Name: stream function for incompressible flow
Uploaded: 2025-02-13T15:43:25.000Z
Duration: 1 min 20 s
Channel: JoVE

Bei der zweidimensionalen inkompressiblen Strömung gibt es horizontale und vertikale Geschwindigkeitskomponenten.
Die Kontinuitätsgleichung stellt die Massenerhaltung sicher, indem sie verlangt, dass die Summe der Änderungsraten dieser Komponenten Null ist, was bedeutet, dass der Nettozufluss und -abfluss in jeder Region ausgeglichen sind.
Um diese Bedingung zu erfüllen, wird die Stream-Funktion eingeführt.
Die horizontale Geschwindigkeit ist die partielle Ableitung der Strömungsfunktion in Bezug auf die vertikale Richtung, während die vertikale Geschwindigkeit die negative partielle Ableitung in Bezug auf die horizontale Richtung ist. Dadurch wird sichergestellt, dass die Massenkonservierung automatisch aufrechterhalten wird.
Ein wesentliches Merkmal der Strömungsfunktion besteht darin, dass sie entlang der Stromlinien, die die Wege von Fluidpartikeln darstellen und tangential zu den Geschwindigkeiten im Strömungsfeld sind, konstant bleibt.
Stromlinien, oft als Höhenlinien dargestellt, ermöglichen eine intuitive Interpretation des flüssigen Verhaltens.
Die Strömungsfunktion vereinfacht die Berechnung von Fluidbewegungen. Die Differenz der Stromfunktionswerte zwischen zwei Stromlinien stellt den Volumenstrom zwischen ihnen dar.
Dadurch entfällt die Notwendigkeit, Geschwindigkeitskomponenten separat zu lösen, was es zu einem effizienten Werkzeug für die Analyse zweidimensionaler Strömungen macht.

stromfunktion

Stream Function

Bei zweidimensionalen inkompressiblen Flüssigkeitsströmen ist die Kontinuitätsgleichung für die Erhaltung der Masse unerlässlich, was bedeutet, dass jede Änderung der Flüssigkeit, die in einen Bereich eintritt oder aus ihm austritt, durch eine entsprechende Änderung an anderer Stelle ausgeglichen wird. Bei inkompressiblen Strömungen, bei denen die Dichte konstant bleibt, vereinfacht sich diese Anforderung zu der Bedingung, dass die Divergenz des Geschwindigkeitsfelds Null sein muss. Mathematisch wird dies wie folgt ausgedrückt:

<img alt="Equation 1" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;">


wobei u und v jeweils die horizontalen und vertikalen Geschwindigkeitskomponenten darstellen.

Um diese Gleichung inhärent zu erfüllen, führen wir die Stromfunktion ψ ein, die es ermöglicht, die Geschwindigkeitskomponenten in Bezug auf ψ wie folgt zu definieren:

<img alt="Equation 2" src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;">

Diese Definitionen erfüllen automatisch die Kontinuitätsgleichung, da sich die gemischten partiellen Ableitungen von ψ gegenseitig aufheben.

Die Stromfunktion ψ ist entlang der Stromlinien konstant, also der Pfade, denen Flüssigkeitspartikel innerhalb der Strömung folgen. Stromlinien können daher als Konturlinien von ψ visualisiert werden, was eine klare Darstellung der Flüssigkeitsbewegung liefert. Jede Stromlinie ist an jedem Punkt entlang ihres Pfads tangential zum Geschwindigkeitsvektor und veranschaulicht die Strömungsrichtung, ohne dass die Geschwindigkeiten an jedem Punkt einzeln berechnet werden müssen.

Darüber hinaus stellt der Unterschied in den ψ\psiψ-Werten zwischen zwei Stromlinien die volumetrische Strömungsrate pro Tiefeneinheit zwischen ihnen dar, was eine direkte Berechnung der Strömungsraten ermöglicht. Diese Eigenschaft vereinfacht die Analyse zweidimensionaler inkompressibler Strömungen erheblich, da keine separaten Gleichungen für u und v gelöst werden müssen. Daher ist die Stromfunktion ψ ein leistungsstarkes Werkzeug zum Modellieren und Interpretieren des Flüssigkeitsverhaltens in Anwendungen mit inkompressibler, gleichmäßiger Strömung.

Stromfunktion

Bei der zweidimensionalen inkompressiblen Strömung gibt es horizontale und vertikale Geschwindigkeitskomponenten.
Die Kontinuitätsgleichung gewährleistet die Massenerhaltung durch ...

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Unit 1

Einheit 1

Differential Analysis of Fluid Flow

Differentielle Analyse der Fluidströmung

KEY TERMS AND CONCEPTS

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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 to satisfy this equation automatically. The velocity components can be defined in terms of ψ. When they are written this way, the mixed partial derivatives of ψ cancel, so the continuity equation is already met.

The stream function is also linked to streamlines. A streamline is the path a fluid particle follows in the flow, and ψ stays constant along each one. Because of this, streamlines can be viewed as contour lines of ψ. Each streamline is tangent to the velocity vector at every point, so it shows flow direction without finding the velocity at every location.

Differences in ψ between two streamlines also have meaning. They represent the volumetric flow rate per unit depth between those lines. This makes it easier to analyze steady, two-dimensional incompressible flows. In many cases, the stream function reduces the need to solve separate equations for u and v, so it is a useful tool for modeling and interpreting fluid motion.

Stream Function for Incompressible Flow

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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

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...

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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...

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key terms and concepts

Learn how the stream function describes two-dimensional incompressible flow, keeps continuity satisfied, and shows streamlines and flow rate in fluid motion.

Stream Function for Incompressible Flow | 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.

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Stream Function for Incompressible Flow

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