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

2차원 비압축성 흐름에는 수평 및 수직 속도 구성 요소가 있습니다.
연속성 방정식은 이러한 구성 요소의 변화율의 합이 0이 되도록 요구함으로써 질량 보존을 보장하며, 이는 모든 지역의 순 유입과 유출이 균형을 이룬다는 것을 의미합니다.
이 조건을 충족하기 위해 stream 함수가 도입되었습니다.
수평 속도는 수직 방향에 대한 스트림 함수의 편도함수이고, 수직 속도는 수평 방향에 대한 음의 편도함수입니다. 이렇게 하면 대량 보존이 자동으로 유지됩니다.
스트림 함수의 주요 기능은 유체 입자의 경로를 나타내고 유동장의 속도에 접하는 유선을 따라 일정하게 유지된다는 것입니다.
종종 등고선으로 묘사되는 유선은 유체 거동을 직관적으로 해석할 수 있도록 합니다.
스트림 기능은 유체 모션 계산을 단순화합니다. 두 유선 사이의 스트림 함수 값의 차이는 두 유선 사이의 체적 유량을 나타냅니다.
이렇게 하면 속도 구성 요소를 개별적으로 해결할 필요가 없으므로 2차원 흐름을 분석하기 위한 효율적인 도구가 됩니다.

스트림 함수

Stream Function

2차원 비압축성 유체 흐름에서 연속 방정식은 질량 보존을 보장하는 데 필수적입니다. 즉, 영역에 들어오거나 나가는 유체의 모든 변화는 다른 곳에서의 해당 변화로 균형을 이룹니다. 밀도가 일정하게 유지되는 비압축성 흐름의 경우 이 요구 사항은 속도장의 발산이 0이어야 한다는 조건으로 단순화됩니다. 수학적으로 이는 다음과 같이 표현됩니다.

<img alt="Equation 1" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;">

여기서 u와 v는 각각 수평 및 수직 속도 성분을 나타냅니다.

이 방정식을 본질적으로 만족시키기 위해 스트림 함수 ψ를 도입합니다. 이를 통해 속도 성분을 다음과 같이 ψ에 따라 정의할 수 있습니다.

<img alt="Equation 2" src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;">

이러한 정의는 ψ의 혼합 편미분이 상쇄되므로 연속 방정식을 자동으로 만족시킵니다.

스트림 함수 ψ는 유체 입자가 흐름 내에서 따르는 경로인 유선을 따라 일정합니다. 따라서 유선은 ψ의 등고선으로 시각화할 수 있어 유체 운동을 명확하게 표현할 수 있습니다. 각 유선은 경로를 따라 모든 지점에서 속도 벡터에 접하여 각 지점에서 개별적으로 속도를 계산할 필요 없이 흐름 방향을 보여줍니다.

또한 두 유선 사이의 ψ\psiψ 값 차이는 두 유선 사이의 단위 깊이당 체적 유량을 나타내므로 유량을 직접 계산할 수 있습니다. 이 속성은 u와 v에 대한 별도의 방정식을 풀 필요성을 제거하여 2차원 비압축성 흐름의 분석을 크게 간소화합니다. 결과적으로 스트림 함수 ψ는 비압축성 정상 흐름과 관련된 응용 프로그램에서 유체 거동을 모델링하고 해석하는 강력한 도구입니다.

2차원 비압축성 흐름에는 수평 및 수직 속도 구성 요소가 있습니다.
연속성 방정식은 다음을 통해 질량 보존을 보장합니다.

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

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...

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

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 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

Stream Function for Incompressible Flow

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