Name: pressure changes across curved streamlines
Uploaded: 2025-02-13T15:43:14.000Z
Duration: 1 min 16 s
Channel: JoVE

L'équation de Bernoulli pour l'écoulement normal à un courant explique comment la pression varie le long des lignes de courant courbes en raison des forces centrifuges vers l'extérieur.
La pression est plus élevée sur le côté intérieur de la courbe, près du centre de courbure, et diminue vers l’extérieur pour équilibrer ces forces centrifuges.
La différence de pression dépend de la vitesse du fluide et de la netteté de la courbe.
La différence de pression est faible pour les lignes de courant presque droites, mais devient plus prononcée dans les écoulements fortement incurvés, tels que les tourbillons ou les coudes de tuyaux.
Dans les applications pratiques, ce concept aide à concevoir des structures hydrauliques, des canaux courbes et des systèmes de tuyauterie, où des calculs de pression incorrects peuvent entraîner une défaillance structurelle.
Il s’applique également aux flux de vent autour des structures civiles comme les bâtiments et les ponts, où les différences de pression causées par les lignes de courant courbes doivent être prises en compte pour assurer la stabilité.
Dans la conception de barrages et de réservoirs incurvés, où l’eau s’écoule le long de surfaces courbes, les différences de pression à travers les lignes de courant garantissent que la structure peut résister aux forces exercées par l’eau, en particulier dans les écoulements à grande vitesse.

Équation de bernoulli pour un écoulement normal à une ligne de courant

L'équation de Bernoulli pour un écoulement normal à une ligne de courant explique comment la pression varie le long des lignes de courant courbes en raison des forces centrifuges vers l'extérieur induites par la courbure du fluide. La pression est plus élevée sur le côté intérieur de la courbe, près du centre de courbure, et diminue vers l'extérieur pour équilibrer ces forces centrifuges.

La différence de pression dépend de la vitesse du fluide et du rayon de courbure. La variation de pression est minimale dans les écoulements avec des lignes de courant presque droites. Cependant, la différence de pression devient significative dans les écoulements fortement courbés, tels que les tourbillons, les coudes de tuyaux ou autour de structures pointues. Cet effet est exprimé comme suit:

<img alt="Equation 1" src="/files/ftp_upload/18034/18034_Equation_1.svg" style="text-align: center; height: 30px;">

En intégrant cette relation et en supposant que l'écoulement est stable, non visqueux et incompressible, l'équation de Bernoulli normale à la ligne de courant est dérivée:

<img alt="Equation 2" src="/files/ftp_upload/18034/18034_Equation_2.svg" style="text-align: center; height: 30px;">

Les structures hydrauliques telles que les déversoirs et les canaux courbes s'appuient sur des prévisions de pression précises pour garantir l'intégrité structurelle sous des écoulements à grande vitesse. De même, les calculs de pression guident la conception des systèmes de canalisations avec des coudes, pour éviter les défaillances dues à des forces excessives. Le concept s'applique également aux écoulements éoliens autour des structures civiles telles que les bâtiments et les ponts, où les différences de pression causées par les lignes de courant courbes doivent être prises en compte pour éviter l'instabilité ou les dommages lors de vents violents.

Équation de Bernoulli pour un écoulement normal à une ligne de courant

L'équation de Bernoulli pour l'écoulement normal à un courant explique comment la pression varie le long des lignes de courant courbes en raison des forces centrifuges vers l'extérieur.
...

bernoulli-s-equation-for-flow-normal-to-a-streamline

Education

Enseignement

JoVE Core

Civil Engineering

Génie civil

Unit 1

Unité 1

Fluid Dynamics

Dynamique des fluides

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Bernoulli's Equation for Flow Normal to a Streamline

pressure changes across curved streamlines

Bernoulli's equation for flow normal to a streamline describes how pressure changes across curved streamlines. When a fluid follows a curved path, the outer side of the curve experiences centrifugal force, which pushes the fluid outward. The pressure is therefore higher on the inner side, near the center of curvature, and lower farther out.

The size of this pressure difference depends on the fluid's velocity and the radius of curvature. In flows with nearly straight streamlines, the pressure variation is very small. In sharply curved flows, the difference becomes much larger. Vortices, pipe bends, and flow around sharp structures are common examples.

This pressure relationship can be written as an equation and then integrated. Under the assumptions of steady, inviscid, and incompressible flow, it leads to Bernoulli's equation normal to the streamline. The result helps predict pressure in systems where flow bends strongly.

Engineers use these pressure calculations in many settings. They help design spillways and curved channels so high-velocity water does not damage hydraulic structures. They also guide pipe system design at bends to reduce failure from large forces. The same ideas apply to wind flow around buildings and bridges, where curved streamlines can create pressure differences that affect stability.

Pressure Changes Across Curved Streamlines

pressure-changes-across-curved-streamlines

bernoulli's equation for flow along a streamline

bernoulli-s-equation-for-flow-along-a-streamline

Pressure and Speed in Bernoulli's Equation

Bernoulli's equation links pressure, speed, and height in a moving fluid along a streamline. It is used for incompressible, inviscid fluids under steady flow. The idea comes from applying Newton's second law to a small fluid element. That element feels forces from pressure differences, gravity, and changes in velocity. The resulting equation shows how fluid energy is shared. In the equation, P is pressure, ρ is fluid density, v is velocity, g is the acceleration due to gravity, and h is...

bernoulli's equation for flow normal to a streamline

Bernoulli's equation for flow normal to a streamline describes how pressure changes across curved streamlines. When a fluid follows a curved path, the outer side of the curve experiences centrifugal force, which pushes the fluid outward. The pressure is therefore higher on the inner side, near the center of curvature, and lower farther out. The size of this pressure difference depends on the fluid's velocity and the radius of curvature. In flows with nearly straight streamlines, the pressure...

bernoulli's equation: problem solving

Venturi Meter Flow Rate from Pressure

A Venturi meter measures fluid flow rate in a pipeline by using pressure changes and Bernoulli's equation. In a sewage system, it can be used to determine wastewater flow rate by reading the pressure difference across the meter. The narrowed throat makes the fluid speed up, and that change in speed is linked to pressure drop. The first step is to find the cross-sectional areas of the pipe and the Venturi throat. These areas are needed to interpret the pressure difference shown on the pressure...

static, stagnation, dynamic and total pressure

static-stagnation-dynamic-and-total-pressure

Pitot-Static Measurement in Bernoulli Flow

Static pressure, dynamic pressure, stagnation pressure, and total pressure are key ideas in fluid dynamics. They are often explained with Bernoulli's equation. In that equation, p is static pressure, 1/2ρV^2 is dynamic pressure, γz is hydrostatic pressure or potential energy pressure, and pT is total pressure. For incompressible flow, the total pressure stays constant along a streamline. Static pressure is the pressure a fluid exerts when it is at rest, or when flowing fluid is brought to rest...

free jet

Liquid Speed from Reservoir Outlets

Free jets show how liquid moves after leaving a reservoir through an opening and flowing into the air without resistance. The jet speed is found with Bernoulli’s principle, and it depends on the height of liquid above the opening. This relationship assumes atmospheric pressure at both the reservoir surface and the jet exit. The liquid column’s height controls the speed at the nozzle because the fluid’s potential energy changes into kinetic energy during flow. As the liquid leaves the...

continuity equation

Fluid Flow and Area Changes

Fluid flow and area changes are linked by the continuity equation. For a steady flow of an incompressible fluid in a confined system, the mass flow rate stays constant. This idea is useful in nozzles, syringes, and pipes where the cross-sectional area changes. The mass flow rate is written as an equation that connects density, area, and velocity. For incompressible flow, the density stays the same, so the equation becomes simpler. When the area gets smaller, the fluid speed must increase. When...

energy line and hydraulic gradient line

Fluid Head Profiles in Pipes and Channels

Fluid head profiles in pipes and channels show how pressure, speed, and height change in a flowing fluid. These profiles come from Bernoulli's equation for steady, incompressible, inviscid flow. In that equation, total head includes pressure head, velocity head, and elevation head. The energy line (EL) shows the total head available in the flow. It sits above the datum and includes all three energy parts. Under ideal conditions, the EL stays horizontal because the total energy does not change...

design example: designing water slide

Water Slide Speed and Bernoulli's Principle

Water slide speed is controlled by Bernoulli's principle and the height of the slide. Engineers use these ideas to keep the ride exciting while making sure the water stays safe for riders. As water moves downhill, gravity makes it speed up. The water at the bottom depends on the height it starts from. A taller slide gives the water more potential energy at the top. That energy changes into kinetic energy as the water descends, so the flow becomes faster. Bernoulli's principle helps describe...

key terms and concepts

Learn how curved streamlines change pressure in fluids, from pipe bends and vortices to spillways, buildings, and bridges in high-velocity flow.

Pressure Changes Across Curved Streamlines | 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

Pressure Changes Across Curved Streamlines

Description

Transcript

Tags