Lecture # 16: Review for Final Exam

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1 AerE 344 Lecture Notes Lecture # 16: Review for Final Exam Hui Hu Rye M Waldman Department of Aerospace Engineering, Iowa State University Ames, Iowa 511, U.S.A

2 Final exam First contact Tues., 9:-1: a.m. Hoover 131 Next week: Final exam on Tuesday December 13 7:3-9:3 a.m. in room Hoover 131 AERE 344 final exam specifics Open book and open class notes multiple-choice problems ( points each) 4 short answer problems related to lab design (15 points each) Exam time is 1 minutes You may use a calculator You cannot use any computer, smartphone, tablet, or network connected device Don t forget to fill out course evaluations! Scheduled final exam Tues. Dec. 13 7:3-9:3 a.m.. Hoover 131

3 Buckingham - Theorem Step 1: List all the variables that are involved in the problem. Step : Express each of the variables in terms of basic dimensions. Basic dimension: M, L,T, F Force - F=MLT -, density - =ML -3 ; or =FL -3 T. Step 3: Determine the required number of pi-terms. Number of pi-terms is equal to k-r, where k is the number of variables in the problem, r is the number of reference dimensions required to described the variables. Step 4: Select a number of repeating variables, where the number required is equal to the number of reference dimensions. Step 5: Form a pi-term by multiplying one of the non-repeating variables by the product of repeating variables, each raised to an exponent that will make the combination dimensionless. Step 6: Repeat Step 5 for each of the remaining non-repeating variables. Step 7: Check all the resulting pi terms to make sure they are dimensionless Step 8: Express the final form as a relationship among the pi-terms, and think about what it means. 1 (, 3, k r )

4 Dimensional analysis example What determines the period of a pendulum? Parameters (5): m [M] l [L] g [L T - ] t [T] θ [-] Dimensions: M, L, T (3) Number of Pi terms: 5-3 = Choose independent variables: m, l, g Create Pi terms from remaining variables t and θ [-] = t * m a l b g c = [T] * [M] a * [L] b * [L T - ] c T: = 1 c c = 1/ M: = a a = L: = b + c b = -1/ [-] = θ = [-] Therefore Pi groups are θ and t g l g m Therefore, the only possible explanation for the period of the pendulum must have a form: g t f ( ) t f ( ) l Compare to the actual period of a pendulum: l t, 1 g The mass does not affect the period, ruled out by purely dimensional arguments! l l g θ t

5 Dimensional Analysis and Similitude Commonly used nondimensional parameters: Euler Reynolds number, Eu number, p V V Mach Number, M c pressure force inertial force VL Re V Froude Number, Fr lg inertial gravity inertial force viscous force force force inertial force compressib lity force l centrifuga l force Strohal Number, Str V inertial force V l inertial force Weber Number, We surface tension force Similitude: Geometric similarity: the model has the same shape as the prototype. Kinematic similarity: the velocity ratio is a constant between all corresponding points in the flow field. Dynamic similarity: Forces which act on corresponding masses in the model flow and prototype flow are in the same ratio through out the entire flow

6 Measurement Uncertainties Error is the difference between the experimentally-determined value and its true value; therefore, as error decreases, accuracy is said to increase. Total error, U, can be considered to be composed of two components: a random (precision) component, a systematic (bias) component, We usually don t know these exactly, so we estimate them with P and B, respectively. Repeatability Precision Error U A error B A measured P A true Reproducibility E A m A true E relative A A error true Both Bias and Precision Errors Bias error True value X=1 precision error measured value X=11 X

7 Measurement Uncertainties Uncertainty in derived quantities: U R B R P R Total uncertainties in multivariate are computed like a vector norm R R X X, X,, R R R dr dx dx dx X X X 1 1 J 1, 3 J X J S B R J i1 R X i B i ; P R J i1 R X i Uncertainties in a single measurement can be estimated from the standard deviation of the ensemble measurements, S. Pi Mean X X P S i i 1 N N 1 1 Si X i X i ; X i X i k k N k1 N k1

8 Measurement uncertainty example Suppose a measurement has a true value of.5 and a measurement uncertainty of +/-.1. >> x =.5 +.1*randn([1,5]); Compute the standard deviation of the data >> std(x).93 Compute the mean of the first n samples >> n = 1:length(x); >> mean_n_x = cumsum(x)./ n; Compare the error to the standard deviation divided by the sqrt(number of samples) for the first n samples >> for i = 1:length(x), >> er(i) = mean_n_x(i)-mean(x); >> u(i) = std(x(1:i))/sqrt(n(i)); >> end

9 Components of a Wind Tunnel Test section Contraction section Diffuser section Setting chamber Screens and similar structures Cooling system / radiators Motors /fans

10 Function of Contraction V a A a Mass cons: V A a a b b V b V A cv a ΔV a Bernoulli: 1 px Vx const 1 1 pa Va pb Vb 1 pb pa Va ( c 1) c A A a A b b ΔV b =? ΔV b At the perturbation: pa ( Va Va ) pa Va ( c 1) ( Vb Vb ) 1 1 ( c Va VaVa ) ( Vb VbVb ) c V V V c V cv V a a a a a b V V cv V a a a b V V cv V 1 V V a a a b a b cva Vb c Va Vb

11 Pressure Measurement Techniques Deadweight gauges: Elastic-element gauges: Electrical Pressure transducers: Wall Pressure measurements Remote connection Cavity mounting Flush mounting Pressure Measurements inside Flow Field:

12 Velocity measurement techniques Pitot Static Probe Advantages: Simple Cheap Disadvantages: Averaged velocity only (slow response) Single point measurements High measurement uncertainties, especially at low velocity p V p stat ( p 1 V p stat,( Bernoulli ) )

13 Velocity measurement techniques Hotwire Probe V Advantage: Flow Field High accuracy High dynamic response The rate of which heat is removed from the sensor is directly related to the velocity of the fluid flowing over the sensor Disadvantage: Single point measurements Fragile, easy to break Much more expensive compared with pitot-static probe. Requires frequent calibration. Current flow through wire dt mc dt w i R q ( V, T Constant-current anemometry Constant-temperature anemometry w w )

14 The nature of light Light as electromagnetic waves. Light as photons. Color of light Index of reflection n c / v 1 c 3x1 8 m / s

velocity in the direction of the fringe pattern Can be configured for or 3 components Better temporal

15 Laser Doppler velocimetry -component LDV Fringe spacing: D sin( / ) f sin( / ) Frequency of the scattering light : T V sin( / ) f V Frequency shift according to Doppler theory: f sin( / ) V T Measures velocity in the direction of the fringe pattern Can be configured for or 3 components Better temporal resolution than pressure transducer Non-intrusive measurement Assumes particles accurately follow local flow velocity

$intensity distribution: For weak refraction, and applying the Gladstone-Dale formula reveals a$

16 Shadowgraphy In shadowgraphy, as light rays pass through the measurement region, the deflection of the light rays as they interact with variations in the optical index lead to an intensity distribution: For weak refraction, and applying the Gladstone-Dale formula reveals a dependence on the second partial derivatives of density. Shadowgraph of a bullet (by Andrew Davidhazy )

edge: For small angles of deflection, and applying the Gladstone-Dale formula reveals a

17 Schlieren In Schlieren, as light rays pass through index variations in the measurement region, the deflection of the light rays cause them to be either blocked or avoid a knife edge: For small angles of deflection, and applying the Gladstone-Dale formula reveals a dependence on the partial derivatives of density. Shadowgraph of a bullet (by Andrew Davidhazy )

18 Visualization of a shockwaves using Schlieren technique After turning on the Supersonic jet Before turning on the Supersonic jet

19 Schlieren vs. Shadowgraphy Shadowgraphy Displays a shadow Shows light ray displacement Contrast level responds to No knife edge used n y Schlieren Displays a focused image Shows ray refraction angle, n Contrast level responds to y Knife edge used for cutoff

Y (mm) Particle Image Velocimetry (PIV) Advantage: Whole flow field measurements Non-intrusive measurements Disadvantage: Low temporal resolution Very expensive compared with

Seed fluid flows with small tracer particles (~µm), and assume the tracer particles moving with the same velocity as the low fluid flows.

20 Y (mm) Particle Image Velocimetry (PIV) Advantage: Whole flow field measurements Non-intrusive measurements Disadvantage: Low temporal resolution Very expensive compared with hotwire anemometers and pitot-static probes. Seed fluid flows with small tracer particles (~µm), and assume the tracer particles moving with the same velocity as the low fluid flows. Measure the displacements (L) of the tracer particles between known time interval (t). The local velocity of fluid flow is calculated by U= L/t. t=t L L U t t= t +t 1 8 spanwise vorticity (1/s) m/s 6 4 GA(W)-1 airfoil - -4 shadow region -6 C. Derived Velocity field A. t=t B. t=t +1 s Classic -D PIV measurement X (mm)

PIV system setup Particle tracers: Illumination system: Camera: Synchronizer: Host computer: processing. to track the fluid movement. to illuminate the flow field in the interest region.

21 PIV system setup Particle tracers: Illumination system: Camera: Synchronizer: Host computer: processing. to track the fluid movement. to illuminate the flow field in the interest region. to capture the images of the particle tracers. the control the timing of the laser illumination and camera acquisition. to store the particle images and conduct image seed flow with tracer particles Illumination system (Laser and optics) camera Synchronizer Host computer

22 Laminar Flows and Turbulent Flows Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. Viscosity determines momentum diffusion. In nonscientific terms laminar flow is "smooth," while turbulent flow is "rough." Turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. Turbulent motion dominates diffusion of momentum and other scalars. The flow is characterized by rapid variation of pressure and velocity in space and time. Flow that is not turbulent is called laminar flow

23 Laminar Flows and Turbulent Flows ' ' ; ' w w w v v v u u u... ),,, ( 1 T t t dt t z y x u T u ' ' ; ' w v u ') ( ') ( ; ') ( 1 ') ( w v dt u T u T t t o

24 Reynolds number: Re LU At the large scales, the flow is determined by the reference length scale, L, and the reference time scale, τ = L/U. The small length scales are governed by the Kolmogorov scales η = (ν 3 / ε) 1/4, and τ η = (ν / ε) 1/. Turbulence ε is the turbulent energy dissipation rate. Scaling: For a flow with Re ~ 1,: Flow scales span 3 orders of magnitude in length and orders of magnitude in time. Turbulent flows contain a vast range of length and time scales that must be resolved!!! Difficult!!

25 Boundary Layer Flow at y, u. 99U Displacement thickness: y Momentum thickness:

26 Low Reynolds number aerodynamics (e.g., MAVs) macro-scale aircraft:rec = 16 ~18 1. MAVs: ReC = 14 ~ CL CL Scale-down of conventional airfoils could not provide sufficient aerodynamic performance for MAV applications. 1. Flat Plate Profiled Airfoil Corrugated Airfoil.4.6 Flat Plate Profiled Airfoil Corrugated Airfoil Angle of Attack (Deg.) X/C* X/C*1 4 Velocity (m/s) X/C* T.K.E 1 3 X/C* T.K.E Y/C*1 Y/C*1 U C 6 5 Velocity (m/s) 1 Y/C*1 Y/C* Re 18 Spanwise vorticity (1*1/s) rough surface airfoil MAX Streamlined airfoil 1 CL CD Y/C*1 1 interest for MAV applications 1 15 Spanwise vorticity (1*1/s) Angle of Attack (Deg.) 15 Y/C*1 It is very necessary and important to establish novel airfoil shape and wing planform design paradigms for MAVs X/C* X/C*1 4 5

27 Review of Quasi-1D Nozzle Flow Assumptions: Steady-state Inviscid No body forces X Quasi-1D: Area is allowed to vary but flow variables are considered a function of x only Mass conservation V dv t U nds S Momentum conservation t UdV U n UdS pds fdv Fviscous V S S V Energy conservation t U U q e dv e U nds pu nds dv f U dv t V S S V V

28 Review of Quasi-1D Nozzle Flow da A ( M 1) du u Throat M=1 u increasing M<1 M>1 Throat M=1 u decreasing M>1 M<1

29 Review of Quasi-1D Nozzle Flow Isentropic relations (Thermodynamics) : Energy Equation : P P P C T T T P P T ( ) ( ) T 1 V T V 1 V CPT 1 1 T C T RT 1 1 M 1 1 (1 M ) (1 M ) P

30 Review of Quasi-1D Nozzle Flow M<1 M>1 Throat A* P*,T** M*=1 u*=a*=(krt*) 1/ A P,T, M u * u * A* ua u* at A* section: M 1 u* a* a* isentropic relation T 1 T ( ) ( ) ( ) ( ) ( ) ( ) ( ) A* u u a P (1 M ) 1 1 y1 P [ (1 M )] M (1 M ) 1 M A * a* * a a* A ( ) A* 1 M 1 [ (1 M 1 )] 1 y1

31 P increasing P increasing 1st, nd and 3 rd critic conditions st critic condition 1st critic condition 3st critic condition

32 Pressure Distribution Prediction within a De Laval Nozzle by using Numerical Approach Throat, A* or A t P T1 P 1 P M<1 M>1 Shock A S P T M<1 Ae Pe P atm Pe P atm Using the area ratio, the Mach number at any point up to the shock can be determined: A * A M M 1 After finding Mach number at front of shock, calculate Mach number after shock using: M 1 1 M1 1 M1 Then, calculate the A * s 1 A M A M 1 which allows us calculate the remaining Mach number distribution A * A M M 1

33 Flow visualization by using smoke wind tunnel Path line Streak lines Streamline Streamlines (experiment)

34 Wind Tunnel Calibration A E Settling chamber Contraction section V Test section p A p E p C * q T 1 C * V q =.5*V Linear curve fitting Experimental data P = P A -P E (Pa)

Pressure Sensor Calibration and Uncertainty Analysis Task #1: Pressure

with a range of +/- 5 inho It has two pressure ports: one for total

A computer data acquisition system to measure the output voltage from the

A manometer of known accuracy Mensor Digital Pressure Gage, Model 11,

Tubing to connect pressure sensors and plenum Lab output: Calibration

35 Pressure Sensor Calibration and Uncertainty Analysis Task #1: Pressure Sensor Calibration experiment A pressure sensor Setra pressure transducer with a range of +/- 5 inho It has two pressure ports: one for total pressure and one for static (or reference) pressure. A computer data acquisition system to measure the output voltage from the manometer. A manometer of known accuracy Mensor Digital Pressure Gage, Model 11, Range of +/- 1 inho A plenum and a hand pump to pressurize it. Tubing to connect pressure sensors and plenum Lab output: Calibration curve Repeatability of your results Uncertainty of your measurements tubing Setra pressure transducer (to be calibrated) Mensor Digital Pressure Gage A computer A plenum hand pump

36 cp --> Cp Measurements of Pressure Distributions around a Circular Cylinder Y P Incoming flow R Cp distribution around a cylinder P P V C p 1 1 V V ( V sin ) 1 1 4sin V 1 X EFD CFD AFD theta (rad ) --> Angle, Deg.

Airfoil Pressure Distribution Measurements.4 Lift Coefficient, C l 1.4 1. 1..8.6.

37 Airfoil Pressure Distribution Measurements.4 Lift Coefficient, C l L 1 V C l c C L = Experimental data Drag Coefficient, C d D 1 V C d c Experimental data Angle of Attack (degrees) Angle of Attack (degrees)

38 y x 8 mm Pressure rake with 41 total pressure probes (the distance between the probes d=mm) Airfoil Wake Measurements and Hotwire Anemometer Calibration y x )] ) ( (1 ) ( [ 1 )] ) ( (1 ) ( [ 1 dy U y U U y U C C C U da U y U U y U U C U D C D D

39 Hot wire measurements in the wake of an airfoil y Pressure rake with 41 total pressure probes (the distance between the probes d=mm) x 8 mm Lab#3 Lab#4 Hotwire probe

U N i1 u i / N V v i / N N i1 Turbulent velocity fluctuations: u' N i1 (

40 PIV in an airfoil wake U Experimental setup for the PIV measurements Instantaneous velocity field Mean velocity components in x, y directions: U N i1 u i / N V v i / N N i1 Turbulent velocity fluctuations: u' N i1 ( ui U) / N v' N i1 ( v i V ) Turbulent Kinetic energy distribution: TKE 1 ( u' v' )

41 Visualization of Shockwaves using Schlieren technique Underexpanded flow Flow close to 3 rd critical Overexpanded flow nd critical shock is at nozzle exit Tank with compressed air Test section 1 st critical shock is almost at the nozzle throat.

42 Set up a Schlieren and/or Shadowgraph System to Visualize a Thermal Plume Point Source Candle plume

43 Pressure Measurements in a de Laval Nozzle Tank with compressed air Test section Tap No. Distance downstream of throat (inches) Area (Sq. inches)

44 Aerodynamic forces on an icing airfoil Force transducer wing model configuration

45 Final exam First contact Tues., 9:-1: a.m. Hoover 131 Next week: Final exam on Tuesday December 13 7:3-9:3 a.m. in room Hoover 131 AERE 344 final exam specifics Open book and open class notes multiple-choice problems ( points each) 4 short answer problems related to lab design (15 points each) Exam time is 1 minutes You may use a calculator You cannot use any computer, smartphone, tablet, or network connected device Don t forget to fill out course evaluations! Scheduled final exam Tues. Dec. 13 7:3-9:3 a.m.. Hoover 131

Lecture # 16: Review for Final Exam

AerE 344 Lecture Notes Lecture # 16: Review for Final Exam Dr. Hui Hu Department of Aerospace Engineering, Iowa State University Ames, Iowa 511, U.S.A Dimensional Analysis and Similitude Commonly used