SYLLABUS FOR THE WRITTEN PRELIMINARY EXAMINATION IN THECONCENTRATION OF AEROHYDRODYNAMICS.

The exam will be open book and notes. Students are also allowed to use a computer, calculator, any references orwritten materials as they see fit. However, students' solutions must be strictly their own. No communication of anytype, 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 theirknowledge from across the area of concentration to familiar and unfamiliar situations. Questions may address asingle topic, or integrate material from different areas. Questions will draw on material contained within therecommended texts listed below.

1. LIST OF RECOMMENDED TEXTS

Text: Karamcheti K, 1980, Principles of Ideal Fluid Aerodynamics, 2nd 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 11.1-11.7, 14.1-14.2, 15.1-15.4, 16.8-16.12, 17.1-17.6

Text: Bertin, J.J., and Smith, M.L., 1998, Aerodynamics for Engineers, 3rd Edition Prentice-Hall,Englewood Cliffs.

Sections: Chapters 1-7

Text: Schetz J. A., 1993, Boundary Layer Analysis, 2nd 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, 5-12and 5-14), 6 (except 6-6-6 and 6-6-8), 7 (except material on suction and injection, and 7-10through 7-15), 8 (except 8-2-3, 8-4-3, 8-4-4 and 8-5), 10 (except 10-4 and 10-5).

Text: Hill P G and Peterson C R, 1992, Mechanics and thermodynamics of propulsion, 2nd edition,Addison-Wesley

Sections: Chapters, 1, 2, 3, 4, 5, 6, 7.1-7.5, 10, 11, 12.1-12.3, 14.

Final 5/18/01

2. SAMPLE QUESTIONS

Consider a (2–D) rocket motor on a test stand whose geometry is sketched below. The rocket has a chamberpressure of pc =107, a chamber temperature of Tc =500 K , and a nozzle area ratio of Ae /At =4. After the nozzle exit,there is a base region denoted by the line A - B which is inclined at 15o to the axis. The external housing O - B maybe considered semi-infinite and is parallel to the axis. For the purpose of this problem, you may assume that theinternal and external flow is air, with γ =1.4.a) For ambient conditions of pb =4 .9 ×104 N/m2 and Tb =220K determine the flow in the vicinity of A -B.

i. Find the pressure along A - B.ii. Determine the initial plume angle θi.iii. Carefully sketch the flow in this region, indicating all waves and contact surfaces.

b) Determine the value of the ambient pressure pb such that the initial plume angle θi will be zero. Sketch the flowfor this condition.c) We wish to make a crude estimate of the surface temperature along A - B, including the effect of a laminarboundary 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 Tw , 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 Stantonnumber?

______________________________

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 anorbit inclination of i =28.5o .The inclination for both the PO and the GEO is 0o. Assume all three orbits are circularorbit. The radiuses with respect to the enter of the earth are RLEO =6678 km, RPO =11378 km, and RGEO =42378 kmfor LEO, PO,and GEO respectively. (Gravitational parameter of the Earth: µ=3 .986 ×105 km3 /s2 )

a) The transfer from LEO to PO is via a ∆V1 at LEO, which sends the spacecraft to a Hohmann ellipse, anda ∆V2 , which combines the plane Change with a tangential burn at the apogee of the transfer orbit to send thespacecraft into the PO. Calculate ∆V1 and ∆V2 .

b) The transfer from PO to GEO is a low-thrust spiral (semi-circular) orbit. Calculate ∆VLT.c) Both stages of the chemical rocket have the same Isp =500s and the same structural coefficient =0.1. If

the 3rd 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 =Ms /Pe =0 .030 kg/W, a power conversion

efficiency of η=0 .8, and a specific impulse of Isp =5000s. If the satellite mass is 200 kg (the initial mass of the 3rd

stage is 1000kg), calculate the following for the 3rd 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 anairfoil with each panel having a vortex of unknown strength at its quarter chord and a “control point” where therecan be no flow perpendicular to the panel at each panel three-quarter chord. With one equation for each control

point one can then proceed to solve for the strengths of the quarter chord vortices for any given flow angle of attackand 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 (dCL/dα)(b) Find the zero lift angle of attack (αL0).(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 -Ayjwhere i and j are unit vectors in the directions of the Cartesiancoordinates x and y and A is a dimensional constant. The velocityfluctuations associated with the turbulence are weak enough so asnot to significantly disturb this velocity field. We wish todetermine the fate of turbulent eddies approaching the stagnationpoint. Consider a small segment of an eddy initially a distance hfrom the stagnation point of length δ , angle θ to the horizontaland average vorticity Ω. Using inviscid vortex theorems,determine expressions for the angle and average vorticity of theeddy as a function of time. Make clear what assumptions you are making.______________________________

The camber line of a thin airfoil is given by the expression

−= 2

2

1cx

cxk

cyc where x is chordwise distance, c is

the chordlength and k is a constant. Estimate the lift coefficient and moment coefficient about the airfoil quarterchord at Mach numbers of 0.1, 0.6 and 2. Over what range of Mach numbers would you expect the airfoil dragcoefficient to reach its maximum value, and why?______________________________

(a) For a typical aerodynamic analysis, how wouldyou idealize the geometry of the P-51 wing? Whatis the analysis method you would use to determinethe spanload distribution? How would youdetermine the twist distribution to obtain a desiredspanload? What are the considerations inspecifying a design spanload?(b) If the taper ratio is 0.5, where does the wingstall first? An explicit expression is required. Whatis the ratio of the section lift coefficient to wing liftcoefficient at this location?(c) Show (starting with fundamentals of theproblem formulation) when and why you canneglect thickness in considering the winganalysis/design of the surface load distribution.(d) What is the lift coefficient at 437 mph and25,000 ft. altitude?

P-51: Span – 37 ft., Wing area, 233. ft2, Weight:10,000 lb normal gross.

z

r

Ue

Ae

______________________________

(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 thesurface. Reformulate the analysis in A above to incorporate the effects of circulation. What are the new valuesof the stagnation point velocity gradient and the momentum thickness at the stagnation point? Interpret thisresult.

(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 thedischarge 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 cm3, the temperature of air in the reservoir before discharge is 300 Kand 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 thedischarge?

(b) A nozzle is attached to the reservoir with a throat area of 1 cm2 and an exit area of 20 cm2. After discharge avalve between the reservoir and the nozzle is opened. Determine the velocity, pressure and temperature at thenozzle 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 ofsuitable 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) gasturbine engines. The basic concept is shown below. In the unmixed configuration (Fig. a), the engine is run withoutthe mixer. The exhaust from the fan and the “core”, which have the same stagnation pressure PT but quite differentstagnation 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 PT/Pexit across theunmixed and mixed nozzles are the same since the actual mixing process occurs at very low Mach number so thatlosses in stagnation pressure are negligible. The mixing process can thus be considered to be carried out as if it

PT

PT

idealy expanded

Pexit

Pexit

TT fan

TT core

m.

core

m.

fan

1

PT

mixing

TT core

TT fan

mixedm.

TT mixed

m.

fan

m.

core

1 2

(a) (b)

occurred in a large chamber. In addition, assume that the nozzles are supersonic but ideally expanded, the flow isthat of a perfect gas, and that the condition at station 1 are the same for the mixer nozzle and for the unmixedsituation. Write the ratio of thrust for mixed nozzle to thrust for unmixed nozzle as a function of the stagnationtemperature ratio, TT core /T T fan , and the bypass ratio, corefan mm /

b) If the temperature ratio TT core /T T fan=4 and the mass flow rate ratio corefan mm / =1, what is the ratio ofthrust for the mixed nozzle to that for the unmixed?

1

2

3

4

5

6

7

8

WRITTEN Ph.D. PRELIMINARY EXAMINATION IN THE CONCENTRATION OFAEROHYDRODYNAMICS.

Department of Aerospace and Ocean EngineeringFebruary 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 isallowed during the exam. The honor code will be strictly enforced.

Students are required to answer four of seven questions. Start each question on a newsheet of paper. Write your name at the top of all answer sheets. Do not hand in solutionsto more than four Questions. Complete and sign the honor code pledge below. Hand inthis completed cover page with your solutions.

Consider the YB-49, shown here frm Lloyd Jones, U.S. Bombers, Aero Publishers, 1974.

Y8.49 and YR8-49A FLYING WING 1.67

NORTHROP YB-49a YRB-49A

b = 172 ftS = 4000 sq. ft.CrocI = 37.33 ftCliP = 9.33 ftW TO = 200,000 lbsFlight conditions: 340 mph at 35,000 ft.

a.b.

c.

d

What is the cruise CL? Comment on this value.What is C DO if the sole contribution is skin friction, andi. if the flow is all laminarii. if the flow is all turbulentiii. comment on the likely state of the boundary layer at the prescribed flight

conditionWhat is the cruise UD if the flow is laminar?, turbulent? -commentAssuming turbulent flow, what is UDma.x and CL for I/Dmax. Comment.

Fit- 1.2. "o"hmp V"'4." VRII-4.A Flyi"t WI"t-

Aerodynamics Prelim questionSpring 2002

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.

The angle of attack needed to generate this same lift coefficient on ideal (elliptical liftdistribution) 3-D wings with the same airfoil section and with aspect ratios of:

2.

6810

a.b.c.

Show how you found the relationships needed to answer part 2 above and how they account forinduced angle of attack, effective angle of attack, and the zero lift angle of attack on a 3-D wing.

The atmosphere on a remote planet is mostly hydrogen. Consider a probe with surfaceslike a flat plate flying at Mach 3.0 in the planet's atmosphere. The static temperature is500oK, and the static pressure is 0.1 atm. The flat plate surfaces are 30 cm long and 20cm 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 wereincreased to 1.0 atm? Why? How would you estimate the effects

quantitatively?

QUALIFYING EXAM QUESTION

The equation of the mean camber line for an infinite span thin airfoil is given by:

~ == 2.659~J 3 -0.607S( ~J 2 + 0.1147\ ~J

for 0.0 ~ (x /c) ~ 0.2025 and

~ = 0.02208\ 1- ~

xfor 0.2025 ~ -~ 1.0.

cWhat is the section lift coefficient C1 as a function of angle of attack? What is the zero lift angleof attack? What angle of attack is required to develop the design lift coefficient ofO.3? Calculatethe 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?

Consider a gaseous mixture of O2 and 0 at slightly elevated temperatures. Thetemperature is high enough so that vibrational energy modes are excited, but electronicexcitation may be neglected. The conditions are such that we may assume that no chemicalreactions 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 vi-

bration.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, theinitial mass of O2 is 3.5 kg, and the temperature T = 15000 K, calculate: Xl, X2, il,

p, Cp and a.

Consider a circular shock tube 0.1 m in diameter in which both driver and driven gases areair, initially at 300K. The driven section of the tube is 3m long. A pressure probe ismounted in the driver section. The probe has a stem, 5mm in diameter, that projects 5cminto the tube. The probe initially records a pressure of 500kPa. Exactly five millisecondsafter the diaphragm ruptures the probe pressure starts to drop. The drop in pressurefollowed 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.

DRIVER SECTIONDRIVEN SECTION

5cm3m

PROBE

DIAPHRAGM LOCATION

An astronaut on extravehicular activities (EV A) suddenly finds the safety tether thatconnects him to the space station is broken. The astronaut quickly decides to open anexternal valve of his air tank so to utilize gas flow from pressurized tank to vacuum toget 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 tankis Tco. The mass flow rate of air breathed by astronaut is ma (kg/s), which is then leakedout of the system (without producing any thrust). The mass flow rate of air used forpropulsion purpose is m(kg/s). Both ma and m are constant during the entireoperation. Consider the gas as a perfect gas, with known specific heat ratio 'Y. The initialdistance 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 theastronaut as a function of time V(t). (Note: you may NOT assume the total mass is aconstant.)

b) Obviously, the fastest way to get back is to adjust ni such that all oxygen inside thetank is consumed just when astronaut reaches space station. Show how you may findthis ni using your result from a). (Ignore the regulator pressure and residual gas inthe tank).

c) Now, if the gas inside the tank undergoes an isentropic process, find the velocity ofthe 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 thecharacteristic velocity c* and thrust coefficient CP. In b), you may define constants usingMo,Mgo Tco, '¥, m'a, L, R, and M to simplify the expression. There is no need to calculate

any numerical value.