Name: pressure and speed in bernoulli's equation
Uploaded: 2025-02-13T15:43:14.000Z
Duration: 1 min 30 s
Channel: JoVE

伯努利方程通过沿流线的压力、速度和高程关系，描述了流体在运动中的行为。
该原理基于能量守恒，假设流体为稳定流动、不可压缩且无黏性，即不存在摩擦力。
压力项表示流体在特定点施加的力。
速度项代表流体的动能，表明流体速度增加时压力会降低。
高程项反映了流体相对于参考点的垂直位置所具有的势能。
这一原理适用于供水系统、大坝溢洪道和文丘里流量计等实际应用场景。
在输水系统中，伯努利方程可预测水流经不同管径管道时压力的下降情况，从而确保城市管网中的高效供水。 
伯努利方程解释了水在大坝溢洪道中向下流动时流速增加、压力降低的原理，有助于工程师设计能够安全应对不同流量的结构。
文丘里流量计通常用于污水处理系统，通过检测管道不同截面之间的压力差来测量流量，从而确保有效的废水管理。

伯努利流线流动方程

Bernoulli's Equation for Flow Along a Streamline

伯努利方程与沿流线流动的流体中的能量守恒有关。该方程适用于稳定流动的不可压缩和无粘性流体。对于这种流动，将牛顿第二定律用于小流体元素，该元素会受到压力差、重力和速度变化的影响。力平衡导致伯努利方程的以下形式： <img alt="Equation 1" src="/files/ftp_upload/18033/18033_Equation_1.svg" style="text-align: center; height: 26px;"> 其中，P 表示压力，ρ 表示流体密度，v 表示速度，g 表示重力加速度，高度用 h 表示。该方程中的每个项表示流体单位体积的能量。 考虑流过具有两个不同直径截面的水平管道的水。在点 1处，管道直径较大，而在点 2处，管道变窄。伯努利方程告诉我们，如果水流在较窄部分（点 2处）的速度增加，则压力必须降低以使能量沿流线守恒。将伯努利方程应用于管道截面可得出： <img alt="Equation 2" src="/files/ftp_upload/18033/18033_Equation_2.svg" style="text-align: center; height: 26px;"> 伯努利原理对于理解和设计流体流动系统至关重要。例如，在供水网络中，管道直径的变化会导致压力变化，从而影响流动效率。正如方程所示，较窄的管道会增加速度，同时降低压力，从而确保整个城市系统的供水稳定。在水坝泄洪道中，随着水下降，流体因海拔而具有的势能会转化为动能。这会导致速度增加，压力相应下降，这种现象用于设计能够安全管理不同排放率的泄洪道。

伯努利方程通过沿流线关联压力、速度和高程，描述流体在运动中的行为。  
它依赖于……

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

Education

教育

JoVE Core

Civil Engineering

土木工程

Unit 1

第一单元

Fluid Dynamics

流体力学

KEY TERMS AND CONCEPTS

关键术语与概念

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 elevation. Each term represents energy per unit volume of the fluid. This makes Bernoulli's equation a useful way to track how energy changes as fluid moves.

A simple pipe example shows the main effect. Water flows through a horizontal pipe with two sections. Point 1 has a large diameter, and point 2 has a narrower diameter. When the water speeds up at the narrow section, the pressure must drop to keep energy conserved along the streamline.

Bernoulli's principle also helps explain real flow systems. In water distribution networks, changes in pipe diameter create pressure changes that affect flow efficiency. A narrower pipe raises velocity and lowers pressure, which helps support a steady supply across urban systems. In dam spillways, water loses potential energy as it falls. That energy turns into kinetic energy, so velocity increases and pressure decreases. Engineers use this effect to design spillways that can handle changing discharge rates safely.

Pressure and Speed in Bernoulli's Equation

pressure-and-speed-in-bernoullis-equation

bernoulli's equation for flow along a streamline

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

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...

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 Bernoulli's equation links pressure, speed, and height in flowing fluids, with pipe and dam spillway examples for high school science.

Pressure and Speed in Bernoulli's Equation | 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 and Speed in Bernoulli's Equation

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