Name: velocity potential in pipe flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

단면이 균일한 길고 곧은 파이프를 통해 물이 꾸준히 흐르는 것을 고려하십시오.
흐름이 회전하지 않는 파이프의 중앙 영역에서는 속도 전위가 흐름을 정의하고 속도는 이 전위의 기울기로 표현됩니다.
속도 구성 요소를 직접 대체하면 흐름이 이 비회전 조건을 충족하는지 확인할 수 있습니다.
속도 전위는 이 파이프 흐름과 같은 3차원 흐름에 적용되는 반면, 흐름 함수는 질량 보존으로 인한 2차원 흐름에만 적용됩니다.
비압축성 흐름에서 연속성 방정식은 속도 벡터의 발산이 0이어야 합니다.
속도 전위를 대체함으로써 이 조건은 비점성 흐름, 비압축 흐름 및 비회전 흐름을 제어하는 핵심 미분 방정식인 라플라스 방정식이 됩니다.
원통형 좌표에서 라플라스 방정식은 방사형, 각도형 및 축 변형을 조정합니다. 파이프 벽의 경계 조건으로 이 방정식을 풀면 흐름에 대한 잠재적 속도가 정의됩니다.
이로부터 전체 속도 필드는 공간 도함수를 취하여 결정되어 파이프 내부의 흐름 패턴을 나타냅니다.

속도 퍼텐셜

Velocity Potential

균일한 단면을 가진 길고 직선적인 파이프를 통과하는 정상적이고 압축 불가능한 흐름에서, 파이프 벽에서 멀리 떨어진 중앙 영역의 흐름은 비회전적입니다. 이 비회전적 특성은 유체 입자가 축을 중심으로 회전하지 않으며, ϕ로 표현되는 속도 퍼텐셜이라는 스칼라 함수를 사용하여 움직임을 설명할 수 있음을 의미합니다. 비회전 흐름에서 속도장 V는 속도 퍼텐셜의 기울기로 정의됩니다.

<img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;">

기울기의 컬은 항상 0이므로, ϕ에 따라 속도를 정의하면 흐름에 와도가 없으므로 비회전이 됩니다.

비압축성 흐름의 경우 연속 방정식은 속도장의 발산이 0이어야 한다고 명시합니다.

<img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;">

이 방정식에 V를 대입하면 방정식은 다음과 같습니다.

<img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;">

이것은 라플라스 방정식으로 알려져 있습니다. 흐름이 ​​비회전적이고 비압축성인 영역에서 속도 퍼텐셜 ϕ는 라플라스 방정식을 충족해야 합니다.

파이프 흐름의 경우 원통형 좌표(r, θ, z)는 흐름을 설명하는 데 자연스러운 선택입니다. 여기서 r은 파이프 중심선으로부터의 반경 방향 거리이고, θ는 각도 위치이며, z는 파이프를 따라 축 방향 거리입니다. 반경, 각도 및 축 변화를 고려한 원통 좌표의 라플라스 방정식은 다음과 같이 표현됩니다.

<img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;">

파이프에서 ϕ의 정확한 형태를 찾으려면 경계 조건이 필요합니다. 예를 들어, 파이프 벽에서 미끄럼 방지 조건은 접선 속도가 0임을 의미하지만 중앙 비회전 영역에서 퍼텐셜 함수는 파이프의 대칭을 따릅니다. 이러한 조건으로 라플라스 방정식을 풀면 흐름장에서 속도 퍼텐셜이 정의됩니다.

ϕ가 결정되면 ϕ의 편미분을 취하여 원통 좌표의 속도 성분을 도출할 수 있습니다. 이러한 구성 요소는 다음과 같습니다.

방사 속도:

<img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;">

각속도

<img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;">

축속도

<img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;">

속도 구성 요소에 대한 이러한 표현식은 기울기에서 비롯되므로 유동이 비회전 조건을 충족한다는 것을 확인합니다.

단면이 균일한 길고 곧은 파이프를 통해 물이 꾸준히 흐르는 것을 고려하십시오.
흐름이 회전하지 않는 파이프의 중앙 영역에서는 ...

Education

교육

JoVE Core

Civil Engineering

토목공학

Unit 1

제1단원

Differential Analysis of Fluid Flow

유체 흐름의 미분 분석

KEY TERMS AND CONCEPTS

핵심 용어 및 개념

velocity potential in pipe flow

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 gives the flow no vorticity and keeps it irrotational.

For an incompressible fluid, the continuity equation requires the divergence of the velocity field to be zero. When V is written in terms of φ, this condition becomes Laplace's equation. In regions where the flow is both incompressible and irrotational, the velocity potential must satisfy this equation.

Pipe flow is often described with cylindrical coordinates: r, θ, and z. Here, r is the radial distance from the pipe centreline, θ is the angular position, and z is the distance along the pipe. In these coordinates, Laplace's equation includes radial, angular, and axial variation.

To find the exact form of φ in the pipe, boundary conditions are needed. At the pipe walls, the no-slip condition means the tangential velocity is zero. In the central irrotational region, the potential function must also match the symmetry of the pipe. Solving Laplace's equation with these conditions gives the velocity potential across the flow field.

Once φ is known, the velocity components can be found by taking partial derivatives. These components are the radial velocity, the angular velocity, and the axial velocity. Together, they show how the flow field comes from a gradient and therefore satisfies the irrotational condition.

Velocity Potential in Pipe Flow

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...

stream function

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...

irrotational flow

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...

velocity potential

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...

plane potential flows

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...

navier&#8211;stokes equations

Fluid Stress and Flow Equations

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...

steady, laminar flow between parallel plates

steady-laminar-flow-between-parallel-plates

Laminar Flow Profile Between Parallel Plates

Laminar flow between parallel plates shows how fluid moves through a narrow, flat channel. This model helps describe flow in water conveyance and drainage systems, where the space is limited and the motion stays orderly. It is also useful for analyzing flow in rectangular channels used in many engineering settings. The flow is studied with the Navier-Stokes equations, which govern fluid motion. These equations are often difficult to solve because they are nonlinear. In this case, the problem...

couette flow

Linear Shear Between Moving Plates

Linear shear between moving plates describes fluid motion in a simple channel with two parallel plates. One plate stays fixed, and the other moves at a constant speed. This setup helps show how viscosity and incompressible flow shape fluid motion under the Navier-Stokes equations. The model uses a steady, laminar, incompressible flow with no pressure gradient in the flow direction. The x-axis points along the moving direction of the upper plate, and the y-axis is perpendicular to both plates.

steady, laminar flow in circular tubes

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...

design example: flow of oil through circular pipes

design-example-flow-of-oil-through-circular-pipes

Pressure Drop in Oil Pipe Design

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

Explore how velocity potential describes irrotational, incompressible flow in a straight pipe, and see how Laplace's equation and cylindrical coordinates apply.

Velocity Potential in Pipe Flow | JoVE K12

JoVE Core Civil Engineering offers a foundational exploration of essential construction materials such as bricks, stones, steel, wood, and concrete, along with fluid mechanics topics like flow over immersed bodies, open channel flow, and dimensional modeling. It also covers key surveying concepts, including leveling, traverse adjustments, curve design, topographic mapping, and geospatial technologies (GPS & GIS). Science-in-action videos showcase laboratory research on materials, fluids, and surveying practices in real-world engineering systems.

civil engineering

Velocity Potential in Pipe Flow

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