User:Dolphin51/Sandbox

J Doug McLean Crowsnest Mr swordfish Ahunt Cagliost Ariadacapo Deeday-UK Steelpillow Eric Kvaalen Completeaerogeek

More than a year ago I volunteered to do a substantial re-write of our article on Downwash. Progress has been slow but I have now reached the stage where it will be beneficial if some Users knowledgeable about the subject, such as yourselves, take a look at what I have done, make comments and possibly offer suggestions. My draft re-write is available below. The draft is not yet mature enough for me to announce it at Talk:Downwash and invite the whole world to peruse it.

You will notice that my in-line citations are not simply citations. For my benefit, and yours, I have inserted quotations of the actual passage from the cited source. Naturally the final version of the article will present citations in the required format.

Thanks in advance for taking the time to peruse my draft. Please make any comments on the Talk page. Dolphin (t) 12:22, 28 May 2023 (UTC)

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Downwash is a term used in aeronautics to refer to a region within a flowfield in which the velocity vector has a downwards component. It can also refer to the magnitude of the vertically downwards component of velocity. It is usually associated with an airfoil that is generating lift. Downwash is given both a common meaning and a scientific meaning, and each is used to explain one of the forces acting on the airfoil.

In common usage downwash is used to explain the lift acting on airfoils such as the lift on the wing of a fixed-wing aircraft and the lift on the rotor disk of a rotorcraft.^{[Note 1]}^[1]

In scientific usage downwash is defined precisely as the vertically downwards component of velocity in a region of downwash. It is used to explain and quantify the lift-induced drag on the wing of a fixed-wing aircraft.^{[Note 2]}^{[Note 3]}^[2]^[3]^[4]^[5]^[6]

In aeronautics the terms upwash and sidewash are also used, and have meanings similar to downwash. The vertical distribution of mass in the atmosphere does not change so any region of downwash is accompanied by an equivalent region of upwash.

Common usage

In common usage, rising air ahead of an airfoil is called upwash, and descending air behind an airfoil is called downwash. The lift on an airfoil is associated with the change of speed and direction of the air as it transitions from the upwash to the downwash.^[1] At each point in the regions of upwash and downwash the flow speed and direction are unique. The magnitudes of this upwash and downwash, in their common usage, are not readily quantified so do not lead to determination of the magnitude of the lift on the airfoil. (In the scientific application of two-dimensional flow around an airfoil section, the lift per unit of span is readily quantified by applying the Kutta–Joukowski theorem to the circulation around the airfoil.)

In two-dimensional flow around an airfoil section, or around a uniform wing of infinite span, the upwards component of momentum in the upwash approaching a unit of span is equal in magnitude, but opposite in direction, to the downwards component in the downwash retreating from a unit of span. Consequently there is lift acting on the airfoil although its magnitude cannot be determined unless the magnitudes of the momenta in the upwash and downwash are assumed; the lift is directed vertically upwards and no lift-induced drag acts on the airfoil.

In the horseshoe vortex model of the airflow around a wing of finite span the common understanding of upwash and downwash is that both are associated primarily with the bound vortex, but superimposed on both upwash and downwash is a small downwards component due to the trailing vortices; it reduces velocities in the upwash, and increases velocities in the downwash.^[7] The upwards component of momentum in the upwash approaching the wing is less than the downwards component of momentum in the downwash retreating from the wing so the lift is not entirely directed upwards but is canted backwards through an angle equal to the downwash angle (or induced angle of attack); as a consequence, a small component of the lift acts as drag on the wing; this component of the lift is called lift-induced drag. This phenomenon is due entirely to the presence of flow associated with trailing vortices.

Rotorcraft

and Momentum theory

Scientific usage

In scientific usage downwash is a vertically downwards component of the airflow velocity vector. It is the component that can be attributed to the trailing vorticity ultimately manifested in the vortices trailing from the wingtips and other locations on the wing.^[2]

In two-dimensional flow, and flow over a uniform wing of infinite span, there are no trailing vortices and downwash is zero.

In the horseshoe vortex model of the airflow around a wing the scientific understanding of downwash is that it is entirely the flow associated with the trailing vortices, and is not attributed to the bound vortex.

The downwash velocity in the vicinity of an untwisted wing of elliptical planform is uniform across the wingspan and its magnitude is denoted by w. Alternatively the downwash may be denoted by the downwash angle ε which is also uniform across the wingspan of an untwisted wing of elliptical planform.^{[Note 4]}^[3] The assumption of uniform downwash across the span of a wing simplifies the task of determining the spanwise lift distribution and the lift-induced drag.

$tan(\epsilon )={\frac {w}{V_{T}}}$ where V_T is the true airspeed of the wing. Angle ε is small so tan(ε) is approximately equal to ε.

$\epsilon ={\frac {C_{L}}{\pi \lambda }}$ where $\lambda$ is aspect ratio.^{[Note 5]}

At the wing, and a short distance downstream of the wing trailing edge, downwash is uniform, or approximately uniform, across the span of the wing. After a short distance downstream of the trailing edge the downwash on each semi-span consolidates into a large vortex. The result is a pair of trailing vortices. Between the trailing vortices the downwash is no longer uniform across the span.^{[Note 6]}^[2]

Two-dimensional flow around a circular cylinder

Possibility for a suitable diagram courtesy of University of Texas.^[8]

Richard Fitzpatrick Flow past a cylindrical obstacle. 336L

[1]

Induced angle of attack

Downwash angle ε is also known as the induced angle of attack $\alpha _{i}$ .^{[Note 7]}^[2]

The presence of downwash reduces the angle of attack on a wing. The effective angle of attack is equal to the geometric angle of attack reduced by the induced angle of attack:^{[Note 8]}^[2]

\alpha _{e}=\alpha _{g}-\alpha _{i}=\alpha _{g}-\epsilon

If a fixed-wing aircraft is to operate at an effective angle of attack of $\alpha _{1}$ , the pilot must achieve a geometric angle of attack of $\alpha _{1}+\alpha _{i}$ in order to compensate for angle of attack lost due to downwash:^[9]

\alpha _{g}=\alpha _{e}+\alpha _{i}=\alpha _{e}+\epsilon

This is readily observed in aircraft with wings of low aspect ratio such as delta wings, particularly when taking off and landing. Some of these aircraft are equipped with a droop nose to allow the pilots an adequate view of the ground when taking off and landing. Concorde has a droop nose because of the high geometric angle of attack when flying slowly during takeoff and on the approach to landing.^[10]

Lift-induced drag

The presence of trailing vortices and downwash influencing the flow around a wing of finite span causes the flow to approach the wing with a reduced angle of attack.^{[Note 9]} The lift vector is defined to be perpendicular to the local flow direction so downwash causes the lift vector to be canted backwards by the angle $\epsilon$ (or $\alpha _{i}$ ). A small component of this lift vector is directed backwards so must be included in the total drag on the wing. This component is called lift-induced drag D_i.^[2]

$D_{i}=Lsin(\epsilon )$ which is approximately $L(\epsilon )$ if epsilon is measured in radians.

$\epsilon ={\frac {C_{L}}{\pi \lambda }}$ where $\lambda$ is aspect ratio.

For an untwisted wing of elliptical planform:^[6]

D_{i}={\tfrac {1}{2}}\rho V^{2}SC_{L}\left({\frac {C_{L}}{\pi \lambda }}\right)

For such a wing the coefficient of induced drag

C_{D,i}

is therefore

{\frac {(C_{L})^{2}}{\pi \lambda }}

For a wing that does not have an elliptical planform:

C_{D,i}={\frac {(C_{L})^{2}}{\pi e\lambda }}

where

e

is the span efficiency factor.^[14]

In two-dimensional flow, and flow over a uniform wing of infinite span, there are no trailing vortices and no downwash so lift-induced drag is also zero.

Horseshoe vortex

Two-dimensional flow

In two-dimensional flow at velocity $V_{\infty }$ around a stationary airfoil (or a uniform wing of infinite span) the velocity $V$ at any point is the vector sum:

V=V_{\infty }+V_{vortex}

where

V_{\infty }

is the velocity of the free stream, and

V_{vortex}

is the velocity associated with the circulation

\Gamma

induced by the bound vortex.

In the flowfield upstream of the airfoil the flow velocity has an upward component. In the common usage, this region of flow is called upwash.

In the flowfield downstream of the airfoil the flow velocity has a downward component. In the common usage, this region of flow is called downwash.

In two-dimensional flow (and flow around a uniform wing of infinite span) there is no trailing vortex. In scientific usage there is neither upwash nor downwash. There is no lift-induced drag.

Three-dimensional flow about a wing of finite span

In the horseshoe vortex model of a stationary wing immersed in a flowfield moving at a velocity $V_{\infty }$ measured at the remote free stream, the velocity $V$ at any point in the flowfield is the vector sum:

V=V_{\infty }+V_{vortex}+V_{trailing}

where

V_{vortex}

is the velocity associated with the bound vortex, and

V_{trailing}

is the velocity associated with the trailing vortices.

Upstream of the wing

At each point in the flowfield upstream of the wing the velocity has an upwards vertical component $V_{upward}$ : $V_{upward}=V_{vortex}+V_{trailing}$ where $V_{vortex}$ is directed upwards, and $V_{trailing}$ is directed downwards and is smaller than $V_{vortex}$ .

In the common usage, $V_{upward}$ shows the region of upwash. In scientific usage, the downward component $V_{trailing}$ is called downwash w.

Downstream of the wing

At each point in the flowfield downstream of the wing the velocity has a downward vertical component $V_{downward}$ :

V_{downward}=V_{vortex}+V_{trailing}

where

V_{vortex}

and

V_{trailing}

are both directed downwards.

In the common usage, $V_{downward}$ shows the region of downwash. In scientific usage, the downward component $V_{trailing}$ is called downwash w.

The velocity vectors adjacent to every point on the trailing edge are always parallel to the wing surface near the trailing edge so the flow leaves the trailing edge smoothly as explained by the Kutta condition.

In three-dimensional flow around a wing of finite span there is significant trailing vorticity and at least one pair of trailing vortices. Lift-induced drag is associated with the trailing vortices and the downwash w.

When a wing is generating lift it is constantly operating in the downwash w induced by its own trailing vortices.^[3] This causes the lift vector to be canted backwards through an angle equal to the downwash angle ε measured at the aerodynamic center. Consequently the lift vector has a small component directed backwards so it functions as part of the total drag. This component of the lift vector is called the lift-induced drag.

Lifting-line theory

The horseshoe vortex model of the vortex system around a wing as it generates lift attributes the downwash to one or more pairs of trailing vortices. Prandtl’s lifting-line theory is more complex than the horseshoe model and attributes the downwash to a continuous distribution of trailing vorticity embedded in a vortex sheet trailing downstream of the wing.^[15]

Lifting-line theory shows that the minimum induced drag on a wing occurs when the downwash is a uniform value, w, across the span of the wing. If this occurs the downwash angle is also a uniform value, ε, across the span of the wing. Uniform downwash across the span causes the spanwise lift distribution to be elliptical.^{[Note 10]} Lifting-line theory shows that uniform downwash and elliptical lift distribution can be achieved by an untwisted wing only if the wing has an elliptical planform.^[2]

A short distance downstream of the trailing edge the vortex sheet consolidates into a pair of trailing vortices. The pressure in the core of each trailing vortex can be low enough that the relative humidity reaches 100% and condensation occurs, rendering the core of the vortex visible as a white filament.

Notes

^ Downwash (aerodynamics). Air forced down by aerodynamic action below and behind the wing of an airplane or rotor of a helicopter. Aerodynamic lift is produced when air is deflected downward. The upward force on the aircraft is the same as the downward force on the air. When the mass (sic) of air in the downwash is equal to the weight (sic) of the aircraft forcing it down, the aircraft rises. Dictionary of Aeronautical Terms, Crane, Dale
^ “These wing-tip vortices downstream of the wing induce a small downward component of air velocity in the neighbourhood of the wing itself. ... this secondary movement induces a small velocity component in the downward direction at the wing. This downward component is called downwash, denoted by the symbol w.” (Anderson, p.231)
^ "Hence, we see that the presence of downwash over a finite wing reduces the angle of attack that each section effectively sees, and moreover, it creates a component of drag – the induced drag D_i." (Anderson, p.232)
^ “the downwash is constant over the span for an elliptical lift distribution.”

“For an elliptic lift distribution, the induced angle of attack is also constant along the span.”

“... both the downwash and induced angle of attack go to zero as the wing span becomes infinite” (Anderson, p.245)
^ ” $\alpha _{i}={\frac {C_{L}}{\pi AR}}$ where AR is aspect ratio.” (Equation 5.42. Anderson, page 246)
^ ”in the 3-D flow over a finite wing, the streamlines leaving the trailing edge from the top and bottom surfaces are in different directions”
“Hence, a sheet of vorticity actually trails downstream from the trailing edge of a finite wing”
“This sheet tends to roll up at the edges and helps to form the wing-tip vortices” (Anderson, p.258)
^ “For an elliptic lift distribution, the induced angle of attack is also constant along the span.”
“both the downwash and induced angle of attack go to zero as the wing span becomes infinite” (Anderson, p.245)
^ "the local relative wind is inclined below the direction of V_∞ by the angle α_i called the induced angle of attack." (Anderson, p.231)
^ ”The effect of a finite wing is to reduce the lift slope” (Anderson, page 255)
^ ”the lift distribution which yields minimum induced drag is the elliptical lift distribution.” (Anderson, page 251)

References

^ ^a ^b Crane, Dale: Dictionary of Aeronautical Terms, 3rd edition, page 172
^ ^a ^b ^c ^d ^e ^f ^g Anderson, John D. (1984) Fundamentals of Aerodynamics, section 5.1, McGraw-Hill ISBN 0-07-001656-9

"These wing-tip vortices downstream of the wing induce a small downward component of air velocity in the neighborhood of the wing itself. ... this secondary movement induces a small velocity component in the downward direction at the wing. This downward component is called downwash, denoted by the symbol w." (page 231)

”the local relative wind is inclined below the direction of V_∞ by the angle α_i called the induced angle of attack.” (page 231)

"Hence, we see that the presence of downwash over a finite wing reduces the angle of attack that each section effectively sees, and moreover, it creates a component of drag – the induced drag D_i." (page 232)

”the downwash is constant over the span for an elliptical lift distribution.” “For an elliptic lift distribution, the induced angle of attack is also constant along the span.” “both the downwash and induced angle of attack go to zero as the wing span becomes infinite” (page 245)

” $\alpha _{i}={\frac {C_{L}}{\pi AR}}$ where AR is aspect ratio.” (Equation 5.42, page 246)

”the lift distribution which yields minimum induced drag is the elliptical lift distribution.” (page 251)

”The effect of a finite wing is to reduce the lift slope” (page 255)

”in the 3-D flow over a finite wing, the streamlines leaving the trailing edge from the top and bottom surfaces are in different directions” “Hence, a sheet of vorticity actually trails downstream from the trailing edge of a finite wing” “This sheet tends to roll up at the edges and helps to form the wing-tip vortices” (page 258)
^ ^a ^b ^c ^d Clancy, L.J. (1975), Aerodynamics, section 5.14, Pitman ISBN 0 273 01120 0

"The main consequence of these vortices is that the air in immediate vicinity of the wing, and behind it, acquires a downward velocity component. It may be measured in terms of either downwash velocity, usually denoted by w, or downwash angle, denoted by ε." (page 75) Cite error: The named reference "Clancy" was defined multiple times with different content (see the help page).
^ Batchelor, G.K. (1967), An Introduction to Fluid Dynamics, sub-section 7.8. Cambridge University Press. ISBN:

"Thus here the 'downwash' due to the trailing sheet vortex makes the effective angle of incidence, relative to the no-lift attitude, equal to a fraction of its apparent value, over the whole wing; and the lift is a similar fraction of the value it would have if each section of the wing acted as an isolated two-dimensional aerofoil." (page 588)
^ Abbott, I.H. and von Doenhoff, A.E. (1949), Theory of Wing Sections, Dover Publications:

"The effect of trailing vortices corresponding to a positive lift is to induce a downward component of velocity at and behind the wing. This downward component is called the downwash." (page 9)

"The effect of the downwash is to change the relative direction of the air stream over the section." (page 9)

"The rotation of the flow also causes a corresponding rotation of the lift vector to produce a drag component in the direction of motion. This induced-drag coefficient varies as the square of the lift coefficient because the amount of rotation and the magnitude of the lift vector increase sumultaneously." (page 9)

"A comparatively simple solution was obtained by Prandtl for an elliptical lift distribution. In this case the downwash is constant between the wing tips, and the induced drag is less than that for any other type of lift distribution." (page 9)
^ ^a ^b Irving, F.G. (1966), An Introduction to the Longitudinal Static Stability of Low-Speed Aircraft, Pergamon Press. ISBN:

"The downwash at the wing also reduces the local incidence and hence the local lift coefficients. The simplest case relates to an untwisted wing having an elliptical spanwise lift distribution. The downwash and local lift coefficient are both constant across the span (and hence the planform is also elliptical). The downwash $\epsilon _{w}$ is then
$\epsilon _{w}={\frac {C_{L}}{\pi \lambda }}$ where $\lambda$ is aspect ratio

and the aerodynamic incidence is $\alpha -\epsilon _{w}$ , where $\alpha$ is the angle between the section zero-lift line and the free stream far ahead of the the wing." (page 19)
^ "Shed Vortex". NASA Glenn Research Center. Retrieved March 18, 2023.
^ Fitzpatrick, Richard (2006-02-02). "mks units". Classical Mechanics: An Introductory Course. University of Texas at Austin. Retrieved 2024-03-04.
^ Anderson, John D “... we obtain the fundamental equation of Prandtl’s lifting-line theory; it simply states that the geometric angle of attack is equal to the sum of the effective angle plus the induced angle of attack.” (page 243)
^ "The high [geometric] angle of attack on landing dictates the need for a moveable nose ... So the nose, which must form a streamlined shape for supersonic flying, has to droop for landing." Flying Concorde (p.47) Calvert, Brian (1981), Fontana
^ Hurt, H. H. (1965) Aerodynamics for Naval Aviators, Figure 1.30, NAVWEPS 00-80T-80
^ Kermode, A.C. (1972). Mechanics of Flight, Figure 3.29, Ninth edition. Longman Scientific & Technical, England. ISBN 0-582-42254-X
^ McLean, Doug (2005). Wingtip Devices: What They Do and How They Do It (PDF). 2005 Boeing Performance and Flight Operations Engineering Conference.
^ Anderson, John D. (1984) Fundamentals of Aerodynamics, p.251, McGraw-Hill ISBN 0-07-001656-9
^ Anderson, John D. (1984) Fundamentals of Aerodynamics, section 5.3, McGraw-Hill ISBN 0-07-001656-9

[1] Downwash (aerodynamics). Air forced down by aerodynamic action below and behind the wing of an airplane or rotor of a helicopter. Aerodynamic lift is produced when air is deflected downward. The upward force on the aircraft is the same as the downward force on the air. When the mass (sic) of air in the downwash is equal to the weight (sic) of the aircraft forcing it down, the aircraft rises. Dictionary of Aeronautical Terms, Crane, Dale

[3] “These wing-tip vortices downstream of the wing induce a small downward component of air velocity in the neighbourhood of the wing itself. ... this secondary movement induces a small velocity component in the downward direction at the wing. This downward component is called downwash, denoted by the symbol w.” (Anderson, p.231)

[4] "Hence, we see that the presence of downwash over a finite wing reduces the angle of attack that each section effectively sees, and moreover, it creates a component of drag – the induced drag D_i." (Anderson, p.232)

[11] “the downwash is constant over the span for an elliptical lift distribution.”

“For an elliptic lift distribution, the induced angle of attack is also constant along the span.”

“... both the downwash and induced angle of attack go to zero as the wing span becomes infinite” (Anderson, p.245)

[12] ” $\alpha _{i}={\frac {C_{L}}{\pi AR}}$ where AR is aspect ratio.” (Equation 5.42. Anderson, page 246)

[13] ”in the 3-D flow over a finite wing, the streamlines leaving the trailing edge from the top and bottom surfaces are in different directions”
“Hence, a sheet of vorticity actually trails downstream from the trailing edge of a finite wing”
“This sheet tends to roll up at the edges and helps to form the wing-tip vortices” (Anderson, p.258)

[15] “For an elliptic lift distribution, the induced angle of attack is also constant along the span.”
“both the downwash and induced angle of attack go to zero as the wing span becomes infinite” (Anderson, p.245)

[16] "the local relative wind is inclined below the direction of V_∞ by the angle α_i called the induced angle of attack." (Anderson, p.231)

[22] ”The effect of a finite wing is to reduce the lift slope” (Anderson, page 255)

[25] ”the lift distribution which yields minimum induced drag is the elliptical lift distribution.” (Anderson, page 251)

[Crane-2] Crane, Dale: Dictionary of Aeronautical Terms, 3rd edition, page 172

[Anderson-5] ^ ^a ^b ^c ^d ^e ^f ^g Anderson, John D. (1984) Fundamentals of Aerodynamics, section 5.1, McGraw-Hill ISBN 0-07-001656-9

"These wing-tip vortices downstream of the wing induce a small downward component of air velocity in the neighborhood of the wing itself. ... this secondary movement induces a small velocity component in the downward direction at the wing. This downward component is called downwash, denoted by the symbol w." (page 231)

”the local relative wind is inclined below the direction of V_∞ by the angle α_i called the induced angle of attack.” (page 231)

"Hence, we see that the presence of downwash over a finite wing reduces the angle of attack that each section effectively sees, and moreover, it creates a component of drag – the induced drag D_i." (page 232)

”the downwash is constant over the span for an elliptical lift distribution.” “For an elliptic lift distribution, the induced angle of attack is also constant along the span.” “both the downwash and induced angle of attack go to zero as the wing span becomes infinite” (page 245)

” $\alpha _{i}={\frac {C_{L}}{\pi AR}}$ where AR is aspect ratio.” (Equation 5.42, page 246)

”the lift distribution which yields minimum induced drag is the elliptical lift distribution.” (page 251)

”The effect of a finite wing is to reduce the lift slope” (page 255)

”in the 3-D flow over a finite wing, the streamlines leaving the trailing edge from the top and bottom surfaces are in different directions” “Hence, a sheet of vorticity actually trails downstream from the trailing edge of a finite wing” “This sheet tends to roll up at the edges and helps to form the wing-tip vortices” (page 258)

[Clancy-6] Clancy, L.J. (1975), Aerodynamics, section 5.14, Pitman ISBN 0 273 01120 0

"The main consequence of these vortices is that the air in immediate vicinity of the wing, and behind it, acquires a downward velocity component. It may be measured in terms of either downwash velocity, usually denoted by w, or downwash angle, denoted by ε." (page 75) Cite error: The named reference "Clancy" was defined multiple times with different content (see the help page).

[Batchelor-7] Batchelor, G.K. (1967), An Introduction to Fluid Dynamics, sub-section 7.8. Cambridge University Press. ISBN:

"Thus here the 'downwash' due to the trailing sheet vortex makes the effective angle of incidence, relative to the no-lift attitude, equal to a fraction of its apparent value, over the whole wing; and the lift is a similar fraction of the value it would have if each section of the wing acted as an isolated two-dimensional aerofoil." (page 588)

[Abbott-8] Abbott, I.H. and von Doenhoff, A.E. (1949), Theory of Wing Sections, Dover Publications:

"The effect of trailing vortices corresponding to a positive lift is to induce a downward component of velocity at and behind the wing. This downward component is called the downwash." (page 9)

"The effect of the downwash is to change the relative direction of the air stream over the section." (page 9)

"The rotation of the flow also causes a corresponding rotation of the lift vector to produce a drag component in the direction of motion. This induced-drag coefficient varies as the square of the lift coefficient because the amount of rotation and the magnitude of the lift vector increase sumultaneously." (page 9)

"A comparatively simple solution was obtained by Prandtl for an elliptical lift distribution. In this case the downwash is constant between the wing tips, and the induced drag is less than that for any other type of lift distribution." (page 9)

[Irving-9] Irving, F.G. (1966), An Introduction to the Longitudinal Static Stability of Low-Speed Aircraft, Pergamon Press. ISBN:

"The downwash at the wing also reduces the local incidence and hence the local lift coefficients. The simplest case relates to an untwisted wing having an elliptical spanwise lift distribution. The downwash and local lift coefficient are both constant across the span (and hence the planform is also elliptical). The downwash $\epsilon _{w}$ is then
$\epsilon _{w}={\frac {C_{L}}{\pi \lambda }}$ where $\lambda$ is aspect ratio

and the aerodynamic incidence is $\alpha -\epsilon _{w}$ , where $\alpha$ is the angle between the section zero-lift line and the free stream far ahead of the the wing." (page 19)

[10] "Shed Vortex". NASA Glenn Research Center. Retrieved March 18, 2023.

[14] Fitzpatrick, Richard (2006-02-02). "mks units". Classical Mechanics: An Introductory Course. University of Texas at Austin. Retrieved 2024-03-04.

[17] Anderson, John D “... we obtain the fundamental equation of Prandtl’s lifting-line theory; it simply states that the geometric angle of attack is equal to the sum of the effective angle plus the induced angle of attack.” (page 243)

[18] "The high [geometric] angle of attack on landing dictates the need for a moveable nose ... So the nose, which must form a streamlined shape for supersonic flying, has to droop for landing." Flying Concorde (p.47) Calvert, Brian (1981), Fontana

[19] Hurt, H. H. (1965) Aerodynamics for Naval Aviators, Figure 1.30, NAVWEPS 00-80T-80

[20] Kermode, A.C. (1972). Mechanics of Flight, Figure 3.29, Ninth edition. Longman Scientific & Technical, England. ISBN 0-582-42254-X

[McLean-21] McLean, Doug (2005). Wingtip Devices: What They Do and How They Do It (PDF). 2005 Boeing Performance and Flight Operations Engineering Conference.

[23] Anderson, John D. (1984) Fundamentals of Aerodynamics, p.251, McGraw-Hill ISBN 0-07-001656-9

[Anderson2-24] Anderson, John D. (1984) Fundamentals of Aerodynamics, section 5.3, McGraw-Hill ISBN 0-07-001656-9

[Note 1]

[1]

[Note 2]

[Note 3]

[2]

[3]

[4]

[5]

[6]

[7]

[Note 4]

[Note 5]

[Note 6]

[8]

[Note 7]

[Note 8]

[9]

[10]

[11]

[12]

[13]

[Note 9]

[14]

[15]

[Note 10]