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

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INVISCID FLOW FROM A SLOT INTO A CROSS STREAM by  DAVE STROPKY B.A.Sc. (1983), M.A.Sc. (1988), University of British Columbia  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in  THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April 1993 © Dave Stropky, 1993  ABSTRACT The 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 of gas turbine blades. This thesis describes a new method for predicting inviscid non-isoenergetic flow from an arbitrarily inclined slot into a uniform free stream. In the absence of flow separation the solution depends on three parameters: [i] the slot angle 13, [ii] a parameter, Cp t, describing the difference in total pressure between the free stream and slot fluids, and [iii] the ratio of densities, D, between the two fluids. For given values Cp t and 13, there exists a unique value of M2/D, where M is the ratio of mass flow of the injectant to the main stream. To guide in the development of the present theory, an isoenergetic solution (Cp t=0, D.1) was found using classical theory. Specifically, ]),.  M  –  [  -  1  where, X 0 < X, <1  –  High curvature of the streamlines near the downstream slot lip produces an unrealistic suction for large slot angles. The addition of source/sink singularities to the model to simulate flow separation improves the results, but adds empiricism to the solution. The present non-isoenergetic procedure uses the dividing streamline to separate the flowfield into two zones; an internal flow region containing the slot fluid and an external flow region containing the free stream fluid. The technique involves finding, by approximate means, the shape of the dividing streamline that satisfies continuity of static pressure between the regions. The boundary conditions at the upstream slot lip were pivotal in finding a unique solution. The results presented here extend previous work to include arbitrary values of f3, Cp t, and D. For a vertical slot, mass flow rates and discharge coefficients obtained from experiments and viscous numerical predictions are compared to present inviscid calculations. The results show good agreement at reasonable Reynolds numbers provided that the separation region is not large. The effects of separation can be included in the present theory by incorporating the approximate shape of the separated flow region, and the results show a significant improvement, especially at higher mass flow ratios and slot angles. Calculated discharge coefficients for lower slot angles do not compare as well with 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 to other geometries and flow conditions. II  CONTENTS ABSTRACT ^  ii  LIST OF FIGURES ^  vi  LIST OF TABLES ^  viii  LIST OF SYMBOLS ^  ix  ACKNOWLEDGEMENTS ^  xiii  Chapter 1 INTRODUCTION ^  1  1.1 Problem Description ^  3  1.2 Literature Review ^  5  1.2.1 Isoenergetic Theory ^  5  1.2.2 Non-Isoenergetic Theory ^  6  1.2.3 Experiments and Numerical Work ^  8  Chapter 2 ISOENERGETIC FLOW ^  10  2.1 Non-Separated Flow ^  11  2.1.1 The Stagnation Point Position ^  14  2.1.2 The Dividing Streamline ^  16  2.2 Separated Flow ^  17  2.2.1 The Bubble Model ^  17  2.2.2 The Wake Source Model ^  19  2.2.3 Comparisons ^  21  Chapter 3 NON ISOENERGETIC FLOW ^ -  25  3.1 Conformal Mapping Limitations ^  25  3.2 Vortex Sheets ^  27  3.3 A New Zonal Interaction Method ^  28  ill  CONTENTS^  iv  3.3.1 History ^  28  3.3.2 Application to the Current Problem ^  29  3.4 Defining the Zonal Interface ^ 3.4.1 Applying the Kutta Condition ^ 3.5 The Internal Flow Region ^  31 33 35  3.5.1 Bardina's Method ^  35  3.5.2 Sridhar's Method ^  37  3.6 The External Flow Region ^  38  3.6.1 Linearized Theory, the Hilbert Integral ^  39  3.6.2 Davis's Method ^  40  3.7 Density Ratio Effects ^  42  3.8 Separated Flow ^  43  Chapter 4 TEACH II: Numerical Solution to the Full Equations ^ 45 -  4.1 Governing Equations ^  46  4.2 Numerical Solution Procedure ^  47  4.3 Problem Geometry ^  48  Chapter 5 RESULTS ^  5.1 Present Method Grid Refinement ^  51 51  5.1.1 Computational Domain Size ^  52  5.1.2 Grid Distribution ^  53  5.2 Non-Isoenergetic Theory Results ^  56  5.2.1 Unit Density Ratio ^  57  5.2.2 General Results ^  58  5.2.3 Dividing Streamlines ^  63  5.2.4 NIT Code Benchmarks ^  64  5.3 Vertical Slot Comparisons^ 5.3.1 Mass Flow Rates ^ 5.3.1.1 Boundary Layer Effects ^  65 65 67  5.3.2 Dividing Streamlines ^  68  5.3.3 Slot Exit Velocity Distributions ^  69  CONTENTS^ 5.4 Oblique Slot Comparisions ^ Chapter 6 CONCLUSIONS AND RECOMMENDATIONS ^  v 71 74  6.1 Conclusions ^  74  6.2 Recommendations ^  80  REFERENCES ^  81  Appendix A ISOENERGETIC THEORY ^  83  A.1 Attached Flow - Milne-Thomson's Solution ^  83  A.1.1 The Stagnation Point Position ^  85  A.1.2 The Dividing Streamline ^  86  A.1.3 The Initial Slope ^  86  A.2 Separated Flow ^  87  A.2.1 The Bubble Model ^  87  A.2.2 The Wake Source Model ^  89  Appendix B NON-ISOENERGETIC THEORY ^ B.1 The Zonal Interface ^  92 92  B.2 The Initial Dividing Streamline Angle 0 0 for Dol. ^  93  B.3 Inverting the Hilbert Integral ^  94  B.4 Coefficients for Davis's Method ^  96  Appendix C NON-ISOENERGETIC RESULTS ^  98  C.1 NIT Results ^  98  C.2 Teach-II Results ^  100  CONTENTS^  vi  LIST OF FIGURES Figure 1.1  General Slot Flow ^  3  Figure 1.2  Ideal Slot Flow ^  4  Figure 1.3  Ainslie Geometry ^  6  Figure 2.1  Physical plane (z=x+iy) ^  11  Figure 2.2  Transform plane (--tt+iii) ^  12  Figure 2.3  Mass flow dependence on stagnation position ^  14  Figure 2.4  Mass flow with Kutta condition applied ^  15  Figure 2.5  Dividing Streamlines ^  16  Figure 2.6  Bubble model physical plane ^  18  Figure 2.7  Bubble length influence ^  19  Figure 2.8  Wake source physical plane ^  20  Figure 2.9  Tangential separation ^  21  Figure 2.10 Streamline comparisons ^  22  Figure 2.11 Centerline pressure distribution ^  23  Figure 3.1  Hodograph plane for Cpt<0 ^  Figure 3.2  Zonal Modelling ^  28  Figure 3.3  Oblique slot zones ^  30  Figure 3.4  Spliced cubic polynomials ^  31  Figure 3.5  Kutta condition problems ^  33  Figure 3.6  Bardina's method ^  36  Figure 3.7  The external flow field ^  38  Figure 3.8  A simple separation model ^  43  Figure 4.1  TEACH-II grid cells ^  47  Figure 4.2  TEACH-II geometry ^  49  Figure 5.1  The NIT grid ^  51  26  CONTENTS^  vii  Figure 5.2  Effects the interaction region length ^  Figure 5.3  Effects of grid size, Cp t =0 ^  54  Figure 5.4  Effects of grid size, Cp t ---0.1 ^  54  Figure 5.5  Effect of cubic spacing on dividing streamline shape near Cp t =0 ^ 55  Figure 5.6  Sketch of dividing streamline shape for small values of Cp t^ 56  Figure 5.7  M-Cp t results for linear and non-linear theory ^  58  Figure 5.8  M-Cp t results for various density ratios ^  59  Figure 5.9  M2/D vs Cp t ^  60  52  Figure 5.10 Cd vs Cpt ^  62  Figure 5.11 Dividing streamlines, t3=15° ^  63  Figure 5.12 Density effects on dividing streamlines ^  64  Figure 5.13 Vertical slot mass flow comparisons ^  66  Figure 5.14 Vertical slot mass flow comparisons at low M ^  67  Figure 5.15 Boundary layer effects ^  68  Figure 5.16 Vertical slot dividing streamline comparisions ^  69  Figure 5.17 Slot exit velocity distributions, M=0.5 ^  70  Figure 5.18 Slot exit velocity distributions, M=0.2 ^  71  Figure 5.19 Discharge coefficient comparisions ^  73  Figure A.1  Confined Slot Flow ^  83  Figure A.2  Initial Slope of the Dividing Streamline ^  87  Figure A.3  Bubble model transform plane ^  87  Figure A.4  Wake source transform plane ^  89  Figure B.1  0 0 for Doi ^  93  Figure B.2  Perturbation curves ^  95  Figure C.1  Cd vs M2/D ^  98  Figure C.2  Dividing streamlines, 0.413° ^  99  Figure C.3  Dividing streamlines, 13-65 ° ^  99  Figure C.4  Dividing streamlines, 13-90 ° ^  100  Figure C.5  Teach-II vector plot, 3=90° ^  100  CONTENTS^  viii  LIST OF TABLES Table 2.1^Separation mass flow ratios ^  22  Table 3.1^Non-Isoenergetic solution methodology ^  31  Table 3.2^Upstream slot lip boundary conditions ^  34  Table 4.1 Governing equations ^  46  Table 5.1^Discharge coefficient constants ^  72  LIST OF SYMBOLS  ak...dk^coefficients of spliced cubic polynomials sap ...ed p^coefficients of perturbation function Aki^matrix coefficients for Bardina's method Am-54o^mapping points bn , bm^control point locations for Sridhar's and Davis' methods C 1 , C2, C ii •••^turbulence  model coefficients  Cim•••C3m^coefficients for Davis' method Cd^discharge coefficient (- 2 psUs / liPt s  —  Poo  Cp^pressure coefficient (— [p— pco ]lip.U. 2) Cps^injectant pressure drop coefficient (= [ps^) Cpl^total pressure coefficient (= [Pt. —Pts ]1 -}pcoU00 2)  D^density ratio (= ps/Poo) fE(x)^spliced cubic tail piece fk(x)^kth spliced cubic polynomial F=4:13-Fi1P^complex potential plane coordinates GE , Gk^turbulence model generation terms hE^asymptotic thickness of injectant  H, L^separation bubble dimensions j, k^control point indices  ix  SYMBOLS^ k^turbulent kinetic energy k^separation speed K  Schwarz-Christoffel mapping constant  LI^computational domain inlet length M^mass flow ratio Mo^isoenergetic attached flow ratio Mk^Kutta condition mass flow ratio mk^slope at kth cubic endpoint n, s, e, w^control volume faces N, 5,..., SE, SW^control volume neighbors N  number of spliced cubics  NB^number of control points for Bardina's method NE^number of control points for Davis' method P^static pressure Pt.^main stream total pressure Pts^injectant total pressure  q , q1 , q2^source/sink locations Q  source/sink strength  Rh^slot Reynolds number (= pU cc h/p, h=slot width) S(x)^exact dividing streamline (zonal interface) shape §(x)^approximate dividing streamline shape  So^source term S ns^turbulence model shear strain tensor U,^undisturbed main stream velocity Us^average injectant velocity V + (x), V -(x)^velocities along the upper and lower surfaces of the dividing streamline V o^velocity at the upstream slot lip  x  SYMBOLS^  xi  w=u—iv^hodograph or complex conjugate velocity plane coordinates  Vu ( x )^undisturbed velocity field  V + (x)  perturbation velocity field  x p , yp^spliced cubic endpoint coordinates ey p^perturbation at pth cubic endpoint  z=x+iy^physical plane coordinates zs^stagnation point location  ak^zonal interface tangent angle at kth cubic endpoint slot angle  X^straight channel mapping plane separation model teat height (x)^boundary layer thickness O * (x)^displacement thickness A^adjacent cubic length ratio dissipation rate tangent angle of the dividing streamline, general transport variable constant (=n/13  — 1)  I' 4,^diffusion coefficient y(x)^vortex sheet strength distribution  turbulence model constant constant (-13/3T) laminar viscosity turbulent viscosity lieff^effective viscosity, =g+th O m^finite slope change at element endpoints in Davis' method  SYMBOLS^ 13 o  initial angle of the dividing streamline  Poo  main stream density  Ps  injectant density  °k Q k,p  turbulence model constants perturbation function perturbation constant cubic mapping space  =11-t+irl 2m N3m ,  Schwarz-Christoffel mapping plane coordinates coefficients for Davis' method  xii  ACKNOWLEDGEMENTS  I would like to express my sincere thanks to both of my supervisors, Professor Ian Gartshore and Professor Martha Salcudean. They allowed me once again to delve into the world of theoretical fluid mechanics, and do research that I know does not pay the bills. Both have been far more than good advisors, they have been good friends. I am dedicating this work in memory of my father Joe.  XIII  Chapter 1 INTRODUCTION  The problem of oblique injection of a secondary stream into a main stream of different total pressure has practical application in many physical situations. Consider the case of a gas turbine engine. The performance of such engines increases in general with increasing inlet temperature, however the temperature of the structural components must be restricted to avoid mechanical failure. Film cooling is a widely used method for keeping the skin temperature of the turbine blades and combustion chamber walls within acceptable limits. A coolant is injected into the gas stream through a series of holes or slots in the component surface. The injectant forms a thin layer over the surface, shielding the metal from the hot combustion gases. The injection process must be carefully controlled so as not to disrupt the aerodynamic efficiency of the engine. There are many other examples of secondary injection, including jet flapped wings, VTOL aircraft, and ground effect machines. Owing to the complex nature of the problem, there is at present, a less than adequate understanding of the fluid mechanics of two stream flow. A typical case encompasses several aspects of fluid mechanics such as unsteadiness, compressibility, turbulence, viscosity, vorticity, and flow separation. What is of primary importance in many of these situations however, is the mass flow of the injectant. The operating state of these devices generally lies in a restricted range of mass flows, so that a mathematical model should predict the mass flow accurately. Resolution of all the details of the flow requires a sophisticated Navier-Stokes solver and considerable computational resources, an often impractical and unnecessary  1  2  CHAPTER 1^  procedure for many engineering purposes. As in many other fluids problems, a simplified analysis may provide acceptable (albeit limited) results. Over the last two centuries, many problems have been solved satisfactorily by assuming the fluid to be inviscid. Classical methods give elegant solutions to inviscid flows in and around a variety of shapes. Aerodynamicists commonly use inviscid analysis to predict the performance of streamlined components. In these and many other instances, inviscid theory captures the essential physics of a problem and produces reasonable results. Much of the insight into the physics of fluid mechanics has come from these techniques. In the last few decades, 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 steady incompressible flow from an arbitrarily inclined slot into a main stream of different total pressure. 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 and to compare the results with existing theory, experimental data, and limited viscous computations. The results will also provide a more meaningful way of defining a discharge coefficient for a jet into a cross flow. In the absence of a cross flow, the standard definition of the discharge coefficient is, Cd =  (Actual Flow)^(t7j) (Ideal Flow)^Pt j — pc° 1n  where Uj is the average discharge velocity of the actual flow, Ptj and pi are the injectant total pressure and density respectively, and p. is the static pressure far from the orifice. Using this definition, Cd provides a direct measure of the losses in the injection process. When fluid is injected into a cross flow, the ideal flow is no longer given by the simple expression in the denominator 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  CHAPTER 1^  3  process. The results of this thesis will provide correct values for the ideal flow calculation, so that a modified Cd could be defined to represent the losses directly.  1.1 Problem Description  Figure 1.1 General Slot Flow Figure 1.1 gives a general description of the flowfield for oblique slot injection into a main stream. 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 main  stream fluid to penetrate into the slot. Regardless, a shear layer forms, separating the secondary and main streams. Instability of this shear layer usually causes the flow to become turbulent downstream of the slot, resulting in mixing of the two fluids. If the injectant flow rate 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 following assumptions are implicit in the idealization: • The thicknesses of the shear layer and all boundary layers are small compared 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^  4  In the ideal case, fluid at total pressure Pts and density p s is injected from a slot at an angle  p  into an otherwise uniform free stream of total pressure Pt. and density No . A dividing streamline, beginning at a stagnation point on the upstream wall and extending downstream to infinity, separates the fluids.  Figure 1.2 Ideal Slot Flow  As the injected fluid leaves the slot, it is deflected by the free stream, asymptotically approaching a constant thickness h., and a constant pressure, p.. Unless stated otherwise, we assume 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 a flow separation model) in Chapter 2. In the absence of separation, the following four independent parameters provide for a unique 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 = p s I P.  (13)  5  CHAPTER 1^ [iv] The location, z s , of a stagnation point where the dividing streamline begins. For reasons explained in Chapter 2, the stagnation point is generally fixed at the upstream slot lip. For fixed values of these parameters, there is only one corresponding mass flow ratio, M = p s Us  /  (1.4)  that satisfies the flowfield boundary conditions. Relatively few analyses have been performed because 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 incompressible inviscid solution. From this point forward, we will classify all solutions into two categories: (i) Solutions to the special case Cp t=0, D=1. In this case, and only in this case, is the entire velocity field continuous. Without this restriction there is a discontinuity in tangential velocity across the dividing streamline (the explanation for this is given in Chapter 3). We shall term this the isoenergetic case, although the usual meaning of this term does not include the unit density ratio. (ii) All other values of Cpt and D. These solutions are termed non-isoenergetic.  1.2 Literature Review 1.2.1 Isoenergetic Theory Ehrich (1953) outlined a Helmholtz-Kirchhoff solution for a generalized slot/orifice configuration that includes the present geometry. To eliminate the infinite velocity point at the downstream slot lip, he included, as an option, a free streamline leaving smoothly from the downstream slot wall at the lip. This resulted in an unrealistic constant pressure separation region of ultimately infinite thickness. He solved the equations pertaining to the present geometry for the case 13=n/2 only. Dewynne et al (1989) analyzed the complementary problem of suction from an inviscid channel flow into a slot. They used classical methods directly because the entire flowfield consists of one fluid and is therefore isoenergetic. Their model includes smooth flow separation from the upstream slot lip in the form of a free shear layer. This shear layer divides  6  CHAPTER 1^  a region of constant pressure fluid from the flow into the slot. The amount of mass flow into the slot depends on the value of this pressure and the location of the stagnation point on the downstream wall. For small values of mass flow, the flow pattern is significantly different from the non-separated case. Ainslie (1991) used hodograph methods to study the flow from the aperture shown in the diagram. The flow was assumed to leave smoothly from the upstream wall and remain attached at the downstream corner. He also conducted experiments for ramp angles in the range 13=47-180°. Measurements of M at Reynolds numbers  Figure 1.3 Ainslie Geometry  above 2,000 (based on the free stream velocity and slot width) showed good agreement with the theoretical predictions. The nature of his apparatus made it difficult to check for flow separation, but he did find the flow to be slightly unsteady.  1.2.2 Non-Isoenergetic Theory Ting and Ruger (1965) point out, for the case of unequal total pressures, that the equilibrium of forces on a fluid element reveals a discontinuity in the velocity field across the dividing streamline. The standard method of conformal mapping cannot be used because the line separating the two streams in the complex potential plane is mapped into two unknown curves in the complex velocity plane. They also show that the solution for small Cpt does not approach the solution for Cpt =0 uniformly. In particular, the initial slope of the dividing streamline is a discontinuous function of Cpt near Cp t-0. For Cpt<0, the mainstream cannot support a stagnation point in the secondary flow at the upstream lip, therefore the flow must leave tangent to the slot wall (assuming that the flow separates at the corner). Similarly, the separating streamline must leave tangential to the mainstream wall for Cpt>0. For Cpt=0, both streams must incur a stagnation point, therefore the flow leaves at some angle 0<O 0 43. For the special case 13=n/2, they reduced the problem to two nonlinear singular integral differential equations. This simplification was due in part to their imposed condition that the secondary fluid exits the slot at a uniform angle f. Using numerical methods, they found solutions for Cpt>0, but were unsuccessful for Cpt<0.  CHAPTER 1^  7  Ting (1966) used linearized supersonic theory for the external flow to find analytic solutions for a wedge and flat plate with normal (inviscid) injection. The injection velocity was much lower than the wedge speed (implying Cp t.1), but larger than the value allowed for in boundary layer theory. Using a Cauchy integral formulation for the inner flow, Ting reduced 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—Cp t . The zeroth order solution was discarded because it produced an unrealistic result when transformed to the physical plane. Cole and Aroesty (1968) point out that the boundary layer on a flat plate "blows off' when pw vw p c„,Uce z R;112 for subsonic laminar flow, and when pw vw / pc„,Uce z 0.02 for subsonic turbulent flow (w a. conditions at the wall). They used an inviscid version of the Prandtl 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 theoretical injection 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 principle  was used to obtain numerical solutions. Comparisons with experimental data showed a significantly deeper jet penetration, probably due in part to the assumptions of a constant wake pressure and a uniform injection velocity across the exit. Goldstein and Braun (1975) analyzed nearly isoenergetic flow from two-dimensional nozzles and orifices. The optional addition of a free streamline (see Ehrich above) allowed for flow separation from the downstream lip. The solution involved expanding Bernoulli's equation for values of Cpt . In a manner similar to that used in thin airfoil theory, the boundary conditions on the unknown dividing streamline were transferred to the (known) isoenergetic dividing streamline. Using only cusped leading edges, they avoided the difficulties associated with the discontinuous initial slope of the dividing streamline (see Ting and Ruger above). The velocity discontinuity along the dividing streamline was handled by introducing a new dependent variable which satisfies only jump and symmetry conditions on the boundaries  CHAPTER 1^  8  rather than the combination of jump and boundary conditions which is imposed on the physical variables. 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 vertical slot into a free stream. Their model assumes that the total pressure in the slot exceeds the free stream static pressure by a small amount, implying Cp t .1 and M«1. Analyzing the injection process as a small disturbance to the free stream, they used a perturbation potential scaled on M and the Hilbert transform to express the solution in terms of a single integral equation. The numerical solution to this equation gave M.1.12(1—Cp t) 3/2 . Experiments were conducted using an open-circuit induced-flow wind tunnel to provide the free stream flow. The total pressure 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 downstream of the slot. The static pressure variations without injection were subtracted from the injection results to minimize tunnel wall boundary layer effects. The flow was turbulent with a typical slot Reynolds number of 6.5x10 4 . The upstream boundary layer thickness was approximately  Tiu of the slot width. They report a significant mass flow increase for a factor of three increase in the boundary layer thickness. For the range of mass flows applicable to their theory, they obtained good agreement with experiment.  1.2.3 Experiments and Numerical Work Sinitsin (1989) performed numerical simulations of the Navier-Stokes equations for 20° and 40° slots using a finite-difference Cartesian code. He also conducted wind tunnel experiments for these same geometries. From the experiments, he found distributions of speed across the slot exit, but not the flow direction. He proposed angular variations that would yield the measured flow rate, and then imposed the resulting velocity distribution as a boundary condition for the numerical simulation. This procedure avoids a 'staircased' representation of the slot that would otherwise result from using a Cartesian grid, and was thought to be a significant 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 effect on the flow field downstream of injection. The flow direction can, however, have a large  CHAPTER 1^  9  effect on the shear stress coefficient and the heat flux at the wall. He found good agreement between predicted and measured velocity profiles downstream of the slot for the 20° case, but only fair agreement for the 40° case. Measurements of the flow rate from the slot were slightly unsteady, and the flow was found to separate from the downstream slot lip (for M>0.9) in the 40° 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 an inclined hole. They find for all geometries that, 1 — Pts —Poo — A + B 2 (Cd ) 2 1ps UsM provides a good fit to the data. A and B are constants and Cd is given by (1.1). This simple form was derived by considering the component losses caused by the injected flow. For the slot, B was found to be sensitive to the slot angle and to the state of the boundary layer upstream of the orifice, and A was very nearly constant at 2.5. Numerical simulations using a finite-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 main stream boundary layer at the slot. Both numerical and experimental results were reasonably independent of slot Reynolds number for values above 1,000. The following two chapters describe new progress made in calculating the ideal flow from a slot into a cross flow. The results described in the literature review are compared to the new results in Chapter 5.  Chapter 2 ISOENERGETIC FLOW  In the previous chapter we stated that no comprehensive solutions exist for the non-isoenergetic flow case. During the initial stages of development of the present theory, it was apparent that a complete 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 the dependence 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 slope is 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. This step requires additional modifications of the non-separated flow model. Ehrich (1953) outlines a method for finding a solution to the present geometry. Owing to the complexity of the analysis, however, it seems unlikely that an analytic solution can be found with his method for slot angles other than n/2. We use instead the concept of a velocity potential (see, e.g., Milne-Thomson, 1968) to find a simple yet complete solution. This is a highly developed analytical tool for solving ideal fluid flow problems. The method involves applying 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. In all but the simplest cases however, the desired potential function cannot be specified directly.  10  CHAPTER 2^  11  Fortunately, we can map the geometry to an auxiliary plane in which the velocity potential can be written directly. A subsequent reverse application of the transformation gives the desired physical plane solution. Schwarz-Christoffel and hodograph transformations (see MilneThomson) are the most commonly used conformal mappings for rectilinear flows (i.e., the present geometry).  In many real flows, the presence of viscosity may induce flow separation from a sharp corner. 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 the present geometry. For non-separated flow, described first, the solution is uniquely determined by the geometry and the stagnation point position. In section 2.2, two models, based on Parkinson's wake source theory, allow for flow separation from the downstream slot lip. All solutions are based on a single Schwarz-Christoffel transformation to the upper half plane.  2.1 Non-Separated Flow  Figure 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 scale  12  CHAPTER 2^  all velocities on the free stream value and all lengths on the slot width. As indicated in the diagram, the scaled velocity flow from the slot is equivalent to the mass flow ratio because of the unit density ratio assumed in this analysis. For ideal flow, the most physically realistic location of the stagnation point S is at the upstream slot lip B. This eliminates the infinite velocity point that would otherwise occur at the corner. Aerodynamicists use a similar condition, 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 and therefore a deeper understanding of the problem. The downstream slot lip D is also a singularity 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 generally localized, and the removal of a singularity by a slight 'rounding' of the corner at D does not have an appreciable effect. Using the Schwarz-Christoffel theorem, we open the simple polygon .910o BC.co DE00 into the upper -plane (see Figure 2.2). Three vertices, 91 00 , B, and C o , in the z-plane correspond arbitrarily to -00, -1, and 0 in the -plane, while symmetry considerations place Too at +00. The values 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 at Acc, and an equal sink at Too . Thus in the -plane we must also have a source and sink at the corresponding points so that there is a uniform stream, Vco say. The uniform flow in the slot  13  CHAPTER 2^  transforms to a source of output M at C. Thus, in the z-plane, we have a source that emits the volume M per unit time over an angle  IL  We can now write the equation for the velocity  potential, F() =^+ — The complex velocity in the z-plane is then given by,  Voo + 11t dF dF w(z) - dz^dz K(t +1)1 X (t d) " -  -  ?  The following boundary conditions complete the problem description,  w(z)  0^at^r;= -s at^l;= 00 Me -Ili^at 0  Combining the above equations,  F() =  —+ In S  7t  (+ s)  w( z)  1) 1 A -  =  M  {( +1)-  -  7E  (2.1)  (2.2)  ^-  g  -1  where,  =m (23)  To find the solution, we must find the value of M that scales the geometry correctly. The standard 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, we can, in principle, find a solution. The integration is very involved for a general fraction, and it is further complicated by the presence of a singularity at  14  CHAPTER 2^  Using a completely different approach, taken from the analysis of flow in a branched canal (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 by a solid wall. Appendix A.1 contains the details of the derivation. The result is,  M=M-{ -–-Ai s –{J--1–X x }  (2.4)  '  2.1.1 The Stagnation Point Position To 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. For practical purposes we integrate numerically because the integrand is finite along the path of integration (see Appendix A.1.1 for details). The following diagram indicates the dependence of M on 3 and ;.  --*— n/18 n/4 --A— n/2 --o— 3n/4  M  0 -1 5^-0.5^0^0.5^1^1.5^2 ± !Zs'  Figure 2.3 Mass flow dependence on stagnation position  15  CHAPTER 2^  The vertical line (1;1=0) represents the Kutta condition. For mass flow ratios lower than the Kutta condition value (M=Mk), the stagnation point moves into the slot along the upstream wall. For M>Mk the stagnation point moves upstream along the mainstream wall. The stagnation point position moves infinitely deep into slot as M-0, and infinitely far upstream as M–.00. Variations in M appear to be greatest near Mk. Perhaps some of the observed unsteadiness of the actual flow is a result of this sensitivity. Note also the non-monotonic dependence of M on p for fixed Iz s l. Figure 2.4 shows the M-p relationship with the Kutta condition applied. The nonmonotonic behavior is clearly visible, with M reaching a minimum of about 0.757 at P=39.2°. 2.0  1.5  M 1.0  • Fitt et al 0.5  1^, 0^n/6^n/3  n/2^2.7E/3  R Figure 2.4 Mass flow with Kutta condition applied For (3–.0, the injected fluid does not deflect the free stream, and consequently we expect M--►1 to be a solution, as is shown in the figure. As the slot angle increases, the fluids deflect each other, and an asymmetric pressure gradient forms in the slot. The fluid on the upstream side of the slot experiences an adverse pressure gradient because of the presence of the stagnation point at the upstream slot lip. On the downstream side, the pressure gradient in the slot is favorable due to the acceleration of the fluid towards the singularity at the corner. Both gradients increase as 13 increases, but evidently the favorable gradient is dominant at higher  16  CHAPTER 2^  angles. 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 exists on 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 gradients and the flow from the slot. These arguments are supported by the experimental results of Fitt  et al (see Figure 2.4). They found a significant region of flow separation downstream of the slot, leading to a 35% reduction in the mass flow predicted by the isoenergetic theory. In Section 2.2 we augment the present theory to include the effects of flow separation.  2.1.2 The Dividing Streamline We can directly specify the equation for any streamline by setting the imaginary part of the velocity potential (i.e., the stream function 111) equal to a constant. In deriving (2.1), we tacitly assumed that the downstream wall has the stream function value W=0. Thus the value of the stream function constant is equal to the volume flow contained between the downstream wall and 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 in Appendix A.1.2. Figure 2.5 shows a sample of the results for the case 13-7E/2. 2 1.5 •  tLfi  y 0.5  I  Ni WANNSS^NNNNN^V, ,NNSC NNNNSNNVCWANNW  -0.5 -2^0^2^4^6^8^10 x Figure 2.5 Dividing Streamlines  CHAPTER 2^  17  In 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) for the range of M considered. Except for M=1 (the Kutta condition), the streamlines leave normal to the surface. It can be shown (see Appendix A.1.3) that the streamline leaves at an angle of 13/2 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 Flow Experimental results for a vertical slot (see, e.g., Fitt) show that there is significant flow separation for M of about 0.4 or more at reasonable Reynolds numbers. Past research shows that when the regions of flow separation are significant, classical potential flow methods yield inadequate solutions. Robertson (1965) lists several techniques developed in the 1940's and 1950's (modifications to the classical method) to include the presence of separated flow regions. Parkinson and Jandali (1970) developed a simple method of adding sources to the classical solution to simulate the effects of separation. This technique is the most straightforward of the methods that include separation effects, and is the method of choice here. The following sections describe the development of two possible separated flow models based on Parkinson's wake source model.  2.2.1 The Bubble Model The separation region effectively acts to displace the surrounding streamlines, much like a solid surface of the same shape and size. Thus the effects of separation can be approximated in inviscid flow by adding a separating streamline of about the same shape and size as the separated region. Parkinson employs such a streamline, through the addition of a source or sources of carefully selected strength and location, to simulate the wake of a bluff body. The addition of sources alone to the previous non-separated solution results in a separation region of finite height but infinite length. Experimental and numerical evidence shows that the separation region, if it exists, is finite in both height and length. To simulate a finite length region in potential flow, the separating streamline must reattach to the solid surface. This  CHAPTER 2  ^  18  requires (for a single source) the addition of a sink of equal strength located downstream of the source. This is the basis for the first separated flow model. Figure 2.6 shows the source/sink bubble model in the physical plane. Application of the Kutta condition at the upstream slot lip simplifies the analysis.  Figure 2.6 Bubble model physical plane Details 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 q 2 are the source and sink locations in the transform plane (see Figure A.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 q 2 must somehow be specified. In his work, Parkinson specifies the pressure at separation (the base pressure) to provide one extra condition. This condition cannot be used here however, for the following reasons. In the transform plane, we can show that the separating streamline leaves at D normal to the surface. Through an analysis similar to Appendix A.1.3, we can also show that this streamline leaves at an angle of it/2+13/2 in the physical plane. Thus D is a stagnation point, making it impossible to specify a variable pressure there.  19  CHAPTER 2^  We can, however, specify values for the length and height of the bubble. Using numerical methods, we can then find the corresponding values of q 1 and q 2 . Results for reasonable bubble dimensions show that the stagnation point is weak; the separating streamline bends rapidly to the tangential separation angle. Figure 2.7 illustrates the effect of bubble length for a vertical slot. Isoenergetic results from the numerical calculations in Chapter 4 give a bubble height of roughly one slot width and a length of about ten slot widths. The results shown here are for a fixed bubble height of one 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 for values greater than about 10H. 1 0.75 ^  ^  ^  ^  ^  M 0.5 0.25  o  0 10 20 30 40 50 60 70  L Figure 2.7 Bubble length influence  2.2.2 The Wake Source Model The previous model, although physically realistic because of the finite bubble, requires input consisting of the bubble height and length that is difficult to obtain in practice. The model is further hindered by the unrealistic stagnation point at separation. Figure 2.8 illustrates a second possible separation model, based on a lone source and a small teat (Parkinson, 1991) protruding from the downstream slot lip. The flow separates smoothly from the teat, eliminating the stagnation point and allowing the base pressure to be specified. Addition of the teat makes the analytical treatment more complex; for our purposes  20  CHAPTER 2^  the analysis is restricted to a vertical slot. We also eliminate the sink, to further decrease the complexity and empiricism. The results of the first model show that this should not significantly affect the mass flow.  Figure 2.8 Wake source physical plane In the diagram, k is the magnitude of the velocity at separation, and is directly related to the base pressure through Bernoulli's equation. Details of the analysis are given in Appendix A.2.2. The results are, M=  ri-712c d+  {-1(d 2 —1)(2d —1)  Afc7---71-  k  (2.6)  [V(d 2 —1)(2d —1) + d 2 + d —1}  b= 1 ln a^d^  d2  Again 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 the separation streamline for three different teat heights. The value of k here is 1.18, taken from the calculations in Chapter 4.  21  CHAPTER 2^  0.000^0.005  ^  0.010  ^  0.015  x-1  Figure 2.9 Tangential separation  The reason for having a teat was to force the fluid to separate tangent to the slot wall. If the fluid bends sharply upstream or downstream immediately after separating, this is not really accomplished. Thus we choose the value of d (i.e., 6) that produces zero curvature of the streamline at separation. Using numerical methods, we find 50.071, a reasonably small teat.  2.2.3 Comparisons Figure 2.10 shows a plot of the separating and dividing streamlines from three different models. The curves denoted BUBBLE are from the bubble model described in Section 2.2.1 and 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 with Cpt=0.0, Rh = ptl oo h/g= 10 5 , and a main stream boundary layer thickness at the upstream slot lip of about TiTh (h=slot width). These results were also used to supply both the separation region size (L .10.6, H..1.2) for the bubble model and the separation speed (k=1.18) for the wake-source model. The TEACH-II results show a much wider downstream spacing between the separating and dividing streamlines than either inviscid model. In the TEACH-II case, viscous action  22  CHAPTER 2^  causes 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 must increase. 3.0  2.0  ^  BUBBLE - - - WAKE ^ TEACH-II (Rh=105 )  1.0  0.0 10.0  5.0  0.0  Figure 2.10 Streamline comparisons The mass flow ratios for each model are listed in Table 2.1. Also given are the isoenergetic Kutta condition value from Section 2.1.1 and the experimental value from Fitt et al. The experimental conditions were roughly the same as those used in the TEACH-II calculations.  Model  M  Attached Flow  1.00  Bubble  0.54  Wake-Source  0.63  TEACH-II  0.67  Experiment  0.65  Table 2.1 Separation model mass flow ratios  23  CHAPTER 2^  The results from both separation models show a significant improvement over the nonseparated case. The predicted mass flow ratios from the wake-source and bubble models are in good and fair agreement respectively with the viscous results even though the streamline patterns are significantly different. We can gain some insight into the physics by examining the pressure field encounted by a fluid particle leaving the slot. Figure 2.11 shows a comparison of the pressure distributions along a streamline emanating from the slot centerline deep within the slot. The variable r measures the distance along the streamline starting from a point five slot widths into the slot.  '11-""""  0.5  [0.5,-5.0]  -  1 — pU 2 2 00 00  -0.5 -  -1 - - - ATTACHED BUBBLE WAKE ^ TEACH-II  -1.5 -  5^10  15  20  r Figure 2.11 Centerline pressure distribution For the attached flow case, the graph shows that the static pressure deep in the slot (p s) is equal  to the recovery pressure (p c0 ). This result can be verified by applying Bernoulli's equation along the streamline with M=1. Separation reduces the flow rate and results in the slot static pressure 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 the  CHAPTER 2^  24  downstream region of the slot is accelerated (the fluid is also accelerated as it turns downstream), corresponding to the significant pressure reduction shown in the diagram. As the flow moves downstream, the streamlines widen (see Figure 2.10), and the pressure rises to the free stream value. All three inviscid models produce a nearly uniform pressure in the slot. In the TEACHII case, boundary layers growing on the slot walls cause a slight acceleration of the centerline flow and consequently a slightly favorable pressure gradient. The pressure also recovers more slowly 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 caused by the rounded shape of the downstream portion of the bubble leading to a strong stagnation point. 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 due to a more realistic pressure distribution. This completes the isoenergetic analysis of ideal slot flow. The following chapter describes the development of a new non-isoenergetic analysis, guided in part by the physics revealed in this chapter. In addition, we will show how the preceding isoenergetic solutions can be easily transformed to produce solutions for the non-isoenergetic case, Cpt=0, Dol.  Chapter 3 NON-ISOENERGETIC FLOW  In this chapter we describe a new procedure for calculating non-isoenergetic inviscid slot flow for general values of 3, Cpt, and D. The first part of the chapter explores the limitations of classical theory towards such a solution. We also consider a possible alternative to the current method that makes use of the transform plane from Chapter 2. The central part of the chapter includes details of the current method such as solution methodology, boundary conditions, numerical approximations, and limitations. In the final part of the chapter we examine the possibility of adding a separated flow model.  3.1 Conformal Mapping Limitations The conformal mapping techniques used in the previous chapter provide an elegant method of solution under the restrictions of isoenergetic flow. In the non-isoenergetic case however, the velocity field is discontinuous and we can no longer apply these methods directly. This is best explained 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 Cp t<0 (see Figure 2.1 for a reference to the physical plane). Although the hodograph planes for Cpt>0 and Cpt<0 are 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 dividing streamline is at the upstream slot lip (i.e., the Kutta condition is in effect), and that all  25  CHAPTER 3^  26  velocities are scaled on the free stream value. For this discussion, we also assume that the flow does not separate from the downstream slot lip.  -iv [1-Cpt  1/2 OW.  D main stream  cpt LD  1/2  slot stream  Figure 3.1 Hodograph plane for Cpt<0 The superscripts + and - along the upper and lower surfaces of the dividing streamline correspond to the main stream and slot fluids respectively. At an arbitrary point P along the streamline, the flow is tangent to the streamline, so that a line drawn through P i" from P" must pass through the origin. To find the magnitude of the discontinuity in speed at P, we use Bernoulli's equation to write an expression for the continuity of static pressure at P, i.e., 1 I - -1Ps (V ) 2^- iPco(V + ) 2  where V+ and V- are the magnitudes of the velocities at  e and / " respectively. Using the 7  definitions 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 Cp t-0 and D=1.  CHAPTER 3^  27  In the classical solution procedure, we open up the pie-shaped hodograph plane into the upper half plane using the transformation =-w ' 03 , and then map the plane to the complex potential plane (F=13-i-iT) using a Schwarz-Christoffel transformation. These planes can then be related to the physical plane by,  dF dF dF 13= w or dz = — = —(-) 31 g dz w g In the isoenergetic case, this equation can in principle be integrated to give z as a parametric function of . The difficulty with the non-isoenergetic case is that a single point along the dividing streamline in the z-plane maps to two distinct points in the -plane, violating the basis of 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 the restricted range 1CpA«1. The solution involved transferring the non-isoenergetic boundary conditions on the dividing streamline to the known isoenergetic shape (in a manner similar to thin airfoil theory). They avoided some of the restrictions placed on conformal mapping theory by using sectionally analytic functions and by introducing new independent variables. It does not seem possible to extend their method to the general case because it is based on perturbations of Cp t near the isoenergetic value.  3.2 Vortex Sheets Aware of the difficulties with hodograph methods, we initially explored an alternative method that makes direct use of the -plane (see Figure 2.2) from Chapter 2. The idea was to replace the dividing streamline in the non-isoenergetic case with a vortex sheet. Since a vortex sheet is defined to produce a discontinuity in tangential velocity, it seemed possible, in principle at least, to find a solution if the strength distribution and position of the sheet were known. The introduction of an image vortex sheet would satisfy the solid wall boundary conditions, and we could 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^  28  To find V (x), however, we need to know the position of the vortex sheet and its strength -  distribution y(x). This leads to a non-linear implicit solution formulation. In addition, since V (x) is the physical plane velocity distribution, we must involve the Schwarz-Christoffel -  transformation function when specifying the -plane complex potential, thus increasing the implicit nature of the solution. It seemed unlikely that we could find a solution without the aid of iterative numerical procedures. This method was subsequently dropped in favor of the current method. For the past thirty years, researchers have applied classical methods to the nonisoenergetic cross flow problem, demonstrating the difficulty in finding a solution. In fact, all of the work done in the past two decades has involved numerical procedures, even though the solutions were only valid for restricted ranges of Cpt . The remainder of this chapter describes a new method that yields general solutions to the non-isoenergetic cross flow problem.  3.3 A New Zonal Interaction Method 3.3.1 History Zonal interactions originate from the application of boundary layer theory for predicting more accurate lift and drag coefficients on airfoil sections. To illustrate this concept, consider part of an airfoil section with a corresponding boundary layer.  Figure 3.2 Zonal modelling  CHAPTER 3^  29  The flow field separates into two distinct zones, an internal zone near the body surface where viscous 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 establish themselves to provide a balance of static pressure at the interface. The viscous effects of the boundary layer displace the streamlines outward by a distance O * (x) (the displacement thickness), thus altering the original inviscid flow patterns. To find a solution, we must find the 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 boundary layer. [1]  [2]  Calculate U 0 (x), the tangential inviscid velocity distribution on the body surface without boundary layer effects. Set U(x)=U 0 (x), where U(x) is the external velocity distribution at the edge of the boundary layer Calculate O *(x) using the equations for the boundary layer zone and the  previously calculated distribution U(x). [3]^Using the inviscid zone equations, calculate a new external velocity distribution U(x) based on the original airfoil having a surface displacement of O *(x). Repeat steps [2] and [3] until convergence. In recent years, researchers have applied this method to predict separated flow from stepped surfaces, 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, provided that the governing equations for each zone are known and boundary conditions at the interface can be specified.  3.3.2 Application to the Current Problem The current problem is ideally suited to zonal analysis because the dividing streamline separates the flow field into two distinct zones, and equation 3.1 provides the boundary conditions 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 field into an internal flow region containing the slot fluid, and an external flow region containing  the free stream fluid.  CHAPTER 3^  30  Figure 3.3 Oblique slot zones  Each region is isoenergetic, and can be solved independently using existing inviscid flow solution procedures. Determining the interface shape, S(x), that satisfies the boundary conditions (3.1) is the essence of the problem. To find S(x), we adopt a similar solution methodology as for the airfoil problem above. A few modifications are made to the airfoil algorithm 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 the upstream slot lip (see Figure 3.3), so that the algorithm must produce M=Mk as part of the  CHAPTER 3^  31  solution. The complete slot flow algorithm is given in Table 3.1. The details of the algorithm are discussed in subsequent sections. [1]  Set p, Cp l, 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 velocity along S(x) for the internal region, 1 7-(x).  [5]  Use (3.1) to calculate the distribution of velocity along the interface for the external region, V +(x).  [6]  Using an inviscid external flow solver, find a new shape S(x) that produces 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 methodology  3.4 Defining the Zonal Interface  Figure 3.4 Spliced cubic polynomials  The majority of the internal and external inviscid solvers referred to in Table 3.1 use approximate (panel) methods for solution, replacing curvilinear boundaries with linear or polynomial segments. Considering this, it makes sense to approximate S(x) since it is part of the boundary for each zone. In his M.A.Sc. thesis, Stropky (1988) used a method of  CHAPTER 3^  32  approximating a zonal interface with a series of N spliced cubic polynomials. The application of this procedure to the current problem is shown schematically in Figure 3.4, with the approximate curve S (x) replacing the exact shape. The coefficients of the cubic segments are given in Appendix B.1. S (x) is defined to be continuous in both value and slope, and thus by decreasing the lengths of the cubic segments (i.e., increasing N) we can represent the exact solution more accurately. The true dividing streamline extends downstream to infinity, so obviously we cannot approximate the entire curve with spliced cubics. We can, however, reasonably represent the section 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 the continuity equation from a point deep in the slot (where the flow is uniform) to a point at downstream infinity (where the flow is uniform and the static pressure is 1, 0 ). Bernoulli: P. = Poo +  IP sUE  -  ll  .—  i ''' ch a UE = ^ s  D  M Continuity: — D x 1= UE x hE Combining these equations we have, hE ....^  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 a small error with this approximation. We can estimate the effect of this approximation by changing the location of x E . The region Osxx E, termed the 'interaction region', is the region in which the zonal boundary conditions are satisfied. As an approximation to the exact shape, S (x) cannot satisfy the interface boundary conditions at every point. Following Stropky's method, we choose to satisfy the boundary conditions at the endpoint of each cubic segment. Thus, the cubic endpoints can be concentrated in areas where the velocity gradients are high to ensure solution accuracy. In the next section we will discuss the application of the Kutta condition using the current zonal model.  33  CHAPTER 3^  3.4.1 Applying the Kutta Condition From the isoenergetic results in Chapter 2, we found that (for fixed 3) the Kutta condition is only 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 than M=Mk. One such solution is shown in Figure 3.5 for the case Cp t=0, D=1, 13=7r/2, M=0.5.  \  Analytic -.- -.- Zonal Model ^  Figure 3.5 Kutta condition problems  The analytic theory for this case gives a stagnation point location z s =(0,-0.14), and a Kutta condition mass flow ratio Mk=1. The solution given by the zonal model is obviously invalid because we had preset the Kutta condition in a case where MoMk. To find the value of M that satisfies the Kutta condition, we need one additional boundary condition. The results of Ting and Ruger, together with the isoenergetic results of Chapter 2 show that the initial slope, 0 0 , of the dividing streamline is given by,  0,  cpt >0  0 0 = 0/2, Cpt — 0  0, cpt  <  o  (33)  This equation provides us with the extra condition necessary for a correct solution. When applied, however, this condition produces an unreliable relationship between 0 0 and M. In some cases we observed the same value of 0 0 for two values of M that differed by more than ten 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 transform  34  CHAPTER 3^  (3.3) into a boundary condition involving velocity. The governing equations for velocity are elliptic, thus the value of the velocity at any point depends upon the shape of the entire boundary (i.e., the entire curve §(x)). This transformed condition is precisely the same boundary condition used at each cubic endpoint. For the case Cpi <O, the streamline leaves tangent 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 also non-zero. Table 3.2 illustrates these conditions, and gives the values of V0+ and V0. We can now find the correct solution by satisfying the interface boundary conditions (3.1) at the cubic endpoints and at the stagnation point.  Cpt < 0  Separating ::::::: Vo+  -Cpt D +(V+)2  I^^° , D^  From^V (3.1),^= DI  ..VV >0, ...13 0  00  \  = 1 3 , . . V + = 0, or,  13  o  \i\  v' o-  _ 11-Cpt l io •• D Cpt > 0 From (3.1),^V0+ = .1Cpt +D(V0- ) 2 , :.V>0, : .0 0 - 0, : .V; = 0, or, 170+ = IfCpt-  Table 3.2 Upstream slot lip boundary conditions For the isoenergetic case we know that 0 0 4/2 and IC  -  Vo+ = 0. Because this is true for any  angle 0<0 0 <13, we are forced to use the original boundary condition (3.3) at the upstream slot  CHAPTER 3^  35  lip. Fortunately, in this particular case the M-0 0 relationship is monotonic and well behaved so that we can find the correct solution. For the case Cp t =0, Dol, we cannot use the isoenergetic results from Chapter 2. We know, however, from (3.1) that 170+ = -IT V0-", so again the upstream slot lip must be a stagnation point for both fluids. In Appendix B.2 we show that the initial slope of the dividing streamline is 13/2, regardless of the value of D. Thus we can again use (3.3) as a boundary condition at the upstream slot lip. Having solved the Kutta condition problem, we can now discuss the remaining details of the solution algorithm given in Table 3.1. In steps [4] and [6] of this algorithm, we discuss using inviscid flow solvers for each region of the flow field. In the next two sections, we will outline the theory for these solvers and show how they are applied to the current problem.  3.5 The Internal Flow Region 3.5.1 Bardina's Method In step [4] of the solution algorithm, we only require V - (x), the internal flow velocity distribution along the zonal interface. It is therefore inefficient and unnecessary to compute the entire internal flow field at each iteration, as the intermediate results are not solutions to the problem. Bardina et al (1982) have applied boundary integral methods to compute the inviscid velocity field around the perimeter of confined channel. The theory is based on the Plemelj integral formula and is related to panel methods used in external flow. No coordinate mappings 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 field boundary with NB points, denoted by the complex coordinates zi =xi +iyj , j=1,2,...,NB (see Figure 3.6). Along the dividing streamline, these points correspond with the location of the cubic endpoints. Next, Bardina shows that any analytic function j(z) that is analytic inside and on the boundary satisfies the following set of equations on the boundary. NB  Aki g(z j) =0, k = 1, 2, ..., NB E j= ^(3A) 1  CHAPTER 3  ^  36  where, -zk^zk^zk^zk -zi ^ In ^ Aki - ^ In -zi^zk -zi^zi -zi_i^zk -zj_i  Akk = In { z k -z k+i zk - zk-1  Figure 3.6 Bardina's method  To find the velocity distribution on the boundary, we set g(zj) = In[Vn- iaj, where yr is the magnitude of the velocity and aj is the flow angle. Substituting into (3.4) and taking the imaginary part of the result, NB r^NB  r  E 21A k;^- attA ki ai,^k 1,2,...,N B  J= 1  ^,  (3.5)  The flow angles, oci, are specified as tangent to the boundary, except at the inlet and outlet where they are normal to the boundary. For the problem to be well posed, we must specify the  ^ ^  CHAPTER 3^  37  value of V - at one point on the boundary. Here we set V1 = M, making sure that the entrance length (L1) is long enough for this condition to be accurate. To increase the computational speed of the method, we need only to recompute the AkJ along the dividing streamline during each iteration cycle.  3.5.2 Sridhar's Method Bardina's method, although an efficient algorithm for computing the velocity distribution along the 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 region using a potential field method developed by Sridhar and Davis (1985). The method consists of replacing the channel walls with polygonal surfaces, and then mapping the flow field to a straight channel (the x-plane) using the following Schwarz-Christoffel transformation, NL^am isinh 2(x - b m )1  d.z K m-1^ = K g(x) d^ x^NU [cosh 11(x - b n )] n=1^  n  (3.6)  where bm is the location in the transformed plane of the mth corner on the lower channel wall, a m is the clockwise turning angle at the mth corner in the physical plane. The constants b n and an are similar quantities for the upper channel wall. The constant K for the current problem can be shown to be,  K= where, as before, h E is the asymptotic height of the dividing streamline. The unknown quantities b m and b n are found from the following set of equations, zk.i - zk = K  fXk+1  Xk  g(x)dx  where k refers to either the upper or lower surface. Sridhar has developed an efficient iterative algorithm for finding the solution to this non-linear equation set. Floryan (1985) noted that Sridhar's method contains an error, which was also discovered and corrected in the application  38  CHAPTER 3^  to the current problem. Floryan's method allows for the use of curved elements along the channel boundaries, and can be used in place of Sridhar's method. Once the values of b m and b n are known, we can then use (3.6) to find the velocity field in the physical plane, i.e., dF dF/ dz =— — w(z) = u – iv =— dz dX dX The complex potential in the straight channel x-plane is simply F(x)=Mx, therefore, ^NU^an n[coshs(x_bld 3 M n=1 w z ^Ohg  ()_  n  NI,^am  [sinh 2(x -bm)]  3  m =1  (3.8)  In the next section we will discuss the external flow solvers used in step [6] of the solution algorithm in Table 3.2.  3.6 The External Flow Region  Figure 3.7 The external flow field Figure 3.7 shows the external flow region for the current problem. The lower boundary consists of the mainstream wall and the approximate interface curve, S (x). The control points,  39  CHAPTER 3^  k=1,2,...N, correspond with the endpoints of the spliced cubics, and the tangent angles ak are found 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) that produces the velocity distribution V + (x) computed in step [5]. Most external solvers work in the direct mode; they compute V + (x) given § (x). Stropky (1988) has developed a method for finding the inverse solution using these direct solvers, and in the next two subsections we use his method to find S (x) given V f(x).  3.6.1 Linearized Theory, the Hilbert Integral Significant improvements in computational speed can be realized by using linearized methods in step [6] of the solution algorithm. Fitt et al have successfully applied these methods for the case of a vertical slot with low mass flow ratios. The linearized results are valid for M«1 because 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 mass flow ratios and small slot angles. From linearized potential theory, the interface velocity can be written as the sum of the undisturbed 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 V + (x) is the P  perturbation velocity. For incompressible flow, the Hilbert integral from thin airfoil theory defines the perturbation velocity, i.e., 1i V+^ (x) – — P  +33  0  1{§() 17+ (0 ^4 x–  With Vu+ (x) =1 and VP+(X) << Vu+ (x), the preceding two equations can be combined to give the velocity distribution at the cubic endpoints,  40  CHAPTER 3^  d {§ ( 0  V + (xk) =1+1  ^4 0  xic  (3.8)  To invert (3.8), Stropky (1988) uses the following procedure. Each cubic endpoint p is given a small vertical displacement (ey p). The corresponding change in V+(xk) due to the perturbation of all endpoints is written as, Nr eV + (xk)= EiC2k,p AYp } p=1  (3.9)  The method for calculating the perturbation coefficients, C24, is given in Appendix B.3. An initial guess for §(x) produces an initial velocity distribution Vg+ (x k ). The corresponding endpoint displacements necessary to produce the desired velocity distribution V 4- (xk) are then computed from, Nr AV + (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 values are added to their corresponding endpoint heights and the process is repeated until the desired velocity distribution is found to within a prescribed tolerance. The details of the procedure are given by Stropky (1988), and are repeated in brief in Appendix B.3. The perturbation coefficients, C24, calculated from (3.8) can be used directly to find §(x) for any external inviscid solver.  3.6.2 Davis's Method There 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 of linearized theory are violated, and solution accuracy degenerates. (ii) For Cp r<O, the external flow has a stagnation point at the upstream slot lip. Since linearized theory fails completely at such a point, the  41  CHAPTER 3^  solution near the upstream slot lip may be grossly in error, even for low M. (iii) The Hilbert integral only gives the velocity distribution on the boundary. Davis (1979) has developed a potential field method based on numerical integration of the Schwarz-Christoffel transformation for general curved surfaces. The surface is subdivided into NE 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, is given by, dZ^1 NE^O m 1^ b)dcd} ^bm ) O m + f — = K exp{— [1n(t •g om m=1  where bm is the location in the transform plane of the mth element endpoint,  (3.11)  O m is the  corresponding finite slope change (if one exists), and 4 is the tangent angle of the physical plane curve. For element m, 4 is related to b by, (1) = Ci m + C2 m b + C3 m b 2  Substituting into (3.11), dz^NE 7 =K1-1{(--b m ) u `'^m=1  Om  C2 m C3m } N 2.11 ^N3 n7 (3.12)  The coefficients C2 m, C3 m, N2 m, and Kam are given in Appendix B.4 for reference. Following Davis, the velocity field is then given by, w(z) = NE  Am (^-  111=1  bm bm  )  C2 m C3m } N 2 m N3n 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 Section 3.5.2). Davis's method is directly suited to the current problem because the coefficients given  42  CHAPTER 3^  in Appendix B.4 are essentially derived from a curve made up of continuous cubic polynomial segments. As mentioned above, the coefficients 52 k calculated from the Hilbert integral and given in Appendix B.3, can be used in Equation 3.10 to find the desired shape §(x). This concludes the discussion of the flowfield solvers. In the next section we discuss the effects of the density ratio on the solution.  3.7 Density Ratio Effects In the following discussion we indicate how the solution for a given density ratio, D=D i , can be used to produce the solution for a different density ratio, D=D2 (for fixed 1 and Cp t). For the case D=D i , we can write (3.1) as, (VA)2 — x (VD1 ) 2 + Cpt Similarly, for D=D2 , (Vii; ) 2 D2 x (VE2 ) 2 + Cp t Next, we make the assumption that, by fixing the dynamic head of the free stream (i.e., 2 = constant ), the shape of the dividing streamline is not altered by a change in the  1-p co U„,  density. Since the governing equations for velocity are linear, the scaled external velocity distribution remains unchanged, i.e., Di + = D2^ V +^or,^VD2 = VDi 11D2 VDi  This equation shows that the internal flow velocities along the dividing streamline are simply scaled by a factor (D i /D2)'. Again, because the equations are linear, we can achieve this by scaling the velocity flow rate from the slot by the same factor, i.e.,  1^ 1—  Di ^Uc = U x 1 ^' ^ ' c2 D2  43  CHAPTER 3^  Since M=U s P s , this equation can be manipulated to give the relationship between the mass flow ratios, i.e., D2  M2 = Mlx 11 Di  (3.15)  In summary, the scaled external velocity field is unchanged (including the shape of the dividing streamline), and the entire internal velocity field is simply scaled by (D i /D2) 1/2 . These results can also be used in conjunction with the results of Chapter 2 to find an analytic solution to the non-isoenergetic case, Cpt=0, Dol. The results of Appendix B.2 are also verified because 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 for various values of Cpt, 13, and D, including comparisons with other methods, are presented in Chapter 5. In the next section we discuss the addition of a separated flow model to the present procedure.  3.8 Separated Flow  Figure 3.8 A simple separation model In Chapter 2, we modelled separated flow by adding sources and sinks of suitable strengths and locations. These singularities modified the complex potential, but the solution procedure was essentially unchanged. Using Shridhar's internal flow method, we could, in theory, add sources and sinks in a similar manner to find a solution. The transformation function in this case is  CHAPTER 3^  44  much more complex however; we would have to employ numerical methods from the onset of the solution procedure. This analysis is beyond the scope of the present work. We have developed a much less sophisticated model for approximating the effects of separation. Similar to the bubble model of Chapter 2, this model requires input in the form of the length and width of the separation region. The method essentially consists of modifying the lower boundary of the internal flow region to simulate the effects of the separating streamline. This concept is illustrated in Figure 3.8. The separation bubble is constructed from two spliced cubics; the first separates smoothly from the downstream slot lip and splices into the second which subsequently attaches to the downstream wall. The location of the join between the cubics can be adjusted to create different bubble shapes. The method is easy to implement as the bubble shape (i.e. the boundary) remains fixed throughout each iteration of the solution algorithm. Results and comparisons with other methods are given in Chapter 5.  Chapter 4 TEACH-II  In the previous chapter, we developed a new method for computing the steady inviscid nonisoenergetic flow from an oblique slot into a main stream. The results from previous work are limited to restricted ranges of Cp t, thus we have no basis for a general comparison of results from similar methods. We can, however, obtain comparative results from other sources such as experimental data or numerical Navier-Stokes solvers. Comparison with these methods also reveals the effects of the approximations used in the present theory. In this chapter, we employ a finite-difference Navier-Stokes solver (TEACH-II) developed by Benodekar et al (1983) to produce 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, calculations for density ratios other than D=1 are beyond the scope the current work, therefore the results are also limited to the unit density ratio case. To simulate the conditions found in most secondary injection devices, we assume that the flow is turbulent. The TEACH-II code uses a two equation (k-e) model of turbulence to approximate the true characteristics. In this model, the concept of turbulent viscosity, g t , is used to relate the kinetic energy of turbulence, k, and its dissipation rate, e. The turbulent viscosity is combined with the laminar viscosity to yield an effective viscosity for use in the transport equations, i.e., Reif = Pi + R  45  CHAPTER 4^  46  Following Leschziner and Rodi (1981), the code has been modified to account for the preferential influence of normal stresses in promoting the transfer of turbulent energy from large to small eddies. The following is a short discussion of the methods used by the code. It should be noted that the TEACH-II code was developed for internal flow and is a fully elliptic method. More efficient parabolic-elliptic external flow solvers exist, but they are not considered in this study.  4.1 Governing Equations For steady turbulent flow, the equations to be solved can be summarized by a single equation of the form,  1^  a^a^  (r 4) 21 + — (r a )— so =0 P^ — (4) + P -1 4) – a— 4) -l ax^ay (^ ax^ax^ay^ay^,..„_, convection terms^diffusion terms^source terms )  (4.1)  where u, v, and 4) (a general transport variable) are the mean flow quantities. Table 4.1 lists the expressions for f4, and S4 for the various transport equations.  Transport Equation  4)  ro  So  Mass  1  0  0  x-momentum  u  Refl.  –  ^+ ap^a  ax^ax  tj,  (^  au) + a  eff ax )^ay  i, ff  (^  av)  e^ax  Reif ^ap^ + a^ (veff avj + a ( Reif au) v y-momentum ^– ay^ay ay^ax ay Turbulence Energy  k  Dissipation Rate  E  !Leff ok  tttGk –pe  lieff  c^,^c 2 ci. it Ge L-2 P—  ae  Table 4.1 Governing equations  k  —  –  k  47  CHAPTER 4^  where, lit -  Gk = 2 [(  CIA Pk2 E  au 2 + av ax  2  au + av  2  ay^ax  ay  G E = Ci Gk -  +  C r bqs  C IA = 0.09, C 1 =1.44, C1= 2.24, Cf = 0.08, C2 =1.92 K  2  al( = 1.0, cr e = ^ , K = 0.4187 [(C2 - C 1 ) C 11/ 2 ] and S, is the shear strain in the direction of the streamlines (see Leschziner and Rodi).  4.2 Numerical Solution Procedure  NW  N^NE  W^  P  uP s  SW  E  •  S^SE  Figure 4.1 TEACH II grid cells -  The 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,  48  CHAPTER 4^  etc.), four faces (n, s, e, w), and two corresponding velocity cells (u and v). The velocities are evaluated on the boundaries of the scalar cells, forming a staggered arrangement of control volumes (see Figure 4.1). The scalar cells are located such that any cell in contact with the physical 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 control volume, i.e., Yn  f pu+ Ys  —  xe xe ax dy + f {Pv4) r ad)} x,  Yn  dy  s  ffS4) dx dy  Ys^CV  (4.2)  Details of the control volume formulation can be found in the text by Patankar (1980). The equations 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 scheme also uses a flux blending technique to eliminate unphysical higher harmonics typical of higher order schemes. The resulting finite difference equations are solved iteratively using the pressure implicit split operator (PISO) algorithm which corrects the velocities and pressures using a modified form of the continuity equation. The code uses a line-by-line solution method combined with an efficient tridiagonal matrix solver.  4.3 Problem Geometry Figure 4.2 on the following page is a schematic of the computational domain and associated boundary conditions for the current problem. The mainstream and slot walls are nonpermeable, and can be either free-slip or no-slip surfaces. Flow at the entrance to the slot is uniform, with negligible turbulent kinetic energy. The slot length is assumed to be long enough to be outside the influence of the slot exit. Slot wall boundary layer effects can be eliminated by using free-slip conditions on the walls.  49  CHAPTER 4^  To study the effects of the main stream boundary layer thickness, we have imposed a turbulent boundary layer of thickness 6 0 at the inlet. Fitt et al found that this boundary layer has a significant effect on the mass flow ratio when the thickness is of the same order as the slot width. The top wall is assumed to be far enough from the slot to simulate free stream the  conditions. In addition, by implementing a zero pressure gradient condition, we eliminate any downstream acceleration of the fluid along the top wall and better emulate the free stream condition. At the exit, we use a standard zero streamwise velocity gradient condition to aid convergence. For this boundary condition to be accurate, we must locate the exit sufficiently far downstream of any separation region that may exist. Use of a large solution domain implies more accurate boundary conditions, however, to maintain solution accuracy, the number of cells needed may become prohibitively large.  Figure 4.2 TEACH II geometry -  Staggering of the velocity cells with respect to the scalar cells causes some difficulty at the corners of the slot. The u and v cells that have half-faces in contact with a wall require  CHAPTER 4^  50  special treatment. Following Djilali (1987), the calculation of fluxes for these cells must use half 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 computational domain are given in Figure 4.2. The grid has at least 80 cells in the horizontal direction (with no less than 8 cells across the slot) and 60 cells in the vertical direction. The grid is nonuniform, with more cells concentrated in regions where the gradients are higher (i.e., around the slot exit). Studies show a less than one percent change in the mass flow ratio when the grid was 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 are made for different values of Cpt . The effects of boundary layers and flow separation are also examined.  ^  Chapter 5 RESULTS  This chapter contains a summary of results from the non-isoenergetic theory (NIT) developed in 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 are compared with results from Chapter 4, NIT solutions from other authors, and experiment.  5.1 Grid Refinement ^XE ^ 0 0 0 0 0 o o ^o^o o o 0 0 0 *00000;3 o 0 000 ^ )0000-  e LIN O^.  •  O ...  00000000000000000000000000000000  0  Z  O^o° 0 o 0  0 o:^000  o  0  °o 0 08/  Figure 5.1 The NIT grid The NIT solutions are only an approximation to the exact results because [i] the solution domain is limited in size [ii] part of the zonal boundaries (the dividing streamline) is approximately represented by a finite number of spliced cubic polynomials [iii] the boundary conditions are satisfied only at a discrete number of control points on the zonal boundaries.  51  52  CHAPTER 5^  A typical distribution of control points (referred to as the 'grid') used by the NIT code is shown in Figure 5.1. The external flow solvers require control points only along the dividing streamline, and the internal solvers require the entire array of nodes shown in the figure. The grid is concentrated near the upstream and downstream slot lips for reasons discussed in Section 5.1.2. The solution depends on both the size of the computational domain and the distribution of control points. In the next two sections we examine the effects of these parameters. For the purposes of the following discussions we will assume a unit density ratio for all calculations in Sections 5.1.1 and 5.1.2.  5.1.1 Computational Domain Size The size of the computational domain is a function of both the inlet length, (interaction) length, x E . For the puposes of this discussion  LIN  LIN,  and the outlet  is fixed at three slot widths  (found using a procedure similar to the one discussed here) and x E is free to vary. Figure 5.2 illustrates the effect of the interaction length on the zonal interface shape for a typical case.  0^2  4  6  8  x  Figure 5.2 Effects of the interaction region length  10  CHAPTER 5^  53  To enable comparisons with the exact result, we have chosen an isoenergetic case. The curve labelled 'IT (isoenergetic theory) is the analytic result from Chapter 2. In both NIT cases, the grid is sufficiently fine so that the shape of the dividing streamline shown in the figure does not visually change by adding more points. The effect of a short interaction length is obvious as the dividing streamline is forced to reach the asymptotic height y=h E too soon. The effect appears to be local, however, as the solution near the slot (the downstream slot lip is at x=1/sin40°.=1.56) is nearly identical to the exact solution. Both NIT cases give mass flow ratios within 1% of the exact result. In all cases examined, we found less than a 0.2% variation in the computed mass flow ratio for interaction lengths in the range 10 s x E s 20 (using identical grid spacings in the region Osxs6).  5.1.2 Grid Distribution In sections of the dividing streamline where the curvature is small, a relatively coarse grid provides an accurate representation of the exact shape. Conversely, in regions of high curvature, a fine grid is necessary to accurately model the exact shape. Regions of high slope also require a finer grid than regions of low slope because the grid is defined using the xcoordinates of the cubic endpoints, leading to longer cubic segments for steeper slopes. In addition, 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 distributions used 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 two factors: [i] The upstream slot lip is a stagnation point for either the internal region or the external region or both. [ii] The largest curvature generally occurs at or near the upstream slot lip (see Section 5.1.3). Figures 5.3 and 5.4 show the effect of grid size on the solution. The grid distribution is exponentially expanding/contracting (see Figure 5.1), based on width of the first cubic segment (ex i ) and a final cubic segment width of h E/4. The finest grid (ex i =0.005) involves 40 nodes along the dividing streamline, and the coarsest grid (ex 1 =0.5) has 12 dividing streamline nodes.  54  CHAPTER 5^ 1.2  e•----^  0.8 0.7  0^0.1^0.2^0.3^0.4  0.5  LOC 1  Figure 5.3 Effects of grid size, Cpt=0  0.2 ^ ^ 0 0.1^0.2^0.3 ^0.4 0.5 ex 1  Figure 5.4 Effects of grid size, Cp e--0.1 For the isoenergetic case (see Figure 5.3), the solution is fairly insensitive to the size of the grid. Over the range 0.01s  6.3E15 0.5,  M varies <15% in the 90° case, while for the 15° case, the  variation is about 1%. In addition, M becomes nearly grid independent for ex 1 <0.1 in either  55  CHAPTER 5^  case, and within 2% of the exact value for ex i <0.05. Plots of the dividing streamlines for the isoenergetic 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 stagnation point. The turning angle of the dividing streamline at stagnation is larger for the 90° case, thus the 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. The solution is now a strong function of the grid size, with about 50% variation in M in the range 0.01s ex i s0.5. In addition, unlike the isoenergetic case, M is a stronger function of ex i at smaller grid sizes. This is undesirable as a grid independent solution cannot be found, and unfortunate because M appears to be a strong function of Cpt near Cpt=0 , especially for higher slot angles. Fortunately, these undesirable charactertics only occur for the nearly isoenergetic case; 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 streamlines plotted Figure 5.5 for two different values of ex 1 .  x Figure 5.5 Effect of cubic spacing on dividing streamline shape near CprO  The large differences in asymptotic height are a result of the large difference in calculated mass flows, 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 of  56  CHAPTER 5^  the first cubic element is 15 ° , but this cannot be discerned from the scale of the diagram. The streamlines 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 the coarse grid streamline becomes highly distorted to satisfy these conditions. Figure 5.6 shows sketches of the dividing streamline near the upstream slot lip for values of Cp t just above and below the isoenergetic value. Also shown in the sketch are the non-zero velocity values at the slot lip required to satisfy the interface boundary conditions.  Vo + = e  Cpt= +6 2 £ 2« 1  Figure 5.6 Sketch of dividing streamline shape for small values of Cpt For Cp t---E 2, the streamline leaves tangent to the slot wall, however it must turn sharply to produce the (small) value for V0. Similar reasoning produces the sketch shown for C p As c—.0, these local distortions cannot be revealed by the NIT code, even for grid sizes as fine as ex 1 =0.001. In addition, for ICA <0.1, the solution algorithm would not converge for grid sizes smaller than ex i =0.01, making accurate predictions of M difficult. Fortunately, we can interpolate both from plots like Figure 5.4 and the final M-Cp t curve for more accurate results. For rp ti>0.3, the dividing streamlines do not contain any regions of high curvature and can be resolved accurately with a grid size of ex i -0.05.  5.2 NIT Results The algorithm given in Table 3.1 is valid over the entire range of each independent parameter, i.e., -co s Cp t 51, 0 5D 500, 0 5 f3 s n. To avoid the problem of multivalued coordinates using  CHAPTER 5^  57  the x-coordinate grid system, we limit the range of slot angles to 0 s 13 s n/2. We also limit Cp t and D to values that produce mass flow ratios in the range generally found in film cooling studies.  5.2.1 Unit Density Ratio In Section 3.7 we indicate how, for a given value of Cp t , the solution for a given density ratio can be used to find the solution for a different density ratio. We will elaborate on this in Section 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 of different density ratios. Figure 5.7 shows the NIT results for both external zone solvers, the linearized Hilbert integral (HB), and Davis' Schwarz-Christoffel method (DB). In both cases, Bardina's method was used for the internal solver. Linearized theory should be effective for Cp t >0, because there is no stagnation point in the external flow field and the lower mass flow ratios imply that the main stream streamlines are not deflected appreciably. On the contrary, for Cp t s 0, the presence of the stagnation point and the significant mainstream deflection (for larger slot angles 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 of validity. In the 90° case, results could not be generated for Cpt <0 because the intial slope of the streamline is n/2, which cannot be represented by a cubic polynomial. Davis' method makes indirect use of the cubic segments (only the endpoint slopes are required), so we can generate the desired results without specifying the polynomial coefficients. The trends are as expected, with increasingly negative values of Cpt producing higher mass 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 is reversed at lower mass flow ratios (<-0.6). In Section 2.1.1 we explained the non-monotonic behavior of the M-f3 curve. The same arguments apply here, except the more energy the slot fluid has, the more easily it overcomes the adverse pressure gradient at the upstream side of the slot. In fact for Cp t <0, the slot fluid stagnation point disappears, decreasing the strength of the adverse pressure gradient, and producing a monotonic M-t3 relationship.  58  CHAPTER 5^ 3.5 ^  3.5 ^ -+- HB  3.0 -  -o-- DB  2.5 -  M  1.5 -  ^0.0 ^  2.0 1.5 1.0 -  1.0 0.5 •  1=15 -3.0  ^  0.5 -2.0^-1.0  ^  0.0  -o-DB  2.5 -  2.0  M  -+- HB  3.0 -  ^  0.0^  1.0  1  13= 40°  -3.0^-2.0^-1.0^0.0^1.0  Cpt  Cpt  M  M  -3.0^-2.0  ' - 1.0^0.0  1.0^-3.0^-2.0^-1.0^0.0^1.0  Cp t  Cp t  Figure 5.7 M Cpt results for linear and non linear theory -  -  For Cp t >0, the converse arguments apply; the turning angle at stagnation for the slot fluid is smaller for larger slot angles, thus increasing the adverse pressure gradient on the upstream side, and decreasing the flow from the slot. In the next section we will discuss the M-Cp t relationship in relation to a non-unit density ratio.  5.2.2 General Results In Section 3.7, we showed that, for fixed values of 13, Cp t, and the free stream dynamic head, the following relation holds true,  ^  CHAPTER 5^  59 M2 - - constant  D  (5.1)  We can use this relation to easily develop M-Cp t plots for various density ratios. Figure 5.8 illustrates the results for a typical slot angle. The curves were generated by using the results from the unit density case in conjuction with Equation 5.1.  ^ 0.25 - -0^ 0.5 --A^ 0.75 -0^ 1.0 - -*^ 1.5 - I^ 2.0 --A^ 3.0  ^0  -3.5 -3^-2.5 -2 -1.5 -1^-0.5^0^0.5^1  Cpt Figure 5.8 M Cpt results for various density ratios -  We can also relate Cp t to M2/D by expanding the definition, i.e., 2 Ptco — 1 {Ps — Pco M2^M Cpt = , Pts^= J. "" ,^,^= 1 — CPs D -p.14,^plooU,,,, `^D^  1  (5.2)  60  CHAPTER 5^  Equations 5.1 and 5.2 show that, for fixed 13 and Cp t, the static pressure drop felt by the slot fluid is constant, irrespective of the density ratio. Figure 5.9 summarizes the relationship between M, (3, Cp t , and D, using (5.1) to collapse the data.  M D  2  -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0  ^  0.5  ^  1  Cpt Figure 5.9 M2/D vs Cpt For negative values of Cp t, the curves are nearly linear, implying from (5.2) that Cps is  a linear function of M2/D. In this linear region, we can describe the curves by, M2 D  = K1 — K2 Cpt  where K1 and K2 are positive constants that depend on the slot angle. Combining this expression with (5.2), we arrive at the following, 1 I Cd —^= + 1— j/M2 A +^— M2 1 P s U s2 K2^K2^D^  Pts  —  Pco  D^(53)  61  CHAPTER 5^  This is precisely the empirical relationship developed by Gartshore et al in their paper on discharge coefficients. They developed this simple relationship by considering the component forms 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 Cp t becomes nonlinear, and the losses are are no longer described by (5.3). We can, however, use the discharge coefficient defined by (1.1) to linearize this region. We can write Cd in terms of M2/D as follows,  M2 "2^1 n u2 2 ^'-' s^-2' t's S D Cd = pt —^= 2 —Poo s ^Ps + IP stJ — Poo M + Ps Poo 1^ D 1 Ps^  1p.U! —  -  -  or, using (5.2), M  Cd  —II 1—Cp D2  t  (5.4)  It is interesting to note that the right hand side of this equation is identical to (3.2), so that the discharge coefficient is equal to the asymptotic height of the dividing streamline. Figure 5.10 gives Cd in terms of Cpt for various slot angles. The discharge coefficient is often plotted in terms of M, so we have included a plot of Cd vs M2/D in Appendix C (Figure C.1) for completeness. Plotted in this manner, the region of low mass flow (Cp t—ol) is now linear, and the curves in this region can be described by, Cd — K3 (1 — Cpt ) where, similar to above, K3 is a positive constant that depends on the slot angle. Combining with (5.4), we have,  M — K3 (1— Cpt )T .11-15  (5.5)  CHAPTER 5^  62  This is precisely the relationship developed by Fitt et al (in their case .0'1) using linearized theory to describe the (low M) flow out of a vertical slot into a cross stream. Fitt predicts a value 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 .5 .  Cpt Figure 5.10 Cd vs Cpt For ideal flow from a slot into a stagnant fluid, the discharge coefficient, Cd, is always unity, regardless of the slot angle. Figure 5.10 shows that values both above and below unity occur when a cross stream is present. When the total pressure coefficient is near unity, the main stream flow acts like a lid over the slot, reducing the amount of flow that would be emitted into a stagnant fluid of the same static pressure as the cross stream, and thus reducing Cd below unity. When the Cpt is large and negative, the cross stream does not have a large influence on the injectant, so we might expect Cd to approach unity. With no cross stream, the streamlines 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 pressure  ^  63  CHAPTER 5^  gradients that pull more fluid out of the slot, thus allowing for discharge coefficients larger than one. The suction effect is greater at larger slot angles, as shown in the figure.  5.2.3 Dividing Streamlines 1.0  3.5 -- -- -2.0^3.92 - - - ^1.0^X2.431.46 - - -^ 0.8 -^- 0.0, / . _0.67 - - - - - .... 0.5^/ , - ^ -0-- 0.75' i• _-- - - - ^ -A-- 0.9 , i ^.0.25 — i :• 0.6,  / 2 z /,'' y /,./ /^0.068 0.4 -^, ' ,  0.0  •  •^i^WWWWWWW^i^I  0^2^4^6 x  •  •  8^10  Figure 5.11 Dividing streamlines, 11=15° Of interest in secondary flows is the trajectory of the injectant as it leaves the slot. Figure 5.11 shows the trajectories of the dividing streamlines for various values of Cp t for a slot angle of 15°. 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 into the main stream. Since Cd is equivalent to h E (the asymptotic height of the streamline), Figure C.1 in Appendix C shows clearly that an increase in the injectant energy at higher mass flow ratios does not signficantly increase the jet penetration. Equation 5.3 shows that the pressure drop ps -poo becomes proportional to the dynamic head of the injectant as M 2/D—.00. This makes sense, as the pressure drop caused by the bending of the mainstream streamlines becomes insignificant for large values of M2/D.  64  CHAPTER 5^  Figure 5.12 is an example of the effect of density on the dividing streamline shape for a fixed value of value of the velocity ratio. The velocity ratio is often used in place of M to create a flow parameter that is independent of the density ratio. Again, we can use the fact that M2/D=const at a given Cp t to generate the plots. 1.0  CA D  Us /U.=0.33  0.3 0.4 0.5 0.6 0.7  13---9o° 0.8 -  0.6 -  y  -  ••••••""  -  _  —  I  2.56 1.70 1.00 0.56 0.25  '  ----------------  •  0.4 -  0.2 -  0.0  0^2^4^6^8^10 x Figure 5.12 Density effects on dividing streamlines  For fixed U s/U.,, higher values of D imply higher kinetic energy for the injectant, thus there is more penetration into the main stream. This is consistent with the values of Cpt given in the legend, lower values giving larger penetration. Again, higher values of D for a fixed velocity ratio give diminishing return on the amount of jet penetration.  5.2.4 NIT Code Benchmarks Convergence rates ranged from 15 to 50 iterations of the solution algorithm (see Table 3.1) for ICA >0.2 and Cp t =0. In every case, the initial guess for the shape of the dividing streamline was a single parabola extending from the upstream slot lip to (x,y)=(x E, 1), with zero slope at the endpoint. The NIT code was compiled using Microsoft Fortran and run under Microsoft  CHAPTER 5^  65  Windows on a 50 MHz IBM compatible PC. Solution timings were 100-300 seconds for the Hilbert-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 same conditions, required 8,000-12,000 seconds to find a solution. A direct comparison is not appropriate however, as the TEACH-II code solves a more complex set of equations, and yields more detailed results.  5.3 Vertical Slot Comparisons In the following sections, we compare the NIT to other theoretical, numerical, and experimental results for the case of a vertical slot. Most of the NIT research found in the literature is concerned with this particular geometry.  5.3.1 Mass Flow Rates From the results of the previous section, we see that when the geometry is fixed, the NIT theory shows that M2/D is a unique function of Cpt . All the comparitive results used here are for 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 dimensions used in the model are taken directly from vector plots for the corresponding TEACH-II computations. Also shown in the figure are the theoretical and experimental results from Fitt et al. Both the TEACH-II and Fitt results are for a Reynolds number of 10 5 , based on the free  stream velocity and the slot width. A typical vector plot from the TEACH-II code is shown in Figure C.5 in Appendix C.2. For mass flow ratios greater than about 0.5 the results begin to diverge. The theoretical results from Fitt et al are expected to fail at high M because the results are based on linearized theory. The NIT results are not linearized, and although they give much better results at high  M, are still not particularly accurate. Incorporation of a separation bubble into the NIT code shows (for the range of values calculated at least) that flow separation has a significant effect on the mass flow.  66  CHAPTER 5^  ^NIT ♦ NIT bub - - - - Fitt theor + Fitt exp — - - Teach-II  Cpt Figure 5.13 Vertical slot mass flow comparisons  Figure 5.14 is a magnified view of the lower mass flow range of Figure 5.13. The results clearly show good agreement of all results to a mass flow ratio of about 0.5. Also included in this figure are the analytic separation model results from Chapter 2. The symbol labelled 'Analytic 1' is the wake-source model result and the symbol labelled 'Analytic 2' is the bubble model result. The separation flow models demonstate that inviscid flow models can give accurate representations of the mass flow ratio, provided that the displacement effects of separation are accounted for. We expect the agreement with viscous results to be better at lower slot angles because the region of flow separation will be smaller.  67  CHAPTER 5^  The results presented here show that the injection process is pressure dominated, as the inviscid models do not take viscosity, diffusion, or turbulence into account. This is not always the case, as we will see in the next section. 1.2 ^NIT • NIT bub - - - - Fitt theor + Fitt exp — - - Teach-II O Analytic 1 O Analytic 2  0.8  M 0.6  •  0.4 -  0.2 -  +^  „  13-90° D=1 0.25  0.5  ^  0.75  ^  1  Cp t Figure 5.14 Vertical slot mass flow comparisons at low M  5.3.1.1 Boundary Layer Effects In the viscous results presented in the previous section, the boundary layer thicknesses on both  the slot and main stream walls were less than one-fifth of a slot width. In practical situations such as film cooling, slots or holes located near the rear of the turbine blade encounter a significant main stream boundary layer. In this section, we use the TEACH-II code to study the effects of the main stream boundary layer thickness on the mass flow from the slot.  68  CHAPTER 5^  R 1 = 1 0,000 )  Cp t  0.0  -  ös -0.4 -  0.6  -4.- 0.20 -0- 0.44 - 6 1.33 -o- 3.90 -c.- 8.32 -o- 14.6 -  0.7  01.8  0.9  1.0  M Figure 5.15 Boundary layer effects Figure 5.15 shows the variation of M with the boundary layer thickness at the slot 050 for various values of Cpt. The trends make sense physically; thicker boundary layers have less average kinetic energy, allowing more fluid to escape from the slot. It appears that the part of the boundary layer nearest to wall has the most significant effect, as the largest increases in M occur when the boundary layer is small. These results support the experimental results of Fitt  et a/, who found a significant increase in the mass flow when O s increased from one-tenth to three-tenths of a slot width. A similar study could be made for the boundary layer thickness on the slot wall, but this is beyond the scope of this thesis.  5.3.2 Dividing Streamlines Figure 5.16 is a comparison of the shape of the dividing streamline near the slot for various mass flow ratios. The results labelled 'UM" are the experimental results of Fitt et al. At the two lowest mass flow ratios, no flow separation was present in the experiments or the viscous calculations. For M=0.6, significant flow separation occurred in both, but Fitt used a slightly rounded slot lip to reduce the extent of separation.  69  CHAPTER 5^ 1.0  NIT 0.1 - - - -^0.22^0.8 - --- 0.6  FITT 0.1 - 0.22 -^0.6  TEACH-II A 0.1 - -A - 0.22 -A - 0.6  M  A  0.6 -  y  A^,41(  0.4 -  /  , -  ®/  -.6;  - - 4-  A"^/ ^,'^.-_.i,-- . -  * 7  0.2 -  0.0  ,,  -  r  A fr , -, 4 ", /,- r A 4 -•  0.0^0.4^0.8  ^  1.2  ^  1.6  x Figure 5.16 Vertical slot dividing streamline comparisions  The results are in good agreement at the lower mass flow ratios, with the slight difference in trajectory most likely due to the displacement effects of the boundary layers in the experiment and the viscous calculations. For M=0.6, the results are signficantly different, due mainly to the displacement effects of the separation bubble. Fitt's experimental results lie between the TEACH-II calculations and the NIT results as expected.  5.3.3 Slot Exit Velocity Distributions Many modern Navier-Stokes solvers use a Cartesian coordinate system as a basis for their calculations. For the present geometry, this means a 'staircased' representation of the slot for angles other than 90°. To avoid this, some researchers have prescribed the conditions at the slot exit. Obtaining slot exit velocity distributions from experiment is difficult, so researchers generally prescribe a uniform distribution of velocity parallel to the slot. It seems unlikely that the distribution would be uniform, especially at lower mass flow ratios, owing to the shapes of  CHAPTER 5^  70  the dividing streamlines shown in the previous section. The slot exit velocity profile has a significant effect on the extent of separation downstream of the slot, and thus on the pressure field, 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 exit profiles. The results of the NIT calculations should provide a much improved exit profile for use in oblique slot calculations. To illustrate the deficiency of a unform tangential distribution, Figures 5.17 shows the vertical slot exit profiles from the TEACH-II and NIT calculations for M=0.5.  Figure 5.17 Slot exit velocity distributions, M=0.5 Both calculations show that the speed distribution is asymmetric and the flow is inclined to the slot angle. The NIT calculations are both more inclined and more asymmetric than the TEACH-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 (the downstream slot lip is an infinite velocity point in the NIT). Figure 5.18 shows that the distribution is even more inclined and asymmetric at lower mass flow ratios. The curve marked 'IT (isoenergetic theory) is from the analytic results of Chapter 2. Here, the stagnation point has been moved into the slot to attain the desired mass flow ratio. The results show a remarkable similarity to the NIT results over the downstream half of the slot. The NIT and TEACH-II distributions compare favorably at this mass flow ratio, where the flow remains attached. This favorable comparison should exist at higher mass flow ratios for smaller slot angles, where separation is less likely.  71  CHAPTER 5^  M=0.2  1.7s1  171  I  NIT  TEACH-II  Figure 5.18 Slot exit velocity distributions, M=0.2  5.4 Oblique Slot Comparisions Gartshore et al developed an empirical relationship (see Equation 5.3) for the discharge coefficient, Cd i , by considering the pressure losses in the injection process. The constant A corresponds to the losses in kinetic energy of the injectant and B corresponds to losses due to a local rise in pressure near the slot exit caused by the deflection of the main stream. The related constants K1 and K2 (see Section 5.2.2) can be found by measuring the slope and intercept of the linear sections of the curves in Figure 5.9. These constants can be combined to give an equivalent A and B, and the results are compared in Table 5.1 for f3=20° and (3 = 4 0 ° . In addition, 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 energy losses are less in the presence of a cross stream (because the favorable pressure gradients formed by the injectant turning parallel to the mainstream wall without separating), even for small cross stream flows. The constant B behaves as expected, as the main stream is increasingly deflected for larger slot angles.  CHAPTER 5  ^  72  The values of A and B from Gartshore et al are from a best fit through experimental data. In their experimental arrangement, slot air was supplied from a large plenum where measurements of the injectant total pressure were made. Thus we expect their value of A to be above unity due to the usual viscous losses.  B  A  13  K1  20°  0.62  1.08  40°  0.58  65° 90°  Gartshore  NIT (=1—K1/K2 )  Gartshore  0.93  2.5  0.43  0.09  1.02  0.98  2.5  0.43  0.12  0.66  1.50  0.67  N/A  0.56  N/A  1.0  2.70  0.37  N/A  0.63  N/A  K2  NIT (=1/K2 )  Table 5.1 Discharge coefficient constants The NIT results appear poor in comparision with Gartshore's empirical results. When the resulting discharge coefficient curves are plotted, however, the results appear somewhat more favorable (see Figure 5.19). The curves labelled 'Gartshore 2' are the results of the empirical relation, and the points labelled 'Gartshore 1' are from numerical TEACH-II simulations of their experimental geometry with a zero mainstream boundary layer thickness at the upstream slot lip. The experimental conditions used to determine A and B included a two slot width boundary layer thickness at the slot. From the definitions of Cd i and Cpt, we have the following relation,  1- Cpt Cdi M2  D Using the results from Figure 5.15, we find, for a vertical slot, that the discharge coefficients are lower for thicker boundary layers because Cpt is higher for a given value of M2 /D. If we  73  CHAPTER 5^  use this result to extrapolate to a zero thickness boundary layer for the experimental results, the disagreement at higher mass flows would be even worse than shown in the figure. 10  It 13=20° Gartshore 1 ■ — - — - Gartshore 2 ^ NIT  Gartshore 1 • — - - — Gartshore 2 ^ NIT  Cd 1  0^0.2^0.4^0.6^0.8^1^1.2  M Figure 5.19 Discharge coefficient comparisions The disparity of results may be due to several factors. Gartshore's results included a plenum which is not present in the non-isoenergetic theory. Entrance losses into the plenum and the resulting turbulence energy generation and flow patterns may have increased the discharge coefficient significantly. For higher mass flow ratios, the shearing effect in the viscous case may be greater at lower slot angles, causing greater losses in the viscous case. At low mass flow ratios, this same shearing effect may help draw more fluid out the slot than occurs in the inviscid case for the same value of Cp t. The main stream flow was also bounded. This may have had an effect on the resulting pressure distributions, and ultimately on the discharge coefficient.  Chapter 6 CONCLUSIONS  AND  RECOMMENDATIONS  6.1 Conclusions A new method has been developed for predicting the flow from an arbitrarily inclined slot into a uniform free stream of arbitrarily different density and total pressure (non-isoenergetic). We apply the method here using incompressible potential flow theory, but this is not a necessary restriction. The results provide a basic understanding of the physics involved in practical situations such as in the film cooling of gas turbine blades. The results also provide a more accurate assessment of the "discharge coefficient", defined as the ratio of the actual flow from the slot to the ideal flow, and an important parameter used to describe the operating state of these devices. To guide the development of the non-isoenergetic technique, we initially found an analytic potential solution (based on a single Schwarz-Christoffel transformation) for the special 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 such solution found in the literature was for the case of a vertical slot), the solution was used as a limiting case check of the non-isoenergetic results. From the isoenergetic solution we observed the following: 1. The mass flow ratio (Mv s Us/p.U cc) is uniquely determined by the slot angle 13 and the stagnation point position, z s , where the two streams initially contact. 74  CONCLUSIONS AND RECOMMENDATIONS^  75  2. As the stagnation point moves upstream along the main stream wall, the mass flow increases monotonically to infinity. Conversely, the mass flow ratio decreases monotonically to zero. The most physically realistic location of the stagnation point is at the upstream slot lip, as this eliminates the infinite velocity point that would otherwise occur there. With this 'Kutta' condition in effect, the mass flow ratio is given by the simple expression,  M  x —X  k  k=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-isoenergetic method. 4. As the slot angle increases, a physically unrealistic suction occurs near the downstream slot lip to keep the flow attached to the wall as it exits the slot. This effect causes M to increase to infinity as The previous analysis assumes that the flow remains attached to the wall as it exits the slot. In  the physical situation, the flow separates from the downstream slot lip at higher mass flows and slot angles. To estimate the effects of this flow separation, we constructed two flow separation models by adding appropriate source and sink singularities to the existing Schwarz-Christoffel model. These models are more realistic (for high slot angles at least), but require empirical input to supply the necessary boundary conditions. From the solutions we observed the following: 1. The mass flow ratio decreases when flow separation occurs. This is because the flow no longer makes a sharp turn as it exits the slot, thus reducing the strong pressure 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 the shape of the separation region within 10 slot widths downstream of the slot. The results from both separated flow models compare favorably with experiments and  76  CONCLUSIONS AND RECOMMENDATIONS^ other more sophisticated calculations, where significant regions of separated are found.  In the non-isoenergetic case, the velocity is discontinuous across the dividing streamline that separates the two streams. Classical methods cannot be directly applied in this case because of the discontinuity. The new non-isoenergetic technique involves separating the flowfield along the dividing streamline into two holomorphic regions, an internal region containing the slot fluid and an external region containing the main stream fluid. The solution is obtained by finding the shape of the dividing streamline that provides continuity of static pressure between the streams. After an initial guess is made for the shape of the streamline, we solve for the internal flow field using standard potential methods. We then solve the external flow field in an inverse fashion by finding a new streamline shape to match the interface pressure distribution from the internal flow field. The cycle is repeated until a specified maximum change in the streamline shape occurs between successive iterations. Due to the complex nature of the problem, we approximate the shape of the dividing streamline, and satisfy the interface boundary conditions at a discrete number of points along the curve. With the Kutta condition applied, and in absence of flow separation, the nonisoenergetic 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] the density ratio, D, of the two fluids. For given values Cpt and 13, there exits a unique value of M2/D, where M is the ratio of mass flow of the injectant to the main stream. The results found  from the application of the new technique to the current problem extend previous work to include arbitrary values of 13, Cp t, and D. From the application of the theory and the resulting solutions, we have observed the following: 1. The boundary conditions at the upstream slot lip were pivotal in finding a unique solution. For non-isoenergetic flow, one or both the internal and external flow regions incurs a stagnation point at this point. For Cp r>0, the dividing streamline leaves parallel to the main stream, creating a stagnation point in the slot fluid. For Cp t<0, the dividing streamline leaves parallel to the slot, and the stagnation point is in the main stream fluid. For the special case Ch=0, the analytic results show that  CONCLUSIONS AND RECOMMENDATIONS^ the initial slope of the streamline bisects the slot angle, producing a stagnation point in 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 stagnation point for cases of nearly isoenergetic flow (i.e., IC p tl«1) create difficulties in finding an accurate solution using approximate methods. 4. The use of linearized theory for the external flow field produces reasonable results for 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 that is a solution to the problem. This is because M2/D is a measure of the kinetic energy ratio between the streams, so that, for fixed Cpt, the static pressure is constant if M2/D is constant. This implies that a solution for a given density ratio can be used to directly produce solutions for alternate density ratios for the same values of Cpt and 3. It also implies that higher values of D produce lower values of the mass flow ratio. This result can also be used to find analytic solutions for density 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 is a result of the local pressure rise in the external flow near the slot becoming proportional 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  ^Us  Poo 1P  is a linear function of Cpt  s  .  8. In the presence of a cross flow at higher values of M2/D, discharge coefficients greater than unity can occur because the sink effect caused by the flow bending to  77  CONCLUSIONS AND RECOMMENDATIONS^  78  remain attached to the wall pull more fluid out of the slot than would otherwise occur. In this region Cd is greater for larger slot angles. 9. At lower values of M2/D, the discharge coefficients are below unity because the main stream acts like a 'lid' over the slot, reducing the amount of flow. In this region 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 jet penetration is limited, i.e., y max .1.64 for 13=90° . 11. For a given velocity ratio between the slot and mainstream fluids, larger values of D produce deeper jet penetration. 12. Slot exit velocity distributions show that the flow is significantly deflected by the main stream, especially at lower mass flow ratios. The distribution is also asymmetric, 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-difference Navier-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 slot angles 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 the upper boundary. In addition, mainstream boundary layers of various thickness were introduced upstream of the slot to study the effects on the mass flow ratio. The following trends were revealed in the (turbulent flow) results. 1. The results are nearly independent of Reynolds number in the range, 3 pU oo h^5 5x10 5 ^ 55x10 where 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 same total pressure difference (at least for boundary layers at the slot thicker than about 0.3 slot widths).  CONCLUSIONS AND RECOMMENDATIONS^  79  3. Most of the increase in mass flow occurs for boundary layers thickness between 0.4 and 3 slot widths. Comparisons of mass flow ratios with experiments, the TEACH-II results, and other inviscid results for the vertical slot case show good agreement provided that the separation region is not large. The effects of separation can be included in the present theory by incorporating a separating streamline into the model. The dimensions of the separated region are supplied from the TEACH-II calculations, and the results show a significant improvement. The shapes of the dividing streamlines and slot exit velocity distributions are in good agreement, again provided that separation, if present, is accounted for. Limited comparisons were made to numerical viscous predictions and experimental results for 20 ° and 40° slots. Non-isoenergetic predictions of a discharge coefficient defined by, _ Poo Cdi = Pts 1^2  f PsUs  -  are only in fair agreement with previous experimental results. Since no separation was observed in the results used for comparison, we suggest the following reasons for the differences, 1. The comparative results included a plenum which is not present in the nonisoenergetic theory. Entrance losses into the plenum and the resulting turbulence energy generation and flow patterns may have increased the discharge coefficient significantly. 2. For higher mass flow ratios, the shearing effect in the viscous case may be greater at lower slot angles, causing greater losses in the viscous case. At low mass flow ratios, this same shearing effect may help draw more fluid out the slot than occurs in the inviscid case for the same value of Cp t. 3. The main stream flow was bounded. The may have had an effect on the resulting pressure distributions.  CONCLUSIONS AND RECOMMENDATIONS^  80  6.2 Recommendations 1. Extend the isoenergetic wake-source model to include oblique slot angles. Use analytic methods to find the teat height that produces zero curvature of the streamline at separation. 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 near isoenergetic flow case to perhaps provide the correct streamline shape for the region of high 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 compressible inviscid flow theory. 6. Apply the solution algorithm to other inviscid flow problems such as jet-flapped wings or ground effect machines. 7. Use viscous-inviscid boundary layer theory for the external flow in conjunction with the 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 the internal flow region. 9. Modify the solution algorithm to solve the flow from two or more slots in succession.  REFERENCES Ackerberg, R.C. & Pal, A. (1968), "On the Interaction of a Two-Dimensional Jet with a Parallel 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 Low Mach Number Incorporating an Improved Correlation for Turbulent Detachment and Reattachment", Rept. PD-22, Thermosciences Div., Dept. of Mech. Eng., Stanford University. Benodekar, R.W., Gosman, A.D., and Issa, R.I. (1983), "The TEACH-II Code for the Detailed Analysis 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 Surface Injection", Intl. J. Heat Mass Transfer, 11, 1167-1183. Davis, R.T. (1979), "Numerical Methods for Coordinate Generation Based on SchwarzChristoffel Transformations", AIAA Paper 79-1463, 4th Computational Fluid Dynamics Conference. Dewynne, J.N., Howison, S.D., Ockendon, J.R., Morland, L.C., and Watson, E.J. (1989), "Slot Suction 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 Channel Flows", J. Comp. Phys., 58, 229-245. 81  REFERENCES^  82  Gartshore, I.S., Salcudean, M., Riaha, A., and Djilali, N. (1991), "Measured and Calculated Values 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 with Unequal Total Pressures", J. Fluid Mech., 70, 481-507. Kirchhoff, G. (1869) "Zur Theorie Frier Fliissigkeitsstrahlen" (On the Theory of Free Fluid Jets), 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 using Various 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, Englewood Cliffs, N.J. Roshko, A. (1954), "A New Hodograph for Free-Streamline Theory", National Advisory Committee for Aeronautics, Tech. Note 3168. Sinitsin, D.M. (1989), "A Numerical and Experimental Study of Flow and Heat Transfer from a 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 TwoDimensional Flow Grids", J. Fluids Eng., 107, 330-337. Stropky, D.M. (1988), "A Viscous-Inviscid Interaction Procedure", M.A.Sc. Thesis, University of 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.  Appendix A ISOENERGETIC THEORY  A.1 Attached Flow - Milne-Thomson's /4 Solution Figure A.1 shows the current geometry (bounded from above by a solid wall) in relation to Milne-Thomson's geometry. In his analysis, Milne-Thomson assumed that the stagnation point was at the corner. This corresponds to for the present analysis (see equation 2.3).  Uc> h  Present Geometry with Bounding Wall  ^  c>U1  Milne-Thomson Geometry  Figure A.1 Confined Slot Flow From the diagram we deduce the following, a=13,  h=h 1 = H, h 2 =1, U1  83  — —  1, U2 = M -  APPENDIX A^  84  Uh =Ui h i +U2h2^U =U1 P-!+U,-12-2- - - 11+} h^h^H The negative velocities are admissible because potential solutions are valid with velocities run in reverse. Substituting these relations into Milne-Thomson's equation 7 (pg. 292) and rearranging, we obtain an equation relating the mass flow from the slot to the height of the bounding wall and the slot angle. [  1 M 1 7013 H- il -  -  1 MH'^=1  We will expand this equation in terms of a small parameter E=1/H, and the limit c-.0 gives the desired result. With, f = --1^(0<k<1) -:  we have,  — e^{1 +^=1 -■^E m -4 + E {r  -  r } {i+ rEm + o(E 2 )} -1  m - Ar r } + 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 invariant in terms of if for any stagnation point position s, including s-1 (i.e.,  M  1-X  M=M). (A.1)  85  APPENDIX A^  A.1.1 The Stagnation Point Position Equation 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 is unnecessarily involved. To integrate, we separate (2.3) into real and imaginary parts, 12,1  z= ff(Od= fic(µm)-Fitcti,rodoi+iTo 11,1 Rol^ x^{6dµ -^and^y = {oth" + Tdtt} where,  .-1f^91{(1-t + 1 +^- r  +iyox  n (1-k j ^µ+irl  To=  '"51 ( + 1 + 61) 1-k ot -r lf k^1-t 11 +il  The results are path independent, provided that the path of integration does not include the singularity 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,  2 _11. k ki s {(11+ tt bl k(r I zs l = Vxs2 + y s^ -  -  dtst  -1  This equation can be integrated using any standard numerical method. Iz s l is the distance from the upstream slot lip to the stagnation point. For s>1 the stagnation point lies along the upstream main wall; for s<1 it is positioned along the upstream slot wall.  86  APPENDIX A^  A.1.2 The Dividing Streamline To find the dividing streamline shape in the physical plane we must first find the corresponding shape in the transform plane. Substituting tlf=,:'s(F)=M into (2.1),  M=  a{ -111 (+ 71 S  or, with .=-R+iri, = f1 + tan -1 — } 11 -1 tan{ri /s}  OsTi<n -1sR<co  To integrate (A.2), we make the following substitution, =^  - sin^cos sin2  ,  d1  (111  dll^S2  di^1  11  thus, xd f {crg2 t}dri^and^yd f{cr+ TC2}di 0^ 0 where 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 equations  A.1.3 The Initial Slope From (A.5), the slope of the dividing streamline in the -plane is, dri^sine dR^- sin CI cos The initial slope is therefore given by, cLril^sin2^112 +00,14)^1 = co lim { 2 ' C I— sin cos CI^ f: 2 i 1:0^- +^+ 0(i15)^-71-•0 3 T A 1 1=0 1-00 }  87  APPENDIX A^  This implies that the dividing streamline leaves the wall in the -plane at an angle of n/2. One of the properties of conformal mapping is that, away from critical points, angles are preserved between the planes. Therefore for sal, the dividing streamline leaves normal to the surface in the z-plane. For s=1 (the Kutta condition), the origin of the dividing streamline lies at a critical point of the transformation function. Examination of (2.3) shows that is a zero of order 1-13/n. At critical points with zeros of order n, angles in the ..plane are increased by a factor of n+1 when transformed to the z-plane. Thus the angle between lines intersecting at the upstream slot lip (---1) is increased by a factor 2--(3/n in the z-plane. Figure A.2 shows that when the Kutta condition is applied, the initial angle of the dividing streamline is (3/2. Dividing Streamline  c  (2-13hr) ni2 = Jr-13/2  mimmtwwwwww n12 aletwwwwwwwwwtme  Figure A.2 Initial Slope of the Dividing Streamline  A.2 Separated Flow A.2.1 The Bubble Model  Figure A.3 Bubble model transform plane  A  0/2  88  APPENDIX A^  Figure 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, ( + 1) 14' (dz -K d l;  nx  where 13/7c=n/(n+m). The constant K is different from the non-separated case because the scaling of points at infinity is based on the ratio of free stream velocities between the physical and transform planes. This ratio is changed in the bubble model due to the addition of the source and sink. From the diagram, F() = Vcc, +  M Q Q ln( - q2) ln + — ln( - q i ) - — n^n^7E  —  and therefore,  w(z) ...  M Q { 1^1 voc, + —+ n n^- (11^-C12 K(+1)1-?" (-  nx  We use the same boundary conditions as in the non-separated case, i.e., 0^at t = -1 1^at t = co Me-i I3 at^--, 0  w(z)  Next, locating the front of the separation region at the corner D, we have, w(t) = 0^at = d = From above boundary conditions we get, 1 1 -0 V. -M -g-{ 1^ n n 1+q i 1+q 2  where,  r  89  APPENDIX A^  Q{ vo. + r1 M a a ch1  _r  1 C12  -  }_0  r  Solving,  M^ii-r' -1 ^ A/0^1+ (q1 +1)(q2 +1) ( cil – r)(c12 -r)  (A.3)  A.2.2 The Wake Source Model  Figure A.4 Wake source transform plane Figure A.4 shows the Schwarz-Christoffel transform plane for the wake source separation model (see section 2.2.2). The addition of the teat (see Figure 2.8) changes the mapping function which is now given by, dz i. , –ozl lilt+1 dc –e From the diagram the velocity potential is,  Q F() . Vc„, + In + — ln( – q) It^TE  and therefore,  (A.4)  90  APPENDIX A^  {lio,-1- 7 M+  : Qn t( 1 q )} it—e  w(z)— ^  C(t  —  d)^1 t +1  Again, we have the normal three boundary conditions, 0  at  t = —1  w( z) = 1  at  t = co  —iM at t --> 0 and finally, we prescribe the separation speed (i.e., the base pressure) at D using, w(t) — 0 w(z) =k  at t=d  From these five boundary conditions we get,  M — Q { 1 =0 Vco—^ It t^1+q  V= C  C=  nd  le — d k C V. Q { q } n (q — d) 2  Vl+d  V + M Q { 1 }-0 c° nci n q — d These 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 =0 Combining the six equations above, and rearranging in terms of k and d,  APPENDIX A^  91  M= =  ITci -1  N  d+  47k  (A.5)  Finally, 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.6)  Appendix B NON-ISOENERGETIC THEORY  B.1 The Zonal Interface To compute the coefficients for the spliced cubic polynomials comprising §(x), we need to specify the value (yk) and slope (mk) at the kth cubic segment endpoint. The values are found iteratively as part of the solution, and we use the simple central difference formula to calculate the slopes, mk  Yk+1 Yk-1  ^  xk+1 xk-1  k -1,2...,N  If 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 +dk  11)  X-  Xk_.1  xk -xk_i  where, ak = Yk-1 m mk bk^ xk  mk +2mk-1 ck = 3 (Yk Yk-1) ( xk  92  ^Os s1  (B.1)  93  APPENDIX B^  dk =2(Yk-1-Yio+  mk +mk-1) xk  In the region x N <x<x E , S (x) progresses smoothly from the end of the last cubic segment to x E, where it reaches the asymptotic height hE. For simplicity, we can construct this segment from another spliced cubic polynomial, i.e.,  v = X -XN  fE( v) = aE + bE V + cE V 2 +d E v 3 ,  XE -XN  0s s1  (B.2)  By constructing fE(v) in this manner, we can use the above definitions of ak...dk, where, in this case:^k-1=N, and m E m0.  B.2 The Initial Dividing Streamline Angle 0 0 for DA. Figure B.1 shows a highly magnified region around the upstream slot lip. The dividing streamline leaves at an angle of 0 0 to the x-axis.  P 0 -  00 KV,  Figure B.1 O. for Dol From Milne-Thomson (pg. 155), the speed of the main stream fluid very near the stagnation point along the dividing streamline is given by, 80  114. = C + R 7776-c"  where C + is a constant, and R is the distance along the dividing streamline from the upstream slot lip. Similarly, for the slot fluid,  ^  94  APPENDIX B^  P- 0 0 V = C R '1-13+6 ° -  We know from equation 3.1 that V + /V - =1/5 = const. Combining these equations we have, 0 0^p-cl o  °  n-13+0 = constant R^0 R 0 "  For 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 2  Note that the results are independent of the density ratio D.  B.3 Inverting the Hilbert Integral We can write the perturbation to the velocity at cubic endpoint xk due to a perturbation of the dividing streamline  A§ (x) as,  ^n  ixE 1{ A (0 ^d AV+(xk ) = -1--^d ^xk 0  Next, we can write A (x) as the sum of individual perturbations eyp to each cubic endpoint p, i.e.,  N  AV + (Xk ) =  E p=i  xE -1c 1.fp(d { al 0 4xk _  Qic,p  4x  op  -^(B.3)  where fp (x) xey p = A p(x) is the linear perturbation to S (x) due to the vertical displacement of a single cubic endpoint p. Following Stropky (1988), we construct the perturbation curve fp(x)  95  APPENDIX B^  from two spliced cubics, as shown as the shaded area in Figure B.2. The curve is constrained to have zero slope at each end and a minimum area under the curve. This type of perturbation curve was shown by Stropky to provide stable numerical convergence.  Figure B.2 Perturbation curves The function4(x) is given by, 0  X < X,  Y-  1  x-x,,,-1  Aap + thp V + ACOP 2 + Adp V 3  r^  xp-x p .4  fp (x)  , 051)51  X-XP  Aap +1 + tibp+ iii) + ACp+ i '11.3 2^3 + 4p+1 11)^1p = xp+i-xp , osys1  0  X > X p+ i  where, eaP = 0 ' ebP= 0 ec., = E, edp = 1 '^r Aap+1 = 1 thp+i = (3 - 8)Ap+1 ecp+ i = -3(1+ 2 A p+ i) + 2E A p+ i edp+ 1 = (2 +3 Ap +1) - E Ap + 1 and,  (B.4)  APPENDIX B^  z... -  96  6,2P+1 (11+ 66 )+ 1)-5 2(1+ A3p+i )^'  A p+ 1 -  xn+i -x p r  Xp -  Xp_11  ,  For the case Cp t<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 corrections at this point (the function f i (x) given in equation set B.3 is not capable of altering 0 0 ). We have devised the following coefficients for fi (x) by degenerating the first cubic polynomial to a parabolic segment with variable initial slope,  ea t = 0, Abi = 2, ec i = -1, Adi = 0 Aa2 =1, Ab2 = 0, Ac2 = - 3, ari2 - 2  From B.3, we obtain the equations for the influence functions C2k ,p , i.e.,  ci--f.fp . (0 1 xE I 4 4 C2 k,p - — a xk 0 Care must be taken to evaluate the integral, as it is singular at x=xk.  B.4 Coefficients for Davis' Method From Davis, the functions C2 m, C3 m, N2 m, and X3 m are given by, 4 [0m+1 +Om (pm ] 2 C3 m = [b m+ 1 - b in ? C2 m = . n1+1- (1)In [b m+ 1 + b m 1C3m bm+i - bm b m rs- _ bm  1.  2m -  p-b m ] e^LL, e bm+1 N-bm+1P-bm+11  97  APPENDIX B^  eR+b,] 2 /2  83m  e [4- 13.+11 2 / 2  N_ bm p 2 -b.2 ]  N _ b m4.1 y 2 - b2,,,+1  where q is defined as the angle of the straight line connecting the m and m+1 points on the curve.  Appendix C NON-ISOENERGETIC RESULTS  C.1 NIT Results 2-^  1.5 •  Cd  1-  0.5 -^  --- 15° - - 40 ° - - 65° ^ 90°  0 0  1  2^3  m 2/ D  4  Figure C.1 Cd vs M2ID  5  6  APPENDIX C  99  13=40° I 0.8 -3'5 — -2.0 ^ 1.0 0.0 - 0.5 —0— 0.75 —6— 0.9  0.6 -  Cpr 0 0 0 0  A  0  2  4  x  A A  6^8^10  Figure C.2 Dividing streamlines, 13=40°  13=65 ° 1  -33 -2.0 ^ 1.0 — - - 0.0 — - - 0.5 —0-- 0.75 —6-- 0.9  M =0.0012 ^ 2  I^•  0^2^4^6^8^ • 10 X  Figure C3 Dividing streamlines, 13=65°  APPENDIX C  1 00 1.6  13.90°  -6.60'  ^  -  3.5 - -2.0 -  - - - 0.0 - - - 0.5 -0- 0.75 0.9 rpt  2  ^ A^=0.0009 A A A A 0.0 ^..^ . D^.. ...^.  0^2^4^6^8  Figure C.4 Dividing streamlines, 13=90°  C.2 Teach-II Results  10  

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