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

Представьте себе воду, равномерно текущую по длинной прямой трубе с равномерным поперечным сечением.
В центральных областях трубы, где поток является ирвращательным, потенциал скорости определяет поток, а скорость выражается как градиент этого потенциала.
Подстановка составляющих скорости напрямую подтверждает, что поток удовлетворяет этому условию невращения.
Скоростной потенциал применяется к трехмерным потокам, таким как этот трубопроводный поток, в то время как функция потока специфична для двумерных потоков, возникающих в результате сохранения массы.
В несжимаемом потоке уравнение непрерывности требует, чтобы дивергенция вектора скорости была равна нулю.
Подставляя потенциал скорости, это условие становится уравнением Лапласа, ключевым дифференциальным уравнением, управляющим невязкими, несжимаемыми и ирротационными потоками.
В цилиндрических координатах уравнение Лапласа корректируется с учетом радиальных, угловых и осевых вариаций. Решение этого уравнения с граничными условиями на стенках трубы определяет потенциал скорости потока.
Исходя из этого, полное поле скоростей определяется путем взятия пространственных производных, выявляющих характер течения внутри трубы.

Потенциал скорости

Velocity Potential

В стационарном несжимаемом потоке через длинную прямую трубу с равномерным поперечным сечением, поток в центральной области (вдали от стенок трубы) является безвихревым. Эта безвихревая природа означает, что частицы жидкости не вращаются вокруг своих осей, и скалярная функция, называемая потенциалом скорости, обозначенная ϕ, может использоваться для описания их движения. В безвихревых потоках поле скорости V определяется как градиент потенциала скорости:

<img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;">


 Поскольку взятие ротора градиента всегда даёт ноль, определение скорости в терминах ϕ гарантирует, что поток не имеет завихренности, что делает его безвихревым.

 Для несжимаемых потоков уравнение непрерывности гласит, что дивергенция поля скорости должна быть равна нулю:

<img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;">


 При подставлении V в это уравнение, уравнение принимает вид:

<img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;">


 Такая форма известна как уравнение Лапласа. В областях, где поток является безвихревым и несжимаемым, потенциал скорости ϕ должен удовлетворять уравнению Лапласа.

 В случае течения в трубе естественным выбором для описания движения являются цилиндрические координаты (r, θ, z), где r — радиальное расстояние от осевой линии трубы, θ — угловое положение, а z — осевое расстояние вдоль трубы. Уравнение Лапласа в цилиндрических координатах, учитывающее радиальные, угловые и осевые изменения, выражается как:

<img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;">


 Нахождение точной формы ϕ в трубе требует граничных условий. Например, на стенках трубы условие отсутствия проскальзывания будет подразумевать нулевую тангенциальную скорость, тогда как в центральной безвихревой области потенциальная функция будет следовать симметрии трубы. Решение уравнения Лапласа с этими условиями определяет потенциал скорости поперек поля потока.

 После определения ϕ компоненты скорости в цилиндрических координатах могут быть получены путем взятия частных производных от ϕ. Эти компоненты следующие:

 Радиальная скорость:

<img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;">


 Угловая скорость

<img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;">


 Осевая скорость

<img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;">

 Эти выражения для компонентов скорости подтверждают, что поток удовлетворяет безвихревому условию, поскольку они получаются из градиента.

Представьте себе воду, равномерно текущую по длинной прямой трубе с равномерным поперечным сечением.
В центральных областях трубы, где поток является ирротационным, ...

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JoVE Core

Civil Engineering

Unit 1

Раздел 1

Differential Analysis of Fluid Flow

Дифференциальный анализ потока жидкости

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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

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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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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...

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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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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.

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Velocity Potential in Pipe Flow

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