Lecture # 16: Review for Final Examhuhui/teaching/2019-08Fx/... · Dimensional Analysis and Similitude force Commonly used non-dimensional parameters: " surface tension force inertial

Copyright © by Dr. Hui Hu @ Iowa State University. All Rights Reserved!

Hui Hu

Rye M Waldman

Department of Aerospace Engineering, Iowa State University

Ames, Iowa 50011, U.S.A

Lecture # 16: Review for Final Exam

AerE 344 Lecture Notes


Dimensional Analysis and Similitude

Commonly used non-

dimensional parameters:

force tension surface

force inertial WeNumber,Weber

force inertial

force lcentrifugaStr Number, Strohal

forcelity compressib

force inertialM Number,Mach

forcegravity

force inertial

lgFr Number, Froude

force viscous

force inertialRe number, Reynolds

force inertial

force pressureEu number,Euler

2

2

lV

V

l

c

V

V

VL

V

p

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


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.

222 PBU

X

True value

X=100

Bias error

precision error

measured value

X=101

Repeatability Precision Error

Reproducibility Both Bias and Precision Errors

truemtruemeasurederror AAEAAA true

errorrelative

A

AE


Measurement Uncertainties

222

RRR PBU

Uncertainty in derived quantities:

J

i

i

i

R

J

i

i

i

R PX

RPB

X

RB

1

2

2

1

2

2 ;

Total uncertainties in multivariate are computed like a vector norm

12

2

1 1

1 1;

N N

i i i i ik kk k

S X X X XN N

JXXXXRR ,,,, 321

1 2

1 2

J

J

R R RdR dX dX dX

X X X

Uncertainties in a single measurement can be estimated from the standard deviation of the

ensemble measurements, S.

2i iP S

X

S

Mean X


Components of a Wind Tunnel

• Test section

• Contraction section

• Diffuser section

• Setting chamber

• Screens and similar structures

• Cooling system / radiators

• Motors /fans


Function of Contraction

Va

ΔVa

ΔVb =?

a

b

Ac

A

2

2 2

2 2

1

2

1 1

2 2

1( 1)

2

x x

a a b b

b a a

p V const

p V p V

p p V c

a a b b

b a

V A V A

V cV

Aa

Ab

ΔVb

Mass cons:

Bernoulli:

At the perturbation:

2 2 2 2

2

2 2

1

a a a a a b

a a a b

a a a b a b

a b a b

c V V V c V cV V

V V cV V

V V cV V V V

cV V c V V

2 2 2 2

2 2 2

1 1 1( ) ( 1) ( )

2 2 2

1 1( 2 ) ( 2 )

2 2

a a a a a b b

a a a b b b

p V V p V c V V

c V V V V V V


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:


)(2

)(,2

1

0

2

0

stat

stat

ppV

BernoulliVpp

• Advantages:

• Simple

• Cheap

• Disadvantages:

• Averaged velocity only (slow response)

• Single point measurements

• High measurement uncertainties, especially

at low velocity

Velocity measurement techniques – Pitot –Static Probe


Velocity measurement techniques – Hotwire Probe

• Advantage:

• High accuracy

• High dynamic response

• Disadvantage:

• Single point measurements

• Fragile, easy to break

• Much more expensive compared

with pitot-static probe.

• Requires frequent calibration.

Flow Field

Current flow

through wire

The rate of which heat is removed from

the sensor is directly related to the

velocity of the fluid flowing over the

sensor

V

• Constant-current anemometry

• Constant-temperature

anemometry

),(2

www TVqRi

dt

dTmc


The nature of light

• Light as electromagnetic waves.

• Light as photons.

• Color of light

• Index of reflection

1/ 0

vcn

sm /3x10c 8

0


Laser Doppler velocimetry

Fringe spacing: 2sin( / 2)

2 sin( / 2)

Frequency of the scattering light :

2sin( / 2)

Frequency shift according to Doppler theory:

2sin( / 2)

T TD f

Vf V

f V

• Measures velocity in the direction of the fringe pattern

• Can be configured for 2 or 3 components

• Better temporal resolution than pressure transducer

• Non-intrusive measurement

• Assumes particles accurately follow local flow velocity

2-component LDV


• 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.

Shadowgraphy

Shadowgraph of a bullet (by Andrew

Davidhazy )


• 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.

Schlieren

Shadowgraph of a bullet (by Andrew

Davidhazy )


Visualization of a shockwaves using Schlieren technique

Before turning on the Supersonic jet

After turning on the Supersonic jet


Schlieren vs. Shadowgraphy

Shadowgraphy

• Displays a shadow

• Shows light ray displacement

• Contrast level responds to

• No knife edge used

Schlieren

• Displays a focused image

• Shows ray refraction angle,

• Contrast level responds to

• Knife edge used for cutoff

yn

2

2

y

n


Particle Image Velocimetry (PIV)

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 .

A. t=t0 B. t=t0+10 s Classic 2-D PIV measurement

C. Derived Velocity field X (mm)

Y(m

m)

-50 0 50 100 150

-60

-40

-20

0

20

40

60

80

100

-0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9

5.0 m/sspanwisevorticity (1/s)

shadow region

GA(W)-1 airfoil

t=t0 t

LU

t= t0+t L

• Advantage:

• Whole flow field measurements

• Non-intrusive measurements

• Disadvantage:

• Low temporal resolution

• Very expensive compared with

hotwire anemometers and

pitot-static probes.


PIV system setup

Illumination system

(Laser and optics)

camera

Synchronizer

seed flow with

tracer particles

Host computer

Particle tracers: to track the fluid movement.

Illumination system: to illuminate the flow field in the interest region.

Camera: to capture the images of the particle tracers.

Synchronizer: the control the timing of the laser illumination and

camera acquisition.

Host computer: to store the particle images and conduct image

processing.


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


Laminar Flows and Turbulent Flows

'';' wwwvvvuuu

...),,,(1 0

0

Tt

t

dttzyxuT

u

0'0';0' wvu

0)'(0)'(;0)'(1

)'( 2222

0

wvdtuT

u

Tt

t

o


Turbulence

ReLU

Reynolds number:

At the large scales, the flow is determined

by the reference length scale, L, and the

reference time scale, τ0 = L/U.

The small length scales are governed by

the Kolmogorov scales η = (ν3 / ε)1/4, and

τη = (ν / ε)1/2.

ε is the turbulent energy dissipation rate.

Scaling:

For a flow with Re ~ 10,000:

Flow scales span 3 orders of magnitude in

length and 2 orders of magnitude in time.

Turbulent flows contain a vast range of

length and time scales that must be

resolved!!! Difficult!!


Boundary Layer Flow

y

Uu 99.0

, yat Displacement thickness:

Momentum thickness:


“macro-scale” aircraft:ReC = 106 ~108

MAVs: ReC = 104 ~105

•“Scale-down” of conventional airfoils

could not provide sufficient

aerodynamic performance for MAV

applications.

•It is very necessary and important to

establish novel airfoil shape and wing

planform design paradigms for MAVs

Low Reynolds number aerodynamics (e.g., MAVs)

X/C*100Y

/C*1

00

0 10 20 30 40 50

-5

0

5

10

15

0.50

-0.50

-1.50

-2.50

-3.50

-4.50

-5.50

Spanwisevorticity(1000*1/s)

X/C*100

Y/C

*10

0

0 10 20 30 40 50

-5

0

5

10

15

0.085

0.075

0.065

0.055

0.045

0.035

0.025

0.015

0.005

T.K.E

X/C*100

Y/C

*10

0

0 10 20 30 40 50

-5

0

5

10

15

0.085

0.075

0.065

0.055

0.045

0.035

0.025

0.015

0.005

T.K.E.

X/C*100

Y/C

*10

0

0 10 20 30 40 50

-5

0

5

10

15

10.0

8.0

6.0

4.0

2.0

0.0

Velocity(m/s)

X/C*100

Y/C

*10

0

0 10 20 30 40 50

-5

0

5

10

15

0.50

-0.50

-1.50

-2.50

-3.50

-4.50

-5.50

Spanwisevorticity(1000*1/s)

X/C*100

Y/C

*10

0

0 10 20 30 40 50

-5

0

5

10

15

10.0

8.0

6.0

4.0

2.0

0.0

Velocity(m/s)

0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12 14 16 18 20

Flat Plate

Profiled Airfoil

Corrugated Airfoil

Angle of Attack (Deg.)

CL

0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12 14 16 18 20

Flat Plate

Profiled Airfoil

Corrugated Airfoil

Angle of Attack (Deg.)

CL

interest for

MAV

applications

310 610410 510

010

110

210

310

710

CURe

MAXD

L

C

C

Streamlined airfoil

rough

surface

airfoil


Review of Quasi-1D Nozzle Flow

0V S

dV U ndSt

viscous

V S S V

UdV U n UdS pdS fdV Ft

2 2

2 2V S S V V

U U qe dV e U ndS pU ndS dV f U dV

t t

Assumptions:

• Steady-state

• Inviscid

• No body forces

Quasi-1D:

• Area is allowed to vary but flow variables

are considered a function of x only

X

Mass conservation

Momentum conservation

Energy conservation



u

duM

A

dA)1( 2

M<1 M>1

Throat

M=1

u increasing

M>1 M<1

Throat

M=1

u decreasing



12 2 2

1 1 1

2 2 2

00

20

2 10

1

2 10

(Thermodynamics) :

( ) ( )

:

11 1

2 2 2

11

2

1(1 )

2

1(1 )

2

P P

P

Isentropic relations

P T

P T

Energy Equation

TV V VC T C T

T C T RT

TM

T

PM

P

M



20

2 2 2 2 2 2 20

0

2 110

2 1

2

1

2 10

* * *

u** section: M 1 * *

*

11 * * * *

2 ( ) ( ) ( ) ( ) ( ) ( ) ( )*

1(1 ) 1 2 1

2 [ (1 )]1 2

1(1 )

2

y

u A uA

at A u aa

isentropic relation

TM A a a a

TA u u a

PM

P MM

M

M<1 M>1

Throat

A*

P*,T**

M*=1

u*=a*=(KRT*)1/2

A

P,T,

M

u

1

1

2

2

2 )]2

11(

1

2[

1)

*(

yMMA

A


1st, 2nd and 3rd critic conditions

1st critic

condition

2st critic

condition

3st critic

condition

P0 in

crea

sin

g

P0 in

crea

sin

g


Pressure Distribution Prediction within a De Laval Nozzle

by using Numerical Approach

atme PP

M<1 M>1

PT1 P1 P

T2

P2

M<1

Shock

AS

Ae

Throat ,

A* or At

atme PP

1

1

2

2

2

* 2

11

1

21

M

MA

A

• Using the area ratio, the Mach number

at any point up to the shock can be

determined:

• After finding Mach number at front of

shock, calculate Mach number after

shock using: 2

12

22

1

11

21

2

MM

M

• Then, calculate the A2*

1

122 2 2

2 2 2

2 11

1 2sA M A M

which allows us calculate the remaining

Mach number distribution

1

1

2

2

2

* 2

11

1

21

M

MA

A

2


Streamlines (experiment)

Flow visualization by using smoke wind tunnel

• Path line

• Streak lines

• Streamline


2

2

1*

*

VC

qC

ppp

T

EA

Test section

Contraction

section

A

E

V

Settling

chamber

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

Linear curve fittingExperimental data

P = PA-P

E (Pa)

q =

0.5

*V

2

Wind Tunnel Calibration


Pressure Sensor Calibration and Uncertainty Analysis

• Task #1: Pressure Sensor Calibration experiment

– A pressure sensor – Setra pressure transducer with a range of +/- 5 inH2O

• 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 2101, Range of +/- 10 inH2O

– 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

Setra pressure

transducer

(to be calibrated)

Mensor Digital

Pressure Gage A computer A plenum hand pump

tubing


Measurements of Pressure Distributions around a Circular

Cylinder

2

2

2

2

2

2

sin41)sin2(

1

1

2

1

V

V

V

V

V

PPCp

R

P

Incoming flow

X

Y

-3

-2

-1

0

1

0 100 200 300 400

Angle, Deg.

Cp

Cp distribution around a cylinder

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

theta (rad ) -->

cp

-->

EFD

CFD

AFD


Airfoil Pressure Distribution Measurements

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 10 12 14 16 18 20

CL=2

Experimental data

Angle of Attack (degrees)

Lift

Co

eff

icie

nt,

C

l

cV

LCl

2

2

1

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 2 4 6 8 10 12 14 16 18 20

Experimental data

Angle of Attack (degrees)

Dra

g C

oe

ffic

ien

t,

Cd

cV

DCd

2

2

1


y

x 80 mm

Pressure rake with 41 total pressure probes

(the distance between the probes d=2mm)

Airfoil Wake Measurements and Hotwire Anemometer

Calibration

y

x

2

2

2

2

2

2

)])(

1()(

[2

2

1

)])(

1()(

[

2

1

dyU

yU

U

yU

CC

CU

dAU

yU

U

yUU

CU

DC

D

D


Hot wire measurements in the wake of an airfoil

y

x

80 mm

Pressure rake with 41 total pressure probes

(the distance between the probes d=2mm)

Hotwire probe

Lab#3

Lab#4


PIV in an airfoil wake

Experimental setup for the PIV measurements

U∞

Instantaneous velocity field

• Mean velocity components in x, y directions:

•Turbulent velocity fluctuations:

• Turbulent Kinetic energy distribution:

NUuuN

i

i /)('1

2

N

i

i Vvv1

2)('

N

i

i NuU1

/

N

i

i NvV1

/

)''(2

1 22

vuTKE


Visualization of Shockwaves using Schlieren technique

Under-

expanded

flow

Flow

close to

3rd critical

Over-

expanded

flow

2nd critical –

shock is at

nozzle exit

1st critical –

shock is

almost at the

nozzle throat.

Tank with compressed air Test section


Set up a Schlieren and/or Shadowgraph System to Visualize a Thermal

Plume

Point Source

Candle plume


Tank with compressed air Test section

Pressure Measurements in a de Laval Nozzle

Tap No. Distance downstream of throat (inches) Area (Sq. inches)

1 -4.00 0.8002 -1.50 0.5293 -0.30 0.4804 -0.18 0.4785 0.00 0.4766 0.15 0.4977 0.30 0.5188 0.45 0.5399 0.60 0.56010 0.75 0.58111 0.90 0.59912 1.05 0.61613 1.20 0.62714 1.35 0.63215 1.45 0.634


Aerodynamic forces on an icing airfoil

Force transducer–wing model configuration


Final exam

Next week:

• Final exam on Tuesday December 13 7:30-9:30 a.m. in room

Hoover 1312

AERE 344 final exam specifics

• Open book and open class notes

• 20 multiple-choice problems (2 points each)

• 4 short answer problems related to lab design (15 points each)

• Exam time is 120 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!

First contact Scheduled final exam

Tues., 9:00-10:00 a.m.

Hoover 1312

Tues. Dec. 13 7:30-9:30 a.m..

Hoover 1312

Documents

Lecture # 16: Review for Final Examhuhui/teaching/2019-08Fx/... · Dimensional Analysis and Similitude force Commonly used non-dimensional parameters: " surface tension force inertial