## Density Altitude Effects on Speeds-To-Fly Calculations

### Credits

Several fine pages discuss the general definition and calculation of density altitude. Among the best are:

- USA Today essay "Understanding Density Altitude"
- Airman Mike's Density Altitude Page

I thank Tom Waits for getting me interested in the effect of density altitude on polar calculations. Until he mentioned it, I hadn't really thought much about it. Welch, Welch, and Irving (The Complete Soaring Pilot's Handbook, McKay: New York, 1977), is an absolutely must-read (if you can find it) for anyone interested in the subject of calculations related to soaring.

### The Question

All pilots understand that high density altitude (read high density altitude, not high density altitude, i.e. conditions for which air density is such that it is like flying at high altitude) has a detrimental effect on aircraft performance. The conditions of high altitude, high humidity, and high temperature all result in air that is especially low in density, and together these three conspire to reduce lift.

The question Tom had was this: How does a polar change when density altitude is considered?

I posed the broader question: How does density altitude effect speed-to-fly calculations in general?

### "Equivalent" Speeds

I turned to Welch, Welch, and Irving for the answer to this question and was startled to (re)learn that polars are generally expressed in rather odd units. The drag equation used for calculating a polar includes in both the parasitic drag part and the induced drag part the term, "r V2." Here r is the air density, and V is the true airspeed. To remove air density from consideration, the trick is to replace r V2 with r0Vi2, which involves the equivalent airspeed. Equivalent airspeed is given the subscript i because it turns out to be the indicated airspeed for a "perfect" airspeed indicator. Here r0 is the "standard atmosphere" sea level air density.

The problem is that do make this work, the true rate of sink, Vs, must also be replaced by the equivalent rate of sink, using

Vsi=sqrt(r/r0)Vs

The polar is then plotted as Vsi vs. Vi.

This leads Welch, Welch, and Irving to conclude (page 261) that

> The performance curve of the glider applies at all altitudes provided that both the forward speed and the rate of sink are "equivalent" speeds.

It would appear that polars are unaffected by density altitude.

### The Problem

Unfortunately, while the equivalent airspeed, Vi, has a very practical meaning (indicated airspeed), the equivalent rate of sink, Vsi, has no such use. No vario reports Vsi. In fact, according to Welch, Welch, and Irving, mechanical varios accurately report true sink rate, and electrical varios display "something like" the true sink rate times r/r0 (which may be considerably different from sqrt(r/r0).

Furthermore, if wind and actual lift or sink are taken into account, those values are always in terms of true speeds. This makes polars plotted using equivalent speeds essentially useless if one wants to consider the effects of density altitude on glider performance.

### The Solution

The solution, it seems to me, is to go ahead and let the polar be a function of air density. That is, a truly useful polar—one that can be used to calculate speeds to fly—must be plotted in terms of true speeds, not equivalent speeds. Of course, that requires on-the-fly graphing, because different density altitudes will produce different polars. And that is exactly what the Speed-To-Fly Calculator does. It allows us to create polars for a given aircraft under a given set of conditions of wind, angle of bank, wing loading, and density altitude.

### The Findings

In a nutshell, Tom's intuition was right on target. High density altitude clearly diminishes glider performance, although not necessarily the way one might expect. Several examples are provided below, all based on data for specific sailplanes.

1. In the absence of wind, lift, and sink, speeds to fly are unchanged at high density altitude. This includes the speed for best L/D and the speed for minimum sink. These speeds are unchanged because they are indicated speeds. Note, however, that ground speed is increased, because true speed must be higher at high density altitude to achieve the same indicated speed.

>     Standard Libelle 15m
>     Wing Loading = 350 N/m2
>     76 (km/h,m/s) data points
>     Angle of Bank: 45 degrees
>     Wind from 000 @ 0 knots; Lift 0 ft/s

    Density Altitude: 0 ft
    Minimum Sink 3.4 ft/s at 45 knots
    Best L/D 26:1 at 60 knots
    Best Vgrd=60 knots (Vsink=3.8 ft/s)

    Density Altitude: 6000 ft
    Minimum Sink 3.7 ft/s at 49 knots true (45 indicated)

    Best L/D 26:1 at 66 knots true (60 indicated)
    Best Vgrd=66 knots (Vsink=4.2 ft/s)

2. In the absence of wind, lift, or sink, L/D is not affected by density altitude. This is because Vi/Vsi = V/Vs.

>     Schweizer 1-35
>     Wing Loading = 293 N/m2
>     151 (knots,ft/min) data points
>     Angle of Bank: 0 degrees
>     Wind from 000 @ 0 knots; Lift 0 ft/s

    Density Altitude: 0 ft
    Minimum Sink 1.9 ft/s at 34 knots
    Best L/D 38:1 at 48 knots
    Best Vgrd=48 knots (Vsink=2.1 ft/s)

    Density Altitude: 6000 ft
    Minimum Sink 2.0 ft/s at 38 knots true (34 indicated)
    Best L/D 38:1 at 53 knots true (48 indicated)
    Best Vgrd=53 knots (Vsink=2.3 ft/s)

3. Minimum sink and sink at the speed for best L/D are both higher at higher density altitude. This is simply a consequence of the y-axis of the polar graph being "stretched" in switching from Vsi to Vs.

>     PW-5
>     Wing Loading = 1.00 rel to 1
>     76 (km/h,km/h) data points
>     Angle of Bank: 0 degrees
>     Wind from 030 @ 20 knots; Lift 0 ft/s

    Density Altitude: 0 ft
    Minimum Sink 1.9 ft/s at 35 knots
    Best L/D 20:1 at 49 knots
    Best Vgrd=30 knots (Vsink=2.5 ft/s)

    Density Altitude: 6000 ft
    Minimum Sink 2.1 ft/s at 38 knots true (35 indicated)
    Best L/D 21:1 at 52 knots true (48 indicated)
    Best Vgrd=34 knots (Vsink=2.7 ft/s)

4. Headwinds and tailwinds have less effect at high density altitude. (This should make sense. The wind has less oomph, right? In addition, you are flying faster to stay at the best speed to fly.) Note that for both minimum sink and best L/D, you don't have to fly as fast (as indicated on your altimeter) at high density altitude as you would at a lower altitude.

>     Nimbus II
>     Wing Loading = 1.00 rel to 1
>     76 (km/h,m/s) data points
>     Angle of Bank: 0 degrees
>     Wind from 005 @ 30 knots; Lift 0 ft/s

    Density Altitude: 0 ft
    Minimum Sink 1.1 ft/s at 31 knots
    Best L/D 19:1 at 59 knots
    Best Vgrd=29 knots (Vsink=2.5 ft/s)

    Density Altitude: 6000 ft
    Minimum Sink 1.1 ft/s at 32 knots true (29 indicated)
    Best L/D 21:1 at 61 knots true (56 indicated)
    Best Vgrd=31 knots (Vsink=2.4 ft/s)

5. In sink or with a headwind, L/D is less affected at high density altitude than under standard conditions. This is probably due to the fact that although the indicated airspeeds aren't much different for optimal performance at different density altitudes, the true speeds can be dramatically higher at high density altitude. That means that penetration is better at high density altitude, leading to a higher ground speed.

>     Ximango Polar Data
>     Gross Wt = 89728 Pa
>     101 (knots,ft/min) data points
>     Angle of Bank: 0 degrees
>     Wind from 000 @ 0 knots; Lift -5 ft/s

    Density Altitude: 0 ft
    Minimum Sink 7.9 ft/s at 53 knots
    Best L/D 12:1 at 62 knots
    Best Vgrd=62 knots (Vsink=8.6 ft/s)

    Density Altitude: 6000 ft
    Minimum Sink 8.2 ft/s at 58 knots true (53 indicated)
    Best L/D 13:1 at 66 knots true (60 indicated)
    Best Vgrd=66 knots (Vsink=8.7 ft/s)

6. Turning radii and the resultant altitude loss on turning are both increased at high density altitude. This is a direct consequence of the increased true speeds necessary for proper flight at high density altitude. Note that in any case, the optimum bank angle for a turn in order to minimize height loss is 45 degrees.

>     Discus
>     Wing Loading = 350 N/m2
>     76 (km/h,m/s) data points
>     Angle of Bank: 5 degrees
>     Wind from 000 @ 0 knots; Lift 0 ft/s
>     Speeds are knots indicated
>     (height loss, radius) are in feet

    Density Altitude: 0 ft
    Bank  V_Best_L/D        V_MinSink
    Angle    (loss, radius)    (loss,radius)
    30    60 (47.7, 568)    49 (35.1, 368)
    45    67 (42.1, 395)    54 (30.3, 258)
    60    79 (47.7,322)     64 (34.8,209)

    Density Altitude: 6000 ft
    Bank  V_Best_L/D        V_MinSink
    Angle    (loss, radius)    (loss,radius)
    30    60 (57.0, 668)    48 (41.5, 431)
    45    67 (50.3, 472)    54 (36.2, 308)
    60    79 (57.0, 386)    64 (41.7, 250)

7. MacCready calculations are largely unchanged taking density altitude into account. Still, the trend is to lower INDICATED speeds between thermals and higher TRUE cruising speeds.

>     Crosscountry (MacCready) Table

>     76 (km/h,m/s) data points for Janus B
>     Wing Loading = 350 N/m2

>     Vi (Vcruise)
>     where Vi is the INDICATED speed to fly between thermals and
>     Vcruise is the TRUE cruising speed
>     (true ground speed, taking account of time used to thermal).

>     Density Altitude: 0 ft

> Climb | Lift Encountered During Glide (ft/s)
> --- | ---
> ft/s | -3 | -3.5 | -4
> 2 | 65 (15) | 66 (14) | 68 (13)
> 1.5 | 63 (12) | 65 (11) | 66 (11)
> 1 | 62 (9) | 63 (8) | 65 (8)
> .5 | 59 (5) | 62 (4) | 63 (4)

>     Density Altitude: 6000 ft

> Climb | Lift Encountered During Glide (ft/s)
> --- | ---
> ft/s | -3 | -3.5 | -4
> 2 | 64 (16) | 65 (15) | 66 (14)
> 1.5 | 62 (13) | 64 (12) | 65 (11)
> 1 | 60 (9) | 62 (8) | 64 (8)
> .5 | 58 (5) | 60 (4) | 62 (4)

### The Nitty-Gritty

If you aren't interested in how this is done, just go back to the calculator and try it out for yourself. What follows is a detailed description of how the speed-to-fly calculator was modified to incorporate density altitude. Correct me if I'm wrong....

1. All data were organized in arrays Vi[ ] and Vsi[ ] as equivalent speeds in knots.
2. All data were converted to true speeds in knots when needed for any calculation.
3. All speeds derived from the data, such as VbestLD and Vminsink, were calculated first as equivalent speeds, then translated to true speeds by multiplying by sqrt(r/r0).
4. The standard temperature and pressure for a density altitude in feet (densalt) and the density ratio r/r0 were calculated using the following formulas. These values were checked against standard atmosphere tables and found to be accurate. Note that these equations are more precise than the "standard lapse rate" devices used by most pilots for rough calculations.

> stdtemp=stdtemp0 * (1 - densalt/145442)
> stdpres=stdpres0 * (1 - densalt/145442)5.255876
> r/r0 = s = (1 - densalt/145442)4.255876

5. Vtrue was calculated in each case from Vequiv/sqrt(s). (Thus, Vtrue is generally greater than Vequiv.)
6. All data were converted to true speeds in user units prior to plotting or listing.

Happy soaring,

Bob Hanson

hansonr@stolaf.edu
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