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

断面が均一な長くてまっすぐなパイプを通って着実に流れる水を考えてみましょう。
流れが非回転であるパイプの中央領域では、速度ポテンシャルが流れを定義し、速度はこのポテンシャルの勾配として表されます。
速度成分を直接代入すると、流れがこの非回転条件を満たすことが確認されます。
速度ポテンシャルは、このパイプ流のような 3 次元流れに適用され、流流関数は質量保存から生じる 2 次元流れに特有です。
非圧縮性流れでは、連続方程式により、速度ベクトルの発散がゼロである必要があります。
速度ポテンシャルを代入することにより、この条件はラプラスの方程式、つまり非粘性、非圧縮性、および非回転の流れを支配する重要な微分方程式になります。
円筒座標では、ラプラスの方程式は半径、角度、および軸の変動を調整します。この方程式をパイプ壁の境界条件で解くと、流れのポテンシャル速度が定義されます。
このことから、空間微分を取ることによって全速度場が決定され、パイプ内の流れパターンが明らかになります。

速度ポテンシャル

Velocity Potential

均一な断面を持つ長くてまっすぐなパイプを通る定常の非圧縮性の流れでは、(パイプの壁から遠い) 中央領域の流れは非回転です。この非回転の性質は、流体粒子が軸の周りを回転しないことを意味し、ϕ で表される速度ポテンシャルと呼ばれるスカラー関数を使用して、それらの動きを記述できます。非回転の流れでは、速度場 V は速度ポテンシャルの勾配として定義されます:

<img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;">


 勾配の回転は常にゼロであるため、ϕ で速度を定義すると、流れに渦がなく、非回転になります。

 非圧縮性の流れの場合、連続方程式は速度場の発散がゼロでなければならないと規定しています:

<img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;">


 この方程式に V を代入すると、方程式は次のようになります:

<img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;">


 これはラプラス方程式として知られています。流れが非回転で非圧縮性の領域では、速度ポテンシャル ϕ はラプラス方程式を満たす必要があります。

 管内流の場合、流れを記述するには円筒座標 (r、θ、z) が自然な選択です。ここで、r はパイプの中心線からの半径距離、θ は角度位置、z はパイプに沿った軸方向距離です。半径、角度、軸方向の変化を考慮した円筒座標のラプラス方程式は、次のように表されます:

<img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;">


 パイプ内の ϕ の正確な形を見つけるには、境界条件が必要です。たとえば、パイプの壁では、滑りなしの条件は接線速度がゼロであることを意味しますが、中央の非回転領域では、ポテンシャル関数はパイプの対称性に従います。これらの条件でラプラス方程式を解くと、流れ場全体の速度ポテンシャルが定義されます。

 ϕ が決定したら、ϕ の偏微分を取ることで円筒座標の速度成分を導き出すことができます。これらの成分は次のとおりです。

 半径方向の速度:

<img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;">


 角速度

<img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;">


軸方向の速度

<img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;">

 速度成分のこれらの式は、勾配から生じるため、流れが非回転条件を満たしていることを確認します。

断面が均一な長くてまっすぐなパイプを通って着実に流れる水を考えてみましょう。
パイプの中央領域では、流れが非回転の場合、...

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

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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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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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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civil engineering

Velocity Potential in Pipe Flow

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