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# Canard Aerodynamics

[[!toc levels=3]]

We are currently working on a 60° delta wing canard.

The aerodynamic characteristics of the delta wing depend on the free stream
velocity and the angle of attack.

The velocity characteristics are usefully divided into low speed
(incompressible), medium speed (compressible), transonic, and supersonic flow.

At sufficiently low angles of attack, often linear theory is sufficient. For
high angles of attack either experimental data or computational fluid dynamics
must be used.

## Low speeds, Incompressible flow7687

### Lift Coefficient at low speeds

At low speeds and low angles of attack, the suction analogy of Polhamus can be
used [1][2].

For higher angles of attack simple theories don't provide accurate values.
For our initial attempt we will look for experimental data to apply [3].


#### Theory of Polhamus

For angles of attack less than about 20° the lift coefficient is
aproximated by a fairly simple expression

[[!teximg code="C_L(\alpha) = \
K_P \cos^2{\alpha}\ \sin{\alpha} +\
K_V \cos{\alpha}\ \sin^2{\alpha}"]]

Where (α) is the angle of attack in radians.

The constants (K<sub>P</sub>) and (K<sub>V</sub>) are lift and suction
coefficients respectively derived from linear lifting surface theory. For our
purposes we can get them from reference [1] page 12.

Since ours is a 60&deg; delta wing, our aspect ratio (_AR_) is

[[!teximg code="\begin{array}{ll}\
 AR &\overset{\underset{\mathrm{def}}{}}{=} b^2/S\\\
  &= b^2/(c b/2)\\\
  &= 4/\tan{\Lambda}\
\end{array}"]]

Where (_b_) is the length of the root chord, (_c_) is the maximum wing span, and
(_S_) is the total wing surface area, and (&Lambda;) is the leading edge sweep angle
as defined for the arrow wing below.

For the present case of a 60&deg; delta wing our aspect ratio is ~2.31
and a reasonable approximation is
(K<sub>P</sub>&nbsp;=&nbsp;2.45) and (K<sub>V</sub>&nbsp;=&nbsp;3.21)

### Aerodynamic Center at low speeds

As a very rough approximation the aerodynamic center of a delta wing in
incompressible flow is located at 50% of the root chord. The aerodynamic center
will typically shift backwards slightly with increasing angle of attack shifting
a total of about 4% when the angle of attack reaches of order 20&deg;.

The aerodynamic center will remain near 50% of the root chord until the Mach
number nears about 0.8, at which point, assuming reasonably low angles of attack,
it should gradually shift toward the supersonic position in a reasonably smooth
manner.

## Medium speeds, Compressible flow

In many cases compressible flow can computed from incompressible flow using an
approximate scaling law such as Prandtl-Glauert or G&ouml;thert similarity.


## Transonic flow

Transonic flow is usually more complex analytically. Initially we will look for
experimental data.


## Supersonic flow

### Lift Coefficient at supersonic speeds

For low angles of attack there is an exact linear theory [4].

For higher angles of attack, initially we can look for experimental data.
However, higher angles of attack may not be too useful initially because
the higher forces per angle at supersonic speeds may restrict our control
envelope to low angles of attack anyway.

---

Restating the formula given in reference [4]

The _arrow wing_ is a slight generalization of the delta wing geometry

[[!img arrowing.png link=arrowing.svg]]

The parameter (&zeta;) shown in the diagram is the _notch ratio_ of the arrow
wing.

For _supersonic_ flow with a _subsonic_ leading edge, the lift coefficient
is given by

[[!teximg code="\beta \, C_{L_\alpha} = \
\frac{4m}{E'(m)}\
\left [\frac{\zeta}{1+\zeta}+\
 \frac{1-\zeta}{(1-\zeta^2)^{3/2}}\
 \cos^{-1}(-\zeta)\
\right ], \qquad m \le 1"]]

For supersonic flow with a _supersonic_ leading edge, the relation is

[[!teximg code="\beta \, C_{L_\alpha} = \
\frac{8m}{\pi(1+\zeta)}\
\left [\frac{1}{\sqrt{m^2-\zeta^2}}\
 \cos^{-1}\left ( -\frac{\zeta}{m} \right )\
 + \frac{\zeta}{\sqrt{m^2-1}} \cos^{-1}\left ( \frac{1}{m} \right )
\right ], \qquad m \ge 1"]]

Where for Mach number (_M_)

[[!teximg code="\begin{array}{lcl}\
\beta &= &\sqrt{M^2-1}\\\
 m    &= &\beta \cot( \Lambda_{LE}\ )
\end{array}"]]

### Aerodynamic center at supersonic speeds

As a rule of thumb the aerodynamic center for low angles of attack in supersonic
flow assuming thin wings is located at 50% of the mean chord. Accounting for the
triangular shape of the delta wing this means that the approximate location of
the aerodynamic center for our supersonic canard is at 2/3 of the root chord.
This result should reasonably accurate for small angles of attack and Mach
numbers of 1.1 or greater.

## Numeric Examples

The PSAS LV2c rollcontrol fins are 30 degree right triangle half-delta wings
with a root chord length of 2.5 inches. What is the lift produced by these fins
at Mach 1.1, at 5000 feet altitude, with a fin deflection of 1 degree?

Calculate the lift coefficient for a full 60 degree delta wing, which would be
two of our rollcontrol fins. Ignore body interference and boundary layer
effects. This is a pretty rough approximation, but serves as a good starting
place.

The basic lift formula gives the force perpendicular to the direction of flight
using the small angle approximation
(cf. the lift and drag diagram on the [[CoordinateSystem]] page)

[[!teximg code="L = \alpha C_{L_\alpha} \frac{1}{2} \rho v^2 A"]]

Where (L) is the lift and (&alpha;) is the angle of attack.

Using the calculator at <http://www.digitaldutch.com/atmoscalc/>, at 5000 feet
in the standard atmosphere the temperature is 5.1&nbsp;C, density
1.055&nbsp;kg/m<sup>3</sup>, and the speed of sound is 334.4&nbsp;m/s.

The Mach angle at Mach = 1.1 is

[[!teximg code="\mu = Arcsin(1/M) = 65.4^\circ"]]

This angle is larger than the fin angle, so the rollcontrol fins have a
subsonic leading edge, and an (m) value < 1.

[[!teximg code="\begin{array}{lcl}\
\beta &= &\sqrt{M^2-1} = 0.4583\\\
 m    &= &\beta \cot( \Lambda_{LE}\ ) = 0.2646
\end{array}"]]

To evaluate CL<sub>&alpha;</sub>, the elliptic integral *E*<sup>'</sup>(m) must
be evaluated. *E*<sup>'</sup>(0)&nbsp;=&nbsp;&pi;/2, for m&nbsp;<&nbsp;1, a
polynomial approximation can be used (See Abramowitz and Stegun [5], 17.3.36.
Note that versus Abramowitz and Stegun, m-squared is used in the series here.
This is intentional and consistent with the aerodynamic formula given above.)

[[!teximg code="\begin{array}{lcl}\
 E^\prime(m) &\approx 1 +\
 \sum_{k=1}^4 a_k m^{2k} +\
 \ln(1/m^2) \sum_{j=1}^4 b_k m^{2j} + \epsilon\\\
 &|\epsilon| < 2 \times 10^{-8}\ , \quad 0 \leqslant m < 1\
\end{array}\
"]]

Where the coefficients are

[[!teximg code="\begin{array}{lcl}\
a_1 &= 0.44325141463\\\
a_2 &= 0.06260601220\\\
a_3 &= 0.04757383546\\\
a_4 &= 0.01736506451\\\
b_1 &= 0.24998368310\\\
b_2 &= 0.09200180037\\\
b_3 &= 0.04069697526\\\
b_4 &= 0.00526449639\\\
\end{array}"]]

Applying this approximation to m = 0.2646, yields
*E*<sup>'</sup>&nbsp;=&nbsp;1.079.

The coefficient of lift per alpha can now be calculated using the formula for
m&nbsp;<&nbsp;1.
In this example (&zeta;&nbsp;=&nbsp;0), so everything in the square brackets
evaluates to arcosine(0)&nbsp;=&nbsp;&pi;/2.
So

[[!teximg code="\beta C_{L_\alpha} = 2 \pi \frac{m}{E^\prime(m)} = 1.5405"]]

Dividing by (&beta;), the lift coefficient is found to be
(C<sub>L<sub>&alpha;</sub></sub>&nbsp;=&nbsp;3.3616).
Taking the fin area as 11.64&nbsp;cm<sup>2</sup>,
which is the area (A) of a single rollcontrol fin,
the lift of one fin under the specified conditions is

[[!teximg code="\begin{array}{ll}
 L &= \alpha C_{L_\alpha} \frac{1}{2} \rho v^2 A\\\
 &= 0.01745 \cdot 3.3616 \cdot 0.5\
    \cdot 1.055 [\rm{\frac{kg}{m^3}}]\\\
    &\quad \cdot(1.1 \cdot 334.4 [\rm{\frac{m}{s}}])^2\
    \cdot 11.64\times10^{-4} [\rm{m}^2]\\\
 &= 4.87 [\rm{N}]\
\end{array}"]]

## References

[1] <http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680022518_1968022518.pdf>
<br>Application of the Leading-Edge-Suction Analogy of Vortex Lift to the Drag Due
to Lift of Sharp-Edge Delta Wings

[2] <http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19670003842_1967003842.pdf>
<br>A Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge-Suction Analogy

[3] <http://www.kfupm.edu.sa/ae/fspub/2006-44asm-saeed-aiaa2006-0063-deltaw.pdf>
<br>Contains experimental lift coefficient for 65&deg; delta wing at low speeds and high angles of attack

[4] <http://www.aoe.vt.edu/~mason/Mason_f/ConfigAeroSupersonicNotes.pdf>
<br>Page 13 has theory on arrow wings at supersonic velocities.

[5] <http://en.wikipedia.org/wiki/Abramowitz_and_Stegun>