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

バッキンガム円周率の定理を使用して円周率の項を決定するために、飛行機の翼の揚力を分析する例を考えてみます。
ここで関連する変数には、翼によって生成される揚力、空気密度、翼面積、航空機の速度、および空気の粘度が含まれます。
各変数は、質量、長さ、時間などの基本的な次元で表現され、無次元の項を構築するための一貫した基礎を提供します。
Pi項の数は、合計変数から基本次元を差し引くことによって決定されます。
 この場合、5 つの変数と 3 つの次元から 2 つの Pi 項が生成されます。次に、空気密度、速度、翼面積の3つの変数が、すべての次元をカバーする反復変数として選択され、それらが独立していることが保証されます。
残りの変数 (揚力と粘性) は、繰り返し変数と組み合わされて、無次元の Pi 項を形成します。
各 Pi 項は、すべての単位が相殺されることを確認することにより、無次元であることが確認されます。最終的な Pi 項は、元の変数を、リフト システムの本質的な特性を捉える式に単純化します。

π 項の決定

Determination of Pi Terms

バッキンガムの π 定理は次元解析における有用な方法であり、変数間の複雑な関係を無次元項に簡略化します。飛行機の翼の揚力の解析に関連する変数には、揚力、空気密度、翼面積、航空機の速度、空気粘性などがあります。各変数を基本次元 (質量、長さ、時間) で表すことで、これらの無次元項を作成するための一貫した基盤が得られます。

 定理は、π 項の数は、変数の合計数から基本次元の数を引いた数に等しいことを示しています。この例では、5 つの変数と 3 つの次元から 2 つの π 項が生成されます。空気密度、速度、翼面積は、すべての次元を独立して網羅するため、繰り返し変数として選択されます。これらの繰り返し変数は、揚力および粘性と組み合わされて無次元 π 項を形成します。

<img alt="Equation 1" src="/files/ftp_upload/18069/18069_Equation_1.svg" style="text-align: center; height: 100px;">

 最初の π 項は揚力生成の効率を表し、揚力が密度、速度、面積にどのように依存するかを表します。 2 番目の π 項は粘性の相対的な影響を考慮し、慣性に関連して粘性力が翼とどのように相互作用するかを明らかにします。

 各 π 項は無次元として検証されているため、複雑な空力挙動を簡略化された式で表すことができます。このアプローチにより、さまざまな条件下での揚力の予測と最適化が可能になり、各要素を個別に調べる必要なく、重要なダイナミクスを捉えることができます。

バッキンガム円周率の定理を使用して円周率の項を決定するために、飛行機の翼の揚力を分析する例を考えてみます。
ここで関連する変数には、次のものが含まれます。

Education

JoVE Core

Civil Engineering

土木工学

Unit 1

ユニット1

Dimensional Analysis, Similitude, and Modeling

次元解析、相似則、およびモデリング

dimensional-analysis-similitude-and-modeling

KEY TERMS AND CONCEPTS

主要用語と概念

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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civil engineering

Buckingham Pi Terms for Wing Lift

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