Name: stream function for incompressible flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

В двумерном несжимаемом потоке существуют горизонтальная и вертикальная составляющие скорости.
Уравнение непрерывности обеспечивает сохранение массы, требуя, чтобы сумма скоростей изменения этих компонентов была равна нулю, что означает, что чистый приток и отток в любой области сбалансированы.
Для выполнения этого условия вводится функция потока.
Горизонтальная скорость является частной производной функции потока по отношению к вертикальному направлению, в то время как вертикальная скорость является отрицательной частной производной по отношению к горизонтальному направлению. Это обеспечивает автоматическое поддержание сохранения массы.
Ключевой особенностью функции потока является то, что она остается постоянной вдоль линий тока, которые представляют собой траектории частиц жидкости и касательны к скоростям в поле потока.
Линии тока, часто изображаемые в виде контурных линий, позволяют интуитивно интерпретировать поведение жидкости.
Функция потока упрощает расчет движения жидкости. Разница в значениях функции потока между двумя линиями тока представляет собой объемный расход между ними.
Это избавляет от необходимости отдельно решать составляющие скорости, что делает его эффективным инструментом для анализа двумерного потока.

Функция потока

Stream Function

В двумерном потоке несжимаемой жидкости уравнение непрерывности необходимо для обеспечения сохранения массы, что означает, что любое изменение в жидкости, входящей или выходящей из области, уравновешивается соответствующим изменением в другом месте. Для несжимаемого потока, где плотность остается постоянной, это требование упрощается до условия, что дивергенция поля скорости должна быть равна нулю. Математически это выражается как,

<img alt="Equation 1" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;">


 где u и v представляют собой горизонтальную и вертикальную компоненты скорости, соответственно.

 Чтобы автоматически удовлетворить этому уравнению, мы вводим функцию потока ψ, которая позволяет определять компоненты скорости в терминах ψ как:

<img alt="Equation 2" src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;">

 Эти определения автоматически удовлетворяют уравнению непрерывности, поскольку смешанные частные производные ψ сокращаются.

 Функция потока ψ постоянна вдоль линий тока, которые являются путями, по которым частицы жидкости следуют в потоке. Поэтому линии тока можно визуализировать как контурные линии ψ, что обеспечивает четкое представление движения жидкости. Каждая линия тока касается вектора скорости в любой точке вдоль ее пути, иллюстрируя направление потока без необходимости вычисления скоростей в каждой точке по отдельности.

 Более того, разница в значениях ψ\psiψ между двумя линиями тока представляет объемный расход на единицу глубины между ними, что позволяет напрямую рассчитывать расход. Это свойство значительно упрощает анализ двумерных несжимаемых потоков, устраняя необходимость решать отдельные уравнения для u и v. В результате функция тока ψ является мощным инструментом для моделирования и интерпретации поведения жидкости в приложениях, рассматривающих несжимаемый, стационарный поток.

В двумерном несжимаемом потоке существуют горизонтальная и вертикальная составляющие скорости.
Уравнение непрерывности обеспечивает сохранение массы за счет ...

Education

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

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 to satisfy this equation automatically. The velocity components can be defined in terms of ψ. When they are written this way, the mixed partial derivatives of ψ cancel, so the continuity equation is already met.

The stream function is also linked to streamlines. A streamline is the path a fluid particle follows in the flow, and ψ stays constant along each one. Because of this, streamlines can be viewed as contour lines of ψ. Each streamline is tangent to the velocity vector at every point, so it shows flow direction without finding the velocity at every location.

Differences in ψ between two streamlines also have meaning. They represent the volumetric flow rate per unit depth between those lines. This makes it easier to analyze steady, two-dimensional incompressible flows. In many cases, the stream function reduces the need to solve separate equations for u and v, so it is a useful tool for modeling and interpreting fluid motion.

Stream Function for Incompressible 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

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

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...

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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.

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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

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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

Learn how the stream function describes two-dimensional incompressible flow, keeps continuity satisfied, and shows streamlines and flow rate in fluid motion.

Stream Function for Incompressible Flow | 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

Stream Function for Incompressible Flow

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