Name: buckingham pi terms for wing lift
Uploaded: 2025-02-13T15:43:30.000Z
Duration: 1 min 15 s
Channel: JoVE

Pour déterminer les termes Pi à l’aide du théorème de Buckingham Pi, considérons l’exemple de l’analyse de la force de portance sur une aile d’avion.
Les variables pertinentes ici incluent la force de portance générée par l’aile, la densité de l’air, la surface de l’aile, la vitesse de l’avion et la viscosité de l’air.
Chaque variable est exprimée en dimensions fondamentales, telles que la masse, la longueur et le temps, afin de fournir une base cohérente pour la construction de termes sans dimension.
Le nombre de termes Pi est déterminé en soustrayant les dimensions fondamentales des variables totales.
 Dans ce cas, cinq variables et trois dimensions donnent deux termes Pi. Trois variables, la densité de l’air, la vitesse et la surface de l’aile, sont ensuite choisies comme variables répétitives pour couvrir toutes les dimensions, en veillant à ce qu’elles soient indépendantes.
Les variables restantes (force de levage et viscosité) sont combinées avec les variables répétitives pour former des termes Pi sans dimension.
Chaque terme Pi est vérifié pour s’assurer qu’il est sans dimension en confirmant que toutes les unités s’annulent. Les termes finaux de Pi simplifient les variables d’origine en expressions qui capturent les caractéristiques essentielles du système d’ascenseur.

détermination des termes pi

Determination of Pi Terms

Le théorème de Buckingham Pi est une méthode précieuse dans l'analyse dimensionnelle, réduisant les relations complexes entre les variables en termes sans dimension. Les variables pertinentes dans l'analyse de la force de portance sur une aile d'avion comprennent la force de portance, la densité de l'air, la surface de l'aile, la vitesse de l'avion et la viscosité de l'air. L'expression de chaque variable en termes de dimensions fondamentales (masse, longueur et temps) fournit une base cohérente pour la construction de ces termes sans dimension.

Le théorème indique que le nombre de termes Pi est égal au total des variables moins le nombre de dimensions fondamentales. Dans cet exemple, cinq variables et trois dimensions donnent lieu à deux termes Pi. La densité de l'air, la vitesse et la surface de l'aile sont sélectionnées comme variables répétitives, car elles couvrent indépendamment toutes les dimensions. Ces variables répétitives se combinent avec la force de portance et la viscosité pour former des termes Pi sans dimension.

<img alt="Equation 1" src="/files/ftp_upload/18069/18069_Equation_1.svg" style="text-align: center; height: 100px;">

Le premier terme Pi exprime l'efficacité de la génération de portance, représentant la façon dont la portance dépend de la densité, de la vitesse et de la surface. Le deuxième terme Pi rend compte de l'impact relatif de la viscosité, révélant comment les forces visqueuses interagissent avec l'aile par rapport à l'inertie.

Chaque terme Pi est vérifié comme étant sans dimension, ce qui permet de représenter un comportement aérodynamique complexe dans des expressions simplifiées. Cette approche permet de prédire et d'optimiser la portance dans diverses conditions, en capturant la dynamique essentielle sans avoir à examiner chaque facteur individuellement.

Détermination des termes Pi

Pour déterminer les termes Pi à l’aide du théorème de Buckingham Pi, considérons l’exemple de l’analyse de la force de portance sur une aile d’avion.
Les variables pertinentes ici incluent ...

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Dimensional Analysis, Similitude, and Modeling

Analyse dimensionnelle, similitude et modélisation

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buckingham pi terms for wing lift

Buckingham Pi terms help simplify the study of lift force on an airplane wing. Dimensional analysis is used to turn several related variables into dimensionless terms, which makes the problem easier to work with.

The wing-lift example uses five variables: lift force, air density, wing area, aircraft velocity, and air viscosity. Each variable is written in fundamental dimensions of mass, length, and time. The Buckingham Pi theorem then shows that the number of Pi terms equals the total number of variables minus the number of fundamental dimensions, so this case gives two Pi terms.

Air density, velocity, and wing area are chosen as repeating variables. They are selected because they independently cover all the needed dimensions. These repeating variables are then combined with lift force and viscosity to build the dimensionless Pi terms.

The first Pi term shows how lift depends on density, velocity, and area. It represents the efficiency of lift generation. The second Pi term shows the effect of viscosity. It compares viscous forces with inertia and helps describe how the wing interacts with the air.

Each Pi term is checked to make sure it has no units. That step confirms that the terms are truly dimensionless. With this method, complex aerodynamic behavior can be written in a simpler form, which helps predict and optimize lift under different conditions.

Buckingham Pi Terms for Wing Lift

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Learn how Buckingham Pi terms simplify airplane wing lift by using density, velocity, area, lift force, and viscosity to build dimensionless terms.

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Buckingham Pi Terms for Wing Lift

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