SYLLABUS FOR THE WRITTEN PRELIMINARY EXAMINATION IN THE CONCENTRATION OF AEROHYDRODYNAMICS.

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1 Final 5/18/01 SYLLABUS FOR THE WRITTEN PRELIMINARY EXAMINATION IN THE CONCENTRATION OF AEROHYDRODYNAMICS. The exam will be open book and notes. Students are also allowed to use a computer, calculator, any references or written materials as they see fit. However, students' solutions must be strictly their own. No communication of any type, implicit or explicit is allowed during the exam. The honor code will be strictly enforced. Students are required to answer four of seven questions. Questions will require the student to effectively apply their knowledge from across the area of concentration to familiar and unfamiliar situations. Questions may address a single topic, or integrate material from different areas. Questions will draw on material contained within the recommended texts listed below. 1. LIST OF RECOMMENDED TEXTS Text: Karamcheti K, 1980, Principles of Ideal Fluid Aerodynamics, 2 nd Edition, Kreiger, Malabar. Sections: All chapters except 7 and 20. Text: Grossman B., 2000, Fundamental Concepts Of Real Gasdynamics, Lecture notes version 3. Sections: All. Text: Anderson J. D., 1990, Modern Compressible Flow with Historical Perspective, Second Edition, McGraw-Hill, New York. Sections: Chapters 1-10 and , , , , Text: Bertin, J.J., and Smith, M.L., 1998, Aerodynamics for Engineers, 3 rd Edition Prentice-Hall, Englewood Cliffs. Sections: Chapters 1-7 Text: Schetz J. A., 1993, Boundary Layer Analysis, 2 nd Edition, Prentice Hall, Englewood Cliffs. Sections: Chapters 1 (except 1-8), 2 (except 2-9), 3, 4 (except 4-6 and 4-7-5), 5 (except 5-7, 5-9, 5-10, 5-12 and 5-14), 6 (except and 6-6-8), 7 (except material on suction and injection, and 7-10 through 7-15), 8 (except 8-2-3, 8-4-3, and 8-5), 10 (except 10-4 and 10-5). Text: Hill P G and Peterson C R, 1992, Mechanics and thermodynamics of propulsion, 2 nd edition, Addison-Wesley Sections: Chapters, 1, 2, 3, 4, 5, 6, , 10, 11, , 14.

2 2. SAMPLE QUESTIONS Consider a (2 D) rocket motor on a test stand whose geometry is sketched below. The rocket has a chamber pressure of p c =10 7, a chamber temperature of T c =500 K, and a nozzle area ratio of A e /A t =4. After the nozzle exit, there is a base region denoted by the line A - B which is inclined at 15 o to the axis. The external housing O - B may be considered semi-infinite and is parallel to the axis. For the purpose of this problem, you may assume that the internal and external flow is air, with γ =1.4. a) For ambient conditions of p b = N/m 2 and T b =220K determine the flow in the vicinity of A -B. i. Find the pressure along A - B. ii. Determine the initial plume angle θ i. iii. Carefully sketch the flow in this region, indicating all waves and contact surfaces. b) Determine the value of the ambient pressure p b such that the initial plume angle θ i will be zero. Sketch the flow for this condition. c) We wish to make a crude estimate of the surface temperature along A - B, including the effect of a laminar boundary layer. (You may neglect the effect of the boundary layer inside the nozzle up to point A). i. For an insulated wall, estimate the wall temperature for Prandtl number 1and 0.7. ii. If we fix the wall temperature at T w, write a relationship which can be used to estimate the Stanton number in terms of the skin friction coefficient. How can we determine the heat transfer rate from the Stanton number? A satellite is to be moved first from low Earth orbit (LEO) to a parking orbit (PO) using a 2-stage chemical rocket, and then from the parking orbit (PO) to a geosynchronous orbit (GEO) using an electric thruster. The LEO has an orbit inclination of i =28.5 o.the inclination for both the PO and the GEO is 0 o. Assume all three orbits are circular orbit. The radiuses with respect to the enter of the earth are R LEO =6678 km, R PO =11378 km, and R GEO =42378 km for LEO, PO,and GEO respectively. (Gravitational parameter of the Earth: µ= km 3 /s 2 ) a) The transfer from LEO to PO is via a V 1 at LEO, which sends the spacecraft to a Hohmann ellipse, and a V 2, which combines the plane Change with a tangential burn at the apogee of the transfer orbit to send the spacecraft into the PO. Calculate V 1 and V 2. b) The transfer from PO to GEO is a low-thrust spiral (semi-circular) orbit. Calculate V LT. c) Both stages of the chemical rocket have the same I sp =500s and the same structural coefficient =0.1. If the 3 rd stage has a total mass of 1000kg, calculate the spacecraft mass for each stage. d) The electric propulsion system has a specific mass of α =M s /P e =0.030 kg/w, a power conversion efficiency of η=0.8, and a specific impulse of I sp =5000s. If the satellite mass is 200 kg (the initial mass of the 3 rd stage is 1000kg), calculate the following for the 3 rd stage: the structure mass, propellant mass, power requirement, and total burn time. A simple 2-D vortex method can be created by placing a series of flat plates or panels along the camber line of an airfoil with each panel having a vortex of unknown strength at its quarter chord and a control point where there can be no flow perpendicular to the panel at each panel three-quarter chord. With one equation for each control

3 point one can then proceed to solve for the strengths of the quarter chord vortices for any given flow angle of attack and velocity and can find the lift coefficient for the airfoil by adding the lifts due to the vortices. Apply this method to a symmetrical airfoil with a flap of 1/3 its chord deflected to 20 degrees, using three panels, each 1/3 of the original airfoil chord. For the airfoil with deflected flap: (a) Find the lift coefficient as a function of angle of attack (dc L /dα) (b) Find the zero lift angle of attack (α L0 ). (c) Find the pitching moment coefficient at the quarter chord. (d) Find the location of the center of pressure. Turbulence is convected towards a stagnation point (see figure). The velocity vector field of the convecting flow is V = Axi -Ayj where i and j are unit vectors in the directions of the Cartesian coordinates x and y and A is a dimensional constant. The velocity fluctuations associated with the turbulence are weak enough so as not to significantly disturb this velocity field. We wish to determine the fate of turbulent eddies approaching the stagnation point. Consider a small segment of an eddy initially a distance h from the stagnation point of length δ, angle θ to the horizontal and average vorticity Ω. Using inviscid vortex theorems, determine expressions for the angle and average vorticity of the eddy as a function of time. Make clear what assumptions you are making. The camber line of a thin airfoil is given by the expression y c c x x k 1 c c 2 = 2 where x is chordwise distance, c is the chordlength and k is a constant. Estimate the lift coefficient and moment coefficient about the airfoil quarter chord at Mach numbers of 0.1, 0.6 and 2. Over what range of Mach numbers would you expect the airfoil drag coefficient to reach its maximum value, and why? (a) For a typical aerodynamic analysis, how would you idealize the geometry of the P-51 wing? What is the analysis method you would use to determine the spanload distribution? How would you determine the twist distribution to obtain a desired spanload? What are the considerations in specifying a design spanload? (b) If the taper ratio is 0.5, where does the wing stall first? An explicit expression is required. What is the ratio of the section lift coefficient to wing lift coefficient at this location? (c) Show (starting with fundamentals of the problem formulation) when and why you can neglect thickness in considering the wing analysis/design of the surface load distribution. (d) What is the lift coefficient at 437 mph and 25,000 ft. altitude? U e A e r z P-51: Span 37 ft., Wing area, 233. ft 2, Weight: 10,000 lb normal gross.

4 (a) Analyze the flow over a 2 ft. circular cylinder traveling at 100 mph at sea level. i) Write down the edge velocity distribution in a coordinate system appropriate for boundary layer calculations. ii) What is the stagnation point velocity gradient? iii) What is the best approximate method for this problem in the region of favorable pressure gradient? iv) What is the value of the momentum thickness at the stagnation point? At a point 60 deg around the cylinder from the stagnation point? v). Estimate the skin friction, displacement thickness, and shape factor for the caes in part iv. vi). What criteria will be used to determine separation? Where do you expect it to occur? (b) Suppose a circulation is imposed around the cylinder such that the stagnation point is moved 20 deg. around the surface. Reformulate the analysis in A above to incorporate the effects of circulation. What are the new values of the stagnation point velocity gradient and the momentum thickness at the stagnation point? Interpret this result. (c) Suppose this infinitely long cylinder is swept 45 deg. Explain how this changes the problem. In a so-called hotshot wind tunnel, the air in a closed reservoir is heated to a high temperature and pressure by the discharge of electrical energy (which has been stored in a bank of capacitors) across an arc gap in the reservoir. Suppose that the volume of the reservoir is 400 cm 3, the temperature of air in the reservoir before discharge is 300 K and the density in the reservoir is 50 times standard atmospheric density. (a) If 250,000 joules of energy are stored in the capacitor bank and if the effective efficiency of delivery of this energy is 80%, what are the temperature, the enthalpy and the pressure of the air in the reservoir after the discharge? (b) A nozzle is attached to the reservoir with a throat area of 1 cm 2 and an exit area of 20 cm 2. After discharge a valve between the reservoir and the nozzle is opened. Determine the velocity, pressure and temperature at the nozzle exit when the flow is supersonic. Be sure to justify whatever gas model (perfect gas, equilibrium air, etc.) that you use to solve this problem. Use of suitable software tools as needed is encouraged. A so-called Mixer Nozzle is a device used to improve the thrust of two stream (i.e., bypass ratio or turbofan) gas turbine engines. The basic concept is shown below. In the unmixed configuration (Fig. a), the engine is run without the mixer. The exhaust from the fan and the core, which have the same stagnation pressure P T but quite different stagnation temperatures are ducted separately. In the mixer configuration (Fig b), they are mixed before ducting. a) Calculate the increase in gross thrust due to this mixing. Assume that the pressure ratio P T /P exit across the unmixed and mixed nozzles are the same since the actual mixing process occurs at very low Mach number so that losses in stagnation pressure are negligible. The mixing process can thus be considered to be carried out as if it P T T T fan ṁ fan P exit idealy expanded ṁ T fan T fan P T T ṁ T core core mixing P T T T core ṁ core P exit T T mixed ṁ mixed 1 (a) 1 2 (b)

5 occurred in a large chamber. In addition, assume that the nozzles are supersonic but ideally expanded, the flow is that of a perfect gas, and that the condition at station 1 are the same for the mixer nozzle and for the unmixed situation. Write the ratio of thrust for mixed nozzle to thrust for unmixed nozzle as a function of the stagnation temperature ratio, T T core /T T fan, and the bypass ratio, m / m b) If the temperature ratio T T core /T T fan=4 and the mass flow rate ratio fan core thrust for the mixed nozzle to that for the unmixed? fan core m / m =1, what is the ratio of

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14 WRITTEN Ph.D. PRELIMINARY EXAMINATION IN THE CONCENTRATION OF AEROHYDRODYNAMICS. Department of Aerospace and Ocean Engineering February 18th, 2002 This exam is open book and notes. Students are also allowed to use a computer, calculator, any references or written materials as they see fit. However, students' solutions must be strictly their own. No communication of any type, implicit or explicit is allowed during the exam. The honor code will be strictly enforced. Students are required to answer four of seven questions. Start each question on a new sheet of paper. Write your name at the top of all answer sheets. Do not hand in solutions to more than four Questions. Complete and sign the honor code pledge below. Hand in this completed cover page with your solutions.

15 c. Consider the YB-49, shown here frm Lloyd Jones, U.S. Bombers, Aero Publishers, Y8.49 and YR8-49A FLYING WING 1.67 NORTHROP YB-49 a YRB-49A Fit "o"hmp V"'4." VRII-4.A Flyi"t WI"t- b = 172 ft S = 4000 sq. ft. CrocI = ft CliP = 9.33 ft W TO = 200,000 lbs Flight conditions: 340 mph at 35,000 ft. a. b. d What is the cruise CL? Comment on this value. What is C DO if the sole contribution is skin friction, and i. if the flow is all laminar ii. if the flow is all turbulent iii. comment on the likely state of the boundary layer at the prescribed flight condition What is the cruise UD if the flow is laminar?, turbulent? -comment Assuming turbulent flow, what is UDma.x and CL for I/Dmax. Comment.

16 Spring 2002 Aerodynamics Prelim question A two dimensional airfoil section has a zero lift angle of attack of minus 5.12 degrees. Find the following: 1. The theoretical lift coefficient of the 2-D airfoil at an angle of attack of 4 degrees. 2. The angle of attack needed to generate this same lift coefficient on ideal (elliptical lift distribution) 3-D wings with the same airfoil section and with aspect ratios of: a. b. c Show how you found the relationships needed to answer part 2 above and how they account for induced angle of attack, effective angle of attack, and the zero lift angle of attack on a 3-D wing.

17 The atmosphere on a remote planet is mostly hydrogen. Consider a probe with surfaces like a flat plate flying at Mach 3.0 in the planet's atmosphere. The static temperature is 500oK, and the static pressure is 0.1 atm. The flat plate surfaces are 30 cm long and 20 cm wide, and the plate temperature must be kept at 300oK. What is the drag and heat transfer on each plate? What would you expect to change if the static pressure were increased to 1.0 atm? Why? How would you estimate the effects quantitatively?

18 QUALIFYING EXAM QUESTION The equation of the mean camber line for an infinite span thin airfoil is given by: ~ == 2.659{ ~J S( ~J \ ~J for 0.0 ~ (x /c) ~ and ~ = \ 1- ~ x for ~ -~ 1.0. c What is the section lift coefficient C1 as a function of angle of attack? What is the zero lift angle of attack? What angle of attack is required to develop the design lift coefficient ofo.3? Calculate the section moment about the quarter chord location. When the geometric angle of attack is 3, what is the section lift coefficient? What is the X/C location of the center of pressure?

19 Consider a gaseous mixture of O2 and 0 at slightly elevated temperatures. The temperature is high enough so that vibrational energy modes are excited, but electronic excitation may be neglected. The conditions are such that we may assume that no chemical reactions occur. Initially there are Xl moles of 0 and X2 moles of O2. a) Show that p = prt for this mixture. Develop a formula for R in terms of X I, X 2, the molecular masses MI, M2, and the universal gas constant R. b) Develop a relationship for Cp for the mixture in terms of: the temperature T, the variables mentioned in part a), and any relevant characteristic temperatures for vibration. c) Develop a formula sound speed in terms of the parameters given in b). d) Given that the initial total mass of the mixture is 5.0 kg, the volume is 0.6 m3, the initial mass of O2 is 3.5 kg, and the temperature T = K, calculate: Xl, X2, il, p, Cp and a.

20 Consider a circular shock tube 0.1 m in diameter in which both driver and driven gases are air, initially at 300K. The driven section of the tube is 3m long. A pressure probe is mounted in the driver section. The probe has a stem, 5mm in diameter, that projects 5cm into the tube. The probe initially records a pressure of 500kPa. Exactly five milliseconds after the diaphragm ruptures the probe pressure starts to drop. The drop in pressure followed by a short period during which the pressure remains constant at 295kPa. Estimate, as best you can (a) The distance from the diaphragm location to the probe (b) The initial mass flow rate through the fractured diaphragm. (c) The time taken for the shock to reach the closed end of the driven section. (d) The drag on the probe stem, during the short period while it reads a pressure of 295kPa. DRIVEN SECTION DRIVER SECTION 3m 5cm DIAPHRAGM LOCATION PROBE

21 An astronaut on extravehicular activities (EV A) suddenly finds the safety tether that connects him to the space station is broken. The astronaut quickly decides to open an external valve of his air tank so to utilize gas flow from pressurized tank to vacuum to get him back to the space station (along a straight line, see attached Figure). Assume the following: at start of the operation, the total mass of astronaut plus gas is Mo (kg), the total mass of the gas inside the tank is Mgo(kg), the temperature inside the tank is Tco. The mass flow rate of air breathed by astronaut is ma (kg/s), which is then leaked out of the system (without producing any thrust). The mass flow rate of air used for propulsion purpose is m(kg/s). Both ma and m are constant during the entire operation. Consider the gas as a perfect gas, with known specific heat ratio 'Y. The initial distance between astronaut and space station is L, and there is no initial relative velocity. a) If the gas inside the tank undergoes an isothermal process, find the velocity of the astronaut as a function of time V(t). (Note: you may NOT assume the total mass is a constant.) b) Obviously, the fastest way to get back is to adjust ni such that all oxygen inside the tank is consumed just when astronaut reaches space station. Show how you may find this ni using your result from a). (Ignore the regulator pressure and residual gas in the tank). c) Now, if the gas inside the tank undergoes an isentropic process, find the velocity of the astronaut as a function of time V(t). Please note: In a) and c), the following variables are known: Mo,Mgo Tco, ', fila' m, L, universal gas constant R, and molecular mass M. Please first write down the characteristic velocity c* and thrust coefficient CP. In b), you may define constants using Mo,Mgo Tco, ', m'a, L, R, and M to simplify the expression. There is no need to calculate any numerical value.

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