Namespaces
Variants
Actions

Difference between revisions of "Prandtl equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(latex details)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
<!--
 +
p0742501.png
 +
$#A+1 = 17 n = 0
 +
$#C+1 = 17 : ~/encyclopedia/old_files/data/P074/P.0704250 Prandtl equation
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742502.png" /> is the unknown function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742505.png" /> are given functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742507.png" />, 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742508.png" /> is the span of the wing which is assumed to be symmetric with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p0742509.png" />-plane and the direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p07425010.png" />-axis coincides with that of the flow of air at infinity; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p07425011.png" /> denotes the chord of the profile which corresponds to the abscissa <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p07425012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p07425013.png" /> is the circulation of the air flow about this profile; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p07425014.png" /> is a constant; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p07425015.png" /> is the velocity of the air flow at infinity; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p07425016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074250/p07425017.png" />.
+
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]] (cf. [[#References|[3]]]).
  
 
The Prandtl equation is named after L. Prandtl.
 
The Prandtl equation is named after L. Prandtl.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Golubets,   "Lectures on the theory of wings" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> T. von Kármán,   J.M. Burgers,   "General aerodynamic theory - Perfect fluid" , Springer  (1936)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N.I. Muskhelishvili,   "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Golubets, "Lectures on the theory of wings" , Moscow-Leningrad  (1949)  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> T. von Kármán, J.M. Burgers, "General aerodynamic theory - Perfect fluid" , Springer  (1936)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR>
 +
</table>

Latest revision as of 20:00, 19 January 2024


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)
How to Cite This Entry:
Prandtl equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prandtl_equation&oldid=19061
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article