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Paragliding Aerodynamics

Once you have tasted flight, you will forever walk the earth with your eyes turned skyward, for there you have been, and there you will always long to return. — Leonardo da Vinci

A paraglider are just a couple layers of fabric (nylon) connected internally in such a way to form a row of cells, closed in the rear edge and open in the front so that incoming air keeps the wing inflated and shapes cross-sections as airfoil teardrops.

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Flight can be depicted in terms of weight (gravity), lift, drag and thrust.

Aero forces

When flying, an upward force on the wing, referred to as Lift, has to be generated from the (relative) airflow around the wing balancing downward gravitacional force, and is perpendicular to relative wind; while the air resistance to motion, or Drag, should be balanced by forward thrust, parallel to relative wind.

Lift is produced as the airfoil deflects the airflow downward. For a given angle of attack, the airfoil produces more lift as the air speed increases. The amount of lift created is proportional to the square of the air speed Lift ~ velocity^2·angle-of-attack. For instance, if the air speed doubles and the angle of attack is held constant, the lift increases by a factor of four.

As the angle of attack increases, the incoming air is deflected more, resulting in increased lift. The amount of lift created at a constant air speed is proportional to the angle of attack. For example, if you double the angle of attack while holding the airspeed constant, you double the amount of lift.

In other words, as the velocity increases, the angle of attack decreases. Conversely, as the air speed decreases, the angle of attack must increase if the lift is to remain constant.

As the glider cannot generate thrust, descent hover at a certain forward speed will fit. Then the Lift/Drag ratio, which represents flight performance, may be captured by Lift·sin(descent_angle) = Drag·cos(descent_angle), where the descent angle relates descent direction and horizon. Thus, Lift / Drag = cos(descent_angle) / sin(descent_angle) = 1 / tg(descent_angle) = horizontal_distance_advanced / vertical_height_lost. An average paper plane may achieve a 3:1 ratio, a paraglider should get to at least 10:1.

The drag force can be expressed as Fd = 1/2·density·velocity^2·Cd·area = dynamic_pressure·Cd·area, where density is the fluid's (air), velocity is referred to the object (wing) speed through the fluid, Cd is the drag coefficient (that will depend on the shape of the wing, and angle of attack), and the projected area. Similarly, the lift force Fl = 1/2·density·velocity^2·Cl·area, being Cl the lift coefficient. Thus, Lift / Drag ratio = Cl / Cd.

Lift/Drag ratio and coefficients vs angles-of-attack

Usually, it is assumed that the projected area and air density are constant. In a paraglider, however, the area is not constant, and not just because of wing span or internal air pressure changes affecting aerofoil shape.

One of the secrets of flight is to exchange resistance to lift. To quantify, drag coefficient measures shape exposure to air. A Cd = 1 is given by a 1m^2 plate.

The paraglider moves through a fluid (relative air) at a slight angle of attack, between the chord of the aerofoil (line connecting the leading and trailing edges) and the direction of the airflow, creating 2 different flows (above and under the wing) not simmetric any more.

Airfoil terms

The angle of attack will affect both the lift and drag forces. Zero angle implies minimum drag, but no lift as well. As seen, lift increases along with the angle of attack, until a critical angle is reached (around 20º), when drag increase exceeds lift decrease and the glider stalls, and eventually acts like a parachute in free fall. When the airfoil drops off, airflow is no longer deflected, so the airfoil does not produce nearly as much lift. The drag also increases dramatically. Dangerous.

Lift/Drag ratio

An airfoil is more likely to stall at low air speeds, because in order to create the required lift, the airfoil must operate at a high angle of attack. The critical angle of attack is determined primarily by the shape of the airfoil. However, it can also be affected by the surface condition of the airfoil.

That wings should be curved on top (air forced to move a little quicker, as it has a longer path to the trailing edge) and flatter on the bottom is related with the misconception that air spends the same time to travel above and below the wing.

Airflow vs angles-of-attack

However, even the air flowing above the wing has a longer path, it gets to the back edge earlier than the one below. And as soon as it reaches the trailing edge, then quickly returns to its original speed, so this change in relative position will be permanent, and the air that flowed under the wing will never catch up the one on top.

According to Bernoulli's principle, which states that faster (air)flow (over the top) means lower pressure or energy, and slower (air)flow (underneath) implies higher fluid's pressure, the result of that pressure difference is then an upward force, lift.

Airflow and pressure

Notice that the air ahead of the wing, at a certain angle of attack, is redirected upwards (upwash). Also, the trailing air will move downwards (downwash). Depression above sucks air and produces 2/3 of lift force, lower wing surface pushing air is responsible for the remaining 1/3, with a resulting lift/drag quotient over 10.

But Bernoulli’s principle apply to inviscid (no viscosity, no friction, thus no resistance) incompressible fluids along a streamline, where there is a steady flow.

Viscosity is a physical property that quantifies how much adjacent molecules "stick" together. Air has viscosity. Air flows around objects, and the air in touch with the aerofoil is actually carried along, thus it (air) has zero speed with respect to the surface (shear stress).

Boundary layer

At the surface of the object (wing) air is stagnant due to friction. Velocity increases from the surface to a maximum in the upstream of the airflow. This region of velocity profile in the airflow, due to the shear stress, is the boundary layer. Commonly, the thickness of this boundary layer extends up to the point where speed reaches 99% of the free-flow stream; the remainder of the airflow is known as the external inviscid flow.

Adverse pressure gradient

The above figure shows the formation of the separation process. The fluid (air) accelerates from the leading edge to turn round the sphere (or aerofoil), as it has further to go than the surrounding fluid (when there is a pressure decrease in the direction of flow, the fluid will accelerate and the boundary layer will become thinner, equally as a narrowing pipe). It reaches a maximum velocity, still with positive pressure gradient (pressure drops with distance, dp/dx>0) and lift force occurs.

As air travels over the surface and gets to the other side of the sphere, fluid will immediately start to slow down (when pressure goes up, so does the potential energy of the fluid, leading to a reduced kinetic energy and a deceleration of the fluid, in a similar way as a widening pipe), hence the thickness of the slower layer increases (smaller velocity gradient) and becomes less stable, and eventually turbulent. When the pressure rises in the direction of flow (adverse pressure gradient, downstream side of the sphere/aerofoil), fluid outside the boundary layer has though enough momentum to overcome this pressure that is trying to push it backwards, but the fluid within the boundary layer has so little momentum due to friction that it will be quickly brought to rest, and likely reversed in direction. When this reversal occurs, the fluid flow becomes detached from the surface of the object at some point before the trailing edge, and instead takes the forms of eddies, whirls and vortices (flow separation and vortex shed), and forces the flowfield into an unstable wake, with the flow being redirected downwards according to the angle of attack. This happens because fluid to either side is moving in the opposite direction.

Vortices result in very large energy losses in the flow. This has quite important consequences in aerodynamics, as flow separation significantly modifies the pressure distribution along the surface and hence the lift and drag characteristics. Such flow separation causes a large increase in the pressure drag (pressure differential between the front and rear), since it greatly increases the effective size of the wing section, which also result in loss of lift and stall.

Flow separation

Whether or not the boundary layer separates against an increase in pressure (adverse gradient) depends upon the character of the boundary layer: laminar (viscous forces higher, lower drag) or turbulent (higher inertial forces, higher drag).

To calculate whether a particular flow is laminar or turbulent use the Reynolds number. The Reynolds number (Re) is a dimensionless equation measuring the ratio of inertial forces (rho·v^2/l) to viscous forces (mu·v/l^2), that allows for the cross referencing of aerodynamic characteristics (including drag coefficient) between objects of different sizes that may move at different rates of speed. In sum, two different types of objects going at different speeds that experience the same Re will have the same drag coefficient.

Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion; while turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities:

Re = (rho·v^2/l) / mu·v/l^2 = rho·v·l/mu

where v is the mean fluid velocity; l the traveled length of fluid; rho the density of the fluid [Kg/m3]; and mu the dynamic viscosity of the fluid [N·s/m2=Kg/m/s].

Turbulent boundary layers tend to be able to sustain an adverse pressure gradient better (delaying flow separation and keeping it attached for as long as possible) than an equivalent laminar boundary layer. The more efficient mixing which occurs in a turbulent boundary layer, transports kinetic energy from the edge of the boundary layer to the low momentum flow at the solid surface, often preventing the separation which would occur for a laminar boundary layer under the same conditions. Thus, although the skin friction (parasitic drag) is increased, overall drag is greatly reduced, and performance likewise gained, while the usable angle of attack can be larger, thereby dramatically improving lift at slow speeds. This is the principle behind the dimpling on golf balls, or the fuzz on a tennis ball.

Smooth surface Rough surface

Air is nonlinear, with no simple math solutions that explain turbulence or flow-attachment, obtained by computation of the basic model of fluid dynamics: the Navier-Stokes/Euler equations, with force boundary conditions. The Reynolds number can be obtained when one uses the nondimensional form of the incompressible Navier-Stokes equations:

rho·(dv/dt + v·Delta(v)) = −Delta(p) + mu·Delta^2(v) + f

where Delta is the gradient (rate of change with position); p is static pressure; and f other body forces.

By elementary Newtonian mechanics, upward lift must be accompanied by downwash with the wing redirecting air downwards. The main issue of flight is how a wing may generate substantial downwash; with downwash there is lift. Real wings fly because of vortex-shedding, and are lifted upwards as they fling air mass downwards.

Air flows all over the wing, not just above or under, but also transversally to both side ends and backwards, due to wing displacement, in the intrados (pressure surface below), and over the extrados (above) flow is inverted from sides towards the centre of the wing, affecting negatively its performance (increase of induced drag). Flow tend to neutralize pressure difference between intrados and extrados, and ends up forming the usual vortex on the ends of the wing that increase its wake.

A way to reduce the induced drag is placing vertical appendices (stabilos) in both ends, whose additional advantage is an increase upon the stabilizer effect on the rotating axis. Likewise, the larger the wingspan (keeping the same surface) and the farer the ends, the lesser influence they will have upon aerodynamic performance.

On the other hand, profile characteristics (torsion) are modified along the wingspan, through aerodynamic torsion while changing profile shapes keeping the angle of incidence; and geometric torsion varying the incidence of those profiles positively, avoiding that way aerofoils run into very low or negative incidence when central angle of attack changes.

Polar curve falcon vs paraglider

The wing is depicted by a range in velocities and configurations: maximum and minimum velocities, best glide ratio, minimum sink rate, etc, that are graphically plotted in a speed (both horizontal/straight and vertical/level) polar curve. Each wing has a different one.

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