Name: stream function for incompressible flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

Nel flusso incomprimibile bidimensionale, ci sono componenti di velocità orizzontale e verticale.
L'equazione di continuità garantisce la conservazione della massa richiedendo che la somma dei tassi di variazione di questi componenti sia zero, il che significa che l'afflusso e il deflusso netti in qualsiasi regione sono bilanciati.
Per soddisfare questa condizione, viene introdotta la funzione stream.
La velocità orizzontale è la derivata parziale della funzione di flusso rispetto alla direzione verticale, mentre la velocità verticale è la derivata parziale negativa rispetto alla direzione orizzontale. Ciò garantisce che la conservazione di massa venga mantenuta automaticamente.
Una caratteristica chiave della funzione di flusso è che rimane costante lungo le linee di flusso, che rappresentano i percorsi delle particelle fluide e sono tangenti alle velocità nel campo di flusso.
Le linee di flusso, spesso rappresentate come linee di contorno, consentono un'interpretazione intuitiva del comportamento dei fluidi.
La funzione di flusso semplifica i calcoli del movimento del fluido. La differenza nei valori della funzione di flusso tra due linee di flusso rappresenta la portata volumetrica tra di esse.
Ciò elimina la necessità di risolvere separatamente i componenti della velocità, rendendolo uno strumento efficiente per l'analisi del flusso bidimensionale.

funzione di corrente

Stream Function

Nel flusso di un fluido incomprimibile bidimensionale, l'equazione di continuità, è essenziale per garantire la conservazione della massa, il che significa che qualsiasi cambiamento nel fluido, che entra o esce da una regione, è bilanciato da un corrispondente cambiamento altrove. Per il flusso incomprimibile, dove la densità rimane costante, questo requisito si semplifica nella condizione che la divergenza del campo di velocità, debba essere pari a zero. Matematicamente, questo viene espresso come,

<img alt="Equation 1" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;">

dove u e v, rappresentano rispettivamente le componenti di velocità orizzontale e verticale.

Per soddisfare intrinsecamente questa equazione, introduciamo la funzione di corrente ψ, che consente di definire le componenti di velocità in termini di ψ come:

<img alt="Equation 2" src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;">

Queste definizioni, soddisfano automaticamente l'equazione di continuità, poiché le derivate parziali miste di ψ, si annullano.

La funzione di corrente ψ, è costante lungo le linee di corrente, che sono i percorsi che le particelle di fluido seguono all'interno del flusso. Le linee di corrente, possono quindi essere visualizzate come linee di contorno di ψ, fornendo una chiara rappresentazione del movimento del fluido. Ogni linea di corrente è tangente al vettore velocità, in qualsiasi punto lungo il suo percorso, illustrando la direzione del flusso, senza la necessità di calcolare le velocità in ogni punto individualmente.

Inoltre, la differenza nei valori ψ\psiψ tra due linee di corrente, rappresenta la portata volumetrica per unità di profondità tra di esse, consentendo il calcolo diretto delle portate. Questa proprietà, semplifica notevolmente l'analisi dei flussi incomprimibili bidimensionali, eliminando la necessità di risolvere equazioni separate per u e v. Di conseguenza, la funzione di corrente ψ, è un potente strumento per modellare e interpretare il comportamento del fluido, in applicazioni che coinvolgono flussi incomprimibili e costanti.

Funzione di Corrente

Nel flusso incomprimibile bidimensionale, ci sono componenti di velocità orizzontale e verticale.
L'equazione di continuità garantisce la conservazione della massa ...

Education

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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

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 to satisfy this equation automatically. The velocity components can be defined in terms of ψ. When they are written this way, the mixed partial derivatives of ψ cancel, so the continuity equation is already met.

The stream function is also linked to streamlines. A streamline is the path a fluid particle follows in the flow, and ψ stays constant along each one. Because of this, streamlines can be viewed as contour lines of ψ. Each streamline is tangent to the velocity vector at every point, so it shows flow direction without finding the velocity at every location.

Differences in ψ between two streamlines also have meaning. They represent the volumetric flow rate per unit depth between those lines. This makes it easier to analyze steady, two-dimensional incompressible flows. In many cases, the stream function reduces the need to solve separate equations for u and v, so it is a useful tool for modeling and interpreting fluid motion.

Stream Function for Incompressible 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

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

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

Learn how the stream function describes two-dimensional incompressible flow, keeps continuity satisfied, and shows streamlines and flow rate in fluid motion.

Stream Function for Incompressible 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

Stream Function for Incompressible Flow

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