Dr WCDK Fernando1

LEARNING OUTCOMES

• Identify the importance of ideal fluid flow analysis

• Discuss various ways to visualize flow fields

• Explain fundamental kinematic properties of fluid motion and deformation

• Discuss the concepts of vorticity, rotationality& irrotationality

• Describe simple ideal flows

• Describe and sketch combined flow patterns2WCDKF-KDU

WHAT IS AN IDEAL FLUID?

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Ideal fluid

Imaginary

Incompressible

Non-viscous

Real fluid

Real or practical

Compressible

Viscous

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INTRODUCTION• Fluid Kinematics deals with the motion of fluids

without considering the forces and momentswhich create the motion.

According to thecontinuum hypothesisthe local velocity of fluidis the velocity of aninfinitesimally small fluidparticle/element at agiven instant t. It isgenerally a continuousfunction in space andtime.

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FLUID FLOW• Lagrangian Description

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FLUID FLOW

• Eulerian Description

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FLUID FLOW

• Eulerian Description

– Pressure field p = p(x,y,z,t)

– Velocity field

– Acceleration field

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VELOCITY• In the rectangular coordinate system,

Directions Velocity components

X u dx/dt

y v dy/dt

z w dz/dt

zkyjxir

wvuV

wkvjuiV

21

222

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Ex 1

• The velocity components expressed in m/s in a fluid flow are known to be u = (6xy2+t),

v = (3yz+t2+5), w = (2+3ty) where x, y, z are given in metres and time t in seconds.

Set up an expression for the velocity vector at point P (4, 1, 2) m at T = 3 S. Also determine the magnitude of velocity for this flow field at the given location and time.

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ROTATIONAL & IRROTATIONAL FLOWS

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Kinematic Description

• In fluid mechanics, anelement may undergo fourfundamental types ofmotion.a)Translationb)Rotationc)Linear straind)Shear strain

• Because fluids are inconstant motion, motionand deformation is bestdescribed in terms of rates.

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TRANSLATION

dx

dy

A

B C

D

y

x

+

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TRANSLATION

dx

dy

A

B C

D

A’

B’ C’

D’

udt

vdt

y

x

+

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ROTATION

dx

dy

A

B C

D

y

x

+

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ROTATION• Angular rotation of element about z-axis is

defined as the average counterclockwise rotation of the two sides BC and BA

dx

dy

A

B C

DA’

B’

C’

D’

y

x

+

da

db

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ROTATION

dydty

u

y

u

x

v

dt

d

dtx

v

dx

dxdtx

v

d

dty

u

dy

dydty

u

d

ddd

z

z

2

1

tan

tan

2

1

1

1

a

b

ba

A’

B’

C’

D’

da

db

y

x

+

dxdtx

v

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EXTENSIONAL STRAIN (DILATATION)

dx

dy

A

B C

D

y

x

+

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EXTENSIONAL STRAIN (DILATATION)

dx

dy

A

B C

D

A’

B’ C’

D’• Extensional strain in x-direction is defined as the fractional increase in length of the

horizontal side of the elementy

x

+

dtx

u

dx

dxdxdtx

udx

dtxx

dxdtx

udx

Extensional strain rate in x-directionWCDKF-KDU

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SHEAR STRAIN

dx

dy

A

B C

D

y

x

+

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SHEAR STRAIN

dx

dy

A

B C

D

y

x

+

db

da

• Defined as the average decrease of the angle between two lines which are initially perpendicular in the unstrained state (AB and BC)

dt

d

dt

d

dd

xy

ba

ba

2

1

2

1Shear-strain increment

Shear-strain rateWCDKF-KDU

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DISTORTION OF A MOVING FLUID ELEMENT

dxdtx

v

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DISTORTION• Average angular displacement

• Mean rate of rotation

• The quantity is known as the Vorticity (Ω ).

• ω = ½ Ω

• For irrotational flow, ω = 0

dt

y

udt

x

v

2

1

y

u

x

v

2

1

y

u

x

v

y

u

x

v

0y

u

x

v

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A STEADY IRROTATIONAL FLOW IS CLASSIFIED AS POTENTIAL FLOW.

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CIRCULATION• Circulation is the line integral of tangential

velocity around a closed contour in the flowfield.

A measure of the rotation within a finite element of a fluid

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CIRCULATION

ldcosVldV

a

Circulation is considered positive in an anticlockwisedirection.

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yvxyy

uuyx

x

vvxuvdyudx

Calculate the circulation within a small fluid element with area yx

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yvxyy

uuyx

x

vvxuvdyudx

yxy

u

x

v

vorticityrelativey

u

x

v

yxlim

0 yx

A

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Circulation per unit area equals the vorticity in flow.

Ex 2

• Determine the circulation Τ around a rectangle defined by x=1, y=1, x=5 and y=4 for the velocity field u = 2x + 3y and v = -2y.

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FLOW VISUALIZATION• Flow visualization is the visual examination of

flow-field features.• Important for both physical experiments and

numerical (CFD) solutions.• Numerous methods

– Streamlines and streamtubes– Pathlines– Streaklines– Timelines– Refractive techniques– Surface flow techniques

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STREAMLINES

• A line in the fluid whose tangent is parallel to at a given instant t.

• Steady flow : the streamlines are fixed in space for all time.

• Unsteady flow : the streamlines are changing from instant to instant.

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STREAMLINES• A Streamline is a

curve that is everywhere tangent to the instantaneous local velocity vector.

• Equation of a general streamline

dt

dzw,

dt

dyv,

dt

dxu

w

dz

v

dy

u

dx

STREAMLINES

• For 2-D flow,

• Streamlines do not cross, otherwise the fluidparticle will have two velocities at the point ofintersection.

• The flow is only along the streamline and notcross it.

u

v

dx

dy

v

dy

u

dx

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STREAM-TUBE• is the surface formed instantaneously by all

the streamlines that pass through a given closed curve in the fluid.

Since no fluid can penetrate the streamlines, the flow passing through each of the sections would be same.

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PATHLINE• A line traced by an individual fluid particle

• For a steady flow the pathlines are identical with the streamlines.

A Pathline is the actual path traveled by an individual fluid particle over a time period.

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STREAKLINE

• A streakline consists of all fluid particles in aflow that have previously passed through acommon point. Such a line can be producedby continuously injecting marked fluid (smokein air, or dye in water) at a given location.

• For steady flow : The streamline, the pathline,and the streakline are the same.

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STREAKLINES

• A Streakline is thelocus of fluid particlesthat have passedsequentially througha prescribed point inthe flow.

• Easy to generate inexperiments: dye in awater flow, or smokein an airflow.

COMPARISON

• For steady flow, streamlines, pathlines, andstreaklines are identical.

• For unsteady flow, they can be very different.

– Streamlines are an instantaneous picture of theflow field

–Pathlines and Streaklines are flow patterns thathave a time history associated with them.

– Streakline: instantaneous snapshot of a time-integrated flow pattern.

–Pathline: time-exposed flow path of anindividual particle. 39WCDKF-KDU

Ex 3• Determine the equation of streamline for a two

dimensional flow field for which the velocitycomponents are given by

i. u = a and v = a where a is a non-zeroconstant. The streamline passes through thepoint (1, 3).

ii. u = y/b2 and v = x/a2. The streamline passesthrough the point (a, 0).

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Ex 4

• In a steady fluid flow, the velocity components are u = 2kx, v = 2ky, w = -4kz. Find the equation of streamline passing through the point (1, 0, 1).

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VELOCITY POTENTIAL FUNCTION

• Imagine that a function φ exist such that itsderivative in any direction gives the velocity inthat direction

• The function φ is called the velocity potentialfunction and lines of constant potentialfunction are termed equipotential lines.

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yv

xu

VELOCITY POTENTIAL FUNCTION

• Since φ is a function of x and y alone,

• For an equipotential line (φ = constant), dφ = 0

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dy.vdx.udyy

dxx

d

v

u

dx

dy

dx.udy.v

0dy.vdx.u

VELOCITY POTENTIAL FUNCTION

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xyyx

0

xyyx

y

u

x

v

22

Hence the velocity potential function, φexists when the flow is irrotational.

Differential Equation of Continuity

• The fluid is continuous both in space & time.

• For an incompressible fluid, the density ρwould be constant.

• For 3-D incompressible flow

• For 2-D incompressible flow

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u

x

v

y

w

z0

• When φ exists,

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2

2

2

2

yx

0yyxx

Φ satisfies the Laplace Equation

Ex 4

• Which of the following velocity fields pertain to the motion of steady, two-dimensional flow of an incompressible fluid

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STREAM FUNCTION

• Mathematically, the stream function for a flow in the x – y is defined as a function of x and y such that the velocity components are given by,

where ψ is the value of stream function.

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xv

yu

STREAM FUNCTION

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Considering the continuity of flow

xyyx

0xyyx

0xyyx

0y

v

x

u

22

22

STREAM FUNCTION

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Show that ψ satisfies the Laplace Equation for irrotational flow

0y

u

x

v

Ex 5

A fluid flows along a flat surface parallel to the x-direction. The velocity u varies linearly with y,the distance from the flat surface and u=Ay

a) Find the stream function of the flow

b) Determine whether the flow is irrotational

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Ex 6

If φ=3xy, find x and y components of velocity at(1, 3) and (3, 3). Determine the dischargepassing between streamlines.

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PROPERTIES OF φ AND ψ

Property ψ φ

Continuity equation

Automatically satisfied

Satisfied if …………………

Irrotationality condition

Satisfied if ………..

Automatically satisfied

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PROPERTIES OF φ AND ψ

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• Streamlines and equipotential lines are orthogonal to each other.

The gradient of the equipotential line = -u/vthe gradient of a stream line = v/u

FLOW THROUGH A BEND

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