Classical Inflow Models

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Classical Inflow Models
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Background

• In Part II and Part III of our lecture notes, we
  used Glauerts inflow model to compute the
  induced velocity in forward flight.
• This model gives acceptable results for
  performance, but does not give good results for
  blade dynamics or vibratory loads.
• It fails miserably for Blade-Vortex-Interactions.

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Why does the model fail in forward
flight?Because most of the wake is the first and
second quadrants.We can not expect the induced
velocities to be uniform!
Freestream
Tip Vortices
Rotor Disk
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Experiment by Harris

• Because most of the tip vortices are in the first
  and fourth quadrants, the induced velocity
  associated with these vorticies (i.e. the inflow)
  is very high in the first and fourth quadrants.
• The resultant reduced angle of attack leads to
  smaller lift forces in the aft region, compared
  to uniform inflow model.
• The blade responds to this reduced loads 90
  degrees later.
• It flaps down (leading to a more negative b1s)
  more than expected.
• Harris experimentally observed this, in a paper
  published in 1972 in the Journal of the American
  Helicopter Society (See Wayne Johnson, page 274,
  figure 5-39).

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Effects of Non-Uniform Inflow on Lateral Flapping
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Effects of Non-Uniform Inflow on
Blade-Vortex-Interaction Loads

• During descent the tip vortices are pushed up
  against the rotor disk by the freestream, leading
  to a very close spacing between the blades and
  the vortices.
• According to Biot-Savart law, this leads to very
  high, and very rapidly varying, induced velocity.
• This affects the airloads dramatically.
• Ignoring this rapid variation in the inflow and
  using a Glauert inflow model will lead to a
  severe underprediction of vibratory loads, and
  associated aerodynamically generated noise.

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Blade Vortex Interaction
The velocity field experienced by the blade
changes dramatically in a matter of mili-seconds!
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Efforts to Improve Inflow Models

• Engineers and researchers recognized very quickly
  that there is need for improvements in the inflow
  model.
• Until the mid 1960s, the work was analytical or
  semi-empirical. This approach is called classical
  vortex theory. See pages 134-141 of text, and our
  web site.
• Starting 1960s, numerical approaches based on
  Biot-Savart law became popular.

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Colemans Model
Coleman divided the Continuous helical vortex
into a Series of circular rings. He assumed that
the vortex strength is constant.
Real Tip Vortex is helical, and distorted. Its
strength varies along its length.
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Side View
Coleman used Biot-Savart law, and numerical
integration to compute the Induced velocity at a
few points on the rotor disk.
Induced velocity v Normal component of
freestream l WR
Rotor Disk
Edgewise component of Freestream Velocity m WR
Successive Rings of Vortices
Gconstant in Colemans Model
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Castles et als Model

• This model is discussed in NACA report 1184. A
  pdf file may be found at the web site
  http//www.ae.gatech.edu/lsankar/AE6070.Fall2002/
  castles.pdf
• Castles replaced the numerical integration in
  Colemans model with analytical integration.
• These authors also replaced the individual rings
  by a continuous sheet of vorticity.

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Castles Model
The individual rings were replaced by a skewed
cylindrical surface. Vorticity strength was
uniform on the cylinder surface. Biot-Savart Law
was used. Results are supplied as charts and
graphs.
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Heyson et als Model

• This model is discussed in the NACA Report 1319.
  An electronic version may be found at
  http//www.ae.gatech.edu/lsankar/AE6070.Fall2002/
  heyson.pdf
• This model rectifies one of the assumptions in
  Castles and Colemans models, namely that the
  rotor only sheds tip vortices.

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Radial Variation of Loads causes a continuous
shedding of vorticity
Heyson assumed that the Circulation G varies
linearly With radius.
G
r/R
Heyson modeled each of the trailing vortices as a
vortex sheet whose shape is in the form of a
skewed cylinder. The effects of each of the
cylinders may be superposed, using the analytical
formulas developed by Castles et al.
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Heysons Model
At each radial location A vortex sheet of
shape Similar to a circular Cylinder is
shed. The strength is constant, Both along the
axis of the Cylinder and around the azimuth.
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Advanced Inflow Models
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Background

• Around 1965, more powerful digital computers
  became readily available.
• Engineers began to model the tip vortex as a
  skewed, distorted, helix.
• The strength of the tip vortex was allowed to
  vary with azimuth, acknowledging the fact that
  the blade loading changes with the azimuthal
  position of the blade.

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Background (Continued)

• Sadler at Bell Helicopter, and Scully at MIT
  developed some of the earliest techniques.
• Sadler used a rigid helical wake, while Scully
  allowed for the wake to deform due to
  self-induced velocities.
• The induced velocity at the strips on the blade
  were computed using Biot-Savart Law.
• Modern methods (e.g. CAMRAD-II) not only model
  the tip vortex, but also the inboard vortices,
  and shed vortices.
• See our web site for several publications related
  to CAMRAD-II.

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Shed Wake
When the lift (or bound circulation) around an
airfoil changes with time, Circulation that is
equal in magnitude to the bound circulation but
opposite In strength is shed into the wake.
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Trailing and Shed Wake Representation
The blade is divided into a number of strips, as
in HW1 and 2
Hub
Strip j
Tip
The end points of vortex segments are Called
markers.
Shed Vortex
Tip Vortex
Inboard Trailing Vortex
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Trialing Vortex Strength
Bound Circulation On next strip GdG
Bound Circulation On this strip G
dG
Right hand screw rule is applied to assign the
orientation of the vortex.
The strength of the most recently shed trailing
vortex segment may be thought of as the change in
the bound vortex strength between adjacent
strips.
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Shed Vortex Strength
Blade position at next time step n1
Strip j
Gjn1
Trailing vortices
Tip vortex
Blade position At time step n
Strength of shed Vortex segment Gjn1-Gjn
The strength of the shed vortex segments Will
vary from one strip to the next.
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Induced Velocity Calculation

• The calculations are done in a time marching
  mode.
• The strength of the shed and trailing vortices
  are known from the previous time steps.
• The geometry of the wake is assumed to be a helix
  (rigid), prescribed (deformed wake, curve fitted
  from experiments) or free wake.
• Free wake geometry is computed by allowing the
  wake to move at the freestream velocity plus the
  induced velocity computed at the junction points
  (markers).

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Rigid Wake Model
where the ? is the azimuthal angle of the
reference blade, and f is the vortex wake age.
mx, mx and mz are advance ratio Components along
x-, y-, and z- directions. The Glauert uniform
inflow model is used to estimate the inflow
components ?x, ?y and ?z , the three components
of l.
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Wake Age
f
Tip Vortex
x
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Prescribed Wake Model
In this model, the x and y coordinates of the tip
vortex are prescribed from a rigid wake model.
The vertical displacements of the tip vortices
are given as
where E is an envelop function given by
Here, A0, A1, M, B, Cn and Dn are all empirical
constants listed in Egolf, A. and Landgrebe, A.
J., Helicopter Rotor Wake Geometry and Its
Influence in Forward Flight, Vol. I, NASA
CR-3726.
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Free Wake Geometry