Name: irrotational flow near boundaries
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 28 s
Channel: JoVE

非回転流とは、流路に沿って回転しない流体粒子の動きを指し、カールがゼロで渦度のない速度場を特徴としています。
速度の非ゼロ成分が1つだけある均一な流れ、定数u = Uは、速度勾配がないため、非回転流れの一例です。
固体物体が均一な流れに導入されると、境界付近を除いて流れは非回転のままです。表面付近では、速度はゼロから急激に増加し、急な勾配と大きなせん断応力が生じます。
真に非粘性の流れの場合、流れは境界を越えて滑り、非回転運動を維持します。しかし、実際の流体では、粘度が固定サーフェスの近くに境界層を形成し、回転効果を生み出し、時には下流の分離を引き起こします。
パイプ流では、入口領域は最初は均一な流れで非回転コアを示しますが、境界層がパイプ壁に沿って形成されて厚くなると、粘性力が支配する完全に発達した領域に移行します。
流れが非回転性、非圧縮性、定常性、非粘性であると仮定すると、ベルヌーイ方程式を流れ場に適用できます。

非回転流

Irrotational Flow

非回転流は、粒子が軸の周りを回転せず、渦度がゼロになる流体運動を特徴とします。流れが非回転であるためには、速度場の回転がゼロでなければなりません。これにより、速度勾配に特定の条件が課せられます。たとえば、z 軸の周りの回転をゼロに維持するには、勾配条件:

<img alt="Equation 1" src="/files/ftp_upload/18059/18059_Equation_1.svg" style="text-align: center; height: 27px;">

 を満たす必要があります。また、他の軸についても同様の条件を満たす必要があります:

<img alt="Equation 2" src="/files/ftp_upload/18059/18059_Equation_2.svg" style="text-align: center; height: 27px;">

 u が一定で、v と w が両方ともゼロである均一な流れは、これらの条件を満たし、非回転流の簡単な例として役立ちます。ただし、固体の物体が導入されると、非回転性は物体から離れた場所でのみ持続します。境界付近では、速度はゼロ（滑りなし条件のため）から自由流値まで急激に増加し、粘性流体に大きなせん断応力と境界層が生じます。

 流体が大きな貯留層からパイプに入ると、初期の流れはほぼ均一で、速度分布が平坦な非回転コアが形成されます。流体が下流に移動するにつれて、粘性によりパイプ壁に沿って境界層が形成され、速度プロファイルに勾配が生じ、最終的に完全に発達した放物線プロファイルになります。この完全に発達した領域では、壁の近くで回転効果が支配的になりますが、中心コアはほぼ非回転のままです。

 管内流の初期の非回転領域では、非圧縮性、非粘性流、非回転性の条件が満たされているため、流線に沿った点間にベルヌーイの方程式

<img alt="Equation 3" src="/files/ftp_upload/18059/18059_Equation_3.svg" style="text-align: center; height: 27px;">

 を適用できます。しかし、境界層が大きくなり、壁の近くで回転効果が顕著になると、ベルヌーイの方程式の仮定は崩れます。つまり、ベルヌーイの方程式は入口領域と非回転コアにのみ適用でき、粘性効果が支配的な完全に発達した回転領域には適用できません。

非回転流とは、流路に沿って回転しない流体粒子の動きを指し、カールがゼロの速度場と...

Education

JoVE Core

Civil Engineering

土木工学

Unit 1

ユニット1

Differential Analysis of Fluid Flow

流体流れの微分解析

KEY TERMS AND CONCEPTS

主要用語と概念

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 far from the object. Near the surface, the fluid speed rises sharply from zero because of the no-slip condition. This creates strong shear stress and a boundary layer in viscous fluids.

A similar change happens when fluid enters a pipe from a large reservoir. At first, the flow is nearly uniform, so the center region forms an irrotational core with a flat velocity profile. Farther downstream, viscosity causes boundary layers to grow along the pipe walls.

As the flow continues, the velocity profile develops more gradient and becomes fully developed and parabolic. In that region, rotational effects are strongest near the walls, while the center may still be approximately irrotational. Bernoulli's equation can apply in the entrance region and in the irrotational core, where the flow is incompressible, inviscid, and irrotational. It no longer applies well in the fully developed region, where viscosity and rotation dominate.

Irrotational Flow Near Boundaries

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...

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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 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...

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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...

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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 irrotational flow behaves near solids and in pipe entrance flow, including boundary layers, rotational effects, and when Bernoulli's equation applies.

Irrotational Flow Near Boundaries | 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

Irrotational Flow Near Boundaries

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