AE 2303 AERODYNAMICS-II

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AE 2303AERODYNAMICS-II

• Dr.S.Elangovan

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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

Entropy
Entropy changes?
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Review of prerequisite elements Cont.

• Forms of the 1st law

The second law
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Review of prerequisite elements Cont.
For an isentropic flow
If dso
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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

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Review of prerequisite elements Cont.
Conservation of momentum (steady flow)
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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.

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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.

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Review of prerequisite elements Cont.
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Review of prerequisite elements Cont.
Conservation laws
Conservation of mass (compressible
flow) Conservation of momentum (frictionless
flow) Conservation of energy (adiabatic)
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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 gas constant
   for air. Ans R287.13J/kgK
2. The value of Cv for air is 717J/kgK. The value of
   R287 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
   h294.2kJ/kgK,Cp1.004kJ/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 V0.086m2, T1740C.

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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 moving through stationary gas
Gas moving through stationary sound wave
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Derivation of speed of sound
Speed of sound cont.
Combination of mass and momentum
Conservation of mass
For isentropic flow
Conservation of momentum
Finally
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Mach Number
MV/a
Mlt1 Subsonic M1 Sonic Mgt1 Supersonic Mgt5 Hyperson
ic
Distance traveled speed x time 4at
Distance traveled at
Source of disturbance
Zone of silence
Region of influence
If M0
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Mach Number cont.
Original location of source of disturbance
Source of disturbance
If M0.5
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Mach Number cont.
Original location of source of disturbance
ut
ut
ut
ut
Direction of motion
Source of disturbance
Mach wave
If M2
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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.

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• 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.

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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 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.

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• 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.

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• 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.

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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.

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• 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.

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• It is always possible to convert an oblique shock
  into a normal shock by a Galilean transformation.

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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.

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• 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.

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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 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.

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Rayleigh flow
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• 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.

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• 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.

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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.

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Fanno flow
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• 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.

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DIFFERENTIAL EQUATIONS OF MOTION 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 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

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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, 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.

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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.

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• 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.

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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 at supersonic speeds have a large
  amount of thrust.

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• 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.

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• 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

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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 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 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 transonic Mach number,
  aircraft with the same longitudinal distribution
  of cross-sectional 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.
•        

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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
  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.

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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.

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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.

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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 .

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• 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.

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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 shock made visible by the halo,
  is seen to be close to arcsine (1) 90 degrees.