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

Dans l’écoulement incompressible bidimensionnel, il existe des composantes de vitesse horizontales et verticales.
L’équation de continuité assure la conservation de la masse en exigeant que la somme des taux de variation de ces composantes soit nulle, ce qui signifie que les entrées et les sorties nettes dans n’importe quelle région sont équilibrées.
Pour satisfaire à cette condition, la fonction stream est introduite.
Le vecteur vitesse horizontale est la dérivée partielle de la fonction d’écoulement par rapport à la direction verticale, tandis que le vecteur vitesse vertical est la dérivée partielle négative par rapport à la direction horizontale. Cela garantit que la conservation de masse est automatiquement maintenue.
Une caractéristique clé de la fonction d’écoulement est qu’elle reste constante le long des lignes de courant, qui représentent les trajectoires des particules de fluide et sont tangentes aux vitesses dans le champ d’écoulement.
Les lignes de courant, souvent représentées sous forme de courbes de niveau, permettent une interprétation intuitive du comportement des fluides.
La fonction stream simplifie les calculs de mouvement des fluides. La différence de valeurs de fonction de flux entre deux lignes de courant représente le débit volumique entre elles.
Il n’est donc plus nécessaire de résoudre séparément les composantes de vitesse, ce qui en fait un outil efficace pour l’analyse de l’écoulement bidimensionnel.

fonction de flux

Stream Function

Dans un écoulement de fluide incompressible bidimensionnel, l'équation de continuité est essentielle pour assurer la conservation de la masse, ce qui signifie que tout changement dans le fluide entrant ou sortant d'une région est équilibré par un changement correspondant ailleurs. Pour un écoulement incompressible, où la densité reste constante, cette exigence se simplifie en la condition selon laquelle la divergence du champ de vitesse doit être nulle. Mathématiquement, cela s'exprime comme suit:

<img alt="Equation 1" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;">

où u et v représentent respectivement les composantes de vitesse horizontale et verticale.

Pour satisfaire intrinsèquement à cette équation, nous introduisons la fonction de flux ψ, qui permet de définir les composantes de vitesse en termes de ψ comme:

<img alt="Equation 2" src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;">

Ces définitions satisfont automatiquement l'équation de continuité, car les dérivées partielles mixtes de ψ s'annulent.

La fonction de flux ψ est constante le long des lignes de courant, qui sont les chemins que suivent les particules de fluide dans l'écoulement. Les lignes de courant peuvent donc être visualisées sous forme de courbes de niveau de ψ, offrant une représentation claire du mouvement du fluide. Chaque ligne de courant est tangente au vecteur de vitesse en tout point de son trajet, illustrant la direction de l'écoulement sans qu'il soit nécessaire de calculer les vitesses à chaque point individuellement.

De plus, la différence de valeurs ψ\psiψ entre deux lignes de courant représente le débit volumétrique par unité de profondeur entre elles, ce qui permet un calcul direct des débits. Cette propriété simplifie considérablement l'analyse des écoulements incompressibles bidimensionnels en supprimant la nécessité de résoudre des équations séparées pour u et v. Par conséquent, la fonction d'écoulement ψ est un outil puissant pour la modélisation et l'interprétation du comportement des fluides dans les applications impliquant un écoulement incompressible et stable.

Fonction de flux

Dans l’écoulement incompressible bidimensionnel, il existe des composantes de vitesse horizontales et verticales.
L’équation de continuité assure la conservation de la masse en ...

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Civil Engineering

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Differential Analysis of Fluid Flow

Analyse différentielle de l'écoulement des fluides

KEY TERMS AND CONCEPTS

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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

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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...

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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...

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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...

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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...

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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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