8. The basic variables representing the thermodynamics state of
the gas are the Density, ρ, Temperature, T and Pressure, p.
SOLO
8
Earth Atmosphere
• The Density, ρ, is defined as the mass, m, per unit volume, v,
and has units of kg/m3
.
v
m
v ∆
∆
=
→∆ 0
limρ
• The Temperature, T, with units in degrees Kelvin ( ͦ K). Is a
measure of the average kinetic energy of gas particles.
• The Pressure, p, exerted by a gas on a solid surface is defined
as the rate of change of normal momentum of the gas particles
striking per unit area.
It has units of N/m2
. Other pressure units are millibar (mbar),
Pascal (Pa), millimeter of mercury height (mHg)
S
f
p n
S ∆
∆
=
→∆ 0
lim
kPamNbar 100/101 25
==
( ) mmHginHgkPamkNmbar 00.7609213.29/325.10125.1013 2
===
The Atmospheric Pressure at Sea Level is:
9. Speed of Sound (a)
This is the speed of sound waves propagation in ambient
air. The speed of sound is given by
SOLO
9
Earth Atmosphere
Sa TRa ⋅⋅= γ
γ air = 1.4
Ra =287.0 J/kg--ͦ
K
TS – Static Air Temperature
True Airspeed (TAS)
The True Airspeed is the speed of the aircraft’s center of
mass with respect to the ambient air through which is
passing.
Indicated Airspeed (IAS)
The Indicated Airspeed is the speed indicated by a
differential-pressure airspeed indicator.
10. Mach Number (M)
Is the ratio of the TAS to the speed of sound at the
flight condition.
SOLO
10
Earth Atmosphere
aTASM /=
Dynamic Pressure (q)
The force per unit area required to bring an ideal
(incompressible) fluid to rest: q=1/2∙ρ∙VT
2
(where VT is
True Air Speed-TAS, and ρ is the density of the fluid).
Impact Pressure (QC)
The force per unit area required to bring moving air to
rest. It is the pressure exerted at the stagnation point on
the surface of a body in motion relative to the air.
PT – Total Pressure, PS – Static Pressure
2
2/1 TSTC VPPQ ⋅⋅=−= ρ
11. SOLO
Aircraft Avionics
11
Air Data Computer
Air Data Computer uses Total and Static Pressure and Static Temperature
of the external Air Flow, to compute Flight Parameters.
12. 12
Earth Atmosphere
Atmospheric Constants
Definition Symbol Value Units
Sea-level pressure P0 1.013250 x 105
N/m2
Sea-level temperature T0 288.15 ͦ K
Sea-level density ρ0 1.225 kg/m3
Avogadro’s Number Na 6.0220978 x 1023
/kg-mole
Universal Gas Constant R* 8.31432 x 103
J/kg-mole -ͦ K
Gas constant (air) Ra=R*/M0 287.0 J/kg--ͦ
K
Adiabatic polytropic constant γ 1.405
Sea-level molecular weight M0 28.96643
Sea-level gravity acceleration g0 9.80665 m/s2
Radius of Earth (Equator) Re 6.3781 x 106
m
Thermal Constant β 1.458 x 10-6
Kg/(m-s-ͦ K1/2)
Sutherland’s Constant S 110.4 ͦ K
Collision diameter σ 3.65 x 10-10
m
13. SOLO
Aircraft Avionics
13
Flight Instruments
Air Data Calculation (Collison)
Geopotential Pressure Altitude
• Low Altitude (Troposphere) : H< 11000 m (36.089 ft ),
( ) kPaHPS
255879.55
1025577.21325.101 ⋅⋅−⋅= −
• Medium Altitude: 11000 m ≤ H ≤ 20000m (36.089 ft - 65.617 ft )
( )
kPaeP H
S
000,1110576885.1 4
6325.22 −⋅⋅− −
⋅=
Air Density Ratio ρ/ρ0
S
S
T
P
⋅
=
35164.00ρ
ρ
14. SOLO
Aircraft Avionics
14
Flight Instruments
Air Data Calculation (Collison)
Mach Number
• Subsonic Speeds (M ≤ 1),
( ) 2/72
2.01 M
P
P
S
T
⋅+=
• Supersonic Speeds (M ≥ 1),
Static Air Temperature TS ͦ K
10
2.01 2
<<
⋅⋅+
= r
Mr
T
T m
S
( ) 2/52
7
17
9.166
−⋅
⋅
=
M
M
P
P
S
T
True Airspeed (TAS) VT m/s
smTMV ST /0468.20 ⋅⋅=
15. SOLO
Aircraft Avionics
15
Flight Instruments
Air Data Calculation (Collison)
Speed of Sound a m/s
• Subsonic Speeds (VC ≤ a),
• Supersonic Speeds (VC ≥ a),
Sa TRa ⋅⋅= γ γ air = 1.4, Ra =287.0 J/kg--ͦ
K
Calibrated Airspeed (CAS) VC m/s
kPa
V
Q C
C
−
⋅+⋅= 1
294.340
2.01325.101
2/72
kPa
V
V
Q
C
C
C
−
−
⋅
⋅
⋅= 1
1
294.340
7
294.340
92.166
325.101
2/7
2/52
2
25. SOLO
Aircraft Avionics
25
Flight Instruments
Airspeed Indicators
2
2
1
vpp StatTotal ⋅+= ρ
The airspeed directly given by the differential pressure is called
Indicated Airspeed (IAS). This indication is subject to positioning errors of the pitot
and static probes, airplane altitude and instrument systematic defects.
The airspeed corrected for those errors is called Callibrated Airspeed (CAS).
Depending on altitude, the critic airspeeds for maneuvre, flap operation etc change
because the aerodynamic forces are function of air density. An equivalent airspeed
VE (EAS) is defined as follows:
0ρ
ρ
VVE =
V – True Airspeed
ρ – Air Density
ρ0 – Air Density at Sea Level
26. SOLO
Aircraft Avionics
26
Flight Instruments
Airspeed Indicators
2
2
1
VPQPP StatCStatTotal ⋅+=+= ρ
V – True Airspeed
ρ – Air Density
ρ0 – Air Density at Sea Level
Air Density changes with altitude. Assuming an Adiabatic Flow, the
relation between Pressure and Density is given by
constC
P
==γ
ρ
γ = Cp/CV= 1.4 for air
Momentum differential equation for the Air Flow is
VdV
C
P
PdVdVPd
C
P
γρ
γ
ρ
/1
/1
0
+=+=
=
Subsonic Speeds
SoundofSpeed
P
a S
ρ
γ ⋅
=
27. SOLO
Aircraft Avionics
27
Flight Instruments
Airspeed Indicators
In the free stream P = PS and V = VT,
At the Probe face P = PT and V=0
0
1 0
/1
/1
=+ ∫∫ T
T
S V
P
P
VdV
C
PdP γ
γ
Subsonic Speeds (continue)
2
1
1
2
/1
11
T
ST
V
C
PP γ
γ
γ
γ
γ
γ
γ
=
−
−
−−
γγ
ρ
/1/1
1
SPC
=
1
2
12
2
1
1
2
1
1
2
−
=
−
⋅
−
+=
⋅⋅
−
+= γ
γ
ρ
γ
γ
γ
γ
γ
ργ
a
V
V
PP
P T
P
a
T
SS
T
S
−
⋅
−
+=
−=−= −
1
2
1
11 1
2
γ
γ
γ
a
V
P
P
P
PPPQ T
S
S
T
SSTC
28. SOLO
Aircraft Avionics
28
Flight Instruments
Airspeed Indicators
In the free stream P = PS and V = VT,
At the Probe face P = PT and V=0
Supersonic Speeds
1
1
2
12
1
1
1
2
2
1
−
−
+
−
−
⋅
+
⋅
+
=
γ
γ
γ
γ
γ
γ
γ
γ
a
V
a
V
P
P
T
T
S
T
−
+
−
−
⋅
+
⋅
+
=
−=−=
−
−
1
1
1
1
2
2
1
1
1
1
2
12
γ
γ
γ
γ
γ
γ
γ
γ
a
V
a
V
P
P
P
PPPQ
T
T
S
S
T
SSTC
Assume Supersonic Adiabatic Air Flow
we obtain
29. SOLO
Aircraft Avionics
29
Flight Instruments
Airspeed Indicators
Mach Number
1
1
2
1
2
1
1
1
2
2
1
−
−
+
−
−⋅
+
⋅
+
=
γ
γ
γ
γ
γ
γ
γ
γ
M
M
P
P
S
T
Subsonic Speeds (M ≤ 1)
γ
γ
γ
1
1
2
1
−
⋅
−
−==
S
TT
P
P
a
V
M
12
2
1
1
2
−
=
⋅
−
+= γ
γρ
γ
γ
M
P
P
SP
a
S
T
From
Supersonic Speeds (M ≥ 1)
30. SOLO
Aircraft Avionics
30
Flight Instruments
Airspeed Indicators (Calibrated Airspeed)
Calibrated Airspeed is obtained by substituting
the Sea Level conditions, that is PS = PS0 ,
VT = VC , a0 = 340.294 m/s.
Subsonic Speeds (VC < a0=340.294 m/s)
−
⋅
−
+= −
1
2
1
1 1
2
0
0
γ
γ
γ
a
V
PQ C
SC
2
0
00
2
0
2
0
2
1
/2
1
2
1 C
S
C
S
C
S
aV
C V
P
V
P
a
V
PQ
C
⋅⋅=
⋅
⋅⋅=
−
⋅+≈
<<
ρ
ργ
γγ
Supersonic Speeds (VC > a0=340.294 m/s)
−
+
−
−
⋅
+
⋅
+
=
−
−
1
1
1
1
2
2
1
1
1
2
0
12
0
0
γ
γ
γ
γ
γ
γ
γ
γ
a
V
a
V
PQ
C
C
SC
( ) mmHginHgkPamkNmbarPS 00.7609213.29/325.10125.1013 2
0 ====
γ air = 1.4
31. SOLO
Aircraft Avionics
31
Flight Instruments
Airspeed Indicators
By measuring (TT) the Temperature of Free
Airstream TS, we can compute the local Speed
of Sound
Sa TRa ⋅⋅= γ
True Airspeed (TAS)
By using the Mach Number computation we
can calculate the True Airspeed (TAS)
M
M
T
RMTRMaV T
aSaT ⋅
⋅
−
+
⋅⋅=⋅⋅⋅=⋅=
2
2
1
1
γ
γγ
32. SOLO
Aircraft Avionics
32
Flight Instruments
Goodrich Air Data Handbook – Basic Air Data Calculation
Altitude
• Low Altitude: h<36.089 ft = 11000 m, PS > 6.6832426 in Hg
( ) ( ) 190255.0
1
190255.0
190255.0190255.0
140000131252.092126.29
140000131252.0
92126.29
hP
P
h S
S
⋅−=
−
=
• Medium Altitude: 36.089 ft = 11000 m ≤ h ≤ 65.617 ft = 20000m
6.6832426 in Hg> PS > 1.6167295 in Hg
( )h
S
S
eP
P
h ⋅−
⋅=
−
= 30000480635.07345726.1
6832426.6
30000480635.0
6832426.6
ln7345726.1
163156.34
163156.34
1
96.710793
16.645177
6167295.116.645177
6167295.1
96.710793 −
−
+
⋅=−
⋅=
h
P
P
h S
S
• High Altitude: h >65.617 ft = 20000m, PS < 1.6167295 in Hg
34. SOLO
Aircraft Avionics
34
Flight Instruments
Goodrich Air Data Handbook – Basic Air Data Calculation
Mach Number M = TAS/a
−
⋅=
−
+⋅= 15115
7/27/2
S
T
S
C
P
P
P
Q
M
Subsonic Flight (M ≤ 1)
Supersonic Flight (M ≥ 1)
1
17
2.7
2.11
2/5
2
2
2
−
−⋅
⋅
⋅=−=
M
M
M
P
P
P
Q
S
T
S
C
where:
TAS = True Airspeed in knots
a = Speed of Sound in knots
where:
QC=½∙ ρ∙V2
= Impact Pressure in Hg
PT = Total Pressure in Hg
PS = True Static Pressure in Hg
35. SOLO
Aircraft Avionics
35
Flight Instruments
Goodrich Air Data Handbook – Basic Air Data Calculation
Mach Number M = TAS/a
Altitude
(feet)
75 KIAS
(Qc=0.2701
In Hg)
100 KIAS
(Qc=04814
In Hg)
200 KIAS
(Qc=1.9589
In Hg)
300 KIAS
(Qc=4.5343
In Hg)
400 KIAS
(Qc=8.3850
In Hg)
500 KIAS
(Qc=13.7756
In Hg)
600 KIAS
(Qc=21.0749
In Hg)
700 KIAS
(Qc=30.7642
In Hg)
S.L. .113 .151 .302 .454 .605 .756 .907 1.058
10,000 .137 .182 .363 .541 .716 .888 1.057 1.230
20,000 .167 .222 .440 .651 .854 1.047 1.242 1.453
30,000 .207 .276 .541 .791 1.023 1.248 1.489 1.754
40,000 .262 .347 .672 .965 1.236 1.520 1.829 2.171
50,000 .331 .438 .831 1.171 1.509 1.875 2.276 2.717
60,000 .418 .549 1.014 1.426 1.862 2.335 2.852 3.419
70,000 .524 .684 1.230 1.754 2.318 2.928 3.592 4.318
80,000 .653 .842 1.497 2.172 2.897 3.678 4.526 5.450
36. SOLO
Aircraft Avionics
36
Flight Instruments
Goodrich Air Data Handbook – Basic Air Data Calculation
Static Temperature
2
2.01 M
T
T T
S
⋅+
=
True Airspeed (TAS)
where:
TS = Static Temperature ͦK
TT = Total Temperature ͦK
⋅+
⋅⋅== 2
2.01
96695.38
M
T
MMTAS T
a
where:
TAS = True Airspeed
M = Mach
a = Speed of Sound
TT = Total Temperature ͦK
39. SOLO
39
Aircraft Avionics
Flight Instruments
Airspeed Indicator (ASI)
White Arc – Flaps Operation Range
VSO – Stalling Speed Flaps Down
VSI - Stalling Speed Flaps Up
VFE – Maximum Speed Flaps Down (Extendeed)
Green Arc – Normal Operation Range
VNO – Maximum Speed Normal Operation
Yellow Arc - Caution Range
VNE – Not to Exceed Speed
Private Pilot Airplane – Flight Instruments ASA, Movie
48. SOLO
48
Aircraft Avionics
Flight Instruments
Heading Indicator
The Magnetic Compass is sensitive
to Inertia Forces. It is a reliable
Heading Instrument in the long
yerm, but during maneuvers it may
swing and be hardly reliable. To
provide a more precise Heading
Instrument a Directional Gyro is
used.
50. SOLO
50
Aircraft Avionics
Flight Instruments
Flux Gate Compass System
The Gate Compass System is connected to Radio Magnetic Indicator (RMI)
and to Heading Situation Indicator (HSI).
Heading Situation Indicator (HSI).Radio Magnetic Indicator (RMI)
76. 76
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA
77. 77
SOUND WAVESSOLO
Disturbances propagate by molecular collision, at the sped of sound a,
along a spherical surface centered at the disturbances source position.
The source of disturbances moves with the velocity V.
-when the source moves at subsonic velocity V < a, it will stay inside the
family of spherical sound waves.
-when the source moves at supersonic velocity V > a, it will stay outside the
family of spherical sound waves. These wave fronts form a disturbance
envelope given by two lines tangent to the family of spherical sound waves.
Those lines are called Mach waves, and form an angle μ with the disturbance
source velocity:
a
V
M
M
=
= −
&
1
sin 1
µ
78. 78
SOUND WAVESSOLO
Sound Wave Definition:
∆ p
p
p p
p1
2 1
1
1=
−
<<
ρ ρ ρ2 1
2 1
2 1
= +
= +
= +
∆
∆
∆
p p p
h h h
For weak shocks
u
p
1
2
=
∆
∆ρ
1
1
11
1
1
1
1
1
2
1
2
1
1
uuuuuu
ρ
ρ
ρ
ρρρ
ρ
ρ
ρ ∆
−≅
∆
+
=
∆+
==)C.M.(
( ) ( ) ppuuupuupu ∆++
∆
−=+=+ 11
1
11122111
2
11
ρ
ρ
ρρρ)C.L.M.(
Since the changes within the sound wave are small, the flow gradients are small.
Therefore the dissipative effects of friction and thermal conduction are negligible
and since no heat is added the sound wave is isotropic. Since
au =1
s
p
a
∂
∂
=
ρ
2
valid for all gases
79. 79
SPEED OF SOUND AND MACH NUMBERSOLO
Speed of Sound is given by
0=
∂
∂
=
ds
p
a
ρ
RT
p
C
C
T
dT
R
C
p
T
dT
R
C
d
dp
d
R
T
dT
Cds
p
dp
R
T
dT
Cds
v
p
v
p
ds
v
p
γ
ρ
ρ
ρ
ρ
ρ
===
⇒
=−=
=−=
=00
0
but for an ideal, calorically perfect gas
ρ
γγ
ρ p
RTa
TChPerfectyCaloricall
RTpIdeal
p
==
=
=
The Mach Number is defined as
RT
u
a
u
M
γ
==
∆
1
2
1
1
111
−−
=
=
=
γ
γ
γ
γ
γ
ρ
ρ
a
a
T
T
p
p
The Isentropic Chain:
a
ad
T
Tdd
p
pd
sd
1
2
1
0
−
=
−
==→=
γ
γ
γ
γ
ρ
ρ
γ
80. 80
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Mach Number Relations (1)
( )
( )
( )
( )
12
2
2
2
1
2
1
2
2
22
2
2
1
22
1
2
2
2
2
22
1
1
2
1
12
22
2
11
1
2
2
221
2
11
2211
2
1
2
1
2
1
2
1
*
12
1
2
1
12
1
1
4..
...
..
uu
u
a
u
a
uaa
uaa
au
h
a
u
h
a
EC
uu
u
p
u
p
pupuMLC
uuMC
p
a
−=−
−
−
+
=
−
−
+
=
→
−
+
=+
−
=+
−
→−=−→
+=+
=
∗
∗
=
γγ
γγ
γγ
γ
γ
γγ
ρρρρ
ρρ ρ
γ
Field Equations:
122
2
2
1
1
2
2
1
2
1
2
1
2
1
uuu
u
a
u
u
a
−=
−
+
+
−
−
−
+ ∗∗
γ
γ
γ
γ
γ
γ
γ
γ
u u a1 2
2
= ∗
u
a
u
a
M M1 2
1 21 1∗ ∗
∗ ∗
= → =
Prandtl’s Relation
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
( )
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
2
1
2
1
1
2
1
2
1
2
1
21
2
1212
2
21
12 +
=
−
−=
+
→−=−
−
+
−+ ∗
∗
uu
a
uuuua
uu
uu
Ludwig Prandtl
(1875-1953)
81. 81
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Mach Number Relations (2)
( ) ( ) ( ) ( )
( )
( )
( )
( )
( )[ ]
( )( ) ( )
M
M
M
M
M
M
M
M
M
2
2
2
2
1
1
2
1
2
1
2
1
2
1
2
2
1
1
2
1 1
2
1
1
1 2
1
2 1 2
1 1 1 1 1
1
2
=
+
− −
=
+ − −
=
+
+
− +
− −
=
− +
+ / + − / / + − / + − −
∗
=
∗
∗
∗
γ
γ γ γ
γ
γ
γ
γ
γ
γ γ γ γ γ
or
( )
M
M
M
M
M
H H
A A
2
1
2
1
2
1
2
1
21 2
1 2
1
1
2
1
2
2
1
1
1
2
1
2
1
1
=
+
−
−
−
=
+
+
−
+
+
−
=
=
γ
γ
γ γ
γ
γ
γ
( )
( )
ρ
ρ
γ
γ
2
1
1
2
1
2
1 2
1
2
2 1
2 1
2
1
2
1 2 1
1 2
= = = = =
+
− +
=
∗
∗
A A u
u
u
u u
u
a
M
M
M
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
82. 82
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q= =0 0,
Mach Number Relations (3)
( )
( )
( ) ( )
( )
p
p
u
p
u
u
u
a
M
M
M
M
M M
M
2
1
1
2
1
1
2
1
1
2
1
2
1
2
1
2 1
2
1
2 1
2 1
2
1
2
1
2
1 1 1 1
1 1
1 2
1
1
1 1 2
1
= + −
= + −
= + −
− +
+
= +
/ + − / − −
+
ρ
γ
ρ
ρ
γ
γ
γ
γ
γ γ
γ
or
(C.L.M.)
( )
p
p
M2
1
1
2
1
2
1
1= +
+
−
γ
γ
( )
( )
( )
h
h
T
T
p
p
M
M
M
a
a
h C T p RTp
2
1
2
1
2
1
1
2
1
2 1
2
1
2
2
1
1
2
1
1
1 2
1
= = = +
+
−
− +
+
=
= = ρ ρ
ρ
γ
γ
γ
γ
( )
( )
( )
s s
R
T
T
p
p
M
M
M
2 1 2
1
1
2
1
1
1
2
1
1
1
2
1
2
1
1
2
1
1
1 2
1
−
=
= +
+
−
− +
+
−
−
− −
ln ln
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
( )
( )
( )
( )
s s
R
M M
M
2 1
1 1
2 1
2 3
2
2 1
2 41
2
2
3 1
1
2
1
1
−
≈
+
− −
+
− +
− << γ
γ
γ
γ
K Shapiro p.125
u
p
ρ
T
e
u
p
ρ
T
e
τ 11
q
1
1
1
1
1
2
2
2
2
2
1 2
83. 83
STEADY QUASI ONE-DIMENSIONAL FLOWSOLO
STAGNATION CONDITIONS
)C.E.( constuhuh =+=+ 2
22
2
11
2
1
2
1
The stagnation condition 0 is attained by reaching u = 0
2
/
21202
020
2
1
1
1
2
1
2
1
22
1
2
M
TR
u
Tc
u
T
T
c
u
TTuhh
TRa
auM
Rc
pp
Tch p
p
−
+=
−
+=+=→+=→+=
=
=
−
=
=
γ
γ
γ
γγ
γ
Using the Isentropic Chain relation, we obtain:
2
1
0102000
2
1
1 M
p
p
a
a
h
h
T
T −
+=
=
=
==
−
−
γ
ρ
ρ γ
γ
γ
Steady , Adiabatic + Inviscid = Reversible, , ( )
q Q= =0 0, ( )~ ~
τ = 0 ( )
G = 0
∂
∂ t
=
0