Blasius' formulas

The 1st and 2nd Blasius formula indicate the dynamic lift and torque that an elongated body (e.g. hydrofoil ) experiences in a flowing medium if certain requirements for the flow type are met. The formulas are named after the engineer and university professor Heinrich Blasius , who taught at the Hamburg University of Applied Sciences (then the Hamburg Engineering School) from 1912 until his death in 1970 .

Overview

See also : dynamic lift

The forces and torques on a body in a flowing medium are generally composed in a complicated way and can only be calculated with the help of computers. However, under conditions that can be approximated in aircraft, there are analytical formulas that originate from the pioneering days (around 1900). For the dynamic lift these are the Kutta-Joukowski formula and the 1st Blasius formula, for the pitching moment it is the 2nd Blasius formula. The 1st Blasius formula allows an exact derivation of the Kutta-Joukowski formula, which is often used in physics textbooks.

The physical basis of the formulas is the pressure distribution in a flowing medium according to Bernoulli's equation . The compressive forces act on the surface elements of the body and are added up, resulting in integrals over the profile curve of the body. Only in special cases , such as the Joukowski profile , are these integrals easy to evaluate.

The prerequisites for the applicability of the Blasius formulas are as follows: The body around the flow must be elongated (like a wing) and have a uniform profile ; strictly speaking, it must be a generalized cylinder with the profile as the base and infinite extent in the third dimension. The formulas therefore indicate forces and torques per unit of length in the third dimension . The body flow must be exactly in the transverse direction, and the flow must be stationary , incompressible , smooth , eddy-free and essentially two-dimensional (2D). The 2D assumption means that in the direction of the third dimension the component of the speed and all variations of sizes are small and negligible.

Real formulas

The mass density of the flowing medium is (constant because incompressible), the profile curve ("contour") of the body is . Then the following applies for the real components and the force per unit length on the body, expressed by the real components and the local flow velocity: ${\displaystyle \rho }$ ${\displaystyle C}$ ${\displaystyle F_{x}}$ ${\displaystyle F_{y}}$ ${\displaystyle v_{x}}$ ${\displaystyle v_{y}}$

1. Blasius' formula

${\displaystyle {\begin{array}{c}\displaystyle F_{x}=-{\frac {\rho }{2}}\oint _{C}[(v_{x}^{2}-v_{y}^{2})\mathrm {d} y-2v_{x}v_{y}\mathrm {d} x]\\\\\displaystyle F_{y}=-{\frac {\rho }{2}}\oint _{C}[(v_{x}^{2}-v_{y}^{2})\mathrm {d} x+2v_{x}v_{y}\mathrm {d} y]\end{array}}}$

For the torque in the longitudinal direction, again per unit length, the following applies

2. Blasius' formula

${\displaystyle M_{0}=-{\frac {\rho }{2}}\oint [(v_{x}^{2}-v_{y}^{2})(x\mathrm {d} x-y\mathrm {d} y)+2v_{x}v_{y}(x\mathrm {d} y+y\mathrm {d} x)]}$

The index 0 stands for the coordinate origin as the reference point for the torque.

Complex formulation

The complex notation is particularly appropriate to the problem.

A point in the plane of the profile is called a complex number

${\displaystyle \!\,z=x+iy}$

shown; analogously a speed and a force (each at the point ) ${\displaystyle z}$

${\displaystyle v=v_{x}+iv_{y}\qquad \qquad F=F_{x}+iF_{y}}$

In the Blasius formulas, the complex conjugate of speed and analog of force appears. Under the given conditions there is a holomorphic function ( it is not itself). ${\displaystyle v^{*}\!\,}$ ${\displaystyle v^{*}(z)\!\,}$ ${\displaystyle v(z)}$

For the force per unit of length, the

1. Blasius' formula

${\displaystyle F^{*}=i{\frac {\rho }{2}}\oint _{C}(v^{*})^{2}\mathrm {d} z}$

Here is the mass density of the flowing medium and the profile curve of the body. ${\displaystyle \rho }$ ${\displaystyle C}$

For the torque in the longitudinal direction, again per unit length, the following applies

2. Blasius' formula

${\displaystyle M_{0}=-{\frac {\rho }{2}}\operatorname {Re} \oint _{C}(v^{*})^{2}z\,\mathrm {d} z}$

The index 0 stands for the coordinate origin as the reference point for the torque.

The integration curve can be deformed at will because of the holomorphism of in the area of ​​the flow field ( Cauchy's integral theorem ). ${\displaystyle v^{*}(z)\!\,}$

Derivation in the complex

The argument here is formulated in a complex manner throughout. Since friction is neglected, there are no shear forces. On one element of the profile curve, the pressure generates a force perpendicular to and equal in amount . If you go around the profile in an anti-clockwise direction, you get the direction of the force in the complex plane by multiplying it with the imaginary unit (corresponding to a 90 ° rotation). Therefore the following applies to a line element: ${\displaystyle \mathrm {d} z}$ ${\displaystyle p}$ ${\displaystyle \mathrm {d} z}$ ${\displaystyle p|\mathrm {d} z|}$

${\displaystyle \mathrm {d} F=ip\mathrm {d} z\!\,}$

According to Bernoulli's equation , the local pressure is given by the square of the absolute value of the speed. Along a streamline that runs from infinity in front of the body to infinity behind the body, applies

${\displaystyle p_{\infty }+{\frac {\rho }{2}}|v_{\infty }|^{2}=p(z)+{\frac {\rho }{2}}|v(z)|^{2}.}$

For the physics of flight, the case is interesting that the medium flows at a great distance from the body at constant speed (viewed from the airplane) and the (air) pressure is also constant at infinity. Then and are the same for all streamlines, and the following applies to local pressure: ${\displaystyle v_{\infty }}$ ${\displaystyle v_{\infty }}$ ${\displaystyle p_{\infty }}$

${\displaystyle p(z)=-{\frac {\rho }{2}}v(z)v^{*}(z)+\mathrm {const} }$

Summing up all the pressure forces acting on the profile thus initially results

${\displaystyle F=i\oint _{C}p\mathrm {d} z=-i{\frac {\rho }{2}}\oint _{C}vv^{*}\mathrm {d} z,}$

taking into account that the constant does not contribute because the sum of all along a closed line is zero. ${\displaystyle \mathrm {d} z}$

Now the speed must be parallel everywhere on the profile curve , i.e. H. it and have the same argument and the following product is real: ${\displaystyle C}$ ${\displaystyle v}$ ${\displaystyle \mathrm {d} z}$

${\displaystyle v^{*}\mathrm {d} z=v\mathrm {d} z^{*}\!\,}$

If one substitutes accordingly in the integral and one forms the complex conjugate on both sides, one obtains the 1st Blasius formula.

The following applies to the torque with reference point 0 on a line element of the profile:

${\displaystyle \mathrm {d} M_{0}=x\mathrm {d} F_{y}-y\mathrm {d} F_{x}=\operatorname {Im} (z^{*}\mathrm {d} F)=\operatorname {Re} (z^{*}p\mathrm {d} z)}$

The expression from Bernoulli's equation can again be used for the pressure and the substitution can be carried out in the integral over the profile curve as above. Here, too, the contribution of the constants vanishes because it leads to an integral over a complete differential:

${\displaystyle \operatorname {Re} \,z^{*}\mathrm {d} z={\frac {1}{2}}\mathrm {d} (zz^{*})}$

By complex conjugating all factors under the real part, one obtains the 2nd Blasius formula.

literature

• Hermann Schlichting , Erich Truckenbrodt : Aerodynamics of the Airplane , Volume 1. Springer 1967

Individual evidence

1. ↑ H. Blasius, Functional Theoretical Methods in Hydrodynamics , Journal for Mathematics and Physics 58 (1910) 90-110.
2. ↑ Dieter Meschede : Gerthsen Physik , 21st edition, Springer 2002, section 3.3.9
3. ↑ Schlichting / Truckenbrodt, Section 6.212
4. ↑ ^{a b c} Schlichting / Truckenbrodt, Section 6.211