potential flow - chapter 6_3 (2012)

8/2/2019 Potential Flow - Chapter 6_3 (2012)

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At the surface of the cylinder :

The maximum velocity occurs at the top and bottom of the cylinder,

magnitude of 2U.

Lift and Drag for a Cylinder without CirculationLift and Drag for a Cylinder without Circulation

Liftand drag are defined as forces per unit length on the cylinder in directions

normal and parallel respectively to the uniform flow.

Pressure for the combined flows becomes uniform at large distances from the

cylinder, where the effectsof the doublet become diminishingly small.

If this pressureis known,p0 as wellas thevelocityV 0, Bernoullis equation may

be used between infinity and points on the boundary of the cylinder in order to

determine the pressure at the boundary. Potential energy changes are

regarded neglible.

Subscript b refers to the cylindrical boundary.

2 2

0 0

far away from cylinder boundary of cylinder

2 2

b bV p V p

g g g g

Proof:

Show that the Drag for a

cylinder without circulation isequal to zero


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FluidFluid cannotcannot penetratepenetrate thethe boundary,boundary, thereforetherefore VVbb mustmust bebeinin thethe transversetransversedirection,direction, thereforetherefore iti t isis equalequal toto VV..

FromFromthethe equationequation forforVV (previously(previously defined)defined) atat thethe boundaryboundary

SubstitutingSubstituting intointo equationequation KK andand solvingsolving forforppbb

ProofProof::

0

rV

sin2

sinsin

sinsin

1

0

0

0

2

0

0

V

VV

V

V

rVVb

2 2 2

0 0 04 sin

2 2b

V p Vp g

g g g

To compute the DRAG, the force components in the xTo compute the DRAG, the force components in the x--direction are integrateddirection are integrated

which stems from the pressure on the boundary:which stems from the pressure on the boundary:

Drag = pressure (inDrag = pressure (in xxdirection)direction) circumference:circumference:

2

0cosbD p rd

2 2 2

20 0 0

00

4 sincos

2 2

V p VD g d

g g g V

2 2 22 2 2

0 0 0

0 0 00

sincos cos cos

2 2

V p VD g d d d

V g g g

22 3

0

0 0

sin0 0

6

VD g

V g

D = 0

Similarly, the lift L will also compute to zero.Similarly, the lift L will also compute to zero.

(Lift = pressure (in(Lift = pressure (in yydirection)direction) circumference)circumference)

Theoretical and experimental agree wellon the front of the cylinder.

Flow separation on the back-half in the

real flow due to viscous effect s causes

differences between the theory and

experiment.

Proof: (Self)

Show that the Lift for a

cylinder without circulation is

equal to zero


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Real experience Drag and lift do exist, called DAlembert

paradox. Zero drag and lift was found because viscous effectswere disregarded. There are viscous effec ts close to the

boundary. This region is called the boundary layer.

Dependant on

Reyno0lds Number:Jean le RonddAlembert(1717-1783)

Example 2Example 2

A potential flow with a free stream velocity of 5m/s

flows over a long semicircular bump. If the free

stream pressure is 101.325kPa and the

temperature is 60C, what is the force from the

flow on the bump per unit length of the bump? The

radius of the bump is 2m.

Case of the Rotating CylinderCase of the Rotating Cylinder

Experiment:Experiment:

Use a light cardboard cylinder, set it in motion in such that (see below):Use a light cardboard cylinder, set it in motion in such that (see below):

The axis of the cylinder is perpendicular to the direction of motion.The axis of the cylinder is perpendicular to the direction of motion.

The cylinder is set spinning rapidly about its own axis as the axis moves.The cylinder is set spinning rapidly about its own axis as the axis moves.

Trajectory is illustrated to the

left. Clearly lift is also present,

associated with the rotation of

the cylinder.

This experiment involves an

actual fluid with viscous action

in regions of high veloci ty

gradients, called the boundary

layer.As a result of the cylinder rotating, a certain

amount of rotary motion is developed of the air

about the cylinder. This aspect of the fluid motion

causes the lift.


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This rotary effec t may be simulated inThis rotary effec t may be simulated in irrotationalirrotational analysis by superposing aanalysis by superposing a

vortex onto a doubletvortex onto a doublet from the previous analysis.from the previous analysis.

Stream Function and Velocity Potential for the combination ofStream Function and Velocity Potential for the combination of doublet, vortexdoublet, vortex

and uniform flowand uniform flow are:are:

Need to find a streamline which encloses the region of physical interest.Need to find a streamline which encloses the region of physical interest.

To locate the stagnation points, set velocity components equal to zero.To locate the stagnation points, set velocity components equal to zero.

is the strength of

the vortex

Setting Vrequal to zero:

A zero radial component may occur at or along a circle with radius

Setting the transverse velocity component to zero:

For zero transverse velocity:

0cos20

rV

2

0V

0

2

sin20

rr

V

21

0

1

20

1

4sin

2sin

V

rV

r

Thus,Thus, for a stagnation point both radial and transverse components mustfor a stagnation point both radial and transverse components must

be zerobe zero, therefore the stagnation points occur at:, therefore the stagnation points occur at:

andand

There will be two stagnation points, since there are two angles for a given sine.There will be two stagnation points, since there are two angles for a given sine.

To determine the streamline through these po ints, evaluate at these points.To determine the streamline through these po ints, evaluate at these points.

Substitute the above coordinates into the stream function:Substitute the above coordinates into the stream function:

To obtain the streamlines, set the streamline equation equal to the aboveTo obtain the streamlines, set the streamline equation equal to the above

constant and rearrange:constant and rearrange:

0

rV

1

0

sin4 V

2

1

0

ln2

Vstag

0lnln2

sin2

1

0

0

Vr

rrV


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If we choose the interior of the circlet to represent the solid cylinder, the outerIf we choose the interior of the circlet to represent the solid cylinder, the outerstreamline pattern is shown below.streamline pattern is shown below.

The effects of the vortex and doublet becomes neglibly small when one

goes to large distances from the cylinder, where the flow becomes uniform

at infinity.

The vortex has one singular po int, and the circulation about this point equals ,The vortex has one singular po int, and the circulation about this point equals ,

the strength of the vortex.the strength of the vortex. Therefore the circulation about the cylinder mustTherefore the circulation about the cylinder must

also be equal to .also be equal to .

Lift and Drag for a Cylinder with CirculationLift and Drag for a Cylinder with Circulation

Presence of a vortex results in a definite lift fo r the cylinder

ABOVE the cylinder, the vortex motion addsto the velocity of flow from the

uniform flow and doublet

BELOW the cylinder, the vortex takes away from the velocity f rom the

uniform flow and doublet.

From Bernoullis equation, the pressure on the upper half of the

cylinder is lower than the pressure on the lower half resulting in the lift

upward.

The strength of the circulation determines the flow patterns, but does

not have any influence on the radius of the cylinder.

Pressure at large distances from cylinder becomes uniform, at infinity p0

Using Bernoulli between points on boundary and infinity (disregard potentialUsing Bernoulli between points on boundary and infinity (disregard potential

energy change):energy change):

Velocity at boundary must be in the transverse direction (cannot penetrateVelocity at boundary must be in the transverse direction (cannot penetrateboundary)boundary)

Inserting the velocity into Bernoulli yields the pressure at the boundary.Inserting the velocity into Bernoulli yields the pressure at the boundary.

Lift:Lift:

Proof (self):Proof (self): (lift per unit length)

2 2

0 0

2 2

bb

V p Vp g

g g g

0

0

0

12 s in

2b

rV

VV V

r

2

00

sinb

L p dV

The cylinder in this case is rotating clockwise (vortex is rotating clockwise with

< 0), and the uniform velocity is in the positive x-direction. Therefore the lift is

upward. If the cylinder was rotating counter clockwise (vortex is rotating

clockwise with > 0), the lift is downward.


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Thus, for a cylinder with ci rculation, lift is equal to the product of the fluidThus, for a cylinder with ci rculation, lift is equal to the product of the fluiddensity, the upstream velocity and the circulation.density, the upstream velocity and the circulation.

This force, which acts perpendicular to the direction of the approach velocityThis force, which acts perpendicular to the direction of the approach velocitycausescauses golfgolf balls to curve when they spin as they are propelled through theballs to curve when they spin as they are propelled through theair.air.

The development of lift on rotating bodies is called theThe development of lift on rotating bodies is called the MagnusMagnus--effect.effect.

The above equation gives the lift per unit length for a right cylinder of anyThe above equation gives the lift per unit length for a right cylinder of anycrosscross--sectional shape placed in a uniform,sectional shape placed in a uniform, inviscidinviscid streamstream..

TheThe generalgeneral equationequation thatthat givesgives thethe liftlift asas aa functionfunction ofof fluidfluid density,density, velocityvelocity

andand circulationcirculation isis calledcalled thethe KuttaKutta--JoukowskiJoukowski lawlaw(This(This isis thethe samesame lawlaw thatthat isis

soso widelywidely usedused toto determinedetermine liftlift onon airfoilsairfoils forfor aerospaceaerospace applicationsapplications..))

The above equation is valid for two-dimensional, incompressible, steady flow

about a boundary of ANY cross-sectional shape

MajorMajor DisadvantagesDisadvantages::

1.1. AssumptionAssumption ofof inviscidinviscid flowflow makesmakes thethe modelmodel unrealisticunrealistic

2.2. UsingUsing simplesimple solutionssolutions forfor constructingconstructing solutionssolutions forfor moremore complexcomplex flowflow

fieldsfields isis limitinglimiting

3.3. ItIt cannotcannot modelmodel flowflow separationseparation whichwhich isis thethe majormajor causecause ofof dragdrag andand liftlift

MajorMajor AdvantagesAdvantages

1.1. ForFor lowlow viscosityviscosity fluids,fluids, quickquick andand reasonablereasonable approximateapproximate solutionssolutions cancanbebe

determineddetermined

2.2. EvenEven forfor viscousviscous flow,flow, effecteffect ofof frictionfrictionisis limitedlimited toto thethe boundaryboundary layerlayer andand

hencehence potentialpotential flowflow analysisanalysis cancanbebe usedused forfor quickquick approximationsapproximations

3.3. CanCanbebe usedused toto determinedetermine pressurepressure distributionsdistributions whichwhich cancan thenthen bebe usedused inin

moremore accurateaccurate modelsmodels

potential flow - chapter 6_3 (2012)

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