Angle of climb at Vx and glide angle when "overweight": five questions

GC: I'd recommend a copy of *Aerodynamics for Naval Aviators* It's a
good start at explaining it all without getting too involved in the
math.

KL: Thanks for the tip. I seem to collect every book except the right
one!
------------------------------------------------------

You don't seem to have any trouble with the math, so I'm not sure what
value that AFNA would have for you. You're looking for more intuitive
explanations than what the math provides, and that's really hard to
come by. Most aerodynamics books don't have a lot of interest in
providing what you want.

On Thu, 27 Nov 2003 at 15:34:14 in message
.net, cddb
wrote:

So somehow, the magic equations seem to explain my confusion...but I
am not quite at peace. Au contraire. I've only talked about L/D,
weight, and thrust. What about drag? I am still claiming that you'll
experience an increase in drag with a weight increase REGARDLESS of
whether or not you have thrust. Thus, due to the increase in drag and
REGARDLESS of thrust or not, your angle THETA will be affected (i.e.
theta will get larger, more negative, airplane will pitch DOWN). In
the case of gliding flight, everybody but me agrees that your glide
angle won't be affected. Obviously, I am wrong but I don't know why.
To reiterate, that's why I've been writing all this stuff, 'cause I
don't get why THETA happens to change only when you have thrust but
doesn't when you don't have any and are gliding.

By rearranging the simple equations I may be able to help you with that
one.

W= weight, L = Lift, D = Drag, t = Thrust, theta = angle of climb (if
negative implies descent, V = forward speed

You started with L = W*cos (theta)
and T
= D + W*sin(theta)

For a change eliminate W from the above
by taking W = L/(cos(theta)

Then T = D + L*tan(theta)

Let K = 0.5*density* wing area

Then T = K*V^2*Cd + K*V^2*Cl Divide both sides by K*V^2

T/(K*V^2) = (Cd + Cl*tan(theta))..........................[1]

Now if T= 0 the left hand side must be 0 and it follows that

Cd = - Cl*tan(theta) this means that theta must be negative and

tan(theta) = -Cd/Cl as one would expect.

The reverse of this is that positive or negative thrust is always needed
_except_ when Tan(theta) = -Cd/Cl

At any other climb angle, thrust or extra drag would be needed for
balance. To glide more steeply at the same Cl and Cd negative thrust
would be needed

Perhaps it does not help after all?

Also in [1] if theta = 0 then

T = K*V^2*Cd which is a necessary condition for steady level flight..




--
David CL Francis

On Fri, 28 Nov 2003 at 16:59:43 in message
, Greg Esres
wrote:

I vote with Peter. Minimum sink is least POWER required, not least
THRUST.

But surely minimum sink rate is only relevant when there is no power or
thrust? It requires finding a minimum value of V*sin(theta) where theta
is the angle of climb (negative when descending).

As I recall for a gliding aircraft minimum sink comes roughly at the AoA
where (Cl^(3/2)/Cd is a maximum. This is normally at a higher AoA than
maximum Cl/Cd and is some cases is quite close to the stall.

Power is drag (or thrust) times velocity.
--
David CL Francis

But surely minimum sink rate is only relevant when there is no power
or thrust?

I suppose so. If power were constant with airspeed, then minimum sink
would also be Vy.

It requires finding a minimum value of V*sin(theta) where theta is
the angle of climb (negative when descending).

Sounds good.

As I recall for a gliding aircraft minimum sink comes roughly at the
AoA where (Cl^(3/2)/Cd is a maximum.

I'd have to look it up to be sure, but it looks right.

This is normally at a higher AoA than maximum Cl/Cd and is some
cases is quite close to the stall.

Agreed, except, according to the books, the velocity of minimum power
is ALWAYS less than least drag.

But surely minimum sink rate is only relevant when there is no power
or thrust?

I suppose so. If power were constant with airspeed, then minimum sink
would also be Vy.

Yes, if power available was constant with airspeed, Vy would also be
your min. sink rate speed. The books indicate that power for a
piston-engine propeller combination increases with velocity. The
increase is not linear though. The derivative dP/dV looks to be of
the form y = -mx + b. Kinda like an upside down smiley face with only
the left side showing.

Thrust available looks fairly constant w.r.t. velocity.



It requires finding a minimum value of V*sin(theta) where theta is
the angle of climb (negative when descending).

Sounds good.

As I recall for a gliding aircraft minimum sink comes roughly at the
AoA where (Cl^(3/2)/Cd is a maximum.

I'd have to look it up to be sure, but it looks right.


That's correct. Min sink rate would occur at the AOA where
1/[(Cl^3/2)/cd] is a minimum as it is the point of least power
required. Min. thrust required is proportional to 1/(Cl/Cd).



This is normally at a higher AoA than maximum Cl/Cd and is some
cases is quite close to the stall.

Agreed, except, according to the books, the velocity of minimum power
is ALWAYS less than least drag.


What book are you reading Greg? My references indicate the same for a
Cessna Skylane. If you assume no power available, then, your min.
sink (best endurance) speed is less than your best glide speed.

My Cessna 172SP POH lists a best glide speed of 68 kts. Best
endurance speed isn't exactly mentioned but I've heard it's close to
stall. I just recall someone saying "If you want to stay up for as
long as you can, fly close to stall". Do you know why the POH doesn't
mention that speed? Would that be giving away too much?

Oh, by the way, could you please give a shot at answering my questions
that are contained in my Nov. 27 reply to Gerry Caron? (in this same
overall post)

Thanks,
Alex

Dunno if this has already come up in these threads, but John Denker has a
great geometric presentation of the interdependency of weight, power, speed,
and flight angle at http://www.av8n.com/how/htm/power.html#sec-power-curve.

On Sat, 29 Nov 2003 at 01:19:53 in message
, Greg Esres
wrote:
This is normally at a higher AoA than maximum Cl/Cd and is some
cases is quite close to the stall.

Agreed, except, according to the books, the velocity of minimum power
is ALWAYS less than least drag.

Sounds reasonable and I don't think it is inconsistent.
--
David CL Francis