Name: velocity potential in pipe flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

Considera l'acqua che scorre costantemente attraverso un tubo lungo e dritto con una sezione trasversale uniforme.
Nelle regioni centrali del tubo, dove il flusso è irrotazionale, un potenziale di velocità definisce il flusso e la velocità è espressa come il gradiente di questo potenziale.
La sostituzione delle componenti di velocità verifica direttamente che il flusso soddisfi questa condizione di irrotazione.
Il potenziale di velocità si applica a flussi tridimensionali, come questo flusso di tubi, mentre la funzione di flusso è specifica per flussi bidimensionali, risultanti dalla conservazione della massa.
In un flusso incomprimibile, l'equazione di continuità richiede che la divergenza del vettore velocità sia pari a zero.
Sostituendo il potenziale di velocità, questa condizione diventa l'equazione di Laplace, un'equazione differenziale chiave che governa i flussi inviscidi, incomprimibili e irrotazionali.
Nelle coordinate cilindriche, l'equazione di Laplace si adatta alle variazioni radiali, angolari e assiali. La risoluzione di questa equazione con le condizioni al contorno sulle pareti del tubo definisce il potenziale di velocità per il flusso.
Da ciò, il campo di velocità completo viene determinato prendendo le derivate spaziali, rivelando il modello di flusso all'interno del tubo.

potenziale di velocità

Velocity Potential

In un flusso stazionario e incomprimibile, attraverso un tubo lungo e dritto con una sezione trasversale uniforme, il flusso nella regione centrale (lontano dalle pareti del tubo) è irrotazionale. Questa natura irrotazionale, indica che le particelle di fluido non ruotano attorno ai loro assi e può essere utilizzata una funzione scalare chiamata potenziale di velocità, rappresentata da ϕ, per descrivere il loro movimento. Nei flussi irrotazionali, il campo di velocità V è definito come il gradiente del potenziale di velocità:

<img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;">


Poiché il rotore di un gradiente è sempre zero, definire la velocità in termini di ϕ, assicura che il flusso non abbia vorticità, rendendolo irrotazionale.

Per i flussi incomprimibili, l'equazione di continuità, afferma che la divergenza del campo di velocità deve essere zero:

<img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;">


Sostituendo V in questa equazione, l'equazione diventa:

<img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;">


Questa è nota come equazione di Laplace. Nelle regioni in cui il flusso è irrotazionale e incomprimibile, il potenziale di velocità ϕ, deve soddisfare l'equazione di Laplace.

Nel caso del flusso in un tubo, le coordinate cilindriche (r, θ, z) sono una scelta naturale per descrivere il flusso, dove r è la distanza radiale dalla linea centrale del tubo, θ è la posizione angolare e z è la distanza assiale lungo il tubo. L'equazione di Laplace secondo le coordinate cilindriche, che tiene conto delle variazioni radiali, angolari e assiali, è espressa come:

<img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;">


Trovare la forma esatta di ϕ nel tubo, richiede condizioni al contorno. Ad esempio, alle pareti del tubo, la condizione di non slittamento, implicherebbe una velocità tangenziale pari a zero, mentre nella regione irrotazionale centrale, la funzione potenziale, seguirebbe la simmetria del tubo. Risolvendo l'equazione di Laplace con queste condizioni, si definisce il potenziale di velocità attraverso il campo di flusso.

Una volta determinato ϕ, le componenti di velocità secondo le coordinate cilindriche, possono essere calcolate prendendo derivate parziali di ϕ. Queste componenti sono:

Velocità radiale:

<img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;">


Velocità angolare

<img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;">


Velocità assiale

<img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;">

Queste espressioni per le componenti di velocità, confermano che il flusso soddisfa la condizione irrotazionale, poiché derivano da un gradiente.

Potenziale di velocità

Considera l'acqua che scorre costantemente attraverso un tubo lungo e dritto con una sezione trasversale uniforme.
Nelle regioni centrali del tubo, dove il flusso è irrotazionale, un ...

Education

Didattica

JoVE Core

Civil Engineering

Ingegneria Civile

Unit 1

Unità 1

Differential Analysis of Fluid Flow

Analisi differenziale del flusso di fluidi

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

velocity potential in pipe flow

Velocity potential in pipe flow describes the motion of fluid in the central region of a long, straight pipe with a uniform cross-section. In this region, the flow is steady, incompressible, and irrotational. Irrotational means the fluid particles do not spin around their own axes.

A scalar function called the velocity potential, written as φ, is used to describe this motion. The velocity field V is defined as the gradient of φ. Because the curl of a gradient is always zero, this form of V gives the flow no vorticity and keeps it irrotational.

For an incompressible fluid, the continuity equation requires the divergence of the velocity field to be zero. When V is written in terms of φ, this condition becomes Laplace's equation. In regions where the flow is both incompressible and irrotational, the velocity potential must satisfy this equation.

Pipe flow is often described with cylindrical coordinates: r, θ, and z. Here, r is the radial distance from the pipe centreline, θ is the angular position, and z is the distance along the pipe. In these coordinates, Laplace's equation includes radial, angular, and axial variation.

To find the exact form of φ in the pipe, boundary conditions are needed. At the pipe walls, the no-slip condition means the tangential velocity is zero. In the central irrotational region, the potential function must also match the symmetry of the pipe. Solving Laplace's equation with these conditions gives the velocity potential across the flow field.

Once φ is known, the velocity components can be found by taking partial derivatives. These components are the radial velocity, the angular velocity, and the axial velocity. Together, they show how the flow field comes from a gradient and therefore satisfies the irrotational condition.

Velocity Potential in Pipe Flow

euler&#39;s equations of motion

Pressure and Velocity in Inviscid Flow

Pressure and velocity in inviscid flow are linked by Euler's equations of motion. In fluid mechanics, viscosity creates shear stress, which is a force that acts along a surface. For low-viscosity fluids like water, these shear stresses are very small. That lets the fluid be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent. Only normal stresses remain, and these act perpendicular to a fluid element. Pressure is the negative of the normal stress. It stays the...

stream function

Stream Function for Incompressible Flow

The stream function helps describe two-dimensional incompressible flow. In this type of flow, the continuity equation keeps mass conserved. That means fluid entering a region must be balanced by fluid leaving it. For incompressible flow, the density stays constant. The continuity requirement then becomes a zero-divergence condition for the velocity field. In two dimensions, the horizontal and vertical velocity components are usually written as u and v. A stream function, written as ψ, is used...

irrotational flow

Irrotational Flow Near Boundaries

Irrotational flow is fluid motion in which particles do not spin around their own axes. It has zero vorticity, which means the curl of the velocity field is zero. That condition places limits on the velocity gradients in the flow. A simple example is uniform flow. In this case, the velocity u stays constant, while v and w are both zero. This satisfies the irrotational conditions and gives a basic model of non-rotating flow. When a solid object is added, irrotational flow usually remains only...

velocity potential

Velocity potential in pipe flow describes the motion of fluid in the central region of a long, straight pipe with a uniform cross-section. In this region, the flow is steady, incompressible, and irrotational. Irrotational means the fluid particles do not spin around their own axes. A scalar function called the velocity potential, written as φ, is used to describe this motion. The velocity field V is defined as the gradient of φ. Because the curl of a gradient is always zero, this form of V...

plane potential flows

Plane Potential Flow Patterns and Models

Plane potential flows describe fluid motion using simple flow patterns and models. They assume the fluid is irrotational and incompressible. These assumptions let the flow be written with a velocity potential function, φ, which gives the flow speed in a direction, and a stream function, ψ, which shows the flow path. Both are governed by Laplace's equation and are useful for estimating flow patterns, velocity distributions, and pressure fields around hydraulic structures. Uniform flow is the...

navier&#8211;stokes equations

Fluid Stress and Flow Equations

Fluid stress and flow equations describe how incompressible Newtonian fluids move under pressure, shear, and gravity. In these fluids, density stays constant, and stress changes linearly with the rate of deformation. The main stress types are normal stress and shear stress. Normal stress depends on pressure acting on the fluid and on deformation rate in specific directions. Shear stress acts along the surface of a fluid layer. It shows how neighboring layers slide past one another, so velocity...

steady, laminar flow between parallel plates

steady-laminar-flow-between-parallel-plates

Laminar Flow Profile Between Parallel Plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings. The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem...

couette flow

Linear Shear Between Moving Plates

Linear shear between moving plates describes fluid motion in a simple channel with two parallel plates. One plate stays fixed, and the other moves at a constant speed. This setup helps show how viscosity and incompressible flow shape fluid motion under the Navier-Stokes equations. The model uses a steady, laminar, incompressible flow with no pressure gradient in the flow direction. The x-axis points along the moving direction of the upper plate, and the y-axis is perpendicular to both plates.

steady, laminar flow in circular tubes

Parabolic Flow in Circular Tubes

Parabolic flow in circular tubes describes the steady, incompressible motion of a viscous fluid through a tube with a constant radius. This Hagen-Poiseuille flow is a classic example of laminar flow. It is often used to study fluid transport in narrow channels, including capillaries. Cylindrical coordinates are used to describe the tube. The radial direction is r, the angular direction is θ, and the axial direction is z. In this flow, the velocity is purely axial. That means the fluid moves...

design example: flow of oil through circular pipes

design-example-flow-of-oil-through-circular-pipes

Pressure Drop in Oil Pipe Design

Pressure drop in oil pipe design depends on pipe size, oil properties, and the flow rate needed through a circular pipe. For steady, incompressible, laminar flow, Hagen-Poiseuille's law gives an exact solution from the Navier-Stokes equations. It is used to find the pressure difference needed to keep oil moving through a pipeline section. The pressure drop equation uses the oil's dynamic viscosity, the pipeline length, the desired volumetric flow rate, and the pipe's inner radius. This...

key terms and concepts

Explore how velocity potential describes irrotational, incompressible flow in a straight pipe, and see how Laplace's equation and cylindrical coordinates apply.

Velocity Potential in Pipe Flow | 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

Velocity Potential in Pipe Flow

Description

Transcript

Tags