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

Considérons l'écoulement dans un tuyau, où les variables clés incluent le diamètre D du tuyau, la vitesse du fluide V, la densité ρ et la viscosité μ.
Pour analyser le système à l’aide du théorème de Buckingham Pi, ces variables peuvent être combinées en produits sans dimension comme le nombre de Reynolds, qui caractérise le comportement de l’écoulement.
Dans ce cas, le théorème de Buckingham Pi est appliqué en identifiant d’abord le nombre de variables impliquées et les dimensions de référence, qui sont la masse M, la longueur L et le temps T.
Selon le théorème de Buckingham Pi, le système à quatre variables et trois dimensions de référence (M, L, T) donne un terme Pi sans dimension, qui est le nombre de Reynolds.
Ici, le nombre de Reynolds est un produit sans dimension qui relie les forces d’inertie aux forces visqueuses dans le fluide, aidant à classer l’écoulement comme laminaire ou turbulent.
Le théorème de Buckingham Pi reformule le système en exprimant les variables sous forme de termes sans dimension, ce qui facilite la généralisation et l’analyse de l’écoulement dans des conditions variables.

le théorème de buckingham pi

The Buckingham Pi Theorem

Le théorème de Buckingham Pi fournit une méthode structurée pour simplifier les problèmes de dynamique des fluides en réduisant les systèmes complexes de variables à des termes sans dimension.

<img alt="Equation 1" src="/files/ftp_upload/18068/18068_Equation_1.svg" style="text-align: center; height: 34px;">

Lors de l'analyse de l'écoulement dans un tuyau, les variables qui définissent le système comprennent le diamètre du tuyau, la vitesse du fluide, sa densité et sa viscosité. Ceux-ci peuvent être combinés en un seul terme sans dimension, le nombre de Reynolds (Re), qui capture leur relation essentielle.

<img alt="Equation 2" src="/files/ftp_upload/18068/18068_Equation_2.svg" style="text-align: center; height: 38px;">


La première étape de l'application du théorème de Buckingham Pi consiste à identifier les dimensions fondamentales de chaque variable. En mécanique des fluides, il s'agit généralement de la masse, de la longueur et du temps. Ici, le diamètre représente la longueur, la vitesse combine la longueur et le temps, la densité englobe la masse et le volume, et la viscosité incorpore la masse, la longueur et le temps.

<img alt="Equation 3" src="/files/ftp_upload/18068/18068_Equation_3.svg" style="text-align: center; height: 140px;">


Le théorème suggère que le nombre de groupes indépendants sans dimension, appelés termes Pi, est la différence entre le nombre de variables k et le nombre de dimensions fondamentales r. Un seul terme sans dimension est nécessaire avec quatre variables et trois dimensions, ce qui simplifie l'analyse.

<img alt="Equation 4" src="/files/ftp_upload/18068/18068_Equation_4.svg" style="text-align: center; height: 32px;">

Ce terme unique, le nombre de Reynolds, compare les forces d'inertie aux forces visqueuses dans le fluide. L'établissement de ce rapport révèle si l'écoulement est susceptible d'être régulier et ordonné, appelé écoulement laminaire, ou chaotique et irrégulier, appelé écoulement turbulent. Des valeurs plus faibles indiquent un écoulement laminaire, où les forces visqueuses dominent, et le fluide se déplace en couches parallèles. Des valeurs plus élevées reflètent un écoulement turbulent, où les forces d'inertie conduisent à un mouvement chaotique. L'application du théorème de Buckingham Pi dans ce contexte réduit les interactions complexes à un seul terme, ce qui facilite la généralisation du comportement de l'écoulement dans des conditions variables.

Le théorème de Buckingham Pi

Considérons l'écoulement dans un tuyau, où les variables clés incluent le diamètre D du tuyau, la vitesse du fluide V, la densité ρ et la viscosité μ.
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Dimensional Analysis, Similitude, and Modeling

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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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Reynolds Number from Buckingham Pi

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