NUMERICAL ANALYSIS OF ETHYLENE INJECTION IN

Transcription

1 NUMERICAL ANALYSIS OF ETHYLENE INJECTION IN THE INLET OF A MACH SIX SCRAMJET Jonathan Philip West A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science, Graduate Department of Aerospace Science and Engineering, University of Toronto. Copyright by Jonathan Philip West 11

2 NUMERICAL ANALYSIS OF ETHYLENE INJECTION IN THE INLET OF A MACH SIX SCRAMJET by Jonathan Philip West Masters of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto, 11 j.west@utoronto.ca Abstract A scramjet inlet was designed for use on a small scale, Mach six, ethylene-fuelled vehicle. The inlet used strut-based cantilevered fuel injectors and a well-defined mixing duct to mix fuel prior to the combustor. Designed using theoretical and numerical analyses, the resulting inlet configuration consisted of a single body shock inlet with vertical fuel injector struts and four cantilevered injectors per strut side. This inlet was 8 cm long and 4 cm high. Numerical analysis of the vehicle was conducted with computational fluid dynamics by solving the Favre- Averaged Navier-Stokes equations; turbulence was simulated using the Wilcox k-ω model. Multispecies simulations in two and three dimensions were used to evaluate the design. Analysis of the simulated flow features, thrust potential and mixing efficiency demonstrated favourable vehicle performance. In particular, the inlet allowed for complete combustion when lean equivalence ratios of less than.7 were used. ii

3 Acknowledgments I would like to sincerely thank all my professors and colleagues at the University of Toronto Institute for Aerospace Studies who enriched my time at the Institute every day. Particularly, I would like to thank my supervising professor J.P. Sislian for his unwavering guidance and assistance completing this thesis. His passion for air breathing propulsion and scientific research is second to none. I would also like to thank my colleagues Jonathan, Julian, Maciej, Yen and Yu for their continuous friendship. In particular I wish to thank Jonathan, Yen and Yu for sharing their expertise in hypersonic vehicles and numerical methods whenever it was required. Finally, I would like to thank my family and Leanna for their encouragement, patience and support. iii

4 Table of Contents Abstract... ii Acknowledgments... iii List of Tables and Figures... vi Nomenclature... viii 1. Introduction Scramjet Background Literature Review Scramjet Propulsion Systems Ethylene Combustion Computational Fluid Dynamics Objective Flight Conditions Vehicle Configuration Method of Analysis Solution Methodology Theoretical Solutions Combustion Equations Theoretical Performance Calculator Program Function Program Method Warp Code Theory Convective Equations Viscous Equations Compressibility Effects Equations Turbulence Model Time Marching Method Convergence Criteria Post Simulation Integration Mixing Efficiency Thrust Potential Inlet and Compression Configuration Two-Shock Inlet Theoretical Performance Simulated Performance One-Shock Inlet and Strut Theoretical Performance Simulated Performance Theoretical After Strut Conditions... 6 iv

5 5. Cantilevered Injector Configuration Design Process Chosen Configuration D Simulation Methodology D Inlet Simulations Simulation for Data Interpolation Thrust Potential Accurate Simulation D Strut Simulations Simulated 3D Inlet Injection Strut Simulation D Mixing Duct Simulations Fuel Injection Simulation Mixing Duct Simulation Simulation Results D Inlet Simulation Body Wall Boundary Layer D Strut Simulation Re-Circulation Zone Boundary Layer Bleed-off Strut Simulation Results D Mixing Duct Results Injection Cases Considered Temperature Field Pressure Field Mach Number Field Equivalence Ratio Field Mixing Efficiency Results Thrust Potential Results Conclusions Vehicle Configuration and Overall Performance Suggestions for Future Research Bibliography Appendix A.1 Additional Details on Theoretical Solutions A.1.1 Gas Dynamics Equations A.1. Combustion Equations A. Inlet Theoretical Performance Program Input and Output... 8 A.3 CFL Auto Adjustment A.4 Mixing Duct Contours for φ =.3,.7 and A.4.1 Mixing Duct Contours for φ = A.4. Mixing Duct Contours for φ = A.4.3 Mixing Duct Contours for φ = v

6 List of Tables and Figures. Objective... 5 Figure.1: Vehicle Configuration Solution Methodology... 7 Figure 3.1: Theoretical Inlet Performance Program Modelled Zones Inlet and Compression Configuration... 1 Figure 4.1: -Shock Inlet Configuration... 1 Table 4.1: -Shock Inlet Geometry Configurations... Figure 4.: Temperature Contours of 1K -Shock Inlet Case... Figure 4.3: Pressure Contours of 1K -Shock Inlet Case... Figure 4.4: 1 Shock Inlet Side View... 3 Figure 4.5: Injection Strut Top View... 3 Table 4.: 1 Shock Inlet and Strut Theoretical Performance... 4 Figure 4.6: Temperature Field of D Viscous 1 Shock Inlet... 5 Figure 4.7: D Viscous Strut Simulations... 6 Table 4.3: Final Strut Geometry... 6 Table 4.4: Theoretical After Strut Conditions Cantilevered Injector Configuration... 8 Figure 5.1: Injector Configuration... 9 Table 5.1: Injector Configuration Variables D Simulation Methodology... 3 Table 6.1: Simulated Domains within Vehicle... 3 Figure 6.1: Vehicle Simulation Cartesian Coordinate System Figure 6. Interpolated D Simulation Showing Copied Data Region Figure 6.3: Initial Air Capture Area Including Streamtrace Intersecting Cowl Figure 6.4: Thrust Potential Accurate Air Capture Area Figure 6.5: 3D Inlet and Strut Simulation Domain Figure 6.6: 3D Injection Strut Only Domain Figure 6.7: Fuel Injection Simulation Domain Figure 6.8: Mixing Duct Simulation Domain Simulation Results Figure 7.1: Mach Number Contours of Vehicle Body Inlet Figure 7.: Pressure Contours of Vehicle Body Inlet Figure 7.3: Temperature Contours of Vehicle Body Inlet Table 7.1: Properties within Boundary Layer Figure 7.4: Air Velocity Based Boundary Layer Profile Figure 7.5: Temperature Based Boundary Layer Profile Figure 7.6: Strut Re-circulation Zone without Boundary Layer Bleed-off... 5 Figure 7.7: Examined X Planes within 3D Strut Domain Figure 7.8: Temperature Contours of 3D Strut... 5 Figure 7.9: Pressure Contours of 3D Strut Figure 7.1 Mach Number Contours of 3D Strut vi

7 Table 7.: Fuel Injection Cases Figure 7.11: Examined X Planes within 3D Mixing Duct Figure 7.1: Temperature Contours of X Planes in φ=.5 3D Mixing Duct Simulation Figure 7.13: Pressure Contours of X Planes in φ=.5 3D Mixing Duct Simulation... 6 Figure 7.14: Mach Number Contours of X Planes in φ=.5 3D Mixing Duct Simulation... 6 Figure 7.15: Equivalence Ratio Contours of X Planes in φ=.5 3D Mixing Duct Simulation. 64 Figure 7.16: Mixing Efficiency Results from 3D Mixing Duct Simulations Figure 7.17: Thrust Potential of the Full Vehicle Inlet Appendix Figure A.1: Inlet Theoretical Performance Program Input Figure A.: Inlet Theoretical Performance Program Output... 8 Figure A.3: Temperature Contours of X Planes in φ=.3 3D Mixing Duct Simulation Figure A.4: Pressure Contours of X Planes in φ=.3 3D Mixing Duct Simulation Figure A.5: Mach Number Contours of X Planes in φ=.3 3D Mixing Duct Simulation Figure A.6: Equivalence Ratio Contours of X Planes in φ=.3 3D Mixing Duct Simulation.. 87 Figure A.7: Temperature Contours of X Planes in φ=.7 3D Mixing Duct Simulation Figure A.8: Pressure Contours of X Planes in φ=.7 3D Mixing Duct Simulation Figure A.9: Mach Number Contours of X Planes in φ=.7 3D Mixing Duct Simulation... 9 Figure A.1: Equivalence Ratio Contours of X Planes in φ=.7 3D Mixing Duct Simulation 91 Figure A.11: Temperature Contours of X Planes in φ=1. 3D Mixing Duct Simulation... 9 Figure A.1: Pressure Contours of X Planes in φ=1. 3D Mixing Duct Simulation Figure A.13: Mach Number Contours of X Planes in φ=1. 3D Mixing Duct Simulation Figure A.14: Equivalence Ratio Contours of X Planes in φ=1. 3D Mixing Duct Simulation 95 vii

8 Nomenclature Note: Symbols listed in order of appearance * subscript k indicates species ** subscript i indicates dimension ϕ = equivalence ratio α = number of carbon atoms β = number of hydrogen atoms, shock angle M k = molar mass * m k = mass flow rate * s c k = stoichiometric concentration * Q = conservative variables J = metric Jacobian ρ = density c k = concentration * v i = velocity ** E = total specific energy k = turbulent kinetic energy ω = dissipation rate per unit of k T = temperature R = residual τ = pseudo time F i = convective flux terms ** K ij = diffusion matrix G = diffusion terms S = compressibility effects terms X i = curvilinear coordinate ** V i = contravariant velocity ** viii p = effective pressure e = specific internal energy q = total velocity magnitude (speed) p = pressure α β i, j r, s i, j = 1 nd J m= 1 = α δ i, j X r, s i, m X j, m 1 + X J δ ij = Kronecker delta X i, j X = x j i i, s X j, r D k = mass diffusion term for µ = effective viscosity 3 1 X J K ij i, r µ k = turbulent diffusion coefficient for k X j, s µ ω = turbulent diffusion coefficient for ω κ = thermal conductivity term for h k = enthalpy * µ T = turbulent viscosity µ = viscosity D k = mass diffusion * κ = thermal conductivity K ij c p = specific heat at constant pressure Pr T = turbulent Prandtl number Sc T = turbulent Schmidt number

9 P k = turbulent kinetic energy production M T = turbulent Mach number ~ k = modified turbulent kinetic energy σ k = turbulent closure coefficient for k σ ω = turbulent closure coefficient for ω k div = user specified minimum turbulent kinetic energy value where ^ i k div ωµ ρ A = Jacobian of F i with respect to Q ** B = Jacobian of G with respect to Q ^ C = Jacobian of S with respect to Q I = identity Matrix CFL = Courant-Friedrichs-Lewy condition a = speed of sound σ = user specified CFL variation parameter ξ = convergence condition residual R c k = concentration of reacting fuel * η m = mixing efficiency A exit = predicted end nozzle area F pot = thrust potential δ = wedge angle M = Mach number ix

10 1 1. Introduction 1.1 Scramjet Background Like any air-breathing engine, a scramjet operates by compressing incoming air, increasing the pressure and temperature of the air through combustion, and then expanding the product gasses in a manner that provides the energy needed to accomplish the task the engine was designed for. In the case of propulsion applications, air breathing engines are designed to eject the expanded gasses with greater momentum than the incoming air providing the vehicle they are attached to with net thrust [1]. A scramjet operates at hypersonic speeds (Mach number greater than six). At these speeds the dynamic pressure is used to compress air entering the engine through inlet shocks. All the stages of an air breathing engine including combustion and expansion occur at supersonic speeds within a scramjet. A scramjet is usually also specified as using diffusive fuel burning within a well defined combustor [1]. A type of scramjet known as a shcramjet [7,8,4] is distinguished by its use of shock-induced combustion to detonate fuel. Once created, practical scramjets would be a breakthrough technology in aerospace propulsion. Unlike a conventional rocket which must include all of its oxidizer on-board, an air breathing scramjet would be capable of using oxygen from the atmosphere and would only need to bring fuel. This advantage gives the scramjet the potential to become a practical re-usable vehicle for extraordinarily quick travel between distant locations on earth. The highest predicted velocities that can be reached by a scramjet approach orbital velocity, because of this another potential application of scramjet propulsion is space travel as a stage for a re-usable orbital transport vehicle [1]. 1. Literature Review There are many aspects that must be considered for scramjet design. Three main topics which were researched in detail in this thesis included scramjet propulsion systems, ethylene combustion, and computational fluid dynamics (CFD).

11 1..1 Scramjet Propulsion Systems Scramjet technology has been researched worldwide for over 4 years and although initial research suggested scramjets could be developed using the same design methodologies as subsonic combustors, the complexity of the supersonic shock interactions and flow fields made the task significantly more complex []. Many notable scramjet research efforts have taken place over the last decade. In 1998 CIAM and NASA [3] conducted a Mach 6.5 flight test program in which a dual mode axial scramjet engine was designed and flown tethered to a rocket. In 4 as part of its Hyper X program, NASA [4] successfully flew X-43A test vehicles twice at Mach seven and Mach 1 demonstrating scramjet-powered hypersonic flight. Most aspects of scramjet performance are still under investigation, including exterior aerodynamics such as the waverider design [5]. However, the biggest problems facing scramjet propulsion are fuel injection, fuel mixing, and fuel ignition [,6,7,8]. These problems result from the short amount of time (milliseconds) in which the fuel can combust within the scramjet [9,1] and complex fuel-shock and fuel-boundary layer interactions [1,11]. As a consequence of the problems surrounding fuel injection, mixing, and ignition, scramjet performance is highly dependent on injector design and placement. Various injector designs are widely available in literature including variations on strut based injectors [11,1], cantilevered ramp injectors [6,7,8,13], and wall injectors [9,11]. Permutations of common fuel injectors exist with fuel injected perpendicular [11,1], parallel [6,7,8,13,1], and at angles into the flow [9] with many designs incorporating multiple types of injectors [11]. Placement of injectors also plays a crucial role with some designs injecting fuel in the inlet [6,7,8,13] and some injecting fuel just before or in the combustor [9,11,1]. Typically, strut-based injectors are only studied at the entrance to the combustor and rarely in the inlet section [1]. Fuel injection velocity [14] and thermodynamic properties [15] are also important design constraints. Of the many challenges faced by scramjet propulsion systems perhaps the most critical is proper fuel-air mixing [16]. This is made particularly difficult due to a need to balance thrust recovery, mixing performance, and vehicle cooling. The most thrust is recovered from fuel injection if the injection is parallel to the flow [16,17]. However, a high convective Mach number is insufficient as the only mixing method due to the reduction in growth of the compressible turbulent mixing

12 3 layer [17]. To increase mixing many configurations use staggered cantilevered rectangular injectors which improve mixing efficiency for parallel injections due to the generation of counter rotating axial vortices that help drive mixing [16]. A trailing shock after the injectors typically also improves mixing efficiency assuming it does not prematurely detonate the fuel [16]. To prevent premature detonation and avoid vehicle damage it is also particularly important to avoid fuel-boundary layer interactions as this can cause fuel ignition and overheat adjacent structures [18]. 1.. Ethylene Combustion Ethylene (C H 4 ) is a highly reactive hydrocarbon [19] that has been considered as a possible fuel for scramjet engines in many studies [,1,]. Although it is not commonly used as a fuel because it is gaseous at room temperature [19,3] it is a common and important intermediate product in the combustion of other hydrocarbons and an additive in many fuel mixtures [19]. One of its most appealing aspects as a possible scramjet fuel is its short ignition time when compared to other hydrocarbon fuels []. Despite many chemical-kinetic modelling schemes [19,4] of varying accuracy and complexity available in the open literature, investigators studying ethylene often find poor agreement between different studies and experimental results. It is often concluded that additional work modelling ethylene is required [19,,4]. Published values for ethylene autoignition temperatures can vary between 698K [3] at the low end to 1133K [5] at the high end, depending on conditions. Although generally considered fast, ethylene autoignition times vary significantly with conditions [,4] Computational Fluid Dynamics Members of the UTIAS Hypersonic Vehicle Propulsion Systems Group (HSVPSG) use an inhouse CFD code known as the Windows Allocatable Resolver for Propulsion (WARP) [6] code for the majority of their numerical research [7,8,13]. The WARP code uses a type of active domain method known as the marching window method [6]. The WARP code solves the Favre-averaged Navier-Stokes (FANS) equations while utilizing the Wilcox k-ω turbulence model [7,8,13,14,6,3].

13 4 In order to produce meaningful results the HSVPSG produces three-dimensional viscous steadystate simulations of the entire vehicle under investigation [7,8,13]. Particular attention is paid to fuel injection simulation. Simulations without chemistry are used to study mixing efficiency [14] and expanded to include chemistry to study fuel combustion [8]. Boundary layer interactions with shocks and fuel are also particularly important. Nodes adjacent to a wall should be between 1-3 µm wide [8] to correctly model turbulent boundary layers in flows characteristic of scramjet engines.

14 5. Objective This thesis project aims to research the mixing performance and feasibility of a Mach six ethylene-fuelled scramjet engine inlet utilizing strut-based cantilevered ramp fuel injectors. Cantilevered ramp fuel injectors allow for various advantages in fuel mixing due to the generation of counter rotating axial vortices which can effectively mix fuel injected more or less parallel to the flow. However, cantilevered ramp injectors are usually designed to be integrated into an engine wall since they must stick out of a receding plane. Strut-based injectors allow for more even injection of fuel across the whole flow field. A strut-based cantilevered injector configuration may be very advantageous for fuel mixing but has not been well researched previously since the flow is inherently three-dimensional and the geometry is difficult to simulate using a structured mesh. By simulating this type of inlet and evaluating its mixing performance and drop in thrust potential a significant contribution can be made to the area of scramjet design, particularly for small scale lower Mach number hydrocarbon-fuelled vehicles..1 Flight Conditions Flight conditions are similar to those used previously by Wang et al. [8] with a dynamic pressure of 14 Psf. At Mach six this results in an approximate cruise altitude of 4.8 km and a ground velocity of 1791 m/s or 6448 km/h. The atmospheric temperature and pressure at this altitude is 647 Pa and 1 K respectively, with stagnation conditions of 4186 kpa and 1811 K.. Vehicle Configuration As stated, the fuel is ethylene injected just within the cowl of the engine using strut-based cantilevered injectors before a mixing region separate from the combustor. The intention is to produce internally an ethylene air mixture ideal for lean diffusive burning between equivalence ratios of.3 and one. The injection struts span the height of the air passageway from the vehicle body to the internal surface of the cowl. Square cantilevered injectors extending outwards from the strut shoulder inject the ethylene parallel to the airflow. The struts also serve as a secondary stage of air compression due to the shape of the struts and the blockage they cause in the passageway. After a mixing period the internal air flow is turned parallel to the vehicle where combustion takes place before exhaust gasses are expanded out of the nozzle. The mixing region after the strut injector for a hydrocarbon fuel is a unique feature being evaluated as part of this thesis. An approximate sketch of the vehicle configuration is shown in figure.1.

15 6 Figure.1: Vehicle Configuration Although no numerical analysis is performed after the mixing duct it is assumed that the air fuel mixture will be turned parallel to the vehicle and from the mixing duct into the combustor through two equal strength shocks..3 Method of Analysis Computational Fluid Dynamics (CFD) using the UTIAS WARP code [6] was conducted in order to design the scramjet and verify its performance. Nodes adjacent to walls were limited to 3 μm wide. Three-dimensional viscous simulations without chemistry were the primary tool for conducting research. The WARP code was used to verify and test configurations in order to develop a detailed design. The inlet performance was quantitatively evaluated based on overall thrust potential losses and mixing efficiency. Property fields such as temperature, pressure, Mach number, and equivalence ratio were also examined to evaluate qualitative performance.

16 7 3. Solution Methodology The results presented in this thesis were largely obtained through the use of numerical experiments conducted with computational fluid dynamics. These simulations were conducted using parameters guided by theoretical gas dynamics equations and theoretical combustion equations. 3.1 Theoretical Solutions In general, gas dynamics equations were used to estimate the shock structures produced by the vehicle geometry and the conditions behind them. Combustion equations were not used in any estimate of possible thrust since the mixing efficiency and drag of the inlet was of interest in this project. However, combustion equations were important for determining the required fuel for simulating injection and the stoichiometric concentrations of fuel. A detailed description of the specific gas dynamics equations used can be found in the appendix section A.1.1 Gas Dynamics Equations Combustion Equations To determine the stoichiometric equation for ethylene combustion the process displayed in equation 3.1 [3] was used. Given the equivalence ratio (φ) for lean combustion the actual combustion equation was determined using equation 3. [3]. The number of carbon and hydrogen atoms in the molecule are indicated by α and β respectively. β β β Cα H β + α + ( O N ) αco + H O α + N (3.1) 4 4 C α H αco β 1 β + α + ϕ 4 β + H ( O N ) 1 β O α + N ϕ 4 1 β + 1 α + O ϕ 4 (3.) The stoichiometric air/fuel ratio was determined using equation 3.3 [3] and the lean air/fuel ratio was determined using equation 3.4 [3]. Molar mass is indicated by M and mass flow rate by m.

17 8 H C N O S M M M M F A β α β α β α = (3.3) ϕ S Fuel Air F A m m F A = = (3.4) The stoichiometric concentrations for ethylene and oxygen were determined using equations 3.5 and 3.6 respectively N O H C O S O M M M M M c = β α β α β α β α (3.5) N O H C H C S Fuel M M M M M M c = β α β α β α β α (3.6) To determine fluid properties for setting boundary conditions for fuel injection the NASA [15] thermodynamic polynomials for ethylene were used to calculate the specific heat capacity at constant pressure. In general, injection temperature and injection mass flow rate were treated as design constraints. Injection pressure was assumed to be equal to that of the surrounding air which was determined using the theoretical performance program as described in section 3. Theoretical Performance Calculator. Additional details about the NASA polynomials and the combustion equations used can be found in appendix section A.1. Combustion Equations. 3. Theoretical Performance Calculator Later in the thesis project it became evident that a program that would predict conditions and geometry resulting from different design choices for a vehicle inlet with strut-based cantilevered injectors using theoretical equations would be beneficial.

18 Program Function The calculator assumes that air after the strut will be turned parallel to vehicle into the combustor using two equal pressure ratio shocks. The inlet design consists of seven different states caused by various shocks and expansions; 1. Atmospheric conditions before vehicle impact. Conditions resulting from inlet shock 3. Conditions resulting from the first strut shock 4. Conditions resulting from the second strut shock 5. Effect of expansion behind strut 6. Conditions resulting from first combustor entrance shock 7. Conditions resulting from second combustor entrance shock These various zones are displayed in figure 3.1. Figure 3.1: Theoretical Inlet Performance Program Modelled Zones A theoretical performance calculation program was written in the D programming language [33] in which the inlet designer can specify the following variables;

19 1 - Vehicle Mach number - Atmospheric temperature and pressure - Desired air mass flow through intake - Vehicle body wedge angle - Spacing between struts and blockage ratio - Slope of backward facing strut face - Number and area of fuel injectors The program will then output the air conditions and pressure ratios for each state within the vehicle inlet as well as the required vehicle geometry to attain the following; - Cowl position and strut height for required mass flow intake - Strut geometry for correct blockage ratio, strut spacing, backward strut face angle, and the condition that the second strut shock intersects the shoulder point - Wedge angles required for two equal strength shocks to turn the flow into the combustor An example of the program input and output can be found in appendix section A. Inlet Theoretical Performance Program Input and Output. 3.. Program Method Gas dynamics equations were used in combination with the bisection method to calculate shock angles and determine air conditions after oblique shocks. Conditions after the strut were assumed to result from an isentropic supersonic expansion where the first area was the total domain multiplied by one minus the blockage ratio of the strut and the second area was the total domain minus the total area of the injectors. The required geometry for the second strut shock to intersect the shoulder point and the two combustor entrance shocks to be equal strength were solved iteratively using the bisection method.

20 Warp Code Theory As previously mentioned the CFD solver used for the completion of this thesis was the WARP [6] code. For the work of this thesis it was used for no-chemistry multi-species simulations in two and three dimensions. This code has been written about in great detail and a thorough description is available in the associated reference so what follows is merely a brief summary as it relates to this project. Descriptions are also available in thesis documents written by previous students within the HSVPSG group [4,43]. The WARP code solves the Favre averaged Navier-Stokes equations in curvilinear coordinates for the chemical species specified. For the simulations in this thesis only three chemical species were used; O, N, and C H 4. The conservative variables solved are displayed in equation 3.7. Q = ρco ρcn ρcc ρv1 1 J ρvd ρe ρk ρω H 4 (3.7) The thermodynamic properties for these species including enthalpy, internal energy, and specific heat were determined by the code using polynomial curve fits from McBride et al. [34], which should allow the values to be valid between K<T<6K. Turbulence is modeled using the Wilcox k-ω turbulence model with the Wilcox dilatational dissipation correction [35,36]. This solution method is shown in generalized coordinates as equation 3.8 where the residual R is reduced. Q = R τ (3.8) The residual is defined in equation 3.9.

21 1 S X G K X X F R d i d j j j i i i i = = = 1 1 (3.9) Convective Equations The convective fluxes for the conservation variables are demonstrated by equation 3.1. The partial derivatives of the fluxes used in calculating the residual are evaluated using a second order conservative Roe scheme with the symmetric minmod limiter of Yee [37] = ω ρ ρ ρ ρ ρ ρ ρ ρ i i i i d i i i i H C i N i O i i V V k p V V E p X V v p X V v V c V c V c J F * *, 1 *, (3.1) The contravarient velocity, total energy, and effective pressure were determined using equations 3.11, 3.1, and 3.13 respectively. Pressure was determined using the ideal gas law. = = d m m m i i v X V 1, (3.11) 1 q k e E + + = (3.1) k p p ρ 3 * + = (3.13)

22 Viscous Equations To calculate the diffusion contribution to the residual the diffusion terms as shown in equation 3.14 are multiplied by the diffusion matrix shown in equation The diffusion partial derivatives are derived using the second order centered finite difference stencil. = ω k T v v c c c G d H C N O 1 4 (3.14) = = = *, *, *, *, 3 1,, * 3 1,1, * * 4 4, *, *,,, *,1, * 1,, * 1,1, * * 4, *, *,, µ ω α µ α µ α κ α β µ β µ α α α β µ β µ β µ β µ α α α j i k j i k j i j i k d k j i k k k j i k H C H C j i N N j i O O j i d d j i d j i d j i j i H C j i N j i O j i j i v v D h D h D h D D D K (3.15) The parameters for effective viscosity, thermal conductivity, and mass diffusion are calculated using equations 3.16, 3.17, and 3.18 respectively. µ T µ µ + = (3.16) T T c p Pr µ κ κ + = (3.17)

23 14 T T H C H C T T N N T T O O Sc D D Sc D D Sc D D µ µ µ + = + = + = 4 4,, (3.18) Compressibility Effects Equations The vector used in the calculation of compressibility effects is displayed in equation For the WARP [6] code this vector typically includes chemical species production terms however since chemistry was not included in the simulations they are neglected. However, compressibility terms utilized for the turbulence model were included. ( ) ( ) + + = T k T k M f k P k M f k k P J S ~ ρω ω ρ ω ω ρ ρω (3.19) Turbulence Model The WARP code employs the Wilcox k-ω turbulence model with the Wilcox dilatational dissipation correction [36]. This two equation model defines turbulence with equations to determine turbulent kinetic energy (k) and turbulent length scale (ω). The turbulent diffusion coefficients used in the diffusion matrix shown in equation 3.15 are calculated using equations 3. and 3.1 respectively. k T k σ µ µ µ + = (3.)

24 15 µ T µ ω = µ + σ ω (3.1) The turbulent viscosity, closure coefficients for k and ω, and the turbulent Schmidt number are defined in equations 3., 3.3, 3.4, and 3.5 respectively. These values have been previously utilized by the HSVPSG group [18]. µ T.9ρk = (3.) ω σ =.9 (3.3) k σ ω =. (3.4) Sc = 1. (3.5) T To model the effect of the convective Mach number on shear layer growth [38,39] the Wilcox dilatational dissipation model [36] is used. It is incorporated into the compressibility effects vector of equation 3.19 using equation f ( M ) = 3 max, T M T (3.6) 16 The turbulent kinetic energy production term calculated using equation 3.7 in generalized coordinates and the modified turbulent kinetic energy ( k ~ ) calculated in equation 3.8 are also used in the compressibility effects vector of equation The turbulent kinetic energy is substituted with k ~ merely to prevent a divide by zero error in the compressibility vector when simulating the free stream where the real turbulent kinetic energy (k) would presumably be zero. The value k div is user specified and set as shown in equation 3.9.

25 16 P k = d d d d j mn m n ρ kx + i, j Jµ β ij (3.7) i= 1 j= 1 3 X i m= 1 n= 1 X i X j v v v ~ ωµ k = maxk, min k div, (3.8) ρ k =.1 (3.9) div Time Marching Method The conservative variables displayed in equation 3.7 are iteratively advanced to steady state using an implicit Euler pseudo time marching scheme [6] in which the difference in conservative variables at each iteration (n) are calculated as shown in equation 3.3. This is solved implicitly as a system of linear equations as shown in equation 3.31 which allows the residual at the current time step to be used. The Jacobians of convective flux, diffusive flux, and source terms related to compressibility effects are shown in equations 3.3, 3.33, and 3.34 respectively. The linear system of equations is solved using approximate factorization [4,41] as shown in equation 3.35 where δ 1,i is the Kronecker delta. n n+1 Q = τ R (3.3) I ^ n i d d d A + τ τ X i= 1 i i= 1 j= 1 X i K n ij B X n j τ C ^ n Q n = τr n (3.31) ^ n i A F Q n i n (3.3) B n n G = (3.33) n Q

26 17 ^ n C S = Q n n (3.34) d A I + τ X ^ n i τ i= 1 i j= 1 d X i K n ij B X n j δ i i τ C, ^ n Q n = τr n (3.35) The WARP [6] code uses a local psedo-time step for convergence acceleration. This time step is based on the CFL condition which results in the time step being proportional to the time required for the wave speed to cross one node per iteration multiplied by a CFL variable. In multiple dimensions the wave speed may be different in each dimension. To strike a balance between these different possible time steps the local time step for each node is calculated for each dimension and the parameter (σ) is used to vary the local time step between the largest and lowest local time steps calculated. This process is displayed effectively in equation The parameter (σ) is set as shown in equation σ 1 σ d 1 d 1 τ = CFL max = min ^ = ^ (3.36) i 1 i 1 Vi + a X i Vi + a X i σ =.5 (3.37) In order to make the results of a simulation less dependent on starting CFL conditions a set of new conditional programming loops were specified within the marching window loop of the WARP [6] control file. These conditions compare the current iteration s maximum residual with the previous iteration s maximum residual and reduce or increase the CFL condition accordingly. A more detailed description of these loops can be found in appendix section A.3 CFL Auto Adjustment Convergence Criteria To reduce computational effort and increase solution speed the WARP [6] code uses a type of active domain cycle known as the marching window method. The active domain is gradually

27 18 moved upstream until the entire domain has converged to below the user specified convergence condition. The upstream boundary of the marching window active domain is set the same as the upstream boundary of the total domain but the downstream boundary of the marching window will only advance past nodes that satisfy the convergence condition as shown in equation The convergence condition is essentially the maximum between the discretized continuity and energy residuals. continuity energy R R ξ = 4 max, 1 1 J ρ J ρe s (3.38) 3.4 Post Simulation Integration The most useful information extracted from the simulations are the vehicle performance parameters of mixing efficiency and thrust potential. These parameters are integrated over planes perpendicular to the vehicle direction of travel using simulation data for flow engulfed by the scramjet engine. At least one data point per centimetre was obtained over the entire length of the engine where mesh constraints allowed Mixing Efficiency Mixing efficiency is a parameter that provides a quantity to the total fuel/air mixing process. Essentially for a streamwise plane where all the intended air and fuel to take part in combustion is available the mixing efficiency is equivalent to the fraction of a reacting species that is mixed such that its concentration is lower than or equal to the stoichiometric concentration relative to the other species. Since the cases utilized use lean diffusive burning a fuel-based mixing efficiency was used. Once the fuel has a lower concentration throughout the plane analysed then the stoichiometric concentration the mixing efficiency will be at one. In other words either stoichimetric conditions or fuel lean conditions exist everywhere with the implication that all injected fuel can be burned. However, this does not imply that the fuel is homogenously mixed since the vehicle is globally lean. Parts of the domain can have different concentrations of fuel as long as that concentration

28 19 is at or lower than stoichiometric. As a result it is possible to obtain fuel-based mixing efficiencies of one under globally lean conditions since more mixing can still take place. The fuel based mixing efficiency calculation for a given plane is based on equation This calculation uses the mass fraction of reacting fuel which can be determined using equation 3.4 where the superscript S denotes stoichiometric concentration. R cc H 4d m plane η mc H 4 = (3.39) mc H 4 c S c O min c C H 4 cc H 4 (3.4) co R C H 4 =, S The stoichiometric concentrations for ethylene and oxygen were calculated using equations 3.5 and 3.6. The resulting stoichiometric concentrations for ethylene and oxygen are shown in equations 3.41 and 3.4 respectively. S c.6375 (3.41) C H 4 = S c =.1815 O (3.4) 3.4. Thrust Potential The thrust potential is a measure of how much thrust the air flow within the analysed plane could provide to a vehicle if ideally expanded to some exit condition plane. The properties of the theoretical exit plane are iteratively determined by reversibly expanding the flow in the plane of interest to an area equal to the assumed outlet nozzle of the scramjet. This outlet area is assumed to be equal to the total height of the air flow features simulated assuming the mixing duct is 5 cm long, the exact value is displayed in equation For the purposes of this project it is a good measure of how much drag is generated internally by the air flow. The thrust potential calculation process is exemplified in equation 3.44.

29 A =.41793* vehicle width (3.43) exit F pot = F + plane d exit Vi, exit + i= 1 ρ ρ exit pot, initial m d i= 1 V air, initial P i, exit exit d m (3.44) The thrust potential used in analysis is normalized by the air mass flow and the initial unnormalized thrust potential in the most upstream plane of the vehicle intake. This ensures that the trust potential shown is initially at zero and that any mass flow lost is assumed to sap all its momentum from the vehicle thrust. The assumption that any air lost from the domain loses its momentum is important when considering features such as air bleed-offs. However, this means that when analysing a vehicle it is necessary to consider only the air that will enter the engine in regions such as the inlet before the cowl since air can still miss the vehicle and exit the domain in a manner that will not sap momentum. In the inlet of the vehicle the area encompassing air that will enter the engine but before the cowl is known as the air capture area. This air capture area can be determined by following a streamline backwards from the cowl.

30 1 4. Inlet and Compression Configuration Initially two types of inlet designs were considered for the scramjet. One inlet uses two shocks which converge on the cowl lip and the other uses one inlet shock and the fuel injector struts to compress the incoming air. A one shock inlet with injection struts was eventually chosen based on theoretical and simulated analysis. 4.1 Two-Shock Inlet It was initially planned to use a two shock inlet to compress the air prior to the cowl. The cowl lip would have been placed where the two shocks intersect. The cowl would then follow the slipline between the regions generated by the two shocks in unison and the super shock created by the intersection of the two shocks Theoretical Performance A WARP [3] tools program called inlet_realgas was used to calculate the required wedge angles for cowl inlet temperatures of 1 K, 9 K, and 8 K, the wedge angles were determined to provide equal compression ratios across both shocks. A schematic of this inlet configuration is shown in figure 4.1. Figure 4.1: -Shock Inlet Configuration 4.1. Simulated Performance A series of two-dimensional invicid simulations were carried out to determine the required inlet lengths to allow for a large enough cowl opening between.8-.1 m. The required wedge

31 angles and resulting shock angles calculated are displayed in table 4.1 along with inlet lengths determined through simulation. Table 4.1: Two-Shock Inlet Geometry Configurations Cowl Temperature (K) Wedge Angles ( o ) Shock Angles ( o ) Inlet Lengths (m) δ1 δ β1 β H=.8m H=.1m The two-dimensional invicid simulations revealed that a weak reflected shock [5] that occurs when the two shocks intersect is too strong under the conditions of interest. The weak shock creates an unacceptable pressure variation in the desired cowl intake. The shock would also likely be reflected which would interfere with fuel injection further downstream. As a result, the two shock inlet configuration was deemed unusable in this case and a one shock inlet was pursued instead. The temperature and pressure contours of the 1 K cowl entrance case are displayed in figures 4. and 4.3 respectively Figure 4.: Temperature Contours of 1K -Shock Inlet Case Note: Cowl drawn in after simulation for reference only Figure 4.3: Pressure Contours of 1K -Shock Inlet Case Note: Cowl drawn in after simulation for reference only

32 3 4. One-Shock Inlet and Strut The one shock inlet and strut uses one large initial wedge angle coupled with wider fuel injector struts to achieve the required air compression. Sketches of the two main components of this configuration are displayed in figures 4.4 and 4.5. The struts are arranged side by side, in-line, with fixed spacing. The strut compression occurs over two shocks. The first shock is generated by the leading edge of the strut. As the leading edge shocks move outwards they intersect each other and reflect back creating the second trailing edge shock. Ideally the second shock will intersect the strut at the shoulder point and be cancelled out by the trailing edge of the strut which is ideally designed to have the same angle. Figure 4.4: 1 Shock Inlet Side View Figure 4.5: Injection Strut Top View 4..1 Theoretical Performance In total the properties of air leaving the compression zone are the result of three shocks; the inlet shock, the leading edge strut shock, and the trailing edge strut shock. The performance of possible inlet and strut geometry configurations were evaluated theoretically using gas dynamics equations [5] and compared as shown in table 1. The most ideal configuration proved to have an inlet shock of.5 o and an initial 8.5 o injector strut wedge angle. This was because the configuration was theoretically predicted to produce an acceptable temperature of 876 K after the second strut shock with a reasonable blockage ratio of.5.

33 4 Table 4.: 1 Shock Inlet and Strut Theoretical Performance Inlet Angle of 17.5 o Inlet Angle of.5 o Inlet Angle of o Wedge Angle Shock Angle 1 Shock Angle Combustor Entrance Properties Blockage Ratio Wedge Angle Shock Angle 1 Shock Angle Combustor Entrance Properties Blockage Ratio δ (degrees) β1 (degrees) β (degrees) M3 P3 (Pa) T3 (K) w/(s+w) δ (degrees) β1 (degrees) β (degrees) M3 P3 (Pa) T3 (K) w/(s+w) Inlet Angle of 5 o Wedge Angle Shock Angle 1 Shock Angle Combustor Entrance Properties Blockage Ratio Wedge Angle Shock Angle 1 Shock Angle Combustor Entrance Properties Blockage Ratio δ (degrees) β1 (degrees) β (degrees) M3 P3 (Pa) T3 (K) w/(s+w) δ (degrees) β1 (degrees) β (degrees) M3 P3 (Pa) T3 (K) w/(s+w) Simulated Performance The one shock inlet and strut was simulated two-dimensionally in two phases, one simulation for the one shock inlet and a series of two-dimensional simulations for the strut. These simulations were viscous and allowed for the verification and finalization of the strut geometry. It was important to use viscous simulations to allow for consideration of boundary layers and recirculation zones. The simulation of the one shock inlet was used to verify the predictions of the theoretical analysis and to determine the cowl position and strut height. A desired air intake mass flow rate of kg per meter vehicle width was set as a design constraint and the required strut height determined to be.77 m. As explained later in section 7. 3D Strut Simulation a 5 mm boundary layer bleed-off was additionally placed between the vehicle body wall and the injection strut. The cowl tip was placed.451 m lengthwise from the tip of vehicle such that the initial body shock just misses entering the vehicle cowl. The temperature field of the one shock inlet is shown in figure 4.6.

34 5 Figure 4.6: Temperature Field of D Viscous 1 Shock Inlet From the two-dimensional viscous simulations of the strut it was discovered that at the planned backward strut angle equal to the second shock (~ o ) the simulations predicted boundary layer separation and a large re-circulation zone. By reducing the backward strut angle to 15 o the recirculation zone was eliminated. Two-dimensional simulations of the final injector strut geometry used for the three-dimensional simulations and the original strut geometry including the re-circulation zone is displayed in figure 4.7.

35 6 Figure 4.7: D Viscous Strut Simulations Although useful for finalizing geometry the two-dimensional viscous strut simulations are unusable for other analysis because the cantilevered fuel injectors produce a uniquely threedimensional flow. Using the geometric dimensions shown in figure 4.5 the finalized geometry is displayed in table 4.3. The overall strut height (depth in figure 4.5) is also shown in table 4.3 for reference. Table 4.3: Final Strut Geometry Geometric Variable Length (mm) a b w s Strut Height Theoretical After Strut Conditions Using the theoretical performance program as described in section 3. Theoretical Performance Calculator the conditions that exist after the fuel injection strut and after the combustor entrance

36 7 shocks were predicted. The conditions after the fuel injection strut were assumed to result from isentropic expansion of the flow after the second strut shock. The wedge angles that turn the flow into the combustor parallel to the vehicle s direction of travel were designed to produce equal strength shocks. The results of this analysis are displayed in table 4.4. Table 4.4: Theoretical After Strut Conditions Flow Feature Expansion After Fuel Injector Strut First Combustor Entrance Shock Second Combustor Entrance Shock Wedge Angle ( o ) Shock Angle ( o ) Resulting Mach Number Resulting Temperature (K) Resulting Pressure (Pa) Overall Pressure Ratio (P/P atm ) N/A N/A The theoretical pressure after the fuel injection strut was used to set fuel inlet boundary conditions. Although only the vehicle inlet is considered in detail by this project the combustor entrance shock results suggest that the final vehicle would have a favourably high total pressure ratio (>5) within the combustor.

37 8 5. Cantilevered Injector Configuration The configuration of the cantilevered injectors was determined to best suit the geometry and injection parameters for the desired range of fuel mass flow rate. Much of the geometry was chosen based on best practices determined from previous experience within the HSVPSG research group and verified to produce acceptable results in the vehicle under investigation using theoretical equations. Additional details about the theoretical analysis are available in section 3.1 Theoretical Solutions. 5.1 Design Process The number and size of the injectors was chosen based on best practices within the HSVPSG group and how well they fit the required strut geometry. The injector geometry was then analysed theoretically. Injection temperature, pressure, and fuel mass flow rate were treated as design constraints and the required injection stagnation temperature and pressure were checked to ensure feasibility. The injector lengths were chosen to equal the length of the backward facing strut (value b in table 4.3) plus 1 cm for simulation of internal injection. The 1 cm space for internal injection is necessary to allow for the initial development of a boundary layer in the fuel jet to prevent a numerical singularity where it meets the boundary layer produced by the air stream over the injector. This internal injection region was separated from the air stream by an one cell wall region 3 µm wide. 5. Chosen Configuration The final injector configuration is displayed in figure 5.1. Four injectors were chosen per strut side. The injectors were sized and spaced as displayed in table 5.1.

38 9 Figure 5.1: Injector Configuration Table 5.1: Injector Configuration Variables Injector Configuration Variable Length (mm) Injector Height 5 Injector Injector Gap 1 Injector Wall Gap 13.5 Injector Width 5 Injector Strut Midpoint Gap 5 Internal Injection Length 1 Injector Length 47.31

39 3 6. 3D Simulation Methodology Due to computational constraints the vehicle simulation had to be conducted in several section domains with overlapping regions. These domains were simulated in order from upstream to downstream and when an upstream domain was converged to steady state the data was interpolated or copied into the overlapping region of the domain immediately downstream and the vehicle simulation was continued. Table 6.1: Simulated Domains within Vehicle Order Simulated Vehicle Section Dimensions Initial Conditions Source Total Nodes 1 Vehicle Body Shock Atmospheric Properties 5 Inlet Cowl Inlet and Front 3 Interpolated from Body of Strut Shock Simulation 3 Full Injection Strut 3 Copied from Cowl Inlet Without Inlet Simulation 4 Back Strut Face with 3 Copied from Full Strut Fuel Injection Simulation 5 Mixing Duct 3 Copied from Fuel Injection Simulation This staged simulation methodology is acceptable for the scramjet problem since the main flow is almost always supersonic. Because the flow is always supersonic information can be assumed to not propagate upstream very far, if at all, in fact the marching window method makes great use of this since only a small section of the whole domain is simulated at any given time. As a result any simulated region passed by the marching window is assumed to have fully converged. These assumptions however would not be valid if a subsonic zone was found to occur downstream. The marching window would become stuck at the subsonic region which would need to be fully simulated and converged in a computationally expensive manner. It would then be important to ensure that the subsonic zone does not expand into other domains and that supersonic conditions dominate in any copied region. For mesh generation a standardized Cartesian coordinate system was used. For ease of mesh generation the origin was situated at the very tip of the vehicle. The X axis extends in the positive direction along the vehicle opposite the proposed direction of travel. The Z axis accounts for domain thickness and as a result is not considered in the two-dimensional inlet

40 31 simulations. Both domain boundaries perpendicular to the Z axis in the three-dimensional simulations are symmetry conditions since only one half of the strut was required for simulation. The Y axis is reversed such that it extends downwards in the positive dimension, is a sense the vehicle is simulated upside down, however, since gravity was not considered a significant factor in the simulation it is of no consequence to the results. This Cartesian coordinate system as it relates to the vehicle is displayed in figure 6.1. Figure 6.1: Vehicle Simulation Cartesian Coordinate System To avoid simulation of the whole vehicle symmetry planes were used perpendicular to the Z axis and placed half way between adjacent struts and through the center of one strut. Since the vehicles struts would all be in line with fixed spacing between them this reduced threedimensional domain is still accurate. Since the inlet utilizes a blockage ratio of.5 and both the strut width and space between strut shoulder points is cm the mesh domain width is cm and covers from the middle of one strut to the middle of the spacing between a theoretically adjacent strut. To accurately simulate boundary layers nodes immediately adjacent to wall boundaries could not be more that 3 µm away from the wall boundary with gradually increasing distance allowed between subsequent nodes. However, since the mesh is structured these small cells propagate beyond where they are necessary in some parts of the mesh. Wall conditions are constant temperature and no slip. The vehicle is assumed to have actively cooled walls and wall conditions are set to have a constant temperature of 5 K.

41 3 6.1 D Inlet Simulations The vehicle inlet prior to the cowl is geometrically two-dimensional and can be simulated accurately in two dimensions. Two different types of two-dimensional inlet simulations were conducted, one simulation for interpolation into the three-dimensional domains and one simulation specialized for determining thrust potential Simulation for Data Interpolation This one shock inlet two-dimensional simulation was conducted such that any chosen cowl position would be well within its boundaries. Information from this simulation was then copied into a small external inlet section in three dimensions preceding the injection strut. The temperature contours of this simulation including an outline of the three-dimensional inlet mesh and interpolated zone is visible in figure 6..

42 33 Figure 6. Interpolated D Simulation Showing Copied Data Region 6.1. Thrust Potential Accurate Simulation Although useful for producing the initial conditions required for the three-dimensional domains the interpolated two-dimensional simulation does not reflect only the air capture area of the inlet and as such cannot be used for computing thrust potential since not all simulated air enters the engine. To accurately determine the change in thrust potential due to the inlet prior to the cowl a two-dimensional simulation incorporating only the engine s air capture area was required. Obtaining this simulation was a two step process. A guess of the air capture area was simulated by bounding the two-dimensional domain with a horizontal outlet boundary parallel to the direction of travel extending from the cowl. This simulation is shown in figure 6.3. This method

43 34 would produce the air capture area under the assumption that the inlet shock terminates exactly on the cowl but in reality the shock slightly misses the cowl to prevent the inlet shock from propagating internally. The fact that the shock misses the cowl results in some mass flow exiting the two-dimensional domain without entering the engine. This mass flow discrepancy results in excessive thrust potential losses since the calculation method assumes mass losses sap their entire momentum from the vehicle. Figure 6.3: Initial Air Capture Area Including Streamtrace Intersecting Cowl A more accurate air capture area domain was created by changing the horizontal boundary to a symmetry condition and adjusting it to follow a streamline intersecting the cowl in the initial simulation. This air capture area simulation is shown in figure 6.4. Although the symmetry condition produced some minor flow discrepancies near the cowl intersection when compared to

44 35 the initial simulation it produced a reasonable and consistent thrust potential result since it allowed the inlet domain to be mass conservative. Figure 6.4: Thrust Potential Accurate Air Capture Area 6. 3D Strut Simulations The three-dimensional strut was simulated in two steps using two different meshes. Although both meshed domains contained all strut geometry the first simulation was extended with a small exterior inlet section which allowed for accurate simulation using initial conditions from the two-dimensional shock simulations. The second simulation lacked this inlet section but was computational far less expensive. To most efficiently use computational resources the strut was simulated using the mesh with the exterior inlet until just before the shoulder point of the

45 36 injection strut. The conditions at the beginning of the front face of the strut were then copied into the domain of the simple mesh for continued simulation. Both strut simulation domains also contain cut out sections below the front strut face to provide adequate grid lines for meshing of the fuel injectors on the back face of the strut. Although computationally wasteful it was learned through experience that they are unavoidable since other possible mesh configurations produce excessively skew cells on the back face of the strut Simulated 3D Inlet In order to use real inlet conditions a small exterior inlet section was meshed just in front of the cowl. The small exterior inlet section was extended below the cowl and bounded by an outlet plane perpendicular to the cowl. The required data was interpolated into the inlet section as an initial condition. The cowl was placed to avoid swallowing the inlet shock which exits the simulation domain through the outlet below the cowl. Unfortunately to mesh the exterior inlet section using a structured mesh the domain beyond the outlet for the shock is actually a cut out section of the simulation complete with a large number of nodes. Although the cut out section has no effect on simulation results it is computationally expensive in terms of memory which also increases required simulation time due to data saving and processing. Initially the inlet section was bounded by a flat body wall only prior to the injection strut. However, due to results elaborated upon in section 7. 3D Strut Simulation a slanted outlet was added to the inlet section at the body wall to bleed-off the first 5 mm of the boundary layer generated by the vehicle body wall. An example of the three-dimensional inlet domain is displayed in figure 6.5.

46 37 Figure 6.5: 3D Inlet and Strut Simulation Domain Note: Domain only shown up to point where simulation was user ended 6.. Injection Strut Simulation The domain used for the continuation of the injection strut simulation spans from the beginning of the front face of the strut to the end of the injectors. The end of the injectors also corresponds to the farthest point where ethylene injection is not necessary to simulate the main flow. It is easier and less computationally expensive to simulate than the strut domain which included the

47 38 inlet section and boundary layer bleed-off. The entire back face of the injection strut was then copied into the next domain to simulate fuel injection. The primary regions of this simulation domain are visible in figure 6.6. The actual simulated geometry used in this domain varied slightly from the planned geometry in three ways for computational reasons. The tip of the shoulder point was flattened into a plane.5 m wide perpendicular to the Z axis to ease simulation of the shock intersection and turning of the flow around the strut shoulder. In addition.5 m drops parallel to the Z axis were added to the back face of the strut at the top of the injectors and just below the injectors to prevent the mesh grid lines that are introduced to the simulation at that point from being squeezed together beyond 3 µm in width. The drop below the injectors also eased simulation of the pinch point between the strut face and the bottom of the injectors.

48 39 Figure 6.6: 3D Injection Strut Only Domain 6.3 3D Mixing Duct Simulations Like the strut domain the mixing duct domain is cm wide with wall boundaries where the vehicle body and cowl wall would be and symmetry boundaries in line with the midpoint of the strut and halfway between struts. The mixing simulation is conducted in sections with overlapping regions for data copy. The first section is from the shoulder point of the strut until

49 cm past the injectors. The second section is from the beginning of injection within the injectors until more than 5 cm into the mixing duct Fuel Injection Simulation The fuel injector domain encompasses the entire back face of the strut, the internal injection domain of the fuel injectors, and approximately cm of mixing duct length past the end of the injectors. The conditions for the entire back face of the strut were read into this domain and everything past the back face of the strut was copied into the next mixing duct domain. In essence no section of the geometry in this domain was unique. However, this simulation domain was useful for simulating fuel injection as it was not as computationally expensive as the long fuel duct and could be used to trouble shoot problems associated with the injection simulation at a lower computational cost. This approach also ensured good overlap between simulation domains concerning ethylene injection. The relevant sections of this domain are displayed in figure 6.7.

50 41 Figure 6.7: Fuel Injection Simulation Domain 6.3. Mixing Duct Simulation The mixing duct simulation domain is the largest domain used in this project. The first 3 cm of the duct s initial conditions are copied from the fuel injection domain starting from the plane of injection within the fuel injector and mixing is simulated for 5 cm past the end of the injectors. The relevant sections of the mixing duct simulation domain are displayed in figure 6.8.

51 Figure 6.8: Mixing Duct Simulation Domain 4

52 43 7. Simulation Results 7.1 D Inlet Simulation The two-dimensional inlet mesh used for the vehicle simulation and data interpolation produced results that agreed well with theoretical predictions. The Mach number, pressure, and temperature contours of the inlet are displayed in figures 7.1, 7. and 7.3 respectively. Figure 7.1: Mach Number Contours of Vehicle Body Inlet

53 Figure 7.: Pressure Contours of Vehicle Body Inlet 44

54 45 Figure 7.3: Temperature Contours of Vehicle Body Inlet Body Wall Boundary Layer The most important flow feature revealed by the two-dimensional inlet simulation is the body wall boundary layer. This boundary layer was studied in the X plane along the closest grid line to the cowl tip. In particular the air velocity parallel to the body wall and the temperature were studied. The free stream conditions were determined a safe distance of 5 cm from the wall where the air velocity was 1544 m/s and the temperature was 68 K. The boundary layer thickness when described by air velocity was determined using the 95% of freestream criteria since the air velocity is zero starting at the wall and asymptotically recovers to freestream conditions. However, when considering the boundary layer thickness described by

55 46 temperature a different profile is observed. The temperature is fixed at 5 K at the wall since the wall is assumed to be actively cooled and rapidly increases greater then freestream due to turbulent dissipation of kinetic energy before asymptotically cooling to freestream conditions. The temperature based boundary layer thickness was therefore determined by using a 15% of freestream criteria along the asymptotically cooling section of the curve. The properties of nodes closest to these thickness heights are displayed in table 7.1 along with the node that displayed the highest temperature. The air velocity and temperature boundary layer profiles are displayed in figures 7.4 and 7.5 respectively. It should be noted that the boundary layer analysis was conducted in an X plane to remain consistent with the way most of the data in this thesis was analysed. However since boundary layers are often examined in planes perpendicular to the wall and knowing the body wall has a wedge angle of.5 o a correction factor of.94 which is equal to the cosine of the body wall wedge angle can be applied for an estimate of the height values for boundary layer features perpendicular to the wall. Table 7.1: Properties within Boundary Layer Corresponding Node Y Distance From Body Wall (cm) Y Distance From Wall *cos(.5 o ) (cm) Air Velocity Parallel to Body Wall (m/s) Temperature (K) Body Wall 5 Maximum Temperature End of Velocity Boundary Layer End of Temperature Boundary Layer Freestream Conditions

56 47 Figure 7.4: Air Velocity Based Boundary Layer Profile Note: Lines show node best corresponding to.95*u criteria

57 48 Figure 7.5: Temperature Based Boundary Layer Profile Note: Lines show node best corresponding to 1.5*T criteria Based on standard definitions and using the X plane based analysis the boundary layer can be classified as approximately 1 to 1.5 cm in height. However, by examining the boundary layer profiles in figures 7.4 and 7.5 it can be seen that the most extreme conditions produced by the boundary layer are very close to the wall and more or less within the first 5 mm from the wall. 7. 3D Strut Simulation As described in detail previously in section 6. 3D Strut Simulations the strut was simulated in a two step process involving one mesh with an external inlet and a mesh without an external inlet. The mesh with the external inlet has initial conditions at flow entrances interpolated into it from

58 49 the two-dimensional inlet simulation and contains two outlets. One outlet allows the body shock to leave the simulation domain past the cowl and the second outlet allows for the body wall boundary layer to be bled off and reduced. The second domain mesh encompasses just the strut internal to the engine. Data interpolated from the simulation with an inlet is used to determine the flow features produced by the fuel injection strut and injectors at steady state prior to fuel injection Re-Circulation Zone It was not known initially that a boundary layer bleed-off outlet would be required prior to the commencement of three-dimensional simulation using the strut mesh with an inlet. As a result the first simulations were conducted assuming that the strut is flush against the body wall and the boundary layer is entirely engulfed by the engine. Unfortunately, the boundary layer entering along the vehicle body wall was shown to create a large unsolvable re-circulation zone when it interacts with the second strut shock and the strut face boundary layer at the shoulder point. This re-circulation zone covers the body wall of the vehicle beside the front face of the strut and can be seen in a plane parallel to the strut crosssection with temperature contours and stream traces only 1 nodes from the body wall as shown in figure 7.6. This re-circulation zone is large and not only difficult to simulate but also a poor design feature since re-circulation zones can excessively heat the vehicle walls, effectively dissipate kinetic energy and potentially cause pre-mature fuel ignition downstream due to air heating.

59 5 Figure 7.6: Strut Re-circulation Zone without Boundary Layer Bleed-off Note: Slice is in same plane as strut cross section 1 nodes (.44mm) from body wall 7.. Boundary Layer Bleed-off To prevent the occurrence of the strut re-circulation zone a boundary layer bleed-off on the body wall prior to the strut was used. In this method a small outlet between the strut and the body wall was introduced to remove some of the large inlet boundary layer. The benefit of this method is that the same strut geometry can be used which minimized the amount of backtracking required. However, by sucking away the boundary layer some momentum was sacrificed. Since the bleedoff saps momentum it was made as small as feasibly possible. It was designed to remove only the first 5 mm of the body wall boundary layer since the analysis presented in section Body Wall Boundary Layer indicated that the most extreme features of the boundary layer were within the first 5 mm from the wall. The 5 mm bleed-off height was also conservatively applied perpendicular to the body wall instead of perpendicular to the X axis.

60 Strut Simulation Results The temperature, pressure, and Mach number results produced by the three-dimensional strut simulation are presented in this thesis on X planes spaced 1 cm apart. The main direction of flow in this domain is not parallel to the X axis but more or less parallel to the body wall. However, the X plane based convention was chosen to remain consistent with the thrust potential and mixing efficiency analysis which must be conducted through integration of X planes since the vehicle direction of travel is parallel to the X axis. The planes examined and their positions within the mesh domain are shown in figure 7.7 where the cowl wall, body wall, injection strut and injectors are shown meshed. Figure 7.7: Examined X Planes within 3D Strut Domain

61 5 The examined temperature, pressure and Mach number contours within the three-dimension strut domain are displayed in figures 7.8, 7.9 and 7.1 respectively. No simulation occurred within the injectors however the nodes within them geometrically display zero values for the temperature, pressure and Mach number within figures 7.8, 7.9 and 7.1. Figure 7.8: Temperature Contours of 3D Strut Note: Temperature in injectors set to in simulation

62 53 Figure 7.9: Pressure Contours of 3D Strut Note: Pressure in injectors set to in simulation

63 54 Figure 7.1 Mach Number Contours of 3D Strut Note: Mach number in injectors set to in simulation As can be seen in figures 7.8, 7.9 and 7.1 the fuel injector strut effectively generates the two strut shocks and an expansion zone after the strut without significant shock propagation after the strut. Importantly it can be seen that there is no significant re-circulation zones within the domain. In this sense the fuel injection strut is shown to effectively allow for fuel injection within the main air flow. However, a small hot region does develop below the fuel injectors. As can be seen in section 7.3 3D Mixing Duct Results this hot region will not propagate far downstream but to help safeguard this region from causing premature ignition the fuel injection temperature was set to 5 K which is lower than the autoignition temperature D Mixing Duct Results The mixing duct was distinctly simulated four times to study vehicle performance for a range of lean equivalence ratio s. A simulation with an equivalence ratio of unity was also conducted as a control. The results of these simulations were important to determine vehicle mixing efficiency.

64 55 In the same way as the three-dimensional strut the results were analysed in X planes perpendicular to the vehicle s direction of travel even though the main air flow is actually parallel to the body wall. As explained previously this was to remain consistent with the way thrust potential and mixing efficiency was analysed. The required fuel mass flow rate was calculated assuming that the mass flow rate of air entering the engine is.3886 kg/s given the domain width of cm. The air mass flow rate was determined through integration of the last X plane analysed from the three-dimensional strut simulation. This mass flow rate would result in an air intake of kg/s for each meter of vehicle width which is exceptionally close to the desired mass flow intake of kg/s as described in section 4..1 Theoretical Performance. The difference can be explained by the mass lost through the boundary layer bleed-off. Since the numerical method used a finite difference stencil mass flow is not necessarily conserved but no significant discrepancies were observed in planes after the boundary layer bleed-off Injection Cases Considered Ethylene injection was simulated with equivalence ratios of.3,.5,.7 and one. Since the injector geometry was fixed for all cases the fuel injection properties were varied to adjust the mass flow. The fuel injection temperature and pressure was fixed at 5 K and 3179 Pa respectively. The temperature was chosen as described in section 7..3 Strut Simulation Results while the pressure was calculated based on theoretical predictions for the after strut expansion as described in section 4.3 Theoretical After Strut Conditions. To achieve the required injection conditions the fuel velocity and other conditions are displayed in table 7.. Table 7.: Fuel Injection Cases Equivalence Ratio Fuel Speed (m/s) Fuel Mach Number Stagnation Temperature (K) Stagnation Pressure (Pa) Critical Area Ratio (A/A*)

65 56 As can be seen in table 7. the required stagnation conditions required to inject fuel are reasonable, especially when you consider only the lean injection cases and ignore the equivalence ratio of unity control. It should be noted however that the stagnation temperature and pressure were predicted assuming isentropic injection. Overall the same flow features were observed developing in each simulated injection case and as such only the property contour results of the simulation where the equivalence ratio is.5 are displayed in this section to remain concise. However, the same figures derived for the other injection cases are available in appendix A.4 Mixing Duct Contours for φ =.3,.7 and 1. For each property contour four X planes of varying distance down the mixing duct are displayed. These planes are displayed within the mixing duct mesh in figure 7.11 for reference. Figure 7.11: Examined X Planes within 3D Mixing Duct

66 Temperature Field The temperature contours of the mixing duct section for equivalence ratio.5 are displayed in figure 7.1. As can be seen the hot section adjacent to the injectors are sucked towards the symmetry plane bisecting the space between struts and cools down. Other noticeable temperature features that can be seen are the hot boundary layers that develop on the vehicle body and cowl wall. It can also be seen that the boundary layers are disproportionately large adjacent to the symmetry boundary that bisects the strut, this is likely due to the action of the counter rotating vortices.

67 Figure 7.1: Temperature Contours of X Planes in φ=.5 3D Mixing Duct Simulation 58

68 Pressure Field The pressure contours of the mixing duct section for equivalence ratio.5 are displayed in figure As can be seen in the first plane the chosen fuel injection pressure is reasonably close to the actual conditions. It can also be seen that as you progress down the mixing duct there are fairly distinct regions of differing pressure. This is the result of trailing shocks that extend from the back tip of the injection strut. The curved and odd appearance of these trailing shocks is simply due to the X plane perspective and variations caused by the fuel injectors. However, these trailing shocks are reasonably weak and would merely aid in fuel mixing. The pressure is actually very uniform in the mixing duct with a difference of about 1 kpa between the pressure minimum and maximum in the last plane of the mixing duct.

69 Figure 7.13: Pressure Contours of X Planes in φ=.5 3D Mixing Duct Simulation 6

70 Mach Number Field The Mach number contours of the mixing duct section for equivalence ratio.5 are displayed in figure The mach number was calculated based on total air speed and not velocity. As can be seen in the first X plane the flow is slowest just after the backward tip of the strut but still supersonic. This suggests the mechanism by which the re-circulation zone behind the strut as described in section 4.. Simulated Performance developed when the backward strut face angle was to steep. It is also visible that the boundary layers on the body and cowl walls grow and then deform near the strut bisection as also shown in 7.3. Temperature Field. Aside from the boundary layers however the overall air speed throughout the duct has more or less equalized by the end of the duct.

71 Figure 7.14: Mach Number Contours of X Planes in φ=.5 3D Mixing Duct Simulation 6

72 Equivalence Ratio Field The equivalence ratio contours of the mixing duct section for equivalence ratio.5 are displayed in figure Since equivalence ratio ranges from infinity where there is only fuel to zero where there is only air the minimum and maximum contour levels considered were.1 and 1 respectively. As can be seem it is extremely evident that the counter rotating vortices drive fuel mixing in this simulation. In the first displayed X plane the fuel is pushed up into a characteristic horseshoe shape. In the second displayed X plane the vortices themselves from each injector can even be seen interacting with the fuel at the ends of each horseshoe shape. The fuel then quickly disperses from these flow features. In the last plane of the duct the highest equivalence ratio can be seen to be practically one which visually indicates a total mixing efficiency near unity. Interestingly the fuel appears to have been pushed more toward the gaps between struts then the strut bisection despite injection taking place entirely behind the struts. The fuel can also be seen to have avoided the boundary layers on the cowl and body walls which is very beneficial from a design perspective since boundary layers can cause premature ignition.

73 64 Figure 7.15: Equivalence Ratio Contours of X Planes in φ=.5 3D Mixing Duct Simulation Note: Minimum and maximum contour levels considered were.1 and 1 respectively

74 Mixing Efficiency Results The mixing was analysed based on the process described in section Mixing Efficiency for all mixing duct cases considered. The mixing efficiency was analysed by integrating properties over planes perpendicular to the X axis. The first and last planes analysed correspond to the first and last planes analysed using property contours within the mixing duct and are visible relative to the mesh in figure Data points between the first and last planes were derived every centimetre. The mixing efficiency results are displayed in figure Figure 7.16: Mixing Efficiency Results from 3D Mixing Duct Simulations As can be seen in figure 7.16 the lean fuel injection cases effectively mix to full mixing efficiency within the mixing duct indicating that the fuel can be completely burned. These results correspond well with the analysed property contours. Not surprisingly the leanest cases reach maximum mixing efficiency quicker since air is globally in greater excess. However, the fuel injection case with an equivalence ratio of.7 just barely reached full mixing efficiency which based on the observed trends suggest that any case where fuel is injected with a greater equivalence ratio would not reach the maximum mixing efficiency within the mixing duct.

75 66 As can also be seen in figure 7.16 the stoichiometric fuel injection case with an equivalence ratio of unity does not reach maximum mixing efficiency. This is not surprising since total mixing efficiency would require the fuel and air to be homogeneously mixed which is an ideal situation in scramjet applications. This is especially true since analysis of the property contours indicated that the fuel tended to be pushed away from air behind the strut and the boundary layers making that air unavailable for mixing. The flow s tendency to favourably mix flow in certain regions suggests why the mixing efficiency results appear to be asymptotically converging to a value less than unity despite the case being globally stoichiometric. 7.4 Thrust Potential Results The thrust potential was analysed for the entire vehicle inlet using the process described in section 3.4. Thrust Potential with the air capture area simulation described in section 6.1. Thrust Potential Accurate Simulation. The 3D simulations analysed for thrust potential were the full strut without external inlet as described in section 6.. Injection Strut Simulation and the full mixing duct as described in section 6.3. Mixing Duct Simulation. The thrust potential was analysed for all mixing duct cases however there is no variation between thrust potential results prior to the mixing duct. In the same way the mixing efficiency was determined thrust potential results were integrated over planes perpendicular to the X axis. Where possible the planes analysed were at most 1 cm apart. However, because a cross section of the entire flow through the engine is required to produce accurate data, X planes could not be analysed if the entire cross section passes a domain boundary. As such there are gaps in the data between simulation domains. The thrust potential results obtained are displayed in figure 7.17.

76 67 Figure 7.17: Thrust Potential of the Full Vehicle Inlet As can be seen in figure 7.17 the thrust potential drops along a constant slope within the D external inlet. This drop in thrust potential would be the result of a combination of wall drag and the turning of flow by the body inlet shock. The slope is straight since the wall drag would reduce thrust at a constant rate and the geometry of the shock would turn the same amount of flow right up until the cowl. The thrust potential drop between the external inlet and injection strut is almost entirely due to the boundary layer bleed-off since it is assumed that all air exiting the domain through the bleed-off will lose its entire momentum to the vehicle. In other words it is assumed that air exiting through the bleed-off will be accelerated to the same speed as the scramjet before exiting the scramjet. This assumption is conservative but necessary since the internal characteristics of the bleed-off were not designed, as such it must be assumed to function as poorly as possible. The drop in thrust potential over the fuel injection strut is due to the extreme flow features the geometry generates which includes two strut shocks, an expansion region, and four obstacles in the form of the fuel injectors. This varied and complex geometry explains not only why the thrust potential loss is so severe by why it is not as constant as observed in the other regions.

77 68 However, Even though the thrust potential is lost in a very short distance a lot of valuable tasks are performed by the strut including significant flow compression and the generation of counter rotating axial vortices. Although the fuel injectors themselves could not be analysed for thrust potential at the moment of injection it is reasonable to postulate that the rise in thrust potential between the fuel injection strut and the mixing duct is the result of fuel injection. The injection of fuel would add additional momentum to the flow increasing the thrust potential. Since the fuel injection area was unchanged between studied equivalence ratio cases not only was more mass flow injected with increased equivalence ratio but it was injected at a faster speed. This explains why the increase in thrust potential between the.5 and.7 cases is greater than the.3 and.5 cases despite the change in mass flow being theoretically the same. The thrust potential as the flow moves down the mixing duct decreases at a constant rate which is similar for all injection cases. The driving force behind the thrust potential decrease is likely drag from the body and cowl walls. However, you can see a slight staircase pattern in the mixing duct results. This staircase pattern is likely due to the weak shocks propagating in the mixing duct that were described in section Pressure Field. Overall the thrust potential results are modest with the inlet s drag resulting in a thrust potential drop of 11 N*s per kg of ingested air for the characteristic lean fuel injection case of equivalence ratio.5. Although further vehicle design and simulation would be required to ensure that this thrust potential would be recovered through combustion previous experience within the HSVPSG group suggests that the inlet drag is not unreasonable.

78 69 8. Conclusions The results of this study add to the base knowledge of scramjet design by describing not only the process by which an engineer can design a small scale ethylene fuelled scramjet inlet but by suggesting the use of cantilevered strut injectors and providing an initial design. 8.1 Vehicle Configuration and Overall Performance The vehicle inlet ultimately designed through the course of this project successfully intakes kg/s of air for combustion per meter width of the vehicle. The inlet is 8 cm long 4 cm high and uses a.5 o one shock inlet preceding the cowl. Within the cowl a.5 blockage ratio 7.7 cm high cm wide vertical fuel injector strut and a 5 cm long mixing duct are internally closed but not parallel to the direction of travel. A 5 mm high boundary layer bleed-off preceding the fuel injection strut is necessary to prevent a re-circulation zone due to boundary layer/shock interactions. Four 5 mm square cantilevered fuel injectors per strut side stick out from the strut shoulders in the general direction of internal airflow and inject ethylene at 5 K. This inlet has been shown to successfully mix almost all fuel for lean equivalence ratios less than.7 by successfully utilizing counter rotating axial vortices generated by the cantilevered fuel injectors. By utilizing the fuel injection strut the fuel flow has been shown to have been injected within the flow cross section successfully avoiding injection into boundary layers. In addition, thrust potential losses were reasonable with the greatest losses produced by the boundary layer bleed-off and the fuel injection strut. For the characteristic lean fuel injection case of equivalence ratio.5 the total thrust potential losses were 11 N*s/kg. It is predicted that a scramjet utilizing this inlet will have a total compression ratio greater than 5 assuming it utilizes two equal strength shocks to turn flow into the combustor. 8. Suggestions for Future Research Although very important the inlet itself is not a full vehicle design and a natural direction for the research to proceed in would be to finish the design and simulation process for the full vehicle. In order to properly simulate the combustor chemically active simulations involving ethylene will be required. To simulate ethylene combustion an ethylene chemical-kinetic combustion scheme will need to be researched, chosen, and programmed into the WARP [6] code. The process of researching an ethylene chemical-kinetic combustion scheme has been started within

79 7 this project as alluded to in section 1.. Ethylene Combustion but much more work is still required. In addition, if more computational resources were available more accurate data for this inlet design could be obtained if larger section domains could be utilized. Although enough overlap between domains was used in this study to accurately simulate flow features there are gaps in mixing efficiency and thrust potential data that are directly due to the need for these separate domains.

80 71 Bibliography Note: References in ASME Format [1] Heiser, W. H., Pratt, D. T., Daley, D. H., & Mehta, U. B. (1994). Hypersonic Airbreathing Propulsion, American Institute of Aeronautics and Astronautics. [] Curran, E. T., 1, "Scramjet Engines: The First 4 Years," Journal of Propulsion and Power, 17(6) pp [3] Voland, R. T., Auslender, A. H., Smart, M. K., 1999, "CIAM/NASA Mach 6.5 Scramjet Flight and Ground Test," American Institute of Aeronautics and Astronautics,. [4] Voland, R. T., Huebner, L. D., and McClinton, C. R., 6, "X-43A Hypersonic Vehicle Technology Development," Acta Astronautica, 59 pp [5] Wu, D., and Xiao, H., 9, "Aerodynamics Simulation of Hypersonic Waverider Vehicle," Canadian Centre of Science and Education: Modern Applied Science, 3() pp [6] Schumacher, J., and Sislian, J. P.,, "Evaluation of Hypersonic Fuel/Air Mixing Performance by Cantilevered Ramp Injectors," 36th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, American Institute of Aeronautics and Astronautics,. [7] Schwartzentruber, T. E., Sislian, J. P., and Parent, B., 5, "Suppression of Premature Ignition in the Premixed Inlet Flow of a Shramjet," Journal of Propulsion and Power, 1(1) pp [8] Wang, Y. -W., and Sislian, J. P., 8, "Numerical Investigation of Methane and Air Mixing in a Shcramjet Inlet," 15th AIAA International Space Planes and Hypersonic Systems and Technologies Conference, American Institute of Aeronautics and Astronautics,. [9] Drummond, J. P., and Diskin, G. S.,, "Fuel-Air Mixing and Combustion in Scramjets," American Institute of Astronautics and Aeronautics,.

81 7 [1] Gruber, M. R., Donbar, J. M., and Carter, C. D., 4, "Mixing and Combustion Studies using Cavity-Based Flameholders in a Supersonic Flow," Journal of Propulsion and Power, (6) pp [11] Tomioka, S., Murakami, A., Kudo, K., 1, "Combustion Tests of a Staged Supersonic Combustor with a Strut," Journal of Propulsion and Power, 17() pp [1] Northam, G. B., and Anderson, G. Y., 1986, "Supersonic Combustion Ramjet Research at Langley," AIAA 4th Aerospace Sciences Meeting, American Institute of Aeronautics and Astronautics,. [13] Alexander, D. C., and Sislian, J. P., 8, "Computational Study of the Propulsive Characteristics of a Shramjet Engine," Journal of Propulsion and Power, 4(1) pp [14] Parent, B., and Sislian, J. P., 4, "Hypersonic Mixing Enhancement by Compression at a High Convective Mach Number," American Institute of Aeronautics and Astronautics, 4(4) pp [15] NASA. Chemical Equilibrium with Application, Retrieved July 17, 9 from [16] Parent, B., Sislian, J. P., and Schumacher, J.,, "Numerical Investigation of the Turbulent Mixing Performance of a Cantilevered Ramp Injector," American Institute of Aeronautics and Astronautics, 4(8) pp [17] Parent, B., and Sislian, J. P., 3, "Effect of Geometrical Parameters on the Mixing Performance of Cantilevered Ramp Injectors," American Institute of Aeronautics and Astronautics, 41(3) pp [18] Sislian, J. P., Martens, R. P., Schwartzentruber, T. E., 6, "Numerical Simulation of a Real Shramjet Flowfield," Journal of Propulsion and Power, (5) pp

82 73 [19] Varatharajan, B., and Williams, F. A.,, "Ethylene Ignition and Detonation Chemistry, Part 1: Detailed Modeling and Experimental Comparison," Journal of Propulsion and Power, 18() pp [] Colket, M. B. I., and Spadaccini, L. J., 1, "Scramjet Fuels Autoignition Study," Journal of Propulsion and Power, 17() pp [1] Taha, A. A., Tiwari, S. N., and Mohieldin, T. O.,, "Combustion Characteristics of Ethylene in Scramjet Engines," Journal of Propulsion, 18(3 Technical Notes) pp [] Ortwerth, P. J., Mathur, A. B., Vinogradov, V. A., 1996, "Experimental and Numerical Investigation of Hydrogen and Ethylene Combustion in a Mach 3-5 Channel with a Single Injector," 3nd AIAA, ASME, SAE, and ASEE, Joint Propulsion Conference and Exhibit, American Institute of Aeronautics and Astronautics,. [3] Linde Gas. 5, Safety Data Sheet Ethylene,. [4] Varatharajan, B., and Williams, F. A.,, "Ethylene Ignition and Detonation Chemistry, Part : Ignition Histories and Reduced Mechanisms," Journal of Propulsion and Power, 18() pp [5] Smyth, K. C., and Bryner, N. P., 1997, "Short-Duration Autoignition Temperature Measurements for Hydrocarbon Fuels Near Heated Metal Surfaces," Combustion Science and Technology, 16 pp [6] Parent, B., and Sislian, J. P.,, "The use of Domain Decomposition in Accelerating the Convergence of Quasihyperbolic Systems," Journal of Computational Physics, 179 pp [7] Sislian, J. P., Dudebout, R., Schumacher, J.,, "Incomplete Mixing and Off-Design Effects on Shock-Induced Combustion Ramjet Performance," Journal of Propulsion and Power, 16(1) pp

83 74 [8] Oosthuizen, P.H., and Carscallen, W.E., 8, "Mech 448 An Introduction to Compressible Fluid Flow," P&CC Course Package, Kingston, Ontario,. [9] Turns, S.R.,, "An Introduction to Combustion: Concepts and Applications," McGraw- Hill Higher Education,. [3] Parent, B., and Sislian, J. P., 4, "Validation of Wilcox k-ω Model for Flows Characteristic to Hypersonic Airbreathing Propulsion," American Instutute of Aeronautics and Astronautics, 4() pp [31] Wilcox, D. C., 1988, "Multiscale Model for Turbulent Flows," American Institute of Aeronautics and Astronautics, 6(11) pp [3] Ciccarelli, G., 9, Mech435 Section 5 Combustion Theory, Queen s University, Kingston [33] Digital Mars., D Programming Language, Retrieved December 8, 1 from [34] McBride, B. J., and Reno, M. A., 1993, Coefficients for Calculating Thermodynamic and Transport Properties of Individual Species, NASA, TM [35] Wilcox, D. C., 1988, "Reassessment of the Scale-Determining Equation for Advanced Turbulence Models," American Institute of Aeronautics and Astronautics, 6(11) pp [36] Wilcox, D. C., 199, Dilatational-Dissipation Corrections for Advanced Turbulence Models, American Institute of Aeronautics and Astronautics, 3(11) pp [37] Yee, H. C., 199, High-Resolution Shock-Capturing Schemes for Invicid and Viscous Hypersonic Flows, Journal of Computational Physics, 88 pp

84 75 [38] Dimotakis, P. E., 1991, Turbulent Free Shear Layer Mixing and Combustion, American Institute of Aeronautics and Astronautics, Washington, DC, pp [39] Papamoschou, D., and Roshko, A., 1988, The Compressible Turbulent Shear Layer: And Experimental Study, Journal of Fluid Mechanics, 197 pp [4] Briley, W. R., and McDonald, H., 1977, Solution of the Multidimensional Compressible Navier-Stokes Equations by a Generalized Implicit Method, Journal of Computational Physics, 4(11) pp [41] Beam R., and Warming, R. F., 1978, An Implicit Factored Scheme for the Compressible Navier-Stokes Equations, American Institute of Aeronautics and Astronautics, 16(4) pp [4] Wang, Y. -W., 7, Numerical Investigation of Gaseous Hydrocarbon Fuel Preinjection in the Inlet of a Hypersonic Engine, University of Toronto Institute for Aerospace Studies. [43] Chan, J., 1, Numerically Simulated Comparative Performance of a Scramjet and Shcramjet at Mach 11, University of Toronto Institute for Aerospace Studies.

85 76 Appendix A.1 Additional Details on Theoretical Solutions A.1.1 Gas Dynamics Equations Gas dynamic equations for supersonic flows generally allow for the prediction of downstream conditions from a particular perturbation based on upstream conditions and the particular characteristics of that perturbation. The subscript 1 is used in the following equations to represent conditions upstream of a perturbation and the subscript is used to represent conditions downstream of a perturbation. These equations were useful for providing an initial guess of inlet performance. Using these equations initial design choices for the scramjet were made for simulation and further refinement using CFD. Equation A.1 [8] is a theoretical equation that determines the wedge angle (δ) of an obstacle that turns a supersonic flow in a way that produces an oblique shock. The required inputs are the initial Mach number (M 1 ) of the flow, the angle of the shock produced relative the direction of initial flow (β), and the ratio of specific heats (γ) which was assumed to be as shown in equation A. for air. However, it is almost always more useful to determine the shock angle produced due to a known wedge angle. Since the equation was complex is was not re-arranged and instead iterative methods were used in conjunction with the equation in its current form. In Microsoft excel the goal seek function was used. When incorporating the equation into a program the bisection method was used. Better estimates are possible by altering the ratio of specific heats based on actual conditions. However, the standard value used was deemed sufficient for the type of exploratory analysis performed using the affected equations. ( M sin β 1) 1 cot β 1 δ = tan + M1 ( γ + cosβ ) (A.1) γ =1.4 (A.) Equations A.3, A.4, and A.5 [8] were used to determine the conditions encountered after an oblique shock with a known shock and wedge angle.

86 77 M1 sin β + ( γ 1) γm 1 sin β 1 M = (A.3) sin ( γ 1) ( β δ ) ( M sin β ) ( γm 1 sin β ( γ 1) ) + ( γ 1) ( γ + 1) M sin β 1 T = T1 (A.4) 1 β ( γ 1) ( γ + 1) γm 1 sin P = P1 (A.5) In was also necessary to theoretically determine conditions after an isentropic expansion. To do this known upstream conditions were used to determine initial density, initial speed of sound, initial velocity, mass flow rate, stagnation pressure, and stagnation density using equations A.6, A.7, A.8, A.9, A.1 and A.11 [8] respectively. P 1 ρ 1 = (A.6) RT1 a = γ (A.7) 1 RT 1 V = (A.8) 1 a1m 1 m = ρ1v 1A1 (A.9) γ γ 1 γ 1 P = P1 1+ M1 (A.1)

87 = γ γ ρ ρ M (A.11) Knowing these conditions and knowing that stagnation conditions and mass flow rate are constant through an isentropic expansion equation A.1 [8] was used in conjunction with an iterative method such as the bisection method to determine the resulting pressure = γ γ γ ρ γ γ ρ P P P P P A m (A.1) Once the after expansion pressure is determined the after expansion Mach number and temperature can be determined using equations A.13 and A.14 [8] respectively. The after expansion density, speed of sound, and velocity can be determined in the same way as the initial values of those properties using equations A.6, A.7, and A.8 respectively as long as after expansion conditions are used as inputs. a V M = (A.13) R P T ρ = (A.14) A.1. Combustion Equations The combustion equations for ethylene are dependent on the number of carbon and hydrogen atoms in the molecule which are indicated by α and β respectively in equations A.15 and A.16. Also required are the molar masses of the atoms in the hydrocarbon and the molecules in air as shown in equations A.17, A.18, A.19, and A.. 4 = H C α (A.15)

88 β 4 (A.16) C H 4 = 79 M C =1.17 (A.17) M H =1.79 (A.18) M N = *14.67 (A.19) M O = * (A.) To determine fluid properties for setting boundary conditions for fuel injection the NASA [15] thermodynamic polynomials for ethylene as displayed in equation A.1 were used with equation A. to calculate the specific heat capacity at constant pressure where R is the universal gas constant. In general, injection temperature and injection mass flow rate were treated as design constraints. Equation A.3 was used to calculate the ratio of specific heats. Injection pressure was assumed to be equal to that of the surrounding air which was determined using gas dynamic theoretical calculations. Equations A.6, A.7, A.9, and A.1 could also be used in the context of fuel injection to determine the ethylene injection density, speed of sound, injection velocity, and stagnation pressure respectively. Stagnation temperature in the fuel flow was determined using equation A.4 [8]. a1 a a3 a4 a5 a6 a7 C H 4 C H 4 C H 4 C H 4 C H 4 C H 4 C H 4 = = = = = = = *1-11 (A.1) 1 3 ( 1( T ) + a( T ) + a3 + a4( T ) + a5( T ) + a6( T ) a7( T 4 ) c p = R a + (A.)

89 c p γ = (A.3) c R p 8 T = T 1+ ( γ 1) M (A.4) A. Inlet Theoretical Performance Program Input and Output An example of the program input is displayed in figure A.1 and an example of the program output is displayed in figure A..

90 Figure A.1: Inlet Theoretical Performance Program Input 81

91 Figure A.: Inlet Theoretical Performance Program Output 8

92 83 A.3 CFL Auto Adjustment In order to make the results of a simulation less dependent on starting CFL conditions a set of new conditional programming loops were specified to be placed within the marching window loop of the WARP [6] control file. These conditions compare the current iteration s maximum residual with the previous iteration s maximum residual. If the residual is decreasing the CFL is gradually increased by multiplying it by a preset factor (cflvh) which is greater than one. If the residual is increasing the CFL is decreased by multiplying it by the factor (cflvl) which is less than one. Maximum and minimum CFL conditions (maxcfl) and (mincfl) are preset for the simulation and will act as the limits on CFL increase and decrease. The required commands implemented into the marching window section of the control file are displayed in equation A.5. if (ximax <= ximaxnm1, CFL = CFL * cflvh; ); if (ximax > ximaxnm1, CFL = CFL * cflvl; ); (A.5) CFL = min(max(mincfl,cfl),maxcfl); ximaxnm1= ximax; The variables (cflvl), (cflvh), (mincfl), and (maxcfl) must be specified at the beginning of the control file. An initial (ximaxnm1) should be set at the same time to avoid error during the first iteration.

93 84 A.4 Mixing Duct Contours for φ =.3,.7 and 1 A.4.1 Mixing Duct Contours for φ =.3 Figure A.3: Temperature Contours of X Planes in φ=.3 3D Mixing Duct Simulation

94 Figure A.4: Pressure Contours of X Planes in φ=.3 3D Mixing Duct Simulation 85

95 Figure A.5: Mach Number Contours of X Planes in φ=.3 3D Mixing Duct Simulation 86

96 87 Figure A.6: Equivalence Ratio Contours of X Planes in φ=.3 3D Mixing Duct Simulation Note: Minimum and maximum contour levels considered were.1 and 1 respectively

97 88 A.4. Mixing Duct Contours for φ =.7 Figure A.7: Temperature Contours of X Planes in φ=.7 3D Mixing Duct Simulation

98 Figure A.8: Pressure Contours of X Planes in φ=.7 3D Mixing Duct Simulation 89

99 Figure A.9: Mach Number Contours of X Planes in φ=.7 3D Mixing Duct Simulation 9

100 91 Figure A.1: Equivalence Ratio Contours of X Planes in φ=.7 3D Mixing Duct Simulation Note: Minimum and maximum contour levels considered were.1 and 1 respectively

101 9 A.4.3 Mixing Duct Contours for φ = 1. Figure A.11: Temperature Contours of X Planes in φ=1. 3D Mixing Duct Simulation

102 Figure A.1: Pressure Contours of X Planes in φ=1. 3D Mixing Duct Simulation 93

103 Figure A.13: Mach Number Contours of X Planes in φ=1. 3D Mixing Duct Simulation 94

104 95 Figure A.14: Equivalence Ratio Contours of X Planes in φ=1. 3D Mixing Duct Simulation Note: Minimum and maximum contour levels considered were.1 and 1 respectively