Aerodynamics

Mathematical model

Aerodynamics is a science which belongs to the mechanics of the fluids, applied in the particular case of the air. For this reason, the mathematical models which apply are:

the Navier-Stokes equations when the viscous effects are not negligible. The principal parameter quantifying these effects is the Reynolds number.
the equations of Euler or true fluid, when the viscous effects are negligible.
equations of Stokes when the viscous effects are dominating.
the equation of state of the gas (ideal gas for the air).

Aerodynamic efforts

Forces and Coefficients

The field of pressure being exerted on an obstacle induces overall a Torseur of efforts where one generally considers:

a force of Trailed: F_x, parallel with the average direction of the flow
a force of drift: F_y, perpendicular to the average direction of the flow, in the horizontal plane
a force of Bearing pressure: F_z, perpendicular to the average direction of the flow, in the vertical plan

The expression of the force is general form: F = Q. S.C

with:

dynamic Pressure 1/2. rho. V ²

Surface of reference

aerodynamic Coefficient

The aerodynamic coefficients are adimensional coefficients being used to quantify the forces in X, there, Z:

C_x: the Coefficient of drag
C_y: the coefficient of drift
C_z: the Coefficient of bearing pressure

The forces being measured in experiments (out of blower), the coefficients are given by posing C = F/(Q. S)

$C_ {X, there, Z} = \ frac {F_ {X, there, Z}} {\ frac {1} {2} \ times \ rho_ {air} \ times V^2 \ times S}$

where S is a surface of reference of the object concerned and V the relative wind on the object.

Definition of the Surface of reference:

for an airfoil generally quite shaped, S is the surface projected in the horizontal plane (or the vertical plan for a vertical stabilizer or a Dérive),
for an object with strong form drag (Traînée pressure) like a car, of which the Cx are 5 to 8 times that of a fuselage of plane, one takes the Master-couple rather (frontal surface).

The trail

The coefficient of trailed is the report/ratio of the trail of the object studied with that of an of the same body surfaces which would have one Cx of 1.

In Aviation, the coefficient of resistance is indicated by the coefficient of drag, reported in the case of the wing to its projected surface.

One can qualify the total trail by a total coefficient brought back to the surface of the wing or the total wet surface of the plane.

In automobile aerodynamics, to know the Cx is not sufficient, it is necessary to also know the frontal surface of the vehicle.

In an assessment of compared trails, one uses the product S. Cx. A " is obtained; surface traînée" equivalent which would have one Cx = 1.

the force of trail is: F = Q. S. Cx = 1/2. ρ_air. V ². S. Cx

ρ_air: density of the air (1,225 kg/m³ with 15°C with the sea level)

V: rate of travel (in m/s)

S: frontal surface of the vehicle (main couple)

C_x: drag coefficient

the fundamental equation F = my makes it possible to calculate this force of trail:

the mass of air concerned is (except for a characteristic coefficient):

m = \ rho_ {air} \ times S \ times V \ times T

acceleration is (except for another characteristic coefficient):

a = \ frac {1} {2} \ times \ frac {V} {T}

The force of trail is:

F = \ frac {1} {2} \ times \ rho_ {air} \ times S \ times C_x \ times V^2

Bearing pressure

The equation of the bearing pressure is similar to that of resistance with C_x replaced by C_z or Cy for a side bearing pressure.

In the Anglo-Saxon literature the C_x coefficient is indicated by C_d (drag) and C_z by C_l (top spin) the bearing pressure.

In the German literature, C_x and C_z are indicated respectively by C_w (Widerstand) and C_a (Achsauftrieb).

the terms C_x and C_z are without dimension (they do not have a unit).

Trailed bearing pressure/

The " finesse" of a wing is defined by his ratio bearing pressure/is trailed C_z/C_x.

Assessment of the trails and Power of flight

We will consider here only aerodynamics in subsonic mode (not of compressibility). The knowledge of the forces acting and resulting on a profile from wing makes it possible to deduce from it the behavior in the various phases from the flight.

The total trail

In aerodynamics, it is of use to break up the total trail of a Avion into three large catégories :

1. the induced trail (by the bearing pressure)
2. the trail parasitizes that one breaks up itself into:

. trail of friction

. form drag or trail of pressure

. trail of interference

3. the trail of compressibility, or trailed wave.

This multiplicity of denomination is a practical cutting aiming at proposing the contribution to the trail of such or such aerodynamic phenomenon. For example, the induced trail returns to the notion of the effort induced by the bearing pressure of the wing. The trail of wave returns to the idea of dissipation on the level of the shock wave. (See also Trailed)

Consequently, It is advisable to keep in memory that in physical terms, only two mechanisms contribute to the trail: the assessment of Pressure and parietal friction (tangential). Thus, if one considers an elementary element of surface of the plane $dS$ at the point $M$ provided with a normal $\ tilde {N}$ and with a tangent $\ tilde {T}$ , the elementary effort on this surface is written:

$\ tilde {F} = (p (M) \ tilde {N} + T_ {W} \ tilde {T}) dS$

It is seen that if one knows in any point of the surface of the plane the pressure $p (M)$ and friction $T_ {W} (M)$ , one is able to express the whole of the aerodynamic efforts being excerçant on this one. With this intention, it is enough to integrate $\ tilde {F}$ on all the surface of the plane. In particular, the trail is obtained by projecting $\ tilde {F}$ on an unit vector $\ tilde {U}$ opposed at the speed of the plane. One obtains then:

$F= \ int_ {S} \ tilde {F}. \ tilde {U} = \ int_ {S} p (M) \ tilde {N}. \ tilde {U} dS + \ int_ {S} T_ {W} \ tilde {T}. \ tilde {U} dS$

In this expression of the trail, the first term gives the contribution of the pressure. It is in this term that intervenes, via a deterioration of the field of pressure, the induced trail and the trail of wave. The second term gathers the trail of friction, due to the phenomenon of Boundary layer

Induced trail

The complete expression is induced Traînée by the Portance . It is proportional to the square of the coefficient of bearing pressure (Cz in French, English Cl), and inversely proportional to the lengthening of the wing. The shape in plan of the wing also plays: the minimal induced drag is obtained in theory by an elliptic distribution of bearing pressure in scale.

Calculation of induced resistance IH

IH = 1/2 rho. V ². S. Ci

with S surfaces reference and Ci coefficient of drag induced

Ci = Cz ²/(pi. lambda. E)

lambda = effective lengthening of the wing (corrected geometrical lengthening)

Oswald Factor, lower than 1 (variable value, approximately 0.75 to 0.85), to take account of a distribution of bearing pressure in nonoptimal scale.

The induced trail is maximum in raised Cz, therefore at low speed and/or high Altitude (until more 50 % of the total trail). The mechanism of the induced trail was theorized by Ludwig Prandtl (1918) in the following way : To have a bearing pressure, one needs an overpressure for the under-surface of the wing and/or a depression with the suction face of the wing. Under the effect of this difference in pressure, the air passes directly from the under-surface to the suction face by circumventing the end of the wing. It results from it that, under the under-surface, the flow of general air is deviated of a few degrees towards the end of the wing, and that on the suction face the flow of air is deviated towards the center of the wing. When respective flows of the under-surface and the suction face end up meeting at the edge of escape of the wing, their directions diverge, which causes at the same time the induced trail and of the swirls behind of the trailing edge.

The power of these swirls is maximum at the end of the wing (marginal swirls). The invisible energy contained in these masses of air in rotation constitutes a danger to the air Navigation. It imposes a distance from minimal separation between planes, especially for light planes following of the airliners.

The induced trail is an important component of the total trail, in particular at the low speeds (forts coefficients of bearing pressure, and of the same for the veils of boats). To reduce the induced trail supposes to decrease Cz of flight (to decrease the wing load), to increase effective lengthening and to distribute the bearing pressure in a decreasing way in scale (elliptic distribution).

Concretely, it is to decrease the induced trail that:

the sailplanes have wings with great lengthening,
the fast planes have wings whose form in plan gives a distribution of bearing pressure close to the ellipse:

either a trapezoid of tapering close to 0.5,

or an ellipse like the wing of the Spitfire. It seems nevertheless that the plan in ellipse does not bring a really significant advantage; it was not taken again since.

the airliners which fly at high Mach (0.85) present a higher tapering, about 0.3, because of the angle of wing sweepback (approximately 25-30°) which causes to overload the wing tips.
tips of wings of the Airbus, and certain Boeing recent, carry vertical wings or Winglet S which increases effective lengthening by recovering part of the energy of the marginal swirl.

Trail of friction

In the flow of a Fluide on a plan one notes with the immediate vicinity of the plan a deceleration of the fluid. The thickness where the fluid is slowed down calls the boundary layer and varies few tenth of mm in laminar flow to more or less 10 mm in turbulent flow. In the boundary layer the Molécule S of air are slowed down, which is translated into a loss of energy which must be compensated by the energy provided by the propulsion of the plane.

Reynolds number (to be developed) Re = V. L/naked

with

V: speed in m/s

L: length of the body or airfoil chord in m,

naked: kinematic viscosity of the fluid (variable with the temperature, approximately 1.15 10th-6 with 15°C).

Form drag

The aerodynamic resistance of an object depends on its form. If one compares a plan perpendicular to the flow with a Sphère and the shape in water drop, one notes that the sphere presents 50% of the resistance of the plan, and the water drop hardly 5% of the resistance of the plan. The form drag is minimal when the flow is not taken down. The sharp variations of section of the body bring separations, turbulence and thus of the trail. In order to reduce these Turbulence S, it is necessary " profiler" the body.

Trail of profile

The coefficient of drag of a profile, valid for a Incidence, a Lengthening and a Reynolds number]] given, is the sum of the trail of friction and the form drag (separations). A quite shaped body has a drag component of form definitely weaker than its trail of friction. The planes the best shaped ones (sailplanes) have a total coefficient of drag brought back to their wet surface hardly higher than the coefficient of friction of a plate planes of the same surface.

Trail of interference

The trail of interference appears for example with the intersections of the airfoils and the fuselage. The distribution of bearing pressure in scale is locally disturbed and present peaks (at the Emplanture) and lacks (on the level of the fuselage).

Trail of compressibility

Trail generated by specific phenomena met when the flows impose a variation of density on the fluid, such as for example the shock waves in aerodynamics Transsonique and supersonic.

Total power of flight

The power of flight is the product of the sum of the trails by speed:

Rtot. V

with Rtot in newton and P in Watt

The resistant power (the energy spent per unit of time) is in Watt S:

P = F \ times V = \ frac {1} {2} \ times \ rho_ {air} \ times S \ times C_x \ times V^3

(On the other hand, the power spent for the maintenance in the air is null : without displacement there is no work. It is thus C_x alone which intervenes in the formula of power).

Minimum capacity of flight

The trail of friction varies (and increases) with few things close (influence of the Reynolds) with the square speed. On the other hand the induced trail decreases with speed and tends towards zero at very high speed. There exists a speed, higher than the stalling speed but lower at the speed of smoothness max where the power of flight is minimal.

Terms of the aerodynamics of the wing

; Lengthening: Lengthening, on a Aerodyne with nonrevolving aerofoil, is the relationship between the scale and the Depth or " average Cord " ; it is also the report/ratio of the square of the scale on the surface. It is one of the factors which contribute to the increase in the smoothness. The larger lengthening is, the more the smoothness of the wing is large (more the angle of planed is weak). The Pente of bearing pressure depends on lengthening. ; Angle of chock: Angle formed by the cord of the wing and the reference axis of the fuselage. ; Angle of incidence: Angle formed by the cord of profile of the wing and the Flight Path Vector, also called angle of attack. ; Angle of planed: Angle ranging between the downward trajectory and the horizontal one. ; Leading edge: In the direction of the flow, left before profile. It is generally of round form, more important ray on the subsonic machines and more end on the supersonic machines. ; Trailing edge: In the direction of the flow, back and thinned part profile. ; Twist profile: Right-hand side connecting the leading edge (left round in front of the wing) at the edge of escape (pleasure party with the back the wing) (see also Profile (aeronautical)). ; Boundary layer: Lay down air in contact with the surface of the wing. The particles with the immediate vicinity of the wing are equipped a clean speed lower than those located in the more external layer. Recent studies show that within this framework the aerodynamic trail of a surface very finely striated can be lower than that of a smooth surface. ; Unhooking of the profile: When, at constant speed of the fluid one increases the value of the angle of incidence, the bearing pressure generated by the profile increases, passes by a maximum (between 15 and 18 degrees, approx.) and decreases more or less brutally. The characteristics of this phenomenon depend on the profile considered, as well as conditions (Reynolds numbers, of Mach, state of the layer-limit) of the flow. ; Unhooking of the wing: unhooking starts locally at the place aerodynamically charged, and extends more or less abruptly on all surface from the wing. The assymetry of the unhooking (which can bring a loss of control in rolling) is more dangerous than unhooking him even. ; Dihedron: to see Dihedral (plane) ; Root: Part of the wing in contact with the fuselage. ; Scale: Outdistance between the two ends of wing. ; Relative thickness: Report/ratio the thickness (maximum distance between under-surface and suction face) to the airfoil chord. ; Suction face: Upper surface of the wing. ; Smoothness: Relationship between the coefficient of bearing pressure and the coefficient of drag. It is also the speed ratio of the machine on the falling speed: for a flying equipment to 180 km/h (that is to say 50 m/s) and a falling speed of 5 m/s the value of the report/ratio is of 10. It is also the relationship between the distance covered and the loss of altitude: when the plane traverses 10 m, it goes down from 1 Mr. the maximum smoothness is independent of the weight but the speed of smoothness maximum increases with the weight for the same plane. The smoothness depends on the coefficient of bearing pressure and thus on the incidence of the wing. ; Under-surface: Lower surface of the wing. ; High-lift devices: The high-lift devices are mobile surfaces whose function is to modify the curve of profile of the wing in order to increase the bearing pressure by it. They generally consist of nozzles of leading edge and camber flaps laid out at the edge of escape. The nozzle of leading edge prolongs towards before and to the bottom the curve of the profile to increase the maximum incidence and thus the maximum bearing pressure of the profile. The camber flaps are directed downwards to increase the bearing pressure, but that increases also the aerodynamic trail (this effect of braking is required with the landing, but not on takeoff). They are used for the phases of flight at low speed (takeoff, landing, in-flight refueling of a jet fighter Supersonique by a Tanker subsonic). The camber flaps are sometimes directed to the top at high speed to reduce and adapt the Cambrure (curve) of the profile to Cz of flight, which reduces slightly the trail (sailplanes). ; Aerodynamic moments: In fact the couples apply to the three axes of an aircraft. One distinguishes the moments from Tangage, Roulis and lace. ; Bearing pressure: Force perpendicular to the flow of the air and directed towards the suction face (surface external of the wing located on the top). To include/understand the bearing pressure, it is necessary to remind our courses of Newtonian physics. Any body at rest remains at rest, and any animated body of a rectilinear continuous motion preserves this momentum until it is subjected to the application of an external force. If one observes a deviation in the flow of the air, or if the air at the origin at rest is accelerated, then a force was printed there. Newtonian physics stipulates that for each action there exists an opposite reaction of equal force. Thus, to generate a bearing pressure, the wing must create an action on the air which generates a reaction called bearing pressure. This bearing pressure is equal to the modification of the quantity of movement of the air which it ducts downwards. The momentum is the bulk product by speed. The bearing pressure of a wing is thus proportional to the quantity of air ducted to the bottom multiplied by the vertical speed of this air. To obtain more bearing pressure, the wing can either duct more air, or to increase the vertical speed of this air. This vertical speed behind the wing is downward flow. ; Profile: to see Profile (aeronautical). ; Reynolds number: Number without dimension representing the ratio enters the inertias and the forces viscous. For a given viscosity and a geometry it gives also the transition between a laminar flow and a turbulent flow. ; Salmon: Variable careenage of form, generally rounded, laid out at the end of the wing. A wing can however be crossed Net, without presenting salmon. ; wing Surface: It is the projected surface of the wing in the horizontal plane, including the surface included in the fuselage. ; marginal Swirl: Swirl present at the end of the wing, generated by the difference in pressure between the under-surface and the suction face. This swirl little to be very marked in the case of wing with weak lengthening and strong incidence (Harmony on takeoff). This swirling effect can be used by prolonging the wing by Ailette S (or Winglet S). ; Trail: The aerodynamic trail is a force which is opposed to the movement of a mobile in a gas; it is resistance to advance. She is exerted in the direction opposed at the speed of the mobile and increases with the square speed, except for the drag component induced by the bearing pressure which decreases with speed. The aerodynamic trail depends on the smoothness: from 2 to 3% of the bearing pressure for a sailplane of competition, of 12% up to 20 to 25% for a machine with weak lengthening (Harmony) or not very shaped (pendular ULM). At constant speed, the trail is balanced by a driving power (plane with engine) or by a loss of potential energy (loss of altitude in the case of a sailplane). ; Winglet: They are small vertical extensions fixed at the end of the wing with an aim of increasing the effective length of the wing (and thus the effective Allongement) to decrease the induced trail. The winglets recover part of the energy of the marginal swirls .

Actions of the wind on the works

August 1st

Aerodynamics out of blower

See Blower

Numerical aerodynamics

The wind tunnel tests are generally inaccessible to the private individuals from their very high cost. Since the years 1980, several Logiciel S was developed making it possible to treat the aerodynamics of tapering bodies numerically (in flow little or not taken down) and is now available on Internet. The computing power of the personal computers returned some of this easily exploitable software, with very short computing times (what was not the case a few years ago). The majority take again the Xfoil software established per Mark Drela of MIT in the USA. It is mainly:

Xfoil , which calculates the flow on a profile in 2D (infinite lengthening). The flow can be selected of type perfect or viscous, with in this case the taking into account of a boundary layer in conformity with the reality and the calculation of the laminar-turbulent site of the transition which is essential to establish the coefficient of drag.

Profile. The seizure of the file of profile requires to comply with certain rules, in particular a good geometrical definition at the edge of attack (density of the points and regularity of the variation of curve). The profiles available on Internet (NASG or UIUC database) often miss definition, which makes plant calculation: the iterations of balance do not converge. This can be corrected (but not always) by a mathematical smoothing in Xfoil or a graphic smoothing, by using a drawing tool managing the curves (curve radius in Rhino for example).

Shutters. The studied profile can be cambered locally (deflection of part of surface), but remains monoprofil. Xfoil does not treat the configurations multi-profiles like the profiles equipped with nozzles of leading edge and shutters with slit. The polar ones obtained are with infinite lengthening, and must be corrected for a real application to finished lengthening.

Validity. The slope of bearing pressure (Cz relation/incidence) calculated is slightly higher than that given by the wind tunnel tests. The level of coefficient of drag calculated is generally lower by 15 to 20% with that given by the wind tunnel tests. It should be noted that the conditions of flow in wind tunnel are inevitably more or less turbulent, which increases the measured trail (more advanced turbulent transition). Visual measurements in bearing flight on the site of the transition confirmed the calculations made with Xfoil.

Xfoil, discussion forum in English, in Yahoo Groups.
XFLR5 , based on the engine of Xfoil, and adding possibilities 3D in software a Windows CE interface makes it possible to study the behavior of the profiles and the wings. It takes again the prototype developments in MIAReX as regards the aspect 3D in " mode; bearing line non-linéaire".
Glider3d, Profili2 , is also based on Xfoil, in a Windows interface.

AVL ( Athena Vortex Lattice ), calculates balance, the bearing pressure and the induced trail of a complete configuration on several airfoils and in 3D. Moving parts (shutters, ailerons, etc…) are defined by a deformation of the average line of the profile. This time Ci, calculation considers a flow of the fluid type true:

- not of viscosity, therefore not of friction (given external which must be calculated separately),

- not of separation; the slopes of bearing pressure are linear: the indications obtained are really valid only with the weak angles of attack, except aerodynamic setbacks (let us say less 7°),

- the flow is incompressible; calculation remains valid into subsonic subcritical (Mach < 0.7) The detailed definition of the masses makes it possible to calculate inertias and the clean modes (Eigenmode) in pitching and rolling-lace. Visualization of the behavior in dynamics.

The main interest of AVL is to be able to explore quickly flying conditions balanced or not. One can seek an unhooking (an excess of Cz) local, the distribution of bearing pressure in scale, the values of deflection of the surfaces mobile, the values defining stabilities (moments and derived in pitching, lace), etc… If a problem is highlighted, the configuration can be modified in the file of definition and re-examined very quickly. To try to make the same thing with the hand, or with a worksheet (spreadsheet), or with a tool which require a complete grid of surface (as Fluent) would take much more time: hours instead of seconds. Comparisons AVL-Flowing show that AVL, by its flexibility and its speed, is adapted better to the initial phases of project (see bonds 5 and 6).

AVL was not conceived to treat the interactions of the airfoils with the fuselage, which requires delicate corrections. Comparisons between AVL and of the wind tunnel tests give an idea of the corrections necessary. The effects of propulsion (blast of propeller) are not treated either.

AVL was written and developed as from 1988 per Harold Youngren and Mark Drela, and was related to Windows in 2004 (AVL 3.26).

Xrotor , also writes by Mr. Drela, relates to in particular the propellers. The software takes into account propellers traditional, contrarotating, rotors of helicopters, wind mills.

Bonds

Xfoil : The site of Xfoil
: the site of discussion of Xfoil
: The site of MIAReX
Xflr5: The site of Xflr5
: The site to charge AVL
: an Australian university site which presents Xrotor
Boeing BWB Final Carryforward, aerodynamic project including a comparison AVL-Flowing
AVL capitolo 4, aerodynamic project including a comparison AVL-Flowing

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