User:
Lift
Storyboard 
The flow around a wing leads to the formation of whirlpools that, depending on the shape and angle of the wing with respect to the flow, can cause whirlpools in a section of these. If volume elements around the wing are considered and it is assumed that energy conservation can be madly assumed, different speeds will have to lead to different pressures (Bernoulli) on the surface.The sum of all pressures on the surface in the vertical direction, both on the wing (downward force) and under the wing (upward force) leads to a total force that we call lift. If this is positive we can overcome gravity and make the body (plane / bird) rise.
ID:(463, 0)
Wing generating lift
Description
Upon observing the average flow around a wing, one can notice that the lines above the wing are longer than those below it. In simplified terms, it is argued that due to this longer path, the speed at the top ($v_t$) is expected to be greater than the speed at the bottom ($v_b$), although both are higher than the speed with respect to the medium ($v$).
If Bernoulli's law is applicable, the difference in velocities would result in a difference in pressures acting on the wing. In particular, if the speed at the top ($v_t$) is greater, its corresponding the pressure on top of wing ($p_t$) would be lower than with the speed at the bottom ($v_b$) and its corresponding the pressure on the bottom of the wing ($p_b$). This would imply the existence of a the lift force ($F_L$) due to the effect of this pressure difference.However, as seen towards the end of the wing profile (right side), turbulence forms, limiting the applicability of Bernoulli's principle. Specifically, it should be considered that in a certain portion of the wing's perimeter, it may not be applicable, and there will be no contribution to lift.
ID:(11075, 0)
Circulation around an object
Concept
To define circulation, we must first establish the path that will be followed around the object/wing in a counterclockwise direction, as indicated in the following image:
Circulation is defined as the product of the perimeter around the object and the projection of velocity onto the surface. Since this velocity projection can vary along the perimeter, we must sum it through infinitesimal elements of the perimeter, where the velocity projection is calculated using the dot product between it and the perimeter element. Graphically, this is represented as follows:
Mathematically, this is expressed through the closed line integral of the aforementioned dot product:
| $ \Gamma =\displaystyle\oint_C \vec{v} \cdot d\vec{l} $ |
Since the sum is performed counterclockwise, in the upper part, the direction in which the perimeter elements point is opposite to the direction of velocity. In the lower part, both point in the same direction, leading to the upper part partially canceling out the lower part.
ID:(1167, 0)
KuttaJoukowski theorem
Concept
The relationship between the aerodynamic circulation ($\Gamma$) and the flow around the object is established through the Kutta-Joukowski theorem, enabling the calculation of the lift force ($F_L$) using the wing span ($L$), the density ($\rho$), and the speed with respect to the medium ($v$) as follows:
| $ \displaystyle\frac{ F_L }{ L } = - \rho v \Gamma$ |
By simplifying the modeling of the flow around the object, it becomes feasible to estimate circulation using the surface that generates lift ($S_w$) and the coefficient of lift ($C_L$) with the equation:
| $ \Gamma = \displaystyle\frac{ S_w }{2 L } C_L v ^2$ |
Consequently, the lift force ($F_L$) can be approximated with the equation:
| $ F_L =\displaystyle\frac{1}{2} \rho S_w C_L v ^2$ |
Here, the coefficient of lift ($C_L$) encapsulates the aerodynamic effects of the object.[1] "Über die Aufgabe der Flügeltheorie und ein neues Verfahren zur Herleitung derselben." (On the task of wing theory and a new method for its derivation.), Martin Wilhelm Kutta, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1902)[2] "Über die Erhaltung des Luftkreises um ein Profil." (On the conservation of the air circle around a profile.), Nikolai Zhukovsky, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1904)
ID:(1168, 0)
Lift coefficient
Description
The lift coefficient is a function of the angle of attack and typically follows the trend indicated in the following figure:
In the illustrated case, the slope is approximately 1.5 per 15 degrees, which corresponds to 0.1 1/degree or 5.73 1/radian.
ID:(7148, 0)
Lift force in the flow
Concept
The difference in pressure between the lower and upper surfaces of the wing generates the lift force, indicated by an arrow perpendicular to the wing's surface. This force opposes the gravitational force acting downward:
Birds or aircraft are able to fly when the lift force exceeds the gravitational force.
ID:(7036, 0)
The takeoff of a plane
Concept
The difference in pressure between the lower and upper surfaces of the wing generates the lift force, indicated by an arrow perpendicular to the wing's surface. This force opposes the gravitational force acting downward:
Birds or aircraft are able to fly when the lift force exceeds the gravitational force.
ID:(15157, 0)
Lift
Model
The flow around a wing leads to the formation of whirlpools that, depending on the shape and angle of the wing with respect to the flow, can cause whirlpools in a section of these. If volume elements around the wing are considered and it is assumed that energy conservation can be madly assumed, different speeds will have to lead to different pressures (Bernoulli) on the surface. The sum of all pressures on the surface in the vertical direction, both on the wing (downward force) and under the wing (upward force) leads to a total force that we call lift. If this is positive we can overcome gravity and make the body (plane / bird) rise.
Variables
Calculations
to 
,
then, select the variable: 
to 
Calculations
Variable
Given
Calculate
Target :
Equation
To be used
Equations
(ID 4416)
The lift force ($F_L$), along with the wing span ($L$), the density ($\rho$), the wing top speed factor ($c_t$), the wing bottom speed factor ($c_b$), the upper wing length ($l_t$), the bottom wing length ($l_b$), and the speed with respect to the medium ($v$), is found in
| $ F_L = \rho L ( c_b l_b - c_t l_t ) v ^2$ |
If we consider the surface that generates lift ($S_w$), given by the wing span ($L$), the upper wing length ($l_t$), and the bottom wing length ($l_b$),
| $ S_w = \displaystyle\frac{1}{2} L ( l_t + l_b )$ |
and for the coefficient of lift ($C_L$), defined as
| $ C_L = 4\displaystyle\frac{ c_t l_t - c_b l_b }{ l_t + l_b }$ |
we obtain
| $ F_L =\displaystyle\frac{1}{2} \rho S_w C_L v ^2$ |
(ID 4417)
(ID 4441)
The lift force ($F_L$) along with the density ($\rho$), the surface that generates lift ($S_w$), the coefficient of lift ($C_L$), and the speed with respect to the medium ($v$) is represented as
| $ F_L =\displaystyle\frac{1}{2} \rho S_w C_L v ^2$ |
which, along with the body mass ($m$) and the gravitational Acceleration ($g$), must be equal to:
| $ F_g = m g $ |
that is:
$\displaystyle\frac{1}{2}\rho S_wC_Lv^2=mg$
resulting in:
| $ C_L =\displaystyle\frac{2 m g }{ \rho S_w }\displaystyle\frac{1}{ v ^2}$ |
(ID 4442)
The coefficient of lift ($C_L$) is calculated with the body mass ($m$), the gravitational Acceleration ($g$), the surface that generates lift ($S_w$), the density ($\rho$), and the speed with respect to the medium ($v$) as follows:
| $ C_L =\displaystyle\frac{2 m g }{ \rho S_w }\displaystyle\frac{1}{ v ^2}$ |
Therefore, with the proportionality constant coefficient sustainability ($c$) and the angle of attack of a wing ($\alpha$),
| $ C_L = c \alpha $ |
we have
| $ \alpha =\displaystyle\frac{2 m g }{ c \rho S_w }\displaystyle\frac{1}{ v ^2}$ |
(ID 4443)
(ID 14515)
(ID 15152)
(ID 15153)
The lift force ($F_L$) depends on the surface that generates lift ($S_w$) and the pressure difference on an object ($\Delta p$) as per
| $ F_L = S_w \Delta p $ |
in the expression for the lift force ($F_L$) with the wing span ($L$), the density ($\rho$), the wing top speed factor ($c_t$), the wing bottom speed factor ($c_b$), the upper wing length ($l_t$), the bottom wing length ($l_b$), and the speed with respect to the medium ($v$)
| $ F_L = \rho L ( c_b l_b - c_t l_t ) v ^2$ |
contains the factor the wing span ($L$) that is associated with the surface that generates lift ($S_w$). However, both can be associated if we consider the wing's width as the average of the upper wing length ($l_t$) and the bottom wing length ($l_b$). This leads us to obtain
| $ S_w = \displaystyle\frac{1}{2} L ( l_t + l_b )$ |
(ID 15154)
The lift force ($F_L$) along with the wing span ($L$), the density ($\rho$), the wing top speed factor ($c_t$), the wing bottom speed factor ($c_b$), the upper wing length ($l_t$), the bottom wing length ($l_b$), and the speed with respect to the medium ($v$) is found in
| $ F_L = \rho L ( c_b l_b - c_t l_t ) v ^2$ |
If we consider the surface that generates lift ($S_w$) given by the wing span ($L$), the upper wing length ($l_t$), and the bottom wing length ($l_b$)
| $ S_w = \displaystyle\frac{1}{2} L ( l_t + l_b )$ |
we can rewrite the equation for the lift force ($F_L$) as
$F_L =\displaystyle\frac{1}{2} \rho S_w \displaystyle\frac{4(c_bl_b-c_tl_t)}{l_b+l_t} v^2$
which allows us to introduce the lift coefficient:
| $ C_L = 4\displaystyle\frac{ c_t l_t - c_b l_b }{ l_t + l_b }$ |
(ID 15155)
The lift force ($F_L$) is related to the aerodynamic circulation ($\Gamma$), the wing span ($L$), the density ($\rho$), and the speed with respect to the medium ($v$) as follows:
| $ \displaystyle\frac{ F_L }{ L } = - \rho v \Gamma$ |
Since the aerodynamic circulation ($\Gamma$) is related to the wing top speed factor ($c_t$), the wing bottom speed factor ($c_b$), the upper wing length ($l_t$), and the bottom wing length ($l_b$) as follows:
| $$ |
We can conclude that:
| $ F_L = \rho L ( c_b l_b - c_t l_t ) v ^2$ |
(ID 15156)
The aerodynamic circulation ($\Gamma$) is defined in terms of the lengths the upper wing length ($l_t$) and the bottom wing length ($l_b$) along with the velocities the speed at the top ($v_t$) and the speed at the bottom ($v_b$), as follows:
$\Gamma = -l_t v_t + l_b v_b$
If the speed at the top ($v_t$) is proportional to the wing top speed factor ($c_t$) with respect to the speed with respect to the medium ($v$):
| $ v_t = c_t v $ |
and the speed at the bottom ($v_b$) is proportional to the wing bottom speed factor ($c_b$) with respect to the speed with respect to the medium ($v$):
| $ v_b = c_b v $ |
we can express it as:
$\Gamma = -l_t c_t v + l_b c_b v$
This leads us to the following equation:
| $ \Gamma = ( c_b l_b - c_t l_t ) v $ |
(ID 15193)
When relating the aerodynamic circulation ($\Gamma$) to the wing bottom speed factor ($c_b$), the wing top speed factor ($c_t$), the bottom wing length ($l_b$), and the upper wing length ($l_t$), we have:
| $ \Gamma = ( c_b l_b - c_t l_t ) v $ |
By estimating the surface that generates lift ($S_w$) with the wing span ($L$) using:
| $ S_w = \displaystyle\frac{1}{2} L ( l_t + l_b )$ |
and calculating the coefficient of lift ($C_L$) with:
| $ C_L = 4\displaystyle\frac{ c_t l_t - c_b l_b }{ l_t + l_b }$ |
The result is:
| $ \Gamma = \displaystyle\frac{ S_w }{2 L } C_L v ^2$ |
(ID 15195)
Examples
(ID 15181)
Upon observing the average flow around a wing, one can notice that the lines above the wing are longer than those below it. In simplified terms, it is argued that due to this longer path, the speed at the top ($v_t$) is expected to be greater than the speed at the bottom ($v_b$), although both are higher than the speed with respect to the medium ($v$).
If Bernoulli's law is applicable, the difference in velocities would result in a difference in pressures acting on the wing. In particular, if the speed at the top ($v_t$) is greater, its corresponding the pressure on top of wing ($p_t$) would be lower than with the speed at the bottom ($v_b$) and its corresponding the pressure on the bottom of the wing ($p_b$). This would imply the existence of a the lift force ($F_L$) due to the effect of this pressure difference.However, as seen towards the end of the wing profile (right side), turbulence forms, limiting the applicability of Bernoulli's principle. Specifically, it should be considered that in a certain portion of the wing's perimeter, it may not be applicable, and there will be no contribution to lift.
(ID 11075)
To define circulation, we must first establish the path that will be followed around the object/wing in a counterclockwise direction, as indicated in the following image:
Circulation is defined as the product of the perimeter around the object and the projection of velocity onto the surface. Since this velocity projection can vary along the perimeter, we must sum it through infinitesimal elements of the perimeter, where the velocity projection is calculated using the dot product between it and the perimeter element. Graphically, this is represented as follows:
Mathematically, this is expressed through the closed line integral of the aforementioned dot product:
| $ \Gamma =\displaystyle\oint_C \vec{v} \cdot d\vec{l} $ |
Since the sum is performed counterclockwise, in the upper part, the direction in which the perimeter elements point is opposite to the direction of velocity. In the lower part, both point in the same direction, leading to the upper part partially canceling out the lower part.
(ID 1167)
The relationship between the aerodynamic circulation ($\Gamma$) and the flow around the object is established through the Kutta-Joukowski theorem, enabling the calculation of the lift force ($F_L$) using the wing span ($L$), the density ($\rho$), and the speed with respect to the medium ($v$) as follows:
| $ \displaystyle\frac{ F_L }{ L } = - \rho v \Gamma$ |
By simplifying the modeling of the flow around the object, it becomes feasible to estimate circulation using the surface that generates lift ($S_w$) and the coefficient of lift ($C_L$) with the equation:
| $ \Gamma = \displaystyle\frac{ S_w }{2 L } C_L v ^2$ |
Consequently, the lift force ($F_L$) can be approximated with the equation:
| $ F_L =\displaystyle\frac{1}{2} \rho S_w C_L v ^2$ |
Here, the coefficient of lift ($C_L$) encapsulates the aerodynamic effects of the object.[1] " ber die Aufgabe der Fl geltheorie und ein neues Verfahren zur Herleitung derselben." (On the task of wing theory and a new method for its derivation.), Martin Wilhelm Kutta, Nachrichten von der Gesellschaft der Wissenschaften zu G ttingen, Mathematisch-Physikalische Klasse (1902)[2] " ber die Erhaltung des Luftkreises um ein Profil." (On the conservation of the air circle around a profile.), Nikolai Zhukovsky, Nachrichten von der Gesellschaft der Wissenschaften zu G ttingen, Mathematisch-Physikalische Klasse (1904)
(ID 1168)
The lift coefficient is a function of the angle of attack and typically follows the trend indicated in the following figure:
In the illustrated case, the slope is approximately 1.5 per 15 degrees, which corresponds to 0.1 1/degree or 5.73 1/radian.
(ID 7148)
The difference in pressure between the lower and upper surfaces of the wing generates the lift force, indicated by an arrow perpendicular to the wing's surface. This force opposes the gravitational force acting downward:
Birds or aircraft are able to fly when the lift force exceeds the gravitational force.
(ID 7036)
The difference in pressure between the lower and upper surfaces of the wing generates the lift force, indicated by an arrow perpendicular to the wing's surface. This force opposes the gravitational force acting downward:
Birds or aircraft are able to fly when the lift force exceeds the gravitational force.
(ID 15157)
(ID 15184)
ID:(463, 0)