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

Уравнение Бернулли для потока перпендикулярно линии тока объясняет, как давление изменяется по кривым линиям тока из-за внешних центробежных сил.
Давление выше на внутренней стороне кривой, вблизи центра кривизны, и уменьшается наружу, чтобы уравновесить эти центробежные силы.
Разница давлений зависит от скорости жидкости и остроты кривой.
Перепад давления невелик для почти прямых линий тока, но становится более выраженным в резко изогнутых потоках, таких как вихри или изгибы труб.
На практике эта концепция помогает при проектировании гидротехнических сооружений, изогнутых каналов и трубопроводных систем, где неправильный расчет давления может привести к разрушению конструкции.
Это также относится к ветровым потокам вокруг гражданских сооружений, таких как здания и мосты, где для обеспечения устойчивости необходимо учитывать перепады давления, вызванные изогнутыми линиями тока.
При проектировании изогнутых плотин и водохранилищ, где вода течет по изогнутым поверхностям, перепады давления на линиях тока гарантируют, что конструкция может выдерживать силы, оказываемые водой, особенно при высокоскоростных потоках.

Уравнение Бернулли для потока, нормального к линии тока

Уравнение Бернулли для потока, перпендикулярного линии тока, объясняет, как давление изменяется поперек изогнутых линий тока из-за внешних центробежных сил, вызванных кривизной жидкости. Давление выше на внутренней стороне кривой, вближе к центру кривизны, и уменьшается по направлению наружу, чтобы уравновесить эти центробежные силы.

 Разница давления зависит от скорости жидкости и радиуса кривизны. Изменение давления минимально в потоках с почти прямыми линиями тока. Однако, разница давления становится значительной в резко изогнутых потоках, таких как вихри, изгибы труб, или вокруг острых структур. Этот эффект выражается как:

<img alt="Equation 1" src="/files/ftp_upload/18034/18034_Equation_1.svg" style="text-align: center; height: 30px;">


 Интегрируя это соотношение и предполагая, что поток является устойчивым, невязким и несжимаемым, выводится уравнение Бернулли, перпендикулярное линии тока:

<img alt="Equation 2" src="/files/ftp_upload/18034/18034_Equation_2.svg" style="text-align: center; height: 30px;">

 Гидравлические сооружения, такие как водосбросы и изогнутые каналы, полагаются на точные прогнозы давления для обеспечения структурной целостности при высокоскоростных потоках. Аналогично, расчеты давления используются при проектировании систем труб с изгибами, чтобы предотвратить разрушение из-за избыточных нагрузок. Эта концепция также применима к ветровым потокам вокруг гражданских сооружений, таких как здания и мосты, где необходимо учитывать перепады давления, вызванные изогнутыми линиями тока, чтобы избежать нестабильности или повреждений при сильном ветре.

Уравнение Бернулли для потока перпендикулярно линии тока объясняет, как давление изменяется по кривым линиям тока из-за внешних центробежных сил.
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bernoulli-s-equation-for-flow-normal-to-a-streamline

Education

JoVE Core

Civil Engineering

Unit 1

Раздел 1

Fluid Dynamics

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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.

Pressure Changes Across Curved Streamlines

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