AKM

BÖLÜM 3

AKIŞKANLARIN KİNEMATİGİ

Engineering Fluid Mechanics 8/E by Crowe, Elger, and RobersonCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

Dr . Ercan Kahya

LAGRANGIAN & EULERIAN DESCRIPTIONS Lagrangian Approach:

Akışkanın hareketini Lagrange bakış açısı ile belirlemek demek, her bir akışkan parçasının yaptığı hareketi teker teker belirlemek demektir

Describe the fluid particle’s motion with time.

The path of a particle: r(t) = x(t) i + y(t) j + z(t) k i, j, k: unit vectors

Velocity of a particle : V(t) = dr(t) / dt = u i + v j + w k

Eulerian Approach:

■ Akım alanının her noktasında, hareket ile ilgili büyüklüklerin (hız, basınç, vb.) zamanla nasıl değiştiklerini belirleyelim

Imagine an array of windows in the flow field: Have information for the fluid particles passing each window for all time.

In this case, the velocity is function of the window position (x, y, z) and time.

u = f1 (x, y, z, t) v = f 2 (x, y, z, t) w = f 3 (x, y, z, t)

Eulerian approach is generally favored

Streamlines & Flow Patterns

Flow Pattern: Construction of streamlines showing the flow direction

Streamlines (light blue): Local velocity vector is tangent to the streamline at every point along the line at a single instant.

Flow through an opening in a tank & over an airfoil section.

Streamline & Pathline

Streamline: line drawn through flow field such that local velocity vector is tangent at every point at that instant

– Tells direction of velocity vector

– Does not directly indicate magnitude of velocity

• Pathline: shows the movement of a particle over time

► In unsteady flow, all can be distinct lines.

► The latter two tells us the history of flow as the former indicates the current flow pattern.

3.3. AKIM ÇiZGiSi VE AKIM BORUSU

■ Hız vektörlerine teğet olarak çizilen eğrilere akım çizgileridenir.

Akım çizgisi ile Şekil 3.1 de tanımlanan yörünge aynı şey midir?

Akım çizgisi ve yörünge ancak, zamanla-değişmeyen akım halinde üst üste düşerler. Zamanla-

değişen akım halinde bunlar farklı farklı şeylerdir.

Examples...

Predicted streamline pattern over the Volvo ECC prototype.

Pathlines of floating particles.

TYPES OF FLOW

Uniform: Velocity is constant along a streamline(Streamlines are straight and parallel)

0

s

V

0

s

V

Non-uniform: Velocity changes along a streamline (Streamlines are curved and/or not parallel)

Express velocity V = V(s,t)

Vortex flow

Steady: streamline patterns are not changing over time

zamanla-değişmeyen akım (permanan akım)

Unsteady: velocity at a point on a streamline changes over time

0

t

V

0

t

V

Flow patterns can tell you whether flow is uniform or non-uniform, but not steady vs. unsteady… Why?

Because streamlines are only instantaneous representation of the flow velocity.

TYPES OF FLOW

LAMINAR & TURBULENT FLOW

(a) Experiment to illustrate the type of the flow (b) Typical dye streaks for different cases

(a) (b)

Engineering Fluid Mechanics 8/E by Crowe, Elger, and RobersonCopyright © 2005 by John Wiley & Sons, Inc. All rights reserved.

LAMINAR & TURBULENT FLOW

DIMENSIONALITY OF FLOW FLIED

→ Characterized by the number of spatial dimensions needed to describe velocity field.

1-D flow:

Axisymmetric uniform flow in a

circular duct

2-D flow:

Uniform flow in a square duct

3-D flow:

Uniform flow in an expanding

square duct

FLOW ACCELERATION (rate of change of velocity with time)

• Consider a fluid particle moving along a pathline...

• There are two components of acceleration:

Tangential to pathline

at : the time-dependent acceleration related to change in speed.

Normal to pathline

an : the centripetal acceleration related to motion along a curved pathline.

Flow Acceleration

Local acceleration – occurs when flow is unsteady (direction or magnitude is changing with respect to time)

Convective acceleration – occurs when flow is nonuniform (acceleration can depend on position in a flow field)

Local acceleration – occurs when flow is unsteady

Centripetal acceleration – occurs when the pathline is curved(normal to the pathline & directed toward the center of rotation)

Example: Convective Acceleration

The nozzle shown below is 0.5 meters long. Find the convective acceleration at x = 0.25 m. The equation describing velocity variation is provided below.

Problem 4.17:

Problem 4.17: (Solution)

Example: