ABSTRACT
Trailing Edge
Chord Line
Camber Thickness
Chord, c
Mean Camber Line
Leading Edge
Trailing Edge
Chord Line
Thickness
Chord, c
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ܿ௧
b
ܿ
b/2
ܣܴ ൌ ܾ ଶ
ܵ ൌ 2 · ܾ
ܿ · ሺ1 ߣሻ
Aspect Ratio
cത ൌ 23 · ൬ܿ ܿ௧ െ ܿ · ܿ௧
ܿ ܿ௧൰ Mean Aerodynamic Chord
ߣ ൌ ܿ௧ܿ Taper Ratio
ܿଵ/ସquarter chord,
cത
ܵ ൌ ܾ ଶ
ܣܴ Area
ܿ௧
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ܯஶ ൏ 0.3
Subsonic Compressible Flow 0.3 ൏ ܯஶ ൏ 0.7
Transonic Flow 0.7 ൏ ܯஶ ൏ 1.2
Supersonic Flow 1.2 ൏ ܯஶ ൏ 5
Hypersonic Flow 5 ൏ ܯஶ
Continuity Equation (Mass Flow Rate) ܣଵ · ଵܸ ൌ ܣଶ · ଶܸ ߩଵ ൌ ߩଶ
Bernoulli’s Equation ܲ 12 · ߩ · ܸ
ଶ ൌ ܲ ൌ constant along stream line
ଵܲ 12 · ߩ · ଵܸ ଶ ൌ ଶܲ 12 · ߩ · ଶܸ
ܣଶ ܣଵ ଵܲ
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For flow that is isentropic (adiabatic and reversible)
State Equation
ଵܲ ൌ ߩଵ · ܴ · ଵܶ ଶܲ ൌ ߩଶ · ܴ · ଶܶ
Continuity Equation (Mass Flow Rate) ߩଵ · ܣଵ · ଵܸ ൌ ߩଶ · ܣଶ · ଶܸ
ൌ ൬ߩଵߩଵ ൰ ఊ
ሺఊିଵሻ ൌ ൬1 ߛ െ 12 · ܯଵ ଶ൰
ൌ ൬ߩଶߩଶ ൰ ఊ
ሺఊିଵሻ ൌ ൬1 ߛ െ 12 · ܯଶ ଶ൰
൫ ܲଵ ଵܲ⁄ ൯ ିଵ ൌ ൭
൫ߩଶ ߩଶ⁄ ൯ ିଵ
൫ߩଵ ߩଵ⁄ ൯ ିଵ൱
ఊ ൌ ൭൫ ܶଶ ଶܶ⁄ ൯
ൌ ൬ߩଶߩଵ൰ ఊ
ሺఊିଵሻ ൌ ቌ1 ߛ െ 1
2 · ܯଵଶ
1 ߛ െ 12 · ܯଶଶ ቍ
Energy Equation ܿ · ଵܶ 12 · ଵܸ
ଶ ൌ ܿ · ଶܶ 12 · ଶܸ ଶ
ܣଶ ܣଵ
ߩଵ ଵܸ
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ܥ, ൌ 2ߛ · ሺܯሻଶ · ቌቆ 2 ሺߛ െ 1ሻ · ሺܯሻଶ
ߛ 1 ቇ ఊ
ሺఊିଵሻ െ 1ቍ
Supersonic Flow Speed of Sound Mach Number a ൌ ඥߛ · ܴ · ܶ M ൌ ܸa
Density, Pressure, and Temperature Ratios (see Appendix E) ߩଶ ߩଵ ൌ
ሺߛ 1ሻ · ܯଵଶ ሺߛ െ 1ሻ · ܯଵଶ 2
ൌ 1 2 · ߛߛ 1 · ൫ܯଵ ଶ െ 1൯
ൌ ቀ2 · ߛ · ܯଵ ଶ െ ሺߛ െ 1ሻቁ · ቀሺߛ െ 1ሻ · ܯଵଶ 2ቁ
ሺߛ 1ሻଶ · ܯଵଶ
Mach Number after Shockwave (see Appendix E)
ܯଶ ൌ ඨ ሺߛ െ 1ሻ · ܯଵଶ 2
2 · ߛ · ܯଵଶ െ ሺߛ െ 1ሻ
ܯଵ ܯଶ
ܯ ൏ 1 Normal Shock
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Deflection Angle (Radians) ߠ ൌ ߚ െ arctan ቆݐܽ݊ሺߚሻ · ሺߛ െ 1ሻ · ܯଵ
ଶ · ݏ݅݊ଶሺߚሻ 2 ሺߛ 1ሻ · ܯଵଶ · ݏ݅݊ଶሺߚሻ ቇ
Density, Pressure, and Temperature Ratios ߩଶ ߩଵ ൌ
ሺߛ 1ሻ · ܯଵଶ · ݏ݅݊ଶሺߚሻ ሺߛ െ 1ሻ · ܯଵଶ · ݏ݅݊ଶሺߚሻ 2
ଶܲ ଵܲ ൌ 1 2 · ߛߛ 1 · ൫ܯଵ
ଶ · ݏ݅݊ଶሺߚሻ െ 1൯
ଶܶ ଵܶ ൌ 12·ሺߛ െ 1ሻ·൫ܯଵ
ଶ·ݏ݅݊ଶሺߚሻ െ 1൯ ሺߛ 1ሻଶ·ܯଵଶ·ݏ݅݊ଶሺߚሻ ·൫ߛ · ܯଵ
ଶ·ݏ݅݊ଶሺߚሻ1൯
Mach Number after Shockwave
ܯଶ ൌ 1ݏ݅݊ሺߚ െ ߠሻ · ඩ 1 ߛ െ 12 · ܯଵଶ · ݏ݅݊ଶሺߚሻ ߛ · ܯଵଶ · ݏ݅݊ଶሺߚሻ െ ߛ െ 12
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Isentropic Ratios (see Appendix D)
ߩଶ ߩଵ ൌ
൫ߩଶ ߩଶ⁄ ൯ ିଵ
൫ߩଵ ߩଵ⁄ ൯ ିଵ ൌ ቌ
1 ߛ െ 12 · ܯଵଶ
1 ߛ െ 12 · ܯଶଶ ቍ
൫ ܲଵ ଵܲ⁄ ൯ ିଵ ൌ ቌ
1 ߛ െ 12 · ܯଵଶ
1 ߛ െ 12 · ܯଶଶ ቍ
1 ߛ െ 12 · ܯଵଶ
1 ߛ െ 12 · ܯଶଶ
Solving for Mach Number after Expansion Fan
ߞൌඨߛ 1ߛ െ 1 ·ݐܽ݊ ିଵ ቌඨߛ െ 1ߛ 1 · ൫ܯଵ
ଶ െ 1൯ቍ െ ݐܽ݊ିଵ ቆට൫ܯଵଶ െ 1൯ቇ ߠ
ߞൌඨߛ 1ߛ െ 1 ·ݐܽ݊ ିଵ ቌඨߛ െ 1ߛ 1 · ൫ܯଶ
ଶ െ 1൯ቍ െ ݐܽ݊ିଵ ቆට൫ܯଶଶ െ 1൯ቇ
Determine M2 by inputting different values of M2 until both parameters are equal
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Lift Coefficient ܥ ൌ ܮ1
2 · ߩஶ · ஶܸଶ · ܵ
Parasite Drag Coefficient ܥௗ, ൌ ܦ1
2 · ߩஶ · ஶܸଶ · ܵ
ܥௗ, ൌ ܦ1 2 · ߩஶ · ஶܸଶ · ܵ
ൌ ܮ · ݏ݅݊ሺߙሻ1 2 · ߩஶ · ஶܸଶ · ܵ
ߨ · ݁ · ܣܴ
Wave Drag Coefficient (approx. for flat-plate and low angles of attack)
ܥௗ,௪ ൌ 4 · ߙ ଶ
ඥܯஶଶ െ 1 Note: α is in Radians
Total Drag Coefficient ܥௗ ൌ ܦ1
2 · ߩஶ · ஶܸଶ · ܵ ൌ ܥௗ, ܥௗ, ܥௗ,௪
α ஶܸ
αi Finite Wings
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ܥ ൌ ܲ െ ஶܲ1
2 · ߩஶ · ஶܸଶ
Pitching Moment Coefficient ܥ, ൌ ݉1
2 · ߩஶ · ஶܸଶ · ܵ · cത
Rolling Moment Coefficient ܥ, ൌ ݉1
2 · ߩஶ · ஶܸଶ · ܵ · ܾ
Yawing Moment Coefficient ܥ,௬ ൌ ݉௬1
2 · ߩஶ · ஶܸଶ · ܵ · ܾ
Subsonic Compressibility Correction to Lift, Pressure, and Moment Coefficients ܥ, ൌ ܥඥ1 െ ܯஶଶ
ܥ, ൌ ܥඥ1 െ ܯஶଶ
ܥ, ൌ ܥඥ1 െ ܯஶଶ
Supersonic Pressure Coefficient
ܥ,௦௦ ൌ 2ߛ · ܯଵଶ · ۉ ۇቌ1
1 2 · ሺߛ െ 1ሻ · ܯଵଶ
1 12 · ሺߛ െ 1ሻ · ܯଶଶ ቍ
െ 1 ی ۊ
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Flat Plate Approximation for Incompressible Flow
Local Reynolds Number ܴ݁௫ ൌ ߩஶ · ஶܸ · ݔߤஶ
Boundary Layer Thickness for Laminar Flow ߜ ൌ 5.2 · ݔඥܴ݁௫
Local Skin Friction Coefficient for Laminar Flow ܥ௫ ൌ 0.664ඥܴ݁௫
Transition Conditions ܴ݁௫ ൎ 3.5 · 10ହ ՜ 4 · 10
Laminar flow Turbulent flow Transitionx
δ
Velocity Profile for laminar flow
ߜݕ ݔ
Laminar flow Turbulent flow
Boundary Layer (viscous flow)
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True Airspeed
௧ܸ ൌ ඨ2 · ሺ ܲ െ ஶܲሻ
ߩஶ
Equivalent Airspeed
ܸ ൌ ඨ2 · ሺ ܲ െ ஶܲሻ
ߩ௦ ൌ ௧ܸ · ඨ ߩஶ ߩ௦
Subsonic Compressible Flow
True Airspeed
௧ܸ ൌ ඩ2 · aஶ ଶ
ߛ െ 1 · ቌ൬ ܲ
ఊିଵ ఊ െ 1ቍ
ൌඩ2 · aஶ ଶ
ߛ െ 1 · ቌ൬ ܲ െ ஶܲ
ஶܲ 1൰
ఊିଵ ఊ െ 1ቍ
Calibrated Airspeed
ܸ ൌ ඩ2 · a௦ ଶ
ߛ െ 1 · ቌ൬ ܲ െ ஶܲ
௦ܲ 1൰
ఊିଵ ఊ െ 1ቍ
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