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Inviscid flow from a slot into a cross stream Stropky, Dave 1993

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INVISCID FLOWFROM A SLOTINTO A CROSS STREAMbyDAVE STROPKYB.A.Sc. (1983), M.A.Sc. (1988), University of British ColumbiaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Mechanical EngineeringWe accept this thesis as conforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1993© Dave Stropky, 1993ABSTRACTThe problem of the oblique injection of a secondary stream into a free stream of different total pressure(non-isoenergetic) has practical application in many physical situations such as in the film cooling ofgas turbine blades. This thesis describes a new method for predicting inviscid non-isoenergetic flowfrom an arbitrarily inclined slot into a uniform free stream. In the absence of flow separation thesolution depends on three parameters: [i] the slot angle 13, [ii] a parameter, Cpt, describing thedifference in total pressure between the free stream and slot fluids, and [iii] the ratio of densities, D,between the two fluids. For given values Cpt and 13, there exists a unique value of M2/D, where M isthe ratio of mass flow of the injectant to the main stream.To guide in the development of the present theory, an isoenergetic solution (Cpt=0, D.1) wasfound using classical theory. Specifically,M – [-1–]),. where, X 0 < X, <1High curvature of the streamlines near the downstream slot lip produces an unrealistic suction for largeslot angles. The addition of source/sink singularities to the model to simulate flow separation improvesthe results, but adds empiricism to the solution.The present non-isoenergetic procedure uses the dividing streamline to separate the flowfieldinto two zones; an internal flow region containing the slot fluid and an external flow region containingthe free stream fluid. The technique involves finding, by approximate means, the shape of the dividingstreamline that satisfies continuity of static pressure between the regions. The boundary conditions atthe upstream slot lip were pivotal in finding a unique solution. The results presented here extendprevious work to include arbitrary values of f3, Cpt, and D.For a vertical slot, mass flow rates and discharge coefficients obtained from experiments andviscous numerical predictions are compared to present inviscid calculations. The results show goodagreement at reasonable Reynolds numbers provided that the separation region is not large. The effectsof separation can be included in the present theory by incorporating the approximate shape of theseparated flow region, and the results show a significant improvement, especially at higher mass flowratios and slot angles. Calculated discharge coefficients for lower slot angles do not compare as wellwith measured values, but this may be due to the differences in geometries used for the comparison.The solution algorithm used in the present problem is quite general, and may be applied toother geometries and flow conditions.IICONTENTSABSTRACT^  iiLIST OF FIGURES  viLIST OF TABLES^ viiiLIST OF SYMBOLS  ixACKNOWLEDGEMENTS^ xiiiChapter 1 INTRODUCTION  11.1 Problem Description ^  31.2 Literature Review  51.2.1 Isoenergetic Theory^  51.2.2 Non-Isoenergetic Theory  61.2.3 Experiments and Numerical Work^  8Chapter 2 ISOENERGETIC FLOW^  102.1 Non-Separated Flow  112.1.1 The Stagnation Point Position^  142.1.2 The Dividing Streamline  162.2 Separated Flow^  172.2.1 The Bubble Model^  172.2.2 The Wake Source Model  192.2.3 Comparisons^  21Chapter 3 NON -ISOENERGETIC FLOW^  253.1 Conformal Mapping Limitations  253.2 Vortex Sheets^  273.3 A New Zonal Interaction Method^  28illCONTENTS^ iv3.3.1 History^  283.3.2 Application to the Current Problem^  293.4 Defining the Zonal Interface^  313.4.1 Applying the Kutta Condition^  333.5 The Internal Flow Region^  353.5.1 Bardina's Method  353.5.2 Sridhar's Method^  373.6 The External Flow Region  383.6.1 Linearized Theory, the Hilbert Integral^  393.6.2 Davis's Method^  403.7 Density Ratio Effects  423.8 Separated Flow^  43Chapter 4 TEACH-II: Numerical Solution to the Full Equations^ 454.1 Governing Equations^  464.2 Numerical Solution Procedure^  474.3 Problem Geometry^  48Chapter 5 RESULTS^  515.1 Present Method Grid Refinement^  515.1.1 Computational Domain Size  525.1.2 Grid Distribution^  535.2 Non-Isoenergetic Theory Results  565.2.1 Unit Density Ratio^  575.2.2 General Results  585.2.3 Dividing Streamlines^  635.2.4 NIT Code Benchmarks  645.3 Vertical Slot Comparisons^  655.3.1 Mass Flow Rates  655.3.1.1 Boundary Layer Effects^  675.3.2 Dividing Streamlines^  685.3.3 Slot Exit Velocity Distributions^  69CONTENTS^ v5.4 Oblique Slot Comparisions ^  71Chapter 6 CONCLUSIONS AND RECOMMENDATIONS^  746.1 Conclusions^  746.2 Recommendations  80REFERENCES^  81Appendix A ISOENERGETIC THEORY^  83A.1 Attached Flow - Milne-Thomson's Solution^  83A.1.1 The Stagnation Point Position  85A.1.2 The Dividing Streamline^  86A.1.3 The Initial Slope^  86A.2 Separated Flow^  87A.2.1 The Bubble Model^  87A.2.2 The Wake Source Model  89Appendix B NON-ISOENERGETIC THEORY^  92B.1 The Zonal Interface^  92B.2 The Initial Dividing Streamline Angle 0 0 for Dol.^  93B.3 Inverting the Hilbert Integral^  94B.4 Coefficients for Davis's Method  96Appendix C NON-ISOENERGETIC RESULTS^  98C.1 NIT Results^  98C.2 Teach-II Results 100CONTENTS^ viLIST OF FIGURESFigure 1.1Figure 1.2Figure 1.3Figure 2.1Figure 2.2Figure 2.3Figure 2.4Figure 2.5Figure 2.6Figure 2.7Figure 2.8Figure 2.9Figure 2.10Figure 2.11Figure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 3.6Figure 3.7Figure 3.8Figure 4.1Figure 4.2Figure 5.1General Slot Flow^  3Ideal Slot Flow  4Ainslie Geometry^  6Physical plane (z=x+iy)  11Transform plane (--tt+iii)^  12Mass flow dependence on stagnation position^  14Mass flow with Kutta condition applied  15Dividing Streamlines^  16Bubble model physical plane^  18Bubble length influence  19Wake source physical plane^  20Tangential separation  21Streamline comparisons^  22Centerline pressure distribution^  23Hodograph plane for Cpt<0  26Zonal Modelling^  28Oblique slot zones  30Spliced cubic polynomials^  31Kutta condition problems  33Bardina's method^  36The external flow field  38A simple separation model^  43TEACH-II grid cells  47TEACH-II geometry^  49The NIT grid  51CONTENTS^ viiFigure 5.2Figure 5.3Figure 5.4Figure 5.5Figure 5.6Figure 5.7Figure 5.8Figure 5.9Figure 5.10Figure 5.11Figure 5.12Figure 5.13Figure 5.14Figure 5.15Figure 5.16Figure 5.17Figure 5.18Figure 5.19Figure A.1Figure A.2Figure A.3Figure A.4Figure B.1Figure B.2Figure C.1Figure C.2Figure C.3Figure C.4Figure C.5Effects the interaction region length^  52Effects of grid size, Cpt=0^  54Effects of grid size, Cpt---0.1  54Effect of cubic spacing on dividing streamline shape near Cpt=0^ 55Sketch of dividing streamline shape for small values of Cpt  56M-Cp t results for linear and non-linear theory^  58M-Cpt results for various density ratios  59M2/D vs Cpt^  60Cd vs Cpt  62Dividing streamlines, t3=15°^  63Density effects on dividing streamlines^  64Vertical slot mass flow comparisons  66Vertical slot mass flow comparisons at low M^  67Boundary layer effects^  68Vertical slot dividing streamline comparisions^  69Slot exit velocity distributions, M=0.5^  70Slot exit velocity distributions, M=0.2  71Discharge coefficient comparisions^  73Confined Slot Flow^  83Initial Slope of the Dividing Streamline^  87Bubble model transform plane^  87Wake source transform plane  8900 for Doi^  93Perturbation curves^  95Cd vs M2/D  98Dividing streamlines, 0.413°^  99Dividing streamlines, 13-65 °  99Dividing streamlines, 13-90 °^ 100Teach-II vector plot, 3=90° 100CONTENTS^ viiiLIST OF TABLESTable 2.1^Separation mass flow ratios^  22Table 3.1^Non-Isoenergetic solution methodology^  31Table 3.2^Upstream slot lip boundary conditions  34Table 4.1 Governing equations^  46Table 5.1^Discharge coefficient constants^  72LIST OF SYMBOLSak...dk^coefficients of spliced cubic polynomialssap...edp^coefficients of perturbation functionAki matrix coefficients for Bardina's methodAm-54o^mapping pointsbn, bm^control point locations for Sridhar's and Davis' methodsC1 , C2, Cii•••^turbulence model coefficientsCim•••C3m^coefficients for Davis' methodCd^discharge coefficient (- 2 psUs / liPts —PooCp pressure coefficient (— [p— pco ]lip.U.2 )Cps^injectant pressure drop coefficient (= [ps^)Cpl^total pressure coefficient (= [Pt. —Pts ]1 -}pcoU002 )D density ratio (= ps/Poo)fE(x)^spliced cubic tail piecefk(x) kth spliced cubic polynomialF=4:13-Fi1P^complex potential plane coordinatesGE , Gk turbulence model generation termshE^asymptotic thickness of injectantH, L separation bubble dimensionsj, k^control point indicesixSYMBOLS^ xk^turbulent kinetic energyk separation speedK Schwarz-Christoffel mapping constantLI^computational domain inlet lengthM mass flow ratioMo^isoenergetic attached flow ratioMk Kutta condition mass flow ratiomk^slope at kth cubic endpointn, s, e, w^control volume facesN, 5,..., SE, SW^control volume neighborsN number of spliced cubicsNB^number of control points for Bardina's methodNE number of control points for Davis' methodP^static pressurePt. main stream total pressurePts^injectant total pressureq , q1 , q2^source/sink locationsQ source/sink strengthRh^slot Reynolds number (= pU cc h/p, h=slot width)S(x) exact dividing streamline (zonal interface) shape§(x)^approximate dividing streamline shapeSo source termSns^turbulence model shear strain tensorU, undisturbed main stream velocityUs^average injectant velocityV+(x), V-(x)^velocities along the upper and lower surfaces of the dividing streamlineVo^velocity at the upstream slot lipSYMBOLS^ xiw=u—iv^hodograph or complex conjugate velocity plane coordinatesVu ( x ) undisturbed velocity fieldV+ (x) perturbation velocity fieldxp , yp^spliced cubic endpoint coordinateseyp^perturbation at pth cubic endpointz=x+iy physical plane coordinateszs^stagnation point locationak^zonal interface tangent angle at kth cubic endpointslot angleX^straight channel mapping planeseparation model teat height(x)^boundary layer thicknessO * (x) displacement thicknessA^adjacent cubic length ratiodissipation ratetangent angle of the dividing streamline, general transport variableconstant (=n/13 — 1)I'4,^diffusion coefficienty(x) vortex sheet strength distributionturbulence model constantconstant (-13/3T)laminar viscosityturbulent viscositylieff^effective viscosity, =g+thOm^finite slope change at element endpoints in Davis' methodSYMBOLS^ xii13oPooPs°kQ k,p=11-t+irl2m, N3minitial angle of the dividing streamlinemain stream densityinjectant densityturbulence model constantsperturbation functionperturbation constantcubic mapping spaceSchwarz-Christoffel mapping plane coordinatescoefficients for Davis' methodACKNOWLEDGEMENTSI would like to express my sincere thanks to both of my supervisors, Professor Ian Gartshoreand Professor Martha Salcudean. They allowed me once again to delve into the world oftheoretical fluid mechanics, and do research that I know does not pay the bills. Both have beenfar more than good advisors, they have been good friends.I am dedicating this work in memory of my father Joe.XIIIChapter 1INTRODUCTIONThe problem of oblique injection of a secondary stream into a main stream of different totalpressure has practical application in many physical situations. Consider the case of a gasturbine engine. The performance of such engines increases in general with increasing inlettemperature, however the temperature of the structural components must be restricted to avoidmechanical failure. Film cooling is a widely used method for keeping the skin temperature ofthe turbine blades and combustion chamber walls within acceptable limits. A coolant isinjected into the gas stream through a series of holes or slots in the component surface. Theinjectant forms a thin layer over the surface, shielding the metal from the hot combustiongases. The injection process must be carefully controlled so as not to disrupt the aerodynamicefficiency of the engine. There are many other examples of secondary injection, including jetflapped wings, VTOL aircraft, and ground effect machines.Owing to the complex nature of the problem, there is at present, a less than adequateunderstanding of the fluid mechanics of two stream flow. A typical case encompasses severalaspects of fluid mechanics such as unsteadiness, compressibility, turbulence, viscosity,vorticity, and flow separation. What is of primary importance in many of these situationshowever, is the mass flow of the injectant. The operating state of these devices generally liesin a restricted range of mass flows, so that a mathematical model should predict the mass flowaccurately. Resolution of all the details of the flow requires a sophisticated Navier-Stokessolver and considerable computational resources, an often impractical and unnecessary1CHAPTER 1^ 2procedure for many engineering purposes. As in many other fluids problems, a simplifiedanalysis may provide acceptable (albeit limited) results.Over the last two centuries, many problems have been solved satisfactorily by assumingthe fluid to be inviscid. Classical methods give elegant solutions to inviscid flows in andaround a variety of shapes. Aerodynamicists commonly use inviscid analysis to predict theperformance of streamlined components. In these and many other instances, inviscid theorycaptures the essential physics of a problem and produces reasonable results. Much of theinsight into the physics of fluid mechanics has come from these techniques. In the last fewdecades, inviscid methods have been coupled with boundary layer theory (known as 'zonal'modelling) to greatly extend the range of useful solutions.In this spirit of simplicity, we apply inviscid analysis to the problem of steadyincompressible flow from an arbitrarily inclined slot into a main stream of different totalpressure. Existing inviscid solutions to this problem are not general, due in part to the (still)complex nature of the problem. The aim of this thesis is to find a general inviscid solution andto compare the results with existing theory, experimental data, and limited viscouscomputations.The results will also provide a more meaningful way of defining a discharge coefficientfor a jet into a cross flow. In the absence of a cross flow, the standard definition of thedischarge coefficient is,Cd =(Ideal Flow)^Pt j — pc°1 nwhere Uj is the average discharge velocity of the actual flow, Ptj and pi are the injectant totalpressure and density respectively, and p. is the static pressure far from the orifice. Using thisdefinition, Cd provides a direct measure of the losses in the injection process. When fluid isinjected into a cross flow, the ideal flow is no longer given by the simple expression in thedenominator of (1.1), however this expression usually is retained. It is important to note that,for the cross flow situation, Cd is no longer a direct measure of the losses in the injection(Actual Flow)^(t7j)CHAPTER 1^ 3process. The results of this thesis will provide correct values for the ideal flow calculation, sothat a modified Cd could be defined to represent the losses directly.1.1 Problem DescriptionFigure 1.1 General Slot FlowFigure 1.1 gives a general description of the flowfield for oblique slot injection into a mainstream. In addition to the features shown, boundary layers generally exist on the solid surfaces.Viscous effects near the upstream lip may allow the slot fluid to penetrate upstream or mainstream fluid to penetrate into the slot. Regardless, a shear layer forms, separating thesecondary and main streams. Instability of this shear layer usually causes the flow to becometurbulent downstream of the slot, resulting in mixing of the two fluids. If the injectant flowrate is high, separation and subsequent reattachment may also occur downstream of the slot.Figure 1.2 on the following page is an idealized representation of the flowfield. The followingassumptions are implicit in the idealization:• The thicknesses of the shear layer and all boundary layers are smallcompared to the thickness of the injected fluid.• The shear layer is stable.• The flowfield is inviscid and irrotational.• The fluids are incompressible.• Bouyancy forces are negligible.CHAPTER 1^ 4In the ideal case, fluid at total pressure Pts and density ps is injected from a slot at an angle pinto an otherwise uniform free stream of total pressure Pt. and density No . A dividingstreamline, beginning at a stagnation point on the upstream wall and extending downstream toinfinity, separates the fluids.Figure 1.2 Ideal Slot FlowAs the injected fluid leaves the slot, it is deflected by the free stream, asymptoticallyapproaching a constant thickness h., and a constant pressure, p.. Unless stated otherwise, weassume that the ideal injectant fluid does not separate from the downstream slot lip. It cannot,however, turn the sharp corner with finite velocity, resulting in a singularity in the flowfield.We will explore the consequences of this singularity (and its subsequent removal through aflow separation model) in Chapter 2.In the absence of separation, the following four independent parameters provide for aunique solution:[i] The slot angle 3[ii] A parameter relating the total pressures of the slot and mainstream fluids,Cpt - [Pt. —Pts ]I 1-p.U.2,^ (1.2)[iii] The density ratio,D = ps I P. (13)CHAPTER 1^ 5[iv] The location, zs , of a stagnation point where the dividing streamline begins.For reasons explained in Chapter 2, the stagnation point is generally fixedat the upstream slot lip.For fixed values of these parameters, there is only one corresponding mass flow ratio,M = psUs / (1.4)that satisfies the flowfield boundary conditions. Relatively few analyses have been performedbecause the problem is a non-linear boundary value problem with an unknown boundary shape.The following section is a synopsis of the progress made towards a steady incompressibleinviscid solution. From this point forward, we will classify all solutions into two categories:(i) Solutions to the special case Cpt=0, D=1. In this case, and only in this case, is the entirevelocity field continuous. Without this restriction there is a discontinuity in tangential velocityacross the dividing streamline (the explanation for this is given in Chapter 3). We shall termthis the isoenergetic case, although the usual meaning of this term does not include the unitdensity ratio. (ii) All other values of Cpt and D. These solutions are termed non-isoenergetic.1.2 Literature Review1.2.1 Isoenergetic TheoryEhrich (1953) outlined a Helmholtz-Kirchhoff solution for a generalized slot/orificeconfiguration that includes the present geometry. To eliminate the infinite velocity point at thedownstream slot lip, he included, as an option, a free streamline leaving smoothly from thedownstream slot wall at the lip. This resulted in an unrealistic constant pressure separationregion of ultimately infinite thickness. He solved the equations pertaining to the presentgeometry for the case 13=n/2 only.Dewynne et al (1989) analyzed the complementary problem of suction from an inviscidchannel flow into a slot. They used classical methods directly because the entire flowfieldconsists of one fluid and is therefore isoenergetic. Their model includes smooth flowseparation from the upstream slot lip in the form of a free shear layer. This shear layer dividesFigure 1.3 Ainslie GeometryCHAPTER 1^ 6a region of constant pressure fluid from the flow into the slot. The amount of mass flow intothe slot depends on the value of this pressure and the location of the stagnation point on thedownstream wall. For small values of mass flow, the flow pattern is significantly differentfrom the non-separated case.Ainslie (1991) used hodograph methods to studythe flow from the aperture shown in the diagram. Theflow was assumed to leave smoothly from the upstreamwall and remain attached at the downstream corner. Healso conducted experiments for ramp angles in the range13=47-180°. Measurements of M at Reynolds numbersabove 2,000 (based on the free stream velocity and slot width) showed good agreement withthe theoretical predictions. The nature of his apparatus made it difficult to check for flowseparation, but he did find the flow to be slightly unsteady.1.2.2 Non-Isoenergetic TheoryTing and Ruger (1965) point out, for the case of unequal total pressures, that the equilibrium offorces on a fluid element reveals a discontinuity in the velocity field across the dividingstreamline. The standard method of conformal mapping cannot be used because the lineseparating the two streams in the complex potential plane is mapped into two unknown curvesin the complex velocity plane. They also show that the solution for small Cpt does notapproach the solution for Cpt=0 uniformly. In particular, the initial slope of the dividingstreamline is a discontinuous function of Cpt near Cpt-0. For Cpt<0, the mainstream cannotsupport a stagnation point in the secondary flow at the upstream lip, therefore the flow mustleave tangent to the slot wall (assuming that the flow separates at the corner). Similarly, theseparating streamline must leave tangential to the mainstream wall for Cpt>0. For Cpt=0, bothstreams must incur a stagnation point, therefore the flow leaves at some angle 0<O043. For thespecial case 13=n/2, they reduced the problem to two nonlinear singular integral differentialequations. This simplification was due in part to their imposed condition that the secondaryfluid exits the slot at a uniform angle f. Using numerical methods, they found solutions forCpt>0, but were unsuccessful for Cpt<0.CHAPTER 1^ 7Ting (1966) used linearized supersonic theory for the external flow to find analyticsolutions for a wedge and flat plate with normal (inviscid) injection. The injection velocitywas much lower than the wedge speed (implying Cpt.1), but larger than the value allowed forin boundary layer theory. Using a Cauchy integral formulation for the inner flow, Tingreduced the problem to a single non-linear integral differential equation in the transform plane.The solution was found by expanding the integral in terms of a small parameter, 1—Cpt. Thezeroth order solution was discarded because it produced an unrealistic result when transformedto the physical plane.Cole and Aroesty (1968) point out that the boundary layer on a flat plate "blows off'when pw vw pc„,Uce z R;112 for subsonic laminar flow, and when pw vw / pc„,Uce z 0.02 forsubsonic turbulent flow (w a. conditions at the wall). They used an inviscid version of thePrandtl boundary layer equations for the inner flow to analyze the same problem as Ting.They extended the range of applicability to include hypersonic and axisymmetric flows.Experimental verification was unsuccessful because they could not simulate the theoreticalinjection distributions in wind tunnel tests.Ackerberg and Pal (1968) analyzed the opposite limit of strong secondary injection(Cpt, 00) from an infinite plate into a uniform stream. A Ritz-Galerkin variational principlewas used to obtain numerical solutions. Comparisons with experimental data showed asignificantly deeper jet penetration, probably due in part to the assumptions of a constant wakepressure and a uniform injection velocity across the exit.Goldstein and Braun (1975) analyzed nearly isoenergetic flow from two-dimensionalnozzles and orifices. The optional addition of a free streamline (see Ehrich above) allowed forflow separation from the downstream lip. The solution involved expanding Bernoulli'sequation for values of Cpt. In a manner similar to that used in thin airfoil theory, the boundaryconditions on the unknown dividing streamline were transferred to the (known) isoenergeticdividing streamline. Using only cusped leading edges, they avoided the difficulties associatedwith the discontinuous initial slope of the dividing streamline (see Ting and Ruger above). Thevelocity discontinuity along the dividing streamline was handled by introducing a newdependent variable which satisfies only jump and symmetry conditions on the boundariesCHAPTER 1^ 8rather than the combination of jump and boundary conditions which is imposed on the physicalvariables. No comparison to experimental data or to other theoretical results was made.Fitt et al (1985) proposed a simple model of irrotational inviscid flow from a verticalslot into a free stream. Their model assumes that the total pressure in the slot exceeds the freestream static pressure by a small amount, implying Cpt .1 and M«1. Analyzing the injectionprocess as a small disturbance to the free stream, they used a perturbation potential scaled on Mand the Hilbert transform to express the solution in terms of a single integral equation. Thenumerical solution to this equation gave M.1.12(1—Cp t) 3/2. Experiments were conductedusing an open-circuit induced-flow wind tunnel to provide the free stream flow. The totalpressure of the injectant was varied by placing gauze resistances over the slot entrance.Suction removed the tunnel floor boundary layer 0.1 m from the leading edge of the slot.Static pressure measurements were made along the tunnel wall both upstream and downstreamof the slot. The static pressure variations without injection were subtracted from the injectionresults to minimize tunnel wall boundary layer effects. The flow was turbulent with a typicalslot Reynolds number of 6.5x10 4. The upstream boundary layer thickness was approximatelyTiu of the slot width. They report a significant mass flow increase for a factor of three increasein the boundary layer thickness. For the range of mass flows applicable to their theory, theyobtained good agreement with experiment.1.2.3 Experiments and Numerical WorkSinitsin (1989) performed numerical simulations of the Navier-Stokes equations for 20° and40° slots using a finite-difference Cartesian code. He also conducted wind tunnel experimentsfor these same geometries. From the experiments, he found distributions of speed across theslot exit, but not the flow direction. He proposed angular variations that would yield themeasured flow rate, and then imposed the resulting velocity distribution as a boundarycondition for the numerical simulation. This procedure avoids a 'staircased' representation ofthe slot that would otherwise result from using a Cartesian grid, and was thought to be asignificant impovement over the uniform distribution generally used. His calculations indicate,for three different profiles of flow direction, that the flow direction does not have a gross effecton the flow field downstream of injection. The flow direction can, however, have a largeCHAPTER 1^ 9effect on the shear stress coefficient and the heat flux at the wall. He found good agreementbetween predicted and measured velocity profiles downstream of the slot for the 20° case, butonly fair agreement for the 40° case. Measurements of the flow rate from the slot were slightlyunsteady, and the flow was found to separate from the downstream slot lip (for M>0.9) in the40° case only. The mass flow ratios used in the study were in the range 0.4-1.5.Gartshore et al (1991) measured discharge coefficients through 20 and 40 ° slots and aninclined hole. They find for all geometries that,1 — Pts —Poo — A + B2(Cd )2 1ps UsMprovides a good fit to the data. A and B are constants and Cd is given by (1.1). This simpleform was derived by considering the component losses caused by the injected flow. For theslot, B was found to be sensitive to the slot angle and to the state of the boundary layerupstream of the orifice, and A was very nearly constant at 2.5. Numerical simulations using afinite-volume Navier-Stokes solver were in good agreement with the experimental results.Numerical studies showed that the value of B was most sensitive to the thickness of mainstream boundary layer at the slot. Both numerical and experimental results were reasonablyindependent of slot Reynolds number for values above 1,000.The following two chapters describe new progress made in calculating the ideal flowfrom a slot into a cross flow. The results described in the literature review are compared to thenew results in Chapter 5.Chapter 2ISOENERGETIC FLOWIn the previous chapter we stated that no comprehensive solutions exist for the non-isoenergeticflow case. During the initial stages of development of the present theory, it was apparent that acomplete analytic isoenergetic solution would be useful for the following reasons:• To provide a basic understanding of the flowfield.• To use as a limiting case check of the non-isoenergetic results.• To analyze the dependence of M on the slot angle 0, and to verify thedependence of M on the stagnation point position, z s .• To find the initial slope of the dividing streamline for the case Cp t=0. The slopeis required as an input boundary condition for the non-isoenergetic method.• To consider the effects of the infinite velocity point at the downstream slot lip.• To estimate the effects of flow separation from the downstream slot lip. Thisstep requires additional modifications of the non-separated flow model.Ehrich (1953) outlines a method for finding a solution to the present geometry. Owing to thecomplexity of the analysis, however, it seems unlikely that an analytic solution can be foundwith his method for slot angles other than n/2. We use instead the concept of a velocitypotential (see, e.g., Milne-Thomson, 1968) to find a simple yet complete solution. This is ahighly developed analytical tool for solving ideal fluid flow problems. The method involvesapplying the principle of superposition to known potential functions (i.e., sources, sinks,vortices, and doublets) to solve Laplace's equation for a given set of boundary conditions. Inall but the simplest cases however, the desired potential function cannot be specified directly.10CHAPTER 2^ 11Fortunately, we can map the geometry to an auxiliary plane in which the velocity potential canbe written directly. A subsequent reverse application of the transformation gives the desiredphysical plane solution. Schwarz-Christoffel and hodograph transformations (see Milne-Thomson) are the most commonly used conformal mappings for rectilinear flows (i.e., thepresent geometry).In many real flows, the presence of viscosity may induce flow separation from a sharpcorner. Techniques developed by Kirchhoff (1869), Roshko (1954), Parkinson and Jandali(1970), and others allow for the presence of such separated flow regions.The remainder of this chapter describes isoenergetic potential flow solutions to thepresent geometry. For non-separated flow, described first, the solution is uniquely determinedby the geometry and the stagnation point position. In section 2.2, two models, based onParkinson's wake source theory, allow for flow separation from the downstream slot lip. Allsolutions are based on a single Schwarz-Christoffel transformation to the upper half plane.2.1 Non-Separated FlowFigure 2.1 Physical plane (z=x+iy)Figure 2.1 is a schematic of the physical plane. The slot is infinitely deep and the(undisturbed) main stream is a uniform free stream. With no loss of generalization, we scaleCHAPTER 2^ 12all velocities on the free stream value and all lengths on the slot width. As indicated in thediagram, the scaled velocity flow from the slot is equivalent to the mass flow ratio because ofthe unit density ratio assumed in this analysis. For ideal flow, the most physically realisticlocation of the stagnation point S is at the upstream slot lip B. This eliminates the infinitevelocity point that would otherwise occur at the corner. Aerodynamicists use a similarcondition, the Kutta condition, to determine a unique value for the lift on a wing. We choose,however, to arbitrarily locate S, the motivation being, to allow for a more flexible solution andtherefore a deeper understanding of the problem. The downstream slot lip D is also asingularity in the flow field (for the attached flow case at least). Infinite velocity points,although unrealistic, occur in many potential flow solutions. Their effects are generallylocalized, and the removal of a singularity by a slight 'rounding' of the corner at D does nothave an appreciable effect.Using the Schwarz-Christoffel theorem, we open the simple polygon .910oBC.coDE00 intothe upper -plane (see Figure 2.2). Three vertices, 9100, B, and Co , in the z-plane correspondarbitrarily to -00, -1, and 0 in the -plane, while symmetry considerations place Too at +00. Thevalues of s and d are found from the analysis.Figure 2.2 Transform plane (=u+iri)The function which transforms the physical plane to the upper half plane is given by,dz K  (+ 1)1-k ( -- d) 1" where k=r3ht (0<).<1). In the z-plane, the uniform stream may be taken to imply a source atAcc, and an equal sink at Too . Thus in the -plane we must also have a source and sink at thecorresponding points so that there is a uniform stream, Vco say. The uniform flow in the slotCHAPTER 2^ 13transforms to a source of output M at C. Thus, in the z-plane, we have a source that emits thevolume M per unit time over an angle IL We can now write the equation for the velocitypotential,F() =^+ —The complex velocity in the z-plane is then given by,dF dFw(z) - -dz^dz K(t +1)1-X (t - d) ?"The following boundary conditions complete the problem description,w(z)0^at^r;= -sat^l;= 00Me-Ili^at 0Combining the above equations,F() = —+ In7t S(+ s)w(z)1)1-A^-M= -+1)-{(g where,7E-1To find the solution, we must find the value of M that scales the geometry correctly. Thestandard procedure is to integrate (2.3) to a location known in both the z and planes (i.e., to=c1 in the .-plane, corresponding to z=1/sin (3 in the z-plane). If X. is a rational fraction, wecan, in principle, find a solution. The integration is very involved for a general fraction, and itis further complicated by the presence of a singularity atVoo + 11t(2.1)(2.2)= m(23)--*— n/18n/4--A— n/2--o— 3n/4M0CHAPTER 2^ 14Using a completely different approach, taken from the analysis of flow in a branchedcanal (see Milne-Thomson, pp. 289-292), we find a general solution in a more elegant manner.The present geometry is equivalent to a branched canal if the free stream is bounded above bya solid wall. Appendix A.1 contains the details of the derivation. The result is,M=M-{- -Ai {J--- } x'– s – 1–X (2.4)2.1.1 The Stagnation Point PositionTo find a solution, we input I (i.e., n, m) and s. M and M are given by (2.4), and integrating(2.3) to =–s gives ;. Analytic integration of (2.3) is very involved for general n and m. Forpractical purposes we integrate numerically because the integrand is finite along the path ofintegration (see Appendix A.1.1 for details). The following diagram indicates the dependenceof M on 3 and ;.-1 5^-0.5^0^0.5^1^1.5^2± !Zs'Figure 2.3 Mass flow dependence on stagnation positionCHAPTER 2^ 15The vertical line (1;1=0) represents the Kutta condition. For mass flow ratios lower than theKutta condition value (M=Mk), the stagnation point moves into the slot along the upstreamwall. For M>Mk the stagnation point moves upstream along the mainstream wall. Thestagnation point position moves infinitely deep into slot as M-0, and infinitely far upstream asM–.00. Variations in M appear to be greatest near Mk. Perhaps some of the observedunsteadiness of the actual flow is a result of this sensitivity. Note also the non-monotonicdependence of M on p for fixed Izs l.Figure 2.4 shows the M-p relationship with the Kutta condition applied. The non-monotonic behavior is clearly visible, with M reaching a minimum of about 0.757 at P=39.2°.2.01.5M1.0•Fitt et al0.5 1^,0^n/6^n/3Rn/2^2.7E/3Figure 2.4 Mass flow with Kutta condition appliedFor (3–.0, the injected fluid does not deflect the free stream, and consequently we expect M--►1to be a solution, as is shown in the figure. As the slot angle increases, the fluids deflect eachother, and an asymmetric pressure gradient forms in the slot. The fluid on the upstream side ofthe slot experiences an adverse pressure gradient because of the presence of the stagnationpoint at the upstream slot lip. On the downstream side, the pressure gradient in the slot isfavorable due to the acceleration of the fluid towards the singularity at the corner. Bothgradients increase as 13 increases, but evidently the favorable gradient is dominant at higherCHAPTER 2^ 16angles. In reality, M obviously does not approach infinity as 13->n, as the calculations show.At higher slot angles, a real fluid cannot support the large adverse pressure gradient that existson the wall just downstream of the slot exit. The fluid separates from the downstream slot lip,reducing the streamline curvature near the corner, which reduces the nearby pressure gradientsand the flow from the slot. These arguments are supported by the experimental results of Fittet al (see Figure 2.4). They found a significant region of flow separation downstream of theslot, leading to a 35% reduction in the mass flow predicted by the isoenergetic theory. InSection 2.2 we augment the present theory to include the effects of flow separation.2.1.2 The Dividing StreamlineWe can directly specify the equation for any streamline by setting the imaginary part of thevelocity potential (i.e., the stream function 111) equal to a constant. In deriving (2.1), we tacitlyassumed that the downstream wall has the stream function value W=0. Thus the value of thestream function constant is equal to the volume flow contained between the downstream walland the corresponding streamline. The dividing streamline contains the volume flow M,therefore klf—,Z(F)=M gives the streamline equation. The details of the calculations are inAppendix A.1.2. Figure 2.5 shows a sample of the results for the case 13-7E/2.21.5y0.5tLfiI•Ni WANNSS^NNNNN^V, ,NNSC NNNNSNNVCWANNW-0.5-2^0^2^4^6^8^10xFigure 2.5 Dividing StreamlinesCHAPTER 2^ 17In each case the asymptotic height of the streamline is M, the value required by continuity.The variation of stagnation point position is quite small (the y-axis has an expanded scale) forthe range of M considered. Except for M=1 (the Kutta condition), the streamlines leave normalto the surface. It can be shown (see Appendix A.1.3) that the streamline leaves at an angle of13/2 when the Kutta condition is applied. This value supports the arguments of Ting and Ruger.This completes the analysis of non-separated flow.2.2 Separated FlowExperimental results for a vertical slot (see, e.g., Fitt) show that there is significant flowseparation for M of about 0.4 or more at reasonable Reynolds numbers. Past research showsthat when the regions of flow separation are significant, classical potential flow methods yieldinadequate solutions. Robertson (1965) lists several techniques developed in the 1940's and1950's (modifications to the classical method) to include the presence of separated flowregions. Parkinson and Jandali (1970) developed a simple method of adding sources to theclassical solution to simulate the effects of separation. This technique is the moststraightforward of the methods that include separation effects, and is the method of choicehere. The following sections describe the development of two possible separated flow modelsbased on Parkinson's wake source model.2.2.1 The Bubble ModelThe separation region effectively acts to displace the surrounding streamlines, much like asolid surface of the same shape and size. Thus the effects of separation can be approximated ininviscid flow by adding a separating streamline of about the same shape and size as theseparated region. Parkinson employs such a streamline, through the addition of a source orsources of carefully selected strength and location, to simulate the wake of a bluff body. Theaddition of sources alone to the previous non-separated solution results in a separation regionof finite height but infinite length. Experimental and numerical evidence shows that theseparation region, if it exists, is finite in both height and length. To simulate a finite lengthregion in potential flow, the separating streamline must reattach to the solid surface. ThisCHAPTER 2^18requires (for a single source) the addition of a sink of equal strength located downstream of thesource. This is the basis for the first separated flow model. Figure 2.6 shows the source/sinkbubble model in the physical plane. Application of the Kutta condition at the upstream slot lipsimplifies the analysis.Figure 2.6 Bubble model physical planeDetails of the analysis are similar to the non-separated case, and are given in Appendix A.2.1.The results are,M .1 ^Mo^1+  (qi + 1)(q2 ÷ 1) (q1 r)(q2 —1") (2.5)where, f—it/f1-1, q 1 and q2 are the source and sink locations in the transform plane (see FigureA.3 in Appendix A.2.1), and Mo is the non-separated mass flow ratio for the same slot angle.To find M for a given slot angle, q 1 and q2 must somehow be specified. In his work, Parkinsonspecifies the pressure at separation (the base pressure) to provide one extra condition. Thiscondition cannot be used here however, for the following reasons. In the transform plane, wecan show that the separating streamline leaves at D normal to the surface. Through an analysissimilar to Appendix A.1.3, we can also show that this streamline leaves at an angle of it/2+13/2in the physical plane. Thus D is a stagnation point, making it impossible to specify a variablepressure there.^ ^ ^ ^ ^CHAPTER 2^ 19We can, however, specify values for the length and height of the bubble. Usingnumerical methods, we can then find the corresponding values of q 1 and q2. Results forreasonable bubble dimensions show that the stagnation point is weak; the separating streamlinebends rapidly to the tangential separation angle.Figure 2.7 illustrates the effect of bubble length for a vertical slot. Isoenergetic resultsfrom the numerical calculations in Chapter 4 give a bubble height of roughly one slot widthand a length of about ten slot widths. The results shown here are for a fixed bubble height ofone slot width, and indicate that the mass flow is nearly independent of the length for L>10.Calculations for other slot angles and bubble heights confirm that M is independent of L forvalues greater than about 10H.10.75M 0.50.25o 0 10 20 30 40 50 60 70LFigure 2.7 Bubble length influence2.2.2 The Wake Source ModelThe previous model, although physically realistic because of the finite bubble, requires inputconsisting of the bubble height and length that is difficult to obtain in practice. The model isfurther hindered by the unrealistic stagnation point at separation.Figure 2.8 illustrates a second possible separation model, based on a lone source and asmall teat (Parkinson, 1991) protruding from the downstream slot lip. The flow separatessmoothly from the teat, eliminating the stagnation point and allowing the base pressure to bespecified. Addition of the teat makes the analytical treatment more complex; for our purposesCHAPTER 2^ 20the analysis is restricted to a vertical slot. We also eliminate the sink, to further decrease thecomplexity and empiricism. The results of the first model show that this should notsignificantly affect the mass flow.Figure 2.8 Wake source physical planeIn the diagram, k is the magnitude of the velocity at separation, and is directly related to thebase pressure through Bernoulli's equation. Details of the analysis are given in AppendixA.2.2. The results are,2cri-71-M =d+ Afc7--71-k (2.6)b= 1 {-1(d2 —1)(2d —1) ln [V(d 2 —1)(2d —1) + d 2 + d —1}a^d^ d2Again we need two extra boundary conditions for a solution. The base pressure provides one,and the following is an explanation of the second. Figure 2.9 shows the initial region of theseparation streamline for three different teat heights. The value of k here is 1.18, taken fromthe calculations in Chapter 4.CHAPTER 2^ 210.000^0.005^0.010^0.015x-1Figure 2.9 Tangential separationThe reason for having a teat was to force the fluid to separate tangent to the slot wall. If thefluid bends sharply upstream or downstream immediately after separating, this is not reallyaccomplished. Thus we choose the value of d (i.e., 6) that produces zero curvature of thestreamline at separation. Using numerical methods, we find 50.071, a reasonably small teat.2.2.3 ComparisonsFigure 2.10 shows a plot of the separating and dividing streamlines from three differentmodels. The curves denoted BUBBLE are from the bubble model described in Section 2.2.1and the curves denoted WAKE are from the wake-source model described in Section 2.2.2.The TEACH-II results are from the Navier-Stokes calculations described in Chapter 4 withCpt=0.0, Rh = ptloo h/g= 105, and a main stream boundary layer thickness at the upstream slotlip of about TiTh (h=slot width). These results were also used to supply both the separationregion size (L .10.6, H..1.2) for the bubble model and the separation speed (k=1.18) for thewake-source model.The TEACH-II results show a much wider downstream spacing between the separatingand dividing streamlines than either inviscid model. In the TEACH-II case, viscous action3.0^ BUBBLE- - - WAKE^ TEACH-II(Rh=105 ) 5.0 10.0CHAPTER 2^ 22causes boundary layers to form which act to slow the fluid motion between the streamlines.Thus, to conserve the mass flow between these streamlines, the distance between them mustincrease.Figure 2.10 Streamline comparisonsThe mass flow ratios for each model are listed in Table 2.1. Also given are the isoenergeticKutta condition value from Section 2.1.1 and the experimental value from Fitt et al. Theexperimental conditions were roughly the same as those used in the TEACH-II calculations.Model MAttached Flow 1.00Bubble 0.54Wake-Source 0.63TEACH-II 0.67Experiment 0.65Table 2.1 Separation model mass flow ratios-0.5 -1 p 2— U2 00 0020155^10rFigure 2.11 Centerline pressure distribution'11-""""[0.5,-5.0]--1 -- - - ATTACHEDBUBBLEWAKE^ TEACH-II0.5-1.5 -CHAPTER 2^ 23The results from both separation models show a significant improvement over the non-separated case. The predicted mass flow ratios from the wake-source and bubble models are ingood and fair agreement respectively with the viscous results even though the streamlinepatterns are significantly different. We can gain some insight into the physics by examiningthe pressure field encounted by a fluid particle leaving the slot.Figure 2.11 shows a comparison of the pressure distributions along a streamlineemanating from the slot centerline deep within the slot. The variable r measures the distancealong the streamline starting from a point five slot widths into the slot.For the attached flow case, the graph shows that the static pressure deep in the slot (p s) is equalto the recovery pressure (p c0). This result can be verified by applying Bernoulli's equationalong the streamline with M=1. Separation reduces the flow rate and results in the slot staticpressure being above the recovery value. In all models, the mainstream flow acts like a 'lid'over the slot exit, greatly restricting the effective flow area. As a result, the flow leaving theCHAPTER 2^ 24downstream region of the slot is accelerated (the fluid is also accelerated as it turnsdownstream), corresponding to the significant pressure reduction shown in the diagram. As theflow moves downstream, the streamlines widen (see Figure 2.10), and the pressure rises to thefree stream value.All three inviscid models produce a nearly uniform pressure in the slot. In the TEACH-II case, boundary layers growing on the slot walls cause a slight acceleration of the centerlineflow and consequently a slightly favorable pressure gradient. The pressure also recovers moreslowly in this case because some of the kinetic energy is converted to turbulent energy.In the bubble model, the recovery portion of the curve has a large kink. This is causedby the rounded shape of the downstream portion of the bubble leading to a strong stagnationpoint. The weak stagnation point at the front of the bubble appears to have a negligible effect.Overall, the wake source model gives better agreement with the viscous results, probably dueto a more realistic pressure distribution.This completes the isoenergetic analysis of ideal slot flow. The following chapterdescribes the development of a new non-isoenergetic analysis, guided in part by the physicsrevealed in this chapter. In addition, we will show how the preceding isoenergetic solutionscan be easily transformed to produce solutions for the non-isoenergetic case, Cpt=0, Dol.Chapter 3NON-ISOENERGETIC FLOWIn this chapter we describe a new procedure for calculating non-isoenergetic inviscid slot flowfor general values of 3, Cpt, and D. The first part of the chapter explores the limitations ofclassical theory towards such a solution. We also consider a possible alternative to the currentmethod that makes use of the transform plane from Chapter 2. The central part of the chapterincludes details of the current method such as solution methodology, boundary conditions,numerical approximations, and limitations. In the final part of the chapter we examine thepossibility of adding a separated flow model.3.1 Conformal Mapping LimitationsThe conformal mapping techniques used in the previous chapter provide an elegant method ofsolution under the restrictions of isoenergetic flow. In the non-isoenergetic case however, thevelocity field is discontinuous and we can no longer apply these methods directly. This is bestexplained by considering the hodograph (complex velocity) plane for the current geometry.Figure 3.1 on the following page represents the hodograph plane for the case Cpt<0 (see Figure2.1 for a reference to the physical plane). Although the hodograph planes for Cpt>0 and Cpt<0are different, we have arbitrarily chosen the latter case to illustrate the principles involved.Here, and in subsequent sections of this chapter, we assume that the origin of the dividingstreamline is at the upstream slot lip (i.e., the Kutta condition is in effect), and that all25main stream-iv[1-CptDslot stream1/21/2OW.cptL DCHAPTER 3^ 26velocities are scaled on the free stream value. For this discussion, we also assume that the flowdoes not separate from the downstream slot lip.Figure 3.1 Hodograph plane for Cpt<0The superscripts + and - along the upper and lower surfaces of the dividing streamlinecorrespond to the main stream and slot fluids respectively. At an arbitrary point P along thestreamline, the flow is tangent to the streamline, so that a line drawn through P i" from P" mustpass through the origin. To find the magnitude of the discontinuity in speed at P, we useBernoulli's equation to write an expression for the continuity of static pressure at P, i.e.,1 I - -1Ps (V ) 2^- iPco(V+ )2where V+ and V- are the magnitudes of the velocities at e and /7" respectively. Using thedefinitions of Cp and D this expression becomes,(V+ )2 - D x (V- )2 Cpt^ (3.1)This equation clearly shows that the velocity field is continuous only when Cpt-0 and D=1.CHAPTER 3^ 27In the classical solution procedure, we open up the pie-shaped hodograph plane into theupper half plane using the transformation =-w ' 03, and then map the plane to the complexpotential plane (F=13-i-iT) using a Schwarz-Christoffel transformation. These planes can thenbe related to the physical plane by,dF dF gdF 13-31 g= w or dz = — = —(-)dz w In the isoenergetic case, this equation can in principle be integrated to give z as a parametricfunction of . The difficulty with the non-isoenergetic case is that a single point along thedividing streamline in the z-plane maps to two distinct points in the -plane, violating the basisof conformal mapping theory.One of the most recent works involving hodograph planes is by Goldstein and Braun(1975). They found non-isoenergetic solutions for various cusp-lipped orifices for therestricted range 1CpA«1. The solution involved transferring the non-isoenergetic boundaryconditions on the dividing streamline to the known isoenergetic shape (in a manner similar tothin airfoil theory). They avoided some of the restrictions placed on conformal mappingtheory by using sectionally analytic functions and by introducing new independent variables. Itdoes not seem possible to extend their method to the general case because it is based onperturbations of Cpt near the isoenergetic value.3.2 Vortex SheetsAware of the difficulties with hodograph methods, we initially explored an alternative methodthat makes direct use of the -plane (see Figure 2.2) from Chapter 2. The idea was to replacethe dividing streamline in the non-isoenergetic case with a vortex sheet. Since a vortex sheet isdefined to produce a discontinuity in tangential velocity, it seemed possible, in principle atleast, to find a solution if the strength distribution and position of the sheet were known. Theintroduction of an image vortex sheet would satisfy the solid wall boundary conditions, and wecould use (3.1) to find the distribution of vortex strength along the sheet, i.e.,y(x) = V+ (x) -17- (x) = 11Cpt + DIV- (x)}2 - V-(x)CHAPTER 3^ 28To find V-(x), however, we need to know the position of the vortex sheet and its strengthdistribution y(x). This leads to a non-linear implicit solution formulation. In addition, sinceV-(x) is the physical plane velocity distribution, we must involve the Schwarz-Christoffeltransformation function when specifying the -plane complex potential, thus increasing theimplicit nature of the solution. It seemed unlikely that we could find a solution without the aidof iterative numerical procedures. This method was subsequently dropped in favor of thecurrent method.For the past thirty years, researchers have applied classical methods to the non-isoenergetic cross flow problem, demonstrating the difficulty in finding a solution. In fact, allof the work done in the past two decades has involved numerical procedures, even though thesolutions were only valid for restricted ranges of Cpt. The remainder of this chapter describesa new method that yields general solutions to the non-isoenergetic cross flow problem.3.3 A New Zonal Interaction Method3.3.1 HistoryZonal interactions originate from the application of boundary layer theory for predicting moreaccurate lift and drag coefficients on airfoil sections. To illustrate this concept, consider part ofan airfoil section with a corresponding boundary layer.Figure 3.2 Zonal modellingCHAPTER 3^ 29The flow field separates into two distinct zones, an internal zone near the body surface whereviscous effects dominate, and an external zone where the flow is essentially inviscid. The term'interactive' implies that the two zones influence each other; in steady flow the zones establishthemselves to provide a balance of static pressure at the interface. The viscous effects of theboundary layer displace the streamlines outward by a distance O *(x) (the displacementthickness), thus altering the original inviscid flow patterns. To find a solution, we must findthe location of the zonal interface, b(x), that provides equilibrium of forces between the zones.In this example, the boundary condition at the interface is the continuity of tangential velocity.The following steps are generally used to predict the flow field in the presence of a boundarylayer.[1] Calculate U0 (x), the tangential inviscid velocity distribution on the bodysurface without boundary layer effects. Set U(x)=U 0 (x), where U(x) isthe external velocity distribution at the edge of the boundary layer[2] Calculate O *(x) using the equations for the boundary layer zone and thepreviously calculated distribution U(x).[3]^Using the inviscid zone equations, calculate a new external velocitydistribution U(x) based on the original airfoil having a surfacedisplacement of O *(x). Repeat steps [2] and [3] until convergence.In recent years, researchers have applied this method to predict separated flow from steppedsurfaces, surfaces with large negative pressure gradients, and airfoils at large angles of attack.This concept can be generalized to include the interaction between any two regions, providedthat the governing equations for each zone are known and boundary conditions at the interfacecan be specified.3.3.2 Application to the Current ProblemThe current problem is ideally suited to zonal analysis because the dividing streamlineseparates the flow field into two distinct zones, and equation 3.1 provides the boundaryconditions along this interface (from this point forward, we will use the terms 'zonal interface'and 'dividing streamline' synonymously). Figure 3.3 illustrates the division of the flow fieldinto an internal flow region containing the slot fluid, and an external flow region containingthe free stream fluid.CHAPTER 3^ 30Figure 3.3 Oblique slot zonesEach region is isoenergetic, and can be solved independently using existing inviscid flowsolution procedures. Determining the interface shape, S(x), that satisfies the boundaryconditions (3.1) is the essence of the problem. To find S(x), we adopt a similar solutionmethodology as for the airfoil problem above. A few modifications are made to the airfoilalgorithm because in addition to S(x) being unkown, the mass flow ratio M is also unknown.We have applied the Kutta condition by fixing the origin of the dividing streamline at theupstream slot lip (see Figure 3.3), so that the algorithm must produce M=Mk as part of theCHAPTER 3^ 31solution. The complete slot flow algorithm is given in Table 3.1. The details of the algorithmare discussed in subsequent sections.[1] Set p, Cpl, and D.[2] Guess an initial dividing streamline shape S(x) and mass flow ratio M.[4] Using an inviscid internal flow solver, find the distribution of velocityalong S(x) for the internal region, 17-(x).[5] Use (3.1) to calculate the distribution of velocity along the interface forthe external region, V+(x).[6] Using an inviscid external flow solver, find a new shape S(x) thatproduces the external velocity distribution V+(x) found in step [4].[7] Adjust M to satisfy the Kutta condition at the upstream slot lip.[8]^Repeat steps [3]-[7] until the solution converges.Table 3.1 Non-Isoenergetic solution methodology3.4 Defining the Zonal InterfaceFigure 3.4 Spliced cubic polynomialsThe majority of the internal and external inviscid solvers referred to in Table 3.1 useapproximate (panel) methods for solution, replacing curvilinear boundaries with linear orpolynomial segments. Considering this, it makes sense to approximate S(x) since it is part ofthe boundary for each zone. In his M.A.Sc. thesis, Stropky (1988) used a method ofCHAPTER 3^ 32approximating a zonal interface with a series of N spliced cubic polynomials. The applicationof this procedure to the current problem is shown schematically in Figure 3.4, with theapproximate curve S (x) replacing the exact shape. The coefficients of the cubic segments aregiven in Appendix B.1. S (x) is defined to be continuous in both value and slope, and thus bydecreasing the lengths of the cubic segments (i.e., increasing N) we can represent the exactsolution more accurately.The true dividing streamline extends downstream to infinity, so obviously we cannotapproximate the entire curve with spliced cubics. We can, however, reasonably represent thesection far downstream of the slot (x>x E) by setting §(x) equal to the asymptotic height of S(x)in this region. The asymptotic height, h E, is found by writing Bernoulli's equation and thecontinuity equation from a point deep in the slot (where the flow is uniform) to a point atdownstream infinity (where the flow is uniform and the static pressure is 1,0 ).Bernoulli: P. = Poo + -IP sUEMContinuity: —D x 1= UE x hECombining these equations we have,ill ''' ch.—as UE = ^DhE ....^M2 ^1 D 1–C t(3.2)Since the true flow at x=x E is nearly uniform and parallel to the x-axis, we introduce only asmall error with this approximation. We can estimate the effect of this approximation bychanging the location of x E. The region OsxxE, termed the 'interaction region', is the regionin which the zonal boundary conditions are satisfied.As an approximation to the exact shape, S (x) cannot satisfy the interface boundaryconditions at every point. Following Stropky's method, we choose to satisfy the boundaryconditions at the endpoint of each cubic segment. Thus, the cubic endpoints can beconcentrated in areas where the velocity gradients are high to ensure solution accuracy. In thenext section we will discuss the application of the Kutta condition using the current zonalmodel.^ Analytic-.- -.- Zonal Model\CHAPTER 3^ 333.4.1 Applying the Kutta ConditionFrom the isoenergetic results in Chapter 2, we found that (for fixed 3) the Kutta condition isonly satisfied by a single mass flow ratio M=Mk. By satisfying the boundary conditions (3.1)only at the cubic endpoints, we are able to generate solutions for mass flow ratios other thanM=Mk. One such solution is shown in Figure 3.5 for the case Cpt=0, D=1, 13=7r/2, M=0.5.Figure 3.5 Kutta condition problemsThe analytic theory for this case gives a stagnation point location z s=(0,-0.14), and a Kuttacondition mass flow ratio Mk=1. The solution given by the zonal model is obviously invalidbecause we had preset the Kutta condition in a case where MoMk. To find the value of M thatsatisfies the Kutta condition, we need one additional boundary condition.The results of Ting and Ruger, together with the isoenergetic results of Chapter 2 showthat the initial slope, 0 0, of the dividing streamline is given by,0, cpt >00/2, Cpt — 00, cpt < o (33)This equation provides us with the extra condition necessary for a correct solution. Whenapplied, however, this condition produces an unreliable relationship between 0 0 and M. Insome cases we observed the same value of 0 0 for two values of M that differed by more thanten percent. This behavior is most likely due to the local nature of the boundary condition(3.3), coupled with the solution approximation. Fortunately, using (3.1), we can transform00 =CHAPTER 3^ 34(3.3) into a boundary condition involving velocity. The governing equations for velocity areelliptic, thus the value of the velocity at any point depends upon the shape of the entireboundary (i.e., the entire curve §(x)). This transformed condition is precisely the sameboundary condition used at each cubic endpoint. For the case Cpi<O, the streamline leavestangent to the slot, so that the velocity of the slot fluid at the upstream slot lip, V0, is non-zero.Similarly, for Cpt>0, the velocity of the main stream fluid at the upstream slot lip, V0+ , is alsonon-zero. Table 3.2 illustrates these conditions, and gives the values of V0+ and V0. We cannow find the correct solution by satisfying the interface boundary conditions (3.1) at the cubicendpoints and at the stagnation point.Cpt < 0 Separating  :::::::Vo+ \ 00-CD +(V+)2pt^°II^D^,From^VD-(3.1),^=..VV >0, ...130 = 1 3 , . . Vo+ = 0, or, 13 \i\ o-' v_l io11-Cpt•• DCpt > 0From (3.1),^V0+ = .1Cpt +D(V0- )2 ,:.V>0, : .0 0 - 0, : .V; = 0, or,170+ = IfCpt-Table 3.2 Upstream slot lip boundary conditionsFor the isoenergetic case we know that 0 04/2 and IC - Vo+ = 0. Because this is true for anyangle 0<00<13, we are forced to use the original boundary condition (3.3) at the upstream slotCHAPTER 3^ 35lip. Fortunately, in this particular case the M-0 0 relationship is monotonic and well behaved sothat we can find the correct solution.For the case Cpt=0, Dol, we cannot use the isoenergetic results from Chapter 2. Weknow, however, from (3.1) that 170+ = -IT V0-", so again the upstream slot lip must be astagnation point for both fluids. In Appendix B.2 we show that the initial slope of the dividingstreamline is 13/2, regardless of the value of D. Thus we can again use (3.3) as a boundarycondition at the upstream slot lip.Having solved the Kutta condition problem, we can now discuss the remaining detailsof the solution algorithm given in Table 3.1. In steps [4] and [6] of this algorithm, we discussusing inviscid flow solvers for each region of the flow field. In the next two sections, we willoutline the theory for these solvers and show how they are applied to the current problem.3.5 The Internal Flow Region3.5.1 Bardina's MethodIn step [4] of the solution algorithm, we only require V-(x), the internal flow velocitydistribution along the zonal interface. It is therefore inefficient and unnecessary to compute theentire internal flow field at each iteration, as the intermediate results are not solutions to theproblem. Bardina et al (1982) have applied boundary integral methods to compute the inviscidvelocity field around the perimeter of confined channel. The theory is based on the Plemeljintegral formula and is related to panel methods used in external flow. No coordinatemappings are needed since the problem is solved in the physical plane.To apply Bardina's method to the current problem, we discretize the internal flow fieldboundary with NB points, denoted by the complex coordinates zi=xi+iyj , j=1,2,...,NB (see Figure3.6). Along the dividing streamline, these points correspond with the location of the cubicendpoints. Next, Bardina shows that any analytic function j(z) that is analytic inside and on theboundary satisfies the following set of equations on the boundary.NBE Aki g(z j) =0, k = 1, 2, ..., NBj=1^(3A)CHAPTER 3^36where,-zk^zk^zk^zk -ziAki -^ In   In ^-zi^zk -zi^zi -zi_i^zk -zj_iAkk = In {zk -zk+i zk - zk-1Figure 3.6 Bardina's methodTo find the velocity distribution on the boundary, we set g(zj) = In[Vn- iaj, where yr isthe magnitude of the velocity and aj is the flow angle. Substituting into (3.4) and taking theimaginary part of the result,NB r^NB r^,E 21Ak;^- attAki ai,^k 1,2,...,NBJ=1 (3.5)The flow angles, oci, are specified as tangent to the boundary, except at the inlet and outletwhere they are normal to the boundary. For the problem to be well posed, we must specify theCHAPTER 3^ 37value of V- at one point on the boundary. Here we set V1 = M, making sure that the entrancelength (L1) is long enough for this condition to be accurate. To increase the computationalspeed of the method, we need only to recompute the AkJ along the dividing streamline duringeach iteration cycle.3.5.2 Sridhar's MethodBardina's method, although an efficient algorithm for computing the velocity distribution alongthe dividing streamline, does not directly yield the velocity field interior to the boundary.Once we have computed the final shape of the zonal interface, we can solve the interior regionusing a potential field method developed by Sridhar and Davis (1985). The method consists ofreplacing the channel walls with polygonal surfaces, and then mapping the flow field to astraight channel (the x-plane) using the following Schwarz-Christoffel transformation,d.z^K m-1^d^ = K g(x)x^NUn [cosh 1 (x - bn )]n=1 (3.6)where bm is the location in the transformed plane of the mth corner on the lower channel wall,am is the clockwise turning angle at the mth corner in the physical plane. The constants bn andan are similar quantities for the upper channel wall. The constant K for the current problemcan be shown to be,K =where, as before, hE is the asymptotic height of the dividing streamline. The unknownquantities bm and bn are found from the following set of equations,zk.i - zk = K fXk+1 g(x)dxXkwhere k refers to either the upper or lower surface. Sridhar has developed an efficient iterativealgorithm for finding the solution to this non-linear equation set. Floryan (1985) noted thatSridhar's method contains an error, which was also discovered and corrected in the applicationNL^amisinh 2(x - bm )1CHAPTER 3^ 38to the current problem. Floryan's method allows for the use of curved elements along thechannel boundaries, and can be used in place of Sridhar's method.Once the values of bm and bn are known, we can then use (3.6) to find the velocity fieldin the physical plane, i.e.,dF dF/ dzw(z) = u – iv = =— — —dz dX dXThe complex potential in the straight channel x-plane is simply F(x)=Mx, therefore,^NU^ann[coshs(x_bld 3w (z) _ M n=1 ^Ohg NI,^amn [sinh 2(x -bm)] 3m =1 (3.8)In the next section we will discuss the external flow solvers used in step [6] of the solutionalgorithm in Table The External Flow RegionFigure 3.7 The external flow fieldFigure 3.7 shows the external flow region for the current problem. The lower boundaryconsists of the mainstream wall and the approximate interface curve, S (x). The control points,CHAPTER 3^ 39k=1,2,...N, correspond with the endpoints of the spliced cubics, and the tangent angles ak arefound from the slope of S‘ (x) at x=x k .In step [6] of the solution algorithm, we need to find the boundary shape S (x) thatproduces the velocity distribution V+(x) computed in step [5]. Most external solvers work inthe direct mode; they compute V+(x) given § (x). Stropky (1988) has developed a method forfinding the inverse solution using these direct solvers, and in the next two subsections we usehis method to find S (x) given Vf(x).3.6.1 Linearized Theory, the Hilbert IntegralSignificant improvements in computational speed can be realized by using linearized methodsin step [6] of the solution algorithm. Fitt et al have successfully applied these methods for thecase of a vertical slot with low mass flow ratios. The linearized results are valid for M«1because the injection process does not cause significant curvature of the main flow streamlines.Similarly, linearized theory is valid in the external flow region of the present case for low massflow ratios and small slot angles.From linearized potential theory, the interface velocity can be written as the sum of theundisturbed and perturbation velocities, i.e.,V+ (x) – Vu+ (x) + Vp+ (x)where VI-11- (x) is the velocity distribution in absence of the injection process, and VP+ (x) is theperturbation velocity. For incompressible flow, the Hilbert integral from thin airfoil theorydefines the perturbation velocity, i.e.,+^1V (x) – —Pi +33 1{§() 17+(0x–^4With Vu+ (x) =1 and V+P (X) << V+ (x), the preceding two equations can be combined to giveuthe velocity distribution at the cubic endpoints,0CHAPTER 3^ 40d {§ (0V + (xk) =1+1^4 xic0To invert (3.8), Stropky (1988) uses the following procedure. Each cubic endpoint p is given asmall vertical displacement (ey p). The corresponding change in V+(xk) due to the perturbationof all endpoints is written as,N reV+ (xk)= EiC2k,p AYp }p=1 (3.9)The method for calculating the perturbation coefficients, C24, is given in Appendix B.3. Aninitial guess for §(x) produces an initial velocity distribution Vg+ (xk ). The correspondingendpoint displacements necessary to produce the desired velocity distribution V 4-(xk) are thencomputed from,N rAV+ (xk)= V+ (xk) - Vg+ (xk)= 21.Qk,p AYp}p=1 (3.10)Equations (3.10) are solved by ordinary linear matrix methods to yield the eyp. These valuesare added to their corresponding endpoint heights and the process is repeated until the desiredvelocity distribution is found to within a prescribed tolerance. The details of the procedure aregiven by Stropky (1988), and are repeated in brief in Appendix B.3. The perturbationcoefficients, C24, calculated from (3.8) can be used directly to find §(x) for any externalinviscid solver.3.6.2 Davis's MethodThere are three basic problems with using the Hilbert to compute the external flow.(i ) At higher mass flow ratios and slot angles, the basic assumptions oflinearized theory are violated, and solution accuracy degenerates.(ii) For Cpr<O, the external flow has a stagnation point at the upstreamslot lip. Since linearized theory fails completely at such a point, the(3.8)CHAPTER 3^ 41solution near the upstream slot lip may be grossly in error, even forlow M.(iii) The Hilbert integral only gives the velocity distribution on theboundary.Davis (1979) has developed a potential field method based on numerical integration of theSchwarz-Christoffel transformation for general curved surfaces. The surface is subdivided intoNE curvilinear elements, with the possibility of a finite slope change between each element(i.e., a corner). The transformation, which maps the physical plane to the upper half plane, isgiven by,dZ^1 NE^O—•g = K exp{— [1n(t- ^bm ) Om + fomm 1^b)dcd}m=1 (3.11)where bm is the location in the transform plane of the mth element endpoint, Om is thecorresponding finite slope change (if one exists), and 4 is the tangent angle of the physicalplane curve. For element m, 4 is related to b by,(1) = Cim + C2m b + C3m b2Substituting into (3.11),dz^NE=K1-1{(--bm )Omu`7'^m=1}C2m C3mN 2.11 ^N3n7(3.12)The coefficients C2m, C3m, N2m, and Kam are given in Appendix B.4 for reference.Following Davis, the velocity field is then given by,w(z) = NE Am(^- bm )bm111=1 }C2m C3mN 2m N n3 m(3.13)The unknown quantities K and bm are found using an iterative procedure developed by Davis.This procedure was subsequently modified by Shridhar for use in his method (see Section3.5.2). Davis's method is directly suited to the current problem because the coefficients givenCHAPTER 3^ 42in Appendix B.4 are essentially derived from a curve made up of continuous cubic polynomialsegments.As mentioned above, the coefficients 52k calculated from the Hilbert integral andgiven in Appendix B.3, can be used in Equation 3.10 to find the desired shape §(x). Thisconcludes the discussion of the flowfield solvers. In the next section we discuss the effects ofthe density ratio on the solution.3.7 Density Ratio EffectsIn the following discussion we indicate how the solution for a given density ratio, D=D i , canbe used to produce the solution for a different density ratio, D=D2 (for fixed 1 and Cpt). Forthe case D=Di , we can write (3.1) as,(VA)2 — x (VD1 )2 + CptSimilarly, for D=D2,(Vii; )2 D2 x (VE2 )2 + CptNext, we make the assumption that, by fixing the dynamic head of the free stream (i.e.,21-pco U„, = constant ), the shape of the dividing streamline is not altered by a change in thedensity. Since the governing equations for velocity are linear, the scaled external velocitydistribution remains unchanged, i.e.,Di D2^ 11D2V+ = V+ or,^VD2 = VDiThis equation shows that the internal flow velocities along the dividing streamline are simplyscaled by a factor (D i/D2)'. Again, because the equations are linear, we can achieve this byscaling the velocity flow rate from the slot by the same factor, i.e.,Di^Uc 1^'= Uc2 x 11—' D2DiCHAPTER 3^ 43Since M=Us Ps, this equation can be manipulated to give the relationship between the massflow ratios, i.e.,M2 = Mlx 11D2Di (3.15)In summary, the scaled external velocity field is unchanged (including the shape of thedividing streamline), and the entire internal velocity field is simply scaled by (D i/D2) 1/2. Theseresults can also be used in conjunction with the results of Chapter 2 to find an analytic solutionto the non-isoenergetic case, Cpt=0, Dol. The results of Appendix B.2 are also verifiedbecause the entire shape of the dividing streamline is unchanged, including the initial slope.This completes the discussion of the details of the new solution procedure. Results forvarious values of Cpt, 13, and D, including comparisons with other methods, are presented inChapter 5. In the next section we discuss the addition of a separated flow model to the presentprocedure.3.8 Separated FlowFigure 3.8 A simple separation modelIn Chapter 2, we modelled separated flow by adding sources and sinks of suitable strengths andlocations. These singularities modified the complex potential, but the solution procedure wasessentially unchanged. Using Shridhar's internal flow method, we could, in theory, add sourcesand sinks in a similar manner to find a solution. The transformation function in this case isCHAPTER 3^ 44much more complex however; we would have to employ numerical methods from the onset ofthe solution procedure. This analysis is beyond the scope of the present work.We have developed a much less sophisticated model for approximating the effects ofseparation. Similar to the bubble model of Chapter 2, this model requires input in the form ofthe length and width of the separation region. The method essentially consists of modifyingthe lower boundary of the internal flow region to simulate the effects of the separatingstreamline. This concept is illustrated in Figure 3.8.The separation bubble is constructed from two spliced cubics; the first separatessmoothly from the downstream slot lip and splices into the second which subsequently attachesto the downstream wall. The location of the join between the cubics can be adjusted to createdifferent bubble shapes. The method is easy to implement as the bubble shape (i.e. theboundary) remains fixed throughout each iteration of the solution algorithm. Results andcomparisons with other methods are given in Chapter 5.Chapter 4TEACH-IIIn the previous chapter, we developed a new method for computing the steady inviscid non-isoenergetic flow from an oblique slot into a main stream. The results from previous work arelimited to restricted ranges of Cpt, thus we have no basis for a general comparison of resultsfrom similar methods. We can, however, obtain comparative results from other sources suchas experimental data or numerical Navier-Stokes solvers. Comparison with these methods alsoreveals the effects of the approximations used in the present theory. In this chapter, we employa finite-difference Navier-Stokes solver (TEACH-II) developed by Benodekar et al (1983) toproduce results for a vertical slot. We exclude oblique slots to avoid a 'staircased'representation of the slot produced by the cartesian coordinate system. In addition, calculationsfor density ratios other than D=1 are beyond the scope the current work, therefore the resultsare also limited to the unit density ratio case.To simulate the conditions found in most secondary injection devices, we assume thatthe flow is turbulent. The TEACH-II code uses a two equation (k-e) model of turbulence toapproximate the true characteristics. In this model, the concept of turbulent viscosity, g t, isused to relate the kinetic energy of turbulence, k, and its dissipation rate, e. The turbulentviscosity is combined with the laminar viscosity to yield an effective viscosity for use in thetransport equations, i.e.,Reif = Pi + R45CHAPTER 4^ 46Following Leschziner and Rodi (1981), the code has been modified to account for thepreferential influence of normal stresses in promoting the transfer of turbulent energy fromlarge to small eddies. The following is a short discussion of the methods used by the code. Itshould be noted that the TEACH-II code was developed for internal flow and is a fully ellipticmethod. More efficient parabolic-elliptic external flow solvers exist, but they are notconsidered in this study.4.1 Governing EquationsFor steady turbulent flow, the equations to be solved can be summarized by a single equationof the form,a^a^aP (r 21)  ^(r -la )— so =0— (4) + P-14) – — 4) + — 4)ax^ay ( ax^ax^ay^ay^,..„_,convection terms^diffusion terms^sourceterms (4.1)where u, v, and 4) (a general transport variable) are the mean flow quantities. Table 4.1 lists theexpressions for f4, and S4 for the various transport equations.Transport Equation 4) ro SoMass 1 0 0x-momentum u Refl. ap^a–^+ (^au) + atj, eff ax )^ay(^av)i, ffax^ax e^axy-momentum v Reif + (Reif au)ay^ap^a^avj + a– ay^ay (veff ay^axTurbulence Energy k !Leff tttGk –peokDissipation Rate E lieff c^,^c 2ci. it —k Ge – L-2 P—kaeTable 4.1 Governing equationsuPN^NEW^ EPs•S^SENWSWCHAPTER 4^ 47where,CIA k2P lit - EGk = 2 [(au 2 + av 2 + au + av 2ax ay ay^axG E = Ci Gk - CrbqsC IA = 0.09, C1 =1.44, C1= 2.24, Cf = 0.08, C2 =1.92K 2al( = 1.0, cre =^ , K = 0.4187[(C2 - C1 ) C11/2 ]and S, is the shear strain in the direction of the streamlines (see Leschziner and Rodi).4.2 Numerical Solution ProcedureFigure 4.1 TEACH -II grid cellsThe principle of TEACH-II is to divide the computational domain into a discrete number of'cells' or control volumes. A scalar control volume has eight adjacent neighbors (NW, N, NE,CHAPTER 4^ 48etc.), four faces (n, s, e, w), and two corresponding velocity cells (u and v). The velocities areevaluated on the boundaries of the scalar cells, forming a staggered arrangement of controlvolumes (see Figure 4.1). The scalar cells are located such that any cell in contact with thephysical boundary will have its face(s) coincident with the boundary.The next step is to use the divergence theorem to integrate (4.1) over the controlvolume, i.e.,Ynfpu+ — -axYsxexe ad)}dy + f {Pv4) rx,Yndy s ffS4) dx dyYs^CV (4.2)Details of the control volume formulation can be found in the text by Patankar (1980). Theequations are discretized using a second order accurate bounded-skew hybrid differencing(BSHD) scheme that takes into account the local direction of the flow. This greatly reduces the'skewness error' or 'false diffusion' produced by less sophisticated methods. The BSHD schemealso uses a flux blending technique to eliminate unphysical higher harmonics typical of higherorder schemes.The resulting finite difference equations are solved iteratively using the pressureimplicit split operator (PISO) algorithm which corrects the velocities and pressures using amodified form of the continuity equation. The code uses a line-by-line solution methodcombined with an efficient tridiagonal matrix solver.4.3 Problem GeometryFigure 4.2 on the following page is a schematic of the computational domain and associatedboundary conditions for the current problem. The mainstream and slot walls are non-permeable, and can be either free-slip or no-slip surfaces. Flow at the entrance to the slot isuniform, with negligible turbulent kinetic energy. The slot length is assumed to be longenough to be outside the influence of the slot exit. Slot wall boundary layer effects can beeliminated by using free-slip conditions on the walls.CHAPTER 4^ 49To study the effects of the main stream boundary layer thickness, we have imposed aturbulent boundary layer of thickness 6 0 at the inlet. Fitt et al found that this boundary layerhas a significant effect on the mass flow ratio when the thickness is of the same order as theslot width.The top wall is assumed to be far enough from the slot to simulate free stream theconditions. In addition, by implementing a zero pressure gradient condition, we eliminate anydownstream acceleration of the fluid along the top wall and better emulate the free streamcondition.At the exit, we use a standard zero streamwise velocity gradient condition to aidconvergence. For this boundary condition to be accurate, we must locate the exit sufficientlyfar downstream of any separation region that may exist. Use of a large solution domainimplies more accurate boundary conditions, however, to maintain solution accuracy, thenumber of cells needed may become prohibitively large.Figure 4.2 TEACH -II geometryStaggering of the velocity cells with respect to the scalar cells causes some difficulty atthe corners of the slot. The u and v cells that have half-faces in contact with a wall requireCHAPTER 4^ 50special treatment. Following Djilali (1987), the calculation of fluxes for these cells must usehalf the face area and a normal velocity equal to that at the outer edge of the half-face.For the majority of results presented in this study, the dimensions of the computationaldomain are given in Figure 4.2. The grid has at least 80 cells in the horizontal direction (withno less than 8 cells across the slot) and 60 cells in the vertical direction. The grid is non-uniform, with more cells concentrated in regions where the gradients are higher (i.e., aroundthe slot exit). Studies show a less than one percent change in the mass flow ratio when the gridwas increased from 80x60 cells to 120x80 cells.In the next chapter, we compare the TEACH-II results with the results from Chapter 3.Comparisons of separating streamlines, slot exit velocity distributions, and pressure fields aremade for different values of Cpt. The effects of boundary layers and flow separation are alsoexamined.o 0 000^)0000- ^eO^.LIN 0o°Z o 00 o:^000 0/0o °o0 08^XE^0 0 0 0 0 o o^o^o o o 0 0 0 *00000;3O00000000000000000000000000000000• ...Chapter 5RESULTSThis chapter contains a summary of results from the non-isoenergetic theory (NIT) developedin Chapter 3. In the initial part of this chapter, we assess the accuracy of the NIT solutions.Next, we analyze the results for various values of 13, Cpt , and D. Finally, the NIT results arecompared with results from Chapter 4, NIT solutions from other authors, and experiment.5.1 Grid RefinementFigure 5.1 The NIT gridThe NIT solutions are only an approximation to the exact results because [i] the solutiondomain is limited in size [ii] part of the zonal boundaries (the dividing streamline) isapproximately represented by a finite number of spliced cubic polynomials [iii] the boundaryconditions are satisfied only at a discrete number of control points on the zonal boundaries.51CHAPTER 5^ 52A typical distribution of control points (referred to as the 'grid') used by the NIT code isshown in Figure 5.1. The external flow solvers require control points only along the dividingstreamline, and the internal solvers require the entire array of nodes shown in the figure. Thegrid is concentrated near the upstream and downstream slot lips for reasons discussed inSection 5.1.2.The solution depends on both the size of the computational domain and the distributionof control points. In the next two sections we examine the effects of these parameters. For thepurposes of the following discussions we will assume a unit density ratio for all calculations inSections 5.1.1 and Computational Domain SizeThe size of the computational domain is a function of both the inlet length, LIN, and the outlet(interaction) length, xE. For the puposes of this discussion LIN is fixed at three slot widths(found using a procedure similar to the one discussed here) and x E is free to vary. Figure 5.2illustrates the effect of the interaction length on the zonal interface shape for a typical case.0^2 4 6 8 10xFigure 5.2 Effects of the interaction region lengthCHAPTER 5^ 53To enable comparisons with the exact result, we have chosen an isoenergetic case. The curvelabelled 'IT (isoenergetic theory) is the analytic result from Chapter 2. In both NIT cases, thegrid is sufficiently fine so that the shape of the dividing streamline shown in the figure does notvisually change by adding more points.The effect of a short interaction length is obvious as the dividing streamline is forced toreach the asymptotic height y=hE too soon. The effect appears to be local, however, as thesolution near the slot (the downstream slot lip is at x=1/sin40°.=1.56) is nearly identical to theexact solution. Both NIT cases give mass flow ratios within 1% of the exact result. In allcases examined, we found less than a 0.2% variation in the computed mass flow ratio forinteraction lengths in the range 10 s x E s 20 (using identical grid spacings in the regionOsxs6).5.1.2 Grid DistributionIn sections of the dividing streamline where the curvature is small, a relatively coarse gridprovides an accurate representation of the exact shape. Conversely, in regions of highcurvature, a fine grid is necessary to accurately model the exact shape. Regions of high slopealso require a finer grid than regions of low slope because the grid is defined using the x-coordinates of the cubic endpoints, leading to longer cubic segments for steeper slopes. Inaddition, in regions where the velocity gradient is high (i.e., near a stagnation point or corner),the grid must be concentrated to give an accurate representation of the velocity distributionsused in the calculations.We find that the solution (i.e., the value of M and the shape of the dividing streamline)is most sensitive to the grid spacing near the upstream slot lip. This is due mainly to twofactors: [i] The upstream slot lip is a stagnation point for either the internal region or theexternal region or both. [ii] The largest curvature generally occurs at or near the upstream slotlip (see Section 5.1.3).Figures 5.3 and 5.4 show the effect of grid size on the solution. The grid distribution isexponentially expanding/contracting (see Figure 5.1), based on width of the first cubic segment(ex i) and a final cubic segment width of hE/4. The finest grid (ex i=0.005) involves 40 nodesalong the dividing streamline, and the coarsest grid (ex 1 =0.5) has 12 dividing streamline nodes.CHAPTER 5^ 541.2e•----^0.8 -0^0.1^0.2^0.3^0.4LOC 1Figure 5.3 Effects of grid size, Cpt=00^0.1^0.2^0.3 ^0.4^0.5ex 1Figure 5.4 Effects of grid size, Cpe--0.1For the isoenergetic case (see Figure 5.3), the solution is fairly insensitive to the size of thegrid. Over the range 0.01s 6.3E15 0.5, M varies <15% in the 90° case, while for the 15° case, thevariation is about 1%. In addition, M becomes nearly grid independent for ex 1 <0.1 in either0.70.50.2CHAPTER 5^ 55case, and within 2% of the exact value for ex i<0.05. Plots of the dividing streamlines for theisoenergetic case show that the curvature near the upstream slot lip is small for all slot angles.This implies that the variation in M with grid size is solely due to the presence of the stagnationpoint. The turning angle of the dividing streamline at stagnation is larger for the 90° case, thusthe stagnation point is stronger and requires a finer grid to resolve the velocity gradients.For the nearly isoenergetic case (see Figure 5.4), the situation is quite different. Thesolution is now a strong function of the grid size, with about 50% variation in M in the range0.01s ex is0.5. In addition, unlike the isoenergetic case, M is a stronger function of ex i atsmaller grid sizes. This is undesirable as a grid independent solution cannot be found, andunfortunate because M appears to be a strong function of Cpt near Cpt=0 , especially for higherslot angles. Fortunately, these undesirable charactertics only occur for the nearly isoenergeticcase; we have found grid independent solutions for all cases where ICA Z 0.2.To explain the behavior of the near isoenergetic cases, consider the dividing streamlinesplotted Figure 5.5 for two different values of ex 1 .xFigure 5.5 Effect of cubic spacing on dividing streamline shape near CprOThe large differences in asymptotic height are a result of the large difference in calculated massflows, i.e., M=0.83 for ex 1 =0.005 and M=0.47 for ex 1-0.2. Since Cpt<0, the initial slope ofVo+ = eCpt= +62£2« 1CHAPTER 5^ 56the first cubic element is 15 ° , but this cannot be discerned from the scale of the diagram. Thestreamlines immediately bend downward, followed by a recovery to the asymptotic height.The interface boundary conditions of equal static pressure are satisfied in each case, but thecoarse grid streamline becomes highly distorted to satisfy these conditions.Figure 5.6 shows sketches of the dividing streamline near the upstream slot lip forvalues of Cpt just above and below the isoenergetic value. Also shown in the sketch are thenon-zero velocity values at the slot lip required to satisfy the interface boundary conditions.Figure 5.6 Sketch of dividing streamline shape for small values of CptFor Cpt---E2, the streamline leaves tangent to the slot wall, however it must turn sharply toproduce the (small) value for V0. Similar reasoning produces the sketch shown for C pAs c—.0, these local distortions cannot be revealed by the NIT code, even for grid sizes as fineas ex 1=0.001. In addition, for ICA <0.1, the solution algorithm would not converge for gridsizes smaller than ex i=0.01, making accurate predictions of M difficult. Fortunately, we caninterpolate both from plots like Figure 5.4 and the final M-Cpt curve for more accurate results.For rpti>0.3, the dividing streamlines do not contain any regions of high curvature and can beresolved accurately with a grid size of ex i- NIT ResultsThe algorithm given in Table 3.1 is valid over the entire range of each independent parameter,i.e., -co s Cpt 51, 0 5D 500, 0 5 f3 s n. To avoid the problem of multivalued coordinates usingCHAPTER 5^ 57the x-coordinate grid system, we limit the range of slot angles to 0 s 13 s n/2. We also limit Cp tand D to values that produce mass flow ratios in the range generally found in film coolingstudies.5.2.1 Unit Density RatioIn Section 3.7 we indicate how, for a given value of Cp t , the solution for a given density ratiocan be used to find the solution for a different density ratio. We will elaborate on this inSection 5.2.2, but here we assume D=1. This restriction is implicit in previous NIT work, and,because of its fundamental nature, this ratio is chosen as the basis for finding solutions ofdifferent density ratios.Figure 5.7 shows the NIT results for both external zone solvers, the linearized Hilbertintegral (HB), and Davis' Schwarz-Christoffel method (DB). In both cases, Bardina's methodwas used for the internal solver. Linearized theory should be effective for Cpt >0, becausethere is no stagnation point in the external flow field and the lower mass flow ratios imply thatthe main stream streamlines are not deflected appreciably. On the contrary, for Cpt s 0, thepresence of the stagnation point and the significant mainstream deflection (for larger slotangles at least) would suggest that linearized theory would be inadequate. The results show,however, that the linearized theory is accurate for Cpt s 0 as well. This is not too surprising,considering in aerodynamics how well linearized theory works outside the expected range ofvalidity. In the 90° case, results could not be generated for Cpt <0 because the intial slope ofthe streamline is n/2, which cannot be represented by a cubic polynomial. Davis' methodmakes indirect use of the cubic segments (only the endpoint slopes are required), so we cangenerate the desired results without specifying the polynomial coefficients.The trends are as expected, with increasingly negative values of Cpt producing highermass flow ratios because of the increased energy of the injectant. For higher mass flow ratios(>-1.0), larger slot angles produce larger mass flow ratios for the same Cpt. The trend isreversed at lower mass flow ratios (<-0.6). In Section 2.1.1 we explained the non-monotonicbehavior of the M-f3 curve. The same arguments apply here, except the more energy the slotfluid has, the more easily it overcomes the adverse pressure gradient at the upstream side of theslot. In fact for Cpt <0, the slot fluid stagnation point disappears, decreasing the strength of theadverse pressure gradient, and producing a monotonic M-t3 relationship.3.5 ^2.5 -3.0 - -+-  HB-o-- DB2.0M1.5 -1.0 -0.5 • 1=15^0.0^-3.0^-2.0^-1.0^0.0^1.0CptM3.0 -2.5 --+-  HB-o-DB2.0 -1.5 -1.0 -0.5 - 1 13=40°0.0^-3.0^-2.0^-1.0^0.0^1.0Cpt-3.0^-2.0 ' - 1.0^0.0CptM1.0^-3.0^-2.0^-1.0^0.0^1.0Cp tMCHAPTER 5^ 583.5 ^Figure 5.7 M -Cpt results for linear and non -linear theoryFor Cpt >0, the converse arguments apply; the turning angle at stagnation for the slot fluid issmaller for larger slot angles, thus increasing the adverse pressure gradient on the upstreamside, and decreasing the flow from the slot. In the next section we will discuss the M-Cp trelationship in relation to a non-unit density ratio.5.2.2 General ResultsIn Section 3.7, we showed that, for fixed values of 13, Cpt, and the free stream dynamic head,the following relation holds true,^0 ^ 0.25- -0^ 0.5^--A^ 0.75-0^ 1.0- -*^ 1.5- I^ 2.0--A^ 3.0CHAPTER 5^ 59M2- - constantD (5.1)We can use this relation to easily develop M-Cpt plots for various density ratios. Figure 5.8illustrates the results for a typical slot angle. The curves were generated by using the resultsfrom the unit density case in conjuction with Equation 5.1.-3.5 -3^-2.5 -2 -1.5 -1^-0.5^0^0.5^1CptFigure 5.8 M-Cpt results for various density ratiosWe can also relate Cpt to M2/D by expanding the definition, i.e.,Ptco — Pts 1 {Ps — Pco  M2^M2Cpt =  , ^= J. "" ,^,^= 1 — CPs1-p.14, plooU,,,,`^ DD(5.2)CHAPTER 5^ 60Equations 5.1 and 5.2 show that, for fixed 13 and Cpt, the static pressure drop felt by the slotfluid is constant, irrespective of the density ratio. Figure 5.9 summarizes the relationshipbetween M, (3, Cpt, and D, using (5.1) to collapse the data.M2D -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0^0.5^1CptFigure 5.9 M2/D vs CptFor negative values of Cpt, the curves are nearly linear, implying from (5.2) that Cps isa linear function of M2/D. In this linear region, we can describe the curves by,D= K1 — K2 Cptwhere K1 and K2 are positive constants that depend on the slot angle. Combining thisexpression with (5.2), we arrive at the following,Pts — Pco 1 I+ 1— j/M2—^= A +^— CdP s Us2 K2^K2^D^M2 1M2D (53)CHAPTER 5^ 61This is precisely the empirical relationship developed by Gartshore et al in their paper ondischarge coefficients. They developed this simple relationship by considering the componentforms of the pressure drop experienced by the slot fluid during the injection process.Comparisons to their results are made in Section 5.4.At lower values of M2/D, the variation with Cpt becomes nonlinear, and the losses are are nolonger described by (5.3). We can, however, use the discharge coefficient defined by (1.1) tolinearize this region. We can write Cd in terms of M2/D as follows,M2"2^1 n 2D 2 ^'-' s -2' t'su SCd = pt —Poo — =s ^Ps + -IP stJ — Poo M2 Ps — Poo 1 +1- Ps D 1p.U!or, using (5.2),M2Cd —II  D 1—CptIt is interesting to note that the right hand side of this equation is identical to (3.2), so that thedischarge coefficient is equal to the asymptotic height of the dividing streamline. Figure 5.10gives Cd in terms of Cpt for various slot angles. The discharge coefficient is often plotted interms of M, so we have included a plot of Cd vs M2/D in Appendix C (Figure C.1) forcompleteness.Plotted in this manner, the region of low mass flow (Cpt—ol) is now linear, and the curves inthis region can be described by,Cd — K3 (1 — Cpt )where, similar to above, K3 is a positive constant that depends on the slot angle. Combiningwith (5.4), we have,M — K3 (1— Cpt )T.11-15 (5.5)(5.4)CHAPTER 5^ 62This is precisely the relationship developed by Fitt et al (in their case .0'1) using linearizedtheory to describe the (low M) flow out of a vertical slot into a cross stream. Fitt predicts avalue of 1.12 for K3 which compares favourably the value of 1.0 taken from Figure 5.10.0 ^,-3.5 -3 -i5 -2 -1.5 - .1. -0.5 0^0.5CptFigure 5.10 Cd vs CptFor ideal flow from a slot into a stagnant fluid, the discharge coefficient, Cd, is alwaysunity, regardless of the slot angle. Figure 5.10 shows that values both above and below unityoccur when a cross stream is present. When the total pressure coefficient is near unity, themain stream flow acts like a lid over the slot, reducing the amount of flow that would beemitted into a stagnant fluid of the same static pressure as the cross stream, and thus reducingCd below unity. When the Cpt is large and negative, the cross stream does not have a largeinfluence on the injectant, so we might expect Cd to approach unity. With no cross stream, thestreamlines bounding the injectant remain parallel to the slot as the fluid exits from the slot.With a cross stream, the injected fluid is deflected parallel to the stream, creating pressureCHAPTER 5^ 63gradients that pull more fluid out of the slot, thus allowing for discharge coefficients largerthan one. The suction effect is greater at larger slot angles, as shown in the figure.5.2.3 Dividing Streamlines1.03.5^- -- -2.0^3.92- - - ---^1.0 X2.43- 0.8 -^- 0.0, . 1.46 - - -^- -0.5^/ , - _^0.67 - - - -/....-0-- 0.75'i • _--, ^- - - -0.6- -A-- 0.9 i , :•i  ^.0.25 —/ ,2 z/,''/,./ /^0.0680.4 -^, 'y0.0 • •^i^WWWWWWW^i^I0^2 4^6x• •8^10Figure 5.11 Dividing streamlines, 11=15°Of interest in secondary flows is the trajectory of the injectant as it leaves the slot. Figure 5.11shows the trajectories of the dividing streamlines for various values of Cpt for a slot angle of15°. Plots for 13=40°, 65 °, and 90° are given in Figures C.2-C.4 in Appendix C.The general trend is as expected, with more energetic injections penetrating further intothe main stream. Since Cd is equivalent to hE (the asymptotic height of the streamline), FigureC.1 in Appendix C shows clearly that an increase in the injectant energy at higher mass flowratios does not signficantly increase the jet penetration. Equation 5.3 shows that the pressuredrop ps-poo becomes proportional to the dynamic head of the injectant as M 2/D—.00. Thismakes sense, as the pressure drop caused by the bending of the mainstream streamlinesbecomes insignificant for large values of M2/D.CA D0.3 2.560.4 1.700.5 1.000.6 0.560.250.7••••••"" _ I ---------------- •CHAPTER 5^ 64Figure 5.12 is an example of the effect of density on the dividing streamline shape for afixed value of value of the velocity ratio. The velocity ratio is often used in place of M tocreate a flow parameter that is independent of the density ratio. Again, we can use the fact thatM2/D=const at a given Cp t to generate the plots.1.00.8 -0.6 -y0.4 -0.2 -0.0Us /U.=0.3313---9o°- - — '0^2^4^6^8^10xFigure 5.12 Density effects on dividing streamlinesFor fixed Us/U.,, higher values of D imply higher kinetic energy for the injectant, thus there ismore penetration into the main stream. This is consistent with the values of Cpt given in thelegend, lower values giving larger penetration. Again, higher values of D for a fixed velocityratio give diminishing return on the amount of jet penetration.5.2.4 NIT Code BenchmarksConvergence rates ranged from 15 to 50 iterations of the solution algorithm (see Table 3.1) forICA >0.2 and Cpt =0. In every case, the initial guess for the shape of the dividing streamlinewas a single parabola extending from the upstream slot lip to (x,y)=(xE, 1), with zero slope atthe endpoint. The NIT code was compiled using Microsoft Fortran and run under MicrosoftCHAPTER 5^ 65Windows on a 50 MHz IBM compatible PC. Solution timings were 100-300 seconds for theHilbert-Bardina code, and 250-1000 seconds for the Davis-Bardina code.In contrast the TEACH-II code from Chapter 4, compiled and ran under the sameconditions, required 8,000-12,000 seconds to find a solution. A direct comparison is notappropriate however, as the TEACH-II code solves a more complex set of equations, andyields more detailed results.5.3 Vertical Slot ComparisonsIn the following sections, we compare the NIT to other theoretical, numerical, andexperimental results for the case of a vertical slot. Most of the NIT research found in theliterature is concerned with this particular geometry.5.3.1 Mass Flow RatesFrom the results of the previous section, we see that when the geometry is fixed, the NITtheory shows that M2/D is a unique function of Cpt. All the comparitive results used here arefor a unit density ratio, so we adopt the traditional convention of plotting in terms of M.Results from various sources are compared to the NIT results in Figure 5.13. The 'NIT bub'results are computed using the separation model given in Section 3.8. The bubble dimensionsused in the model are taken directly from vector plots for the corresponding TEACH-IIcomputations. Also shown in the figure are the theoretical and experimental results from Fittet al. Both the TEACH-II and Fitt results are for a Reynolds number of 105 , based on the freestream velocity and the slot width. A typical vector plot from the TEACH-II code is shown inFigure C.5 in Appendix C.2.For mass flow ratios greater than about 0.5 the results begin to diverge. The theoreticalresults from Fitt et al are expected to fail at high M because the results are based on linearizedtheory. The NIT results are not linearized, and although they give much better results at highM, are still not particularly accurate. Incorporation of a separation bubble into the NIT codeshows (for the range of values calculated at least) that flow separation has a significant effecton the mass flow.^NIT♦ NIT bub- - - - Fitt theor+ Fitt exp— - - Teach-IICHAPTER 5^ 66CptFigure 5.13 Vertical slot mass flow comparisonsFigure 5.14 is a magnified view of the lower mass flow range of Figure 5.13. Theresults clearly show good agreement of all results to a mass flow ratio of about 0.5. Alsoincluded in this figure are the analytic separation model results from Chapter 2. The symbollabelled 'Analytic 1' is the wake-source model result and the symbol labelled 'Analytic 2' is thebubble model result.The separation flow models demonstate that inviscid flow models can give accuraterepresentations of the mass flow ratio, provided that the displacement effects of separation areaccounted for. We expect the agreement with viscous results to be better at lower slot anglesbecause the region of flow separation will be smaller.^NIT• NIT bub- - - - Fitt theor+ Fitt exp— - - Teach-IIO Analytic 1O Analytic 2CHAPTER 5^ 67The results presented here show that the injection process is pressure dominated, as theinviscid models do not take viscosity, diffusion, or turbulence into account. This is not alwaysthe case, as we will see in the next section.1.20.8M 0.6•+^„0.4 -0.2 -13-90°D=10.25 0.5^0.75^1Cp tFigure 5.14 Vertical slot mass flow comparisons at low M5.3.1.1 Boundary Layer EffectsIn the viscous results presented in the previous section, the boundary layer thicknesses on boththe slot and main stream walls were less than one-fifth of a slot width. In practical situationssuch as film cooling, slots or holes located near the rear of the turbine blade encounter asignificant main stream boundary layer. In this section, we use the TEACH-II code to studythe effects of the main stream boundary layer thickness on the mass flow from the slot.1.00.901.8R 1) = 10,0000.0 -Cp tM-0.4 -0.6ö s-4.- 0.20-0- 0.44- 6- 1.33-o- 3.90-c.- 8.32-o- 14.60.7CHAPTER 5^ 68Figure 5.15 Boundary layer effectsFigure 5.15 shows the variation of M with the boundary layer thickness at the slot 050 forvarious values of Cpt. The trends make sense physically; thicker boundary layers have lessaverage kinetic energy, allowing more fluid to escape from the slot. It appears that the part ofthe boundary layer nearest to wall has the most significant effect, as the largest increases in Moccur when the boundary layer is small. These results support the experimental results of Fittet a/, who found a significant increase in the mass flow when O s increased from one-tenth tothree-tenths of a slot width. A similar study could be made for the boundary layer thickness onthe slot wall, but this is beyond the scope of this thesis.5.3.2 Dividing StreamlinesFigure 5.16 is a comparison of the shape of the dividing streamline near the slot for variousmass flow ratios. The results labelled 'UM" are the experimental results of Fitt et al. At thetwo lowest mass flow ratios, no flow separation was present in the experiments or the viscouscalculations. For M=0.6, significant flow separation occurred in both, but Fitt used a slightlyrounded slot lip to reduce the extent of separation.NIT FITT0.1 0.1- - - -^0.22^- - 0.22---  0.6 -^0.6TEACH-IIA  0.1- -A - 0.22-A - -0.6 -y0.4 -0.2 -A^,41(, -/ ®/-.6; - - 4-A"^/ ^,'^.-_.i,-- -. -* 7,,A fr , - r -, 4 ", -/ , -  r A 4 - •0.0MACHAPTER 5^ 690.0^0.4^0.8^1.2^1.6xFigure 5.16 Vertical slot dividing streamline comparisionsThe results are in good agreement at the lower mass flow ratios, with the slight difference intrajectory most likely due to the displacement effects of the boundary layers in the experimentand the viscous calculations. For M=0.6, the results are signficantly different, due mainly tothe displacement effects of the separation bubble. Fitt's experimental results lie between theTEACH-II calculations and the NIT results as expected.5.3.3 Slot Exit Velocity DistributionsMany modern Navier-Stokes solvers use a Cartesian coordinate system as a basis for theircalculations. For the present geometry, this means a 'staircased' representation of the slot forangles other than 90°. To avoid this, some researchers have prescribed the conditions at theslot exit. Obtaining slot exit velocity distributions from experiment is difficult, so researchersgenerally prescribe a uniform distribution of velocity parallel to the slot. It seems unlikely thatthe distribution would be uniform, especially at lower mass flow ratios, owing to the shapes ofCHAPTER 5^ 70the dividing streamlines shown in the previous section. The slot exit velocity profile has asignificant effect on the extent of separation downstream of the slot, and thus on the pressurefield, velocity field, and heat transfer characteristics. This is confirmed numerically by Sinitsin(1989), who found large variations in the heat transfer downstream of the slot for different exitprofiles. The results of the NIT calculations should provide a much improved exit profile foruse in oblique slot calculations.To illustrate the deficiency of a unform tangential distribution, Figures 5.17 shows thevertical slot exit profiles from the TEACH-II and NIT calculations for M=0.5.Figure 5.17 Slot exit velocity distributions, M=0.5Both calculations show that the speed distribution is asymmetric and the flow is inclined to theslot angle. The NIT calculations are both more inclined and more asymmetric than theTEACH-II calculations, due in most part to the presence of the upstream boundary layer(allowing more flow to penetrate near the upstream slot lip) and the separated flow region (thedownstream slot lip is an infinite velocity point in the NIT).Figure 5.18 shows that the distribution is even more inclined and asymmetric at lowermass flow ratios. The curve marked 'IT (isoenergetic theory) is from the analytic results ofChapter 2. Here, the stagnation point has been moved into the slot to attain the desired massflow ratio. The results show a remarkable similarity to the NIT results over the downstreamhalf of the slot. The NIT and TEACH-II distributions compare favorably at this mass flowratio, where the flow remains attached. This favorable comparison should exist at higher massflow ratios for smaller slot angles, where separation is less likely.M=0.2 I1.7s1 NIT171 TEACH-IICHAPTER 5^ 71Figure 5.18 Slot exit velocity distributions, M=0.25.4 Oblique Slot ComparisionsGartshore et al developed an empirical relationship (see Equation 5.3) for the dischargecoefficient, Cdi, by considering the pressure losses in the injection process. The constant Acorresponds to the losses in kinetic energy of the injectant and B corresponds to losses due to alocal rise in pressure near the slot exit caused by the deflection of the main stream. The relatedconstants K1 and K2 (see Section 5.2.2) can be found by measuring the slope and intercept ofthe linear sections of the curves in Figure 5.9. These constants can be combined to give anequivalent A and B, and the results are compared in Table 5.1 for f3=20° and (3 = 4 0 ° . Inaddition, values of A and B are calculated from the NIT results for 13=65° and f3=90°.For ideal flow with no cross flow, the value of A is 1, and of course B has no meaning.The fact the A<1 for the NIT reiterates the results discussed in Section 5.2.2; the kinetic energylosses are less in the presence of a cross stream (because the favorable pressure gradientsformed by the injectant turning parallel to the mainstream wall without separating), even forsmall cross stream flows. The constant B behaves as expected, as the main stream isincreasingly deflected for larger slot angles.CHAPTER 5^ 72The values of A and B from Gartshore et al are from a best fit through experimentaldata. In their experimental arrangement, slot air was supplied from a large plenum wheremeasurements of the injectant total pressure were made. Thus we expect their value of A to beabove unity due to the usual viscous losses.A B13 K1 K2 NIT(=1/K2 )Gartshore NIT(=1—K1/K2 )Gartshore20° 0.62 1.08 0.93 2.5 0.43 0.0940° 0.58 1.02 0.98 2.5 0.43 0.1265° 0.66 1.50 0.67 N/A 0.56 N/A90° 1.0 2.70 0.37 N/A 0.63 N/ATable 5.1 Discharge coefficient constantsThe NIT results appear poor in comparision with Gartshore's empirical results. When theresulting discharge coefficient curves are plotted, however, the results appear somewhat morefavorable (see Figure 5.19). The curves labelled 'Gartshore 2' are the results of the empiricalrelation, and the points labelled 'Gartshore 1' are from numerical TEACH-II simulations oftheir experimental geometry with a zero mainstream boundary layer thickness at the upstreamslot lip. The experimental conditions used to determine A and B included a two slot widthboundary layer thickness at the slot. From the definitions of Cdi and Cpt, we have thefollowing relation,1- CptCdi M2DUsing the results from Figure 5.15, we find, for a vertical slot, that the discharge coefficientsare lower for thicker boundary layers because Cpt is higher for a given value of M2/D. If weI t13=20°■ Gartshore 1— - — - Gartshore 2^ NIT• Gartshore 1— - - — Gartshore 2^ NITCHAPTER 5^ 73use this result to extrapolate to a zero thickness boundary layer for the experimental results, thedisagreement at higher mass flows would be even worse than shown in the figure.0^0.2^0.4^0.6^0.8^1^1.2MFigure 5.19 Discharge coefficient comparisionsThe disparity of results may be due to several factors. Gartshore's results included a plenumwhich is not present in the non-isoenergetic theory. Entrance losses into the plenum and theresulting turbulence energy generation and flow patterns may have increased the dischargecoefficient significantly. For higher mass flow ratios, the shearing effect in the viscous casemay be greater at lower slot angles, causing greater losses in the viscous case. At low massflow ratios, this same shearing effect may help draw more fluid out the slot than occurs in theinviscid case for the same value of Cpt. The main stream flow was also bounded. This mayhave had an effect on the resulting pressure distributions, and ultimately on the dischargecoefficient.1 0Cd 1Chapter 6CONCLUSIONS AND RECOMMENDATIONS6.1 ConclusionsA new method has been developed for predicting the flow from an arbitrarily inclined slot intoa uniform free stream of arbitrarily different density and total pressure (non-isoenergetic). Weapply the method here using incompressible potential flow theory, but this is not a necessaryrestriction. The results provide a basic understanding of the physics involved in practicalsituations such as in the film cooling of gas turbine blades. The results also provide a moreaccurate assessment of the "discharge coefficient", defined as the ratio of the actual flow fromthe slot to the ideal flow, and an important parameter used to describe the operating state ofthese devices.To guide the development of the non-isoenergetic technique, we initially found ananalytic potential solution (based on a single Schwarz-Christoffel transformation) for thespecial case where the two streams have equal total pressure and density (isoenergetic).Besides providing a basic understanding of the flowfield for arbitrary slot angles (the only suchsolution found in the literature was for the case of a vertical slot), the solution was used as alimiting case check of the non-isoenergetic results. From the isoenergetic solution we observedthe following:1. The mass flow ratio (Mv sUs/p.Ucc) is uniquely determined by the slot angle 13and the stagnation point position, zs, where the two streams initially contact.74CONCLUSIONS AND RECOMMENDATIONS^ 752. As the stagnation point moves upstream along the main stream wall, the mass flowincreases monotonically to infinity. Conversely, the mass flow ratio decreasesmonotonically to zero. The most physically realistic location of the stagnationpoint is at the upstream slot lip, as this eliminates the infinite velocity point thatwould otherwise occur there. With this 'Kutta' condition in effect, the mass flowratio is given by the simple expression,x kM— Xk=13^(O<X<l)3. The initial slope of the dividing streamline with the Kutta condition in effect is f3/2.This slope is required as an input boundary condition for the non-isoenergeticmethod.4. As the slot angle increases, a physically unrealistic suction occurs near thedownstream slot lip to keep the flow attached to the wall as it exits the slot. Thiseffect causes M to increase to infinity asThe previous analysis assumes that the flow remains attached to the wall as it exits the slot. Inthe physical situation, the flow separates from the downstream slot lip at higher mass flows andslot angles. To estimate the effects of this flow separation, we constructed two flow separationmodels by adding appropriate source and sink singularities to the existing Schwarz-Christoffelmodel. These models are more realistic (for high slot angles at least), but require empiricalinput to supply the necessary boundary conditions. From the solutions we observed thefollowing:1. The mass flow ratio decreases when flow separation occurs. This is because theflow no longer makes a sharp turn as it exits the slot, thus reducing the strongpressure gradients that pull the fluid from the slot.2. For the case of a vertical slot, the mass flow ratio appears to be dependent on theshape of the separation region within 10 slot widths downstream of the slot. Theresults from both separated flow models compare favorably with experiments andCONCLUSIONS AND RECOMMENDATIONS^ 76other more sophisticated calculations, where significant regions of separated arefound.In the non-isoenergetic case, the velocity is discontinuous across the dividing streamline thatseparates the two streams. Classical methods cannot be directly applied in this case because ofthe discontinuity. The new non-isoenergetic technique involves separating the flowfield alongthe dividing streamline into two holomorphic regions, an internal region containing the slotfluid and an external region containing the main stream fluid. The solution is obtained byfinding the shape of the dividing streamline that provides continuity of static pressure betweenthe streams. After an initial guess is made for the shape of the streamline, we solve for theinternal flow field using standard potential methods. We then solve the external flow field inan inverse fashion by finding a new streamline shape to match the interface pressuredistribution from the internal flow field. The cycle is repeated until a specified maximumchange in the streamline shape occurs between successive iterations.Due to the complex nature of the problem, we approximate the shape of the dividingstreamline, and satisfy the interface boundary conditions at a discrete number of points alongthe curve. With the Kutta condition applied, and in absence of flow separation, the non-isoenergetic solution depends on three parameters: [i] the slot angle f3 [ii] a parameter, Cpt,describing the difference in total pressure between the free stream and slot fluids [iii] thedensity ratio, D, of the two fluids. For given values Cpt and 13, there exits a unique value ofM2/D, where M is the ratio of mass flow of the injectant to the main stream. The results foundfrom the application of the new technique to the current problem extend previous work toinclude arbitrary values of 13, Cpt, and D. From the application of the theory and the resultingsolutions, we have observed the following:1. The boundary conditions at the upstream slot lip were pivotal in finding a uniquesolution. For non-isoenergetic flow, one or both the internal and external flowregions incurs a stagnation point at this point. For Cpr>0, the dividing streamlineleaves parallel to the main stream, creating a stagnation point in the slot fluid. ForCpt<0, the dividing streamline leaves parallel to the slot, and the stagnation point isin the main stream fluid. For the special case Ch=0, the analytic results show thatCONCLUSIONS AND RECOMMENDATIONS^ 77the initial slope of the streamline bisects the slot angle, producing a stagnation pointin both fluids.2. The boundary conditions in (1.) are independent of the density ratio.3. Localized regions of high curvature in the dividing streamline near the stagnationpoint for cases of nearly isoenergetic flow (i.e., IC p tl«1) create difficulties infinding an accurate solution using approximate methods.4. The use of linearized theory for the external flow field produces reasonable resultsfor lower slot angles and/or mass flow ratios.5. For given values of Cpt and 13, there is only one corresponding value of M2/D thatis a solution to the problem. This is because M2/D is a measure of the kineticenergy ratio between the streams, so that, for fixed Cpt, the static pressure isconstant if M2/D is constant. This implies that a solution for a given density ratiocan be used to directly produce solutions for alternate density ratios for the samevalues of Cpt and 3. It also implies that higher values of D produce lower values ofthe mass flow ratio. This result can also be used to find analytic solutions fordensity ratios other than unity in the isoenergetic case.6. For values of M2/D greater than about 0.5, M2/D is a linear function of Cpt. This isa result of the local pressure rise in the external flow near the slot becomingproportional to the kinetic head of the free stream in this regime.7. For values of M2/D less than 0.5, M2/D is proportional to (1-Cpt)3. In this region,the discharge coefficient,Cd^UsPoo1 Psis a linear function of Cpt .8. In the presence of a cross flow at higher values of M2/D, discharge coefficientsgreater than unity can occur because the sink effect caused by the flow bending toCONCLUSIONS AND RECOMMENDATIONS^ 78remain attached to the wall pull more fluid out of the slot than would otherwiseoccur. In this region Cd is greater for larger slot angles.9. At lower values of M2/D, the discharge coefficients are below unity because themain stream acts like a 'lid' over the slot, reducing the amount of flow. In thisregion Cd is smaller for larger slot angles (for 3<70° at least).10. For a fixed slot angle, the jet penetration is greater for larger values of M2/D and 13.If the linear region described in (6.) continues to M2/D-4.00, this implies that the jetpenetration is limited, i.e., ymax.1.64 for 13=90 ° .11. For a given velocity ratio between the slot and mainstream fluids, larger values ofD produce deeper jet penetration.12. Slot exit velocity distributions show that the flow is significantly deflected by themain stream, especially at lower mass flow ratios. The distribution is alsoasymmetric, with most of the flow exiting through the downstream half of the slot.Largely as a basis for comparison with the non-isoenergetic results, we used a finite-differenceNavier-Stokes computer code (TEACH-II) to make predictions for the case of a vertical slot.The code uses a higher order differencing scheme to reduce discretization errors. Oblique slotangles were avoided because of the cartesian coordinate system used by the code.Modifications were made to simulate external flow by imposing a zero pressure gradient on theupper boundary. In addition, mainstream boundary layers of various thickness were introducedupstream of the slot to study the effects on the mass flow ratio. The following trends wererevealed in the (turbulent flow) results.1. The results are nearly independent of Reynolds number in the range,3 pU oo h^55x10 5^ 55x10where h is the slot width and U. is the free stream velocity.2. Thicker main stream boundary layers produce larger mass flow ratios for the sametotal pressure difference (at least for boundary layers at the slot thicker than about0.3 slot widths).CONCLUSIONS AND RECOMMENDATIONS^ 793. Most of the increase in mass flow occurs for boundary layers thickness between 0.4and 3 slot widths.Comparisons of mass flow ratios with experiments, the TEACH-II results, and other inviscidresults for the vertical slot case show good agreement provided that the separation region is notlarge. The effects of separation can be included in the present theory by incorporating aseparating streamline into the model. The dimensions of the separated region are suppliedfrom the TEACH-II calculations, and the results show a significant improvement. The shapesof the dividing streamlines and slot exit velocity distributions are in good agreement, againprovided that separation, if present, is accounted for.Limited comparisons were made to numerical viscous predictions and experimentalresults for 20 ° and 40° slots. Non-isoenergetic predictions of a discharge coefficient definedby,Pts _ PooCdi =  1^2-f PsUsare only in fair agreement with previous experimental results. Since no separation wasobserved in the results used for comparison, we suggest the following reasons for thedifferences,1. The comparative results included a plenum which is not present in the non-isoenergetic theory. Entrance losses into the plenum and the resulting turbulenceenergy generation and flow patterns may have increased the discharge coefficientsignificantly.2. For higher mass flow ratios, the shearing effect in the viscous case may be greaterat lower slot angles, causing greater losses in the viscous case. At low mass flowratios, this same shearing effect may help draw more fluid out the slot than occursin the inviscid case for the same value of Cpt.3. The main stream flow was bounded. The may have had an effect on the resultingpressure distributions.CONCLUSIONS AND RECOMMENDATIONS^ 806.2 Recommendations1. Extend the isoenergetic wake-source model to include oblique slot angles. Use analyticmethods to find the teat height that produces zero curvature of the streamline atseparation.2. Add the wake-source model to the internal flow solver for the non-isoenergetic case.3. Use analytical methods to analyze the region near the stagnation point for the nearisoenergetic flow case to perhaps provide the correct streamline shape for the region ofhigh curvature. This will enable a more accurate solution for this case.4. Attain further results to compare with the present theory for slot angles other than 90°.5. Use the solution algorithm to predict the flow from an oblique slot using compressibleinviscid flow theory.6. Apply the solution algorithm to other inviscid flow problems such as jet-flapped wingsor ground effect machines.7. Use viscous-inviscid boundary layer theory for the external flow in conjunction withthe current solution algorithm to simulate the effects of the mainstream boundary layer.8. Include the effects of a plenum. This could easily be done by changing the shape of theinternal flow region.9. Modify the solution algorithm to solve the flow from two or more slots in succession.REFERENCESAckerberg, R.C. & Pal, A. (1968), "On the Interaction of a Two-Dimensional Jet with aParallel Flow", J. Math. Phys., 47, 32-56.Ainslie, B. (1991), "Analytical and Experimental Study of Flow from a Slot into a Freestream",M.A.Sc. Thesis, University of British Columbia.Bardina, J., Kline, S.J., and Ferziger, J.H. (1982) "Computation of Straight Diffusers at LowMach Number Incorporating an Improved Correlation for Turbulent Detachment andReattachment", Rept. PD-22, Thermosciences Div., Dept. of Mech. Eng., StanfordUniversity.Benodekar, R.W., Gosman, A.D., and Issa, R.I. (1983), "The TEACH-II Code for the DetailedAnalysis of Two-Dimensional Turbulent Recirculating Flow", Dept. Mech. Eng.,Imperial College, London, Rept. FS/83/3.Cole, J.D. & Aroesty, J. (1968), "The Blowhard Problem - Inviscid Flows with SurfaceInjection", Intl. J. Heat Mass Transfer, 11, 1167-1183.Davis, R.T. (1979), "Numerical Methods for Coordinate Generation Based on Schwarz-Christoffel Transformations", AIAA Paper 79-1463, 4th Computational FluidDynamics Conference.Dewynne, J.N., Howison, S.D., Ockendon, J.R., Morland, L.C., and Watson, E.J. (1989), "SlotSuction from Inviscid Channel Flow", J. Fluid Mech., 200, 265-282.Djilali, N. (1987), "An Investigation of Two-Dimensional Flow Separation with Reattachment",Ph.D. Thesis, University of British Columbia.Erich, F. F. (1953), "Penetration and Deflection of Jets Oblique to a General Stream", J. Aero.Sci., 20, 99-104.Fitt, A.D., Ockendon, J.R., and Jones, T.V. (1985), "Aerodynamics of Slot-Film Cooling:Theory and Experiment", J. Fluid Mech., 160, 15-27.Floryan, J.M. (1985), "Conformal-Mapping-Based Coordinate Generation Method for ChannelFlows", J. Comp. Phys., 58, 229-245.81REFERENCES^ 82Gartshore, I.S., Salcudean, M., Riaha, A., and Djilali, N. (1991), "Measured and CalculatedValues of Discharge Coefficients from Flush-Inclined Holes", Can. Aero. Space J., 37,9-15.Goldstein, M.E. & Braun, W.H. (1975), "Inviscid Interpenetration of Two Streams withUnequal Total Pressures", J. Fluid Mech., 70, 481-507.Kirchhoff, G. (1869) "Zur Theorie Frier Fliissigkeitsstrahlen" (On the Theory of Free FluidJets), Jour. fur die Reine and Angew. Math. (Crelle's Jour.), 70, 289-298.Leschziner, M.A. & Rodi, W. (1981), "Calculation of Annular and Twin Parallel Jets usingVarious Discretization Schemes and Turbulence-Model Variations", J. Fluids Eng.,103, 353-360.Milne-Thomson, L.M. (1968), Theoretical Hydrodynamics, 5th Ed., MacMillan & Co. Ltd.,London, England.Parkinson, G.V. & Jandali, T. (1970), "A Wake Source Model for Bluff Body Potential Flow",J. Fluid Mech., 40, 577-594.Parkinson, G.V. (1991), A private communication.Patankar, S.V. (1980), Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington,D .C.Robertson, J.M. (1965), Hydrodynamics in Theory and Application, Prentice Hall, EnglewoodCliffs, N.J.Roshko, A. (1954), "A New Hodograph for Free-Streamline Theory", National AdvisoryCommittee for Aeronautics, Tech. Note 3168.Sinitsin, D.M. (1989), "A Numerical and Experimental Study of Flow and Heat Transfer froma Flush, Inclined Film Cooling Slot", M.A.Sc. Thesis, University of British Columbia.Sridhar K.P. & Davis, R.T. (1985), "A Schwarz-Christoffel Method for Generating Two-Dimensional Flow Grids", J. Fluids Eng., 107, 330-337.Stropky, D.M. (1988), "A Viscous-Inviscid Interaction Procedure", M.A.Sc. Thesis, Universityof British Columbia.Ting, L. & Ruger, C.J. (1965), "Oblique Injection of a Jet into a Stream", AIAA J., 3, 534-536.Ting, L. (1966), "Pressure Distribution on a Surface with Large Normal Injection", AIAA J., 4,66-80.Uc> hc>U1Present Geometry with Bounding Wall^Milne-Thomson GeometryAppendix AISOENERGETIC THEORYA.1 Attached Flow - Milne-Thomson's /4 SolutionFigure A.1 shows the current geometry (bounded from above by a solid wall) in relation toMilne-Thomson's geometry. In his analysis, Milne-Thomson assumed that the stagnation pointwas at the corner. This corresponds to for the present analysis (seeequation 2.3).Figure A.1 Confined Slot FlowFrom the diagram we deduce the following,a=13, h=h1 = H, h2 =1, U1 — — 1, U2 = -M83APPENDIX A^ 84Uh =Uihi +U2h2^U =U1 P-!+U,-12-2- - -11+}h^h^HThe negative velocities are admissible because potential solutions are valid with velocities runin reverse. Substituting these relations into Milne-Thomson's equation 7 (pg. 292) andrearranging, we obtain an equation relating the mass flow from the slot to the height of thebounding wall and the slot angle.[1 - M 1- 7013 H-il MH' 1^ =1We will expand this equation in terms of a small parameter E=1/H, and the limit c-.0 gives thedesired result. With,f = -:--1^(0<k<1)we have,— e^{1 +^=1-■^E m- r } {i+ rEm + o(E 2 )} -1-4 + E {r m - Arr } + E 2 ) = 1--+ If — M -r } =In the limit as E-►0,Inspection of (2.3) shows that this relation must be true for M as well because (2.3) is invariantin terms of if for any stagnation point position s, including s-1 (i.e., M=M).M 1 - X (A.1)APPENDIX A^ 85A.1.1 The Stagnation Point PositionEquation 2.3 gives the location in the physical plane of a point in the transform (solution)plane. We use numerical methods here because for general slot angles, analytic integration isunnecessarily involved. To integrate, we separate (2.3) into real and imaginary parts,12,1z= ff(Od= fic(µm)-Fitcti,rodoi+iToRol^ 11,1x^{6dµ -^and^y = {oth" + Tdtt}where,.-1f^91{(1-t + 1 +^- r +iyoxn (1-k j µ+irlTo= lf k^1-t'"51 ( + 1 + 61)1-kot -r 11 +ilThe results are path independent, provided that the path of integration does not include thesingularity at -(0,0).To find the location of the stagnation point, we integrate to -(-s,0) along the real axis.In this case, Ti=dri=0, and, after some algebra, we can write,Izs l = Vxs2 + y2 _11. k ki-s{(11+ ttbl-k(rs^ dtst-1This equation can be integrated using any standard numerical method. Izs l is the distance fromthe upstream slot lip to the stagnation point. For s>1 the stagnation point lies along theupstream main wall; for s<1 it is positioned along the upstream slot wall.APPENDIX A^ 86A.1.2 The Dividing StreamlineTo find the dividing streamline shape in the physical plane we must first find the correspondingshape in the transform plane. Substituting tlf=,:'s(F)=M into (2.1),M = a{ 11- (+71 Sor, with .=-R+iri,= f1 + tan -1 — }1-1 OsTi<ntan{ri /s} -1sR<coTo integrate (A.2), we make the following substitution,, - sin^cosdll^S2 di^1 11=^ (111d1 sin2thus,xd f {crg2 t}dri^and^yd f{cr+ TC2}di0^ 0where xd and yd are the coordinates of the dividing streamline in the physical plane. Again,any standard numerical method can be used to solve these equationsA.1.3 The Initial SlopeFrom (A.5), the slope of the dividing streamline in the -plane is,dri^sinedR^- sin CI cosThe initial slope is therefore given by,cLril^sin2^112 +00,14)^1 = co'CI— sin cos CI^f: i21:0^- +^+ 0(i15)^-lim { -23 TA1 }71-•01=0 1-00Dividing Streamline(2-13hr) ni2 = Jr-13/2A mimmtwwwwwwcaletwwwwwwwwwtmen12 0/2APPENDIX A^ 87This implies that the dividing streamline leaves the wall in the -plane at an angle of n/2. Oneof the properties of conformal mapping is that, away from critical points, angles are preservedbetween the planes. Therefore for sal, the dividing streamline leaves normal to the surface inthe z-plane. For s=1 (the Kutta condition), the origin of the dividing streamline lies at a criticalpoint of the transformation function. Examination of (2.3) shows that is a zero of order1-13/n. At critical points with zeros of order n, angles in the ..plane are increased by a factorof n+1 when transformed to the z-plane. Thus the angle between lines intersecting at theupstream slot lip (---1) is increased by a factor 2--(3/n in the z-plane. Figure A.2 shows thatwhen the Kutta condition is applied, the initial angle of the dividing streamline is (3/2.Figure A.2 Initial Slope of the Dividing StreamlineA.2 Separated FlowA.2.1 The Bubble ModelFigure A.3 Bubble model transform planeAPPENDIX A^ 88Figure A.3 shows the source/sink bubble model in the Schwarz-Christoffel transform plane.The shape of the physical and transform planes are the same as those for the non-separated case(see section 2.1), so we have, as before,dz - K ( + 1)14' (- nx dl;where 13/7c=n/(n+m). The constant K is different from the non-separated case because thescaling of points at infinity is based on the ratio of free stream velocities between the physicaland transform planes. This ratio is changed in the bubble model due to the addition of thesource and sink. From the diagram,F() = Vcc, + —M ln + —Q ln( - q i ) - —Q ln( - q2)n^n^7Eand therefore,M Q {  1^1 voc, + —+n n^- (11^-C12 w(z) ...K(+1)1-?" (- nxWe use the same boundary conditions as in the non-separated case, i.e.,0^at t = -11^at t = coMe-i I3 at^--, 0Next, locating the front of the separation region at the corner D, we have,w(t) = 0^at = d = rFrom above boundary conditions we get,V. --M -g-{ 1^1  1 - 0n n 1+q i 1+q 2w(z)where,APPENDIX A^ 891  } _ 01 M Q {  1vo. + r a a ch _r C12 - rSolving,M^ii-r'-1 ^A/0^1+  (q1 +1)(q2 +1)(cil – r)(c12 -r) (A.3)A.2.2 The Wake Source ModelFigure A.4 Wake source transform planeFigure A.4 shows the Schwarz-Christoffel transform plane for the wake source separationmodel (see section 2.2.2). The addition of the teat (see Figure 2.8) changes the mappingfunction which is now given by,dz i.dc, –ozl lilt+1–eFrom the diagram the velocity potential is,F() . Vc„, + In + —Qln( – q)It^TE(A.4)and therefore,V. Q {  q }k C le — dVl+dn (q — d)2APPENDIX A^ 90{lio,-1- 7M + Q:n t( 1 q )} it—ew(z)—^C(t — d)^1 t +1Again, we have the normal three boundary conditions,0 at t = —1w(z) = 1 at t = co—iM at t --> 0and finally, we prescribe the separation speed (i.e., the base pressure) at D using,w(t) — 0w(z) =kat t=dFrom these five boundary conditions we get,Vco—^ItM — Q { 1 =0t^1+qV= CC =ndV + M Q { 1 }-0c° nci n q — dThese five equations contain eight unknowns. A further relation is obtained by integrating(A.4) from B to E. The results of the integration give,1+e-2d =0Combining the six equations above, and rearranging in terms of k and d,APPENDIX A^ 91NITci -1M== d + 47- kFinally, integration of (A.4) from D to 'E gives the relation between d and 6,S- 1 {  V(d 2 - 1)(2d - 1) In [  V(d 2 - 1)(2d - 1) + d 2 + d - 1}d^ d2(A.5)(A.6)Appendix BNON-ISOENERGETIC THEORYB.1 The Zonal InterfaceTo compute the coefficients for the spliced cubic polynomials comprising §(x), weneed to specify the value (yk) and slope (mk) at the kth cubic segment endpoint. The values arefound iteratively as part of the solution, and we use the simple central difference formula tocalculate the slopes,mk Yk+1 Yk-1 ^k -1,2...,Nxk+1 xk-1If more accurate results are desired, a higher order scheme can be used. Each cubic polynomial(see Figure 3.4) is written as,fk(IP) = ak + bk + ck 11)2 +dkX - Xk_.111)  ^Os s1xk -xk_i (B.1)where,ak = Yk-1mbk^ k xkck = 3 (Yk Yk-1) ( mk +2mk-1 xk92P - 000KV,APPENDIX B^ 93mk +mk-1)dk =2(Yk-1-Yio+ xkIn the region xN<x<xE, S (x) progresses smoothly from the end of the last cubic segment to xE,where it reaches the asymptotic height hE. For simplicity, we can construct this segment fromanother spliced cubic polynomial, i.e.,fE(v) = aE + bE V + cE V 2 +dE v 3 ,v = X -XNXE -XN0s s1(B.2)By constructing fE(v) in this manner, we can use the above definitions of ak...dk, where, in thiscase:^k-1=N, and mEm0.B.2 The Initial Dividing Streamline Angle 00 for DA.Figure B.1 shows a highly magnified region around the upstream slot lip. The dividingstreamline leaves at an angle of 00 to the x-axis.Figure B.1 O. for DolFrom Milne-Thomson (pg. 155), the speed of the main stream fluid very near the stagnationpoint along the dividing streamline is given by,80114. = C+ R 7776-c"where C+ is a constant, and R is the distance along the dividing streamline from the upstreamslot lip. Similarly, for the slot fluid,APPENDIX B^ 94P-00V = C - R '1-13+6°We know from equation 3.1 that V + /V- =1/5 = const. Combining these equations we have,00^p-cloR " °0 n-13+00 = constantFor the L.H.S. to be a constant the exponent must be equal to zero, therefore,00^13 — eo it -00 — n -13 +00--).0 = p_0 2Note that the results are independent of the density ratio D.B.3 Inverting the Hilbert IntegralWe can write the perturbation to the velocity at cubic endpoint xk due to a perturbation of thedividing streamline A§ (x) as,+ixE 1{A (0^AV (xk ) = -1--^d ^d^n xk -0Next, we can write A (x) as the sum of individual perturbations eyp to each cubic endpoint p,i.e., NAV+ (Xk ) = Ep=i{ xE c-1-1.fp(dal 0 4xk _ 4x opQic,p -^(B.3)where fp(x) xeyp = A p(x) is the linear perturbation to S (x) due to the vertical displacement ofa single cubic endpoint p. Following Stropky (1988), we construct the perturbation curve fp(x)APPENDIX B^ 95from two spliced cubics, as shown as the shaded area in Figure B.2. The curve is constrainedto have zero slope at each end and a minimum area under the curve. This type of perturbationcurve was shown by Stropky to provide stable numerical convergence.Figure B.2 Perturbation curvesThe function4(x) is given by,0Aap + thp V + ACOP 2 + Adp V 3X < X, -1Yx-x,,,-1r^ , 051)51xp-xp .42^3X-XP Aap +1 + tibp+ iii) + ACp+ i '11.3 + 4p+1 11)^1p = xp+i-xp , osys10 X > Xp+ i (B.4)fp (x)where,eaP = 0 ' ebP '^r= 0 ec., = E, edp = 1 -Aap+1 = 1thp+i = (3 - 8)Ap+1ecp+i = -3(1+ 2 A p+ i) + 2E A p+iedp+1 = (2 +3 Ap +1) - E Ap + 1and,APPENDIX B^ 966,2P+1 (11+ 66 )+1)-5z..  -2(1+ A3p+i )^'xn+i -xpA p+ 1 - rXp - Xp_11,For the case Cpt<0, the initial slope of the dividing streamline 0 0 is equal to the slot angle 3(see Figure B.1). The perturbation curve for p=1 must be modified allow for slope correctionsat this point (the function fi(x) given in equation set B.3 is not capable of altering 00). Wehave devised the following coefficients for fi(x) by degenerating the first cubic polynomial to aparabolic segment with variable initial slope,eat = 0, Abi = 2, eci = -1, Adi = 0Aa2 =1, Ab2 = 0, Ac2 = -3, ari2 - 2From B.3, we obtain the equations for the influence functions C2k,p, i.e.,xE --ci f.f.p(01 I 4 C2 k,p - —a 4xk -0Care must be taken to evaluate the integral, as it is singular at x=xk.B.4 Coefficients for Davis' MethodFrom Davis, the functions C2m, C3m, N2m, and X3m are given by,4[0m+1 +Om (pm ]2 C3m =[bm+1 - b in?C2m = .n1+1- (1)In [bm+ 1 + bm 1C3mbm+i - bme^LL, bm rs- _ bm p-bm ]1. 2m -ebm+1 N-bm+1P-bm+11APPENDIX B^ 97eR+b,]2 /2 N_ bm p2 -b.2 ]e[4- 13.+112 / 2 N _ bm4.1 y2 - b2,,,+1where q is defined as the angle of the straight line connecting the m and m+1 points on thecurve.83mAppendix CNON-ISOENERGETIC RESULTSC.1 NIT Results2-^1.5 -•Cd 1- 0.5 -^---  15°- - 40°- - 65°^ 90°00 1 2^3 4 5 6m2/DFigure C.1 Cd vs M2ID99APPENDIX C6^8^104xFigure C.2 Dividing streamlines, 13=40°0 213=40° I0.8 --3'5— -2.0^ 1.00.0- 0.5—0— 0.75—6— 0.9Cpr0 0 0 0A A A0.6 -13=65° 1-33-2.0^1.0— - - 0.0— - - 0.5—0-- 0.75—6-- 0.92M =0.0012^•0^2^4^6^8^10XI^•Figure C3 Dividing streamlines, 13=65°1.613.90°-6.60'2APPENDIX C 1 00^ -3.5- - -2.0- - - 0.0- - - 0.5-0- 0.750.9 rpt. D^..A^^=0.0009 A A A A0.0 ^..^...^.0 2^4 6^8Figure C.4 Dividing streamlines, 13=90°C.2 Teach-II Results10


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