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= Boundary layer and stall =Note that we have not yet considered additional complications such as '''sweep'''. In addition, delta wings work in a completely different manner from conventional wings.
→In two dimensions (aerofoils): Use MathJax for formulas.
For simplicity, this article only considers flight at gliding speed (Mach ~0.1), so any compressible flow effect is ignored.
== Definitions ==
A cross section of a wing has the geometry of an '''aerofoil'''.
In the aeroplane frame of reference, the angle between the chord line and the incident flow is the '''angle of attack'''.
== Fundamental physical concepts == === Force and moment === '''Force''' is the action on a body by a body. If you jump, you fall back onto the ground, and it is the gravitational force that pulls you down. Gravity is the action by the Earth on you. A force combined with a leverage gives a '''moment'''. When you use a wrench to tighten a nut, you exert a force, and, combined with the leverage of the wrench, creates a "twisting force" on the nut which tightens it. The moment is proportional to the leverage, so if you cannot tighten a nut sufficiently, the solution is to use a longer wrench. === Mechanical work and mechanical energy === '''Work''' is a transfer of energy. When you lift a stone, your lifting force does work on it, in which process energy goes from you to the stone (and as a result, you will feel tired). If, as a result of doing work, energy goes into A, we say '''positive work''' is done on A, conversely, if energy leaves B, we say '''negative work''' is done on B. In this case, your lifting force do positive work on the stone, and the weight of the stone does negative work on you. Energy can take many forms, such as '''kinetic energy''' (energy associated with moving with a speed) and '''potential energy'''. In this aerodynamic discussion we consider '''gravitational potential''' and '''pressure potential'''. Gravitational potential is the energy associated with being in a (physically) high place. When you lift the stone, the energy that you give the stone becomes its gravitational potential because it has moved to a higher place. Pressure potential is the energy associated with being in a high pressure: you will feel tired after blowing up a hundred balloons, which is because the air inside a balloon has pressure potential, which ultimately comes from you, if you do not blow, the balloon does not inflate itself. Energy is '''conserved''', which means it cannot be created nor destroyed. It can only go from one body to another body, or go from one form into another form. If you drop the stone, its gravitational potential will become kinetic energy, so it will pick up speed. When the stone hits the ground and comes to a stop, we say its mechanical energy has been '''dissipated''', but it has really gone into heat: we just do not notice. === Viscosity === A fluid has '''viscosity''' if it is "sticky". When you spill ketchup or syrup, it does not come off your clothes voluntarily: it sticks onto it. A fluid which has viscosity is said to be '''viscid'''. If a fluid has zero viscosity, it is said to be '''inviscid'''. Air is viscid, although its viscosity is so small that we seldom notice. === Miscellaneous === Being '''normal''' is being orthogonal (at an angle of 90 degrees) to the tangent line of the curve drawn at the location of interest. A '''coefficient''' is a non-dimensional quantity. It is obtained by multiplying (or dividing) something on the quantity of interest to eliminate the dimensions. If you are 1.8m tall, your height has a unit of metres and a dimension of length. If your height is divided by a quantity (which can be chosen rather randomly), say the wingspan of a K-21 (18 metres), we can say you have a "height coefficient" of 0.1, which has no unit and no dimension. So long as this random quantity is kept constant, the coefficient is an equivalent representation of the actual value. A '''field''' is a spacial distribution of some quantity of interest. If a field is known, effectively the quantity is known at every location within that field (we say the field '''evaluates to''' something at that point). For example, there is a temperature field in your house, which might evaluate to 25 degrees on the armchair in the living room, 20 degrees on the bedroom floor, but only 15 degrees somewhere on the balcony. A '''gradient''' is how quickly something (a field variable, i.e. a quantity that can be represented by a field) changes with respect to distance. The most notable example being the slope of the ground. On a hill, the height varies with position, so we can define a height gradient. If the height gradient is large, this means the height changes quickly with respect to distance, so we say (the slope of) the hill is steep. The '''aspect ratio''' of something is (roughly) the ratio between the length and the width. It describes how slender or stubby something is. A square has the aspect ratio of 1 because its width is the same as its length. A (new) pencil has a larger aspect ratio than a piece of rubber. == What a wing actually does ==
The fundamental purpose of having wings is to produce '''lift''', which balances the gravitational force on the aeroplane so that it can stay airborne.
Wherever forces are involved, a resultant moment can be defined. By definition, an aerofoil produces a constant moment irrespective of angle of attack around its '''aerodynamic centre'''.
== How lift is created ==
Consider an object moving along a curved path. Elementary Newtonian physics dictates that force (at least a component of it) must be exerted on the object in the direction normal to its path pointing into the concave side of the path, i.e. the '''centripetal force'''.
Note that the energy approach is generally not applicable to the explanation of lift: no work is being done nor is any energy being transferred between the aeroplane and the air in the process.
== Creation of lift: fake physics ==
=== An incorrect explanation ===
People with some qualitative aerodynamic knowledge often argues that it is the “Kutta condition” that the air meets at the trailing edge in the same time. However, the Kutta condition, despite a lack of precise mathematical formulation, requires nothing more than the trailing edge being a stagnation point (in 2D). In other words, it requires that the streamlines meet, just like two carriage ways merge into one, but the vehicles on the carriage ways can travel at very different speeds before reaching the junction.
== What does the lift force depend on ==
=== In two dimensions (aerofoils) ===
In 2D, the lift force produced by an aerofoil depends only on two factors: the angle of attack, and the geometry of the aerofoil. To be specific, the only factor about an aerofoil that matters is the '''camber''' of the aerofoil, i.e. how bent it is (the mathematical definition will not be introduced in this elementary article). The thickness of the aerofoil has zero (in theory, and very little in practice) effect on lift, which is not easy to understand without working through the continuum mechanics. However, thickness is a useful design tool to modify the pressure distribution around the aerofoil, thereby improving the stalling characteristics.
Any symmetric aerofoil has a lift coefficient (lift force per unit chord and unit depth) of 2πα\(2\pi\sin(\alpha)\), usually simplified into \(2\pi\alpha\), where α \(\alpha\) is the angle of attack(measured in radians). A cambered aerofoil has an additional lift coefficient at zero angle of attack added to this value. Therefore, the lift-AOA charts all look very much similar for different aerofoil shapes: the gradient is invariant.
=== In three dimensions (wings) ===
This is the first reason why high aspect ratio (slender) wings are aerodynamically desirable.
== Boundary layer and stall == === Introduction ===
Glider pilots usually are introduced the phenomenon of '''stall''' within the first several flights. This section explains the fundamental physics of stall.
Review how lift is created: it is necessary that the streamlines follow the shape of the aerofoil and bends accordingly. If the streamlines cease to follow the aerofoil, i.e. the flow detaches from the aerofoil, lift will be reduced very significantly. This is fundamentally what a stall is. It is observed that wings stall once a critical angle of attack is reached. To understand the physics of stall, the concept of '''boundary layers''' must first be introduced.
=== Boundary layer physics ===
Consider pouring honey out of a jar: can you empty the jar completely? This is not possible because the honey sticks on the inside wall of the jar and it will not come off completely no matter how long you allow the jar to drain. This is the '''no-slip condition''' of viscous flow: honey is a viscous fluid, and whenever it contacts a wall, it will not slip on it but stick onto it.
Things are, unfortunately, further complicated by the fact that, while the wing can remove energy from the boundary layer, the outer flow can help the boundary layer by “dragging it along”. The interacting factors become such a mess that a mathematical description of the precise point of separation is not yet possible. However, it is generally observed that, there are two methods to make a boundary layer separate, namely a very steep adverse pressure gradient, or a less steep adverse pressure gradient over a prolonged distance.
=== Stalling in 2D ===
Here it is necessary to quote without proof that the pressure gradients over the upper surface of an aerofoil is generally proportional to the angle of attack.
A '''trailing edge stall''' happens when the boundary layer separation point near the trailing edge moves forward because the adverse pressure gradient is increased due to an increased angle of attack. This type of stall is more gentle with plenty of warning signs and a gradual loss of lift. This is the stall behaviour observed on training gliders, e.g. K21s.
=== 3D stalling behaviour: planform and washout ===
==== Introduction ====
The stalling of wings in 3D is slightly more complicated than the 2D case. As discussed before, we would expect the spanwise variation of the flow field to have an effect.
In addition, the stall of an aeroplane is almost never symmetric, i.e. the two wings almost always stall differently, this can be a difference of how deep the stall is, or one stalls but the other one does not. The reader probably has been told by an instructor that this is the fundamental cause of a '''wing drop''' and a '''spin'''.
==== Planform effects ====
Tip stalling is a characteristic of strongly tapered wings. A tip stall is unpleasant and difficult to handle. This is because the imbalanced force from the stalling sites (which are at the wing tips) can create a large moment around the roll axis of the aeroplane, because the leverage is large. This means the aeroplane can enter a wing drop very quickly and a spin will readily develop. In addition, because the ailerons are typically located at the wing tip rather than the root, a tip stall means the ailerons will be completely ineffective.
A special case is an elliptical wing, which will stall simultaneously. This is part of the reason why a Spitfire is difficult to handle.
==== Washout ====
An aeroplane designer will usually aim to give the aeroplane the root stall characteristic. When a rectangular wing is not suitable for use, there is another method to move the stall site inboard, which is to add a twist along the spanwise direction, referred to as “'''washout'''”. This makes the tip installation incidence less than the root, so when the aeroplane increases its angle of attack, the critical value is firstly reached at the root instead of the tip. A K-21 has this feature.
Washout can also be used to modify the lift distribution and reduce the induced drag: this is not the primary consideration of this article.
== Drag ==
=== Introduction ===
A classic problem in hydrodynamics is called the “d'Alembert’s Paradox”. It describes the very bizarre result that, despite every care being taken and a wide range of cases and methods considered, theoretically anything moving in a fluid will have precisely, absolutely, identically zero drag. Any child who has waved a tennis bat will be able to tell that this is complete rubbish, but for hundreds of years fluid dynamists could not tell what had gone wrong.
An aeroplane experiences three kinds of drag, namely '''skin-friction drag''', '''form drag''', and '''induced drag'''.
=== Laminar and turbulent boundary layers ===
Unfortunately, before the discussion of drag creation, it is necessary to introduce the concept of laminar and turbulent boundary layers. In short, a flow can be either '''laminar''' or '''turbulent''', and going from laminar to turbulent is called the “'''transition'''”. This is most easily visualised by observing a burning cigarette in still air: the smoke initially goes straight up nicely (laminar), but after some distance the column suddenly goes unstable (transition) and the smoke wraps up into some unpredictable shape (turbulent flow).
A laminar boundary layer can naturally transit into a turbulent one due to inherent instability, typically when the flow speed is sufficiently high. If natural transition does not occur, it is possible to use a “tripping device” to force premature transition at lower speeds.
=== Drag forces explained ===
==== Form drag ====
Form drag, also known as '''pressure drag''', is the drag caused by the pressure in front of the body being higher than the rear pressure. This is most obvious for a “bluff body”, i.e. a body that does not have a streamlined shape. It is difficult to cycle or walk into strong head wind, because a human body is a bluff body with a lot of form drag. The pressure in front of you is greater than the pressure on your back, and this pressure difference tries to push you backwards. Going forward and you have to continuously fight against this form drag.
A wake is created by boundary layer separation. Recall that a turbulent boundary layer is more reluctant to separate compared with a laminar one, so somehow making the boundary layer turbulent can reduce the form drag considerably. This is the reason why golf balls have dimples: the dimples can trip the boundary layer so that it is turbulent.
==== Skin-friction drag ====
Pulling a teaspoon out of a cup of tea can be effortless, but pulling it out of a jar of honey can be more difficult. This is because the teaspoon moving in honey experiences a significant skin-friction drag. Skin-friction drag is created by the viscous fluid sticking onto the solid object, and it largely depends on the '''surface area''' submerged (or in case of an aeroplane, indeed almost all the surface area). Unfortunately, streamlining a body to reduce form drag usually means creating a large surface area, so there is a subtle balance between form drag and skin-friction drag for the aeroplane designer to get right.
Most of the skin-friction drag on a glider comes from the wings. Recall that a laminar boundary layer causes less skin-friction than a turbulent boundary layer, so a lot of gliders use what is called “'''laminar flow aerofoils'''” which are aerofoils designed to keep the boundary layer laminar for as long as possible. The problem being laminar boundary layers separate easily, so the post hoc fix is to add a turbulent trip on the surface of the wing, just before the point where it would otherwise separate. Such a device can be found on the lower surface of the wing of DM (a CGC single seater).
==== Induced drag ====
Induced drag only exists in 3D: there is no 2D equivalent. When a lifting wing flies through the air, it is well known that a pair of '''tip vortices''' will be created behind the aeroplane. This is generally explained by considering the air escaping from the under side to the upper side around the tip: a more precise explanation is very involved and shall not be discussed here. However, by creating these vortices, extra kinetic energy is transferred to the air, which now has an additional swirl motion. This energy must come from the wing, and the result to the aeroplane is known as induced drag.
The magnitude of the induced drag coefficient is proportional to the lift coefficient to the second power, and '''inversely proportional to the aspect ratio of the wing'''. When the aspect ratio tends to infinity, the induced drag is zero. This is the second reason that, for aeroplanes whose drag is critical, high aspect ratio wings are desirable. Almost all gliders have large aspect ratio wings: the most extreme example being the ETA, which has a record-breaking glide ratio of 70:1.
==== Lift-to-drag ratio ====
It can be shown that, for a glider, the aeroplane lift-to-drag ratio is identical to the glide ratio. It just needs to be noted that the aeroplane lift-to-drag ratio is not the same as the wing L/D, because the fuselage contributes only to drag but not lift.
== Downwash (advanced topic) ==
'''Warning''': readers whose working knowledge on fluid mechanics is limited are advised to Google “tip vortices” and jump to the Implications section.
=== Physical introduction ===
Circulation results in lift. For a finite span wing, at each section there must be a defined circulation value which can be determined by a closed line integral around the aerofoil. By the Stokes’s theorem (vector calculus), this implies that there is a '''distribution of vorticity''' within the integration loop. However, the loop encloses potential flow (which, by definition, can have no vorticity), and within the aerofoil contour the flow speed is zero so there cannot be vorticity either (the curl of a constant zero is zero). Therefore, this vorticity is distributed on the surface of the aerofoil. In the real flow, this corresponds to the vorticity in the boundary layer.
We have already argued that at the tip of the wing, the local wing loading is zero. If for an aerofoil section there is zero loading, this implies that the circulation for this section is zero. By the same argument, we find that the vorticity is zero for a section plane taken at the tip of the wing.
According to '''Helmholtz’s second theorem''', a vortex filament cannot end in a fluid. If there is a vortex line going spanwise when a section plane in the middle is examined, but it is not found going outwards on the section plane at the tip, the only possibility is that it has been deflected to some other direction. But in which direction? Again, according to Helmholtz’s theorems, vortex lines move with the fluid. Because the aeroplane flies forward, the flow around it is effectively going backwards. Therefore, the vortex line has been deflected into going backwards by the flow.
In a sense, we have shown that, for a wing of finite span, a '''sheet of vorticity''' is trailed behind it because vorticity is being shed off from the root to the tip of the wing. But this does not remain as a sheet forever. For the third time according to Helmholtz’s theorems, these vortex lines move with the fluid, but do not forget that the vortices can introduce swirling flow themselves. The result is that this vortex sheet rolls up into a single vortex with all the strength there is. For an aeroplane, because of the obvious symmetry, a pair of vortices are formed. These are known as tip vortices. Despite the name, it is good to understand that '''they are not in general produced by flow escaping around the tip''', but it is a result of a finite span wing generating lift. A more detailed examination of the rolling up process reveals that the distance between these tip vortices is less than the geometrical span of the wing, i.e. ''the tip vortices are closer to the fuselage than the tips are''.
=== Implications ===
Tip vortices can be '''very strong''': they are as strong as vorticity required to keep the aeroplane in the air. In a sense they are a pair of tornados trailed behind an aeroplane which contains an enormous amount of kinetic energy (hence the induced drag on the aeroplane). In the real world, air is viscous, and these tornados are eventually dissipated, but this dissipation needs time, and in the world of aviation, time means distance. Therefore, it is generally not a good idea to stay too close behind other aeroplanes, especially the heavier ones, whose tip vortices are strong and generally highly turbulent (with the additional interaction with the wake). Airliners that generate tip vortices of significant strengths have the suffix “heavy” or (for an A380) “super heavy” on their callsigns to warn air traffic controllers to keep other traffic well separated, and you do not want to be anywhere close with your glider.
Downwash is a consequence of having these trailing vortices (and this is used in the '''force approach''' to explain induced drag, contrasted to the energy approach we have been using). Viewing from behind, the left-hand vortex rotates clockwise and the right one counter-clockwise. Therefore, between these vortices there is a region where the flow is effectively going down (apart from the going back component which we take for granted). This is known as the downwash. Similarly, on either side of these vortices there is upwash. It is also important to remember that, because the flying speed we consider is subsonic, vorticity downstream can have an effect upstream, so downwash and upwash exists even in front of an aeroplane, although the strength is questionable.
The understanding of these vortices is important to the competency in aerotowing. You will have a chance to appreciate the strength of these vortices and the degree of turbulence caused when the exercise “boxing the wake” is done. Ask an instructor for more details.
[[Category:Theory]]
