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

Considere el flujo a través de una tubería, donde las variables clave incluyen el diámetro de la tubería D, la velocidad del fluido V, la densidad ρ y la viscosidad μ.
Para analizar el sistema utilizando el teorema Pi de Buckingham, estas variables se pueden combinar en productos adimensionales como el número de Reynolds, que caracteriza el comportamiento del flujo.
En este caso, el teorema de Buckingham Pi se aplica identificando primero el número de variables involucradas y las dimensiones de referencia, que son la masa M, la longitud L y el tiempo T.
Según el teorema Pi de Buckingham, el sistema con cuatro variables y tres dimensiones de referencia (M, L, T) da como resultado un término Pi adimensional, que es el número de Reynolds.
Aquí, el número de Reynolds es un producto adimensional que relaciona las fuerzas de inercia con las fuerzas viscosas en el fluido, lo que ayuda a clasificar el flujo como laminar o turbulento.
El teorema de Buckingham Pi reformula el sistema expresando las variables como términos adimensionales, lo que facilita la generalización y el análisis del flujo en condiciones variables.

el teorema pi de buckingham

The Buckingham Pi Theorem

El teorema Pi de Buckingham proporciona un método estructurado para simplificar los problemas de dinámica de fluidos al reducir sistemas complejos de variables a términos adimensionales. <img alt="Equation 1" src="/files/ftp_upload/18068/18068_Equation_1.svg" style="text-align: center; height: 34px;"> Al analizar el flujo a través de una tubería, las variables que definen el sistema incluyen el diámetro de la tubería, la velocidad del fluido, su densidad y viscosidad. Estas pueden combinarse en un único término adimensional, el número de Reynolds (Re), que captura su relación esencial. <img alt="Equation 2" src="/files/ftp_upload/18068/18068_Equation_2.svg" style="text-align: center; height: 38px;"> El primer paso para aplicar el teorema Pi de Buckingham es identificar las dimensiones fundamentales de cada variable. En mecánica de fluidos, estas suelen ser masa, longitud y tiempo. Aquí, el diámetro representa la longitud, la velocidad combina longitud y tiempo, la densidad abarca masa y volumen, y la viscosidad incorpora masa, longitud y tiempo. <img alt="Equation 3" src="/files/ftp_upload/18068/18068_Equation_3.svg" style="text-align: center; height: 140px;"> El teorema sugiere que la cantidad de grupos adimensionales independientes, conocidos como términos Pi, es la diferencia entre la cantidad de variables k y la cantidad de dimensiones fundamentales r. Solo se necesita un término adimensional con cuatro variables y tres dimensiones, lo que simplifica el análisis. <img alt="Equation 4" src="/files/ftp_upload/18068/18068_Equation_4.svg" style="text-align: center; height: 32px;"> Este único término, el número de Reynolds, compara las fuerzas inerciales con las fuerzas viscosas en el fluido. Establecer esta relación revela si es probable que el flujo sea suave y ordenado, conocido como flujo laminar, o caótico e irregular, conocido como flujo turbulento. Los valores más bajos indican flujo laminar, donde predominan las fuerzas viscosas y el fluido se mueve en capas paralelas. Los valores más altos reflejan flujo turbulento, donde las fuerzas inerciales conducen a un movimiento caótico. La aplicación del teorema Pi de Buckingham en este contexto reduce las interacciones complejas a un solo término, lo que facilita la generalización del comportamiento del flujo en condiciones variables.

El teorema Pi de Buckingham

Considere el flujo a través de una tubería, donde las variables clave incluyen el diámetro de la tubería D, la velocidad del fluido V, la densidad ρ y la viscosidad μ.
Para...

Education

Educación

JoVE Core

Civil Engineering

Ingeniería Civil

Unit 1

Unidad 1

Dimensional Analysis, Similitude, and Modeling

Análisis dimensional, similitud y modelado

dimensional-analysis-similitude-and-modeling

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

dimensional analysis

Reynolds Number and Pipe Flow

Dimensional analysis helps simplify pipe flow problems in fluid mechanics by turning several variables into dimensionless groups. These groups show the key relationships in the system without tracking every variable one by one. They also reduce the number of independent variables, which makes the problem easier to study and understand. A common example is fluid flow through a long, smooth-walled, horizontal circular pipe. This kind of pipe appears in water distribution systems and sewage...

the buckingham pi theorem

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,...

determination of pi terms

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...

dimensionless groups in fluid mechanics

Key Fluid Flow Ratios for Modeling

Dimensionless groups in fluid mechanics are key ratios that make fluid behavior easier to compare without using specific units. These numbers help simplify analysis across different flow situations. The Reynolds number, or Re, compares inertial forces to viscous forces. It helps show whether a flow is laminar or turbulent and is important in pipeline and aerodynamic surface design. Other dimensionless numbers describe how fluid flow responds to gravity, pressure, elasticity, and speed. The...

correlation of experimental data

Using Experimental Data in Dimensional Analysis

Dimensional analysis helps organize experimental data and test physical relationships. It can simplify a complex problem and point to the key dimensionless groups that matter. But it does not give a full solution by itself. Experimental data is needed to find the exact relationship and to check theoretical predictions. In the example of a spherical particle moving through a viscous fluid, dimensional analysis shows that drag force depends on the particle’s diameter, velocity, and the fluid’s...

modeling and similitude

Scaled Models for Testing Dams and Ships

Scaled models help engineers study large systems in a smaller, controlled setup. They are useful when a full-size test would cost too much or take too long. In civil infrastructure and hydrology, a scaled dam model can show how water flow, turbulence, and pressure will behave in the real structure. For a model to give useful results, it must keep similitude. Similitude means the model matches the prototype, or full-size system, in important ways. Geometric similarity keeps the shape and...

typical model studies

River and Spillway Model Design

River and spillway model design uses scaled systems to predict how fluid will behave in full-scale settings. These model studies help engineers study river flow, dam spillways, and structures that interact with open surfaces. For river models, Froude number similarity is important. The Froude number compares inertial and gravitational forces, and matching it helps copy surface flow features such as wave patterns and velocity. Practical limits can cause geometric distortion, which means the...

design example: creating a hydraulic model of a dam spillway

design-example-creating-a-hydraulic-model-of-a-dam-spillway

Testing Spillway Flow with a Scale Model

A scale model of a dam spillway is used to study how water moves over the structure. These hydraulic models are often built at a 1:15 ratio. They help show flow speed, flow patterns, and energy dissipation, which is the loss of water energy as flow moves through the spillway. Researchers use the model to watch how water behaves in a controlled setting. They can examine velocity distribution, wave formation, turbulence, and flow separation. These details help them see how the spillway performs...

key terms and concepts

Learn how the Buckingham Pi theorem reduces pipe flow variables to the Reynolds number and helps distinguish laminar flow from turbulent flow.

Reynolds Number from Buckingham Pi | JoVE K12

JoVE Core Civil Engineering provides a foundational exploration of essential construction materials in civil engineering, focusing on bricks, stones, steel, wood, and concrete through easy-to-understand animated videos. This collection covers the properties, manufacturing processes, and practical applications of these materials. Additionally, the accompanied science-in-action videos demonstrate the research experiments performed in laboratories.

civil engineering

Reynolds Number from Buckingham Pi

Description

Transcript

Tags