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

Düzgün bir kesite sahip uzun, düz bir borudan sürekli olarak akan suyu düşünün.
Akışın irrotasyonel olduğu borunun merkezi bölgelerinde, bir hız potansiyeli akışı tanımlar ve hız, bu potansiyelin gradyanı olarak ifade edilir.
Hız bileşenlerinin değiştirilmesi, akışın bu irrotasyonel koşulu karşıladığını doğrudan doğrular.
Hız potansiyeli, bu boru akışı gibi üç boyutlu akışlar için geçerlidir, akış fonksiyonu ise kütle korunumundan kaynaklanan iki boyutlu akışlara özgüdür.
Sıkıştırılamaz akışta, süreklilik denklemi hız vektörünün ıraksamasının sıfır olmasını gerektirir.
Hız potansiyelini yerine koyarak, bu koşul Laplace denklemi, görünmez, sıkıştırılamaz ve irrotasyonel akışları yöneten anahtar bir diferansiyel denklem haline gelir.
Silindirik koordinatlarda, Laplace denklemi radyal, açısal ve eksenel değişimler için ayarlanır. Bu denklemi boru duvarlarındaki sınır koşullarıyla çözmek, akış için hız potansiyelini tanımlar.
Bundan, tam hız alanı, uzamsal türevler alınarak belirlenir ve boru içindeki akış modeli ortaya çıkar.

hız potansiyeli

Velocity Potential

Uzun, düz bir borudan geçen, düzgün bir kesite sahip, sıkıştırılamaz, sabit akışta, merkez bölgedeki (boru duvarlarından uzaktaki) akış irrotasyoneldir. Bu irrotasyonellik, akışkan parçacıklarının eksenleri etrafında dönmediği anlamına gelir ve ϕ ile gösterilen hız potansiyeli adı verilen bir skaler fonksiyon, hareketlerini tanımlamak için kullanılabilir. İrrotasyonel akışlarda, hız alanı V, hız potansiyelinin eğimi olarak tanımlanır:

<img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;">

 Bir gradyanın eğrisini almak her zaman sıfır olduğundan, hızı ϕ cinsinden tanımlamak, akışın girdap oluşturmamasını ve onu irrotasyonel hale getirmesini sağlar.

 Sıkıştırılamaz akışlar için, süreklilik denklemi hız alanının diverjansının sıfır olması gerektiğini belirtir:

<img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;">


 Bu denkleme V'yi koyduğumuzda denklem şu hale gelir:

<img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;">


 Buna Laplace denklemi denir. Akışın irrotasyonel ve sıkıştırılamaz olduğu bölgelerde, hız potansiyeli ϕ Laplace denklemini sağlamalıdır.

 Boru akışı durumunda, silindirik koordinatlar (r, θ, z) akışı tanımlamak için doğal bir seçimdir, burada r boru merkez hattından radyal mesafe, θ açısal konum ve z boru boyunca eksenel mesafedir. Laplace denklemi, radyal, açısal ve eksenel değişimleri hesaba katarak silindirik koordinatlarda şu şekilde ifade edilir:

<img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;">


 Borudaki ϕ'nin tam formunu bulmak sınır koşullarını gerektirir. Örneğin, boru duvarlarında kaymama koşulu sıfır teğetsel hız anlamına gelirken, merkezi irrotasyonel bölgede potansiyel fonksiyonu borunun simetrisini takip eder. Laplace denklemini bu koşullarla çözmek, akış alanı boyunca hız potansiyelini tanımlar.

 ϕ belirlendikten sonra, silindirik koordinatlardaki hız bileşenleri ϕ'nin kısmi türevleri alınarak türetilebilir. Bu bileşenler şunlardır:

 Radyal hız:

<img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;">


 Açısal hız

<img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;">


 Eksenel hız

<img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;">

 Hız bileşenleri için bu ifadeler, bir eğimden kaynaklandıkları için akışın dönmeme koşulunu karşıladığını doğrular.

Hız Potansiyeli

Düzgün bir kesite sahip uzun, düz bir borudan sürekli olarak akan suyu düşünün.
Akışın irrotasyonel olduğu borunun merkezi bölgelerinde, bir ...

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JoVE Core

Civil Engineering

İnşaat Mühendisliği

Unit 1

Ünite 1

Differential Analysis of Fluid Flow

Akışkan Akışının Diferansiyel Analizi

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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 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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