AE 2303
AERODYNAMICS-II
Dr.S.Elangovan
Introduction

Review of prerequisite elements
–
–
–
–

Speed of sound
–
–

Perfect gas
Thermodynamics laws
Isentropic flow
Conservation laws
Analogous concept
Derivation of speed of sound
Mach number
Review of prerequisite elements

Perfect gas:
Entropy
Equation of state
P  RT
For calorically perfect gas
u  u (T )
h  u  RT
dh  c p dT
du  cv dT
c p  cv  R
cp

cv
ds 
q
T
ds 
du  Pdv dh  vdP

T
T
Entropy changes?
T 
 
s2  s1  cv ln  2   R ln  1 
 T1 
 2 
T 
P 
s2  s1  c p ln  2   R ln  2 
 T1 
 P1 
T2 
s  s   
 exp 2 1  2 
T1 
cv  1 
T2 
s2  s1  P2 
 exp
 
T1 
c p  P1 
R cv
R cp
Review of prerequisite elements
Forms of the 1st law
q  w  e
Tds  de  pd
Tds  dh  dp
The second law
q
ds 
T
Cont.
Review of prerequisite elements
If ds=o
For an isentropic flow
R
cv
 
T2   2 
     2 
T1  1 
 1 
T2  P2 
  
T1  P1 
Cont.
R
cp
P 
  2 
 P1 
 1
 1

T2  P2 
  
T1  P1 
 1

P2   2 
  
P1  1 
P


 2 
  
 1 

 constant
 1
Review of prerequisite elements
Cont.
1  m
2
m
Conservation of mass (steady flow):
1V1 A1   2V2 A2
Rate of mass
Rate of mass
enters control = leaves control
volume
volume
VA  (   d )(V  dV )( A  dA)
VAd  AdV  VdA  0
d

V
A
1
flow
dx
  d
2
V  dV
A  dA


dV dA

0
V
A
If  is constant (incompressible):
dV
dA

V
A
Review of prerequisite elements
Cont.
Conservation of momentum (steady flow):
Net force on
Rate momentum
gas in control = leaves control
volume
volume
Rate momentum
enters control
volume
F  Fp  m V 2  m V 1
p

V
A
1
flow
dx
2
p  dp
  d
V  dV
A  dA
Euler equation (frictionless flow):
V2
dp

 constant
2

Review of prerequisite elements
Cont.
Conservation of energy for a CV (energy balance):
Basic principle:
• Change of energy in a CV is related to
energy transfer by heat, work, and energy in
the mass flow.
2
2




dECV
V
V
i
e
 Q  W  m i  ui 
 gzi   m e  ue 
 gze 
dt
2
2




heat transfer
work transfer
energy transfer due to mass flow
Review of prerequisite elements
Cont.
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.
Fp  pA  W p  FpV
W  WCV  W p
m  AV  m v  AV
W p  m pv
W p  W e  Wi
2
2




dECV
V
V
 Q  WCV  m e pe ve  m i pi vi   m i  ui  i  gzi   m e  u e  e  gze 
dt
2
2




2
2




dECV
V
V
i
e







 Q  WCV  mi  ui  pi vi 
 gzi   me  u e  pe ve 
 gze 
dt
2
2




h  u  pv
2
2




dECV
V
V
i
e
 Q  WCV  m i  hi 
 gzi   m e  he 
 gze  Most important form
dt
2
2



 of energy balance.
Review of prerequisite elements
Cont.
2
2


V
V
e
i
  m i  hi 

Q  W  m e  he 


2 
2 



Ve2  Vi2 
dq  dw  he  hi  
2
For calorically perfect gas (dcp=dcv=0):
T
h
V
1
flow
dx
2
T  dT
h  dh
V  dV
h  c pT
For adiabatic flow (no heat transfer)
and no work:
c p dT  VdV  0
Review of prerequisite elements
Cont.
Conservation laws
Conservation of mass
(compressible flow):
Conservation of momentum
(frictionless flow):
Conservation of energy
(adiabatic):
m 1  m 2 
d
dV dA


0
 V
A
F  Fp  m V 2  m V 1 
V
dq  dw  h  h  
2
e
e
i
dP


 VdV  0
 Vi 2
 c p dT  VdV  0
2
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 gas
constant 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=1740C.
Speed of sound
Sounds are the small pressure disturbances in the gas around us,
analogous to the surface ripples produced when still water is disturbed
Sound wave
Sound wave
P  dP
  d
T  dT
V  dV
P

T
V 0
Sound wave moving
through stationary gas
P  dP
  d
T  dT
V  a  dV
P

T
V a
Gas moving through
stationary sound wave
Speed of sound
cont.
Combination of mass and momentum
Derivation of speed of sound
a
dp
d
Conservation of mass
m  aA    d a  dV A
d
dV


a

P2   2 
For
  
P1  1 
isentropic flow
P
 constant

Conservation of momentum
PA  P  dP  A  m a  dV   m a
dP  adV
Finally
dP P

d 
a
P
 RT

Mach Number
M=V/a
M<1
Subsonic
M=1
Sonic
M>1
Supersonic
M>5
Hypersonic
Distance traveled =
speed x time = 4at
Distance traveled = at
Zone of
silence
If M=0
Source of
disturbance
Region of
influence
Mach Number cont.
Original location
of source of
disturbance
Source of
disturbance
If M=0.5
Mach Number cont.
ut
ut
ut
ut
Original location
of source of
disturbance
Direction
of motion
Source of
disturbance
Mach wave:
If M=2
at 1
sin   
ut M
Normal and Oblique
Shock

A shock wave (also called shock front or
simply "shock") is a type of propagating
disturbance. 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.

Shock waves are characterized by an abrupt,
nearly discontinuous change in the
characteristics of the medium. Across a
shock there is always an extremely rapid rise
in pressure, temperature and density of the
flow. In supersonic flows, expansion is
achieved through an expansion fan. A shock
wave travels through most media at a higher
speed than an ordinary wave.

Unlike solutions (another kind of nonlinear wave),
the energy of a shock wave dissipates relatively
quickly with distance. Also, the accompanying
expansion wave 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
the degradation and merging of the shock wave and
the expansion wave produced by the aircraft.

Thus the sonic boom associated with the
passage of a supersonic aircraft is the sound
wave resulting from the degradation and
merging of the shock wave and the
expansion wave produced by the aircraft.

When a shock wave passes through matter,
the total energy is preserved but the energy
which can be extracted as work decreases
and entropy increases. This, for example,
creates additional drag force on aircraft with
shocks.
Oblique Shock

An oblique shock wave, unlike a normal shock, is
inclined with respect to the incident upstream flow
direction.

It will occur when a supersonic flow encounters a
corner that effectively turns the flow into itself and
compresses.

The upstream streamlines are uniformly deflected
after the shock wave. The most common way to
produce an oblique shock wave is to place a wedge
into supersonic, compressible flow. Similar to a
normal shock wave, the oblique shock wave consists
of a very thin region across which nearly
discontinuous changes in the thermodynamic
properties of a gas occur. While the upstream and
downstream flow directions are unchanged across a
normal shock, they are different for flow across an
oblique shock wave.

It is always possible to convert an oblique
shock into a normal shock by a Galilean
transformation.
EXPANSIONWAVES,RAYLEIGH AND
FANNO FLOW

A Prandtl-Meyer expansion fan is a centered expansion
process, which turns a supersonic flow around a convex corner.

The fan consists of an infinite number of Mach waves, diverging
from a sharp corner. In case of a smooth corner, these waves
can be extended backwards to meet at a point.

Each wave in the expansion fan turns the flow
gradually (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
of thermodynamics. Across the expansion fan, the
flow accelerates (velocity increases) and the Mach
number increases, while the static pressure,
temperature and density decrease. Since the
process is isentropic, the stagnation properties
remain constant across the fan.
Prandtl-Meyer Function

θ2 − θ1 = ν(M2) − ν(M1)
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
often come 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,
unlike Fanno flow, the stagnation temperature is a
variable.
Rayleigh flow

The heat addition causes a decrease in stagnation
pressure, which is known as the Rayleigh effect and
is critical in the design of combustion systems. Heat
addition will cause both supersonic and subsonic
Mach numbers to approach Mach 1, resulting in
choked flow. Conversely, heat rejection decreases a
subsonic Mach number and increases a supersonic
Mach number along the duct. It can be shown that
for calorically perfect flows the maximum entropy
occurs at M = 1. Rayleigh flow is named after John
Strutt, 3rd Baron Rayleigh.



Solving the differential equation leads to the relation shown
below, where T0* is the stagnation temperature at the throat
location of the duct which is required for thermally choking the
flow.
These values are significant in the design of combustion
systems. For example, if a turbojet combustion chamber has a
maximum temperature of T0* = 2000 K, T0 and M at the
entrance to the combustion chamber must be selected so
thermal choking does not occur, which will limit the mass flow
rate of air into the engine and decrease thrust.
For the Rayleigh flow model, the dimensionless change in
entropy relation is shown below.
Fanno flow

Fanno flow refers to adiabatic flow through a constant area
duct where the effect of friction is considered.Compressibility
effects often come into consideration, although the Fanno flow
model certainly also applies to incompressible flow. For this
model, the duct area remains constant, the flow is assumed to
be steady and one-dimensional, and no mass is added within
the duct. The Fanno flow model is considered an irreversible
process due to viscous effects. The viscous friction causes the
flow properties to change along the duct. The frictional effect is
modeled as a shear stress at the wall acting on the fluid with
uniform properties over any cross section of the duct.
Fanno flow

For a flow with an upstream Mach number greater
than 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 Mach number less than 1.0, acceleration
occurs and the flow can become choked in a
sufficiently long duct. It can be shown that for flow of
calorically perfect gas the maximum entropy occurs
at M = 1.0. Fanno flow is named after Gino Girolamo
Fanno.
DIFFERENTIAL EQUATIONS OF
MOTION FOR STEADY
COMPRESSIBLE FLOWS
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.
Critical Mach Number (Mcr)

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 to
travel 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:


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.

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, make
control 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 of accidents involving high-speed military
and experimental aircraft in the 1930s and 1940s.
Drag Divergence Mach Number

The drag divergence Mach number is 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 its
low speed value.

The value of the drag divergence Mach
number is typically greater than 0.6;
therefore it is a transonic effect. The drag
divergence Mach number is usually close to,
and always greater than, the critical Mach
number. Generally, the drag coefficient
peaks at Mach 1.0 and begins to decrease
again after the transition into the supersonic
regime above approximately Mach 1.2.

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 at
supersonic speeds have a large amount of
thrust.

In early development of transonic and supersonic
aircraft, a steep dive was often used to provide extra
acceleration through the high drag region around
Mach 1.0. In the early days of aviation, this steep
increase in drag gave rise to the popular false notion
of an unbreakable sound barrier, because it seemed
that no aircraft technology in the foreseeable future
would have enough propulsive force or control
authority to overcome it. Indeed, one of the popular
analytical methods for calculating drag at high
speeds, the Prandtl-Glauert rule, predicts an infinite
amount of drag at Mach 1.0.

Two of the important technological advancements
that arose out of attempts to conquer the sound
barrier were the Whitcomb area rule and the
supercritical airfoil. A supercritical airfoil is shaped
specifically to make the drag divergence Mach
number as high as possible, allowing aircraft to fly
with relatively lower drag at high subsonic and low
transonic speeds. These, along with other
advancements including computational fluid
dynamics, have been able to reduce the factor of
increase in drag to two or three for modern aircraft
designs
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.
swept wing

Unusual variants of this design feature are
forward sweep, variable sweep wings , and
pivoting wings. Swept wings as a means of
reducing wave drag were first used on jet
fighter aircraft. Today, they have become
almost universal on all but the slowest jets
(such as the A-10), and most faster airliners
and business jets. The four-engine propellerdriven TU-95 aircraft has swept wings.

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).
Transonic Area Rule

Within the limitations of small perturbation theory, at a given transonic
Mach number, aircraft with the same longitudinal distribution of crosssectional area, including fuselage, wings and all appendages will, at
zero lift, have the same wave drag.

Why: Mach waves under transonic conditions are perpendicular to
flow.

Implication:
Keep area distribution smooth, constant if possible.
Else, strong shocks and hence drag result.

Wing-body interaction leading to shock formation:


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.
Transonic Area Rule
Transonic Area Rule

In fluid dynamics, potential flow describes
the velocity field as the gradient of a scalar
function: the velocity potential..

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


In the case of an incompressible flow the velocity
potential satisfies Laplace's equation. However,
potential flows also have been used to describe
compressible flows. The potential flow approach
occurs in the modeling of both stationary as well as
nonstationary flows.
Applications of potential flow are for instance: the
outer flow field for aerofoils, water waves, and
groundwater flow. For flows (or parts thereof) with
strong vorticity effects, the potential flow
approximation is not applicable.
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.
Mach stem or Mach 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 or Mach front.
Mach angle μ

Thus it is possible to have shock less
compression or expansion in a supersonic
flow by having the production of Mach waves
sufficiently spaced (cf. isentropic
compression in supersonic flows). A Mach
wave is the weak limit of an oblique shock
wave (a normal shock is the other limit). They
propagate across the flow at the Mach angle
μ.


where M is the Mach number.
Mach waves can be used in schlieren or
shadowgraph observations to determine the local
Mach number of the flow. Early observations by
Ernst Mach used grooves in the wall of a duct to
produce Mach waves in a duct, which were then
photographed by the schlieren method, to obtain
data about the flow in nozzles and ducts. Mach
angles may also occasionally be visualized out of
their condensation in air, as in the jet photograph
below.

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 shock
made visible by the halo, is seen to be close
to arcsine (1) = 90 degrees.