AE1303 AERODYNAMICS II

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AE1303 AERODYNAMICS -II
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ONE DIMENSIONAL COMPRESSIBLE FLOW

• Continuity Equation
• for steady flow

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ONE DIMENSIONAL COMPRESSIBLE FLOW

• EULER EQUATION

For steady flow
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ONE DIMENSIONAL COMPRESSIBLE FLOW
Momentum Equation
Energy Equation
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Oblique Shock Waves

• The oblique shock waves typically occurs when a
  supersonic flow is turned to itself by a wall or
  its equivalent boundary condition.
• All the streamlines have the same deflection
  angle q at the shock wave, parallel to the
  surface downstream.
• Across the oblique shock, M decreases but p, T
  and r increase.

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Expansion Waves

• The expansion waves typically occur when a
  supersonic flow is turned away from itself by a
  wall or its equivalent boundary condition.
• The streamlines are smoothly curved through the
  expansion fan until they are all parallel to the
  wall behind the corner point.
• All flow properties through an expansion wave
  change smoothly and continuously. Across the
  expansion wave, M increases while p, T, and r
  decreases.

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Source of Oblique Waves

• For an object moving at a supersonic speed, the
  object is always ahead of the sound wave fronts
  generated by the object. This cause the sound
  wave fronts to coalesce into a line disturbance,
  called Mach wave, at the Mack angle m relative to
  the direction of the beeper.
• The physical mechanism to form the oblique shock
  wave is essentially the same as the Mach wave.
  The Mach wave is actually an infinitely weak
  shock wave.

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Oblique Shock Relations

• The oblique shock tilts at a wave angle b with
  respect to V1, the upstream velocity. Behind the
  shock, the flow is deflected toward the shock by
  the flow deflection angle q.
• Let u and w denote the normal and parallel flow
  velocity components relative to the oblique shock
  and Mn and Mt the corresponding Mach numbers, we
  have for a steady adiabatic flow with no body
  forces the following relations

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Oblique Shock Relations (contd.)

• So and Mn1 and Mn2 all satisfy
  the corresponding normal shock relations, which
  are all functions of M1 and b, because

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Oblique Shock Relations (contd.)?-ß-M relation

• For any given free stream Mach number M1, there
  is
• a maximum q beyond which the shock will be
  detached.
• For any given M1 and q lt qmax, there are two bs.
  The
• larger b is called the strong shock
  solution, where M2 is
• subsonic. The lower b is called the weak
  shock solution,
• where M2 is supersonic except for a small
  region near
• qmax.
• 3. If q 0, then b p/2 (normal shock) or b m
  (Mach wave).

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Straight Oblique Shock Relations (contd.)

• For a calorically perfect gas,

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Supersonic Flow Over Cones

• The flow over a cone is inherently
  three-dimensional. The three-dimensionality has
  the relieving effect to result in a weaker shock
  wave as compared to a wedge of the same half
  angle.
• The flow between the shock and the cone is no
  longer uniform the streamlines there are curved
  and the surface pressure are not constant.

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Shock Wave Reflection

• Consider an incident oblique shock on a lower
  wall that is reflected by the upper wall at
  point. The reflection angle of the shock at the
  upper wall is determined by two conditions
• (a) M2 lt M1
• (b) The flow downstream of the reflected shock
  
• wave must be parallel to the upper
  wall. That is, the flow is deflected downward by
  q.

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Pressure-Deflection Diagram

• The pressure-deflection diagram is a plot of the
  static pressure behind an oblique shock versus
  the flow deflection angle for a given upstream
  condition.
• For left-running waves, the flow deflection angle
  is upward it is considered as positive. For
  right-running waves, the flow deflection angle is
  downward it is considered as negative.

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Intersection of Shock Waves of Opposite Families

• Consider the intersection of left- and
  right-running shocks (A and B). The two shocks
  intersect at E and result in two refracted shocks
  C and D. Since the shock wave strengths of A and
  B in general are different, there is a slip line
  in the region between the two refracted waves
  where
• (a) the pressure is continuous but the entropy
  is
• discontinuous at the slip line
  (b) the velocities
  on two sides of the slip line are in the
• same direction but of different
  magnitudes

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Intersection of Shock Waves of the Same Family

• As two left running oblique shock waves A and B
  intersect at C , they will form a single shock
  wave CD and a reflected shock wave CE such that
  there is slip line in the region between CD and
  CE.

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Prandtl-Meyer Expansion Waves

• M2 gt M1. An expansion corner is a means to
  increase the flow mach number.
• P2/p1 lt1, r2/r1 lt1, T2/T1 lt 1. The pressure,
  density, and temperature decrease through an
  expansion wave.
• The expansion fan is a continuous expansion
  region, composed of of an infinite number of Mach
  waves, bounded upstream by m1 and downstream by
  m2.

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Prandtl-Meyer Expansion Waves (contd.)

• Centered expansion fan is also called
  Prandtl-Meyer expansion wave.
• where m1 sin-1(1/M1) and m2 sin-1(1/M2).
• Streamlines through an expansion wave are smooth
  curved lines.
• Since the expansion takes place through a
  continuous succession of Mach waves, and ds 0
  for each wave, the expansion is isentropic.

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Prandtl-Meyer Expansion Waves (contd.)

• For perfect gas, the Prandtl-Meyer expansion
  waves are governed by
• Knowing M1 and q2, we can find
• M2

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Prandtl-Meyer Expansion Waves (contd.)

• Since the expansion is isentropic, and hence To
  and Po are constant, we have

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Shock-Expansion Method-Flow Conditions
Downstream of the Trailing Edge

• In supersonic flow, the conditions at the
  trailing edge cannot affect the flow upstream.
  Therefore, unlike the subsonic flow, there is no
  need to impose a Kutta condition at the trailing
  edge in order to determine the airfoil lift.
• However, if there is an interest to know the flow
  conditions downstream of the T.E., they can be
  determined by requiring the pressures downstream
  of the top- and bottom-surface flows to be equal.

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Conditions Downstream of the T.E.-An Example

• For the case shown, the angle of attack is less
  than the wedges half angle so we expect two
  oblique shocks at the trailing edge.
• In order to know the flow conditions downstream
  of the airfoil, we start a guess value of the
  deflection angle g of the downstream flow
  relative of the free stream.
• Knowing the Mach number and static pressure
  immediately upstream of each shock leads to the
  prediction of the static pressures downstream of
  each shock.
• Then through the iteration process, g is changed
  until the pressures downstream of the top- and
  bottom-surface flow become equal.

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Total and Perturbation Velocity Potentials

• Consider a slender body immersed in an inviscid,
  irrotational flow where
• We can define the (total) velocity potential F
  and the perturbation velocity potential f as
  follows

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Velocity Potential Equation

• For a steady, irrotational flow, starting from
  the differential continuity equation
• we have
• In terms of the velocity potential F(x,y,z), the
  above continuity equation becomes

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Linearized Velocity Potential Equation

• By assuming small velocity perturbations such
  that
• we can prove that for the Mach number ranges
  excluding

the transonic range
the hypersonic range
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Linearized Pressure Coefficient

• For calorically perfect gas, the pressure
  coefficient Cp can be reduce to
• For small velocity perturbations, we can prove
  that
• Note that the linearized Cp only depends on u.

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Prandtl-Glauert Rule for Linearized Subsonic Flow
(2-D Over Thin Airfoils)
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Cp of 2-D Supersonic Flows Around Thin Wings

• For supersonic flow over any 2-D slender airfoil,
  
• where q is the local surface inclination with
  respect to the free stream

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Cl of 2-D Supersonic Flow Over Thin Wings

• For supersonic flow over any 2-D slender airfoil,

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Cm of 2-D Supersonic Flow Over Thin Wings

• For supersonic flow over any 2-D slender airfoil,
  the pitching moment coefficient with respect to
  an arbitrary point xo is
• The center of pressure for a symmetrical airfoil
  in supersonic flow is predicted at the mid-chord
  point.