UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI Date: 17-May-2010 I, Shane Bunnag, hereby submit this original work as part of the requirements for the degree of: Master of Science in Aerospace Engineering It is entitled: Bleed Rate Model Based on Prandtl-Meyer Expansion for a Bleed Hole Normal to a Supersonic Freestream Student Signature: Shane Bunnag This work and its defense approved by: Committee Chair: Awatef Hamed, PhD Awatef Hamed, PhD 8/17/

2 Bleed Rate Model Based on Prandtl-Meyer Expansion for a Bleed Hole Normal to a Supersonic Freestream A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science in the Department of Aerospace Engineering and Engineering Mechanics of the College of Engineering by Shane Bunnag B.S. California State Polytechnic University, Pomona June 2004 Committee Professor Awatef Hamed (Chair) Dr. John W. Slater, NASA Glenn Research Center Professor Prem K. Khosla

3 Abstract The presented work shows that Prandtl-Meyer expansion can be used as a foundation to predict bleed rate for a single bleed hole oriented normal to a supersonic freestream. A CFD study was used to explore flowfield phenomena that can be used in conjunction with Prandtl-Meyer expansion theory to improve model accuracy. Of these phenomena, the shear layer and barrier shock were the best defined and their geometric placement within the bleed hole were the basis for the bleed rate model. Coefficients of variation of the root mean square error between data and predictions were between 0.10 and 0.15 for all but the highest of freestream Mach numbers evaluated. Development of an analytical bleed rate model and recommendations for follow-on activity are presented. i

4 Acknowledgements I am indebted to my advisor, Dr. Awatef Hamed, for providing expertise and instruction through not only the duration of this thesis but also through several enjoyable and worthwhile classes I had the pleasure of attending. This work would not have been possible without her guidance, enthusiasm, and consummate involvement with the propulsion community. I am also grateful to my manager at GE Aviation, Mr. Stephen Ettorre, who gave me opportunity to pursue a graduate education. His encouragement and willingness to endure my hectic and constantly fluctuating schedule made it possible for me to attend classes in the middle of the work day. My professional and academic endeavors would not be possible without my family. I am forever thankful to my mom and dad. Their sacrifices and lifetime of hard work are the foundation for all my accomplishments. I would also like to mention my brother, Charles, whose untiring commitment to his profession is a source of inspiration. Finally, I feel truly fortunate to have experienced graduate school together with Ms. Jacqueline Siegel, whose love and support I could not have done without. i

5 Table of Contents Abstract... i Acknowledgements... i Table of Contents... ii 1.0 Introduction Purpose of an Inlet Subsonic Inlets Supersonic Inlets The (Un)Starting Problem Control of Shockwave/Boundary Layer Interaction by Boundary Layer Bleed Boundary Layer Bleed Modeling Bleed Rate Capability Models In Lieu of Resolved Holes Prior Published Work Motivation and Objectives CFD Simulation of Bleed Through a Single Resolved Hole Rationale Computational Details Quantitative Predictions Observations Acceleration of Boundary Layer Upstream of Bleed Hole Corner Expansion Detached Shock Recirculation Region Jet Plume Bleed Rate Model Based on Prandtl-Meyer Expansion Development Results and Discussion Conclusions and Recommendations References Figures Figure 1. Diameter variation of a streamtube with increasing Mach number ii

6 Figure 2. Subsonic inlet shown in typical underwing installation with capture streamlines at flight Mach numbers above, near, and below design Mach number Figure 3. Cutaway of a General Electric CF6-50 turbofan mounted under a Boeing 747 pylon. Representative of most underwing engine installations for subsonic aircraft. Reproduced from [34] Figure 4. Bow shock formation upstream of a subsonic inlet operating at Mach number much greater than design intent Figure 5. Supersonic two-dimensional ramp inlet with capture streamlines and compression shock structure at design Mach number Figure 6. Cutaway of an inlet installation on a BAe Concorde. Representative of underwing engine installations for supersonic aircraft featuring two-dimensional ramp inlets with variable geometry such as the Grumman F-14 and McDonnell Douglas F-15. Reproduced from [35]. 43 Figure 7. Relationship of corner expansion and barrier shock at a boundary layer bleed hole. Reproduced from [11] Figure 8. Schematic of computational domain [10] Figure 9. Computational grid in bleed region [10] Figure 10. Sonic flow coefficient predictions from CFD study presented in Section Figure 11. Mach number contours for CFD study presented in Section Figure 12. Total pressure contours for CFD study presented in Section Figure 13. Static pressure contours for CFD study presented in Section Figure 14. Streamlines of bleed flow for CFD study presented in Section Figure 15. Decrease in boundary layer thickness in a supersonic flow approaching a corner. Reproduced from [28] Figure 16. Contours of constant streamwise velocity for plenum pressure ratio of with bleed hole located inches from inflow boundary Figure 17. Contours of constant streamwise velocity for plenum pressure ratio of with bleed hole located inches from inflow boundary Figure 18. Flowfield features for a supersonic flow separating off an aft-facing step. Reproduced from [29] Figure 19. Instantaneous streamwise pressure contours for supersonic flow over a cavity. Reproduced from [31] Figure 20. Detached shock and flow regimes for a wedge of large apex angle in supersonic flow Figure 21. Contours of Mach number showing geometric relationship among key flowfield features. Plenum pressure ratio of shown iii

7 Figure 22. Streamlines of bleed flow for CFD study presented in Section 4.0. View is through bleed hole wall, looking toward symmetry plane Figure 23. Static pressure contours for CFD study presented in Section 4.0. Note deflection and weakening of barrier shock due to interaction with boundary layer growth on bleed hole wall. Plenum pressure ratio of shown. View is top looking down into bleed hole Figure 24. Profiles for a jet ensuing from a near-sonic orifice. Reproduced from [33] Figure 25. Flowfield described by bleed rate model Figure 26. Dimensions for definition of flow areas Figure 27. Dimensions for definition of flow areas Figure 28. Isosurface of sonic flow. Three-dimensional nature of shear layer, barrier shock, and sonic line due to boundary layer interaction are clearly depicted Figure 29. Predicted sonic flow coefficients compared to experimental data [7]. Solid lines represent predicted values. Symbols represent data at different Mach numbers Figure 30. Distribution of standoff distances used to close model Figure 31. Distribution of flow turning angles predicted by Prandtl-Meyer theory Figure 32. Coefficient of variation of the root mean square error between data [7] and predictions Appendix: Tabulated Results for Bleed Rate Model iv

8 1.0 Introduction 1.1 Purpose of an Inlet The flight Mach number of an aircraft will rarely match the Mach number required by the engine compressor for efficient operation. Quite often, the flight Mach number will be significantly higher, I particularly on supersonic aircraft. The purpose of the inlet is then to capture air moving at the speed of the aircraft and diffuse it to a speed usable by the engine. If the air approaching the inlet is assumed to be one-dimensional, compressible, and isentropic, then the mass of air contained in the streamtube captured by the inlet can be found by using the mass flow parameter [1]. This relationship reveals that the streamtube area that must be captured by the inlet will vary throughout the operating envelope of the aircraft. The streamtube area will be small at high Mach numbers and will increase toward infinity as Mach number goes to zero. This relationship is shown as a plot in Figure 1 and illustrated conceptually in Figure 2. A successful inlet design will be able to capture the range of flows demanded by the engine throughout the entire speed and maneuver envelope of the vehicle. 1.2 Subsonic Inlets For subsonic aircraft, a well-designed inlet is capable of handling the entire range of streamtube areas in the flight envelope of a given aircraft with very little aerodynamic loss. Inlets designed for subsonic flow are generally uncomplicated I One notable exception is the Lockheed C-141 which features engines with a zero-length inlet. No diffusion was required because the flight Mach number was very close to the required compressor inlet Mach number during cruise operation [36]. 1

9 flowpaths that, at least conceptually, consist of an opening with a rounded lip to allow air to smoothly enter from non-axial directions, followed by a divergent flowpath to diffuse the flow headed toward the engine face (Figure 2 and Figure 3). In fact, treatment of subsonic inlets in most academic texts revolves around compressible, isentropic flow theory [2, 3]. Total pressure at the exit of the inlet can be expected in the range of 97% of ambient conditions in this straightforward and ubiquitous design [3]. 1.3 Supersonic Inlets In the subsonic regime, losses are attributable only to viscous forces. At transonic and supersonic flight speeds, shockwaves create a fundamentally new aerodynamic loss mechanism that limits the versatility of the inlet design shown in Figure 2 and Figure 3. At higher Mach numbers the streamtube containing the flow required by the engine is very small. An inlet of this type, if designed with enough margin to pass the flow demanded by the engine at low flight speed, will have an opening much larger than required for the approaching streamtube at high Mach numbers. As a result, a bow shock will form upstream of the inlet opening to abruptly slow the oncoming flow to subsonic speed, thereby matching the capture streamtube diameter to the inlet opening diameter (Figure 4). Flow that is not contained in the capture streamtube bypasses the inlet and causes spillage drag which is credited against the net thrust of the engine. In addition, the bow shock causes a large total pressure loss that adversely affects the thermodynamic efficiency of the engine. These problems are severe enough to obviate the type of inlet shown in Figure 2 and Figure 3 for supersonic aircraft. 2

10 The most commonly used type of inlet for supersonic aircraft decelerates the flow with a series of lower-loss oblique shocks on an external surface before introducing it into a converging duct to finally decelerate to the flow to subsonic speed with a terminal normal shock (Figure 5 and Figure 6). Because part of the compression process takes place within a duct where the flow is still supersonic, a process called starting is used to ingest the bow shock of Figure 3 into the inlet duct where it becomes the terminal shock, downstream of which the flow becomes subsonic 1.4 The (Un)Starting Problem Starting an inlet usually involves lowering the backpressure on the flow entering the inlet, often through some means of variable geometry. At low enough back pressure, the bow shock is ingested by the inlet duct where it forms the terminal shock in a series of oblique compression shocks. The detailed process for starting an inlet is well described in academic texts [2, 3]. Downstream of this terminal shock, the flow is subsonic and can be further diffused through a divergent duct. Once formed, the shock system, particularly the terminal normal shock, is neutrally stable due to competing flowfield effects. As the terminal shock moves downstream toward the throat of the inlet duct, it will stand at a lower Mach number, and the total pressure loss across it will be lower. As the total pressure loss across the shock decreases, the inlet duct will be capable of passing more flow than is captured by the inlet opening. To satisfy continuity, the shock is discharged downstream of the throat to the divergent region of the inlet duct where flow once again becomes supersonic and the total pressure loss across the shock increases due to higher Mach 3

11 number, thereby reducing the flow capability of the inlet duct. Conversely, the pressure on the downstream side of the terminal shock increases as the flow is diffused through the inlet duct. Inlet geometry and engine control scheduling is designed so that these competing forces balance and the terminal shock finds its final resting point at or just downstream of the inlet throat. Because the terminal shock is neutrally stable, small flowfield perturbations such as a reduction in engine airflow requirement (e.g. pilot-commanded throttle movement) or wind gust can cause it to be expelled out of the inlet in a highly dynamic process called unstart. During an unstart, the pressure recovery of the inlet drops nearly to zero in a matter of milliseconds [4]. Damage to the engine and inlet structure, compressor stall, and engine flameout are all possible outcomes of an inlet unstart. Of more immediate consequence to the pilot, however, is the violent yawing moment that is imparted to twin-engine aircraft that experience an asymmetric unstart while traveling at high Mach numbers. Some of the most extreme accounts of these events were documented during the operational history of the SR-71. The thrust imbalance caused by a single-engine unstart induced yawing moments so severe that the pilot s head was thrown into cockpit canopy with enough force to permanently damage his helmet [4]. 1.5 Control of Shockwave/Boundary Layer Interaction by Boundary Layer Bleed Independent of external factors such as throttle movement and changes in ambient conditions, unstart can arise from the flowfield within the inlet itself. Interaction between the inlet shock system and the boundary layer on the compression surfaces can cause boundary layer separation and reduce the area available to inlet flow. 4

12 The resulting increased backpressure can be enough to expel the neutrally stable shock system out of the inlet opening. Additionally, since the boundary layer is subsonic, the effect of the shockwave on the flowfield has a means to propagate upstream as far as 50 boundary layer thicknesses [5]. If this upstream propagation is disruptive enough, it can interfere with the formation of the compression shock system and cause an unstart. Needless to say, boundary layer growth on the surfaces of an inlet flowpath must be controlled. A common and established method for doing this involves bleeding the boundary layer away from the inlet flowpath through an array of small holes near the region of shockwave impingement. Slater, et al. describe the bleed process in their 2009 paper [6]. Portions of their work, along with others, are paraphrased here. The flow that is bled away from the inlet flowpath is mostly the lower momentum portion of the boundary layer. As a result, the average kinetic energy of the boundary layer is increased [7]. Consequently the ability of the boundary layer to overcome the adverse pressure gradient through the shockwave is significantly enhanced. As noted in Section 1.4, a decrease in engine airflow can cause the terminal shock to migrate upstream due to an increase in static pressure at the engine face. Left unattended, this can precipitate an unstart. To control upstream movement of the terminal shock, boundary layer bleed can also be used to compensate for the decrease in engine airflow. As the terminal shock moves upstream into the bleed region, the bleed flow increases due to the static pressure rise downstream of the shock. This increase in 5

13 bleed flow alleviates some of the back pressure behind the shock and inhibits its upstream motion. The geometry of the bleed hole itself can also provide a degree of stabilization to operating environment of the inlet. Because the inlet freestream II is supersonic, a shockwave forms within the bleed hole near its downstream edge that partly extends into the boundary layer of the freestream. In the vicinity of the bleed hole, this shockwave can prevent upstream communication of the adverse pressure gradient through the subsonic portion of the boundary layer. As the result, the compression shock system is more tolerant to downstream instabilities. This shockwave is often called a barrier shock [8] and will be discussed in detail in Section Bleed regions commonly consist of circular holes with a diameter much smaller than the characteristic dimensions of the inlet flowpath (e.g. inlet diameter or length). Bleed holes are arranged in rows or groups where necessary and can number in the hundreds. The bleed system exhausts flow into a plenum in which the static pressure is lower than that of the local inlet freestream. Bleed flow rate increases as the plenum pressure decreases until the bleed system chokes, after which the flow rate will remain constant despite lower plenum pressures. The plenum is usually sized so that the bleed flow can decelerate and static pressures can equalize throughout the plenum. The bleed flow is ducted away from plenum and can be used by the engine or other aircraft system. However, a common approach is to simply dump the bleed flow overboard to the ambient flow. While less complicated than trying to scavenge the II The flow in the inlet duct is referred to as the freestream. This should not be confused with the flow that surrounds aircraft in flight which is referred to as the ambient stream. 6

14 bleed flow, dumping it overboard adds a component of drag that is due to the loss of momentum of the dumped flow. To reduce the drag penalty associated an overboard dump, the exit flowpath for the bleed flow is usually designed to exhaust at supersonic speed and approximately in line with the ambient flow or at a location which might reduce base drag. Exhausting the flow at supersonic speed also ensures that external disruptions do not enter the plenum through the exit flowpath and affect plenum static pressure. The ducting between the plenum and exit can be a significant part of the inlet structure. For example, a plenum for a bleed region on the centerbody of an axisymmetric inlet may exhaust through a duct that is directed through the centerbody itself and then through a support strut to reach the exit located on the external surface of the inlet cowl. Such a bleed system is used on the inlet spikes of the SR-71. Operation of the bleed system requires that the plenum pressure be maintained at lower static pressures than that of the local inlet freestream. The plenum pressure is obtained by connecting the plenum to the ambient flow through a duct. The interface of the duct to the ambient flow operates as a converging diverging nozzle with choked flow at the throat. Thus, the size of the nozzle throat area controls the plenum pressure and, consequently, the bleed rate. Some bleed systems are designed such that the bleed holes are always choked. However, this can lead to increased levels of bleed drag. The alternative is to maintain a plenum pressure such that the bleed holes are not choked, but are still effective in extracting the desired bleed rate. A control system that adjusts the size of the exit nozzle area can be used to set the plenum pressure. The stability of the inlet against unstart can be improved by 7

15 controlling the plenum pressure to maintain a near-constant plenum pressure as the terminal shock encounters the throat bleed regions and the bleed rate increases. However, such a control system would be complicated, costly, and heavy to implement in an airframe application. A fixed-exit area may be sufficient to extract the desired bleed flow over the range of operations of the inlet. 8

16 2.0 Boundary Layer Bleed Modeling 2.1 Bleed Rate Capability The ability of a bleed system to remove mass from the inlet flowfield is evaluated by the sonic flow coefficient [7]: Q sonic = m bl Eqn. 2.1 m ideal This ratio compares the actual mass flow rate that the bleed system removes m bl by the mass flow rate through an equivalent area if the flow is expanded isentropically to fill the flow area, as described by m ideal = p T A bl γg c T T R 1 + γ 1 2 γ+1 2 γ 1 Eqn. 2.2 where p T and T T are total properties of the inlet freestream and A bl is the flow area of the bleed region in question. Combining Eqns. 2.1 and 2.2 gives: Q sonic = p T A bl m bl γg c T T R 1 + γ 1 2 γ+1 2 γ 1 Eqn. 2.3 The sonic flow coefficient is often plotted against the plenum static pressure normalized to inlet freestream total pressure p S pl p T. For a given inlet freestream Mach number, the bleed system will choke at a particular plenum pressure. At this condition, the bleed system is not capable of higher bleed rates despite lower plenum pressures. 2.2 Models In Lieu of Resolved Holes A stated previously, boundary layers are usually bled away from the inlet freestream through an array of holes. However, in a practical analysis, such as an 9

17 industry CFD simulation of an integrated inlet system, discrete bleed holes are usually not resolved in the computational grid [9, 10]. To do this would be incredibly time intensive and would overburden all but the most powerful computer resources. A surrogate bleed model is used instead to specify mass removal rates and velocity components along a region representing the array of bleed holes in the computational domain. The goal is then to attempt to capture the effect of bleed holes by specifying an appropriate boundary condition over the bleed region rather than incorporating the geometry of the hole into the computational domain. Although the main purpose of a bleed hole is to draw the boundary layer away from the inlet freestream, its effect goes beyond this singular act. In addition to mass removal, a bleed hole can add a host of beneficial and detrimental effects to a flowfield including turbulence generation [11] and vortex shedding [9]. The model that will be presented aims only to model the rate of mass removal. Other aspects of bleed hole interaction with the inlet flowfield are not represented. 2.3 Prior Published Work In their recent paper, Hamed, et al. [9], provided a chronology of prior work in boundary layer bleed modeling. Relevant portions are paraphrased here. Bleed models have historically been based on empirical correlations of mass flux. Harloff et al. [12] derived correlations for sonic flow coefficient based on Bragg s orifice flow model [13] which they validated with experimental data. Computational results obtained using these bleed models [14] under-predicted the static pressure downstream of the shock wave/turbulent boundary-layer interactions when compared to experimental data from 10

18 Willis et al. [15]. This in turn led to inaccuracies between data and computed bleed mass flow. Hamed et al. [10] also reported differences between the numerical predictions bleed mass flow using bleed models and those obtained from three-dimensional simulations which resolved the flow within the bleed-hole passages. Slater [16] reported differences among the mass flux predictions by three bleed models in the WIND-US code through an oblique shock-wave/boundary-layer interaction. The differences between the bleed model predictions were noticeable upstream of the oblique shock where some models did not bleed any flow while others predicted negative bleed (injection) even though the model coefficients were adjusted to provide the same overall bleed. Akatsuka et al. [17] and Slater [18] subsequently proposed modifications to improve predictions at high bleed pressure ratios and to allow blowing into the main stream. Hamed et al. [19-22], Flores et al. [23], and Lin et al. [24] conducted computational studies that resolved the flow within the bleed passages that consisted of different hole geometries including normal, slanted, single and multiple holes and slots. Several numerical studies revealed flow features in the bleed region and within the bleed passages including the barrier shock, which were reported first by Hamed et al. [25, 26] and by Rimlinger et al. [8]. In the works cited above, the mechanism that turns the freestream flow into the bleed hole is assumed to Prandtl-Meyer expansion. Shih diagrams the corner expansion and its relationship to the downstream barrier shock [11, 27]. The diagram is reproduced 11

19 in Figure 7. While supersonic corner expansion is known to occur at forward edge of the bleed hole inlet, it has been largely ignored as a driving element in bleed models. 12

20 3.0 Motivation and Objectives Despite the vast amount of work on the subject, there is a history of inadequate boundary layer bleed capability during in-flight operation when compared to predictive models. This has been documented on many aircraft programs including the B-1, F-15, F-18, and F-22. At first, it may seem that the obvious solution is to simply design a bleed system with excess capability. However, this has its own problems. While inadequate bleed is undesirable, excessive bleed can also be detrimental. Flow that is bled away from the inlet has essentially been stolen from the jet engine and will not participate in the production of thrust. Furthermore, the process of diverting flow into the bleed system creates aerodynamic loss that manifests itself as drag on the aircraft. In light of this, accurate prediction of bleed system performance continues to be a subject of research in the aerospace community. The remainder of this paper will present a bleed rate model that uses Prandtl- Meyer expansion theory as a basis for predicting the rate of boundary layer mass bled away from the freestream through a single bleed hole. III This paper will also present the results of a CFD study and the subsequent literature search that guided the development process. It is also hoped that the material will also aid future efforts in modeling boundary layer bleed from first principles. III At the time of writing, work on using Prandtl-Meyer expansion to predict bleed rate was also presented by Randall Chriss (NASA Glenn Research Center) at the 3rd Annual Shock Wave/Boundary Layer Interaction Workshop in Cleveland, OH. 13

21 4.0 CFD Simulation of Bleed Through a Single Resolved Hole 4.1 Rationale A number of bleed models have been developed over the years. While some are quite successful, they are not based on first principles and rely heavily on experimental data [11]. As the first steps in development of a bleed rate model based on first principles, a CFD study was conducted to better visualize the bleed flowfield. Subsequently, a literature search was carried out to uncover details of the individual phenomena that constitute the bleed flowfield. While not all of the phenomena discussed are incorporated into the development of the bleed model, they will help explain any discrepancies between predicted values and experimental data. 4.2 Computational Details The computational domain used for the CFD study is identical to that used by Hamed, et al. [9] and is shown in Figure 8 and Figure 9. It consists of a freestream with a flat plate as the bottom boundary in which a bleed hole is set. The bleed hole is oriented 90 to the flat plate and exhausts into a plenum region that is many times larger in both length and diameter than the hole itself. The hole is circular so flow symmetry is assumed along the streamwise-normal plane (x-z in Figure 8). Therefore only half of the freestream, bleed hole, and plenum are modeled and a symmetry boundary condition is used simulate the other half of the domain. The fluid used in the computational domain is an ideal gas. The inflow boundary is a supersonic boundary with total pressure and total temperature specified at psi and 527 R respectively. Velocity in the streamwise 14

22 direction was set to correspond to a freestream Mach number of Normal and spanwise components of velocity were set to zero. No initial boundary layer thickness was specified and no velocity, pressure, or temperature profiles were enforced. The flat plate is an adiabatic, no-slip wall inches wide and inches long. The boundary of the domain opposite of the flat plate (i.e. the top of the domain) is an adiabatic, free-slip wall located sufficiently far away from the flat plate surface that it does not influence the flow through the bleed hole. A free-slip wall was specified for computational efficiency. The bleed hole is a 0.25-inch diameter circular hole located inches downstream of the inflow boundary. The length of the bleed hole is also 0.25 inch. The walls of the hole are adiabatic, no-slip walls. The freestream outflow boundary is specified as supersonic outlet and therefore has upstream conditions imposed upon it. The plenum domain consists of no-slip, adiabatic walls and a subsonic outlet. The static pressure at the plenum outlet is specified to attain the desired plenum pressure ratio p S pl p T. A range of plenum pressure ratios from from 0.04 to 0.36 were simulated. Unlike the procedure used by Hamed et al. [9] and Slater [18], the computational domain was not manually divided into subdomains for parallel processing. Instead, a MeTiS partitioning algorithm was used to automate this process. The simulation was run using version 12.0 of the CFX commercial CFD code with the SST turbulence model. 15

23 4.3 Quantitative Predictions The sonic flow coefficients predicted by the CFD model for varying plenum pressure ratios are shown in Figure 10. As stated in Section 4.1, the primary purpose of the CFD study was to visualize the flowfield. Therefore, sonic flow coefficient predictions are presented here only for completeness and are not incorporated into the model presented in Section Observations Contour plots of predicted Mach number, total pressure and static pressure along the symmetry plane at the center of the bleed hole are presented in Figure 11 through Figure 13. Three-dimensional streamlines seeded at the surface of the bleed inlet are presented Figure 14. The following subsections present observations on the mechanisms that are believed to affect bleed rate. Findings from a literature search for each observation are also discussed Acceleration of Boundary Layer Upstream of Bleed Hole Close inspection of the flow upstream of the bleed hole shows that boundary layer thickness is decreasing as it is approaching the hole. This is an interesting observation since the flow is supersonic and should not be affected by hole geometry until it crosses the first Mach line of the corner expansion. However, since the boundary layer is subsonic, the effect of the corner expansion has a means of communicating upstream[5]. Instead of rising abruptly as predicted by inviscid theory, the Mach number of the boundary layer will increase gradually as it approaches the upstream lip of the bleed hole. As a result, boundary layer thickness will decrease. Olssen, et al. describes the 16

24 change in boundary layer thickness to be quite considerable for a high speed flow approaching an aft-facing step [28]. Their work suggests that the boundary layer almost completely vanishes as it meets edge of the corner expansion geometry (Figure 15). A plot of streamwise velocity contours near the upstream edge of the bleed hole for plenum pressure ratio p S pl p T of predicts the same behavior (Figure 16). This begs the question, Does upstream boundary layer thickness affect bleed rate? To answer this, a CFD solution with a different boundary layer thickness was generated. The model described in Section 4.2 was rerun with a plenum pressure ratio p Spl p T of with the inflow and outflow boundary of the freestream swapped. With this arrangement, the freestream flow must travel inches instead of inches before it reaches the bleed hole, thus allowing for more boundary layer growth. A plot of streamwise velocity contours near the upstream edge of the bleed hole for this configuration shows the same behavior as describe above (Figure 17). Note that in this case, the boundary layer thickness is roughly the same as the diameter of the hole. Remarkably, the sonic flow coefficients for the two cases differ by only 3.68% despite the boundary layers thicknesses being an order of magnitude different. While bleed rate is seemingly unaffected, other factors such as the health of the boundary layer downstream of the hole are probably heavily influenced by upstream boundary layer thickness Corner Expansion Most, if not all, of the works cited in Section 2.3 have suggested that the mechanism that turns and accelerates the boundary layer flow into the bleed hole is the well-known Prandtl-Meyer expansion fan. However, Smith indicates that supersonic 17

25 flow is turned through the corner of an aft-facing step through a means not governed by Prandtl-Meyer theory [29]. Smith shows that flow separation off an aft-facing step consist of a supersonic expansion phenomenon that interacts with a region of recirculating flow. The additional complexity associated with this expansion process is most noticeable in the shape of the Mach lines. In Prandtl-Meyer theory, the Mach lines are straight and radiate from a single point, whereas the Mach lines in Smith s description do not originate from a single point and are highly curved (Figure 18). A shear layer is created between the flow that is turned past the corner and the flow in the region of recirculating flow IV behind the step. In the realm of boundary layer bleed, Shih also describes a region of separated flow within a bleed hole oriented 90 to the freestream [11]. The separated flow is analogous to the recirculating flow region behind an aft-facing step, as described by Smith. Furthermore, when Shih plots Mach number contours along the centerline of the bleed hole, the contours that form the border between the recirculation region and the expanding flow are tightly spaced, indicating strong shearing action as predicted by Smith. Shih s CFD prediction of the Mach number contours is similar to those shown in Figure 11. Further corroboration for both Smith s and Shih s findings are provided by Figure 14 in which streamlines of recirculating or near-stagnant flow can be seen in the bleed hole flowfield at lower plenum pressure ratios p S pl p T. The distinction between Prandtl-Meyer expansion and the corner expansion process described by Smith is important. In textbook treatments of Prandtl-Meyer IV In literature related to flows detaching from an aft-facing step, this region is be called the base region. 18

26 expansion [1, 5, 30], the flow is turned over a solid, free-slip boundary. There is no simple means of accounting for what happens when a shear layer takes the place of the solid boundary. Furthermore, when determining the turning angle of the flow, the static pressure downstream of expansion is usually known beforehand when applying Prandtl-Meyer theory. In Smith s work, the pressure downstream of the expansion is a result of the surrounding flowfield (i.e. not known beforehand) and is solved for by an iterative or numerical process. It is difficult to determine if Smith s interpretation of corner expansion phenomena is entirely applicable to the boundary layer bleed process. While a bleed hole is not an aft-facing step, there are undeniable similarities in the recirculation region and shear layer seen in Shih s CFD predictions, the CFD figures of Section 4.3, and Smith s work. Conversely, it could also be the case that Prandtl-Meyer expansion is a reasonably good approximation for the expansion process since pressure downstream of the expansion fan was somewhat enforced by the plenum. In other words, the plenum sink pressure has a greater influence on the corner expansion than the effects described by Smith. In addition to an aft-facing step, work on the subject of supersonic flow over a cavity was reviewed to see if it could be used as an analogy to the flowfield in a bleed hole. A time-unsteady numerical study performed by Aradag [31] indicates the flow in a cavity is considerably different from the flow within a bleed hole. Streamlines of flow and contours of static pressure predicted by his study are presented in Figure 19. Note that the flow has no means to exit the cavity other than through the same opening it 19

27 entered. This causes a highly unsteady flowfield. Additionally, the recirculation region and shear layer form at the opposite end of the cavity than would be expected for a bleed hole. Because of these differences, cavity flows are ignored as a possible analog to boundary layer bleed Detached Shock After the flow is turned through the corner expansion it must be turned again. A majority of the expanded flow is turned into the bleed hole and the balance of it is turned back toward the freestream. This shown in Figure 14 and has been documented by Harloff et al. [12], Hamed et al. [25, 26], and by Rimlinger et al. [8]. The flow, having emerged from an expansion fan, is now at a higher Mach number than it was in the freestream. Near the downstream lip of the bleed hole a shockwave must form to turn the flow. This shockwave has been described as a two-segment shock [18] because part of it turns the flow back toward the freestream and the rest of it turns the flow in the direction of the bleed hole. Indeed, Figure 11 through Figure 13 show a shockwave with a noticeable kink that separates two segments of the shockwave. In literature, this shockwave is often called a barrier shock because it can prevent downstream flow disturbances from being propagated upstream through the boundary layer. This is extremely useful in inlet design where a barrier shock can be used to isolate the effects of shockwave impingement on a boundary layer from the upstream flowfield. Figure 11 through Figure 13 suggest that, at least at the symmetry plane of the bleed hole, all of the flow that enters the bleed plenum crosses the lower segment of the barrier shock. In light of this, barrier shock geometry, particularly the standoff distance 20

28 from the downstream lip, could be useful in predicting the bleed rate from first principles. The author speculates that, at the symmetry plane of the hole, the barrier shock is analogous to a detached bow shock upstream of a wedge with a large apex angle. In this case the wedge would have an apex angle of 90 and would be oriented at some angle of attack to the approaching flow. Unfortunately, very little, if any, data exists for such large wedge angles, let alone wedges with dissimilar backpressures on each side (in this case, p Spl on one side, p S on the other). Analytical work in the area of detached shocks has shown that the standoff distance is dependant of the location of the sonic line downstream of the shock [32]. That is, the location in which the flow becomes supersonic after having decelerated to subsonic speed through the shock. For a wedge or a thick plate with a wedge forebody in supersonic flow, the sonic line always occurs at the aft-most point of the wedge (Figure 20). In the case of the downstream edge of the bleed hole, the wedge geometry extends to infinity relative to the bleed flowfield. Finding the location of the sonic line in the flow that is turned away from the bleed hole (i.e. the flow that passes the through the upper segment of the barrier shock) becomes a problem of accelerating flow along a flat plate. The solution to which is possibly beyond the scope of determining bleed rate. For the flow that enters the hole, however, the sonic line always is always connected to the intersection of the shear layer that is formed by the corner expansion and the lower segment of the barrier shock. This relationship is shown in Figure 21. In this treatment, the location of the sonic line is taken to be the dependent variable in the problem since it can be found if the angle of the shear layer and location of detached 21

29 shock are known. However, short of developing a numerical solver, a simple means of determining the standoff distance of the detached shock from first principles remains to be found. The author considers this a worthwhile endeavor for future work since the sonic line and the lower segment of the barrier shock are well defined geometries that could be used as control surfaces for a first-principles-based model Recirculation Region For lower plenum pressure ratios p S pl p T, particularly those at which the hole is choked, the static pressure of the recirculation region is often not the same as the plenum pressure (Figure 13). Figure 14 shows that, for these conditions, a small but substantial amount of flow is present in the recirculation region. So far the images shown from the CFD study have been through the symmetry plane looking outward toward the bleed hole wall. In Figure 22, the view is through the bleed hole wall looking inward toward the symmetry plane. In this view, some streamlines can be seen making an aggressive turn away from the region behind the barrier shock toward the recirculation region. Note that this only occurs near the walls of the bleed hole. Because of boundary layer interaction, the barrier shock is relatively weak and ineffective at blocking upstream communication near the wall of the bleed hole (Figure 23). Therefore, the high-pressure flow downstream of the shock has a means to escape the region bounded by the barrier shock. The CFD study shows that the recirculation region is predominantly composed of flow that has escaped from downstream of the barrier shock as well as low-momentum flow that was entrained from boundary layer of the hole. This flow forms only a small 22

30 percentage of the total bleed flow however its presence it noteworthy because it adds yet another complication to the corner expansion process. In Prandtl-Meyer expansion theory, there is no mass entering the control volume other than the expanding flow itself. In Smith s more complete interpretation of corner expansion, the recirculation region behind an aft-facing step is exchanging mass only with the shear layer. In a bleed hole, mass exchange along the shear layer takes place along with additional influx from the process describe above Jet Plume Downstream of the sonic line the flow is unimpeded by any flow structures that were created by the hole geometry. The total pressure of the flow at this point is still high enough to accelerate it to supersonic speed as it moves toward the plenum. Since it is not bounded by a surface, the flow takes on the familiar structure of an underexpanded jet (Figure 11). The jet has a considerable amount of momentum compared to the relatively quiescent flow it is exhausting into. As a result, it is not uncommon for the jet to persist several hole diameters into the plenum [18], especially at lower plenum pressure ratios p S pl p T. While vena contracta within the jet has been the subject of existing work [12], the author s interest in the jet plume focuses on its influence on the recirculation region adjacent to it. Using Rayleigh scattering process, Panda, et al. were able to unobtrusively measure the flowfield ensuing of from a near-sonic orifice [33]. His results, shown in Figure 24, indicate that a jet can begin to affect the surrounding quiescent flow after traveling less than four orifice diameters downstream. From the CFD results shown in 23

31 Figure 11 through Figure 14, the orifice from which the jet plume ensues is bounded by the downstream wall of the bleed hole and the lower segment of the barrier shock. Hence the emphasis on standoff distance in Section At lower plenum pressure ratios p S pl p T, the orifice area occupies a significant portion of the bleed hole. It is believed that at these conditions the jet plume influences a large enough volume of the bleed hole to affect the recirculation region. For bleed rate predictions that focus on the jet plume, the transfer of mass and momentum between the jet and recirculation region could be a source or error. Additionally, any influence that the jet plume has on the recirculation region would be reflected in the in the downstream pressure of the corner expansion process, providing further suggestion that plenum pressure is not necessarily the pressure downstream of the corner expansion. Furthermore, in a multiple bleed configuration, the jet flow into the plenum further interferes with fluid communication between the expansion process and the plenum. This is especially true in multiple row configurations with slanted holes [11]. 24

32 5.0 Bleed Rate Model Based on Prandtl-Meyer Expansion 5.1 Development Development of the bleed rate model is guided by the observations from the CFD study presented in Section 4.4. At the symmetry plane of the bleed hole, the lower segment of the barrier shock is treated as a control surface that all flow is assumed to pass through. In other words, the flow area for the bleed flow is not the geometric area of the hole but rather the surface area of the barrier shock bounded by the walls and the inlet surface of the hole. The following assumptions and simplifications are made: 1. Despite evidence to the contrary, the corner expansion process is assumed to be a Prandtl-Meyer expansion fan with the downstream static pressure equal to the plenum static pressure. All of the flow that is turned through the expansion fan emerges parallel to the shear layer that separates the expanded flow from the recirculation region. 2. A barrier shock forms upstream of the downstream lip of the hole. The division between the lower segment and the upper segment of the barrier shock is assumed to be bleed inlet surface. All of the flow that emerges from the expansion process must pass through the lower segment which is approximated as a normal shock. Accordingly, the shear layer and the expanded flow are assumed to intersect the lower segment of the barrier shock at After crossing the shock, the flow emerges with a lower total pressure and passes through a sonic line which defines the beginning of the jet flow that exhausts into the plenum. The sonic line is assumed to be orthogonal to the axis of the hole. 25

33 4. A region of recirculating flow is formed that is aerodynamically bounded by the shear layer and the edge of the jet plume that issues from the sonic line. 5. The model is only valid for a single bleed hole oriented normal to supersonic freestream. A concession is made with respect to the corner expansion and the highly threedimensional flow effects illustrated in Figure 23. While the corner expansion is most likely a combination of Prandtl-Meyer expansion and the expansion process described by Smith, for the sake of simplicity, it is assumed to be governed only by Prandtl-Meyer theory. Furthermore, the notion that all of the flow passes through the lower segment of the barrier shock and then through the sonic line is only made possible by neglecting the flow in the recirculation region and limiting the analysis to the symmetry plane of the bleed hole and assuming that it is representative of the entire flowfield. An illustration of the assumed flowfield is shown in Figure 25. Since this model attempts to predict the bleed rate, the labeling of the numerator of Eqn. 2.3 is changed as follows: Q sonic = p T A bl m predicted γg c T T R 1 + γ 1 2 γ+1 2 γ 1 Eqn. 5.1 Freestream total pressure and total temperature are taken to be the driving gas dynamic values that determine flow across the lower segment of the barrier shock. Recall, that this is the control surface that determines flow rate into the bleed hole. Substituting the proper terms into the mass flow function yields m predicted and Eqn. 5.1 can be rewritten as: 26

34 p T A γg c γ 1 ls Q sonic ls = T T R M M2 p T A γg c bl T T R 1 + γ 1 2 γ+1 2 γ 1 γ+1 2 γ 1 Eqn. 5.2 where A ls is the surface area of the lower segment of the barrier shock. Since Prandtl- Meyer expansion is an isentropic process, neither total pressure or temperature change from their freestream values therefore these terms as well as the ratio of specific heats in the numerator divide with denominator to unity. Eqn. 5.2 then simplifies to: Q sonic ls = A lsm 1 + γ 1 2 M2 A bl 1 + γ 1 2 γ+1 2 γ 1 γ+1 2 γ 1 Eqn. 5.3 The flow downstream of the Prandtl-Meyer expansion fan has a Mach number of M 2, therefore it is substituted in place of M: Q sonic ls = A lsm γ 1 2 M 2 2 A bl 1 + γ 1 2 γ+1 2 γ 1 γ+1 2 γ 1 Eqn. 5.4 Using well-known isentropic relationships [30], the Mach number of the flow that enters the bleed hole M 2 can be found. Knowing the static pressures of both the freestream and the plenum, which are assumed to be the upstream and downstream static pressures of the expansion fan, respectively: p Tpl p Spl p T p S = 1 + γ 1 2 M γ 1 2 M 2 γ γ 1 γ γ 1 Eqn

35 Since total pressure remains constant across a Prandtl-Meyer expansion p T = p Tpl, Eqn. 5.5 can be rewritten: p S = 1 + p Spl γ 1 2 M γ 1 2 M 2 γ γ 1 Eqn. 5.6 With flow properties known, the geometry of the flow features can be found. The angle that flow is deflected θ 2 is the difference between the Prandtl-Meyer angle of the flow upstream and downstream of the expansion [30]: θ 2 = ν M 2 ν M Eqn. 5.7 where: ν M = γ + 1 γ 1 tan 1 γ 1 γ + 1 M2 1 tan 1 M 2 1 Eqn. 5.8 Given the standoff distance of the barrier shock, determination of the surface area of the lower segment (i.e. the flow area of control surface) is a matter of trigonometry. From Figure 26, the length of the shear layer L sl is defined by: L sl = 2r cos θ 2 Eqn where r is the radius of the hole, is the standoff distance of the barrier shock from the downstream lip of the hole measured along the bleed inlet surface. The length of the sonic line is then: L = 2r L sl cos θ 2 = 2r 2r cos 2 θ 2 Eqn = cos 2 θ 2 28

36 With the length of the sonic line known, it is possible to find the surface area of the circular segment that is bound by the sonic line and the lower segment of the barrier shock (i.e. the sonic flow area). From the dimensions in Figure 27, the angle of arc swept by the circular segment φ is defined by: φ = cos 1 r L r Eqn The sonic flow area is then defined by: A = r2 2 2φ sin 2φ Eqn The sonic flow area can projected into the plane of lower segment of the barrier shock. The projection is the segment of an ellipse with a length along its semi-minor axis equal to the radius of the bleed hole. The length along its semi-major axis a is defined by: a = r cos 90 θ 2 Eqn In the same way the area of an ellipse is found if the area of the circle from which it was projected is known, the area of the elliptical segment is found by: A ellipse = A a r Eqn Is interesting to note the elliptical segment approximates the geometry of the barrier shock quite well (Figure 28). However, only the lower segment of the barrier shock is of interest. Recall that lower segment is assumed to be bound by the inlet surface of the bleed hole. Therefore the surface area of the lower segment A ls is found by repeating the process outlined above for the portion of the elliptical segment that extends beyond 29

37 the inlet surface of the bleed hole and subtracting that area from A ellipse. In this case, the standoff distance Δ, which is yet to be found, is used instead of L in Eqn Since all of the flow is that passes through the lower segment of the barrier shock is assumed to pass through the sonic line, Eqn. 5.6 can be reformulated to use the sonic flow area as the control surface: p Tns A γg c Q sonic = T T R M ns 1 + γ 1 2 M 2 ns p T A γg c bl T T R 1 + γ 1 2 γ+1 2 γ 1 γ+1 2 γ 1 Eqn where p Tns and M ns are the total pressure and Mach number, respectively, of the flow after crossing through the barrier shock. They are found by using the well-known relationships for a normal shock: p Tns p T = γ + 1 M 2 γ 1 M 2 2 γ γ 1 γ + 1 2γM 2 γ 1 1 γ 1 Eqn and M 2 ns = 1 + γ 1 2 M 2 2 γm 2 2 γ 1 2 Eqn Since total temperature does not change across a normal shock, Eqn. 5.9 simplifies to: Q sonic = p T ns A M ns 1 + γ 1 2 M 2 ns p T A bl 1 + γ 1 2 γ+1 2 γ 1 γ+1 2 γ 1 Eqn The remaining unknown variable is the standoff distance of the barrier shock from the downstream lip of the bleed hole Δ. It is required that the flow across A ls is 30

38 equal to the flow across A and since both A ls and A depend on Δ, Δ can be iterated on until Eqn. 5.4 and 5.19 are equal. When this condition is satisfied, an appropriate value for Δ has been found and a prediction for sonic flow coefficient has been made. 5.2 Results and Discussion Results for the bleed rate model are shown in Figure 29. For a given freestream Mach number, accuracy improves with higher plenum pressure ratios p S pl p T. This may simply be due to the fact that at lower bleed rates less of the flow phenomena described in this paper are at work. Therefore there is simply less opportunity for error between the actual flowfield and predictive models. Curiously, for all but the lowest freestream Mach number, the predicted values intersect the experimental data. The reason for this is unclear. Figure 30 shows the distribution of standoff distances that were used to close the model. An initial guess of 0.05 in. was used. Figure 31 shows the angle between the flow downstream of the corner expansion and the bleed inlet surface. The agreement between experimental data collected by Willis, et al. [7] and predicted values is described, in aggregate, by the coefficient of variation of the root mean square error (abbreviated CV(RMSE)) between experimental data and predicted values. Lower values of CV(RMSE) indicate greater agreement between data and predictions. CV(RMSE) is defined as: CV RMSE = n i=1 Q sonic data,n Q sonic predicted,n 2 n n i=1 Q sonic data,n n Eqn

39 where n is the number of data points for a given freestream Mach number. The CV(RMSE) values are shown in Figure 32. Disagreement between prediction and experimental data remains generally constant for all but the highest freestream Mach numbers. General inaccuracies at all freestream Mach numbers and plenum pressure ratios p Spl p T can be traced to the assumptions and simplifications made during development of the model. Restricting the analysis to the symmetry plane of the model is likely the greatest source of error in the model. The shape of the hole adds a spanwise component of velocity to the flow that is not captured by the symmetry plane analysis. Additionally, the flow is not neatly bounded by the shear layer and the jet plume, nor does all of it cross the barrier shock and sonic line. Furthermore, the spanwise flow and interaction with the boundary layer along the walls of the bleed hole cause further errors when the barrier shock and the sonic line are treated as two-dimensional control surfaces. Finally, the reliance on Prandtl-Meyer expansion theory, the key component of the model, could itself be an erroneous decision. Since the application of Prandtl- Meyer expansion is done in isolation of the downstream shear layer, detached shock, and recirculation region, it is likely that the Mach number and turning angle predicted are not accurate. The degree of inaccuracy, however, is in question. 32

40 6.0 Conclusions and Recommendations The presented work has shown that Prandtl-Meyer expansion can be used as a foundation to predict bleed rate for a single bleed hole oriented normal to supersonic freestream. A CFD study proved useful in exploring other flowfield phenomena that can be used in conjunction with Prandtl-Meyer expansion theory to improve model accuracy. Of these phenomena, the shear layer and barrier shock were the best defined and their geometric placement within the bleed hole were the basis for the bleed rate model. Deviations from experimental data remain constant for all freestream Mach numbers with the exception of Since this outlier occurs at the upper end of Mach numbers evaluated, the model should be applied to higher Mach numbers to see if the trend continues. The distribution of standoff shock distance (Figure 30) and flow turning angle (Figure 31) should also be compared against standoff distances predicted by CFD studies of resolved bleed holes such those performed by Slater [18] and Hamed et al. [9]. It is recommended that the corner expansion process and subsequent shear layer be studied under greater scrutiny. In light of Smith s work [29], it is possible that purely inviscid Prandtl-Meyer expansion theory does not capture the complexity of the corner expansion process. A process should also be found to determine the standoff distance of the barrier shock from first principles. The iterative method described in the development of the author s model relies heavily on simplifications that do not capture the complete behavior of the flowfield. Finally, a method should be devised to account for differences at spanwise locations of the bleed hole. Since the analysis presented in this paper was constrained to the symmetry plane, spanwise components of velocity are 33

41 neglected. Moving away from the symmetry plane, a different flow field is encountered. It is suggested that the new method either integrates analysis at multiple spanwise locations or accounts for spanwise velocity components at the symmetry plane. 34

42 References [1] Shapiro, A., "The Dynamics and Thermodynamics of Compressible Flow," Vol. 1, The Ronald Press Company, New York, 1953, [2] Kerrebrock, J.L., "Aircraft Engines and Gas Turbines," The MIT Press, Cambridge, MA, 2001, [3] Mattingly, J.D., Heiser, W.H., and Daley, D.H., "Aircraft Engine Design," AIAA Educational Series, AIAA, New York, NY, 1987, [4] Rich, B.R., and Janos, L., "Skunk Works: A Personal Memoir of My Years of Lockheed," Back Bay Books, 1996, [5] Roshko, A., and Liepmann, H.W., "Elements of Gas Dynamics," Dover Edition, Mineola, NY, 2001, [6] Slater, J.W., and Saunders, J.D., "Modeling of Fixed-Exit Porous Bleed Systems for Supersonic Inlets," Journal of Propulsion and Power, Vol. 26, No. 2, 2010, [7] Willis, B.P., D.O., D., and Hingst, W.R., "Flow Coefficient Behavior for Boundary Layer Bleed Holes and Slots," 33rd Aerospace Sciences Meeting and Exhibit, AIAA, Reno, MV, 1995, [8] Rimlinger, M.J., Shih, T.I., and Chyu, W.J., "Three-Dimensional Shock- Wave/Boundary-Layer Interactions with Bleed Through a Circular Hole," 2th AIAA, SAE, ASME, and ASEE Joint Propulsion Conference and Exhibit, Nashville, TN, 1992, [9] Hamed, A., Manavasi, S., Shin, D., "Effect of Reynolds Number on Supersonic Flow Bleed," 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, AIAA, Orlando, FL, 2010, [10] Hamed, A., Li, Z., Manavasi, S., "Flow Characteristics Through Porous Bleed in Supersonic Turbulent Boundary Layers," 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition, AIAA, Orlando, FL, 2009, [11] Shih, T.I., Benson, T.J., Willis, B.P., "Structure of Shock-Wave/Boundary-Layer Interactions with Bleed through Rows of Circular Holes," 35th Aerospace Sciences Meeting & Exhibit, AIAA, Reno, NV, 1997, [12] Harloff, G.J., and Smith, G.E., "On Supersonic-Inlet Boundary-Layer Bleed Flow," 33rd Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, 1995, 35

43 [13] Bragg, S.L., "Effect of Compressibility on the Discharge Coefficient of Orifices and Convergent Nozzles," Journal of Mechanical Engineering Science, Vol. 2, No. 1, 1960, [14] Harloff, G.J., and Smith, G.E., "Numerical Simulation of Supersonic Flow Using a New Analytical Bleed Boundary Condition," 31st Joint Propulsion Conference and Exhibit, AIAA, San Diego, CA, 1995, [15] Willis, B., Davis, D., and Hingst, W., "Flowfield Measurements in a Normal-Hole- Bled Oblique Shock Wave and Turbulent Boundary Layer Interaction," 31st Joint Propulsion Conference and Exhibit, AIAA, San Diego, 1995, [16] Slater, J.W., "Verification Assessment of Flow Boundary Conditions for CFD Analysis of Supersonic Inlet Flows," 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, AIAA, Salt Lake City, 2001, [17] Akatsuka, J., Watanabe, Y., Murakami, A., "Porous Bleed Model for Boundary Condition of CFD Analysis," 3rd AIAA Flow Control Conference, AIAA, San Francisco, CA, 2006, [18] Slater, J., "Improvements in Modeling 90 Bleed Holes for Supersonic Inlets," 47th Aerospace Sciences Meeting and Exhibit, AIAA, Orlando, FL, 2009, [19] Hamed, A., and Li, Z., "Simulation of Bleed-Hole Rows for Supersonic Turbulent Boundary Layer Control," 46th AIAA Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, 2008, [20] Hamed, A., Yeuan, J.J., and Shih, S.H., "Shock-Wave Boundary Layer Interactions with Bleed, Part 1: Effect of Slot Angle," Journal of Propulsion and Power, Vol. 11, No. 6, 1995, [21] Hamed, A., Yeuan, J.J., and Shih, S.H., "Shock-Wave Boundary Layer Interactions with Bleed, Part 2: Effect of Slot Location," Journal of Propulsion and Power, Vol. 11, No. 6, 1995, [22] Shih, T.I., "Control of Shock-Wave/Boundary-Layer Interactions by Bleed," International Journal of Fluid Machinery and Systems, Vol. 1, No. 1, 2008, [23] Flores, A.J., Shih, T.I., Davis, D.O., "Bleed of Supersonic Boundary-Layer Flow Through Rows of Normal and Inclined Holes," 35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, AIAA, Los Angeles, CA, 1999, [24] Lin, Y.-., Stephens, M.A., Shih, T.I., "Effects of Plenum Size on Bleeding a Supersonic Boundary Layer," 35th Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, 1997, 36

44 [25] Hamed, A., and Lehnig, T., "Investigation of Oblique Shock/Boundary-Layer Bleed Interaction," Journal of Propulsion and Power, Vol. 8, No. 2, 1992, [26] Hamed, A., Shih, S.H., and Yeuan, J.J., "An Investigation of Shock/Turbulent Boundary Layer/Bleed Interaction," 28th Joint Propulsion Conference and Exhibit, AIAA, Nashville, TN, 1992, [27] Shih, T.I., Rimlinger, M.J., and Chyu, W.J., "Three-Dimensional Shock- Wave/Boundary-Layer Interactions with Bleed," AIAA Journal, Vol. 31, No. 10, 1993, pp , [28] Olsson, G.R., and Messiter, A.F., "Acceleration of Hypersonic Boundary Layer Approaching a Corner," The University of Michigan, Ann Arbor, MI, [29] Smith, H.E., "The Flow Field and Heat Transfer Downstream of a Rearward Facing Step in Supersonic Flow," USAF Office of Aerospace Research, ARL , Dayton, OH, 1968, [30] Anderson, J.D., "Modern Compressible Flow With Historical Perspective," McGraw-Hill, Boston, MA, 1990, [31] Aradag, S., and Knighty, D.D., "Simulation of Supersonic Flow over a Cavity," 43rd AIAA Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, 2005, [32] Moeckel, W.E., "Approximate method for predicting form and location of detached shock waves ahead of plane or axially symmetric bodies," Glenn Research Center, NACA-TN-1921, 1949, [33] Panda, J., and Seasholtz, R.G., "Velocity and Temperature Measurement in Supersonic Free Jets Using Spectrally Resolved Rayleigh Scattering," 37th Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, 1999, [34] Flightglobal, "Boeing 747 Pylon and GE CF6 Cutaway," Pylon-and-Ge-cf6-Cutaway-5588.Aspx, Accessed 2010, Updated 2007, [35] Flightglobal, "BAe Concorde Modified Intake Cutaway," Concorde-Modified-Intake-Cutaway-8156.Aspx, Accessed 2010, Updated 2008, [36] Hunecke, K., "Jet Engines: Fundamentals of Theory, Design and Operation," Motorbooks International Publishers and Wholesalers, Osceola, WI,

45 Streamtube Diameter per Unit Flow [in/lbm/sec] Figures 100 Sea Level ft ft Mach Number Figure 1. Diameter variation of a streamtube with increasing Mach number. 38

46 M flight < M design M flight M design INLET Figure 2. Subsonic inlet shown in typical underwing installation with capture streamlines at flight Mach numbers above, near, and below design Mach number. 39

47 INLET Figure 3. Cutaway of a General Electric CF6-50 turbofan mounted under a Boeing 747 pylon. Representative of most underwing engine installations for subsonic aircraft. Reproduced from [34]. 40

48 Bow Shock Spillage M flight >1>M design INLET Figure 4. Bow shock formation upstream of a subsonic inlet operating at Mach number much greater than design intent. 41

49 M flight = M design >1 Compression Shock System Terminal Normal Shock Figure 5. Supersonic two-dimensional ramp inlet with capture streamlines and compression shock structure at design Mach number. 42

50 Figure 6. Cutaway of an inlet installation on a BAe Concorde. Representative of underwing engine installations for supersonic aircraft featuring two-dimensional ramp inlets with variable geometry such as the Grumman F-14 and McDonnell Douglas F-15. Reproduced from [35]. 43

51 Figure 7. Relationship of corner expansion and barrier shock at a boundary layer bleed hole. Reproduced from [11]. 44

52 Figure 8. Schematic of computational domain [10]. Figure 9. Computational grid in bleed region [10]. 45

53 Q sonic p Spl /p T Figure 10. Sonic flow coefficient predictions from CFD study presented in Section

54 p Spl p T = p Spl p T = p Spl p T = p Spl p T = p Spl p T = p Spl p T = Figure 11. Mach number contours for CFD study presented in Section

55 p Spl p T = p Spl p T = p Spl p T = p Spl p T = p Spl p T = p Spl p T = Figure 12. Total pressure contours for CFD study presented in Section

56 p Spl p T = p Spl p T = p Spl p T = p Spl p T = p Spl p T = p Spl p T = Figure 13. Static pressure contours for CFD study presented in Section

57 p Spl p T = p Spl p T = p Spl p T = p Spl p T = p Spl p T = p Spl p T = Figure 14. Streamlines of bleed flow for CFD study presented in Section

58 Boundary Layer Edge Expansion Mach Lines Sonic Line Viscous Sublayer Figure 15. Decrease in boundary layer thickness in a supersonic flow approaching a corner. Reproduced from [28]. 51

59 in Figure 16. Contours of constant streamwise velocity for plenum pressure ratio of with bleed hole located inches from inflow boundary in Figure 17. Contours of constant streamwise velocity for plenum pressure ratio of with bleed hole located inches from inflow boundary. 52

60 Figure 18. Flowfield features for a supersonic flow separating off an aft-facing step. Reproduced from [29]. 53

61 Figure 19. Instantaneous streamwise pressure contours for supersonic flow over a cavity. Reproduced from [31]. 54

62 Detached Shock M > 1 Sonic Line M < 1 M > 1 M < 1 M > 1 Figure 20. Detached shock and flow regimes for a wedge of large apex angle in supersonic flow. 55

63 Upper segment of barrier shock Lower segment of barrier shock Shear Layer Approximate location of sonic line Figure 21. Contours of Mach number showing geometric relationship among key flowfield features. Plenum pressure ratio of shown. 56

64 VIEW Figure 22. Streamlines of bleed flow for CFD study presented in Section 4.0. View is through bleed hole wall, looking toward symmetry plane. 57

65 Weakening barrier shock due to boundary layer influence Figure 23. Static pressure contours for CFD study presented in Section 4.0. Note deflection and weakening of barrier shock due to interaction with boundary layer growth on bleed hole wall. Plenum pressure ratio of shown. View is top looking down into bleed hole. 58

66 Figure 24. Profiles for a jet ensuing from a near-sonic orifice. Reproduced from [33]. 59

67 p T, T T, M Prandtl-Meyer Expansion Fan Edge of streamtube captured by bleed hole p S Upper segment of barrier shock Shear Layer Lower segment of barrier shock (Control surface for Q sonic ) p Spl Sonic Line Bleed Hole Wall Recirculation Region Jet Plume Figure 25. Flowfield described by bleed rate model. 60

68 Bleed Hole Inlet 2 Shear Layer L sl Lower segment of barrier shock L * Sonic Line L sl cos 2 Plenum Inlet Figure 26. Dimensions for definition of flow areas. 61

69 CL Semi-minor axis of elliptical segment Semi-major axis of elliptical segment a 90-2 L * Lower segment of barrier shock Sonic Line Sonic Flow Area Figure 27. Dimensions for definition of flow areas. 62

70 Figure 28. Isosurface of sonic flow. Three-dimensional nature of shear layer, barrier shock, and sonic line due to boundary layer interaction are clearly depicted. 63