5 Aerodynamics

Recurring Terminology

Symbol Definition
a slope of lift curve, dCL/dα
ac aerodynamic center, location along the chord where pitching moments about this center do not change with angle of attack (25% MAC for airfoils in subsonic flow, 50% MAC for airfoils in supersonic flow)
AOA angle of attack
AR aspect ratio =[wing span]2/[reference wing area]=b2/S
B wing span
bt horizontal tail span
C coefficient, a non-dimensional representation of an aerodynamic property
c wing chord length Camber maximum curvature of an airfoil, measured at maximum distance between chord line and amber line, expressed in % of MAC. Camber line theoretical line extending from an airfoil’s leading edge to the trailing edge, located halfway between the upper and lower surfaces.
CD drag coefficient
CDi induced drag coefficient
CD0CDPE parasitic drag coefficient
cf friction coefficient
Chord straight-line distance from an airfoil’s leading edge to its trailing edge
CL lift coefficient
Cp pressure coefficient = Δp/q
e Oswald efficiency factor
l distance traveled by flow, or characteristic length of surface
M Mach number
MAC mean aerodynamic chord, chord length of location on wing where total aerodynamic forces can be concentrated.
MGC mean geometric chord, the average chord length, derived only from a plan form view of a wing (similar to MAC if wing has no twist and constant cross section & thickness-to-chord ratio).
P pressure
Preq'd power required
q dynamic pressure = 12ρaV2T=12ρ0V2T
R gas constant
Rn,Re Reynolds number
S reference wing area, includes extension of wing to fuselage centerline.
St horizontal tail surface area
SW wetted area of surface
T temperature
V true velocity
Ve equivalent velocity
α angle of attack
αi induced angle of attack
δ depth of boundary layer, or surface wedge angle
μ viscosity, or wave angle
ν flow turning angle
θ shock wave angle
ρ density
  • Perfect Fluid
    • incompressible, inelastic, and non-viscous
    • used in flow outside of boundary layers at M < .7
  • Incompressible, inelastic, viscous
    • used for boundary layer studies at M < .7
  • Compressible, non-viscous, elastic fluid
    • used outside boundary layers up to M = 5

5.1 Dimensional Analysis Interpretations

(ref 5.2)

Aerodynamic force = F

  • F=f(ρμ,TV, shape, orientation, size, roughness, gravity)
  • For aircraft ignore R, K & hypersonic effects

  • Initially assume similar body orientations, shapes & roughness.
  • Dimensional Analysis reveals four non-dimensional (π) parameters:
    • Force Coefficient π1=FρV2l2
    • Reynolds Number π2=ρVlμ
    • Mach Number π3=Va
    • Froude Number π4=Vlg

A closer look at the force coefficient:

CF=FρV2l2=>F12ρV2S

where 12ρaV2T=12ρ0V2e=dynamic pressureq

Dimensions of reference wing area, S are the same

A feel for q

  • Kinetic energy of a moving object = 12mV2T
  • Block of moving air kinetic energy = 12ρ (volume) V2T
  • Dividing through by volume yields KE per volume of moving air = 12ρV2T
  • “Dynamic pressure” or “q” = potential for converting each cubic foot of the airflow’s kinetic energy into frontal stagnation pressure
  • Feel q by extending your hand out the window of a moving car

A feel for coefficients

  • CF=(F/S)/q = the ratio between the total force pressure and the flow’s dynamic pressure
  • Lift is the component of the total force perpendicular to the free stream flow
  • Drag is the component along the flow
  • Break total into lift and drag coefficients:
    • CL=(L/S)/q
    • CD=(D/S)/q
  • Increasing dynamic pressure generates a larger total force, lift and drag

  • Froude number is not significant in aerodynamic phenomena
  • Recall that forces are also a function of angle of attack, shape & surface roughness, therefore

CL,CD=f[MReα] for a given shape, roughness

Note in the figure above the Reynolds effects are exaggerated

To compare test day and standard day aircraft or to match wind tunnel CF data to actual aircraft; the shape, roughness, M, Rn and α must be equal for both aircraft

LAqASA=CL=LMqMSM

5.2 General Aerodynamic Relations

(refs 5.1, 5.2, 5.10)

Lift & Drag forces can be described using two approaches:

  1. Change in momentum of airstream, F=d[mv]/dt
  2. “Bernoulli” approach which requires the continuity and conservation of energy equations

Continuity Equation

Fluid Mass in = Fluid Mass out

ρ1V1A1=ρ2V2A2

For subsonic (incompressible) flow ρ1=ρ2

V1A1=V2A2

Conservation of Energy (Bernoulli) Equation:

Potential + Kinetic + Pressure = constant (changes in Potential energy are negligible)

Energy per unit volume is pressure then Dynamic Pressure + Static Pressure = Total Pressure

12ρV2+ps=constant 12ρV2+ps=pt

  • This classic approach only applies in the “potential flow” region and not in the boundary layer where energy losses occur
  • Pressures around a surface can be calculated or measured from tests and converted into pressure coefficients,

cp=(plocalpambient)/dynamic pressure=Δp/q

  • cp values can be mapped out for all surfaces

  • Summation of all pressures perpendicular to surface yield the pitching moments and the “Resultant Aerodynamic Force” which is broken into lift and drag components

  • Lift & drag forces are referred to the aerodynamic center (ac) where the pitching moment is constant for reasonable angles of attack.
  • Pitching moments increase with airfoil camber, are zero if symmetric.
  • Aerodynamic center is located at 25% MAC for fully subsonic flow and at 50% MAC for fully supersonic flow.

5.3 Wing Design Effects on Lift Curve Slope

(refs 5.1, 5.2, 5.10)

Aspect Ratio Effect

  • Pressure differential at wingtip causes tip vortex

  • Vortex creates flow field that reduces AOA across wingspan

  • Local AOA reductions decrease average lift curve slope

2D wing = wind tunnel airfoil extending to walls (infinite aspect ratio).

a0=Lift curve slope for an infinite wing

a=Lift curve slope for a finite wing

  • Above relationship estimated as α=dCLdα=a01+57.3a0πAR

Trailing Edge Flap Effects

Leading Edge Flap Effects

Boundary Layer Control Effects

5.4 Elements of Drag

(refs 5.1, 5.2, 5.10)

  • Skin friction shear stress is a function of velocity profile at surface

Shear stress τw=μ(dvdy)y=0

  • Viscosity μ increases with temperature (ref 5.9)

Sutherland law: μ=μ0(TT0)1.5(T0+S)(T+S)

Power law: μ=μ0(TT0)n

Where T0=273.15K=518.67R

For air: S=110.4K=199R; n=0.67

For air at 273 K : μ0=1.717×105[kg/m s]=3.59×107[slug/ft s]

Inserting air values (TK=Kelvin and TR=Rankin) into Sutherland law gives

μ=1.458×106T1.5KTK+110.4[kgs m]=2.2×108T1.5RTR+199[slugs ft]

5.4.1 Reynolds Number Effects

(ref 5.10)

  • Laminar boundary layers have more gradual change in velocity near surface than turbulent boundary layers.
  • High Reynolds numbers help propagate turbulent flow.

Shearing stress: τw=μ(dvdy)y=0

Skin friction coefficient: Cf=τw12ρV2=τwq

Laminar boundary layer: Total Cf=1.328(ReL)1/2

Turbulent boundary layer: Total Cf=0.455(log(ReL))2.580.074(ReL)1/5

ReL based on total length of flat plate

  • Depth of boundary layer (δ) depends on local Reynolds number (Rex) and whether the flow is turbulent or laminar.

Rex=ρVxμInertia ForcesViscous Forces

x= distance traveled to point in question

δlam=5.2xRex

δturb=0.37xRe0.2x

5.4.2 Pressure Drag

  • Ideal frictionless flow has no losses and leads to zero pressure drag
  • Real fluids have friction and energy losses along surface
  • Energy losses negate total pressure recovery, lead to decreasing total pressure along surface

  • Imbalance of pressures on surfaces causes pressure drag

  • Profile streamlining reduces pressure drag

5.4.3 Interference Drag

  • Occurs with multiple surfaces approximately parallel to flow
  • Caused by flow’s interference with itself or by excessive adverse pressure gradient due to rapidly decreasing vehicle cross section
  • Most severe with surfaces at acute angles to each other
  • Effects often reduced by fillets around contracting surfaces

5.4.4 Induced Drag

  • Wingtip vortex reduces local AOA at each station along wing
  • Local lift vector is perpendicular to local AOA
  • Local lift vector is therefore tilted back relative to freestream lift
  • Induced drag defined as rearward component of local lift vector

Induced Drag (Di)=L(αi)

For elliptical lift distributions αi=CLπAR

C_{D_i} = \frac{D_i}{qS} = \frac{C_L^2}{\pi \mathrm{AR}}

Oswald efficiency factor, e, accounts for losses in excess of those predicted above (due to uneven downwash and changing interference drag effects).

\therefore C_{D_i} = \frac{C_L^2}{\pi \mathrm{AR} e}

5.5 Aerodynamic Compressibility Relations

(ref 5.8)

Prandtl/Glauert Approximation

Approximates Mach effects on aerodynamics below critical Mach

C_{P_{\mathrm{compressible}}} = \frac{1}{\sqrt{1-M^2}}C_{P_{\mathrm{incompressible}}}

Total vs Ambient Property Relations for Adiabatic Flow

\frac{T_T}{T} = 1 + \frac{\gamma -1}{2}M^2 \text{ Isentropic flow not required} \frac{P_T}{P} = \left[1 + \frac{\gamma - 1}{2}M^2 \right]^{\frac{\gamma}{\gamma-1}} \text{ Isentropic (shockless) flow required} \frac{\rho_T}{\rho} = \left[1 + \frac{\gamma - 1}{2} M^2 \right]^{\frac{1}{\gamma -1}} \text{ Isentropic flow required}

Normal Shock Relations

Assumes isentropic flow on each side of the shock Assumes flow across shock is adiabatic Property changes occur in a constant area (throat)

\frac{P_2}{P_1} = \frac{1 - \gamma + 2\gamma M_1^2}{1+\gamma} \frac{\rho_2}{\rho_1} = \left[\frac{2 + \left(\gamma - 1\right) M_1^2}{\left(\gamma+1\right) M_1^2} \right]^{-1} \frac{T_2}{T_1} = \left[\frac{1 - \gamma + 2\gamma M_1^2}{1 + \gamma} \right]\left[\frac{2 + \left(\gamma - 1\right) M_1^2}{\left(1 + \gamma\right) M_1^2} \right] M_2^2 = \frac{M_1^2 + \frac{2}{\gamma - 1}}{\frac{2\gamma}{\gamma-1} M_1^2-1}

Normal shock summary

P_{T_1} > P_{T_2}

P_{1} < P_{2}

\rho_{T_1} > \rho_{T_2}

\rho_{1} < \rho_{2}

T_{T_1} > T_{T_2}

T_{1} < T_{2}

M_1 > M_2

s_1 < s_2

5.5.1 Oblique Shocks

Oblique Shock Description

\delta = \text{surface turning angle}

\theta = \text{shock wave angle}

\text{Subscript 1 denotes upstream conditions}

\text{Subscript 2 denotes downstream conditions}

Oblique Shock Relations

  • Calculate P_2/P_1, T_2/T_1, and \rho_2/\rho_1 across oblique shocks by using normal shock equations and substituting M_1 \sin\theta in place of M_1
  • Calculate total pressure loss across oblique shock as

\frac{P_{T_2}}{P_{T_1}} = \left[\left[\frac{\gamma - 1}{\gamma + 1} + \frac{2}{{\left(\gamma + 1\right)M_1^2\sin^2\theta}}\right]^{\gamma} \left[\frac{2\gamma}{\gamma+1} M_1^2\sin^2\theta - \frac{\gamma - 1}{\gamma + 1} \right]\right]^{\frac{1}{1 - \gamma}}

  • Calculate relation between Mach number and angles as

M_2^2\sin^2\left(\delta - \theta\right) = \frac{M_1^2 \sin^2\theta + \frac{2}{\gamma-1}}{\frac{2\gamma}{\gamma - 1} M_1^2\sin^2\theta - 1}

Oblique Shock Turning Angle as a Function of Wave Angle

  • Two \theta solutions exist for every M_1 & \delta combination
    • These represent the strong and weak shock solutions
    • Weak shocks normally occur in nature
  • There is a minimum Mach number for each turning angle
  • The wave angle of a weak shock decreases with increased Mach
  • For a given Mach number, \theta approaches \mu as \delta decreases

Mach Cone Angle

Minimum Wave Angle: \mu = \sin^{-1}\left(1/M\right)

5.5.2 Supersonic Isentropic Expansion Relation

  • The wave angle \mu determines where the lower pressure can be felt and thus where the flow can be accelerated
  • As the flow accelerates, a new wave angle forms and the subsequent lower pressure further accelerates the flow
  • Results in a series of Mach waves forming a “fan” until the flow turns and accelerates so that it is parallel to the new boundary

Prandtl-Meyer Function

Shows flow’s required turning angle (\nu) to accelerate from one Mach number to another

\nu_M = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \left[\tan^{-1}\sqrt{\frac{\gamma - 1}{\gamma + 1} \left(M^2 - 1\right)} \right] - \tan^{-1}\sqrt{M^2-1}

  • If upstream Mach (M_1) = 1, then \nu_1 = 0, and equation directly relates downstream Mach (M_2) to surface turning angle (\Delta \nu)
  • If M_1 > 1, determine M_2 as follows:
    • Calculate upstream ν1 from above equation
    • Calculate \nu_2 = \nu_1 + \Delta \nu
    • Reverse above equation to obtain corresponding M_2
  • Above equation is tabulated in NACA TR 1135 and is plotted below

Example: Flow initially at M_1 = 2.0 accelerates through an expansion corner of 24 deg. Exit Mach number is 3.0

5.5.3 Two-Dimensional Supersonic Airfoil Approximations

  • Determine surface static pressures by calculating changes through obliques shocks and expansion fans

  • Ackert approximations for thin wings are based on

C_p = \frac{\Delta P}{q} \cong \pm\frac{2 \delta}{\sqrt{M^2 - 1}}

  • Double wedge airfoil approximations

C_L \cong \frac{4 \alpha}{\sqrt{M^2 - 1} }

C_D \cong \frac{4 \alpha^2}{\sqrt{M^2 - 1}} + \frac{4}{\sqrt{M^2 - 1}}\left(\frac{t}{c}\right)^2

  • Biconvex wing approximations

C_L \cong \frac{4 \alpha}{\sqrt{M^2 - 1} }

C_D \cong \frac{4 \alpha^2}{\sqrt{M^2 - 1}} + \frac{5.33}{\sqrt{M^2 - 1}}\left(\frac{t}{c}\right)^2

5.6 Drag Polars

(ref 5.2)

5.6.1 Drag Polar Construction and Terminology

C_L = \text{lift coefficient}

C_D = \text{drag coefficient}

C_{D_i} = \text{induced drag coefficient}

C_{D_0} = \text{parasitic drag coefficient}

\mathrm{AR} = \text{aspect ratio}

e = \text{Oswald efficiency factor}

l = \text{length flow has traveled}

S_{\mathrm{wet}} = \text{wetted area of surface}

S = \text{reference wing area}

Simple Drag Polar Equation Limitations

  • No separated flow losses
  • Symmetric Camber
  • Applies at one Mach, Altitude, \mathrm{cg}

C_D = C_{D_0} + \frac{C_L^2}{\pi \mathrm{AR} e} = C_{D_0} + C_{D_i}

“Polar” form of simple drag polar:

Linearized form of simple drag polar:

5.6.2 Complicating Factors

Airflow Separation Effects

Drag Polar Equation Accounting for Flow Separation:

C_D = C_{D_{\mathrm{min}}} + \frac{\left(C_L - C_{L_{\mathrm{min}}}\right)^2}{\pi \mathrm{AR} e} +k_2\left(C_L - C_{L_{\mathrm{break}}}\right)

  • Delete last term if C_L < C_{L_{\mathrm{break}}}
  • Determine k_2 from flight test

Reynolds Number Effects

(refs 5.4, 5.11)

  • Calculate length \mathrm{Re}_L and friction coefficient (c_f) for each surface as

\mathrm{Re}_L = \frac{\rho Vl}{\mu} = 7.101 \times10^6 \left[\frac{\delta}{\theta^2} \right]\left[\frac{T_K + 110}{398}\right] l

T_K = \text{Kelvin}

l = \text{total length, ft}

c_f = \left[\frac{1.328}{\sqrt{\mathrm{Re}_L}}\right] \left[1 + 0.1305 M^2 \right]^{-0.12} \text{ laminar} or c_f = \left[\frac{0.074}{\left(\mathrm{Re}_L\right)^2} - \frac{1700}{\mathrm{Re}_L} \right] \text{ transition} or c_f = 0.455\left[\log \mathrm{Re}_L\right]^{-258} \left[1 + 0.144 M^2\right]^{-0.65} \text{ turbulent}

  • In general, c_f decreases as \mathrm{Rn} increases (unless transitioning from laminar to turbulent flow)
  • Friction drag = c_f q S_{\mathrm{wet}} for each component (S_{\mathrm{wet}} = \text{wetted area})
  • Correct from test day to standard day aircraft drag coefficient by summing differences of each component’s drag change

\Delta C_D = \frac{\sum\left(c_{f_s} - c_{f_t} \right) S_{\mathrm{wet}}}{S}

Wing Camber or Incidence Angle Effects

Note slight increase in drag as lift decreases towards zero

Linearized drag polar for aircraft with wing camber and/or incidence

Revised drag polar equation accounting for wing camber or incidence

C_D = C_{D_{\mathrm{min}}} + \frac{\left(C_L - C_{L_{\mathrm{min}}} \right)^2}{\pi \mathrm{AR} e}

  • Generally not necessary since most flight occurs above C_{L_{\mathrm{min}}}

Mach Number Effects

  • Aircraft with low parasitic drag coefficients and high fineness ratios pay a relatively small “wave drag” penalty.

  • With external stores, same aircraft pays larger Mach penalty

Propeller Slipstream Effects

  • a.k.a “scrubbing” drag
  • Propwash increases flow speed over surface within slipstream
  • More drag is created by higher q and vorticity.
  • Function of prop speed and power absorbed (C_p) or thrust (C_T)
  • Problem should be addressed in airframe or propeller models

Trim Drag Effects

(ref 5.4)

e = \text{wing Oswald efficiency factor}

e_t = \text{tail Oswald efficiency factor}

b = \text{span}

b_t = \text{tail span}

x = \text{wing ac-to-cg distance}

l = \text{wing ac to tail ac distance}

S = \text{Area}

C_{D_{\mathrm{trim}}} = \frac{W^2}{\pi q^2 Sb^2e} \left[\frac{2}{lW}\left[x_0 - x_1\right] + \frac{1}{l^2} \left[1 + \frac{S}{S_t} \frac{e}{e_t} \left(\frac{b}{b_t}\right)^2\right] \left[x_0^2 - x_1^2\right] \right]

Trim drag change relative to total induced drag:

\frac{\Delta C_{D_{\mathrm{trim}}}}{\Delta C_{D_i}} = \frac{x}{l} \left[\frac{x}{l} \left(\frac{b}{b_t}\right)^2 \frac{e}{e_t} - 2 \right]

Plot of above equation

5.6.3 Drag Polar Analysis

D = \bar{q}SC_D = \bar{q}S \left[C_{D_0} + \frac{C_L^2}{\pi \mathrm{AR} e} \right] = \frac{1}{2}\rho_0 V_e^2 S\left[C_{D_0} + \frac{W^2}{\pi \mathrm{AR} e \left(\frac{1}{2} \rho_0 V_e^2 S \right)^2} \right]

  • For a given configuration (C_{D_0} \text{, } S \text{, } \mathrm{AR} \text{, } e)

D = k_1 V_e^2 + k_2 \frac{W^2}{V_e^2}

first term = parasitic drag, second term = induced drag

  • For any given weight, D = f(\text{equivalent airspeed}) only

  • Minimum total drag occurs when D_{\mathrm{induced}} = D_{\mathrm{parasitic}}
    • same as speed where C_{D_i} = C_{D_0}
    • occurs at max C_L/C_D ratio (same as max L/D ratio)
  • Minimum drag/velocity occurs at min slope of Drag vs V curve
    • same as speed where 3C_{D_i} = C_{D_0}
    • occurs at max C_L^{1/2}/C_D ratio

Power required = drag x true airspeed

P_{\mathrm{req}} = DV_T = D\frac{V_e}{\sqrt{\sigma}} = k_1\frac{V_e^3}{\sqrt{\sigma}} + k_2\frac{W^2}{\sqrt{\sigma}V_e}

Minimum total P_{\mathrm{req'd}} occurs when P_{\mathrm{induced}} = P_{\mathrm{parasitic}}

  • same as speed where C_{D_i} = 3C_{D_0}
  • occurs at max C_L^{3/2}/C_D ratio

Minimum power/velocity occurs at min slope of P_{\mathrm{req'd}} vs V curve

  • same as speed where C_{D_i} = C_{D_0}
  • occurs at max C_L /C_D ratio

Optimum Aerodynamic Flight Conditions

Gliders/ Engine-Out Flight

  • Max range (minimum glide slope) occurs at max C_L/C_D
    • same as condition where C_{D_0} = C_{D_i} if drag polar is parabolic
  • Min sink rate (minimum power req’d) occurs at max C_L^{3/2} /C_D ratio same as condition where 3C_{D_0} = C_{D_i} if drag polar is parabolic

Reciprocating Engine Aircraft (assuming constant \mathrm{BSFC} & prop \eta)

  • Max range (minimum power/velocity) occurs at max C_L/C_D ratio
    • same as condition where C_{D_0} = C_{D_i} if drag polar is parabolic
  • Max endurance (minimum power req’d) occurs at max C_L^{3/2} / C_D
    • same as condition where 3C_{D_0} = C_{D_i} if drag polar is parabolic

Turbine Jet Engine Aircraft (assuming constant \mathrm{TSFC})

  • Max range at constant altitude (minimum drag/velocity)
    • occurs at max C_L^{1/2} / C_D ratio
    • same as condition where C_{D_0} = 3C_{D_i} if drag polar is parabolic
  • Best cruise/climb range (maximum \left[M \times L/D \right] ratio)
    • occurs at max C_L/C_D^{3/2} ratio
    • same as condition where C_{D_0} = 2C_{D_i} if drag polar is parabolic
  • Best endurance (minimum drag)
    • occurs at max C_L/C_D ratio
    • same as condition where C_{D_0} = C_{D_i} if drag polar is parabolic

To calculate optimum speed V_2 for configuration2 & weight2 based on optimum speed V_1 at configuration1 & weight1

5.7 References

5.1 Roberts, Sean “Aerodynamics for Flight Testers” Chapter 3, Subsonic Aerodynamics, National Test Pilot School, Mojave, CA, 1999
5.2 Lawless, Alan R., et al, “Aerodynamics for Flight Testers” Chapter 4, Drag Polars, National Test Pilot School, Mojave ,CA, 1999
5.3 Hurt Hugh H., “Aerodynamics for Naval Aviators,” University of Southern California, Los Angeles, CA, 1959.
5.4 McCormick, Barnes W., “Aerodynamics, Aeronautics, and Flight Mechanics,” Wilet &Sons, 1979
5.5 Stinton, Darryl, “The Design of the Aeroplane,” BSP Professional Books, Oxford, 1983
5.6 Roskam, Jan Dr., “Airplane Design, Part VI,” Roskam Aviation and Engineering Corp. 1990
5.7 Anon, “Equations, Tables, and Charts for Compressible Flow” NACA Report 1135, 1953
5.8 Lewis, Gregory, “Aerodynamics for Flight Testers” Chapter 6, Supersonic Aerodynamics, National Test Pilot School, Mojave CA, 1999
5.9 White, Frank M. “Fluid Mechanics” pg 29, McGraw-Hill, 1979, ISBN 0-07-069667-5.
5.10 Anderson, John D. Jr, “Introduction to Flight” pg 142, McGraw-Hill, 1989, ISBN 0-07-001641-0.
5.11 Twaites, Bryan, Editor, “Incompressible Aerodynamics: An Account of the steady flow of incompressible Fluid Past Aerofoils, Wings, and Other Bodies,” Dover Publications, 1960.