"Applied Science, Faculty of"@en . "Mechanical Engineering, Department of"@en . "DSpace"@en . "UBCV"@en . "Ainslie, Bruce"@en . "2010-11-10T17:26:31Z"@en . "1991"@en . "Master of Applied Science - MASc"@en . "University of British Columbia"@en . "The fluid mechanics of a jet issuing from a slot into a freestream is studied analytically and experimentally. The analytic study assumes incompressible, inviscid, irrotational flow for both the jet and freestream as well as equal stagnation pressures. The analysis proceeds by means of potential flow theory, using the Helmholtz-Kirchoff method of mapping the physical flow onto a simpler domain. This method examines the relationships between the geometrical and velocity parameters. Theoretical results for the mass flow out of the slot, shape of the streamline which divides the injectant from the mainstream, the coefficient of pressure across the slot, the velocity field across the slot and the coefficient of discharge from the slot are found for various slot angles. The calculated flow rates and coefficients of discharge are compared with data obtained from experiments performed using air flowing from a plenum, through a slot, into a small wind tunnel.\r\nThe calculated and measured flow rates give similar trends and agree within \u00B117%. Differences are due to experimental uncertainties and the neglect in the theory of downstream flow separation and subsequent reattachment, upstream boundary layer thickness, or turbulent mixing along the dividing streamline. Despite these differences, the gross influence of the mainstream on the mass flow from the jet compares well. The effect of unequal stagnation pressures in the jet and mainstream is investigated experimentally and found to be smaller than the effect of changing geometry.\r\n\r\nThe general agreement between theory and experiment suggests that pressure effects control the overall flow rate to a large extent and that this model can serve as a skeletal study for slot flow."@en . "https://circle.library.ubc.ca/rest/handle/2429/29914?expand=metadata"@en . "ANALYTICAL AND EXPERIMENTAL STUDY OF FLOW FROM A SLOT INTO A FREESTREAM by Bruce Ainslie B.Sc. (Eng.), Queen's University, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE D E G R E E OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering and Institute of Applied Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1991 \u00C2\u00A9 Bruce Ainslie In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Date DE-6 (2/88) ABSTRACT The fluid mechanics of a jet issuing from a slot into a freestream is studied analytically and experimentally. The analytic study assumes incompressible, inviscid, irrotational flow for both the jet and freestream as well as equal stagnation pressures. The analysis proceeds by means of potential flow theory, vising the Helmholtz-Kirchoff method of mapping the physical flow onto a simpler domain. This method examines the relationships between the geometrical and velocity parameters. Theoretical results for the mass flow out of the slot, shape of the streamline which divides the injectant from the mainstream, the coefficient of pressure across the slot, the velocity field across the slot and the coefficient of discharge from the slot are found for various slot angles. The calculated flow rates and coefficients of discharge are compared with data obtained from experiments performed using air flowing from a plenum, through a slot, into a small wind tunnel. The calculated and measured flow rates give similar trends and agree within \u00C2\u00B117%. Differences are due to experimental uncertainties and the neglect in the theory of downstream flow separation and subsequent reattachment, upstream boundary layer thickness, or turbulent mixing along the dividing streamline. Despite these differences, the gross influence of the mainstream on the mass flow from the jet compares well. The effect of unequal stagnation pressures in the jet and mainstream is investigated experimentally and found to be smaller than the effect of changing geometry. -ii-The general agreement between theory and experiment suggests thai pressure effects control the overall flow, rate to a large extent and that this model can serve as a skeletal study for slot flow. - iii -TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS iv LIST OF FIGURES vi LIST OF SYMBOLS viii ACKNOWLEDGEMENTS ix 1. INTRODUCTION AND EXISTING THEORY 1.1 Introduction 1 1.2 Film Cooling 1 1.3 Existing Theory 2 2. PRESENT THEORY 2.1 Reasons for the Particular Model 7 2.2 Helmholtz-Kirchoff Method 11 2.3 Present Analysis 13 2.4 Location of Delta 19 2.5 Shape of Dividing Streamline 19 3. RESULTS 3.1 Slot Width d/h 21 3.2 Shape of Dividing Streamline 24 3.3 Mass Flow Out of Slot 26 3.4 Coefficient of Pressure 27 3.5 Velocity Across Slot 30 3.6 Discharge Coefficient 33 4. EXPERIMENTAL WORK 4.1 Description of Apparatus 35 4.2 Experimental Procedure 37 4.3 Experimental Results 40 - iv -T A B L E S O F CONTENTS (Continued) Page 5. DISCUSSION OF RESULTS 5.1 Experimental Errors 52 5.2 Expected Effects of Reynolds Number 54 5.3 Coefficient of Discharge 55 5.4 Off Equal Energy Results 56 6. RECOMMENDATIONS AND CONCLUSIONS 6.1 Improvement of Experiment 57 6.2 Improvement of Theory 58 6.3 Conclusions 59 7. BIBLIOGRAPHY 60 8. APPENDIX I Calculation of Dividing Streamline 62 9. APPENDIX II Change of Variables 63 - v -LIST O F F I G U R E S Page 2.1 90\u00C2\u00B0 Slot with Finite Depth 8 2.2 Truncated Slot For Arbitrary Angles 8 2.3 Dividing Streamline 9 2.4 Cavity with Lip 10 2.5 Slot Geometry Used In Analysis 11 2.6 Physical or z-plane 13 2.7 Hodograph or \u00C2\u00A3-plane 14 2.8 Potential or w-plane 14 2.9 Upperhalf or T-plane 15 2.10 Power Transformation 16 2.11 Mapping Sequence 18 3.1 Path of Integration 21 3.2 Graph of d/h vs 6 24 3.3 Plots of Dividing Steamlines 25 3.4 Graph of Mass Flow from Slot 26 3.5 Graph of Cp vs Position for Various 6 29 3.6 Velocity Field Across Slot t>=20^ 31 3.7 Velocity Field Across Slot B=40\u00C2\u00B0 31 3.8 Velocity Field Across Slot B=90\" 32 3.9 Velocity Field Across Slot B= 180\" 32 3.10 Discharge Coef for Various Angles 34 4.1 Plenum Cross Section 36 -vi-4.2 Experimental Set-up 37 4.3 Flap Attachment 38 4.4 Mass Flow vs Velocity 6=47\u00C2\u00B0 40 4.5 Mass Flow vs Velocity 6=60.5\u00C2\u00B0 41 4.6 Mass Flow vs Velocity 6=71.5\u00C2\u00B0 41 4.7 Mass Flow vs Velocity 6=90\u00C2\u00B0 42 4.8 Mass Flow vs Velocity 6= 180\" 42 4.9 \u00E2\u0080\u00A2 Slot Velocity vs Reynolds Number 6=47\u00C2\u00B0 43 4.10 Slot Velocity vs Reynolds Number 6=60.5\u00C2\u00B0 44 4.11 Slot Velocity vs Reynolds Number 6=71.5\" 44 4.12 Slot Velocity vs Reynolds Number 6=90\u00C2\u00B0 45 4.13 Slot Velocity vs Reynolds Number 6=180\u00C2\u00B0 45 4.14 Best Fit Data vs Theory 46 4.15 Reynolds No. Dependance on Slot Velocity 47 4.16 Coef of Discharge vs Reynolds No. 6=47\u00C2\u00B0 48 4.17 Coef of Discharge vs Reynolds No. 6=60.5\u00C2\u00B0 48 4.18 Coef of Discharge vs Reynolds No. 6=71.5\u00C2\u00B0 49 4.19 Coef of Discharge vs Reynolds No. 6=90\u00C2\u00B0 49 4.20 Coef of Discharge vs Reynolds No. 6=180\u00C2\u00B0 50 4.21 Best Fit vs Theory for Coef of Discharge 51 4.22 Highest R e s data vs Theory for C D 51 - vii -LIST OF SYMBOLS P t j total presure of injected flow P t 0 0 total pressure of freestream flow Pj static pressure of injected flow Poo static pressure of freestream flow p density of air UQO freestream velocity z physical plane coordinates x + iy t, hodograph plane coordinates u -iv u streamwise velocity component v normal velocity component w potential plane coordinates 4> + hp 4> velocity potential stream function T upper half plane coordinate a +ib d slot width h asymptotic height of injected flow downstream 6 slot angle T lower half plane coordinate A location of downstream corner in upper half plane rj modulus of complex number 6j argument of complex number y path of integration C p coefficent of pressure L'p \ fopU^ Cr> coefficient of discharge CD = V S slot velocity V S = U X h/d -viii-ACKNOWLEDGEMENTS I would like to acknowledge the help and support of my supervisor, Dr. I.S. Gartshore. I would also like to thank Ed Abel for his help in the construction of the experimental apparatus. Finally, I would like to acknowledge the Natural Sciences and Engineering Research Council of Canada for its funding. - ix -1 CHAPTER 1 INTRODUCTION AND EXISTING THEORY 1.1 Introduction The study of a jet injected at an angle into a uniform stream has been investigated. Work of this type has been carried out for many years for different purposes: mixing processes in combustion chambers, mixing of sediment laden streams into reservoirs, blood flow in arteries of mammals [ 1 ] and protection of materials from high temperature gases by the introduction of a secondary flow. It is this last application - called film cooling - that serves as a motivation for this project. 1.2 Film Cooling The design of more compact and efficient gas turbines has led to increased inlet gas temperatures. Unfortunately, gas temperatures over 1300 K can cause extensive wear on turbine blades unless these are properly protected. It has been shown (21 that a variation of 20 K around a mean blade temperature of 1200 K can halve or double the life of a blade. Hence, there is a need for a sophisticated cooling scheme to prevent metallurgical breakdown of the turbine blades. High inlet gas temperatures and the prevention of wear on the blades are achieved by circulating cool air from the compressor through channels inside the blade and then bleeding this cool air through small holes located on the leading edge of the blade surface. This fluid flow over the blade surface results in a thin film of cool gas shielding the blade surface from the hot inlet gases. Film cooling near the leading edge will not only protect this 2 region but it will affect the fluid mechanics and heat transfer over the entire blade surface. For this reason, it is important to study the fluid mechanics of the film cooling process. Of special interest is the flow field around the hole. In particular, the shape of the dividing streamline (which separates the injected flow from the mainstream), the velocity and pressure distribution across the hole surface, the mass flow from the hole, and the coefficient of discharge from the hole (which serves as a measure of the pressure within the blade needed to discharge the required mass How through the hole), are of engineering significance. 1.3 Existing Theory In general, the study of film cooling is one of a viscous fluid flow in a complicated 3-D geometry. Typically, there will be a shear layer separating the freestream and the injectant. The injected flow may separate at the rear of the hole and then reattach. There will also be a viscous interaction ahead of the hole due to the boundary layer of the mainstream. In addition, due to viscosity and the instabilities in the shear layer, the flow will become turbulent downstream, and possibly upstream, of the hole and mixing between the injectant and the freestream will occur. Analysis of the full Navier-Stokes equations, including turbulence terms, is quite difficult and often requires additional empirical information [3),[4j. However, the full details of the flow field are not always necessary in order to examine the interaction of an injected flow and a freesteam. 3 Some insight can be gained by examining a two-dimensional inviscid incompressible flow since the equations involved are analytically tractable. However, this problem is still difficult because the shape of the dividing streamline becomes an unknown boundary and hence gives rise to a nonlinear problem. Correspondingly, there are many ways to analyze this problem. Cole and Areosty (5) attacked the 2-D problem by assuming that the thickness of the injected layer is small compared to all other pertinent dimensions in the problem. By scaling the Navier-Stokes equations with this small parameter and by noting that in the injected layer the normal components of velocity can be considerably larger than the inverse square root of the local Reynolds number, they show that the usual Prandtl boundary layer equations are not valid. Instead the flow within the dividing streamline satisfies the so-called \"inviscid\" boundary layer equations, while outside the flow is assumed to be a potential flow. The dividing streamline becomes a discontinuity in velocity with the pressure being continuous across it. By transferring the problem to stream function coordinates, a relationship between the pressure field, the injected flow distribution across the slot and the shape of the dividing streamline can be found involving an Abel integral. Unfortunately, this analysis requires the knowledge of both the pressure field and the injected flow distribution before the dividing streamline can be found. One aim of this project is to determine the flow distribution at the surface of the slot and not to prescribe it. In a similar approach, Ting and Ruger [61 examined the same problem with the inviscid boundary layer equations holding in the injected layer and the freestream being described by an appropriate 4 theory. They assumed that the flow at the slot surface was everywhere constant and inclined at the same angle as the slot. They used conformal mapping methods on the injected layer to develop an integral equation for the shape of the dividing streamline. Again, the disadvantage of this approach is that the flow from the slot surface is fixed and from computational work the flow does not appear to behave in this way [7]. Finally, Ting [8] has also shown that this approach will not admit to a perturbation expansion for small differences in total pressure between freestream and injected fluid since the shape of the dividing streamline will not depend continuously on the small parameter e = P t 0 0 -Pti-Since the dividing streamline is an unknown, the two-dimensional potential problem becomes a nonlinear free boundary problem. Fitt, Ockendon and Jones (2) examined this problem by means of a perturbation expansion. If the total pressure head of the injected fluid is, again, P t i and the static pressure of the freestream P^, then they considered the case where P t l - P x is small with respect to but also large enough to assume that viscous boundary layer effects may be ignored. They then obtained an integral equation for the velocity potential in terms of the small parameter E where P t i = P^ + 1/2 p U ^ e 2 . Numerical evaluation of the integral gives the shape of the dividing streamline and the mass flow from the slot. In addition, their method imposes, as a constraint, tangential separation of the dividing streamline from the leading edge of the slot, which one would like to have since only the injected layer would have a stagnation point at the leading edge when the total pressure of the injected is less than the total pressure of the 5 freestream. However, it is not practical to extend the theory from the normal injection required in the analysis to arbitrary slot angles. Another approach to finding the shape of the dividing streamline involves taking the nonlinear free boundary problem and turning it into a calculus of variation problem (9). The variation integral is then solved computationally by means of a Ritz-Galerkin method. A simpler way of examining the problem is to consider the case of equal stagnation pressures. This eliminates the discontinuity in velocity across the dividing streamline and allows the methods of conformal mapping to be used. The problem is solved by mapping the slot domain to a simpler region, where the field lines are easy to analyze, and then mapping back to the slot domain. David Stropky [101 used a Schwarz-Christoffel transformation on an infinitely deep slot to calculate the shape of the dividing streamline and mass flow rates for arbitrary slot angles. Unfortunately, his approach does not give tangential separation of the dividing streamline. Also, the flow from the slot does not represent flow from a reservoir or plenum where the fluid is at rest. One way to achieve tangential separation while avoiding the need to prescribe the velocity at the slot surface is by means of hodograph methods. These are conformal mappings, but unlike the Schwarz-Christoffel mappings, these are not concerned with the physical plane but more with the velocity fields. Ehrich [11] examined the flow from an orifice in this manner. His method provided tangential separation but the shape of the dividing streamline was never calculated. Problems with arbitrary slot angles were never explicitly examined. 6 Goldstein and Braun [12] extended Ehrich's work to include small perturbations in total pressure by means of sectionally analytic functions. They, however, calculate the dividing streamlines and mass flow rates only for an orifice. The analytic work developed in this thesis is the explicit extension of Ehrich's work to arbitrary slot angles. 7 CHAPTER 2 PRESENT THEORY 2.1 DESCRIPTION AND JUSTIFICATION OF PRESENT MODEL Since a 2-D model of an inviscid ideal fluid was to be considered, it seemed complex function methods were the obvious way to start. The goal was to examine the problem with a 90\u00C2\u00B0 slot and then eliminate this restriction to include arbitrary angles. For the case of equal stagnation pressures, it was obvious to start with Schwarz-Christoffel transformations to transform the slot flow to the upperhalf plane. The general background theory can be found in ref. (131,(141. The required transformation between planes, for a 90\u00C2\u00B0 slot of finite depth (fig 2.1), is: dz _ K dw (w - a)(w - b) 2. i (w - c)(w \u00E2\u0080\u0094 d) (where z is the physical plane and w is the upper half plane). The variables a.b.c.d represent the positions of the vertices in the upperhalf plane and K is a constant needed to properly orient the map. These must be chosen by trial and error in order to get the slot symmetric in the physical plane. This equation gives an elliptic integral which cannot be solved analytically. The slot was chosen to have a finite depth so that the velocity field could be prescribed at the bottom of the slot. 8 Figure 2.1 90\" slot with finite depth. A computer code was written to solve arid plot the transformation. However, the shape of the dividing streamline in the upperhalf plane did not seem analytically tractable and hence a computer code was needed to generate it. The problem was no longer an analytic model but a rather crude computational model. To simplify matters, the slot was modified. It was suggested that the trailing edge of the slot was probably the most important aspect of the geometry so the leading edge was kept at a 90\u00C2\u00B0 corner and the trailing edge remained angled to form a cavity (see fig. 2.2). v. \ S ^ ^ Figure 2.2. Truncated slot for arbitrary angles. 9 Also, to Include the tangential separation of the mainstream from the slot surface, it was hoped that by adding a source at the bottom of the cavity as well as a sink along the upstream wall, tangential separation, could be modelled ( ref (15] ). However, it was not possible to find the location or the strength of the source that was needed. This becomes obvious when one remembers that the conformal map will preserve the angles between streamlines from plane to plane except at stagnation points. No matter where the source is, the dividing streamline intersects the upperhalf plane at 90\u00C2\u00B0. Now, since the map is not conformal here (the injected flow has a stagnation point here), one does not find the dividing streamline leaving the slot at 90\u00C2\u00B0 but at 90\u00C2\u00B0 or n/2 radians multiplied by the local effect of the map (which turns out to be 3/2). So the dividing streamline separates at 135\u00C2\u00B0 regardless of where the sink lies.(see fig 2.3) a. VJPPERHALF y\j#\u00C2\u00A3 b. ? t ^ S V * L \"PLANE Figure 2.3 Dividing streamline. 10 One way, then, to achieve tangential separation was to have a small lip where the plane Is turned through an angle of 180\u00C2\u00B0 (see fig. 2.4). Hence, the idea was to incorporate a small lip in the cavity and use the computer to generate the shapes of the dividing streamlines. Again, the resulting transformations were difficult and the size of the lip was somewhat arbitrary; It was hard to determine the lip size so that it did not change the flow within the slot unrealistically. David Stropky [10] has vised this idea on an infinitely deep slot but the lip size is still arbitrary. This route no longer seemed analytically feasible so the geometry and Schwarz-Christoffel methods were abandoned. A different technique, called the Helmholtz-Kirchoff method or hodograph method, was then considered. This method is better suited to the task since It does not concern Itself with the shape of the solid boundaries but more with the velocity field. This allowed the model to have 2 pieces which were not connected. To achieve the goal of tangential separation, while not prescribing the velocity at the slot Figure 2.4 Cavity with lip. 11 surface but modelling the important aspects of the slot geometry and the inside of the slot as a plenum, the schematic of figure 2.5 was used. Figure 2.5 Slot geometry used in analysis. 2.2 H e l m h o l t z - K i r c h o f f M e t h o d The Helmholtz-Kirchoff method is a conformal mapping procedure which maps a physical plane of interest into a simplified plane (usually the upperhalf plane) where the flow field can easily be analyzed 116).(17],118]. To map to the upperhalf plane, the physical ( z = x +iy ) plane is examined in terms of its velocity field. Firstly, the physical plane is transferred to the hodograph plane having coordinates t, = u - iv, where u and v represent the the streamwise and normal velocity components. (This is simply the conjugate of the vector velocity.) Secondly, the physical plane is re-written in terms of potential coordinates or w = + iij). For jet flows, this method is especially well suited since along the boundaries, in the physical plane, the flow is 12 inclined at a fixed angle and hence the argument of the complex velocity is constant. So the boundaries get mapped to straight lines in the hodograph plane. Also, lines of constant velocity get mapped to circular straight lines in the potential plane. The potential and hodograph plane are then mapped to the same upperhalf plane. The transformation between the upperhalf plane and the potential plane gives the shape of streamlines in the upperhalf pane. Then the upperhalf plane or T plane is mapped back to the physical plane since by definition: arcs in the hodograph plane. Similarly, streamlines get mapped to dw \u00E2\u0080\u0094 = u \u00E2\u0080\u0094 iv = hence o r 2.2 So a mapping can be found from a complex physical plane to a simplified plane and hence the flow field can be analyzed by means of a simpler intermediate geometry. 13 2.3 Present Analysis In the present analysis the physical plane (z-plane) is mapped to the upperhalf plane (T-plane) by means of the velocity potential (w-plane) and hodograph planes (i;-plane). We consider the following physical plane where all velocities are scaled by the freestream velocity Ux and distances by the asymptotic height of the jet downstream, h. Figure 2.6. Physical or z-plane. Hence at Position (1) velocity is 1 Position (3) velocity is 0 Position (2) velocity is less than 1 Position (4) velocity is arbitrarily large at an angle of 6 Next, we examine the associated hodograph or \u00C2\u00A3-plane 14 The points (1), (3), (4) are easily found in the hodograph plane by inspection but the dividing streamline is unknown in this plane. As well, we examine the flow field in terms of the potential or w-plane. 1% 0 -I Figure 2.8. Potential or w-plane Here, as stated before, the streamlines become straight lines. One has the choice to arbitrarily decide the zero point for the velocity potential, +iji we have, \u00E2\u0080\u0094 = <}>t + tV\u00C2\u00BB. = \u00C2\u00AB - tv = C hence becomes, using\" 2.3 and 2.4 \u00E2\u0080\u00A2K JTo a a If we let z=0 be the leading edge of the slot, point (2), then z=0 corresponds to T=-l so, TT J - i \u00C2\u00AB a The figure 2.11 summaries the mapping sequence. Figure 2.11. Mapping sequence. 19 2.4 Location of Delta To find the value of the constant A , we realize that the image of A in the physical plane is the downward corner, point (4), of the slot. Since the leading edge (point (2)) lies at 0, then for a level slot Z(A) = d/h Is a positive real number. Hence we have the following two equations: Equation 2.6 gives a constraint to find A, once this is known, equation 2.7 will give the value of d/h. 2.5 Shape of Dividing Streamline To find the shape of the dividing streamline we simply follow it through the different mappings from the w-plane to the T-plane and finally to the Z-plane. In the w-plane it is: 2.6 2.7 i> = o,4>>o 20 In the T-plane this becomes r(0 = -4^,<'> \u00E2\u0080\u00A2 r,(j)e*W \u00E2\u0080\u00A2 r3(s)ete^ 3.1 Next a path of integration was required. One could not simply integrate from -1 to T along the x axis, since the integrand has a non-simple pole at x=0. To avoid this pole, the path of integration was changed to a box. Figure 3.1. Path of integration. 22 Since , / f = p.v.[Af (by an application of Cauchy's theorem) where the integrand is analytic Inside the path of integration, YB-Along each section of the box, the path of integration was parametrized and the integral was written in the form: Care was taken to break the intervals of integration up so that the proper branches of the arguments would be respected in each section of the complex plane. Once the integrand's arguments were properly determined and the parametrization determined, the integral was then re-written as a sum of integrals along the box, giving: 3.2 3.3 23 where R(t)= modulus of integrand R(t) = ^(1). r8(<).r,(0 and G(t)= argument of integrand e ( \u00C2\u00AB ) \u00C2\u00AB * i ( \u00C2\u00AB ) + + e3(t) If we consider a function A(A) where A(A) = Im z(A) then to find A one must determine the root(s) of A . To proceed the value of A was incremented from zero upwards. For each increment, the value of A(A) was evaluated for the corresponding path of integration. The incrementation was continued until a value of A* was found which gave a root for A. This procedure was carried out on the Apollo computers using an integration subroutine called DQAG from the Slatec mathematical software library. DQAG is a Gauss-Konrad general purpose globally adaptive integration routine, complete with error control and integrand examinator (a procedure which examines the integrand to determine how many integration points should be chosen on a given interval). It was noted that as a function of A, A(A) was monotonically increasing, so that A* was, in fact, the unique root of A . In addition, A was evaluated On a different path of integration with the exact values being reproduced. Once A was established, then the value of d/h could be found by evaluation of Eq. 2.7. The results are shown in fig. 3.2. 24 8 6-LJ CO O _J CO 2-0 SLOT SIZE VS SLQT ANGLE J I L _ 1 40 80 120 SLOT ANGLE 6' 160 200 Figure 3.2. Graph of d/h versus 0 3.2 Shape of Dividing Streamline Once A is known, the shape of the dividing streamline could be found. To evaluate the integral in Eq. (2.8), the dependence on the parametrlzation variable (t) was eliminated from the upper limit of Integration. By a simple change of variables we obtain (see Appendix II): 3.4 with 25 Again, expressing the functions in their real and imaginary parts allows the evaluation of z(t) = x(t) + iy(t) for 11 (0,\u00C2\u00AB). Starting with t=0 and incrementing in steps the integrals are evaluated until t=n. The resulting values become the coordinates of the dividing streamline in the z-plane. As a check one should find that It was found that in fact x(3.1) was considerably larger than the other values and y(3.1) was around 0.99. For purposes of plotting the streamlines, the variables were rescaled so that the slot width is unity for all cases. The results are shown in figure 3.3. x(ir) = oo and ?(*) = 1 DIVIDING STREAMLINES 2 fis90\ .E 1 ' 0 - 1 0 1 2 3 Position x/d Figure 3.3. Plots of Dividing Streamlines. 26 3.3 Mass Flow Out of Slot To determine the mass flow out of the slot, one must rescale the variables, since in our analysis we scaled all variables such that the mass flow out of the slot was unity. By adjusting the slot width to be unity and keeping the value of d / h fixed, we find that the mass flow out of the slot becomes h / d . Results are shown in fig. 3 .4 . SLOT MASS FLOW VS SLOT ANGLE i i i i 200 SLOT ANGLE 9' Figure 3 .4 . Graph of Mass Flow From Slot 27 3.4 Coefficient o f Pressure To find the coefficient of pressure across the slot surface, we can write Bernoulli's equation which holds everywhere, i.e., Hence and hence P{z) + 1/2^(2)1' = Pm + l/2pUt 2 oo cP = i- ic(r)|5 C p ( * ) \u00C2\u00AB l - K j r ^ ) # / , r r 3.5 Next, one must determine the image of the slot surface in the T-plane. To do this, we assume that the slot surface in the T-plane can be parametrized as some curve T(t) = uft) + ih(t). We know for t=0, T(t) = -1 and since the dividing streamline is tangent to the slot surface at point (2) in the physical plane, the dividing streamline must be tangent to the slot surface in the T-plane. To check whether a point T G has its image on the slot surface, one must check if z{T0) = x + ty has y = 0,*e(0,rf) 28 i.e. we must check has y(0 = 0,*(<)e(0,<0 r ( 0 = Mt) + ih(t) Choosing ii(t) =-1+t we obtain 1 r1 / i = - ( \u00C2\u00AB + \u00C2\u00AB ; ) \u00E2\u0080\u00A2 [ / Ti-ra-rjCosSdr + i / r x \u00E2\u0080\u00A2 r 2 \u00E2\u0080\u00A2 r 3 sin 0drl T Jo JO where -r r3 \u00E2\u0080\u0094 ((<-r - ( l + A)) a + r J/i 2(*))' / 2' = ((t-r-1)7+r,h,(t))-1/i-B/2K e \u00E2\u0080\u0094 01 + 0J + 03 u = 0 0/TT \u00E2\u0080\u00A2 tan'1 r / f f l A . ( 1 fl/irj.ton\"1 r / l ( < ) tJ - fc8(t) V = 2 To proceed, t is incremented from 0 to A+1, h(t) is guessed until z(t) lies on the slot surface or specifically Im z(t) < 0.01 , Re z(t) E (0,d). So for each value of t, a value of h(t) is found and hence the image of the slot surface is constructed in the T-plane. Once the points on the slot surface are found, then the values of Cp = 1 - |C(T)|l are easily calculated. Using the T-plane coordinates, T = a +ib, for the points lying on the slot surface we have CP = 1 - [ ( q ( q - A ) + + fe2A2 ( a - A ) J + i>2 Results are shown in fig. 3.5. COEF. OF PRESSURE VS. POSITION o 0 - \"' 3 CO CL O e = 20\" 6 = 40* e = s o ' - 3 - 1 0 2 Position x/d Figure 3.5. Graph of Cp vs. Position for various 6 3.5 Velocity Across Slot To find the velocity across the/Slot we note that u(z) + tv(z) = C(T) hence \u00C2\u00AB(*) + .*(*) = ( ^ r - 3.7 for T = a + tb where + [(o - A ) J + o 2 ] * / * J ' 8 0 = tan 1 o(o - A) + 6J where the points T = a+ib are those found in the previous section. Inspection of the velocity fields show that these are not constant across the slot as shown in figures 3.6 to 3.9. The velocity vectors have been truncated at x/d = 0.98 to avoid the infinite velocities that occur at x/d = 1. Similar results are also observed in numerical simulations of similiar (but not identical) slot geometries studied at U.B.C. 171,1191. Velocity Field Across 20\u00C2\u00B0 Slot Unit Vector I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I \u00E2\u0080\u00A20.25 -0.00 0.25 0.50 0.75 1.00 1.25 Position x/d Figure 3.6. Velocity Field Across Slot 8=20\u00C2\u00B0 Velocity Reid Across 40\u00C2\u00B0 Slot Unit Vector -0.25 -0.00 0.25 0.50 0.75 1.00 1.25 Position x/d Figure 3.7. Velocity Field Across Slot 6=40\u00C2\u00B0 Velocity Field Across 90\u00C2\u00B0 Slot I \u00E2\u0080\u0094 \u00E2\u0080\u00A2 \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 I \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 I \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i i \u00E2\u0080\u0094 i I \u00E2\u0080\u0094 i i \u00E2\u0080\u0094 i i \u00E2\u0080\u0094 | i i i i 1 i i i i i -0.25 -0.00 0.25 0.50 0.75 1.00 1.25 Position x/d Figure 3.8. Velocity Field Across Slot 6=90\u00C2\u00B0 Velocity Field Across Orifice Slot I \u00E2\u0080\u0094 \u00E2\u0080\u00A2 \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 I \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 J \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 i \u00E2\u0080\u0094 I i \u00E2\u0080\u0094 i i i I i i \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 i f . . \u00E2\u0080\u00A2 i -0.25 0.00 0.25 0.50 0.75 1.00 1.25 Position x/d Figure 3.9. Velocity Field Across Slot fl=180 (orifice) 3 3 3 . 6 Discharge Coefficient The coefficient of discharge from a slot into a freestream can be expressed as (which may differ from other definitions): where V s = the average slot velocity defined as V s = U ^ h / d Pp = the plenum static pressure Px = the freestream static pressure . Now since we have equal stagnation pressures CD P - P *p 1 oo 3.9 Pp Poo \u00E2\u0080\u0094 Ptp ~ Pt oo \u00E2\u0080\u0094 Pt oo \u00E2\u0080\u0094 Pt oo where P tp the plenum total pressure P the freestream total pressure. This gives l/2pUl _ Ul 34 or ui(k/dy hence CD = d7/h7 3.10 This is plotted in fig 3.10. and may be compared with the values found experimentally as: CD = A + B \u00E2\u0080\u00A2 Ul/V7 3.11 for similiar slot geometeries [19]. Presumably the presence of A and B are conditioned by viscous effects and by geometry. COEF. OF DISCHARGE VS SLOT ANGLE i i i i o 9- 0.6-0 . 4 -0 . 2 -0 . 0 0 0=47* a o Theory Least Squares \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 p. - p . - 1/2 p U.\u00C2\u00BB \u00C2\u00B0 p. - p . - U S \u00E2\u0080\u00A2 1/2 p U.\u00C2\u00BB \u00E2\u0080\u00A2 p. - p_ - 0.75 \u00E2\u0080\u00A2 1/2 p U .* 2 5 0 0 Re =U b/v 5 0 0 0 Figure 4.9. Dimensionless Velocity vs Reynolds Number 6=47\u00C2\u00B0 8 => 1.0 0.8 0.6 H 0.4-0.2-0.0 SLOT VELOCITY VS .REYNOLDS NUMBER 0=60.5* 0 o \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 _ Least Squares Theory \u00E2\u0080\u00A2 p. - p. - 1/2 p U.\u00C2\u00AB \u00E2\u0080\u00A2 p. - p. - 125 \u00C2\u00BB 1/2 p U.\u00C2\u00BB - p. - p. - 0.75 \u00E2\u0080\u00A2 1/2 p U.\u00C2\u00BB 2500 Re =U b/v 5000 Fig. 4.10. Dimensionless Velocity vs Reynolds Number 6=60.5\u00C2\u00B0 1.0 0.8 0.6 H > 0.4 0.2 H 0.0 SLOT VELOCITY VS .REYNOLDS NUMBER 0=71.5* 0 Least Squares Theory \u00E2\u0080\u00A2 p. - p. - 1/2 P U.\u00C2\u00AB \u00C2\u00B0 p. - p. - 125 \u00E2\u0080\u00A2 1/2 p U.\u00C2\u00BB \u00E2\u0080\u00A2 p. - p. - 0.75 \u00E2\u0080\u00A2 1/2 p U.\u00C2\u00AB 2500 Re=Umb/u 5000 Fig. 4.11. Dimensionless Velocity vs Reynolds Number 6=71.5\u00C2\u00B0 1.0 0.8-0 . 6 > 0.4-I 0.2 0.0 SLOT VELOCITY VS .REYNOLDS NUMBER 0=90* 0 + 9 + Least Squares * * * * J ! ! \u00E2\u0080\u00A2 . ^Theory \u00E2\u0080\u00A2 p. - p. - 1/2 p U.\u00C2\u00AB 0 p. - p. - t25 \u00E2\u0080\u00A2 1/2 p U.\u00C2\u00AB \u00E2\u0080\u00A2 p. - p. - 0.75 \u00E2\u0080\u00A2 1/2 p U.\u00C2\u00BB 2500 Re =U b / i / 5000 Fig. 4.12. Dimensionless Velocity vs Reynolds Number 6=90\u00C2\u00B0 SLOT VELOCITY VS .REYNOLDS NUMBER > 1.0 0.8-0.6-0.4-0.2 0.0 0=180* 0 Theory Least Squares \u00E2\u0080\u00A2 p. - p_ - 1/2 p UJ 0 p. - p. - t25 \u00E2\u0080\u00A2 1/2 p U.\u00C2\u00AB \u00E2\u0080\u00A2 p. - p . \" 0.75 \u00E2\u0080\u00A2 1/2 p U.\u00C2\u00AB 2500 Re^U.b/z/ 5000 Fig.4.13 Dimensionless Velocity vs Reynolds Number 6=180\u00C2\u00B0 (orifice) 4 6 For each flap angle a line of best fit can be found for the V s / U ^ , ratio and these points are plotted on the theoretical Vg/Ua, curve in figure 4.14. Error bars have been included to show the expected accuracy of the results for the equal energy case (see section 5.1). 1.0 SLOT VELOCITY VS SLOT ANGLE i i i i \u00E2\u0080\u00A2 p.-- P. - V2 PU.* P. - t25 \u00E2\u0080\u00A2 1/2 pU.\u00C2\u00BB P_ - 0.75 \u00E2\u0080\u00A2 1/2 pU.\u00C2\u00BB 80 120 SLOT ANGLE 6' 200 Figure 4.14. Best fit Data Vs Theory As well a Reynolds number dependence is plotted against V g / U o o for various Re s in figure 4.15. The data is from the equal stagnation pressure tests. 47 Figure 4.15. Reynolds Number Dependance on Slot Velocity In addition, the discharge coefficient from the slot can be calculated. Values of C D vs Re s are plotted for each angle in figure 4.16 to 4.20. COEF. OF DISCHARGE VS REYNOLDS NUMBER 12-1 1 1 L o D o CO b 4-0) o o e = 47' 0 Least Squares _i Theory p. - 1/2 pU.\u00C2\u00BB O P , - p. - 1.25 \u00E2\u0080\u00A2 1/2 pU.\u00C2\u00AB \u00E2\u0080\u00A2 p.- p. - 0.75 \u00E2\u0080\u00A2 1/2 p0.\u00C2\u00AB 1000 2000 3000 Re = Umb/v 4000 Fig. 4.16. Coef. of Discharge vs Reynolds Number 0=47\u00C2\u00B0 1 8 o 6-C7> a>9 _ O 0 COEF. OF DISCHARGE VS REYNOLDS NUMBER I I ; I 6 = 60.5* 0 Theory Least Squares \u00E2\u0080\u00A2 p . - p. - 1/2 pU.\u00C2\u00BB o P , - p. - 1.25 \u00E2\u0080\u00A2 1/2 pUm* \u00E2\u0080\u00A2 p , - p. - 0.75 \u00E2\u0080\u00A2 1/2 pUj 1250 2500 3750 Re = U b/v 5000 Fig. 4.17. Coef of Discharge vs Reynolds Number 0=60.5\u00C2\u00B0 COEF. OF DISCHARGE VS REYNOLDS NUMBER 5.0-a \u00E2\u0080\u00A2 O 3,8-OJ cn D -C o 2.5-Q O **-* = o,\u00C2\u00ABs > o or for w = <|> + iip it is w a 0. Intergrating (2.3) we obtain w = -(T + logT + 1 - ITT) (where the constant of integration has been chosen to properly orient the mapping) Rearranging we find *(w + 1 ) - 1 = log(TtT) hence e*(u;+t)-l _ j g T but , . , . e , i r e\u00E2\u0084\u00A2 so that Im(TeT) = 0 With T = | +in we have Jm((\u00C2\u00A3 + \u00C2\u00BBTf)e*(co6TJ + 1 sin T J ) ) = 0 or \u00C2\u00A3 = \u00E2\u0080\u0094TJ cot TJ h e n C e T = r,(-cotv + i) = \u00E2\u0080\u0094: (cos TJ \u00E2\u0080\u0094 i sin 77) sin 77 \" sinr/ APPENDIX II C H A N G E O F V A R I A B L E S Given let and when therefore V K J\u00E2\u0080\u0094l $ 8 j = (r(c) + l)r-- 1 i\u00C2\u00BB = (T(t) + l)dr s= -1 r = 0 s = T(t) r = 1 * w \" ; /.' ((T(t) ? i ) , + + i ) ^ ( ( r ( ( ) + 1 ) r ' ( 1 + A ) ) , \" ( r w + 1 ) , i r or *(\u00C2\u00AB) - -(T(t) +1) yo l ( r m + 1 ) r . _ 1 ] 1 + . / \u00E2\u0080\u009E * "@en . "Thesis/Dissertation"@en . "10.14288/1.0098448"@en . "eng"@en . "Mechanical Engineering"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Analytical and experimental study of flow from a slot into a freestream"@en . "Text"@en . "http://hdl.handle.net/2429/29914"@en .