Name: irrotational flow near boundaries
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 28 s
Channel: JoVE

L’écoulement irrotationnel fait référence au mouvement des particules de fluide qui ne tournent pas le long du chemin d’écoulement, caractérisé par un champ de vitesse sans courbure et sans vorticité.
Un écoulement uniforme avec une seule composante non nulle de la vitesse, une constante u = U, est un exemple d’écoulement irrotationnel car il n’a pas de gradients de vitesse.
Lorsqu’un objet solide est introduit dans un écoulement uniforme, l’écoulement reste irrotationnel, sauf près des limites. Près de sa surface, la vitesse augmente brusquement à partir de zéro, créant des pentes abruptes et des contraintes de cisaillement importantes.
Dans le cas d’un écoulement véritablement inviscide, l’écoulement glisserait au-delà de la limite, préservant ainsi le mouvement irrotationnel. Cependant, dans les fluides réels, la viscosité forme une couche limite près des surfaces fixes, produisant des effets de rotation et provoquant parfois une séparation en aval.
Dans l’écoulement du tuyau, la région d’entrée présente initialement un noyau irrotationnel avec un écoulement uniforme, mais à mesure qu’une couche limite se forme le long de la paroi du tuyau et s’épaissit, elle se transforme en une région pleinement développée dominée par des forces visqueuses.
En supposant que l'écoulement est irrotationnel, incompressible, stable et inviscide, l'équation de Bernoulli peut être appliquée dans le champ d'écoulement.

Écoulement irrotationnel

Irrotational Flow

L'écoulement irrotationnel est caractérisé par un mouvement fluide dans lequel les particules ne tournent pas autour de leurs axes, ce qui entraîne une vorticité nulle. Pour que l' écoulement soit irrotationnel, la courbure du champ de vitesse doit être nulle. Cela impose des conditions spécifiques sur les gradients de vitesse. Par exemple, pour maintenir une rotation nulle autour de l'axe z, la condition de gradient:

<img alt="Equation 1" src="/files/ftp_upload/18059/18059_Equation_1.svg" style="text-align: center; height: 27px;">


doit être satisfaite, ainsi que des conditions similaires pour les autres axes:

<img alt="Equation 2" src="/files/ftp_upload/18059/18059_Equation_2.svg" style="text-align: center; height: 27px;">


Un écoulement uniforme, où u est constant et v et w sont tous deux nuls, satisfait ces conditions et sert d'exemple simple d'écoulement irrotationnel. Cependant, lorsqu'un objet solide est introduit, l'irrotationnalité ne persiste que loin de l'objet. Près de la limite, la vitesse augmente brusquement de zéro (en raison de la condition d'absence de glissement) à la valeur de l'écoulement libre, créant une contrainte de cisaillement importante et une couche limite dans les fluides visqueux.

Lorsque le fluide pénètre dans un tuyau à partir d’un grand réservoir, l’écoulement initial est presque uniforme, ce qui entraîne un noyau irrotationnel où la distribution de vitesse est plate. Au fur et à mesure que le fluide se déplace vers l’aval, une couche limite se forme le long des parois du tuyau en raison de la viscosité, ce qui entraîne le développement de gradients de profil de vitesse qui conduisent finalement à un profil parabolique entièrement développé. Dans cette région entièrement développée, les effets rotationnels dominent près des parois, tandis qu’un noyau central peut rester approximativement irrotationnel.

Dans la région irrotationnelle initiale de l’écoulement dans le tuyau, l’équation de Bernoulli

<img alt="Equation 3" src="/files/ftp_upload/18059/18059_Equation_3.svg" style="text-align: center; height: 27px;">

peut être appliquée entre des points le long d’une ligne de courant lorsque les conditions d’incompressibilité, d’écoulement non visqueux et d’irrotationnalité sont respectées. Cependant, à mesure que la couche limite augmente et que les effets rotationnels deviennent importants près des parois, les hypothèses de l’équation de Bernoulli s’effondrent. Cela signifie que l’équation de Bernoulli n’est applicable que dans la région d’entrée et le noyau irrotationnel, mais pas dans la région rotationnelle entièrement développée où les effets de viscosité dominent.

L’écoulement irrotationnel fait référence au mouvement des particules de fluide qui ne tournent pas le long du chemin d’écoulement, caractérisé par un champ de vitesse avec une courbure nulle et ...

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Civil Engineering

Génie civil

Unit 1

Unité 1

Differential Analysis of Fluid Flow

Analyse différentielle de l'écoulement des fluides

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

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 far from the object. Near the surface, the fluid speed rises sharply from zero because of the no-slip condition. This creates strong shear stress and a boundary layer in viscous fluids.

A similar change happens when fluid enters a pipe from a large reservoir. At first, the flow is nearly uniform, so the center region forms an irrotational core with a flat velocity profile. Farther downstream, viscosity causes boundary layers to grow along the pipe walls.

As the flow continues, the velocity profile develops more gradient and becomes fully developed and parabolic. In that region, rotational effects are strongest near the walls, while the center may still be approximately irrotational. Bernoulli's equation can apply in the entrance region and in the irrotational core, where the flow is incompressible, inviscid, and irrotational. It no longer applies well in the fully developed region, where viscosity and rotation dominate.

Irrotational Flow Near Boundaries

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...

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irrotational flow

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...

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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...

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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...

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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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Laminar Flow Profile Between Parallel Plates

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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 irrotational flow behaves near solids and in pipe entrance flow, including boundary layers, rotational effects, and when Bernoulli's equation applies.

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civil engineering

Irrotational Flow Near Boundaries

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