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Analytical and experimental study of flow from a slot into a freestream Ainslie, Bruce 1991

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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 © 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 ±17%. 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° 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 £-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° 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° 40 4.5 Mass Flow vs Velocity 6=60.5° 41 4.6 Mass Flow vs Velocity 6=71.5° 41 4.7 Mass Flow vs Velocity 6=90° 42 4.8 Mass Flow vs Velocity 6= 180" 42 4.9 • Slot Velocity vs Reynolds Number 6=47° 43 4.10 Slot Velocity vs Reynolds Number 6=60.5° 44 4.11 Slot Velocity vs Reynolds Number 6=71.5" 44 4.12 Slot Velocity vs Reynolds Number 6=90° 45 4.13 Slot Velocity vs Reynolds Number 6=180° 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° 48 4.17 Coef of Discharge vs Reynolds No. 6=60.5° 48 4.18 Coef of Discharge vs Reynolds No. 6=71.5° 49 4.19 Coef of Discharge vs Reynolds No. 6=90° 49 4.20 Coef of Discharge vs Reynolds No. 6=180° 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° 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° slot of finite depth (fig 2.1), is: dz _ K dw (w - a)(w - b) 2. i (w - c)(w — 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° 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°. 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° but at 90° 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° regardless of where the sink lies.(see fig 2.3) a. VJPPERHALF y\j#£ 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° (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 = <J> + 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 — = u — 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 £-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, <j), and this is chosen to be zero at point (2). As well, the zero for the stream function is chosen to be the streamline (1 )-(2)-( 1). The difference between 15 stream function values represents the mass flow between the stream functions. Hence, since the mass flow out of the slot is U^h or 1 when scaled, the value of the stream function (3)-(4) must be -1. Finally, we consider the upperhalf plane. Both the hodograph and potential planes are mapped onto this plane. The points (1), (2), (3), (4) can be arbitrarily chosen in the T-plane to simplify the mappings. Figure 2.9. Upperhalf of T-plane. To map the W-plane to the T-plane we use a Schwarz-Christoffel transformation of the polygon with vertices at points (2) and (3) having angles of 2n and 0, respectively. The desired mapping is then dw = T + 1 dT T 2.3 16 where (2) is chosen to be -1 in the T-plane and (3) is chosen to be 0 in the T-plane. Further analysis of this mapping shows that the constant K has a value of 1 In. Next, to map the hodograph plane to the T-plane, we first open up the plane to the lower half T-plane by means of a power transformation. r Figure 2.10. Power transformation. If we require r = c° then we require the line ae~%9 to be mapped to * e we find te 17 gives IT hence r = <* Next the lower half T-plane is mapped to the same upperhalf T-plane (as defined by the Schwarz-Christoffel mapping). This is achieved by means of a bilinear mapping with: r = 0 T-0 r = i = ^ r = oo r = oo T = A giving and hence C H j T ^ ) ^ 2.4 where A is an unknown constant representing the position of the trailing edge of the slot, point (4), in the T-plane. Finally, we map the upperhalf T-plane back to the physical plane remembering that for w = <j>+iji we have, — = <}>t + tV». = « - tv = C hence becomes, using" 2.3 and 2.4 •K 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 « 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^,<e(0,7r) The derivation of this result can be found in Appendix I. Hence in the Z-plane it is 21 C H A P T E R 3 R E S U L T S 3.1 Slot Width d/h In order to evaluate the integrals of equations 2.6 and 2.7, for arbitrary 0, a computer code was required. Firstly, the integral was re-written so that each expression of the integrand was expressed in terms of its modulus (r^ and its argument (Bj), i.e. it was re-written as. f(s) = ra(«)e"><'> • r,(j)e*W • 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 ( « ) « * i ( « ) + + 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,«). 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 '<L> 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 ( * ) « 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 = - ( « + « ; ) • [ / Ti-ra-rjCosSdr + i / r x • r 2 • r 3 sin 0drl T Jo JO where -r r3 — ((<-r - ( l + A)) a + r J/i 2(*))' / 2' = ((t-r-1)7+r,h,(t))-1/i-B/2K e — 01 + 0J + 03 u = 0 0/TT • 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 «(*) + .*(*) = ( ^ 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° 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 •0.25 -0.00 0.25 0.50 0.75 1.00 1.25 Position x/d Figure 3.6. Velocity Field Across Slot 8=20° Velocity Reid Across 40° 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° Velocity Field Across 90° Slot I — • — i — i — i — I — i — i — i — i — I — i — i i — i I — i i — i i — | 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° Velocity Field Across Orifice Slot I — • — i — i — i — I — i — i — i — i — J — i — i — i — i — I i — i i i I i i • • i f . . • 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 — Ptp ~ Pt oo — Pt oo — 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 • 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-<u cn o -C to 6 -Q o % 3 -o cj OH 1 1 : i 1 • 0 40 80 120 160 200 Slot Angle 6' Figure 3.10. Discharge Coefficient For Various Slot Angles 35 CHAPTER 4 EXPERIMENTAL WORK 4.1 Description of Apparatus Experimental tests on film cooling and its effectiveness involve detailed measurements of turbulent velocity fields, fluctuating pressure and temperature fields as well as mass flow rates ( ref 1201-1221 ). It is beyond the scope of this research project to undertake a full investigation of all the variables to test the previous calculations. However, a simplified model of film cooling, examining only the overall influence of the freestream on the injected flow was investigated to check the validity of the previous results. A series of experiments, modelling" the flow from a plenum, out of a slot, into a freestream were performed using the small blue wind tunnel in the aerodynamics lab at U.B.C. The plenum consisted of a cylindrical tank which was installed beneath the floor of the wind tunnel. The plenum air was supplied from a hose connected, through a flowmeter and a pressure regulator, to a large compressor. Above the floor of the wind tunnel, a 0.16 cm aluminium sheet metal cover was placed over the plenum. Cut into the cover was a 20 cm x 0.4 cm slot through which the plenum flow exited. At the downstream vertex of the slot, a small sheetmetal flap was attached to the floor cover. The flap length was 18 cm and at its end a small piece of dowelling (O.D., 0.5 cm) was glued. Since the plenum was circular in cross-section and flow lengthwise across the slot was to be eliminated (in order to approximate the 2-D nature of the theory), a rectangular plywood box was inserted into the plenum. This gave the plenum a square cross-section of 20 cm x 18 cm. 36 The sides of the flap fit flush against the sides of the plywood box. The sides of this box were fitted with pressure taps, 10 cm from the bottom, to measure static pressure. The apparatus is sketched in fig. 4.1. In the windtunnel, 1 m long side plates were placed vertically in the direction of the freestream, parallel to the sides of the slot. These were installed to assure that the freestream flow would pass directly over the slot and not around it. A pitot static tube was placed ahead of the slot, in between the side plates, to measure the freestream velocity. The pitot static tube was connected to an alcohol manometer. ao N N N N N N N V \ i Flood. — V VfcTTD SCAUP •TLYVtoCD BOX *PL£Mun \VX£T Figure 4.1. Plenum cross section. A second manometer was used to measure the static pressure in the plenum. Both manometers used as the reference pressure the static pressure measured by the pitot tube. 37 wort starve 2 Figure 4.2. Experimental set-up. 4 .2 E x p e r i m e n t a l P r o c e d u r e Before each test the slot flap had to be Installed. A small piece of wood measuring 18 cm x 2 cm was cut so that one of its faces was angled at a specific angle. On the angled face the slot flap was glued. At the leading edge of the flap the small dowelling lip was attached to eliminate or reduce separation of the flow. The other edge of the flap was ground to form a small edge. This edge was then glued onto the angled face of the wood block. Next, the wood/flap piece was glued to the sheet metal cover. Once the epoxy had set, the resulting edge was carefully filed and sanded down to form a slightly rounded corner. 38 Figure 4.3. Flap attachment. Once the flap was ready, this piece was placed so that the flap fit flush in the plenum and the sheetmetal covered the wind tunnel floor. Masking tape was placed along the plywood / flap edge to assure that no flow between these two occurred. Next the upstream sheet metal cover was butted up against the downstream cover. The upstream section had a 4 mm x 20 cm section, centered about its middle, cut from its downstream side. When the two pieces were butted against one another the cut section formed the slot opening. Duct tape was used to hold the covers onto the top of the plywood box and to form a seal, preventing any air from flowing out of the sides of the plenum. The covers were then taped along their sides and lengths to the wind tunnel floor. The width of the slot was measured and recorded by means of a pair of calipers. Several measurements were taken along the length of the slot and an average value was used as the slot width. Next, the two sheet metal side plates were placed parallel to the flow, forming walls at each end of the slot. These were held in place by a lip that was bent into the top of each side plate at about 95°. Hence, when the side plates were placed in the 39 wlndtunnel, the bent sections were forced slightly downward, locking them in place. The top and bottom of the side plates were then taped with masking tape. Care was taken so that the side plates were both parallel to the slot and vertical. The pitot static tube was aligned with the flow; positioned ahead of the slot but between the side plates. The pitot static tube was connected to a manometer which was leveled and zeroed. The static pressure from the plenum was also measured by a second manometer which used the same reference pressure, by means of a T-junction in the pitot tube's, line. This manometer was also zeroed and levelled. To the plenum, a hose was attached from the compressor. The flow meter was connected to this hose close to the plenum. Once all was in place, the wind tunnel speed was increased until a desired speed was reached. Next, the plenum air flow was increased until the excess pressure in the plenum, beyond the static pressure in the freestream, equalled that of the dynamic pressure of the freestream. Once this equilibrium was reached, the height of the bob in the flowmeter was recorded. Then the speed of the freestream velocity was increased and hence the volume flow into the plenum was increased until equality in pressure was again reached. Nine points were recorded for velocities ranging from 4 m s 1 to 15 ms 1 . Values lower than 4 m s 1 resulted in mass flow rates from the flowmeter which were too small to be reliably measured and the 15 ras'1 velocity was close to the maximum that the windtunnel could produce. As well as recording the volume flow for equal energies, five points off equal energy points were recorded. In this case, the freestream velocity was fixed and after the plenum flow had been adjusted to record the equal pressure data, the flow rate was decreased until the plenum 40 pressure was 75% of the dynamic part of the freestream. The mass flow rate was recorded. Similarly, the volume flow was increased until the plenum pressure was 125% of the freestream dynamic pressure. Experiments were conducted for slot angles of 47°, 60.5°, 71.5 90° and 180° (orifice). The two plenum pressure taps were also connected to the same manometer to check if there was any discrepancy in pressure measurements between the two. None was observed. 4.3 Experimental results From the experimental data, plots of mass flow out of the slot versus velocity were made. The mass flow out of the slot has been scaled by the slot length. The theory is also plotted with the data. The theoretical curve is obtained by determining the slot width and the appropriate value of d/h. Results are shown in figures 4.4 to 4.8 where a linear least squares curve through the origin has been plotted through the measured values. SLOT FLOW VS FREESTREAM VELOCrTY .02 CO _o in .00 0 4 8 12 16 Freestreom Velocity m/s Figure 4.4. Mass Flow vs Velocity B=47° SLOT FLOW VS FREESTREAM VELOCITY Freestream Velocity m/s Figure 4.5. Mass Flow vs Velocity 6=60.5° SLOT FLOW VS FREESTREAM VELOCITY Freestream Velocity m/s Figure 4.6. Mass Flow vs Velocity 6=71.5° 42 SLOT FLOW VS FREESTREAM VELOCITY Freestream Velocity m/s Figure 4.7. Mass Flow vs Velocity 6=90° SLOT FLOW VS FREESTREAM VELOCITY Freestream Velocity m/s Figure 4.8. Mass Flow vs Velocity 6=180" (orifice) 4 3 The data is also presented in nondimensional form in figures 4.9 to 4.13 , with the velocity being represented as a slot Reynolds number Re s = bUoo/v, and the mass flow as a ratio of slot velocities to freestream velocities V s / U x (where V s x slot width x slot length = volume flow). Also, the data points for the unequal stagnation pressures are included. SLOT VELOCITY VS .REYNOLDS NUMBER 1.0 J  0 . 8 8 ZD > 0.6-0 . 4 -0 . 2 -0 . 0 0 0=47* a o Theory Least Squares • • • p. - p . - 1/2 p U.» ° p. - p . - U S • 1/2 p U.» • p. - p_ - 0.75 • 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° 8 => 1.0 0.8 0.6 H 0.4-0.2-0.0 SLOT VELOCITY VS .REYNOLDS NUMBER 0=60.5* 0 o • • • _ Least Squares Theory • p. - p. - 1/2 p U.« • p. - p. - 125 » 1/2 p U.» - p. - p. - 0.75 • 1/2 p U.» 2500 Re =U b/v 5000 Fig. 4.10. Dimensionless Velocity vs Reynolds Number 6=60.5° 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 • p. - p. - 1/2 P U.« ° p. - p. - 125 • 1/2 p U.» • p. - p. - 0.75 • 1/2 p U.« 2500 Re=Umb/u 5000 Fig. 4.11. Dimensionless Velocity vs Reynolds Number 6=71.5° 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 ! ! • . ^Theory • p. - p. - 1/2 p U.« 0 p. - p. - t25 • 1/2 p U.« • p. - p. - 0.75 • 1/2 p U.» 2500 Re =U b / i / 5000 Fig. 4.12. Dimensionless Velocity vs Reynolds Number 6=90° SLOT VELOCITY VS .REYNOLDS NUMBER > 1.0 0.8-0.6-0.4-0.2 0.0 0=180* 0 Theory Least Squares • p. - p_ - 1/2 p UJ 0 p. - p. - t25 • 1/2 p U.« • p. - p . " 0.75 • 1/2 p U.« 2500 Re^U.b/z/ 5000 Fig.4.13 Dimensionless Velocity vs Reynolds Number 6=180° (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 • p.-- P. - V2 PU.* P. - t25 • 1/2 pU.» P_ - 0.75 • 1/2 pU.» 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.» O P , - p. - 1.25 • 1/2 pU.« • p.- p. - 0.75 • 1/2 p0.« 1000 2000 3000 Re = Umb/v 4000 Fig. 4.16. Coef. of Discharge vs Reynolds Number 0=47° 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 • p . - p. - 1/2 pU.» o P , - p. - 1.25 • 1/2 pUm* • p , - p. - 0.75 • 1/2 pUj 1250 2500 3750 Re = U b/v 5000 Fig. 4.17. Coef of Discharge vs Reynolds Number 0=60.5° COEF. OF DISCHARGE VS REYNOLDS NUMBER 5.0-a • O 3,8-OJ cn D -C o 2.5-Q O **-* <u o 1.3-o 0.0-e = 71.5* 0 o ' • a + + a ~Theory ~ LeastB Squares p. - 1/2 fUj o P,- p. - 1.25 • 1/2 pMj • p , - p. - 0.73 • 1/2 pU.« 1000 , 2000 3000 Re = U b/v 8 «• ' 4000 Fig. 4.18. Coef of Discharge vs Reynolds Number 6=71.5° COEF. OF DISCHARGE VS REYNOLDS NUMBER 3.0 -* ' 1 L o2.5H o ? 2.0 H D o •2 1.5 H Q 1.0 H o ° 0.5 H 0.0 0 = 90' Theory • Least Squares • p,- p. - 1/2 pU.« o P,- p_ - 125 • 1/2 fUj • p » - p_ - 0.75 • 1/2 pU.« 0 1250 2500 3750 Re = U b/v 5000 Fig. 4.19. Coef of Discharge vs Reynolds Number 0=90° 50 3.0 COEF. OF DISCHARGE VS REYNOLDS NUMBER o o £ 1-5H CD o O 0.0 e = 180*+ 0 " i — * -Least Squares 4 » Theory • p . - P. - 1/2 pU„* a P,- P. - t25 • 1/2 pU.» • p . ~ P. - 0.75 • 1/2 pi)m* 1250 2500 3750 Re = U'b/v 5000 Fig. 4.20. Coef of Discharge vs Reynolds Number 6=180° (orifice) From these curves a line of best fit for the value of Cp} can be obtained. These values (for each angle 6) are plotted against the theoretical curve showing C D as a function of V^U ^in figure 4.21. In addition, the values for the coefficient of discharge corresponding to the highest Reynolds number for each test are plotted against the theory and the emperical relation used by Gartshore et al. (19] with : CD = 2.5 + 0.12- ( ^ ) 2 in figure 4.22. 51 COEF. OF DISCHARGE VS SLOT VELOCITY i i i 0.00 0.25 0.50 0.75 1.00 SLOT VELOCITY VJ \ J m Figure 4.21. Best Fit Data vs Theory for Discharge Coef. Figure 4.22 Highest Re s data vs Theory for C D 52 CHAPTER 5 DISCUSSION OF RESULTS 5.1 Experimental Errors Errors in the experiment can be attributed to uncertainty in the slot width, discrepancy in readings between the two manometers and difficulty in reading the height of the bob in the flowmeter. The slot was cut out of a piece of aluminum sheetmetal. It was difficult to assure that this cut was accurate to more than a few tenths of a millimeter. As well, when the flap was glued to the wood piece, it was difficult to assure that the flap was flush to the wood surface within a few tenths of a millimeter. In addition, the wood face could not be cut perfectly. As a result, the slot width varied by up to 0.5 mm along the length of the slot. This results in a ±12% error in the calculation of the theoretical slot flow. Another source of error was the discrepancy between the two manometers. When these were hooked in parallel to a single pitot static tube, they would show different readings for different freestream velocities. Measurements showed that there was a ±9% error between the two. This resulted in a ±3% error in velocity calculations, used for calculating the theoretical slot flow. This also contributed to an error in the experimental mass flow since the two pressures were to match before a measurement was to be taken. The measurement of the mass flow itself was a bit of a problem. The bob in the flowmeter was not always steady ; it could oscillate up and down about 2 mm resulting in an average error of ±2% . 53 In addition, since the theory did not include flow separation at the rear of the slot, the downstream edge was slightly rounded. It was hard to determine the exact radius of curvature of the lip but it was estimated at about 1 mm . It was not possible to determine if separation was present since the side plates blocked the view of the slot and hence it was not possible to look for separation with a tuft of wool. Most likely, separation was present particularly for the higher slot angles (71.5°, 90°, and 180°). As well, the rounding of the lip made the determination of the slot width more uncertain. Overall experimental uncertainty of Vs/U00 was ±17% as shown by the error bars on fig. 4.14. Also, the data in three of the experiments converges to higher values than calculated. One might suspect that this could be a result of the plenum leaking. This, however, does not seem likely because the plenum was installed and taped several times with more precautions being made each succesive time and with the similar trends being observed. Experimental errors, which should be independent of the slot angle, should even out over the course of the experiment and simply randomly scatter the data about its mean. Another explanation for the larger experimental results could be that the pressure in the plenum is not measured where the velocity is zero. Hence, in order to reach a desired static pressure, the stagnation pressure of the plenum must be raised to a higher than expected value. This is achieved by the addition of a greater mass flow rate. To try to avoid this scenario, two extra screens, with a mesh size of 0.5mm, were added to the plywood box in order to make the flow more uniform in the plenum, and a second plenum pressure tap position was investigated. No conclusions could be drawn from these tests. 54 5.2 Expected Effects of Reynolds Number One would expect that the difference between the theoretical result and the experimental results would show a dependence on the Reynolds number. That is, one would expect the theoretical results to be consistently higher than those obtained during the experiments, since the presence of boundary layers at the slot opening would serve to plug the slot and hence reduce its effective size. As well, flow separation would reduce the flow out of the slot since this would tend to increase the pressure at the slot exit. However, one would expect as the Reynolds number increases the boundary layers would thin and the viscous effects would no longer dominate, bringing the experimental results closer to the theory. For the 47° slot and orifice, this trend was noticed but not for the others. The plot of mass flow versus angle at the different Reynolds number (fig. 4.15) shows that the data points associated with the lowest Reynolds number are consistently lower than the other Reynolds number values which suggest that the effects of viscocity are important at these low Reynolds numbers. However, the higher Reynolds number points are not ordered in any particular way. That is, the slot velocity, for a given angle, is not necessarily lower for the lower Reynolds numbers. The values however, are all very close to one another, showing that the slot velocity is starting to approach a common value independant of Reynolds number. These common values are sometimes higher and other times lower than the calculated values due to the uncertainty in the experimental results. If one examines the plots of slot velocity versus Reynolds number for the various angles (fig 4.9 to 4.13) one sees that the measured values of 55 the slot velocity not only converge to a common value but that this convergence is always from below. Hence the expected decrease in viscous effects is present at higher Reynolds numbers but due to uncertainty in the experiment the data do not always converge to the calculated values. Since viscous effects, while neglected in the theory, should become less important at higher Reynolds numbers a broader range of Reynolds number should be used to better check this theory. Ockendon, Fitt and Jones [2] use a Reynolds number of 6.5xl04, or almost 20 times what is used here, to conduct their experiments. 5.3 Coefficient o f Discharge The coefficient of discharge shows similar trends as the mass flow rates; that is the theoretical value of C D is less than the line of best fit representing the data for the 47° slot and the orifice and it is larger than the line of best fit for the other angles. The values of C D also appear to show a dependence on Reynolds number; as the Reynolds number increases the experimental results tend to a common value from above. This is expected since, as the Reynolds number increases, the plugging of the slot by the presence of boundary layers should diminish and so the pressure in the plenum required to discharge a given mass flow should decrease. Again, the common value that the data converges to is not consistently higher than the theory due to the uncertainties in the experiment. The theoretical analysis shows that C D varies inversely with the square of the dimensionless mass flow out of the slot. This trend is also obtained and confirmed by Gartshore et al. [19] with experimental 56 observations and numerical simulations for a slightly different geometry. Again, then, the inviscid theory gives the correct behaviour for the flow, suggesting that pressure is the most important aspect of this process. 5.4 Off Equal Energy Results Examining the values of V s / U ^ for both the equal energy cases and the non-equal energy cases at the various Reynolds numbers shows that the mass flow rate varies monotonically with changes in total pressures between the jet and the mainstream. When the total pressure in the jet is greater than the freestream a greater mass flow is recorded (for that particular freestream velocity), and when the total pressure in the jet is lower, a corresponding decrease in mass flow rate is observed. The differences are again not strongly affected by changing Reynolds number. This suggests that in fact the dominating effect in this process is pressure and not viscosity. On the other hand, the values of C D do not seem to depend strongly on differences in total pressure between jet and mainstream. They show a Reynolds number effect since the small differences observed for different total pressures vanish as Re s increases. It appears the coefficient of discharge has a stronger viscous dependence than the mass flow rate. 57 CHAPTER 6 RECOMMENDATIONS AND CONCLUSIONS 6.1 Improvement of Experiment The experiment could be improved by constructing a larger slot. This would allow for larger mass flow rates and hence decrease the relative error in reading the flowmeter. It would also allow for higher Reynolds numbers which might show the effects of viscosity on this flow more clearly. Also, a larger slot would reduce the relative error in measuring the slot width. However, a wider slot would make the relative length of the slot and the depth of the plenum smaller. Ideally, the slot would be infinitely long and the plenum infinitely deep. As well, larger mass flows into the plenum would likely result in increased flow within the plenum when this should really be a region of still air. Hence, there is a balance to be achieved in determining the slot size. Precise manufacturing of the slot would again eliminate errors in determining slot size. As well, construction of a shorter flap would reduce the thickness of the boundary layer, along the flap, at the slot opening. Presently, the flap length is 45 times the slot thickness. A slot length of 25 times the slot thickness might be used to decrease the effects of this boundary layer. Acrylic side plates could be used to force the flow over the slot instead of the sheetmetal ones used here. This would allow for observations of flow separation and the use of liquid crystal thermochromic indicators [2] to observe the dividing streamlines. 58 In addition, running some tests at slot angles that are less than 47° might be useful. At lower angles, less vertical momentum is put into the jet and hence it is less likely to have turbulent mixing and also less likely that flow separation would occur. Both these effects would create an experimental set up which is better approximated by the present model. 6.2 Improvement to Theory To improve the theory, it could be extended to include the effects of a finite separation bubble. This might be achieved by modifying the downstream corner geometry, in the physical plane, to include a small bubble. An infinitely long separation bubble can easily be added but this seems unrealistic. Extending the theory to include different densities would be useful since in practice the cooling fluid can be significantly heavier than the freestream. Unfortunately, the addition of unequal energies causes a slip line to occur (at the dividing streamline) which prevents the use of the methods used here. This could be overcome for a small difference between the two densities by means of a perturbation expansion using the present theory as the zeroth order solution. This method could be used for small differences in total pressures too. However, this will then involve the evaluation of a nonlinear integral by some other method. Using the present analysis as a starting point, one could add viscous effects around the separation streamline. Inside the viscous region, a simplified set of the Navier-Stokes equations could be solved and matched to the potential solution outside . 59 Finally, since film cooling is most important at the leading edge of the turbine blade, a model of a jet penetrating into a counterstream might be more useful. An analysis of this type has been carried out by Hopkins and Robertson [23) where they claim that the inclusion of different densities does not pose any additional problems. 6.3 C o n c l u s i o n s The idea of a smooth entry of a jet into a mainstream with no discontinuity in velocity across the dividing streamline is important since such a flow would show little tendency to mix and hence make a good insulating layer. The present theory examines such a model for the purpose of analyzing film cooling processes. This theory provides the desired mass flow from a slot for a given freestream total pressure and velocity, the coefficient of discharge from a slot, the coefficient of pressure across the slot as well as the velocity field along the slot surface. The theory compares favourably with the experimental results but discrepancies between the two are due to inaccuracies in the experiment and the need for some viscous effects in the theory. Although real flows of this nature are three dimensional and include viscous and turbulent effects, the 2-D inviscid model allows one to examine the important parameters. It shows that the flow is mainly pressure dominated. It gives the proper form of the mass flow rate as a function of freestream velocity and the proper form for the discharge coefficient. Such models also provide a means for checking (perhaps simplifying) computer models for phenomena which cannot be easily measured. 60 BIBLIOGRAPHY [I] Sobey, I.J. Laminar Boundary-Layer Flow Past a Two-Dimensional Slot. J . Fluid Mech., Vol. 83, pp. 33-47, (1977). [2] Fitt, A.D. , Ockendon, J.R. Jones, T.V. Aerodynamics of Slot-Film Cooling: Theory and Experiment. J . Fluid Mech., Vol. 160, pp. 15-27, (1985). (3] Kacker, S.C., Pai, B.R. and Whitelaw, J .H . The Prediction of Wall Jet Flows with Particular Reference to Film Cooling Progress in Heat-Mass Transfer. Vol. 2, pp. 163-186, (1969). (4] Stek, J .B. and Brandt, H. Aerodynamic Throttling of a Two-Dimensional Flow by a Thick Jet. Aeronautical Quarterly, pp. 230-242. [5] Cole, J .D and Aroesty, J . The Blowhard Problem - Inviscid Flows with Surface Injection. Intern. J . Heat Mass Transfer., Vol. 11, pp. 1167-1183, (1968). [61 Ting, Lu and Ruger, C. Oblique Injection of a Jet into a Stream. AIAA J . , Vol. 3, pp. 534-536, (1965). [7] Sinitsin.D. A Numerical and Experimental Study of Flow And Heat Transfer From a Flush Inclined Film Cooling Slot., U.B.C. Dept. of Mech. Eng. Thesis, (1989). [8] Ting, Lu. Pressure Distribution on a Surface With Large Normal Injection. AIAA J . , Vol. 4, pp. 1573-1579, (1966). [9] Ackerberg, R.C. and Pal, A. On the Interaction of a Two-Dimensional Jet with a Parallel Flow. J . Math. Phys.,Vol. 47, pp. 32-56, (1968). [10] Stropky,David. Personal Communication, (1991) [II] Ehrich, F.F. Penetration and Deflection of Jets Oblique to a General Stream. Journal of the Aeronautical Sciences, Vol. 20, p. 99-104, (1953). [12] Goldstein, M.E. and Braun, W. Inviscid Interpretation of Two Streams with Unequal Total Pressures. J . Fluid Mech., Vol. 70, pp. 481-507, (1975). [13] Paterson, A.K. A First Course in Fluid Dynamics. Cambridge Univ. Press, (1987). 6 1 114] Bewley, L.V. Two-Dimensional Fields in Electrical Engineering. Dover Publishing Inc., N.Y., (1968). [15] Parkinson, G.V. and Jandali, T. A Wake Source Model for Bluff Body Potential Flow. J . Fluid Mech., Vol. 40, pp. 577-594, (1970). [16] Gurevich. Theory of Jets in Ideal Fluids. Academic Press,N.Y.,(1965). [17] Milne-Thomson. Theoretical Hydrodynamics, Univ. Press, Glasgow, (1972). [18] Birkhoff, E . and Zarantonello, H. Jets Wakes and Cavities. Academic Pres, N.Y., (1957). [19] Gartshore, I., Salcudean, M. , Riaha, A and Djilali, N. Measured and Calculated Values of Discharge Coefficients from Flush-Inclined Holes. Canadian Aeronautics and Space Journal, Vol. 37, pp. 9-15, (1991). [20] Wittig, S. and Scherer, V. Heat Transfer Measurements Downstream of a Two-Dimensional Jet Entering a Cross Flow. J . of Turbomachinery, Vol. 109, pp.572-578. [21] Mick, W.J. and Mayre, R.E. Stagnation Film Cooling and Heat Transfer, Including its Effect within the Hole Pattern. Journal of Turbomachinery, Vol. 110, pp. 67-72, (1988). [22] Gartshore, I., Salcudean, M. and Marys, P. Film Cooling Effectiveness. CSME Mechanical Eng. Forum, (1990). [23] Hopkins, D.F. and Robertson, J . M . Two-Dimensional Incompressible Fluid Jet Penetration. J . Fluid Mech., Vol. 24, pp. 273-287, (1967). 62 APPENDIX I CALCULATION OF DIVIDING STREAMLINE the dividing streamline in the potential plane is: V> = o,«s > 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™ so that Im(TeT) = 0 With T = | +in we have Jm((£ + »Tf)e*(co6TJ + 1 sin T J ) ) = 0 or £ = —TJ cot TJ h e n C e T = r,(-cotv + i) = —: (cos TJ — 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—l $ 8 j = (r(c) + l)r-- 1 i» = (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 *(«) - -(T(t) +1) yo l ( r m + 1 ) r . _ 1 ] 1 + . / „ * 

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