lecture 3 part 1 potential flow

8/18/2019 Lecture 3 Part 1 Potential Flow

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Potential FlowChapter Three

Dr. Hamdy A. Kandil15 March 2016

PART ONE

The Stream Function, ψ (x,y) Consider the continuity equation for an incompressible 2D flow

Substituting the clever transformation, ψ (x,y)

Defined as:

Gives

This is true for any smoothfunction ψ (x,y)

}

satisfied bealways willmass of onconservati

unknow one unknows two functionstream using

c

ψ →

v

u

eq. continuity thesatisfiesalwaysitthatso

0)()( =∂

∂−

∂

∂+

∂

∂

∂

∂=

∂

∂+

∂

∂⇒

x y y x y

v

x

u

0=∂

∂+

∂

∂

y

v

x

u


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The Stream Function, ψ Why do this?

Single variable ψ replaces (u,v ). Once ψ is known, (u,v ) can be determined.

Physical significance

1. Curves of constant ψ are streamlines of the

flow2. Difference in ψ between streamlines is equal

to volume flow rate between streamlines

The Stream Function: Physical Significance

Recall that the streamline equation isgiven by:

∴ Change in ψ alongstreamline is zero

1. Curves of constant ψ are streamlines of the flow


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Let dq represent the volume rate of flow per unit width perpendicularto the x–y plane passing between the two streamlines.

From conservation of mass we know that the inflow, dq , crossing the

arbitrary surface AC must equal the net outflow through surfacesAB and BC . Thus,

or in terms of the stream function

Thus, the volume rate of flow, q ,between two streamlines can bedetermined by integration to yield

2. Difference in ψ between streamlines is equal to volumeflow rate between streamlines

Example Stream Function The velocity components in a steady, incompressible, two-

dimensional flow field are

Determine the corresponding stream function and show on a

sketch several streamlines.Indicate the direction of flow along the streamlines.

4xv2yu ==

(y)f x2(x)f y 22

1

2+−=ψ +=ψ

From the definition of the stream function

x x

v y y

u 42 =∂

∂−==

∂

∂=

C y x ++−=222ψ

For simplicity, we set C = 0


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Example Solution

Streamlines can now be

determined by settingψ =const. and plotting theresulting curve.

With the aboveexpression for ψ , thevalue of ψ at the origin iszero so that the equationof the streamline passing

through the origin is0 = -2x2 + y2

For

22yx2 +−=ψ

Ψ=0

Ψ≠≠≠≠0 12 /

xy 22

=ψ

−ψ

which we recognize as theequation of a hyperbola

Rate of rotation (angular velocity)

Consider the rotation about z-axis of the rectangular element δδδδx-δδδδy

The rotation of the side δδδδx


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Rate of rotation (angular velocity)

Angular Velocity of OA

The rotation of the side δδδδy

Angular Velocity of OB

Rate of rotation (angular velocity) The rotation of the element about the z axis is defined as the

average of the angular velocities of the two mutuallyperpendicular lines OA and OB. If counterclockwise rotation isconsidered to be positive, then:

Average rotation about z-axis

Average rotation about x-axis,

Average rotation about y-axis,

Rotation Vector

∂

∂−

∂

∂=

y

u

x

v z

2

1ω

∂

∂−

∂

∂=

z

v

y

w x

2

1ω

∂

∂−∂

∂= x

w

z

u y

2

1ω

∂

∂−

∂

∂+

∂

∂−

∂

∂+

∂

∂−

∂

∂=++=∴ k

y

u

x

v j

x

w

z

ui

z

v

y

wk ji z y x

ˆˆˆ2

1ˆˆˆ ω ω ω ω r


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Rotational and Irrotational Flows

The vorticity is defined as:

ζ =

k y

u

x

v j

x

w

z

ui

z

v

y

w ˆˆˆ

∂

∂−

∂

∂+

∂

∂−

∂

∂+

∂

∂−

∂

∂

For irrotational flow ζ= ϖ = 0

Examples: Rotational flow:

Solid-Body Rotation(Forced Vortex): uθ = ω r

Rotational and Irrotational Flows Examples: Irrotational flow:

Free Vortex: uθ = K/r


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Mathematical Representation Vorticity is the curl of the velocity vector

For 3-D vorticity in Cartesian coordinates:

∂

∂−

∂

∂+

∂

∂−

∂

∂+

∂

∂−

∂

∂=

∂

∂

∂

∂

∂

∂≡×∇

y

u

x

vk

x

w

z

u j

z

v

y

wi

wvu

z y x

k ji

V vvv

vvv

v

The horizontal relative vorticity (about z axis) is found byeliminating terms with vertical ( ω) components:

( )

∂

∂−

∂

∂=

∂

∂

∂

∂

∂

∂≡×∇•

y

u

x

v

wvu

z y x

k ji

V k

vvv

vv


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α

+

NOTE:The flow is irrotational if ω=0, ξ=0, Г=0

Circulation ( Г ) The circulation ( Г ) is a measure of rotation and is defined as

the line integral of the tangential component of the velocitytaken around a closed curve in the flow field.

ld V ld V

rrr•=•=Γ

∫∫ α cos

vdydxdy y

uudydx

x

vvudxd −

∂

∂+−

∂

∂++=Γ =Γ ∫ )()(

area

dxdy yu

xv

z .

)(

ζ =

∂

∂−

∂

∂=

= ξ . area

For 2-D Cartesian Coordinates


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Potential Function, φ Irrotational approximation:

vorticity is negligibly small

In general, inviscid regions arealso irrotational, but there aresituations where inviscid floware rotational, e.g., solid bodyrotation.

What are the implications of irrotational approximation. Look at

continuity and momentum equations.Use the vector identity where φ is a scalar functionSince the flow is irrotational where

φ is a scalar potential function

Irrotational Flow Approximation

Therefore, regions of irrotational flow are also called regions ofpotential flow.

From the definition of the gradient operator ∇∇∇∇

Substituting into the continuity equation for incompressible flow

gives:

Cartesian

Cylindrical

zw

yv

xu

∂∂=

∂∂=

∂∂= φ φ φ ,,

zu

r u

r u zr

∂

∂=

∂

∂=

∂

∂=

φ

θ

φ φ θ ,

1 ,


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Irrotational Flow Approximation This means we only need to solve 1 linear scalar equation to

determine all 3 components of velocity!

Luckily, the Laplace equation appears in numerous fields ofscience, engineering, and mathematics. This means there arewell developed tools for solving this equation.

Laplace Equation

Momentum equationIf we can compute φ from the Laplace equation (which came fromcontinuity) and velocity from the definition , why do we

need the NSE? ⇒ the answer: To compute Pressure.To begin analysis, apply irrotational approximation to viscous termof the incompressible NSE

= 0

Irrotational Flow Approximation Therefore, the NSE reduces to the Euler equation for

irrotational flow

Instead of integrating to find P, use vector identity to deriveBernoulli equation


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Irrotational Flow Approximation This allows the steady Euler equation to be written as

This form of Bernoulli equation is valid for inviscid and irrotationalflow since we’ve shown that NSE reduces to the Euler equation.

However,

Inviscid

Irrotational (ζ = 0)

Irrotational Flow Approximation Therefore, the process for irrotational flow

1. Calculate φ from Laplace equation (from continuity)

2. Calculate velocity from definition

3. Calculate pressure from Bernoulli equation (derived from

momentum equation)

Valid for 3D or 2D

∞

∞∞ ++=++ gzV p

gzV p

22

22

ρ ρ


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Irrotational Flow Approximation2D Flows For 2D flows, we can also use the stream function ψ Recall the definition of stream function for planar (x-y) flows

Since vorticity is zero for irrotational flow,

This proves that the Laplace equation holds for the streamfunction and the velocity potential

xv

yu

∂

∂−=

∂

∂=

ψ ψ

0=∂

∂−

∂

∂=

y

u

x

v zζ

= ∇2ψ

Irrotational Flow Approximation2D Flows

Constant values of ψ :streamlines

Constant values of φ :

equipotential lines ψ and φ are mutually

orthogonal ψ is defined by continuity;

∇2ψ results fromirrotationality

φ is defined by irrotationality;∇2φ results from continuity

Flow solution can be achieved by solving either ∇2φ or ∇2ψ ,however, BCs are easier to formulate for ψ.


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Relation between ψ and φ lines If a flow is incompressible, irrotational, and two dimensional, the

velocity field may be calculated using either a potential function or astream function.

Using the potential function, the velocity components in Cartesiancoordinates are

And

For lines of constant potential (dφ = 0), which are called

equipotential lines:

Since a streamline is everywhere tangent to the local velocity, theslope of a streamline, which is a line of constant ψ , is

yv

xu

∂

∂=

∂

∂=

φ φ ,

vdyudxdy y

dx x

d +=∂

∂+

∂

∂=

φ φ φ

v

u

dx

dy

c

−=

=φ

Relation between ψ and φ lines

Comparing equations of slopes yields:

The slope of an equipotential line is the negative reciprocal ofthe slope of a streamline.

Therefore, streamlines (ψ = constant) are everywhere

orthogonal (perpendicular) to equipotential lines (φ = constant). This observation is not true, however, at stagnation points,

where the components vanish simultaneously.

u

v

dx

dy

c

=

=Ψ

cc dxdydx

dy

=Ψ=

−=

)(

1

φ


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Irrotational Flow Approximation2D Flows Similar derivation can be performed for cylindrical coordinates

(except for ∇2ψ for axisymmetric flow)

Planar, cylindrical coordinates: flow is in (r,θ ) plane

Axisymmetric, cylindrical coordinates : flow is in (r,z ) plane

Planar Axisymmetric



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Potential flows Visualization

For such a system, dyeinjected upstream reveals anapproximate potential flowpattern around a streamlinedairfoil shape.

Similarly, the potential flowpattern around a bluff body isshown. Even at the rear of the

bluff body the streamlinesclosely follow the body shape.

Generally, however, the flowwould separate at the rear ofthe body, an importantphenomenon not accounted forwith potential theory.

Flow fields for which an incompressible fluid is assumed to be frictionless andthe motion to be irrotational are commonly referred to as potential flows.

Paradoxically, potential flows can be simulated by a slowly moving, viscous

flow between closely spaced parallel plates.


Method of Superposition

1. Since ∇2φ = 0 is linear, a linear combination of two or more

solutions is also a solution, e.g., if φ 1 and φ 2 are solutions,then (Aφ 1), (A+φ 1), (φ 1+φ 2), (Aφ 1+Bφ 2) are also solutions

2. Also true for ψ in 2D flows (∇2ψ =0)3. Velocity components are also additive


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Given the principal of superposition, there are severalelementary planar irrotational flows which can becombined to create more complex flows.

Elementary Planar Irrotational Flows

Uniform stream

Line source/sink

Line vortex

Doublet

Elementary Planar Irrotational FlowsUniform Stream

u = U∞=constant , v = 0, w = 0

In Cartesian coordinates

φ = U∞ x , ψ = U∞ y

Conversion to cylindricalcoordinates can be achievedusing the transformation

φ = U∞ r cos θ , ψ = U∞ r sin θ

U∞


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The flow is an incoming far field flow which isperpendicular to the wall, and then turns its direction near

the wall The origin is the stagnation point of the flow. The velocity

is zero there.

Stagnation Flow

x

ψ ψψ ψ

y

Application: Stagnation Flow

For a stagnation flow,

Hence,

Therefore,

And

Therefore

θ φ 2cos22

222 r B

) y x( B

=−=

Byv y Bxu x −==∂

∂

==∂

∂ φ φ ,

( ) j Byi Bx ˆˆ −=Vr

θ ψ 2sin2

2r

B Bxy ==

Byv x Bxu y −==∂

∂

−==∂

∂ψ ,


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Elementary Planar Irrotational FlowsLine Source/Sink Let’s consider fluid flowing

radially outward from a linethrough the originperpendicular to x-y plane

from mass conservation: The volume flow rate per unit

thickness is K

This gives velocitycomponents

r r v

r r r

K vr

∂

∂−=

∂

∂==

∂

∂=

∂

∂==

ψ

θ

φ

θ

φ

π

θ 0

2

and

0 2

== θ π

vr

K vr and

K

v r

Stream function and potential function

2

0r

K

r r π

φ

θ

φ =

∂

∂=

∂

∂ and

By integration:

rln2π

φ K

=

π θ

ψ

π θ

ψ

2

2

0

K

r

K

r

r

=

∂

∂⇒=

∂

∂

=∂

∂−

and

By integration:

θ π

2

K =Ψ Equations are for a source/sink

at the origin

K


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Elementary Planar Irrotational FlowsLine Source/Sink

If source/sink is movedto (x,y) = (a,b)

K

K K

K K

Elementary Planar Irrotational FlowsLine (potential) Vortex

A potential vortex is defined as asingularity about which fluid flowswith concentric streamlines

Vortex at the origin. First look at

velocity components

These can be integrated to give φ

and ψ Equations are for a line vortexat the origin where thearbitrary integration constantsare taken to be zero at(r,θ)=(1,0)

∅

−

Γ

∅

vθ

&


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The potential represents a flow swirling around origin with aconstant circulation Γ .

The magnitude of the flow decreases as 1/r.

Free Vortex

Line Vortex now we consider situation when the

stream lines are concentric circles i.e. weinterchange potential and streamfunctions:

ln

K

K r

φ θ

ψ

=

= −• circulation

0C C C

ds ds d φ φ Γ = ⋅ = ∇ ⋅ = =∫ ∫ ∫ V • in case of vortex the circulation is zero

along any contour except ones enclosingorigin

2

0

( ) 2

ln2 2

K rd K r

r

π

θ π

φ θ ψ π π

Γ = =

Γ Γ = = −

∫


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Stokes’s Theorem The transformation from a line integral to a surface integral in

three-dimensional space is governed by Stokes’s theorem :

where n dA is a vector normal to the surface, positive when pointingoutward from the enclosed volume, and equal in magnitude to theincremental surface area.

In words, the integral of the normal component of the curl of thevelocity vector over any surface A is equal to the line integral ofthe tangential component of the velocity around the curve C whichbounds A .

Stokes’s theorem is valid when Arepresents a simply connected region inwhich V S is continuously differentiable.

Thus, it is not valid if the area A containsregions where the velocity is infinite.

Potential Vortex The curl of the velocity vector for the potential vortex can be

found using the definition for the curl of V S in cylindricalcoordinates

But and Γ


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which simplifies in two dimensions to:

Although the flow is irrotational ( ζ=0), we must remember that

the velocity is infinite at the origin (i.e., when r = 0 ). In fact, the flow field at the origin is rotational and vorticity

exists there.

We will now calculate the circulation around a closed curve C 1which encloses the origin. We can choose a circle of radius r 1

The circulation is

Recall that Stokes’s theorem, is not valid if the region containspoints where the velocity is infinite, which is true for vortexflow at the origin.

However, if we calculate the circulation around a closed curve C 2,which does not enclose the origin, such as that shown in Fig. 3.13b, we find that

Therefore, the circulation around a closed curve not containingthe origin is zero.

Paths for the calculation of the circulation for a potential vortex: (a)closed curve C 1, which encloses origin; (b) closed curve C 2, which does

not enclose the origin.


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Shape of the free surface

2φ θ

π

Γ =

2

2

p V gz const

ρ + + =

2 2

1 2

2 2

V V z

g g= +

at the free surface p=0:

2

2 28 z

r gπ

Γ = −

∅

Γ

Bernolli’s equation


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Elementary Planar Irrotational FlowsLine Vortex

If vortex is moved to (x,y) =(a,b)

lecture 3 part 1 potential flow

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