Aerodynamic Note

8/3/2019 Aerodynamic Note

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Introduction Review of prerequisite elements

Perfect gas

Thermodynamics laws

Isentropic flow

Conservation laws

Speed of sound Analogous concept

Derivation of speed of sound Mach number


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Review of prerequisite elements

Perfect gas:Equation of state

For calorically perfect gas

T

qds

=

RTP =

v

p

vp

v

p

c

c

Rcc

dTcdu

dTcdh

RTuhTuu

=

+=

=

=+==

)(

Entropy

Entropy changes?

=

+

=

1

2

1

212

2

1

1

212

lnln

lnln

P

PR

T

Tcss

RT

Tcss

p

v

p

v

cR

p

cR

v

P

P

c

ss

T

T

c

ss

T

T

=

=

1

212

1

2

1

212

1

2

exp

exp

T

vdPdh

T

Pdvduds

=

+=


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Review of prerequisite elementsCont.

Forms of the 1st law

dpdhTds

pddeTds

ewq

=+==+

T

qds

The second law


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Review of prerequisite elements Cont.

=

=

=

1

2

1

2

1

1

2

1

1

2

1

2

P

P

P

P

T

T

For an isentropic flow

1

1

2

1

2

1

2

1

1

2

1

2

1

2

=

=

=

=

P

P

P

P

T

T

T

T

p

v

c

R

c

R

If ds=o

constant=

P


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Conservation of mass (steady flow):

Rate of mass

enters control

volume

Rate of mass

leaves control

volume

=

1 2dAA

dVV

d

++

+

A

V

dx

flow

A

dA

V

dV

A

dA

V

dVd

VdAAdVVAd

dAAdVVdVA

AVAV

mm

=

=++

=++

+++=

=

=

0

0

))()((

222111

21

Ifis constant (incompressible):


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Conservation of momentum (steady flow):

Rate momentum

leaves control

volume

Rate momentum

enters control

volume

-

Net force on

gas in control

volume

=

( ) ( )12 VmVmFF p =+

Euler equation (frictionless flow):

=+constant

2

2

dpV

1 2

dAA

dVV

d

dpp

+++

+

A

V

p

dx

flow


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++

+++= e

eeei

iii

CV gzV

umgzV

umWQdt

dE

22

22

heat transfer energy transfer due to mass flowwork transfer

Basic principle:

Change of energy in a CV is related to

energy transferby heat, work, and energy inthe mass flow.

Conservation of energy for a CV (energy balance):


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iep

pCV

WWW

WWW

=

+=

( )

++

+++=

+=

+++

++++=

++

+++=

e

e

eei

i

iiCV

CV

ee

eeeeii

iiiiCVCV

ee

eeii

iiiiieeeCVCV

gzV

hmgzV

hmWQdt

dE

pvuh

gzV

vpumgzV

vpumWQdt

dE

gzV

umgzV

umvpmvpmWQdt

dE

22

22

22

22

22

22

pvmW

AVvmAVm

VFWpAF

p

ppp

=

==

==

Most important form

of energy balance.

Analyzing more about Rate of Work Transfer:

work can be separated into 2 types:

work associated with fluid pressure as mass entering or leaving the CV.other works such as expansion/compression, electrical, shaft, etc.

Work due to fluid pressure:

fluid pressure acting on the CV boundary creates force.


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( )( )

2

22

22

22

ieie

i

ii

e

ee

VVhhdwdq

Vhm

VhmWQ

+=

+

+=

1 2dVVdhh

dTT

++

+

Vh

T

dx

flowTch p=

For adiabatic flow (no heat transfer)

and no work:

For calorically perfect gas (dcp=dcv=0):

0=+VdVdTcp


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Conservation of mass

(compressible flow):

Conservation of

momentum

(frictionless flow):

Conservation of energy

(adiabatic):

021 =++=A

dA

V

dVdmm

( ) ( ) 012 =+=+ VdVdP

VmVmFF p

( ) ( ) 02

22 =++= VdVdTcVVhhdwdq pie

ie


Conservation laws


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Group Exercises 1

1. Given that standard atmospheric conditions for air at 150C are a

pressure of 1.013 bar and a density of 1.225kg, calculate the gasconstant for air. Ans: R=287.13J/kgK

2. The value of Cv for air is 717J/kgK. The value of R=287 J/kgK.

Calculate the specific enthalpy of air at 200C. Derive a relation

connecting Cp, Cv, R. Use this relation to calculate Cp for air using

the information above. Ans: h=294.2kJ/kgK,Cp=1.004kJ/kgK

3. Air is stored in a cylinder at a pressure of 10 bar, and at a room

temperature of 250C. How much volume will 1kg of air occupy

inside the cylinder? The cylinder is rated for a maximum pressure of

15 bar. At what temperature would this pressure be reached? Ans:

V=0.086m2, T=174

0

C.


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Speed of sound

0=VT

P

dVV

dTT

d

dPP

=+++

Sound wave

Sounds are the small pressure disturbances in the gas around us,

analogous to the surface ripples produced when still water is disturbed

aVT

P

=

dVaV

dTT

d

dPP

=+

++

Sound wave

Sound wave moving

through stationary gas

Gas moving through

stationary sound wave


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Derivation of speed of sound

Speed of sound cont.

( ) ( )

adVd

AdVadaAm

=

+==

( ) ( )

adVdP

amdVamAdPPPA

==+

d

dpa =

constant

1

2

1

2

=

=

P

PP

RTP

a

P

d

dP

==

=

Conservation of mass

Conservation of momentum

Combination of mass and momentum

For

isentropic flow

Finally


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Mach Number

M=V/a

Source of

disturbance

Distance traveled =

speed x time = 4at

Zone of

silence

Region of

influence

If M=0

M1 Supersonic

M>5 HypersonicDistance traveled = at


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Mach Numbercont.

Source of

disturbance

If M=0.5

Original location

of source ofdisturbance


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Mach Numbercont.

Source of

disturbance

If M=2

Original location

of source of

disturbanceutut

ut

ut

Mut

at 1sin ==

Mach wave:

Direction

of motion


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Normal and Oblique

Shock

A shock wave (also called shock front or

simply "shock") is a type of propagatingdisturbance. Like an ordinary wave, it carries

energy and can propagate through a medium

(solid, liquid, gas or plasma) or in some

cases in the absence of a material medium,through a field such as the electromagnetic

field.
http://en.wikipedia.org/wiki/File:Schlierenfoto_Mach_1-2_Pfeilfl%C3%BCgel_-_NASA.jpg


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Shock waves are characterized by an abrupt,nearly discontinuous change in thecharacteristics of the medium. Across ashock there is always an extremely rapid risein pressure, temperature and density of theflow. In supersonic flows, expansion is

achieved through an expansion fan. A shockwave travels through most media at a higherspeed than an ordinary wave.


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Unlike solutions (another kind of nonlinear wave), the

energy of a shock wave dissipates relatively quickly

with distance. Also, the accompanying expansionwave approaches and eventually merges with the

shock wave, partially canceling it out. Thus the sonic

boom associated with the passage of a supersonic

aircraft is the sound wave resulting from thedegradation and merging of the shock wave and the

expansion wave produced by the aircraft.


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Thus the sonic boom associated with the

passage of a supersonic aircraft is the soundwave resulting from the degradation and

merging of the shock wave and the

expansion wave produced by the aircraft.


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When a shock wave passes through matter,

the total energy is preserved but the energywhich can be extracted as work decreases

and entropy increases. This, for example,

creates additional drag force on aircraft with

shocks.


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Oblique Shock

An oblique shock wave, unlike a normal shock, isinclined with respect to the incident upstream flow

direction.

It will occur when a supersonic flow encounters acorner that effectively turns the flow into itself andcompresses.
http://en.wikipedia.org/wiki/File:X-15_Model_in_Supersonic_Wind_Tunnel.jpg


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The upstream streamlines are uniformly deflectedafter the shock wave. The most common way to

produce an oblique shock wave is to place a wedgeinto supersonic, compressible flow. Similar to anormal shock wave, the oblique shock wave consistsof a very thin region across which nearlydiscontinuous changes in the thermodynamic

properties of a gas occur. While the upstream anddownstream flow directions are unchanged across anormal shock, they are different for flow across anoblique shock wave.


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It is always possible to convert an oblique

shock into a normal shock by a Galileantransformation.


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EXPANSIONWAVES,RAYLEIGH AND

FANNO FLOW

A Prandtl-Meyer expansion fan is a centered expansionprocess, which turns a supersonic flow around a convexcorner.

The fan consists of an infinite number ofMach waves,diverging from a sharp corner. In case of a smooth corner,these waves can be extended backwards to meet at a point.
http://en.wikipedia.org/wiki/Convex_sethttp://en.wikipedia.org/wiki/Mach_wavehttp://en.wikipedia.org/wiki/Mach_wavehttp://en.wikipedia.org/wiki/Convex_set


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Each wave in the expansion fan turns the flowgradually (in small steps). It is physically impossible to

turn the flow away from itself through a single "shock"wave because it will violate the second law ofthermodynamics. Across the expansion fan, the flowaccelerates (velocity increases) and the Mach numberincreases, while the static pressure, temperature and

density decrease. Since the process is isentropic, thestagnation properties remain constant across the fan.


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Prandtl-Meyer Function

2 1 = (M2) (M1)


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Rayleigh flow

Rayleigh flow refers to diabetic flow through a

constant area duct where the effect of heat addition or

rejection is considered. Compressibility effects oftencome into consideration, although the Rayleigh flow

model certainly also applies to incompressible flow.

For this model, the duct area remains constant and no

mass is added within the duct. Therefore, unlikeFanno flow, the stagnation temperature is a variable.


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Rayleigh flow


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The heat addition causes a decrease in stagnationpressure, which is known as the Rayleigh effect and is

critical in the design of combustion systems. Heataddition will cause both supersonic and subsonicMach numbers to approach Mach 1, resulting inchoked flow. Conversely, heat rejection decreases asubsonic Mach number and increases a supersonic

Mach number along the duct. It can be shown that forcalorically perfect flows the maximum entropy occursat M = 1. Rayleigh flow is named after John Strutt, 3rdBaron Rayleigh.


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Solving the differential equation leads to the relation shownbelow, where T0* is the stagnation temperature at the throatlocation of the duct which is required for thermally choking theflow.

These values are significant in the design of combustionsystems. For example, if a turbojet combustion chamber has amaximum temperature of T0* = 2000 K, T0 and M at theentrance to the combustion chamber must be selected so

thermal choking does not occur, which will limit the mass flowrate of air into the engine and decrease thrust. For the Rayleigh flow model, the dimensionless change in

entropy relation is shown below.


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Fanno flow

Fanno flow refers to adiabatic flow through a constant areaduct where the effect of friction is considered.Compressibilityeffects often come into consideration, although the Fanno flowmodel certainly also applies to incompressible flow. For thismodel, the duct area remains constant, the flow is assumed tobe steady and one-dimensional, and no mass is added withinthe duct. The Fanno flow model is considered an irreversibleprocess due to viscous effects. The viscous friction causes theflow properties to change along the duct. The frictional effect ismodeled as a shear stress at the wall acting on the fluid withuniform properties over any cross section of the duct.


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Fanno flow


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For a flow with an upstream Mach number greaterthan 1.0 in a sufficiently long enough duct,

deceleration occurs and the flow can become choked.On the other hand, for a flow with an upstream Machnumber less than 1.0, acceleration occurs and theflow can become choked in a sufficiently long duct. Itcan be shown that for flow of calorically perfect gas

the maximum entropy occurs at M= 1.0. Fanno flowis named after Gino Girolamo Fanno.
http://en.wikipedia.org/wiki/Mach_numberhttp://en.wikipedia.org/wiki/Mach_number


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DIFFERENTIAL EQUATIONS OFMOTION FOR STEADY

COMPRESSIBLE FLOWS


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TRANSONIC FLOW OVER WING

In aerodynamics, the critical Mach number

(Mcr) of an aircraft is the lowest Mach

number at which the airflow over a small

region of the wing reaches the speed of

sound.


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Critical Mach Number (Mcr)


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For all aircraft in flight, the airflow around the aircraft

is not exactly the same as the airspeed of the aircraft

due to the airflow speeding up and slowing down totravel around the aircraft structure. At the Critical

Mach number, local airflow in some areas near the

airframe reaches the speed of sound, even though the

aircraft itself has an airspeed lower than Mach 1.0.This creates a weak shock wave. At speeds faster

than the Critical Mach number:


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drag coefficient increases suddenly, causing

dramatically increased drag

in aircraft not designed for transonic or

supersonic speeds, changes to the airflow

over the flight control surfaces lead to

deterioration in control of the aircraft.


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In aircraft not designed to fly at the Critical Mach

number, shock waves in the flow over the wing and

tail plane were sufficient to stall the wing, makecontrol surfaces ineffective or lead to loss of control

such as Mach tuck. The phenomena associated with

problems at the Critical Mach number became known

as compressibility. Compressibility led to a number ofaccidents involving high-speed military and

experimental aircraft in the 1930s and 1940s.


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Drag Divergence Mach Number

The drag divergence Mach numberis the

Mach number at which the aerodynamic drag

on an airfoil or airframe begins to increase

rapidly as the Mach number continues to

increase. This increase can cause the drag

coefficient to rise to more than ten times itslow speed value.


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The value of the drag divergence Machnumber is typically greater than 0.6; therefore

it is a transonic effect. The drag divergenceMach number is usually close to, and alwaysgreater than, the critical Mach number.Generally, the drag coefficient peaks at Mach

1.0 and begins to decrease again after thetransition into the supersonic regime aboveapproximately Mach 1.2.


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The large increase in drag is caused by the

formation of a shock wave on the upper

surface of the airfoil, which can induce flow

separation and adverse pressure gradients

on the aft portion of the wing. This effect

requires that aircraft intended to fly atsupersonic speeds have a large amount of

thrust.


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In early development of transonic and supersonicaircraft, a steep dive was often used to provide extraacceleration through the high drag region aroundMach 1.0. In the early days of aviation, this steepincrease in drag gave rise to the popular false notionof an unbreakable sound barrier, because it seemedthat no aircraft technology in the foreseeable futurewould have enough propulsive force or control

authority to overcome it. Indeed, one of the popularanalytical methods for calculating drag at highspeeds, the Prandtl-Glauert rule, predicts an infiniteamount of drag at Mach 1.0.


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Two of the important technological advancements thatarose out of attempts to conquer the sound barrierwere the Whitcomb area rule and the supercriticalairfoil. A supercritical airfoil is shaped specifically tomake the drag divergence Mach number as high aspossible, allowing aircraft to fly with relatively lowerdrag at high subsonic and low transonic speeds.These, along with other advancements including

computational fluid dynamics, have been able toreduce the factor of increase in drag to two or threefor modern aircraft designs


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swept wing

A swept wing is a wing platform with a wing

root to wingtip direction angled beyond

(usually aft ward) the span wise axis,

generally used to delay the drag rise caused

by fluid compressibility.


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swept wing


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Unusual variants of this design feature areforward sweep, variable sweep wings , and

pivoting wings. Swept wings as a means ofreducing wave drag were first used on jetfighter aircraft. Today, they have becomealmost universal on all but the slowest jets

(such as the A-10), and most faster airlinersand business jets. The four-engine propeller-driven TU-95 aircraft has swept wings.


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The angle of sweep which characterizes a

swept wing is conventionally measured along

the 25% chord line. If the 25% chord line

varies in sweep angle, the leading edge is

used; if that varies, the sweep is expressed

in sections (e.g., 25 degrees from 0 to 50%span, 15 degrees from 50% to wingtip).


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Transonic Area Rule

Within the limitations of small perturbation theory, at a given transonicMach number, aircraft with the same longitudinal distribution of cross-sectional area, including fuselage, wings and all appendages will, atzero lift, have the same wave drag.

Why: Mach waves under transonic conditions are perpendicular toflow.


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Implication:

Keep area distribution smooth, constant if possible.

Else, strong shocks and hence drag result.

Wing-body interaction leading to shock formation:


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Observed: cp distributions are such that

maximum velocity is reached far aft at root

and far forward at tip.

Hence, streamlines curves in at the root,

compress, shock propagates out.


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Transonic Area Rule


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Transonic Area Rule


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In fluid dynamics, potential flow describes

the velocity field as the gradient of a scalar

function: the velocity potential..


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As a result, a potential flow is characterized

by an irrotational velocity field, which is a

valid approximation for several applications.

The irrotationality of a potential flow is due to

the curl of a gradient always being equal to

zero


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In the case of an incompressible flow the velocitypotential satisfies Laplace's equation. However,potential flows also have been used to describecompressible flows. The potential flow approachoccurs in the modeling of both stationary as well asnonstationary flows.

Applications of potential flow are for instance: theouter flow field for aerofoils, water waves, and

groundwater flow. For flows (or parts thereof) withstrong vorticity effects, the potential flowapproximation is not applicable.


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Mach wave

In fluid dynamics, a Mach wave is a pressure

wave traveling with the speed of sound

caused by a slight change of pressure added

to a compressible flow.
http://en.wikipedia.org/wiki/File:Schlierenfoto_Mach_1-2_Pfeilfl%C3%BCgel_-_NASA.jpg


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Mach stem orMach front

These weak waves can combine in

supersonic flow to become a shock wave if

sufficient Mach waves are present at any

location. Such a shock wave is called a

Mach stem orMach front.


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Mach angle

Thus it is possible to have shock lesscompression or expansion in a supersonic

flow by having the production of Mach wavessufficiently spaced (cf. isentropiccompression in supersonic flows). A Machwave is the weak limit of an oblique shock

wave (a normal shock is the other limit). Theypropagate across the flow at the Mach angle

.


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where Mis the Mach number. Mach waves can be used in schlieren or

shadowgraph observations to determine the localMach number of the flow. Early observations by ErnstMach used grooves in the wall of a duct to produceMach waves in a duct, which were then photographedby the schlieren method, to obtain data about the flowin nozzles and ducts. Mach angles may also

occasionally be visualized out of their condensation inair, as in the jet photograph below.


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U.S. Navy F/A-18 breaking the sound barrier.

The white halo is formed by condensed water

droplets which are thought to result from an

increase in air pressure behind the shock

wave(see Prandtl-Glauert Singularity). The

Mach angle of the weak attached shockmade visible by the halo, is seen to be close

to arcsine (1) = 90 degrees.

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