POTENTIAL FLOWPOTENTIAL FLOW(Sec. 8.5.1 & 8.5.2)

In this unit you will learn

• Theoretical foundation of fluid dynamics

• Allows complex fluid dynamic phenomenon to be studied by simplified ideal fluid analysis

Hurricane or Cyclone

Tornado

NASA picture of Hurricane Wilma

Cyclone

R

Vθ(r)

Hurricanes• A type of low pressure system, generally formed

in the tropics, is accompanied by thunderstorms and a circulation of wind near the ground.– Can be modeled using potential flow theory

Types of Tropical cyclone depends on the maximum sustained wind speed, u

1. Tropical depression: u < 62 km/h2. Tropical storm: 62 < u < 117 km/s3. Hurricane: u > 119 km/h

Whirlpool

TYPES OF FLUID FLOWTYPES OF FLUID FLOW

Compressible

Viscous

Rotational

Unsteady Steady

Irrotational

Inviscid

Incompressible

Fluid Flow

Property of fluid

Property of Flow

We will study this flow

Potential Flow• Potential or Inviscid flow is a flow where viscosity

can be neglected like in high Reynolds number flows• Flow outside the boundary layer may be treated as

Potential flow• Compressible fluid flow cannot be modeled as

potential flow• Bernoulli Eqn. can be used in Potential flow to

predict pressure distribution on a surface.• It is generally described by two scalar quantities

– Stream function (ψ)– Potential function (φ)

Why Use Potential Flow?• Potential flows helps to simplify complex flow situations so

that a good approximate solution can be found• As potential flow represents an idealized flow solution it

does not exist in real flows• Most CFD (Computational Fluid Dynamics) solvers use

potential flow theory to approximate the flow solution outside the boundary layers in incompressible flows

STREAM FUNCTION• A Stream Line is a line tangential to

instantaneous velocity vector• There is no flow across streamlines• The difference of stream functions ψ

between two stream lines gives the volume of fluid per unit depthflowing between them from the left to right

CARTESIAN COORDINATES:

POLAR COORDINATES:y

u∂∂

=ψ

xv

∂∂

−=ψ

θψ

∂∂

=r

vr1

rv

∂∂

−=ψ

θ

Some streamlines

• A Potential Function, φ is defined such that its derivative in any direction gives the velocity in that direction

X-Y coordinate:

Polar coordinate:

yv

xu

∂∂

=∂∂

=φφ ;

rvr ∂

∂=

φθφ

θ ∂∂

=r

v 1

∫∫ ==A

Bp

A

Bm dsvdsvφ

POTENTIAL FUNCTION

• Stream & potential lines always intersect at right angles

Function OriginsBoth Stream and Potential functions were

developed to satisfy the continuity equations in 2 and 3 planes

Continuity Equations:

Stream and potential functions:

0=∂∂

+∂∂

+∂∂

zw

yv

xu

0=∂∂

+∂∂

yv

xu

xv

∂∂

−=ψ

xu

∂∂

=φ

zw

∂∂

=φ

yv

∂∂

=φ

yu

∂∂

=ψ

Laplace EquationsIn inviscid, incompressible, irrotational the continuity

equation in 3 planes breaks down to:

Likewise the vorticity function in 2 planes (ω=-k∆2ψ=0)breaks down to:

Bottom Line: These are second and third order Laplaceequations for which there are many solutions and analytical techniques known to solve them

02

2

2

2

=∂∂

+∂∂

yxψψ

02

2

2

2

2

2

=∂∂

+∂∂

+∂∂

zyxφφφ

Rotational & Irrotational flow• A fluid particle may rotate while moving from point to

point. For example, If side ‘Aa’ of a fluid block rotates by βand side ‘Ab’ rotates by α, the average rate of rotation of the block (ω) is the average rotation of the two sides

dt)(

21 βαω +

=

Rotational vs. Irrotational

• If both sides rotate symmetrically α = β , rotation is zero:

Such a flow is called Irrotational.

• If the sides rotate at different rate (α ≠β), the fluid is deformed and the flow is called Rotational.

0)(21

=−

=dt

βαω

0)(21

≠−

=dt

βαω

Properties of Irrotational flow

• Ideal fluid (Potential flow) is Irrotational– A fluid particle moving, without rotation cannot

develop a rotation under the action of a force. It requires a shear stress (~ rate of angular deformation) to develop rotation.

– Flow outside the boundary layer, where viscosity effect is negligible is irrotational

– Vorticity is zero in an irrotational flow• Real fluid or Viscous flow is rotational

VORTICITY• Rotation of a fluid particle is defined as the average

angular velocity of any two mutually perpendicular lines of the fluid particle. [Anticlockwise is positive]. Hence the angular velocity about z axis:

• Vorticity (ζ) is twice the angular velocity of a fluid particle.

• For irrotational flow vorticity is zero

ω22

1 ξ=

−=

dydu

dxdv

z

The element ABCD could rotate around its z axis while moving from instant t and at t+dt . Angular velocity wz oftwo edges (AB, BC)

Elemental change in angles is found by taking limitsdt = 0

−=

dtd

dtdw z

βα21

dtxv

dtdxxudx

dtdxxv

d dt ∂∂

=

∂∂

+

∂∂

= −→

.

.tanlim 1

0α

dtyu

dtdyyvdy

dtdyyu

d dt ∂∂

=

∂∂

+

∂∂

= −→

.

.tanlim 1

0β

Rotation

Rotation about z axis, x axis and y axis are

∂∂

−∂∂

=

∂∂

−∂∂

=

∂∂

−∂∂

=xw

zuw

zv

yww

yu

xvw yxz 2

1;21;

21

Twice the value of the rate of rotation is defined as vorticity, ξ. Its value about x, y & z axis are as follows

zzyyxx www 2;2;2 === ξξξFor irrotational flow vorticity is zero. So ξx = ξy= ξz = 0,

For irrotational flow, vorticity, ξ = 0

In polar coordinate, vorticity is written as

DEFORMATION

Angular deformation is total changes in the angle between two mutually perpendicular lines in the fluid. From above figure the rate of angular, γ deformation is

∂∂

−∂

∂==

θξ θ r

zzv

rrv

rw )(12

+=

dydu

dxdvγ

CIRCULATION• Circulation is defined as the line integral of the tangential

velocity component about a closed curve C, of arc length ds. Using the definition of φ we get:

• It can be proved that: Circulation = Vorticity x Area

• K denotes the strength of the vortex

( )

21

2

1

.cos

φφφ

α

−==

++===Γ

∫∫ ∫ ∫c c c

xdzvdyudxdsVdsVv

Kπ2=Γ

Forced Vortex (Man made)• A vortex represents a flow in which the streamlines

are concentric circles. There is no radial flow. So, Vr = 0

• In a Forced Vortexvelocity is proportional to radius

vθ = K .r

Tangential velocity vθ, at any radiusis inversely proportion to r

So,

From definitions of φ and ψ

Solving we get: φ = Κ θ; & ψ = − K ln r

Circulation Γ= Φ1−Φ2 = 2Κ π.

rK

rrv =

∂∂

−=∂∂

=ψ

θφ

θ1

θπ

φπ

ψ2

;ln2

Γ=

Γ−= r

Free Vortex (occurs in nature)

Whirlpool (Free or Forced ?)

Problem- vortex (Munson p-343)

• A liquid drains from a large tank through a small opening. A vortex forms whose velocity distribution away from the tank opening can be approximated as that of a free vortex having a velocity potential φ=(Γθ/2π). Determine an expression relating the surface shape to the strength of the vortex as specified by the circulation

Problem: Tornado

• A tornado may be modeled as the circulating flow with vr = vz = 0 and vρ = ωr for r < R ; = ωR2/r for r>R.

• Determine whether this flow pattern is irrotationalin either the inner or outer region. Using the pressure distribution p(r) in the tornado, assumingp = pα. Find the location and magnitude of the lowest pressure(White 8.14)