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

Si consideri il flusso attraverso un tubo, in cui le variabili chiave includono il diametro D del tubo, la velocità del fluido V, la densità ρ e la viscosità μ.
Per analizzare il sistema utilizzando il teorema di Buckingham Pi, queste variabili possono essere combinate in prodotti adimensionali come il numero di Reynolds, che caratterizza il comportamento del flusso.
In questo caso, il teorema di Buckingham Pi viene applicato identificando prima il numero di variabili coinvolte e le dimensioni di riferimento, che sono la massa M, la lunghezza L e il tempo T.
Secondo il teorema di Buckingham Pi, il sistema con quattro variabili e tre dimensioni di riferimento (M, L, T) risulta in un termine Pi greco adimensionale, che è il numero di Reynolds.
In questo caso, il numero di Reynolds è un prodotto adimensionale che mette in relazione le forze inerziali con le forze viscose nel fluido, aiutando a classificare il flusso come laminare o turbolento.
Il teorema di Buckingham Pi riformula il sistema esprimendo le variabili come termini adimensionali, rendendo più facile generalizzare e analizzare il flusso in condizioni variabili.

il teorema di buckingham

The Buckingham Pi Theorem

Il teorema di Buckingham Pi fornisce un metodo strutturato per semplificare i problemi di dinamica dei fluidi, riducendo sistemi complessi di variabili a termini adimensionali.

<img alt="Equation 1" src="/files/ftp_upload/18068/18068_Equation_1.svg" style="text-align: center; height: 34px;">

Nell'analisi del flusso attraverso un tubo, le variabili che definiscono il sistema, includono il diametro del tubo, la velocità del fluido, la sua densità e viscosità. Questi possono essere combinati in un singolo termine adimensionale, il numero di Reynolds (Re), che esprime la loro relazione essenziale.

<img alt="Equation 2" src="/files/ftp_upload/18068/18068_Equation_2.svg" style="text-align: center; height: 38px;">

Il primo passo nell'applicazione del teorema di Buckingham Pi è identificare le dimensioni fondamentali di ciascuna variabile. Nella meccanica dei fluidi, queste sono in genere massa, lunghezza e tempo. Qui, il diametro rappresenta la lunghezza, la velocità combina lunghezza e tempo, la densità comprende massa e volume e la viscosità incorpora massa, lunghezza e tempo.

<img alt="Equation 3" src="/files/ftp_upload/18068/18068_Equation_3.svg" style="text-align: center; height: 140px;">

Il teorema suggerisce che il numero di gruppi adimensionali indipendenti, noti come termini Pi, è la differenza tra il numero di variabili k e il numero di dimensioni fondamentali r. È necessario un solo termine adimensionale con quattro variabili e tre dimensioni, semplificando l'analisi.

<img alt="Equation 4" src="/files/ftp_upload/18068/18068_Equation_4.svg" style="text-align: center; height: 32px;">

Questo singolo termine, il numero di Reynolds, confronta le forze inerziali con le forze viscose nel fluido. Stabilire questo rapporto, rivela se è probabile che il flusso sia regolare e ordinato, noto come flusso laminare, o caotico e irregolare, noto come flusso turbolento. Valori inferiori indicano flusso laminare, dove dominano le forze viscose e il fluido si muove in strati paralleli. Valori più alti riflettono flusso turbolento, dove le forze inerziali portano a un moto caotico. L'applicazione del teorema di Buckingham Pi in questo contesto, riduce le interazioni complesse a un singolo termine, rendendo più facile generalizzare il comportamento del flusso in condizioni variabili.

Il Teorema di Buckingham

Si consideri il flusso attraverso un tubo, in cui le variabili chiave includono il diametro D del tubo, la velocità del fluido V, la densità ρ e la viscosità μ.
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KEY TERMS AND CONCEPTS

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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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