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

无旋流动是指流体粒子沿流线运动时不发生旋转，其特征是速度场的旋度为零，且不存在涡量。
速度仅有一个非零分量且为常数 u=U 的均匀流动是无旋流动的一个例子，因为它缺乏速度梯度。
当固体物体被引入均匀流场时，除了在边界附近，流场仍保持无旋状态。在其表面附近，流速从零开始急剧增加，形成陡峭的梯度和显著的剪切应力。
对于真正的无黏性流动，流体将沿边界滑过，保持无旋运动。然而，在真实流体中，黏性会在固定表面附近形成边界层，产生旋转效应，有时还会导致下游分离。
在管道流动中，入口区域最初表现出无旋的核心和均匀流动，但随着管壁处边界层的形成并逐渐增厚，流动将过渡到以黏性力为主导的充分发展区域。
假设流动是无旋的、不可压缩的、定常的且无黏性的，则可在流场中应用伯努利方程。

无旋流

Irrotational Flow

无旋流的特征是流体运动中粒子不绕其轴旋转，导致涡度为零。对于无旋流，速度场的旋度必须为零。这对速度梯度施加了特定条件。例如，绕 z 轴保持零旋转，必须满足梯度条件： <img alt="Equation 1" src="/files/ftp_upload/18059/18059_Equation_1.svg" style="text-align: center; height: 27px;"> 以及其他轴的类似条件： <img alt="Equation 2" src="/files/ftp_upload/18059/18059_Equation_2.svg" style="text-align: center; height: 27px;"> 均匀流满足这些条件，其中 u 为常数，v 和 w 均为零，可作为无旋流的一个简单示例。但是，当引入固体物体时，无旋性仅在远离物体的地方继续存在。在边界附近，速度从零（由于无滑移条件）急剧增加到自由流值，在粘性流体中产生显著的剪切应力和边界层。 当流体从大型水库进入管道时，初始流动几乎是均匀的，从而形成速度分布平坦的无旋核心。当流体向下游流动时，由于粘度，边界层沿管壁形成，导致速度分布产生梯度，最终导致完全发展的抛物线轮廓。在这个完全发展的区域，旋转效应在壁附近占主导地位，而中心核心可能保持大致无旋。 在管道流的初始无旋区域中，当不可压缩性、无粘性流动和无旋性条件成立时，伯努利方程 <img alt="Equation 3" src="/files/ftp_upload/18059/18059_Equation_3.svg" style="text-align: center; height: 27px;"> 可应用于沿流线的点之间。然而，随着边界层的形成和壁附近的旋转效应变得显著，伯努利方程的假设就不成立了。这意味着伯努利方程仅适用于入口区域和无旋核心，而不适用于粘度效应占主导地位的完全发展的旋转区域。

无旋流动是指流体粒子沿流路运动时不发生旋转，其特征是速度场的旋度为零，且……

Education

教育

JoVE Core

Civil Engineering

土木工程

Unit 1

第一单元

Differential Analysis of Fluid Flow

流体流动微分分析

KEY TERMS AND CONCEPTS

关键术语与概念

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...

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 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 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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