# Why does the critical speed depend on the density

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### Air resistance [3]

[415]**Air resistance**. Our knowledge of air resistance has expanded significantly in recent years as a result of the rapid development of aviation technology and aviation. Following the in Vol. 6 and Ergbd. What has been said, the following is communicated here.

I. *General*. The amount of air resistance*W.* depends on the speed*v* with which the body is moved through the air, further on the size of the body and on the density*ρ* = *γ*/*G* the air (*γ* = spec. Weight, *G* = Acceleration due to gravity). The size of the body is usually characterized by its projection onto a plane perpendicular to the direction of movement. It is used as the main bulkhead *F.* designated. Resistance is usually represented in the following form: *W.* = *c _{w} F ρ v*

^{2}/ 2. The expression

*ρ v*

^{2}/ 2 is referred to as

*Back pressure*; as an abbreviation for this one often uses the letter

*q. c*is a dimensionless number that depends on the shape of the body and is called the

_{w}*Resistance number*. In the event that the air resistance changes exactly with the square of the speed according to the above equation, is

*c*a constant size. However, this is not the case in all cases. Often the resistance obeys another law; it is then, if the above approach is retained,

_{w}*c*no longer constant but a function of speed. The characterization of the body shape by a single numerical value from which the resistance can be calculated is therefore not possible in many cases.

_{w}The pressure differences occurring in the vicinity of the body cause a change in the density of the air due to the compressibility of the air. As long as you are dealing with speeds that are small compared to the speed of sound, which is usually the case in flight technology (except for very fast rotating propellers, at which speeds of up to 280 m / sec. Are currently reached), the changes are the density is insignificant and can be neglected in most cases. For example, through a speed of 50 m / sec. (= 180 km / h) produced a change in air density of around 1%. The above resistance formula and the following statements apply in general to incompressible media, in particular also to drip liquids.

For the pressure conditions on the surface of the body, it does not matter whether the body moves through the still air or whether the air flows against the resting body. We therefore do not need to distinguish between these two cases. The relationship between the flow velocity at any point in the vicinity of the body and the pressure prevailing there is governed by Bernoulli's equation, which reads: *ρ v*^{2}/2 + *p* = *p*_{0} = const. *p* is called the static pressure, *p*_{0} the total pressure.

The equation says that the sum of dynamic pressure and static pressure is constant. This law, which serves as the basis for the measurement of flow velocities, generally applies to every smooth movement of liquid, in which the velocity at a certain point does not change over time ("stationary" movement).

The flow resistance can be broken down into two different parts, namely the surface friction resistance, which is caused by the friction of the air flowing along the body, and the shape or vortex resistance. The latter is caused by the fact that the flow behind the body does not recombine, but rather creates a "slipstream" (dead water) as a result of the formation of eddies (Fig. 1). The vortex formation creates a pressure reduction on the back of the body opposite the front, which constitutes the form resistance. The eddies are created by the internal friction of the liquid. It turns out that the influence of a very small friction of the liquid in the free flow, where the speed difference between neighboring particles is only very small [415], is quite insignificant, whereas on the body surface, where the speed is within a narrow range of that the value corresponding to the free flow drops to zero on the body surface (the liquid adheres to the surface), friction plays a decisive role. It is the reason that the smooth flow, which would always follow the surface with frictionless movement, detaches itself from it at a certain point (in Fig. 1 approximately at *a*) and thereby gives rise to vortex formation. Since the form resistance depends on the size of the vortex area that forms, the position of the separation point is important for the resistance. In order to obtain a small resistance, one must endeavor to relocate the detachment point as far as possible to the rear end of the body. Their position depends on the one hand on the shape of the body and on the other hand on the so-called Reynolds number *R.* dependent. According to O. *Reynolds* In the case of bodies of geometrically similar shape, there is only a similarity of the flow pattern and thus the same *v d.* Resistance number if the product in the cases to be compared*v d*/*ν* = *R.* = Reynolds number is the same. Here means *d* one in the cases to be compared, but otherwise any length dimension of the body and *ν* = *μ*/*ρ* the kinematic toughness (*μ* = Toughness measure). The drag coefficient therefore generally depends not only on the shape of the body but also on the speed, the absolute size of the body and the kinematic viscosity of the medium in which the body moves.

In the case of bodies with sharp edges (e.g. plate placed perpendicular to the direction of movement), the detachment point is bound to this. In such cases the resistance number is almost independent of the Reynolds number, i.e. the resistance obeys the quadratic law here.

Kinematic toughness values *ν:*

II. *Test results on air resistance*. - To determine the size of the air resistance, one is almost exclusively dependent on experiments. In recent times, these have mostly been carried out in such a way that an air stream that is as uniform as possible in terms of time and space is blown against the test specimen attached to the weighing equipment. The following test results relate to speeds that are small compared to the speed of sound.

1. Resistance from different bodies. - Circular disk perpendicular to the direction of movement *c _{w}* = 1.1; square plate perpendicular to the direction of movement

*c*= 1.1; rectangular plate (aspect ratio 1:50) perpendicular to the direction of movement

_{w}*c*= 1.56. The drag coefficient of these bodies is almost independent of the speed.

_{w}2. Resistance of spheres, ellipsoids and hemispheres. - In the case of balls, after a certain speed ("critical speed") is exceeded, there is a sudden strong reduction in the area of the vortex and thus a sharp decrease in the drag coefficient.

In Fig. 2 is the resistance number *c _{w}* depending on the Reynolds number

*v d*/

*ν*applied. For a sphere with a diameter of 20 cm and for air at 15 ° and 760 mm pressure, for example, [416] the critical speed

*vd*/

*ν*~ 240000 to

*v*~ (240000 x 0.142) / 20 = 1700 cm / sec. = 17 m / sec. In FIG. 2, the resistance numbers of an elongated (axis ratio 1: 1.8) and a flattened (axis ratio 1: 0.75) ellipsoid of revolution are plotted. These bodies show a similar decrease in drag coefficient as the ball. In the case of the flattened ellipsoid, this decrease occurs at a higher Reynolds number, and in the case of the elongated one at a lower Reynolds number than in the case of the sphere. It has been shown that the slimmer the body, the more gradual the decrease in resistance. For an open hemisphere with a diameter of 25 cm, measurements of

*Eiffel*in a speed range from 4 to 32 m / sec. the following resistance figures: Convex side blown

*c*= 0.32, blown on concave side

_{w}*c*= 1.42 to 1.60.

_{w}3. Resistance of balloon bodies. - In the Göttingen research institute, a number of elongated bodies of revolution were investigated, similar to those used in the construction of airships, some of which showed very little resistance. The most favorable of the forms examined (Fig. 3) resulted in a Reynolds number *R.* ~ 200000 the resistance number *c _{w}* = 0.04. The resistance of such a body is accordingly in comparison with the resistance of a circular disc of the same diameter as the main bulkhead, only about the 27th part.

4. Resistance of cylindrical bodies. - In the case of cylinders with a flow perpendicular to the axis, there are also critical velocities and thus a strong variability in the coefficient of resistance. For very long circular cylinders with Reynolds numbers up to about 150,000 the drag coefficient is *c _{w}* mean = 0.95. With larger Reynolds numbers, the drag coefficient decreases to less than half the value. In flight technology, cylindrical bodies with an elongated cross-section (struts) are used as structural parts to reduce the resistance. In Fig. 4 the shapes of three struts and their drag coefficients are shown, the latter depending on the product

*v d*mm m / sec. (

*d*= Strut thickness). It follows from this that the drag coefficient becomes smaller, the narrower the cross-section. However, shapes that are too slim become less favorable again because of the increasing frictional resistance. In the case of slender forms, the critical speed is very small, and the change in the coefficient of resistance is no longer as abrupt as in the case of short bodies.

5. Air resistance of airfoils. - The entire air force acting on a wing is divided into a component perpendicular to the direction of movement (lift) and a component parallel to it (drag). Like drag, lift is defined by *A.* = *c _{a} F (ρ v*

^{2}) / 2. For

*F.*the area of the wing is used here.

*c*is called the lift number. To characterize the properties of an airfoil, it is customary to plot the lift coefficient as a function of the drag coefficient for different angles of attack, whereby the angle of attack is to be understood as the angle of the wing chord with the direction of movement. The curve obtained in this way is called "polar curve". In Fig. 5 a favorable wing cross-section ("profile") is shown with the associated polar curve. This result relates to a wing with a rectangular plan and an aspect ratio of 1: 5, the larger side being perpendicular to the direction of movement (wingspan). The resistance of a wing can be based on the

_{a}*Prandtl*I separate the wing theory into two essential parts, namely the induced drag and the profile drag. In the latter, as before, a distinction can be made between surface friction resistance and eddy resistance. In the theory mentioned, it is assumed that the lift is distributed in the form of a [417] semi-ellipse over the wingspan, which is approximately the case for wings with a rectangular outline and the same angle of attack for all wing cross-sections. The drag coefficients correspond to the two parts of the wing drag

*c*and

_{wi}*c*(see Fig. 5). The induced drag is only dependent on the lift and the wing outline; it is expressed by the equation

_{Where}*c*=

_{wi}*c*

_{a}^{2}/

*π F*/

*b*

^{2}, under

*b*understood the wingspan. In the polar curve, therefore, the resistance number becomes

*c*of the induced drag, if it is plotted as a function of the lift, represented by a parabola (parabola of the induced drag). With a rectangular wing outline is

_{wi}*F.*/

*b*

^{2}as it is easy to stand, identical to the aspect ratio. The profile resistance, expressed by the resistance number

*c*, is primarily dependent on the profile shape and, as already mentioned, caused by surface friction and the formation of eddies. It can be reduced to a low level by suitable shaping of the profile. The most essential result of Prandtl's theory is the realization that even in a completely frictionless medium to generate a certain lift, a certain drag, namely the induced one, is unavoidable. As one can see from the above equation, the larger the wingspan becomes with constant lift and the same wing area. For the theoretical case that the span becomes infinitely large, it assumes the value zero.

_{Where}III. *Movements at supersonic speed*. As mentioned above, the compressibility of the air plays an important role in movements that are faster than the speed of sound. Because of the high pressures, there are considerable changes in density in the vicinity of the body. The pressure effects spread in all directions at the speed of sound. As the closer examination shows, this has the consequence that conical compression waves are formed. Fig. 6 shows these relationships again for an infantry projectile [418] (taken by L. *Do* using the Schlieren method). The angle at the point of the cone *α* has a very simple relationship to the velocity of the bullet *v.* Because it is sin *α*/2 = *a*/*v*, if *a* means the speed of sound (= 333 m / sec.). The drag coefficient increases considerably when the speed of sound is exceeded, but slowly decreases again when the speed is even higher. In Fig. 7, the drag coefficient of the projectile shapes shown in Fig. 8 is dependent on the ratio *v*/*a* applied.

Literature: Eiffel, G., La resistance de l'air et l'aviation, Paris 1910. - Prandtl, L., Liquid and Gas Movement, Jena 1913. - Ders., Some relationships from mechanics that are important for flight technology, Zeitschr . f. flight tech. and engine oil. 1910, p. 3 ff. - v. Mises, R., Fluglehre, Berlin 1918. - Communications from the Göttingen Research Institute for Aerodynamics, Zeitschr. f. flight tech. and engine oil. from 1910 and Techn. Ber. d. Airplane mastery.

C Wieselsberger.

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