Difference between revisions of "Prandtl equation"
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The fundamental integro-differential equation of the wing of an airplane of finite span. In the derivation of the Prandtl equation, assumptions are made which make it possible to consider every element of the wing as if it were in a plane-parallel air flow around the wing. This makes it possible to connect the geometric characteristics of the wing with its aerodynamic properties. The Prandtl equation thus obtained has the form | The fundamental integro-differential equation of the wing of an airplane of finite span. In the derivation of the Prandtl equation, assumptions are made which make it possible to consider every element of the wing as if it were in a plane-parallel air flow around the wing. This makes it possible to connect the geometric characteristics of the wing with its aerodynamic properties. The Prandtl equation thus obtained has the form | ||
| − | + | $$ \tag{1 } | |
| + | |||
| + | \frac{F ( t _ {0} ) }{B ( t _ {0} ) } | ||
| + | + | ||
| + | |||
| + | \frac{1} \pi | ||
| + | \int\limits _ { - } a ^ { a } | ||
| + | |||
| + | \frac{F ^ { \prime } ( t) }{t - t _ {0} } | ||
| + | \ | ||
| + | d t = f ( t _ {0} ) ,\ \ | ||
| + | - a < t _ {0} < a , | ||
| + | $$ | ||
| − | where | + | where $ F $ |
| + | is the unknown function, $ F ^ { \prime } ( t) = d F ( t) / dt $, | ||
| + | $ B $ | ||
| + | and $ f $ | ||
| + | are given functions, $ B ( t) = c b ( t) $, | ||
| + | $ f( t) = \nu \omega ( t) $, | ||
| + | and the improper integral is taken in the sense of the Cauchy principal value. The meanings of the quantities appearing in the equation are as follows: $ 2 a $ | ||
| + | is the span of the wing which is assumed to be symmetric with respect to the $ y z $- | ||
| + | plane and the direction of the $ z $- | ||
| + | axis coincides with that of the flow of air at infinity; $ b ( x) $ | ||
| + | denotes the chord of the profile which corresponds to the abscissa $ x $; | ||
| + | $ F( x) $ | ||
| + | is the circulation of the air flow about this profile; $ c $ | ||
| + | is a constant; $ \nu $ | ||
| + | is the velocity of the air flow at infinity; and $ \omega $ | ||
| + | is a function depending on the curvature of the profile and the overwind of the wing (cf. [[#References|[1]]]). Based on experiments, it is assumed that $ F ( - a ) = F ( a) $. | ||
The Prandtl equation can be solved in closed form only under very stringent assumptions. In the general case, the Prandtl equation can be reduced to a [[Fredholm equation|Fredholm equation]] (cf. [[#References|[3]]]). | The Prandtl equation can be solved in closed form only under very stringent assumptions. In the general case, the Prandtl equation can be reduced to a [[Fredholm equation|Fredholm equation]] (cf. [[#References|[3]]]). | ||
Revision as of 08:07, 6 June 2020
The fundamental integro-differential equation of the wing of an airplane of finite span. In the derivation of the Prandtl equation, assumptions are made which make it possible to consider every element of the wing as if it were in a plane-parallel air flow around the wing. This makes it possible to connect the geometric characteristics of the wing with its aerodynamic properties. The Prandtl equation thus obtained has the form
$$ \tag{1 } \frac{F ( t _ {0} ) }{B ( t _ {0} ) } + \frac{1} \pi \int\limits _ { - } a ^ { a } \frac{F ^ { \prime } ( t) }{t - t _ {0} } \ d t = f ( t _ {0} ) ,\ \ - a < t _ {0} < a , $$
where $ F $ is the unknown function, $ F ^ { \prime } ( t) = d F ( t) / dt $, $ B $ and $ f $ are given functions, $ B ( t) = c b ( t) $, $ f( t) = \nu \omega ( t) $, and the improper integral is taken in the sense of the Cauchy principal value. The meanings of the quantities appearing in the equation are as follows: $ 2 a $ is the span of the wing which is assumed to be symmetric with respect to the $ y z $- plane and the direction of the $ z $- axis coincides with that of the flow of air at infinity; $ b ( x) $ denotes the chord of the profile which corresponds to the abscissa $ x $; $ F( x) $ is the circulation of the air flow about this profile; $ c $ is a constant; $ \nu $ is the velocity of the air flow at infinity; and $ \omega $ is a function depending on the curvature of the profile and the overwind of the wing (cf. [1]). Based on experiments, it is assumed that $ F ( - a ) = F ( a) $.
The Prandtl equation can be solved in closed form only under very stringent assumptions. In the general case, the Prandtl equation can be reduced to a Fredholm equation (cf. [3]).
The Prandtl equation is named after L. Prandtl.
References
| [1] | V.V. Golubets, "Lectures on the theory of wings" , Moscow-Leningrad (1949) (In Russian) |
| [2] | T. von Kármán, J.M. Burgers, "General aerodynamic theory - Perfect fluid" , Springer (1936) |
| [3] | N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1972) (Translated from Russian) |
Prandtl equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prandtl_equation&oldid=19061