Polar diagram (fluid dynamics)

In fluid mechanics, a polar diagram (or polar for short ) is a graphic representation of the forces acting on a body against a flow for different angles of attack . The forces themselves are not shown, as they depend, among other things, on the flow velocity, but dimensionless coefficients . This representation was developed by Otto Lilienthal in order to assess the aerodynamic properties of wings. The polar diagram is still used today for the characterization of profiles and aircraft.

Lilienthal polar

The actual polar diagram, the so-called Lilienthal polar, is a plot of the lift coefficient on the ordinate (vertical axis) over the drag coefficient on the abscissa (horizontal axis). In addition to this resistance polar , there is also the moment polar , in which the moment coefficient is plotted against the angle of attack. The distance between the origin of the coordinates and a point on this curve is called the pole ray . In the case of the resistance polar, the rise in the pole beam is the slip ratio for the respective point. ${\ displaystyle c_ {a}}$ ${\ displaystyle c_ {w}}$ ${\ displaystyle c_ {m}}$ ${\ displaystyle E}$

Resistance poles allow conclusions to be drawn about the aerodynamic quality of a body. For example, in the case of airfoil profiles in glider construction, the area of application, high-speed flight or good slow flight characteristics , can be seen from the curve.

Special points on the Lilienthal polar using the example of a profile

NACA 2412

Lilienthalpolare of NACA 2412. The contact circle is not drawn to scale for the sake of clarity

The figure on the right shows a Lilienthal polar of the NACA 2412 profile calculated with XFOIL . A number of special points can be identified here:

"1" Minimum buoyancy . The profile has the smallest (most negative) lift coefficient here. This point corresponds to the minimum flight speed in horizontal inverted flight.

{\ displaystyle c_ {a, min}}

"2" zero lift. The profile does not generate lift . This point corresponds to parabolic flight . The drag coefficient at this point is denoted by. ${\ displaystyle c_ {a} = 0}$ ${\ displaystyle c_ {w0}}$

"3" smallest total air force . The polar here has the smallest distance to the origin. is only noteworthy for profiles that have a pronounced laminar pelvis above . Here an airplane reaches the greatest aerodynamically possible speed in an almost vertical dive. ${\ displaystyle c_ {r, min}}$ ${\ displaystyle c_ {r, min}}$ ${\ displaystyle c_ {a} = 0}$ ${\ displaystyle c_ {w0}}$

"4" Minimal resistance . For symmetrical profiles it is usually (but not mandatory!) Included .

{\ displaystyle c_ {w, min}}

{\ displaystyle c_ {a} = 0}

"5" Best sliding , the contact point of the steepest Polstrahls. The glide angle is minimal here; an aircraft glides over the longest distance with a given loss of altitude ( ). This point is related to the best glide speed ( ). For jet aircraft, this is also the best climb and minimum thrust speed. For propeller-driven aircraft, it is the minimum drag speed, but not the minimum power speed that is . ${\ displaystyle E_ {max}}$ ${\ displaystyle \ gamma}$ ${\ displaystyle \ tan \ gamma = 1 / E}$ ${\ displaystyle V ^ {*}}$ ${\ displaystyle 0 {,} 76 \ cdot V ^ {*}}$

"6" Best ascent, lowest sink. The so-called climb rate becomes a maximum. Here an airplane has the lowest rate of descent when gliding. The minimum sink speed for propeller and jet aircraft is .

{\ displaystyle c_ {a} ^ {3} / c_ {w} ^ {2}}

{\ displaystyle 0 {,} 76 \ cdot V ^ {*}}

Note: The points "5" and "6" can also be found in the same way for the negative part of the polar.

"7" maximum lift, . The profile reaches its greatest lift, becomes maximum. This corresponds to the lowest flight speed in level flight. ${\ displaystyle c_ {a, max}}$ ${\ displaystyle c_ {a}}$

The points shown are not only found on profile polar, but also on polar for total aircraft.

Resolved polar

Example of a resolved polar: lift polar

With resolved polars, the force coefficients are shown on the ordinate over the angle of attack on the abscissa. The resolved polar diagram of lift coefficients in relation to the angle of attack is widely used . Characteristic is an almost linear course with small angles of attack, with symmetrical wing profiles through the coordinate origin. The course inclines at high angles of attack, runs through the apex and then drops off again in the so-called excessive flight condition. The course around this apex , the maximum achievable lift coefficient, characterizes the tear-off behavior of a wing profile or aircraft. ${\ displaystyle \ alpha}$

Resolved polars illustrate the influence of variables such as the Reynolds number or shape parameters such as buoyancy aids and surface properties on individual coefficients of a body exposed to a flow. With ground-based vehicles, for example, the influence of crosswinds on driving stability is decisive.

Resolved polar for the example above

Resolved polar of NACA 2412

In the resolved polar of the NACA 2412, the course of the lift, drag and moment coefficients is plotted against the angle of attack. To make the moment coefficient unambiguous, its reference point must also be specified. If this information is missing, the moment usually (and also in this example) refers to a point at 25% profile depth (“t / 4”). A few other important parameters can be found here:

${\ displaystyle \ alpha _ {0}}$ : the angle of attack for which zero lift results.
${\ displaystyle c_ {a0}}$ : the lift at zero angle of attack.
${\ displaystyle {\ frac {\ partial c_ {a}} {\ partial \ alpha}}}$ : the increase in lift.
The size of the linear part of the lift polar. Here i. A. no detachment.
${\ displaystyle \ alpha _ {krit}}$ : The angle of attack at maximum lift.
${\ displaystyle c_ {m0}}$ : the moment coefficient at zero lift.
The (almost) horizontal course of the torque curve in the linear part of the example shows that the neutral point is (very close) in the torque reference point (here: t / 4).