Name: reynolds number from buckingham pi
Uploaded: 2025-02-13T15:43:29.000Z
Duration: 1 min 9 s
Channel: JoVE

パイプを通る流れについて考えてみましょう。ここでは、パイプの直径 D、流体速度 V、密度 ρ、粘度μなどの主要な変数があります。
バッキンガム円周率の定理を使用してシステムを解析するために、これらの変数をレイノルズ数のような無次元積に組み合わせて、流れの振る舞いを特徴付けます。
この場合、バッキンガム円周率の定理は、最初に関連する変数の数と参照次元 (質量 M、長さ L、時間 T) を特定することによって適用されます。
バッキンガム円周率の定理によれば、4 つの変数と 3 つの参照次元 (M、L、T) を持つシステムは、レイノルズ数である 1 つの無次元円周率項になります。
ここで、レイノルズ数は、慣性力を流体の粘性力に関連付ける無次元の積であり、流れを層流または乱流に分類するのに役立ちます。
バッキンガム円周率の定理は、変数を無次元項として表現することでシステムを定式化し、さまざまな条件下での流れの一般化と分析を容易にします。

バッキンガムの π 定理

The Buckingham Pi Theorem

バッキンガムの π 定理は、複雑な変数のシステムを無次元の項に縮小することで、流体力学の問題を簡素化する構造化された方法を提供します。

<img alt="Equation 1" src="/files/ftp_upload/18068/18068_Equation_1.svg" style="text-align: center; height: 34px;">

 パイプを通る流れを解析する場合、システムを定義する変数には、パイプの直径、流体の速度、密度、粘度などがあります。これらは、それらの基本的な関係を表すレイノルズ数 (Re) という単一の無次元項にまとめることができます。

<img alt="Equation 2" src="/files/ftp_upload/18068/18068_Equation_2.svg" style="text-align: center; height: 38px;">

 バッキンガムの π 定理を適用する最初のステップは、各変数の基本的な次元を特定することです。流体力学では、これらは通常、質量、長さ、時間です。ここで、直径は長さを表し、速度は長さと時間、密度は質量と体積、粘度は質量、長さ、時間を組み込んでいます。

<img alt="Equation 3" src="/files/ftp_upload/18068/18068_Equation_3.svg" style="text-align: center; height: 140px;">

 この定理は、π 項と呼ばれる独立した無次元グループの数が、変数の数 k と基本次元の数 r の差であることを示唆しています。4 つの変数と 3 つの次元では、無次元項は 1 つだけ必要であり、分析が簡素化されます。

<img alt="Equation 4" src="/files/ftp_upload/18068/18068_Equation_4.svg" style="text-align: center; height: 32px;">

 この単一の項であるレイノルズ数は、流体内の慣性力と粘性力を比較します。この比率を確立すると、流れが滑らかで整然とした層流と呼ばれる流れになるか、または無秩序で不規則な乱流と呼ばれる流れになるかがわかります。値が低い場合は層流で、粘性力が優勢で、流体は平行な層を動きます。値が高い場合は乱流で、慣性力が無秩序な動きにつながります。このコンテキストでバッキンガムの π 定理を適用すると、複雑な相互作用が単一の項に簡略化され、さまざまな条件下での流れの挙動を一般化することが容易になります。

パイプを通る流れについて考えてみましょう。主要な変数には、パイプの直径D、流体速度V、密度ρ、粘度μなどがあります。
宛先。。。

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

reynolds number from buckingham pi

The Buckingham Pi theorem turns fluid flow variables into dimensionless terms. In a pipe flow example, the key variables are the pipe diameter, fluid velocity, density, and viscosity. These variables can be combined into one dimensionless group, the Reynolds number, which captures their relationship.

Applying the theorem starts with identifying the fundamental dimensions of each variable. In fluid mechanics, these dimensions are mass, length, and time. The pipe diameter represents length, velocity combines length and time, density includes mass and volume, and viscosity includes mass, length, and time.

The theorem also shows how many independent dimensionless groups, or Pi terms, are needed. That number is found by subtracting the number of fundamental dimensions, r, from the number of variables, k. With four variables and three dimensions, only one dimensionless term is needed.

That single term is the Reynolds number, often written as Re. It compares inertial forces to viscous forces in the fluid. Low values point to laminar flow, where the fluid moves in smooth, parallel layers. Higher values point to turbulent flow, where motion becomes chaotic and irregular.

This makes the Buckingham Pi theorem useful for simplifying fluid dynamics problems. It reduces several linked variables to one term that can be used to compare flow behavior under different conditions. That simplification makes it easier to generalize how fluids move through a pipe.

Reynolds Number from Buckingham Pi

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key terms and concepts

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civil engineering

Reynolds Number from Buckingham Pi

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