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

İki boyutlu sıkıştırılamaz akışta, yatay ve dikey hız bileşenleri vardır.
Süreklilik denklemi, bu bileşenlerin değişim oranlarının toplamının sıfır olmasını gerektirerek kütle korunumunu sağlar, yani herhangi bir bölgedeki net giriş ve çıkışın dengelenmesi anlamına gelir.
Bu koşulu yerine getirmek için, akış işlevi tanıtıldı.
Yatay hız, akış fonksiyonunun dikey yöne göre kısmi türevidir, dikey hız ise yatay yöne göre negatif kısmi türevdir. Bu, kütle korumasının otomatik olarak sürdürülmesini sağlar.
Akış fonksiyonunun önemli bir özelliği, akışkan parçacıklarının yollarını temsil eden ve akış alanındaki hızlara teğet olan akış çizgileri boyunca sabit kalmasıdır.
Genellikle kontur çizgileri olarak gösterilen akış çizgileri, akışkan davranışının sezgisel olarak yorumlanmasına olanak tanır.
Akış fonksiyonu, akışkan hareketi hesaplamalarını basitleştirir. İki akış çizgisi arasındaki akış fonksiyonu değerlerindeki fark, aralarındaki hacimsel akış hızını temsil eder.
Bu, hız bileşenlerini ayrı ayrı çözme ihtiyacını ortadan kaldırarak onu iki boyutlu akışı analiz etmek için verimli bir araç haline getirir.

akış fonksiyonu

Stream Function

İki boyutlu sıkıştırılamaz akışkan akışında, süreklilik denklemi kütle korunumunu sağlamak için esastır, yani bir bölgeye giren veya bölgeden çıkan akışkandaki herhangi bir değişim başka bir yerdeki karşılık gelen bir değişimle dengelenir. Yoğunluğun sabit kaldığı sıkıştırılamaz akış için, bu gereklilik hız alanının diverjansının sıfır olması koşuluna basitleştirilir. Matematiksel olarak, bu şu şekilde ifade edilir:

<img alt="Equation 1" src="/files/ftp_upload/18058/18058_Equation_1.svg" style="text-align: center; height: 27px;">

burada u ve v sırasıyla yatay ve dikey hız bileşenlerini temsil eder.

 Bu denklemi doğal olarak sağlamak için, hız bileşenlerinin ψ cinsinden şu şekilde tanımlanmasına izin veren akış fonksiyonu ψ'yi tanıtıyoruz:

<img alt="Equation 2" src="/files/ftp_upload/18058/18058_Equation_2.png" style="text-align: center; height: 30px;">

 Bu tanımlar, ψ'nin karışık kısmi türevleri birbirini götürdüğü için otomatik olarak süreklilik denklemini sağlar.

 Akış fonksiyonu ψ, akışkan parçacıklarının akış içinde izlediği yollar olan akış çizgileri boyunca sabittir. Akım çizgileri bu nedenle ψ'nin kontur çizgileri olarak görselleştirilebilir ve akışkan hareketinin net bir temsilini sağlar. Her akım çizgisi, yolu boyunca herhangi bir noktada hız vektörüne teğettir ve her noktadaki hızları ayrı ayrı hesaplamaya gerek kalmadan akış yönünü gösterir.

 Dahası, iki akım çizgisi arasındaki ψ\psiψ değerleri arasındaki fark, aralarındaki birim derinlik başına hacimsel akış hızını temsil eder ve akış hızlarının doğrudan hesaplanmasını sağlar. Bu özellik, u ve v için ayrı denklemleri çözme ihtiyacını ortadan kaldırarak iki boyutlu sıkıştırılamaz akışların analizini önemli ölçüde basitleştirir. Sonuç olarak, akım fonksiyonu ψ, sıkıştırılamaz, sabit akış içeren uygulamalarda akışkan davranışını modellemek ve yorumlamak için güçlü bir araçtır.

Akış Fonksiyonu

İki boyutlu sıkıştırılamaz akışta, yatay ve dikey hız bileşenleri vardır.
Süreklilik denklemi, kütle korunumunu şu şekilde sağlar: ...

Education

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

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

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

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 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...

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 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.

civil engineering

Stream Function for Incompressible Flow

Description

Transcript

Tags