its diffusion for the caseA? = 3 and S\/H = 0.5 at x\/H = 1. The values in Figure 6.14 are non-dimensionalizedby V,et\/H for the vorticity and V,2et\/H for the vorticity production and diffusion.The results shown in Figure 6.14 reveal the general trend that the vorticity coreis strengthened by vorticity stretching while weakened by turbulence diffusion, a phenomenon also observed for the cases studied in the previous chapter. Moreover, themagnitude of the lowest negative diffusion is generally several times greater than thehighest positive production. This is an indication that diffusion due to turbulence is themajor controlling factor for the reduction in the vorticity values shown in Figure 6.13.Returning to Figure 6.13, a comparison of the graphs there with those in FigureChapter 6. Multiple Jet Interactions with a Crossflow(a) S\/H =0.25178IC)C>EDExCu1.0 1.5 2.0 2.5Downstream Positions x\/H(b) S\/H =0.508.0 \u20147.06.05.04.03.02.01.00.00.0D AR=1L. AR=3a AR=60.5 1.0 1.5 2.0 2.5 3.0 3.5Downstream Positions x\/HS\/H=0.75>C)C>EDExCu(c)8.0I- 7.0________>\u2014. 6.05.04.03.00.5 1.0 1.5 2.0 2.5 3.0 3.5Downstream Positions x\/HFigure 6.13: Variation of the maximum streamwise vorticity with x\/H for jets fromorifices that have different aspect ratios and at various lateral spacings.D AR=1L AFI=30 AF1=6Chapter 6. Multiple Jet Interactions with a Crossflow 179PROD DIFF3.02 0.51 0.232.58 0.40 -0.032.15 0.29 -0.281.72 0.18 -0.541.29 0.07 -0.800.86 -0.04 -1.050.43 -0.15 -1.31Figure 6.14: Distributions of streamwise vorticity, production, and diffusion for the caseAR = 3 and S\/H = 0.5 in the cross-stream plane x\/H = 1.0.(a) (b) Production (c) Q Diffusion0.000.00z\/H 0.25Levels7654320.000.00zIH 0.25Levels7654320.000.00zIH 0.25Levels765432Chapter 6. Multiple Jet Interactions with a Crossflow 1806.12 reveals the following observation: at the same jet spacing, a larger reduction in thevorticity value corresponds to more mixing has taken place. This is particularly evidentfor jets with A? = 6 at S\/H = 0.25, where the reduction in vorticity is most rapid (fromFigure 6.13(a)) and the corresponding value of S is least (from Figure 6.12(a)) so thatthe fluid is most well mixed. At S\/H = 0.5, the rates of vorticity reduction are similar forthe three types of jets, and correspondingly the reduction rates of 5 are approximatelythe same. Similarities also hold for the case S\/H = 0.75. This observation lends supportto the role of the streamwise vorticity on jet mixing in the far field.It is also interesting to note that at the same lateral spacing, jets from high aspectratio orifices possess higher absolute values of streamwise vorticity than those from squareorifices. This is due to the fact that slender jets have larger lateral surface areas to beacted upon by the crossflow than blunt jets of the same area. The shearing action of thecrossflow stretches the fluid elements within the jet and consequently a higher value ofthe streamwise vorticity results.6.3 Consequences of Holdeman\u2019s RelationshipTo further check the consistency in the extent of jet penetration for which Eq.(6.1) isexpected to apply, we observe that at a given value of M, a rearrangement of Eqs.(6.1)and (6.2) leads to the following relationships for J and S\/W:CMH2 66)RW2 (.)(6.7)WVMThat is, for a given choice of M, A?, and C, the momentum flux ratio J is proportionalto H2\/W and the spacing ratio S\/W has a unique value. Therefore, it is possible tochoose different values for the jet width W such that desirable jet penetration and mixingChapter 6. Multiple Jet Interactions with a Crossilow 181are achieved in each case. The values of the momentum ratio J and the jet spacing Sare subjected to the choice of W. This observation is used to check the consistency ofHoldeman\u2019s correlation by examining the jet penetration due to different choices of W.The following study is carried out with parametric values that are typical for tertiaryjets in a recovery boiler.6.4 Implications for Tertiary Level Jets in a Recovery BoilerIn an idealization of the tertiary air system of a recovery boiler, we consider an arrayof jets coming from orifices of aspect ratio 4, which is common in recovery boilers. Themass flow input is chosen to be 20% of the crossflow mass, so that M = 0.20.Now, Eq.(6.6) implies that small jets, for which W = 0.10 m (say) require muchhigher velocity to achieve good penetration than do larger jets of W = 0.20 m. Choosingtypical recovery boiler values of Uc = 3 m\/s and H = 10 m, corresponding jet velocityvalues may be calculated easily if C 0.8 is chosen as a value appropriate for jets ofaspect ratio 4. This value of C is obtained by interpolating the values listed in Table 6.3.Since the attention now is on the tertiary air system, the values of the jet widthchosen are W = 0.10, 0.15 and 0.20 m. The corresponding values of J and hence Vet areobtained from Eq.(6.6) and listed in Table 6.4. The penetration in the center-plane forthe three cases are shown in Figure 6.15. It is observed that the penetration for each caseis slightly beyond the center, and this is caused by the high momenta of the jets underconsideration. At such high momenta, the jets merge more rapidly and this causes thetrajectory to be lifted up, as mentioned in the last chapter. The remarkable observationis that the penetration is nearly the same for the three cases. This verifies the validityof using Eq.(6.1) to predict jet penetration.It is further remarked that the trends shown in Eqs.(6.6) and (6.7) provide insightChapter 6. Multiple Jet Interactions with a Crossflow 182(a) W = 0.10 m (b) W = 0.15 m (c) W = 0.20 m6xIH__\\ Tracer5 \\ Levels Conc.7 0.40431 0.05210\u2022y\/H 16x\/H6xIHy\/H 1 \u00b0 y\/H 1Figure 6.15: Jet tracer concentration in the center-plane for uniform crossflow.Chapter 6. Multiple Jet Interactions with a Crossilow 183W (m) J Yt (m\/s)0.10 400 600.15 178 400.20 100 30Table 6.4: Values of W, J and Vjet employed in the simulation of tertiary-type jets.that is useful for the design of air injection systems. The relation that J varies as (H\/W)2for a given shape of jet orifice and required mass input, is of interest, suggesting thatlarge, low speed jets can penetrate as deeply and have similar mixing effects as small,high speed jets. The fact that S\/W is a constant for given AR and M as indicated inEq.(6.7) is also of interest. For the values chosen here (M = 0.20, AR = 4, C = 0.8),S\/W becomes equal to 4, so that small jets must be more closely spaced (i.e. S is small)than large jets (for which S is larger), the ratio of S\/W remaining constant.Considerations for Opposing JetsThe results given above are a great simplification of actual tertiary jets; isothermal jetsfrom one side of a chamber (of size H) have been correlated, and jet penetration to thecenter of the chamber has been identified. For actual furnace tertiary jets which areinterlaced or opposed, other effects may appear, such as oscillation or other forms ofinstability. For instance, it has been observed by Quick et al. (1991) that when planejets are discharged strongly at each other, bifurcation may occur and the flow field mayexhibit steady states that are asymmetric, or may not reach steady state at all. To avoidthis instability, the penetration of the jets must be reduced, as shown in Figure 6.16.Then Eq.(6.1) is useful in finding conditions that establish the required trajectories ifH\/2 is used as the confinement dimension.Chapter 6. Multiple Jet Interactions with a Crossflow 184I tt IFigure 6.16: Schematic representation of opposing jets in a crossflow with penetrationreduced to avoid collision between jets.6.5 Non-uniform CrossflowIn practice, the up-flow approaching actual tertiary jets is not likely to be uniform. Thus,the effects due to crossflow non-uniformity need to be examined. To accomplish this, apeaked crossflow profile with a flat top, shown in Figure 6.17, is chosen for this study.The profile is mathematically described by0 8A+[2(_)]where Uc = Uc, when = so that U0 is the peak velocity and Uc(y\/H) is symmetricalabout = . The values of U0 and A are chosen so that the integrated mean velocityUmean for the profile has the value of 3 m\/s, and that the minimum velocity, found aty\/II = 0 or 1 at the inlet, is 10% of the peak velocity tJ,. With these two criteria, thepeak velocity Uo S 1.65 X Umean, or 4.95 m\/s, and A = . This non-uniform velocityprofile has a value of standard deviation of 1.68 rn\/s.II HIJETJETuc171=Chapter 6. Multiple Jet Interactions with a Crossflow 1851.75ucUc(mean) 1.501.251.000.750.500.250.000.00 0.25 1.00yIHFigure 6.17: Velocity profile of the peaked crossflow.The calculation is performed with the same choices of jet parameters and chamberwidth as in the previous section. The resulting jet fluid concentration profiles in thecenter-plane, obtained with this non-uniform crossflow, are shown in Figure 6.18. Itis observed that the jet fluid distribution, and hence the jet penetration, for the threecases are similar to those obtained when the crossflow was uniform. This provides theassurance that the non-uniformity of the type examined does not affect the applicabilityof the relationships Eqs.(6.1), (6.6) and (6.7) describing the penetration when geometricand operational parameters are varied. This observation will have to be validated for amore complex flow generated by the interaction among primary, secondary, and tertiaryjets in a boiler.1.650.1650.50 0.75Chapter 6. Multiple Jet Interactions with a Crossflow 1866xIH(a) W=O.lOm (b) W=O.15m (c) W=O.20m6xIH5431x\/H5\u20194321021Levels765432TracerConc.0.400.340.280.220.170.110.051Oiy\/H 1 0 y\/H 1 0 y\/H 1Figure 6.18: Jet tracer concentration in the center-plane for non-uniform crossflow.Chapter 6. Multiple Jet Interactions with a Crossflow 187INTERLACED JETSSYMMETRY PLANE-SYMMETRY PLANEFigure 6.19: Plan view of the interlaced jet injection scheme.6.6 Interlaced Jets on Both Sides of the ChamberThe experimental results of Holdeman et al. (1984) suggest that interlaced jets providevery effective mixing. The interlaced arrangement is introduced by placing every secondjet on the opposite wall, so that in our previous simulations where the jet spacing wasS\/W = 4, the interlaced arrangement would lead to S\/W = 8 on either wall. Figure6.19 shows the arrangement.Simulations are carried out with the same three jet sizes and velocities listed in Table 6.4. The jets act on the same non-uniform crossflow profile given in Eq.(6.8). Thecross-sectional profiles of the jet fluid distribution at the downstream location x\/H = 1.0are shown in Figure 6.20. The profiles indicate that, in each case, the maximum concentration of the jet fluid is located at the center of the chamber. This observation verifiesthat the application of Eq.(6.1) results in adequate jet penetration even for interlacedarrangements.Chapter 6. Multiple Jet Interactions with a Crossflow 188\u201456\u2014\u20147o 00.0 0.2 0.4 0.6z(m)\u2014--I,.., I0.0 0.2 0.4 0.6 0.8z(m)Figure 6.20: Jet tracer concentration in the cross-stream plane x\/H = 1.0 due to mixingwith interlaced jets.xIHFigure 6.21: Standard deviation of the streamwise velocity component at various downstream positions for the three sizes of interlaced jets.(b) W=0.15m (c) W=0.20m(a) W=0.lOm10y (m)8 \u2014.3\u2014-6 \u20146\u2014-\u2014\u2014\u2014\u20144 .\u20146\u2014-.3---2 3-\u201400.0 0.2 0.4z (m)10y (m)864210y (m)8642Levels765432TracerConc.0.250.210.170.130.090.050.010>ci)U.Ict3Cc5Cl)ctC0Cl)C1)E-oC0z2.01.5 -1.00.50.0 \u20140.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0W=O.lOm1.1Chapter 6. Multiple Jet Interactions with a Crossfiow 189Although the jet fluid distribution is similar, the velocity distribution shows differences for the different choices of orifice width W. Thus, the uniformity in the velocitydistribution is investigated. The standard deviation of the strearnwise velocity component from its average value is computed for each of the three cases. A low value of thestandard deviation indicates a more uniform flow profile than a high value. For applications to recovery boilers, it is often desirable to have a velocity distribution as uniformas possible in the main flow downstream of the tertiary jets.The results are shown in Figure 6.21 for the variation of the standard deviationof the strearnwise velocity component with the downstream position x\/H for the threetypes of jets. The standard deviation is non-dimensionalized by the value of the standarddeviation of the original non-uniform profile (1.68 m\/s). The largest jets that are operatedat the lowest momentum provide the most uniform flow farther downstream \u2014 smalleststandard deviation at large x\/H. Hence, the following conclusion is made: for interlacedjets interacting with the type of non-uniform crossflow prescribed, large jets at a lowmomentum satisfy the mass flow requirement and create less flow non-uniformity fardownstream.In passing, note that with this interlaced jet injection scheme, the highest momentumratio case (J = 400, Table 6.4) corresponds to the following parametric values: H = 10m, W = 0.10 m, AR = 4, S\/W = 8, and = 60 rn\/s. For the 1000 tons-per-day boilerowned by the Weyerhaeuser company in Kamloops, B.C., the conditions for tertiary jetsare as follows: H = 11 m, W = 0.15 m, AR = 4.3, S\/W = 9, and V = 69 rn\/s.The Kamloops boiler\u2019s tertiary jets are therefore operating under conditions that aresimilar to those studied here. Hence, the jets should penetrate to the middle of the boileradequately. It is worthwhile to note that according to our results, similar penetrationcould be achieved with bigger jets operating at a lower speed. The use of lower speed jetsis more power efficient since at the same mass flow requirement, they need less power toChapter 6. Multiple Jet Interactions with a Crossfiow 190operate than higher speed jets.6.7 Concluding RemarksOur results indicate that Holdeman\u2019s correlation Eq.(6.1) is useful in correlating conditions upon which the jet trajectory in the center-plane is similar. A certain choice of thevalue of C leads to conditions that will give rise to effective transport of jet fluid into thechamber. The value is chosen based on the desire to have the jets to penetrate into themiddle of the chamber without significant over-penetration or under-penetration. Theoptimality of this strategy is verified by our quantitative study on jet mixing. Suitablechoice for the value of C depends on the aspect ratio of the jet orifice, and decreases asaspect ratio increases. From the data listed in Table 6.3, the following empirical formularelating C and AR is obtained:C = 1.83 \u2014 0.36 AR + 0.027 A?2 (6.9)The formula is valid for values of A? in the range of 1 and 6.A value of C = \/1. that is a factor of two larger than the optimal value predictedby Eq.(6.9) would mean over-penetration, aud vice versa. Also, for two-sided interlacedjets, the effective orifice spacing would be half the spacing between neighboring jets onthe same wall. Keeping these items in mind, Eq.(6.1) can be used as a quick referenceto estimate how jets will penetrate into a confined crossflow. To be cautious, it shouldbe remembered that the formula only applies in situations where lateral jet interactionis significant and where the jet spacing is not too small. Also, for cold jets entering ahot crossflow, the Thring-Newby scaling criterion, discussed in Perchanok et al. (1989),should be applied first to determine the effective size of the orifice opening to accountfor the effects of jet expansion. After this is done, the above strategy can be used toestimate the jet penetration for such a case.Chapter 6. Multiple Jet Interactions with a Crossflow 191Our quantitative study of jet mixing reveals that the extent of the dispersion ofjet fluid is related to the decay rate of the vorticity possessed by the counter-rotatingvortices constituting the cross-section of each jet. This demonstrates the important roleof vorticity dynamics for jet entrainment. In addition, our results show that slenderrectangular jets can have a higher mixing capability than square jets with equivalentareas, especially when the jet spacing is small. This observation should be consideredin the design of combustion chambers or boilers where rapid mixing of air with fuel isdesired.In the study of mixing with interlaced jets, the following trend is revealed: at agiven mass flow ratio, higher velocity uniformity in the flow downstream of the jetsis observed with the use of bigger jets operating at a lower momentum ratio. Whenapplying our results, however, we need to realize that the actual flow in a recovery boileris significantly more complex than our idealized conditions. Moreover, our focus hasbeen on the jet penetration, which is only one criterion and does not necessarily reflectthe many parameters which need to be accounted for in the optimization of the designof a boiler. Nevertheless, the present study provides an estimate of the trends withdifferent parametric variations, and these trends can be used as guidelines for air systemdevelopment.Chapter 7Conclusions and RecommendationsOur studies of turbulent jets reveal valuable information on the characteristics of theflow fields. The information is useful for gaining basic understanding about the nature ofturbulent jet interactions, and for simulations and designs of kraft recovery boilers. Thecontributions of the thesis are summarized in the following section.7.1 Results of the Present StudyIn our, simulations of closely spaced jets issuing into a quiescent surrounding, our numerical results agree with the experimental observations that the jets merge rapidly and themerging is complete not far from the jet entrance. These results suggest the possibilityof simulating primary level jets in a kraft recovery boiler using slot equivalences. Twotypes of slot jets are considered: open slots and porous slots. They both give results thatadequately reflect the structures in the velocity field due to the interaction of discreteprimary jets, and this is done with a significant saving in computational costs. However,in our simulations, slot jets produce lower values of turbulent viscosity in the flow field.This may have implications on the diffusive transport of scalar tracer and momentum.Thus, we need to exercise caution in the representation of discrete primary jets by slots.Our parametric studies on a row of square jets in a confined crossflow clarify certainbasic characteristics of multiple jet interactions. In the investigation on the characteristicsof streamwise vorticity associated with each jet, it is found that an increase in lateralconfinement due to a reduction in orifice spacing does not increase the maximum vorticity192Chapter 7. Conclusions and Recommendations 193very much. This result is due to the competing effects of vorticity diffusion across from theneighboring jet, which weakens the vorticity, and the shearing flow around the jet, whichstrengthens the vorticity. In addition, our mathematical model produces the result thatat a high momentum ratio between the jet and the crossflow and at an intermediate valueof jet spacing, the usual vortex core seen in half of the cross-section of a jet modifies intotwo cores. This unexpected result is a consequence of the increase in shear experiencednear the top surface of the jet. It serves to illustrate how the jet structures can modifyin such a row-jet configuration when compared to the structures of an isolated jet.Our parametric studies on row-jet injection into a crossifow verify that, with a minormodification, the semi-empirical relationship (Eq.(2.24) or Eq.(6.1)) due to Holdemanand his co-workers can be applied to jets from rectangular orifices. The relationshipgives conditions on the geometric and operational parameters to yield consistent jetpenetration. The modification is needed because a slender jet with the long side alignedin the direction of the crossflow penetrates deeper than a wider jet that has the sameorifice area. This fact is verified by our flow visualization experiment.In addition to penetrating deeper into the crossflow, slender jets are observed topossess higher absolute values of the streamwise vorticity. This is a consequence of thelarger lateral surface of the jet available to experience the shear by the crossflow, whichstretches the fluid elements within the jet. The decay of the streamwise vorticity isobserved to follow trends that are consistent with the degree of mixing between the jetand the cross-stream. This observation is made in our examination of jets from orificeshaving different aspect ratios. It is found that in the cases where the spacing between eachpair of jets is the same, a larger reduction in the vorticity possessed by each jet indicatesa higher degree of mixing. This observation points to the significance of the counterrotating vortex cores within each jet on jet mixing. Furthermore, our quantitative studyon jet mixing demonstrates that the strategy to have the jet to penetrate mid-way intoChapter 7. Conclusions and Recommendations 194the cross-stream indeed leads to most uniformly mixed fluid.For jets interacting with the non-uniform crossflow defined in Eq.(6.8), the resultsshow that Holdeman\u2019s semi-empirical relationship remains valid in estimating the jetpenetration. The non-uniformity is prescribed as a peaked velocity profile with a flattenedtop, the type of profile that may be found in a kraft recovery boiler having a chimney-typeup-flowing core.An examination of Holdeman\u2019s relationship reveals that at a given jet-to-crossflowmass flow rate, larger jets operating at a lower momentum can penetrate into the crossflowas effectively as smaller jets at a higher momentum. In addition, when jets are placed inan interlaced arrangement, the use of Holdeman\u2019s correlation also yields satisfactory jetpenetration. When these interlaced jets interact with the non-uniform crossflow describedin the last paragraph, our results show that, at a given mass flow requirement, the use oflarger jets leads to a higher degree of flow uniformity in the mainstream. This observationis a consequence of the lower momentum requirement for the larger jets compared tosmaller jets.The numerous calculations required in this study are performed successfully and efficiently with the use of the multigrid solution technique. Our relatively simple problemsof jets in crossflows have provided many useful cases to test the correctness in the implementation and the robustness of the multigrid algorithm. The results are useful invalidating the simulation strategy for kraft recovery boilers and in providing guidelinesfor air system design for the boilers.7.2 Recommendations for Further StudiesThe modelling strategy for primary level jets using slots may need further refinementto account for the lower values in turbulent viscosity produced in slot simulations. OurChapter 7. Conclusions and Recommendations 195results show that a simple increase in the input turbulence intensity does not raise theturbulent viscosity levels. Thus, more studies are needed to examine how a better reproduction of the turbulent viscosity field can be achieved, and to examine the significanceof the turbulent viscosity on the flow field.When this study was commenced, the mathematical model used in the code couldonly allow for the simulations of isothermal flows. Consequently, the use of Thring-Newbyscaling criterion is necessary to study the jet behavior when there are differences in temperature between the jet and the crossflow. The use of Holdeman\u2019s correlation Eq.(6.1)to estimate jet penetration should now be examined further through non-isothermal calculations. The value of the parameter C appearing in the correlation may need to beadjusted when the differences in temperature between the jet and the crossflow are takeninto account.The observation regarding the existence of multiple vortex cores in half of a jet revealsthat there are fundamental differences in the jet structure between an isolated jet and ajet within an array in a crossflow. These differences should be examined more carefullywith the use of more sophisticated turbulence models. Indeed, better models that takeinto account the anisotropy in the turbulence characteristics are needed for more accuratesimulations of jet flows both in quiescent environments and in crossflows. The k \u2014 e modelmay be adequate in simulating gross flow features such as the penetration and spreading rates of jets; however, more refined models, such as the multiple-time-scale modeldiscussed by Kim and Benson (1993) or the Reynolds-stress closure models described byLeschziner (1989), are needed in the detailed examination of the flow field. The implementation of these more sophisticated models will enable more accurate simulations forboth jet flows and complicated recirculatory flows found in a recovery boiler.The use of a more sophisticated turbulence model will also test the computationalefficiency of our multigrid algorithm. At the current stage of development, the algorithmChapter 7. Conclusions and Recommendations 196used in our multigrid code MGFD is termed geometric multigrid; that is, the algorithmoperates on multiple levels of grids constrainted by the physical characteristics of thedomain. Although much improvement in the convergence rate has been achieved throughthe present algorithm, this method has the drawback that the coarsest grid used still hasto be rather fine relative to the size of the domain. This is the case because the coarsestgrid has to resolve fine geometric features such as jet orifices. Even though the use ofdomain segmentation has helped to improve the efficiency, a better approach is to usethe algebraic multigrid method discussed in Ruge and St\u00fcben (1987).The algebraic multigrid method does not consider the geometric constraints of thedomain when coarse grid levels are defined and hence very coarse grid can be reachedwithin the multigrid algorithm. Convergence rate can thus be improved since a largeerror range can now be acted upon by the solution smoother. The extension of this algebraic multigrid technique to the Navier-Stokes equations is not trivial, but this method,when coupled with an adaptation strategy such as the one presented in Thompson andFerziger (1989) for selective local grid refinement, appears to be a very promising solutionalgorithm.Another subject matter that deserves attention is the problem with close jet spacingat a high jet-to-crossflow momentum ratio. In Chapter 5, it is found that for the caseS\/W = 2 at J = 72, the convergence performance of our solution algorithm deterioratesdrastically. In particular, it is observed that large residues for the momentum and turbulence model equations are found at cells adjacent to the jet entrance. These findingssuggest that complex flow structures may be present near the root of the jet and thesestructures may influence the performance of our solver.These complex structures may be caused by the unsteadiness in the phenomena offlow separation and reattachment around the root of the jet. The occurrence of suchunsteadiness at a high Reynolds number at locations where flow separates from andChapter 7. Conclusions and Recommendations 197reattaches to a physical body is common in many problems of engineering interest. Anexample is the flow over a blunt plate discussed by Djilali (1987) and Tafti and Vanka(1991). A similar kind of unsteadiness can occur around the circumference of the jetentrance where the jet meets the mainstream and consequently the structures of thesystem of collar or horse-shoe vortices around the jet root can be affected. It is possiblethat the severity of the unsteadiness increases as the jet spacing is reduced, which mayexplain the difficulty in obtaining convergence for the case S\/W = 2 at J = 72.For this case, if our interest is only in the gross characteristics of the time-averaged flowfield, then the results presented in Chapter 5 may be adequate. However, a more refinedtime-dependent numerical study is needed in order to understand the detailed physicalcharacteristics of the flow field, especially around the jet entrance. At present, the largeeddy simulations (LES) used by Tafti and Vanka appear to be indispensable to yieldinformation on the detailed structures of flow separation and reattachment. However,such a simulation requires a large amount of computing resources. The algebraic multigridmethod described previously may offer substantial saving in computing costs. A detailedexperimental investigation of the flow field for this case of close jet spacing at a highvalue of J will also be needed to validate the numerical results.Appendix AFlow Visualization ExperimentThe flow visualization experiment is performed in the Kamloops water model shown inFigure 6.8. A detailed description of the experimental facility and the instrumentationcan be found in the thesis by Ketler (1993). The model is constructed primarily oftransparent plexiglass with numerous orifices located around the sides to serve as jetinlets. Water is led into the model through many pipes, with each pipe feeding about 7-8orifices at the primary level and a single orifice at the secondary level. For the tertiaryjets, each orifice is also fed by a single pipe located directly behind the orifice to maximizeflow uniformity. The flow rate entering each pipe is adjustable and is monitored by ananalog flow meter.There are 174 primary orifices distributed almost uniformly on all four sides of themodel near the base. A total amount of 122 US gal\/mm or 7.7 x iO m3\/s of water is fedinto the model through these orifices to generate an up-flowing stream within the model.At this flow rate, the overall flow pattern generated by the primary jets is observed tobe quite steady. Jets on the secondary level are not used since it has been observed thatthe interaction caused by flow from this level would cause gross unsteadiness in the flowpattern.The tertiary orifices are located on two opposing walls, with five orifices drilled intoa removable aluminum template on each wall. Only one set of orifices is used since theinterest here is on jet penetration due to single-sided injection. Two templates, shownin Figure 6.9, are made for the experiment, one for square orifices and the other for198Appendix A. Flow Visualization Experiment 199Kam loopsmodelJetFigure A.1: Schematic of the flow pattern in the east-west plane within the model generated by primary jets.rectangular orifices of J11 4.Preliminary visualization study has shown that the flow pattern generated by theprimary jets is quite steady but non-uniform. A main reason for the lack of uniformity isthat the base of the model is sloped to one side. As a result, a gross recirculating patternis formed as shown in Figure A.1.Because the objective here is to study jet penetration into a cross-flowing stream, theexistence of such a pattern causes the choice of the east wall to be used as the locationof the jets as indicated in Figure A.1. It is observed that the extent of the up-flowingregion at the tertiary jet level is slightly larger than half of the width of the model, andwithin the region the flow is quite steady and unidirectional. Thus, this region is to beused as the crossflow on which the tertiary jets act. The flow rate chosen for the tertiaryjets is 4 US gal\/mm or 2.5 x iO m3\/s for each jet.The flow visualization experiment is carried out by the use of a sheet of laser lightshining through the transparent base of the model. The sheet is generated by a prismoptic set-up located below the base. The laser light is generated by a powerful 7 WattsAppendix A. Flow Visualization Experiment 200LaserKamloopsLiJe JetFibre optic cableFigure A.2: Schematic of the experimental set-up for flow visualization.argon laser and is transmitted to the prism optic via a fiber optic cable. The set-up ofthe visualization experiment is shown in Figure A.2.The flow pattern is marked by the displacement of tiny polystyrene balls added tothe flow. The diameter of those halls are about 200 urn and they have density veryclose to that of water, which makes them very suitable for flow visualization. Light willbe reflected from these balls when they are illuminated by the laser sheet. The imageof the flow pattern within the sheet of illumination is captured by a S-VHS camera,which records at 30 frames per second. The recorded images can be processed by imagediscretization for particle image velocimetry analysis. A crude estimate on the velocity ata particular location within the flow may also be obtained simply by studying successivevideo frames on a monitor display, with the knowledge that the time lapse between anytwo frames is seconds.Appendix BA Quantitative Description of Jet MixingConsider mixing between a single jet and a mainstream shown in Figure B.1, wherethe flow is assumed to be incompressible and steady. Let the volume flow rate of themainstream and the jet be Qi and Q2, respectively, and let 4\u2019 and 2 be the respectiveconcentration of tracer per unit volume of each flow. The average concentration of themixture can then be defined as_1Q1+2Q2 BiavQ1+Q2(If the mixing were perfect, then every fluid element will have this 4av amount of tracerconcentration.By continuity of fluid volume and tracer concentration, the following expressions arederived for a cross-section A downstream of the jet:Lu.dA=Q1+Q2 (B.2)andL 4U dA = iQi + 2Q2 (B.3)Combining Eqs.(B. 1-B.3), we obtain= av(Q1+Q2)= avLT-1A (B.4)Therefore,\u2014 av)U . dA = 0 (B.5)201Appendix B. A Quantitative Description of Jet Mixing 20202__Conc. 2(>i)QiConc.Figure B.1: A schematic drawing representing the mixing between a jet and a crossflow.The evaluation of U dA and (4\u2014 av) at every point in the cross-section A will givethe volume per unit time passing each point and the defect from perfect mixing of thatvolume, respectively. The graph of the volume flow rate U . dA versus the concentrationdefect ( \u2014 av) gives a distribution at the cross-section in which Eq.(B.5) dictates thatwhen integrated over A, the distribution has a mean value of zero. Such a distribution isuseful in noting the volume flow rate across the section having tracer concentration lyingwithin a certain range.Instead of working directly with such a distribution, it is more convenient to workwith its normalized form written as a probability density function f(), defined as1 J (U dA)having concentration in range B 6)Q1+Q2 JItttttwhere =\u2014 avAppendix B. A Quantitative Description of Jet Mixing 203To show that the normalization is correct, note thatf f() d =-00\u2014 (U . dA)1 2= 1 (B.7)because taking all values of (U . dA)& over all possible values of j (\u2014co, co) gives thetotal volume flow rate Qi + Q2. Here, the notation (U . dA) refers to the flow rateof fluid elements having concentration in the range of ( \u2014 + ) where Li is asmall increment in..The mean of f() is zero sinceJ f() d = > f()-00\u2014(U. dA)= 0 (B.8)by virtue of Eq.(B.5).A measure of how well the tracer has been mixed is provided by the variance of f(),defined as= J\u2019 2f() d (B.9)where the subscript x denotes that the integral is evaluated with the values of \u00e7 andf() to be computed in the cross-stream plane located at the streamwise coordinate x.Clearly, o = 0 for perfect mixing since then 0 in the cross-stream plane. When f()is specified to be evaluated in the plane at x, it may be denoted as f(), as shown inEq,(6.3). This quantity o measures the spread of the tracer defect\u2014from thevalue of perfect mixing, which is zero. This variance can be normalized by o, which isdefined as the largest possible value of the variance corresponded to the situation beforeAppendix B. A Quantitative Description of Jet Mixing 204any mixing has started. Its value is obtained by considering the distributions of andU. dA in the surface A\u2019 shown in Figure B.1. The surface A\u2019 is defined to extend fromthe jet exit to a cross-sectional plane far upstream from the jet, where the effects of thejet are not felt.To evaluate o, recall that at any section A,= Qi + Q2dA) (B.10)Define i = \u2014 and 2 = 2 \u2014 av, then at the section A\u2019, f(j) is equal to zeroexcept at = i and = 2, where it takes on the values1 Qi 1 Q2= Qi + Q2 \u00e7- and f(2) = Q1 + Q2(B.11)Therefore, the value of u is obtained as= L2fdA, (B.12)= f(.) Li (B.13)1 ,2\u2014 t2 1 2 B14\u2014 Q + Q2 c2 Qi + Q2Finally, to have a normalized measure for mixing, a standard deviation S may bedefined as= Jo\/u (B.15)It has value ranges from 1 (no mixing) to 0 (fully mixed) and should be a good quantitative measure of jet-crossflow mixing.In passing, we remark that with simple modifications such as to take into account thepossible density difference between the jet stream and the cross-stream, the methodologypresented here can be extended to variable density flows. 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