Study of minimum length, supersonic nozzle design using the Method of Characteristics

Murrough Murnaghan Study of minimum length, supersonic nozzle design using the Method of Characteristics Master’s Thesis to achieve the university degree of Master of Science Master’s degree programme: Master’s Degree in Space and Aeronautical Engineering (MASE) submitted to Universitat Polit`ecnica de Catalunya Supervisor Dr. Sara Arbos Torrent Co-supervisor Oriol Lizandra Dalmases Escola Superior d’Enginyeries Industrial, Aeroespacial i Audiovisual de Terrassa (ESEIAAT) Terrassa, June 2019 Declaration of Originality I declare that I have authored this thesis independently, that I have not used other than the declared sources/resources, and that I have explicitly indicated all material which has been quoted either literally or by content from the sources used. The text document uploaded is identical to the present master‘s thesis. Date Signature ii Abstract This report outlines the pertinent theory and methods for supersonic nozzle design using the method of characteristics. The programmes developed design ideal nozzle contours, using numerical methods based on the gas properties, for desired exit velocities. Ideal contours include the most rapid expansion without inducing shock waves followed by the straightening section to provide uniform exit conditions. Then based on the ideal results, the contours can be truncated and the shape modified to induce minor shocks and only sacrifice minimal thrust to save length and thereby weight. iii Acknowledgements I would like to take this opportunity to express my sincere gratitude and appreciation for the time and effort afforded to helping me complete this thesis. Both Sara and Oriol have provided direction and encouragement when it was most needed. Oriol’s help and tuition was patient and effective with the theory. He has also furthered my coding abilities greatly by sharing his experience with me throughout the semester. Most of all, I would like to express my appreciation for his availability. Regular meetings helped to keep the project on track and afforded us the opportunity to modify the goals of the study. iv Contents Abstract iii Acknowledgements iv 1. Introduction 1 1.1. Background ............................. 1 1.1.1. TheIdealRocket ...................... 3 1.2. AimsandObjectives ........................ 4 1.2.1. Limitations ......................... 4 1.3. Methodology ............................ 5 2. Literature Review 6 2.1. Nozzles................................ 6 2.1.1. NozzlePrinciples...................... 7 2.1.2. NozzleDesign ....................... 8 2.1.3. Thermodynamic Relations . . . . . . . . . . . . . . . . 10 2.1.4. Istentropic Nozzle Flow . . . . . . . . . . . . . . . . . . 11 2.1.5. Thrust and Thrust Coefficient . . . . . . . . . . . . . . . 13 2.2. RealNozzles............................. 13 2.2.1. BoundaryLayer....................... 15 2.2.2. MultiphaseFlow ...................... 15 2.2.3. Other Phenomena and Losses . . . . . . . . . . . . . . 17 2.3. Computational Methods . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1. Prandtl-Meyer........................ 17 2.3.2. EulerEquations....................... 18 2.3.3. The Method of Characteristics for Ideal Gas . . . . . . 19 2.4. TruncatedNozzles ......................... 23 2.4.1. Truncated Ideal Contour nozzles (TIC) . . . . . . . . . 23 2.4.2. Compressed Truncated Ideal Contoured Nozzles (CTIC) 24 v Contents 2.4.3. Thrust Optimised Contoured Nozzles (TOC) . . . . . . 25 2.4.4. Simply Truncated Nozzles . . . . . . . . . . . . . . . . 26 3. Methodology 28 3.1. NozzleDesign............................ 28 3.1.1. Two-Dimensional Method of Characteristics . . . . . . 28 3.1.2. Three-Dimensional Method of Characteristics . . . . . 32 3.2. NozzleTruncation ......................... 34 4. Results & Discussion 36 4.1. Two-Dimensional Nozzles . . . . . . . . . . . . . . . . . . . . . 36 4.1.1. IdealContours ....................... 36 4.1.2. Truncated Ideal Contour . . . . . . . . . . . . . . . . . . 45 4.1.3. Compressed Truncated Ideal Contour . . . . . . . . . . 47 4.2. Three-Dimensional Nozzles . . . . . . . . . . . . . . . . . . . . 48 4.2.1. IdealContours ....................... 48 4.2.2. Truncated Ideal Contour . . . . . . . . . . . . . . . . . . 53 4.2.3. Compressed Truncated Ideal Contour . . . . . . . . . . 55 4.3. NozzleParameters ......................... 56 5. Conclusions 61 5.1. Summary............................... 61 5.2. Recommendations for Future Work . . . . . . . . . . . . . . . 62 Bibliography 63 A. Characteristic Directions 66 B. Derivation of the Prandtl-Meyer Function 70 vi Nomenclature ˙ mMass flowrate γIsentropic expansion factor µMach angle ωPrandtl-Meyer function θFlow turning angle AArea aLocal speed of sound CFThrust Coefficient g0Acceleration due to gravity hEnthalpy I+Invariant inclined at θ+µ I−Invariant inclined at θ−µ ItTotal Impulse Isp Specific Impulse kCompression factor m+Length along I+ vii Contents m−Length along I− NNumber of characteristic lines PPressure rtThroat radius VVolume vVelocity xAxial position yRadial position relative to throat radius CTIC Compressed Truncated Ideal Contour F Force/Thrust II Mach line M Molecular mass/Mach number MOC Method of Characteristics PDE Partial Differential Equation R Universal gas constant T Reference temperature t time TIC Truncated Ideal Contour TOC Thrust Optimised Contour TOP Thrust Optimised Parabolic viii List of Figures 2.1. Simplified depictions of exhaust gases from three different rocket nozzles [3]........................... 8 2.2. Method of Characteristics nozzle designs [8]........... 9 2.3. Relationship between area ratio, temperature ratio and pressure ratio as a function of Mach number in a bell shape nozzle. [3] .............................. 12 2.4. Boundary layer flow conditions at the exit of a supersonic nozzle [3]. .............................. 16 2.5. Depiction of the variables associated with the Prandtl-Meyer function. ............................... 18 2.6. Streamlines and Mach lines according to linear theory. The mean Mach line ( II ) is midway between the upstream and downstream flow Mach lines (II1and II2respectively). . . . . 19 2.7. Flow around a curved wall by modified linear theory. . . . . . 20 2.8. Geometry of characteristic lines. . . . . . . . . . . . . . . . . . 21 2.9. Illustration of nozzle truncation constraints. . . . . . . . . . . 24 2.10. Mach diamonds in a nozzle jet stream. [10]........... 25 2.11. TOC nozzle contour [5]. ...................... 26 2.12. Comparison of TIC (Ideal Truncated) and TOC (Rao-Shmyglevsky) nozzle flow structures [5]...................... 27 3.1. Diagram of the zones related to MOC nozzle design [2]. . . . 29 3.2. Conic nozzle with 20◦half angle and 20 characteristic lines. . 30 3.3. Minimum length two-dimensional nozzle with ME=3 and 20 characteristiclines. ....................... 31 3.4. Example of a typical wedge nozzle. . . . . . . . . . . . . . . . 32 3.5. Five subdivisions of the first characteristic. . . . . . . . . . . . 33 3.6. Minimum length three-dimensional nozzle with ME=3 ,20 characteristic lines and 5inlet subdivisions. . . . . . . . . . . . 34 ix 1. Introduction The Method of Characteristics does not deal with the boundary layer of the nozzle flow and thereby, the heat transferred from the gas to the nozzle. However, the energy dissipated through heat loss is generally <1% , meaning a minor loss in Isp. 1.3. Methodology This report comprises five chapters: — Chapter 1Introduction gives an overview of the project by defining the objectives and limitations of the study. — Chapter 2Literature Review lays out the information pertinent to compiling the codes for this thesis. — Chapter 3Method presents the steps taken to develop the two-dimensional and three-dimensional codes and resulting nozzles. — Results and numerical analyses are presented in Chapter 4Results and Discussion. — Chapter 5Conclusions provides the conclusion of the study and recommendations for further study. Appendices follow the report and present detailed results and explanations relevant to the report’s content. 5 2. Literature Review 2.1. Nozzles According to fable, the principles of rocketry were first tested more than 2,000 years ago. Ancient Greek stories tell of Archytas, a philosopher and mathematician, who demonstrated a wooden pigeon that flew around, on wires, by propulsive steam. Heron of Alexandria, believed to have created the first steam engine and the first coin operated vending machine, used a sphere, atop boiling water, with a number of holes in it to create thrust. The steam entered the sphere and was forced through two L-shaped exhaust pipes to make the ball rotate. Historians believe the first modern rockets were created by the Chinese in the first century A.D. and they were akin to modern fireworks. Until the modern era, rockets were largely used for military purposes. Three men are credited as the “fathers of rocketry”. Konstantin E. Tsiolkovsky, Robert Goddard and Hermann Oberth are credited with having defining roles in modern rocketry and spaceflight. Tsiolkovsky published his famous rocket equation in 1903, revolutionising aerospace. Goddard is known for his role in the first liquid-fuelled rocket, while Oberth is the man who created the German V-2rocket. The V-2missile became the first man-made object to cross the K ´ arm ´ an Line into space. Oberth is the only one of the three who lived to see rockets being used for space exploration. Sputnik 1was the first artificial satellite launched into space. The Soviet Union launched Sputnik in 1957 before launching Yuri Gagarin into space, making him the first human to orbit Earth, four years later. The rapid innovation brought on by the space race between the Soviet Union and the United States came to a dramatic climax on July 20,1969 when Neil 6 2. Literature Review Armstrong and Buzz Aldrin became the first people to walk on the moon. The space race ended in the 1970s and aerospace innovation stagnated. Recently there has been a significant resurgence of the space industry thanks to the popularisation and success of private space corporations. Each one of these events endeavoured to maximise thrust and specific impulse through propellant and nozzle design. The aerospace community is continuously trying to develop better nozzles in order to convert the maximum amount of energy from the propellant reaction, to kinetic energy in the engine’s nozzle. 2.1.1. Nozzle Principles As explained in Chapter 1, the modern rocket engine ignites the propellant in the chamber to create pressure. The only escape for the particles is through the nozzle.From the chamber, the nozzle converges to accelerate the particles to sonic velocity. This constricted part of the nozzle, with the smallest cross-section, is known as the throat. The nozzle then expands to accelerate the flow to supersonic speeds, creating thrust. It is the expanding section which is the main focus of modern rocket propulsion studies and this study will focus on. The nozzle’s area ratio, the ration of the exit area to the throat area, determines the exit pressure of the exhaust gas. An idealised nozzle expands exhaust gases, isentropically, to the same pressure as the ambient pressure beyond the nozzle exit. The issue for engineers when designing nozzles, is that ambient pressure is a function of altitude. Conventional nozzles are designed to perform optimally at a single mid-range altitude and have one single, uninterrupted contour. At low altitudes, conventional nozzles are over-expanded and under-expanded at high altitudes (Figure 2.1). A solution to this issue, common in modern spaceflights, is the use of two-stage rockets. Vast amounts of propellant are needed to hoist the huge masses of rockets and their cargo from the Earth’s surface. As the rockets reach high altitudes and low pressures, the first stage detaches, shedding the massive fuel tanks and the engines designed for atmospheric use. The second stage engines then start-up, making use of nozzles specifically 7 2. Literature Review Figure 2.1.: Simplified depictions of exhaust gases from three different rocket nozzles [3]. designed to be used in the vacuum of space. Of course, the problem of optimal design altitudes still exists for first stage engines and there are many solutions offered by the engineering community. Double-bell nozzles, aerospike nozzles and many more offer designs which can adjust their shape or are designed to operate at more than one optimal altitude. However, the Bell nozzle has been ubiquitous in rocket design from the V-2until the SpaceX Falcon Heavy. 2.1.2. Nozzle Design Two primary categories of nozzles exist, Conical Contour Nozzles and Thrust Optimised Contour (TOC) nozzles. Conical nozzles are self explanatory in that they are a cone shape defined by their half angle. Bell shape 8 2. Literature Review (De Laval) nozzles provide faster expansion of the exhaust gases, allowing the desired velocity to be reached in a shorter distance, making the nozzles lighter and shorter than conical nozzles. Bell nozzles entail more manufacturing complexity than conical nozzles. The Method of Characteristics (MOC) is used to design TOC nozzles. The streamlines of the expanding gas are calculated to produce a bell shaped nozzle (Figure 2.2). In the nozzle, the streamlines follow the contours of the nozzle walls, expanding through small expansion waves. Figure 2.2.: Method of Characteristics nozzle designs [8]. MOC designed nozzles make use of the uniform conditions at the throat and at the exit of a minimum length nozzle. The nozzle throat is uniform at Mach singularity and the nozzle exit velocity is imposed by the design dependent on the desired engine thrust. As the bell shape of the nozzle decreases in area, small compression waves may propagate within the flow, presenting as mild shocks. The idealised MOC cancels such shocks, thus minimising energy loss [3]. Minimum length nozzles are ones which end at the point when uniform exit velocity has been met. However, these nozzles are often too long, and thus too heavy for most spaceflight applications. Truncated Ideal Contour (TIC) nozzles are compressed versions of TOC nozzles. Two methods of creating TIC nozzles exist. The first is to specify the length as a fraction of a conical nozzle with a half-angle of 15 ◦ and then create a contour. The second option is to design a minimum length 9 2. Literature Review Bell nozzle and truncate the design using a Rao parabolic approximation. A Thrust Optimised Parabolic (TOP) nozzle is created using the second method and manages to increase the thrust potential of a rocket while reducing the length [3]. 2.1.3. Thermodynamic Relations Mathematical tools for determining rocket engine design parameters are furnished by thermodynamic relations. Such tools are used to evaluate and compare the performance of various rocket systems. Nozzle size and shape can be determined for any system that uses the expansion of gases. Conservation of energy can be applied to the adiabatic process inside a nozzle. As the processes this report considers do not involve shocks or friction, the change in entropy is zero. The work performed by the gas at a velocity v plus the internal energy constitutes the enthalpy in the process. From these assumptions and the expression for the enthalpy of an ideal gas, we get the following equation for stagnation enthalpy per unit mass. h0=h+v2/2J=constant (2.1) It is also important to note the continuity equation of the principle of conservation of mass, which states the mass flowrate at any section x is equal to the mass flowrate at any other section y for steady flow with a single inlet and outlet, such as a nozzle. The mass flowrate can be written in terms of the cross-sectional area A, the velocity vand the volume V. ˙ mx=˙ my=Av/V(2.2) The ideal gas law will also be important to note going forward. PV =nRT (2.3) 10 2. Literature Review For isentropic flow, the following relations hold between any two points x and y , where γ is the constant ratio between the specific heat capacity cp and the specific heat at constant volume cv. Tx/Ty= (Px/Py)(γ−1)/γ= (Vy/Vx)γ−1(2.4) The area ratio for a nozzle with isentropic flow can eventually be expressed in terms of the Mach number at any to given points within the nozzle. Ay Ax =Mx Myv u u u t 1+M2 y(γ−1)/2 1+M2 x(γ−1)/2!(γ+1)/(γ−1) (2.5) Figure 2.3demonstrates the relationship between area ratio, temperature ratio and pressure ratio as a function of Mach number. Figure 2.3shows that the contraction in the chamber may be as small as 3to 5times the throat area. The effect of the nozzle’s rapid expansion is evident then as pressure and temperature drop while Mach increases. The other remarkable aspect of the graph is the minor influence of the constant γ, represented as kin the image. 2.1.4. Istentropic Nozzle Flow As demonstrated in Figure 2.3, the gas pressure and temperature drop dramatically in the nozzle to create the supersonic speeds desired. This process is reversible, “essentially isentropic” [3]. From Equation 2.1, the following expression for the nozzle exit velocity v2can be found. v2=q2J(h1−h2) + v2 1(2.6) With constant γEquation 2.6can be rewritten to give: 11 2. Literature Review Figure 2.3.: Relationship between area ratio, temperature ratio and pressure ratio as a function of Mach number in a bell shape nozzle. [3] v2=v u u u t 2γ γ−1RT1"1− P2 P1!(γ−1)/γ#+v2 1(2.7) v1 refers to the velocity of the chamber while v2 refers to a point in the nozzle. Generally v1 is comparatively small and can be neglected to give an equation frequently used in nozzle analysis. 12 2. Literature Review v2=v u u u t 2γ γ−1RT1"1− P2 P1!(γ−1)/γ#(2.8) 2.1.5. Thrust and Thrust Coefficient Thrust is generated by the reaction force on the rocket structure which comes from the propellant momentum flux. With supersonic flow, the pressure at the nozzle exit plane may be different to the ambient pressure and the pressure thrust component adds to the momentum thrust: F=˙ mv2+ (P2−P3)A2(2.9) It is clear from Equation 2.9the maximum thrust is obtained when the nozzle is operating in a vacuum and P3=0 . Below the vacuum of space, Equation 2.9gives the variation of thrust with altitude. The thrust coefficient is obtained by dividing the thrust by the chamber pressure (P1) and the throat area (At). The thrust coefficient (CF) is then: CF=v u u u t 2γ2 γ−12 γ+1(γ+1)/(γ−1)"1− P2 P1!(γ−1)/γ#+P2−P3 P1 A2 At (2.10) 2.2. Real Nozzles In reality, nozzle flow is two-dimensional but axisymmetric. Temperatures and velocities are not uniform across the section of a simple bell shape nozzle and are usually higher in the centre. For example, in an ideal nozzle, the surface where the Mach number is 1, is a plane but in a real nozzle, the surface is slightly curved downstream of the throat. The average velocity v2 13 2. Literature Review for an axisymmetric nozzle can be determined as a function of the radius r. (v2)average =2π A2Zr2 0v2r dr (2.11) The assumptions related to ideal nozzles listed in Section 1.1.1of this report are approximations that allow relatively simple algorithms and calculations for analysing real rockets and the associated phenomena. Compared to the ideal nozzle, real nozzles have energy losses and not all of the chamber energy can be converted to kinetic energy. [3] provides a list of the following 10 losses: 1. In conical nozzles, the divergence of the flow leaving the nozzle causes a loss which can be reduced for bell shaped contours. 2. Pressure losses in the chamber, as a result of small areas compared to the throat area (nozzle contraction ratios), cause pressure losses in the chamber, marginally reducing the thrust and exhaust velocity. 3. The effect of the boundary layer may reduce the flow velocity, reducing the effective exhaust velocity by 0.5% to 1.5%. 4. As discussed in Section 2.2.2, solid or liquid particles in the gas may result in losses of up to 5%. 5. 0.5% losses can be induced by chemical reactions in nozzle flow, changing the gas properties. 6. Transient operations endure lower performance. 7. During operation, the throat diameter may be eroded, decreasing the nozzle expansion ratio. Decrease in thrust will be proportional to the decrease in area ratio. 8. Non-uniform gas composition may result in incomplete mixing, turbulence or incomplete combustion regions. 9. The value of γ may not remain perfectly constant and could have an effect of losing between 0.2% and 0.7% of thrust. 10. Thrust losses for fixed expansion nozzles may be up to 15% during a portion of the flight, compared to a nozzle with altitude compensation. 14 2. Literature Review following rearrangement making use of standard trigonometric identities [6]. dy dx!I =tan(θ−µ); dy dx!II =tan(θ+µ)(2.15) In conjunction with Figure 2.8, Equation 2.15 shows that the two physical characteristics, I− and I+ , are inclined at the Mach angle µ , to velocity vector ~ v. Figure 2.8.: Geometry of characteristic lines. Aligning the velocity vector parallel to the x-axis, we obtain the values u=V,v=0, M=V/a,θ=0; Equation A.7becomes: dy dx!I+,I− =±qV2 a2−1 1−V2 a2 =∓1 √M2−1=∓tan µ(2.16) 21 2. Literature Review Thus, it is clear that the characteristic I+ is inclined at the Mach angle µ below the velocity vector and the physical characteristic I− is inclined at µ above the velocity vector. Numerical Methods. As we have seen the Mach angle is defined as µ=sin−11 M The Prandtl-Meyer function ωis a function of the Mach number. dω(M) = pM2−1du u(2.17) Appendix B contains the derivation of the Prandtl-Meyer function which is found to be: ω(M) = Karctan √M2−1 K−arctan pM2−1 (2.18) Where Kis a constant defined by: K=sγ+1 γ−1(2.19) The inclination of the characteristics are then defined by the following equations where m+and m−are lengths along the characteristics. ∂ ∂m+(θ+ω) = sin µsin θ r(2.20) ∂ ∂m−(θ−ω) = −sin µsin θ r(2.21) For two-dimensional calculations, r→∞ . Therefore, the following constant relationships hold. 22 2. Literature Review θ+ω=constant along m+≡I+(inclined θ−µ) θ−ω=constant along m−≡I−(inclined θ+µ)(2.22) 2.4. Truncated Nozzles The length of nozzle expansion sections, designed by the MOC for given design Mach numbers, is minimum. The straightening section is responsible for obtaining uniform exit conditions. In order to reduce the nozzle losses, the straightening section controls the interaction of the exhaust gases with the surrounding atmospheric fluid. Thus, the efficiency of the nozzle depends on the straightening section. 2.4.1. Truncated Ideal Contour nozzles (TIC) The ideal nozzle produced by MOC codes is very long and consequently very heavy. The reason for the long length is to obtain uniform exit velocity, producing a one-dimensional exhaust profile. As the slope of the nozzle contour is very small towards the end of the nozzle, the contribution is negligible. To reduce the weight and external drag of the nozzle, the contour can be truncated, producing a TIC nozzle. [1] proposed a graphical technique for optimising nozzle contours. A set of ideal nozzle contours is plotted with lines representing the dimensionless contours along with lines showing constant surface area, exit diameter and thrust coefficient. Figure 2.9shows a set of MOC contours plotted with thrust coefficient. In the figure, lines of constant length, area and radius are included and the thrust coefficient curve is exaggerated for illustrative purposes. Point A shows the largest truncation and represents a nozzle of maximum performance. Point B is found when the constant surface area is tangent to the thrust coefficient and represents the optimum nozzle length for a given surface area. To obtain a nozzle of maximum thrust for a given expansion ratio, point C is the point where the thrust coefficient is tangent to a line of constant radius. 23 2. Literature Review Figure 2.9.: Illustration of nozzle truncation constraints. 2.4.2. Compressed Truncated Ideal Contoured Nozzles (CTIC) As the name states, CTIC nozzles are compressed versions of TIC nozzles. For the same flow profile, it was proposed by Gogish [9] in 1966 that CTIC nozzles would have higher performance than a Rao nozzle of the same flow profile. The procedure to axially compress a TIC nozzle yields more rapid initial expansion, followed by a more severe turn back than the TIC nozzle. The compressed contour results in the propagation of strong compression waves into the flow field. With strong enough compressions, the characteristic lines will converge and create an oblique shock wave, running from left to right. Oblique shocks are often visible in the exhaust jet of jet engines and rocket engines, presenting themselves as Mach/shock diamonds before being dissipated by mixing (see Figure 2.10). 24 2. Literature Review Figure 2.10.: Mach diamonds in a nozzle jet stream. [10] The static pressure will then increase as the flow crosses the shock wave. Inducing a shock near the nozzle wall will increase the pressure on the wall, thus increasing the thrust produced by the nozzle. Gogish considered this method in his proposal but later studies, namely that of Hoffman [4], showed the CTIC nozzle was not more efficient than the Rao nozzle. However, the performance differential was very small for some designs indicating that an optimal CTIC nozzle can be a viable design for certain applications. 2.4.3. Thrust Optimised Contoured Nozzles (TOC) Nozzle contours can be designed using the calculus of variations. A complex method of designing the optimum nozzle contour and exit area for given length and ambient pressure was developed in Germany during the late 1950s. The method was then simplified by Rao before being commonly adapted. As a result, in the west, the resultant nozzle is commonly referred to as a Rao nozzle. To produce a TOC nozzle the procedure is as follows. For given throat curvature rtd and a variety of θN (see Figure 2.11). Given design parameters the points P and N can be found by satisfying the following conditions: 1. Mass flow across PE is equal to the mass flow across NP. 2. The nozzle gives maximum thrust. 25 2. Literature Review Figure 2.11.: TOC nozzle contour [5]. With N and P known, the expansion region TNKO is fixed and the contour line is constructed by selecting points P’, P”, etc along the kernel region NK. A series of control surfaces P’E’, P”E”, etc. can then be generated to compute E’, E”, etc. along the nozzle contour. The TOC nozzle produced by [5] had the same performance as the TIC nozzle produced in the same work, for the same initial conditions and gas properties. The method produces a shock free flow in the upper region NPE. Thus, the effect desired by Gogish in the CTIC nozzle, returning increased wall pressure, is not a feature of the TOC nozzle. By definition, an ideal nozzle is produced when point P is located at point K. Right-running shocks are produced downstream of point P, when P 6= K. This is because a more extreme flow turning is induced and the compression waves will coalesce into a shock wave. 2.4.4. Simply Truncated Nozzles The TIC and TOC nozzles are similar in shape. However, the TOC nozzle has a greater initial expansion and a more severe turn compared to the 26 2. Literature Review TIC nozzle. This means, for the TOC nozzle, a greater wall angle and Mach number downstream of the throat but lower wall angles and Mach number at the nozzle exit, relative to the TIC nozzle (see Figure 2.12). While the performance is unaffected, the difference in flow structure affects the separation and side-load characteristics [5]. Figure 2.12.: Comparison of TIC (Ideal Truncated) and TOC (Rao-Shmyglevsky) nozzle flow structures [5]. 27 3. Methodology 3.1. Nozzle Design As previously discussed, to accelerate flow to supersonic speeds, convergentdivergent nozzles are used. This study specifically deals with the divergent section, beginning at the throat. One-dimensional analyses predict the flow properties as a function of x the distance along the nozzle, through a nozzle of given shape. The one-dimensional analyses predict flow properties as an average for a nozzle cross-section and therefore, cannot deduce wall contours. 3.1.1. Two-Dimensional Method of Characteristics The converging subsonic flow is multidimensional and therefore slightly curved. However, for simplicity, the sonic line is assumed to be straight. Also, due to the symmetry of the flow about the nozzle’s axis, the contour need only be calculated for one side of the nozzle. This symmetry applies to both twoand three-dimensional calculations. For two dimensional analyses the relationships defined as constant in Equation 2.22 holds true along characteristic lines and the constants are referred to as invariants. Conic Nozzle Design In order to gain a better understanding of the MOC and the associated theory, a two-dimensional code to produce the profile of a conic nozzle 28 3. Methodology was created. The algorithm to compute the characteristics in a conic nozzle forms the basis of the algorithms for twoand three-dimensional Bell nozzle designs with the MOC. The methodology implemented is as follows. Initial conditions and gas properties are provided which include the half angle of the nozzle, the number of characteristic lines, the sonic throat conditions and the adiabatic coefficient of the gas. A unit throat radius has been employed in each analysis to non-dimensionalise the resulting radii. From the initial conditions, the invariants ( I+ and I− ) at the first point along the flow axis can be determined by determining omega, the Prandtl-Meyer function. Prandtl-Meyer expansion allows the computation of the Prandtl-Meyer function at the intersection of the first characteristic line and the nozzle wall, giving the invariants at that point. With the first and last invariants I+ 1 and I+ N known, it is assumed that the increase in invariant values I+ is linear and thus all invariants I+ i can be calculated in the kernel zone (see Figure 3.1). With the knowledge that θ=0 at the axis, the invariants I− are simply calculated as I− i=−I+ i at each characteristics’ intersection with the axis. Figure 3.1.: Diagram of the zones related to MOC nozzle design [2]. As the value of θ at the wall is imposed, ω can be obtained as simply ωi=θ−I− i . The positive invariants I+ in the transition zone are then determined from definitions. The invariants throughout the flow field are 29 3. Methodology then known and the remaining values of θ , ω , Mach number and µ can easily be found. The final step is to determine the position of each point in Cartesian coordinates. The inclination of the invariants is defined in Equation 2.22. Thus, the position of the nodes can be found by simple trigonometry. The length of the nozzle is the minimum which provides uniform exit velocity. This condition is met when the final characteristic intersects the wall of the nozzle, line BC in Figure 3.1. An example of the results for the simple conic nozzle design is displayed in Figure 3.2 Figure 3.2.: Conic nozzle with 20◦half angle and 20 characteristic lines. 30 4. Results & Discussion Figure 4.1.: Numerical output for Mach 3and 10 characteristic lines. Figure 4.2.: Numerical output for Mach 3and 30 characteristic lines. 37 4. Results & Discussion Figure 4.3.: Numerical output for Mach 3and 50 characteristic lines. Figure 4.4.: Numerical output for Mach 3and 75 characteristic lines. 38 4. Results & Discussion Figure 4.5.: Numerical output for Mach 3and 100 characteristic lines. Figure 4.6.: Numerical output for Mach 3and 200 characteristic lines. 39 4. Results & Discussion As Mach number increases, the initial expansion becomes more severe. More rapid expansions lead to longer straightening sections to obtain uniform exit conditions. This can be observed in Figures 4.7through 4.10. As the figures show, a greater fraction of the nozzle length is occupied by the straightening section as the Mach number increases while a greater fraction of the nozzle radius is occupied by the expansion section. Figure 4.7.: Numerical output for Mach 2. 40 4. Results & Discussion Figure 4.8.: Numerical output for Mach 3. Figure 4.9.: Numerical output for Mach 4. 41 4. Results & Discussion Figure 4.10.: Numerical output for Mach 5. These relationships lead to the exponential increase between Mach number and nozzle length depicted by the graph in Figure 4.11. 42 4. Results & Discussion Figure 4.11.: Relationship between Mach number and nozzle length. Further validating the value of increased mesh density, the error, as calculated with respect to propellant mass fraction, decreases with the increase in number of characteristic lines (see Figure 4.12). 43 4. Results & Discussion Figure 4.12.: Error in propellant mass fraction calculations versus number of characteristic lines. 44 4. Results & Discussion 4.1.2. Truncated Ideal Contour As explained in Section 2.4, having plotted the nozzle contours with curves of constant thrust coefficient, the nozzles can be truncated. For the ideal two-dimensional contours obtained, this plot is shown in Figure 4.13. Figure 4.13.: Zoomed ideal nozzle contours with lines of constant thrust coefficient. The plot is then used to obtain a TIC nozzle of thrust coefficient 1.65 based on the ideal contour nozzle produced for design ME=3 . Truncating the nozzle at the inflection point of the thrust coefficient curve produces a TIC nozzle of length x/rt≈13. 45 4. Results & Discussion Figure 4.14.: Characteristic lines for a truncated ideal contour (TIC) nozzle obtained by truncating the ideal contour nozzle presented in Figure 4.8. The TIC nozzle produced, outputs a thrust coefficient of CF=1.64 and the exit Mach number at the wall is ME=2.72 rather than 3, as a result of the shortened straightening section. The lowest Mach number at the exit is ME=2.47. The dimensionless surface area of the ideal nozzle was 442.96 . The surface of the TIC nozzle is 231.52 , a reduction in surface area of 47.73% . The nozzle length has reduced from x/rt=23.4 to x/rt=13.4 , a reduction in length of 42.74%. 46 4. Results & Discussion 4.2.2. Truncated Ideal Contour As with the two-dimensional contours the ideal nozzle contours are plotted with lines of constant thrust coefficient, to truncate the ideal nozzle produced by the code. Figure 4.24.: Zoomed ideal nozzle contours with lines of constant thrust coefficient. Truncating the ideal nozzle of design ME=3 at the inflection of CF=1.6 produces the TIC nozzle shown in Figure 4.25. 53 4. Results & Discussion Figure 4.25.: Characteristic lines for a truncated ideal contour (TIC) nozzle, in three dimensions, obtained by truncating the ideal contour nozzle presented in Figure 4.22. The thrust coefficient at the exit of the TIC nozzle in Figure 4.25 is found to be CF=1.58 , compared to CF=1.65 at the exit of the ideal contour nozzle. The truncated nozzle represents a shortening of almost 64% , from x/rt≈9.85 to x/rt≈3.52 . The Mach number suffers a reduction from uniform exit ME=3.00 to ME=2.52 at the wall of the TIC nozzle. The dimensionless surface area of the ideal contour nozzle was 73.24 and the surface area of the TIC nozzle is 19.07 . The total surface area of the nozzle has reduced by 74% by truncating the nozzle. 54 4. Results & Discussion 4.2.3. Compressed Truncated Ideal Contour Making use of Equation 3.1with k=0.75 , the TIC nozzle obtained was compressed to produce the CTIC nozzle displayed against the ideal contour and the TIC in Figure 4.26. Figure 4.26.: Representation of the wall contours for an ideal contour nozzle, a TIC nozzle and a CTIC nozzle based on the ideal contour for uniform exit velocity ME=3. The three-dimensional CTIC nozzle is of length x/rt≈2.77. 55 4. Results & Discussion 4.3. Nozzle Parameters Table 4.1shows the axial length of the six contours produced for ME=3 . The difference in length between the two-dimensional nozzles and the three-dimensional nozzles is significant. On average, the three-dimensional nozzles are 32.13% of the length of the two-dimensional nozzles. Length (x/rt)2D3D Ideal 23.4 9.753 TIC 13.41 3.52 CTIC 12.66 2.77 Table 4.1.: Lengths of the six contours generated for ME=3. The surface area of the nozzles is unit-less as it is calculated in terms of the non-dimensionalised axial length and radius, x/rt and r/rt respectively. The surface areas are relevant as they relate to the mass of the nozzle, although not directly as the nozzle thickness is usually not uniform along the nozzle length. For each of the relevant contours, the surface area is shown in Table 4.2. Surface Area 2D3D Ideal 187.6 75.24 TIC 100.93 22.66 CTIC 72.47 18.79 Table 4.2.: Surface Area of each of the six nozzles defined by the contours generated for ME=3. As the MOC theory collapses for the CTIC nozzles produced, more complex analysis mechanisms are required to compute the flow. Computational Fluid Dynamics models using transient hybrid solvers can be utilised to assess the performance of the CTIC nozzles as well as observing the effect of the oblique shocks propagated and their locations within the nozzle. Unfortunately, such analyses are outside the scope of this work. According to the theory, it is expected that the CTIC nozzles presented here would produce 56 4. Results & Discussion a Mach number greater than the corresponding TIC nozzle, with lesser surface area. The external drag may not be significantly reduced despite the reduction in surface area as the angle at which the surface intersects the external flow is increased. However, the external drag represents a minor contribution to overall nozzle performance. Consequently, the CTIC nozzles included previously will be disregarded hence forward. Mach Number 2D3D Ideal 3.00 3.00 TIC 2.72 2.52 Table 4.3.: Exit Mach number of ideal and TIC nozzles defined by the contours generated for ME=3. The nozzles presented here have been truncated with reference to arbitrarily chosen thrust coefficients, merely as an example to use for analysis, as they have no intended application. In actual design processes, nozzles are generally truncated in accordance with design length or area ratios imposed. Table 4.3details the change in Mach number as a result of the truncation of the ideal contour nozzles. There is a more significant compromise in ME for the axisymmetric nozzle compared to the two-dimensional, wedge nozzle. This is a product of the fact that the ideal contour of the axisymmetric nozzle is significantly shorter than that of the ideal wedge nozzle. To understand the effect of truncating the nozzles more comprehensively, it is worth assessing the change in CF. Thrust Coefficient 2D3D Ideal 1.66 1.65 TIC 1.64 1.58 Table 4.4.: Thrust coefficient at the wall exit of ideal and TIC nozzles defined by the contours generated for ME=3. As with the Mach number, the three-dimensional nozzle suffered greater losses in truncation. This is to be expected as the portion of the length 57 4. Results & Discussion truncated for the axisymmetric nozzle was greater than the fraction of the two-dimensional contour truncated in creating the TIC nozzle. It is important to note that the losses as a result of truncating the ideal contour nozzles is much less than the amount of length and surface area sacrificed. The codes created can be used to analyse the effect of many variables considered when designing a nozzle, whether it is for a wind tunnel, a rocket nozzle or a supersonic aircraft. Using a model for an axisymmetric nozzle with ME=3, the effect of variation in γwas examined. Figure 4.27.: Maximum nozzle height relative to the throat radius, as a function of γ . Results obtained for three-dimensional calculations with ME=3 imposed. Figures 4.27 and 4.28 show an inverse relationship between length and height, as γ increases. While the nozzle’s area ratio decreases, the length 58 4. Results & Discussion increases with γ , meaning the nozzle becomes longer and narrower, with a slower expansion with higher values of γ . This is a result of the gas expanding less rapidly as the specific heat of the gas at constant pressure increases more than the specific heat at constant volume. Figure 4.28.: Nozzle length as a function of γ . Results obtained for three-dimensional calculations with ME=3 imposed. As mentioned in the theory section of this report, the MOC breaks down at hypersonic exit velocities. The error between the thrust coefficient CF calculated for the MOC nozzle and the expected CF calculated with the one-dimensional model begins to increase rapidly when the velocity crosses Mach 5. The increased error is displayed in Table 4.5. 59 4. Results & Discussion MEError (%) 5 5.37 6 10.76 7 17.6 8 25.91 9 35.61 Table 4.5.: Error in Thrust Coefficient for two-dimensional model with 50 characteristic lines. The two dimensional model was used with fifty characteristic lines to assess the hypersonic error as the axisymmetric model cannot compute contours for ME≥8 . At ME≈8 , the code does not solve and at velocities greater than Mach 8, the code returns an error in iterating the value of theta at the nozzle wall. 60 5. Conclusions 5.1. Summary The Method of Characteristics is well defined, having been the preferred method of defining supersonic nozzle contours since the 1960s. In this report, the relevant theory and methods have been outlined before detailing the results obtained for the MOC calculations carried out. It has been explained, that the codes were validated and the methods by which the validations were done. Having solved the system of PDEs to produce nozzle contours, the results have shown the relationships between nozzle design features and the variables involved in the calculations, which are not clear from examination of the equations. The capabilities of the method of characteristics have also been investigated, leading to the confirmation that the method breaks down above and below the supersonic range of exit velocities. As a greater fraction of the flow becomes hypersonic/subsonic, the calculation error increases. After producing ideal contours, the nozzles designed by the programmes developed, were truncated in accordance with the theory laid out. The truncated nozzles were then compressed. However, the compressed truncated nozzles could not be analysed with the same methods as the ideal and simply truncated nozzles. It would have been interesting to investigate the effect of varying the compression factor in the equation proposed. However, as mentioned in the Introduction of this report, a significant constraint placed on the project was the time available to complete it. 61 5. Conclusions 5.2. Recommendations for Future Work The method of characteristics for ideal nozzle design is well understood. However, there are conflicting reports on the validity of CTIC nozzles. The original proposal by Gogish [9] predicted the CTIC nozzle to be superior to the Rao nozzle. This was later debunked by Hoffman [4] although he did find the CTIC nozzle to be a very close alternative to the Rao nozzle with the performance being 0.04% to 0.34% lower for the CTIC nozzle in the parametric analysis carried out. It seems that an optimal CTIC design method has yet to be defined. 62 Appendix A. Characteristic Directions Rearranging Equation A.8with the help of typical trigonometric identities, the following relationships are obtained, as mentioned in Section 2.3.3,The Physical Characteristics. dy dx!I =tan(θ−µ); dy dx!II =tan(θ+µ)(A.9) 69 Appendix B. Derivation of the Prandtl-Meyer Function 70 Appendix B. Derivation of the Prandtl-Meyer Function dω=pM2−1du u=pM2−1 dM M+1 2 dT T!(B.1) But cpT+u2 2=constant (B.2) Thus, T=T0 1+γ−1 2M2(B.3) Leading to dT t=− γ−1 22M dM 1+γ−1 2M2(B.4) dω=√M2−1 M 1− γ−1 2M2 1+γ−1 2M2!dM =√M2−1 M dM 1+γ−1 2M2(B.5) Equation B.5is then integrated with ω=0 at M=1 to give: ω=Karctan √M2−1 K−arctan pM2−1 (B.6) Where Kis a constant defined by: K=sγ+1 γ−1(B.7) 71