Ch 4Fluids in Motion

IntroductionIn the previous chapters we have defined some basic properties of fluids and have considered various situations involving fluids that are at rest.

In general, fluids have a well-known tendency to move or flow. The slightest of shear stresses will cause the fluid to move. Similarly, an appropriate imbalance of normal stresses (pressure) will cause fluid motion.

In this chapter we will discuss various aspects of fluid motion without being concerned with the actual forces necessary to produce the motion. That is, we will consider the kinematics of the motion—(the velocity and acceleration of the fluid, and the description and visualization of its motion. No forces)

The analysis of the specific forces necessary to produce the motion (the dynamics of the motion) will be discussed in detail in the following chapters.

A wide variety of useful information can be gained from a thorough understanding of fluid kinematics. Such an understanding of how to describe and observe fluid motion is an essential step to the complete understanding of fluid dynamics.

Flow Patterns: Pathline

The pathline is the line traced out by a fluid particle.

t = 0t = 1

t = 2 t = 3

Pathline

Flow Patterns: Streamline

The streamline is a curve that is everywhere tangent to the local velocity vector.

Flow Patterns: StreaklineIt is the instantaneous locus of all fluid particles that have passed through a given point.

4 32 1

StreaklineA

If at point A in a flow field, a dye is injected, then the photograph of the dye streak would be a streakline. In other words, if fluid particles 1 through 4 have passed successively through point A, the shown dotted line (joining all these particles at time t) would be the streakline.

Dividing Streamline and Stagnation Point

• When an object divides the flow, then the streamline that follows the flow division is called “dividing streamline”.

• The point of division is called the stagnation point (since the flow is stagnant there).

StagnationPoint

DividingStreamline

Laminar and Turbulent Flow

• In laminar flow, the fluid flows in layers parallel to each other. No mixing.

• In turbulent flow, the velocity is fluctuating with time and a strong mixing occurs between fluid layers.

1 Dimensional Flow

Pipe flow:

Function of radial position, r, only !

Duct flow:

Function of axial distance, x, only !

Not very good assumption but many practical flows could be modeled as 1 D flow

2 Dimensional Flow

3 Dimensional Flow

Methods of Predicting Velocity Field

• Numerical methods:Solving the same set of equations using numerical methods.

Predicted streamline pattern over theVolvo ECC prototype.

•Analytical methods:Solving a set of equations to get the velocity field.

Pathlines of floating particles

Smoke traces about an airfoil

with a large

angle of attack.

• We inject fluid markers (ink or dust) and study the streamlines, pathlines and streaklines.

Experimental methods

Volume Flow Rate

• Flow rate (or discharge, Q) is defined as the rate at which a certain fluid volume passes through a given section in the flow stream.

• V is assumed to be constant.

VAt

tVATime

VolumeQ

)(

Volume Flow Rate, QIf the velocity is constant over the cross section,

AVQ

V

A AdVQAdVdQthen

dAVdQ But if V is not normal to dA

Average Velocity

A

AdV

A

QV

AdVQAV

A

A

We define the average velocity as

averageV max

V

But what is dA?

Average Velocity Continued ..

dyybdA

dybdA

drrdA

triangle

glerec

circle

)(

2

tan

y

dybV= f(y)

V average

y = 0

y = adA

ydy

b

dA

r

R

dr

Mass Flow Rate

Qm

AVm

AdVm

AdVmd

A

Acceleration in Cartesian and Streamline Coordinates

(a) Cartesian coordinates (b) Streamline coordinates

Acceleration in Cartesian Coordinates

u v w

This is a vector result whose scalar components can be written as

1D acceleration 2D acceleration

3D acceleration

Local Acceleration

Convective Acceleration

Streamline CoordinatesIn the streamline coordinate system the flow is described in terms of one coordinate along the streamlines, denoted s, and the second coordinate normal to the streamlines, denoted n.

Tangential acceleration, at

Acceleration in Streamline Coordinates

Local acceleration

Convective acceleration.

Normal acceleration, an

nt er

Ve

s

VV

t

Va

)()(

2

Uniform Flow Patterns

0

s

VV

In the uniform flow, the velocity vector (magnitude + direction) does not change a long a streamline.The streamlines should be straight and parallel to each other.

0convective

a

0

s

VV

In figure a, the streamlines are straight but not parallel. So, a change in the velocity magnitude will occur as we move along the streamline.

In figure b, the streamlines are parallel but they are not straight. So, a change in the velocity direction occurs.

0convective

a

Non-uniform Flow Patterns

System and Control Volume• A fluid system is a given quantity of matter

consisting always of the same matter.• A control volume (CV) is a geometric volume

defined in space and enclosed by a control surface.

Lagrangian Method• The position of a

specific fluid particle traveling along a pathline is recorded with time.

There are two approaches to describe the velocity of a flowing field.

kwjviudt

trdtV

kzjyixtr

)(

)(

)(

Eulerian Method

Traffic Engineer

Eulerian Approach

• The properties of fluid particle passing a given point in space are recorded with time.

• The Eulerian approach is generally used to analyze fluid motion.

Control Volume Equation(Reynolds Transport Equation)

This equation relates the time rate of change of a property of a system to the time rate of change of the property in a control volume plus the net efflux of the property across the control surface.

CV CS

sys AdVbdbdt

d

dt

dB

• Intensive properties are those that are independent of the mass of the system.

• Extensive properties are those that are dependent on the

system mass.

Intensive and Extensive properties of a System

P

T

V

m

P

T

V

m

2121

P

T

V

m

2121

Ext

ensi

vep

rop

erti

esIn

ten

sive

pro

per

ties

The amount of an extensive property that a system possesses at a given instant, can be determined by adding

up the amount associated with each fluid particle in the system.

syssys

sys

dbB

dbdB

Derivation of theControl Volume Equation

(Reynolds Transport Equation)

See also handouts

tVl 11

If the flow is steady, this term is zero

If the velocity is not uniform over the cross section

cs

sys AVbdbdt

d

dt

dB

Uniform Velocity distribution

cs

sys AVbdbdt

d

dt

dB

Application of Reynolds Transport Equation to

Conservation of Mass Principle (Integral Form of the Continuity Equation)

General form of the Integral continuity equation

cs

sys AVbdbdt

d

dt

dB

cs

sys AVddt

d

dt

dM )1()1(

0

cs

AVddt

d

Continuity at a PointDifferential Form of the Continuity Equation

• If the flow is steady

tw

zv

yu

x

)()()(

0)()()(

wz

vy

ux

• If the flow is also incompressible

0

z

w

y

v

x

u

Applications on Continuity

Equation

Rotation• The rotational rate of a fluid element is the average

rotational rate of two initially perpendicular sides of a fluid particle.

)(2

1

y

u

x

vz

)(2

1

z

v

y

wx

)(2

1

x

w

z

uy

Vorticity

ky

u

x

vj

x

w

z

ui

z

v

y

wz

)()()(

For irrotational flow:

z

v

y

w

x

w

z

u

y

u

x

v

The Vorticity of a fluid particle is a vector equal to twice the rotational rate of the particle.

)(2 kjizyxz

Vortices• A forced vortex is a

rotational flow with concentric circular streamlines in which the fluid rotates as a solid body.

• A free (potential) vortex is an irrotational flow in which the velocity varies inversely as the distance from the center.

Separation• Separation in a

flow occurs when the streamlines move a way from the body boundaries and a local re-circulation region occurs.