

Wind Effects on V Speeds
by: John T. Lowry, Ph.D
Introduction
In aircraft performance questions we generally focus on air
speed, our airplane's movement with respect to the air mass, but
there are reasons for paying attention to our airplane's progress
over the earth. During navigation we want to see how we're
progressing from Point A to Point B, on the earth. We
also care, on a smaller scale, about movement with respect to the
ground whenever we deal with two particular V speeds: speed for
best (steepest) angle of climb V_{x}, and speed for best
(longest) glide V_{bg}. With V_{x} we're trying
to clear some approaching earthbound obstacle; with
V_{bg} we're trying to get as far as possible over the
ground before contacting it.
While Robert Jordan, in Ernest
Hemingway's For Whom the Bell Tolls,reluctantly reported
to Pilar, as she questioned him about his making love to Maria,
that "Yes, ... It [the earth] moved," and even though geologists
report, not at all reluctantly, that the earth's tectonic plates
move about an inch a year, we can safely ignore any such
movement. So aircraft movement with respect to the earth is due
to just two phenomena: (1) aircraft movement with respect to the
air; and (2) air movement with respect to the earth. The first
gives air speed; the second is wind.
The purpose of this article is to
quantify the effect wind has on V_{x} and V_{bg}.
We'll cover the exact relationship between speeds with respect to
the air and with respect to the earth, a couple of approximations
to that relation (not limiting for the great majority of general
aviation aircraft) and we'll give graphical and formulabased
methods for coming up with numerical wind effects. We'll use
sample headwinds and tailwinds to demonstrate those effects on a
typical Cessna 172. In the end you'll be able to calculate
correct values of V_{x} and V_{bg}, in reasonable
winds, for your particular airplane.
Background
The rule for computing relative motion gets down to this
simple vector equation:
Here V stands for velocity, speed together
with its direction, stands for the earth, pfor the
airplane, and afor the air. The way to remember this
rule is to notice the collapse of the "interior" repeated symbol
on the right. That's what counts. In English, the rule reads:
velocity of the plane with respect to the earth (ground velocity)
equals velocity of the air with respect to the earth (wind
velocity) plus velocity of the plane with respect to the air (air
speed with heading). Of course Eqn. (1) works for any three
objects. From the physicist's point of view, there's nothing
special about the earth, the air, and airplanes. Aviators know
better.
Here's what Eqn. (1) looks like as
a vector diagram.
Figure 1 gives less pedantic
abbreviations of the some of the velocities and introduces
h,^{}"hdot," as the rate of climb. Symbol is
conventionally used for the flight path angle; there and
elsewhere parentheses take the places of (unavailable) subscripts
to tell us precisely which velocities and angles we're talking
about.
There are lots of symbols in Figure
1. Now what are we after? The fullthrottle air speed (and, in
the gliding case, with a different picture, the nothrottle air
speed) V_{a} (also simply V in the Figure) which, in a
given (here) headwind, will give us the largest flight path angle
with respect to the earth, _{ p}, henceforth simply
_{p}. It makes sense to call that air speed "Vsubx in
the wind," V_{xw}. But remember, since we fly solely by
reference to air speeds, that V_{xw} is an air
speed. In our graphs and calculations it will be a
trueair speed, but it's easy enough to later convert
that to more operational calibrated, or even indicated, air
speed.
So we want to maximize _{p}
with respect to V. To do that, we're going to need an algebraic
relationship between those two variables. One such comes from
briefly staring at Figure 1:
We need V_{p} which, as Figure 1 shows, depends on
V_{a}, V_{w} and _{a}. Application of the
"law of cosines," the Pythagorean theorem for nonright
triangles, gives:
What saves us, in using Eqn. (3), is that _{a} is a
small angle and therefore its cosine is quite close to unity.
Even cosine 8, probably a larger climb or glide angle we'll
encounter or achieve, only differs from unity by 1%. This small
flight path angle approximation is one we've made consistently
and is the one largely responsible for the relative simplicity of
TBA and of the poweravailable/powerrequired
(P_{a}/P_{r}) analysis. With the cosine set equal
to one, the right hand side of Eqn. (3) is a perfect square and
we have the simple relationship:
Of course! All we're saying is that, for general aviation
aircraft, flight path angles are so shallow we'll be using the
gliding version, too, rememberthat there's no practical
difference between the (slant) air speed and its horizontal
component. Figure 1 doesn't support that statement, but that's
because we exaggerated the angles so much. None of these
airplanes climbs steadily at 30, not even a Bonanza!
Combining Eqns. (2) and (4), we
have
This simple but good approximation will be the basis of the
following graphic and analytic calculations of V_{xw}. In
a glide, both h^{} and _{p} (and its sine) will
be negative. But the same relation still holds. It's also true,
for small angles, that:
Those two trigonometric functions differ by less than 0.001 up
until nearly 8.
The "Exact" Bootstrap
Formula for Sin(_{p})
To investigate how well Eqn. (5) approximates the "exact"
formula, combine Eqns. (2) and (3) to get:
If one uses later Eqn. (8), from The Bootstrap Approach, to
evaluate h^{}, our sample Cessna at MSL gives
V_{x} = 56.86 KCAS for a 20 knot headwind, 48.96 KCAS for
a 30 knot headwind. Eqn. (5) gives, instead, 56.81 and 48.82
KCAS, respectively. So that approximation is not a problem. We
placed 'exact' in quotes, in this sidebar, because TBA uses a
small angle approximation itself. Hardly anything is exact in
aerodynamics or in airplane performance calculations.
The Graphical
Approach
From performance flight tests, or otherwise, assume you have a
graph of h^{}, as a function of air speed V, for a given
weight and density altitude (or relative air density):
h^{}(V;W,). The graph peaks, of course, at V_{y}
(for climbs) or at V_{md} (for glides). If you start from
the origin and lay off various lines hitting the h^{}
curve, the one of greatest slope is the one tangent to the curve.
If you go steeper, you miss the h^{} curve. See Figure 2.
(The somewhat oddlysized wind speeds are due to our desire to
plot calibratedair speeds along the horizontal axis. At
4000 ft density altitude, 20 KCAS = 21.2 KTAS.) Using the
approximate "fact" that the slant length from the origin to the
curve is the same as the horizontal distance V, Eqn. (5) tells us
that the point of tangency is at the speed which is V_{x}
in a calm. Furthermore, starting another tangent line from point
(V_{w}, 0) along the V axis, Eqn. (5) tells us that
itspoint of tangency is at the speed for best angle of
climb into a headwind of speed V_{w}, i.e., at
V_{xw}.
So that's all there is to the
purely graphical approach to finding V_{xw}(W,) or
V_{bgw}(W,):
Draw a graph of h^{}(V;
W,);
Lay off a tangent from
(V_{w}, 0) to that graph; and
Come straight down from the point
of tangency, to the V axis, and read off V_{xw} (W,) or
V_{bgw}(W,).
Simply treat tailwinds as negative
headwinds. If you take starting points for your tangent lines
farther and farther to the left, you can see that speed for best
angle, V_{xw}, approaches speed for best rate of climb,
V_{y}.
What about the other way, as
V_{w} gets larger? Eventually our approximationsmall
path angles breaks down. As V_{w} approaches V =
V_{a}, Eqn. (5) blows up. Put into practical terms, in a
60 knot wind you've got bigger problems than worrying about how
steep your climb is.
When you're gliding, P_{a}
= 0 and so h^{} = P_{r}/W and the peak of
P_{r}/W is at V_{md}. See Figure 3. As you back
up the starting point for the tangent, taking larger and larger
tailwinds, V_{bg} approaches V_{md}. As you take
larger headwinds, it looks like V_{bg} keeps increasing.
It does increase, but don't forget that when the wind speed gets
to be comparable to the air speed (say about half its size), our
approximations break down. When wind speed is higher than your
airplane's stall speed you can stay aloft forever.
Speed for best glide V_{bg}
is just the unpowered version of V_{x}, speed for best
angle of climb. Indeed, when gliding, V_{bg} is
your speed for greatest positive (least negative) flight path
angle. In similar fashion, V_{md} is just the unpowered
version of V_{y}. A rate of sink is just a rate of climb
which happens to be negative.
The Formula
Approach
The Bootstrap Approach furnishes us simple formulas for excess
power and for rate of climb:
Here E, K FG, F, G, and H are all "composite" Bootstrap
parameters which depend on the nine Bootstrap Data Plate (BDP)
parameters and the two variables W and . So once we've tied down
the BDP, we're in perfect position to use elementary differential
calculus to come up with expressions for V_{xw} and
V_{bgw}. Solving those expressions will turn out to be
something else again.
For reference, Table 1 gives the
BDP data used to construct the rateofclimb graphs of Figures 2
and 3. Gross weight W = 2400 lbf and relative air density
(corresponding to h_{} = 4000 ft) = 0.8881. The figure
for rated torque M_{0} comes from the Lycoming engine's
rated power 160 HP at rated speed 2700 RPM. Table 2 gives, for
this gross weight and air density, values of the various
Bootstrap composite parameters.
Computing the derivative of the
right hand side of Eqn. (5), using Eqn. (8), and then setting the
resulting expression equal to zero, one gets, for the
fullthrottle climb,
BDP Item  Symbol  Value  Units 
Reference Wing Area  S  174  ft^{2}  Wing Aspect Ratio  A  7.378   Parasite Drag Coefficient  C_{D0}  0.037   Airplane Efficiency Factor  e  0.72  
Rated Engine FullThrottle
Torque  M_{0} 
311.2  ftlbf  Loss Proportion Independent of Air Density  C  0.1137   Propeller Diameter  d  6.25  ft  Slope of Linearized Propeller Polar  m  1.70  
Intercept of Linearized
Propeller Polar  b  0.0564   Table
1. Typical Cessna 172 Bootstrap Data Plate.
In the gliding case, E = F = 0
and so K = G. Then we have the similar equation
Symbol  Value  Units  E  464.7  lbf  F  0.0046508  slug/ft  G  0.0067952  slug/ft  H  1,879,309  ftlbf^{2}/slug  K  0.011446  slug/ft  Table
2. Composite Bootstrap parameters corresponding to the
BDP of Table 1, W = 2400 lbf, relative air density = 0.8881.
While these fifth degree
equations can't be solved algebraically, in this form they are
easy to solve numerically. Just plug values for V into
the right hand sides of either Eqns. (9) or (10) until you get
the value of headwind or tailwind desired. That's how we got the
solutions given in the captions of Figures 2 and 3. For a more
comprehensive look, at three widelyseparated altitudes, see
Figure 4, where we used Quattro Pro's SolveFor facility to get
the numbers.
A common rule of thumb is to adjust
calmwind values of V_{x} or V_{bg} by adding or
subtracting half the speed of the encountered headwind or
tailwind. As the curves (not lines) of Figure 4 show, that rule
of thumb greatly overcompensates for light winds; but those are
not the problem. But picking on a sizeable 20 knot headwind (and
settling on moderate altitude 5000 feet), the graph shows one
should back off from calmwind V_{x} only about four
knots and should add to calmwind V_{bg} about six. In a
20 knot tailwind, it suggests adding to calmwind V_{x}
about two knots and subtracting from calmwind V_{bg}
about four knots. Except for V_{bg} and a headwind, the
conventional rule of thumb is about twice what's needed.
Adjusting by onefourth of the encountered headwind or tailwind
is a better rule.
That revised rule is also supported
by calculating the slopes of the graphs at V_{w} = 0.
Evaluating the derivative of V with respect to V_{w}
there (by taking the inverse of the derivative of V_{w}
with respect to V) one finds, in the glide case, that the slope
there is exactly 1/4. In the climb case, the derivative is even
smaller. But again it's not the very small winds one is concerned
about.
Summary
V_{x}, speed for best (steepest) angle of climb
relative to the earth, in comparison with its calmwind value, is
reduced in a headwind and increased in a tailwind.
V_{bg}, speed for best (longest) glide relative to the
earth, in comparison with its calmwind value, is increased in a
headwind and reduced in a tailwind. The simple Bootstrap Approach
formulas for the aircraft's rate of climb, as a function of air
speed, allow one to obtain polynomial functions whose solutions
are those V speeds in any given wind condition. But those
expressions, of fifth degree, cannot be solved analytically.
While numerical solutions are necessary, those are not hard to
get.
We presented both graphical and
numerical solutions to the relevant equations for a sample Cessna
172 at one weight and at several density altitudes. The
corresponding graphs and formulas support a revised
Rule of Thumb for V Speeds
in Wind: In a headwind (tailwind), reduce
V_{x} (V_{bg}), relative to its calmwind value,
by onefourth of the speed of the wind. In a tailwind
(headwind), increase V_{x} (V_{bg}), relative to
its calmwind value, by onefourth of the speed of the wind. Only
for substantial headwinds (twenty knots or more), double the
foregoing prescription for V_{bg}. 