Density Altitude Effects on SpeedsToFly Calculations
Credits
Several fine pages discuss the general definition and calculation of density
altitude. Among the best are:
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^{ }mustread^{ }(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^{ }speedtofly^{ }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,^{
}"rV^{2}."^{ }Here^{
}r^{ }is^{ }the air^{
}density,^{ }and^{ }V^{ }is^{ }the^{
}true^{ }airspeed.^{ }To^{ }remove^{ }air^{
}density^{ }from^{ }consideration,^{ }the^{
}trick^{ }is^{ }to^{ }replace^{ }rV^{2
}with
r_{0}V_{i}^{2},^{
}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^{ }r_{0}^{
}is^{ }the^{ }"standard^{ }atmosphere"^{
}sea^{ }level^{ }air^{ }density.
The^{ }problem^{ }is^{ }that^{ }do^{
}make^{ }this^{ }work,^{ }the^{ }true^{
}rate^{ }of^{ }sink,^{ }V_{s},^{
}must^{ }also^{ }be^{ }replaced^{ }by^{
}the^{ }equivalent^{ }rate^{ }of^{ }sink,^{
}using
V_{si}=sqrt(r/r_{0})V_{s}
The^{ }polar^{ }is^{ }then^{ }plotted^{
}as^{ }V_{si}^{ }vs.^{ }V_{i}.
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,^{ }V_{i},^{ }has^{ }a^{
}very^{ }practical^{ }meaning^{ }(indicated^{
}airspeed), the^{ }equivalent^{ }rate^{ }of^{
}sink,^{ }V_{si},^{ }has^{ }no^{
}such^{ }use.^{ }No^{ }vario^{ }reports^{
}V_{si}.^{ }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/r_{0}^{
}(which^{ }may^{ }be^{ }considerably^{
}different^{ }from^{ }sqrt(r/r_{0}).
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^{ }onthefly^{
}graphing,^{ }because^{ }different density^{ }altitudes^{
}will^{ }produce^{ }different^{ }polars.^{
}And^{ }that^{ }is^{ }exactly^{ }what^{
}the^{ }SpeedToFly 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.

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)


In^{ }the^{ }absence^{ }of^{ }wind,^{
}lift,^{ }or^{ }sink,^{ }L/D^{ }is^{
}not^{ }affected^{ }by^{ }density^{ }altitude.^{
}This^{ }is^{ }because^{ }V_{i}/V_{si}^{
}=^{ }V/V_{s}.
Schweizer 135
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)


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^{ }yaxis^{
}of^{ }the^{ }polar^{ }graph^{ }being^{
}"stretched"^{ }in^{ }switching^{ }from^{
}V_{si}^{ }to^{ }V_{s}.
PW5
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)


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)


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)


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)


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 NittyGritty
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^{ }speedtofly^{
}calculator^{ }was^{ }modified^{ }to^{
}incorporate^{ }density^{ }altitude.^{ }Correct^{
}me^{ }if^{ }I'm^{ }wrong....

All^{ }data^{ }were^{ }organized^{ }in^{
}arrays^{ }Vi[^{ }]^{ }and^{ }Vsi[^{
}]^{ }as^{ }equivalent^{ }speeds^{ }in^{
}knots.

All^{ }data^{ }were^{ }converted^{ }to^{
}true^{ }speeds^{ }in^{ }knots^{ }when^{
}needed^{ }for^{ }any^{ }calculation.

All^{ }speeds^{ }derived^{ }from^{ }the^{
}data,^{ }such^{ }as^{ }VbestLD^{ }and^{
}Vminsink,^{ }were^{ }calculated^{ }first^{
}as^{ }equivalent^{ }speeds,^{ }then^{
}translated^{ }to^{ }true^{ }speeds^{ }by^{
}dividing^{ }by^{ }sqrt(r/r_{0}).

The^{ }standard^{ }temperature^{ }and^{ }pressure^{
}for^{ }a^{ }density^{ }altitude^{ }in^{
}feet^{ }(densalt) and^{ }the^{ }density^{
}ratio^{ }r/r_{0}^{
}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=stdtemp_{0}^{ }*^{ }(1^{
}^{ }densalt/145442) 
stdpres=stdpres_{0}^{ }*^{ }(1^{
}^{ }densalt/145442)^{5.255876} 
r/r_{0}^{
}=^{ }s^{ }=^{ }(1^{
}^{ }densalt/145442)^{4.255876} 


V_{true}^{ }was^{ }calculated^{ }in^{
}each^{ }case^{ }from^{ }V_{equiv}/sqrt(s).^{
}(Thus,^{ }V_{true}^{ }is^{ }generally^{
}greater^{ }than^{ }V_{equiv}.)

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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