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

Stellen Sie sich vor, dass Wasser stetig durch ein langes, gerades Rohr mit einem gleichmäßigen Querschnitt fließt.
In zentralen Bereichen des Rohrs, in denen die Strömung irrotierend ist, definiert ein Geschwindigkeitspotential die Strömung, und die Geschwindigkeit wird als Gradient dieses Potentials ausgedrückt.
Durch das direkte Ersetzen der Geschwindigkeitskomponenten wird überprüft, ob die Strömung diese Rotationsbedingung erfüllt.
Das Geschwindigkeitspotential gilt für dreidimensionale Strömungen, wie z. B. diese Rohrströmung, während die Strömungsfunktion spezifisch für zweidimensionale Strömungen ist, die sich aus der Massenerhaltung ergeben.
Bei inkompressiblen Strömungen erfordert die Kontinuitätsgleichung, dass die Divergenz des Geschwindigkeitsvektors Null ist.
Durch die Substitution des Geschwindigkeitspotentials wird diese Bedingung zur Laplace-Gleichung, einer Schlüsseldifferentialgleichung, die viskose, inkompressible und irrotationale Strömungen bestimmt.
In zylindrischen Koordinaten passt sich die Laplace-Gleichung an radiale, winkelige und axiale Variationen an. Die Lösung dieser Gleichung mit Randbedingungen an den Rohrwänden definiert das Geschwindigkeitspotential für die Strömung.
Daraus wird das komplette Geschwindigkeitsfeld durch räumliche Ableitungen bestimmt, wodurch das Strömungsmuster innerhalb des Rohres sichtbar wird.

geschwindigkeitspotential

Velocity Potential

Bei einer gleichmäßigen, inkompressiblen Strömung durch ein langes, gerades Rohr mit gleichmäßigem Querschnitt ist die Strömung im zentralen Bereich (weit von den Rohrwänden entfernt) rotationsfrei. Diese Rotationsfreiheit bedeutet, dass Flüssigkeitspartikel nicht um ihre Achsen rotieren, und eine Skalarfunktion namens Geschwindigkeitspotential, dargestellt durch ϕ, kann zur Beschreibung ihrer Bewegung verwendet werden. Bei rotationsfreien Strömungen wird das Geschwindigkeitsfeld V als Gradient des Geschwindigkeitspotentials definiert:

<img alt="Equation 1" src="/files/ftp_upload/18060/18060_Equation_1.svg" style="text-align: center; height: 12px;">

Da die Rotation eines Gradienten immer Null ist, stellt die Definition der Geschwindigkeit in Bezug auf ϕ sicher, dass die Strömung keine Wirbel aufweist und somit rotationsfrei ist.

Für inkompressible Strömungen besagt die Kontinuitätsgleichung, dass die Divergenz des Geschwindigkeitsfelds Null sein muss:

<img alt="Equation 2" src="/files/ftp_upload/18060/18060_Equation_2.svg" style="text-align: center; height: 9px;">


Wird V in diese Gleichung eingesetzt, ergibt sich folgende Gleichung:

<img alt="Equation 3" src="/files/ftp_upload/18060/18060_Equation_3.svg" style="text-align: center; height: 13px;">


Dies ist als Laplace-Gleichung bekannt. In Bereichen, in denen die Strömung drehungsfrei und inkompressibel ist, muss das Geschwindigkeitspotential ϕ der Laplace-Gleichung genügen.

Bei einer Rohrströmung sind Zylinderkoordinaten (r, θ, z) eine natürliche Wahl zur Beschreibung der Strömung, wobei r der radiale Abstand von der Rohrmittellinie, θ die Winkelposition und z der axiale Abstand entlang des Rohrs ist. Die Laplace-Gleichung in Zylinderkoordinaten, die radiale, Winkel- und axiale Variationen berücksichtigt, wird wie folgt ausgedrückt:

<img alt="Equation 4" src="/files/ftp_upload/18060/18060_Equation_4.svg" style="text-align: center; height: 30px;">


Um die genaue Form von ϕ in der Rohrleitung zu finden, sind Randbedingungen erforderlich. Beispielsweise würde die Haftbedingung an den Rohrwänden eine Tangentialgeschwindigkeit von Null bedeuten, während im zentralen Rotationsfreiheitsbereich die Potentialfunktion der Symmetrie des Rohrs folgen würde. Das Lösen der Laplace-Gleichung unter diesen Bedingungen definiert das Geschwindigkeitspotential über das Strömungsfeld.

Sobald ϕ bestimmt ist, können die Geschwindigkeitskomponenten in Zylinderkoordinaten durch partielle Ableitungen von ϕ abgeleitet werden. Diese Komponenten sind:

Radialgeschwindigkeit:

<img alt="Equation 5" src="/files/ftp_upload/18060/18060_Equation_5.svg" style="text-align: center; height: 30px;">


Winkelgeschwindigkeit

<img alt="Equation 6" src="/files/ftp_upload/18060/18060_Equation_6.svg" style="text-align: center; height: 30px;">


Axialgeschwindigkeit

<img alt="Equation 7" src="/files/ftp_upload/18060/18060_Equation_7.svg" style="text-align: center; height: 30px;">

Diese Ausdrücke für Geschwindigkeitskomponenten bestätigen, dass die Strömung die Rotationsfreiheitsbedingung erfüllt, da sie aus einem Gradienten stammen.

Geschwindigkeitspotential

Stellen Sie sich vor, dass Wasser stetig durch ein langes, gerades Rohr mit einem gleichmäßigen Querschnitt fließt.
In zentralen Bereichen des Rohrs, in denen die Strömung unförmig ist, ...

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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

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