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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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Review of prerequisite elements Cont.
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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Review of prerequisite elements Cont.
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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Review of prerequisite elements Cont.
++
+++= 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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Review of prerequisite elements Cont.
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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Review of prerequisite elements Cont.
( )( )
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
Review of prerequisite elements Cont.
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.
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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.