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Title:Potential FlowsAdvisor : Ali R. Tahavvor, Ph.D.By : Hossein Andishgar 87682175 Pouya Zarrinchang 87682177

Islamic AzadUniversity Shiraz Branch Department ofEngineering .Faculty ofMechanical EngineeringYear:2012 June

FLOWS

FLOWS

FLOWS

STREAKLINES

The Streaklines

Streamtubes

Basic Elements for Construction Flow Devices Any fluid device can be constructed using following Basic elements.The uniform flow: A source of initial momentum.Complex function for Uniform Flow : W = UzThe source and the sink : A source of fluid mass.Complex function for source : W = (m/2p)ln(z)The vortex : A source of energy and momentum.Complex function for Uniform Flow : W = (ig/2p)ln(z)Velocity fieldVelocity field can be found by differentiating streamfunction

On the cylinder surface (r=a)

Normal velocity (Ur) is zero, Tangential velocity (U) is non-zero slip condition.10Irrotational Flow ApproximationIrrotational approximation: vorticity is negligibly small

In general, inviscid regions are also irrotational, but there are situations where inviscid flow are rotational, e.g., solid body rotation

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Irrotational Flow Approximation 2D flowsFor 2D flows, we can also use the stream functionRecall the definition of stream function for planar (x-y) flows

Since vorticity is zero,

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

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Elementary Planar Irrotational FlowsUniform StreamIn Cartesian coordinates

Conversion to cylindrical coordinates can be achieved using the transformation

Proof with Mathematica 14

Elementary Planar Irrotational Flows source & sinkA doublet is a combination of a line sink and source of equal magnitudeSource

Sink

17Elementary Planar Irrotational FlowsLine Source/SinkPotential and streamfunction are derived by observing that volume flow rate across any circle is This gives velocity components

18Elementary Planar Irrotational FlowsLine Source/SinkUsing definition of (Ur, U)

These can be integrated to give and

Equations are for a source/sinkat the originProof with Mathematica 19Elementary Planar Irrotational FlowsLine Source/SinkIf source/sink is moved to (x,y) = (a,b)

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Elementary Planar Irrotational FlowsLine VortexVortex at the origin. First look at velocity components

These can be integrated to give and

Equations are for a source/sinkat the origin23Elementary Planar Irrotational FlowsLine VortexIf vortex is moved to (x,y) = (a,b)

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Two types of vortexRotational vortex Irrotational vortex

Cyclonic Vortex in Atmosphere

THE DIPOLE Also called as hydrodynamic dipole.It is created using the superposition of a source and a sink of equal intensity placed symmetrically with respect to the origin.Complex potential of a source positioned at (-a,0): The complex potential of a dipole, if the source and the sink are positioned in (-a,0) and (a,0) respectively is :

Complex potential of a sink positioned at (a,0):

Streamlines are circles, the center of which lie on the y-axis and they converge obviously at the source and at the sink. Equipotential lines are circles, the center of which lie on the x-axis.

Dipole

Elementary Planar Irrotational FlowsDoubletAdding 1 and 2 together, performing some algebra, and taking a0 gives

K is the doublet strength33THE DOUBLET A particular case of dipole is the so-called doublet, in which the quantity a tends to zero so that the source and sink both move towards the origin.The complex potential of a doublet

is obtained making the limit of the dipole potential for vanishing a with the constraint that the intensity of the source and the sink must correspondingly tend to infinity as a approaches zero, the quantity

Doublet 2D

Examples of Irrotational Flows Formed by SuperpositionSuperposition of sink and vortex : bathtub vortex

SinkVortex37

Examples of Irrotational Flows Formed by SuperpositionFlow over a circular cylinder: Free stream + doublet

Assume body is = 0 (r = a) K = Va2

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circular cylinder

Flow Around cylinder

Flow Around cylinder

Flow Around a long cylinder

Streaklines around cylinder & Effect of circulation and stagnation point

V2 Distribution of flow over a circular cylinder

The velocity of the fluid is zero at = 0o and = 180o. Maximum velocity occur on the sides of the cylinder at = 90o and = -90o.Pressure distribution on the surface of the cylinder can be found by using Benoullis equation. Thus, if the flow is steady, and the pressure at a great distance is p,

Cp distribution of flow over a circular cylinder

Combining source and a sink

Compare the flow around golf ball& sphere

Combination Source +sink + uniform flow

Flow around Airfoil

Airfoil anatomy

Airfoil and boundry layer and sepration

Airfoil @ streamlines & stream function

JOUKOWSKIS transformation

INTRODUCTIONA large amount of airfoil theory has been developed by distorting flow around a cylinder to flow around an airfoil. The essential feature of the distortion is that the potential flow being distorted ends up also as potential flow.The most common Conformal transformation is the Jowkowski transformation which is given by

To see how this transformation changes flow pattern in the z (or x - y) plane, substitute z = x + iy into the expression above to get

This means that

For a circle of radius r in Z plane x and y are related as:

Consider a cylinder in z plane

In z plane

C=0.8

C=0.9

C=1.0

Translation TransformationsIf the circle is centered in (0, 0) and the circle maps into the segment between and lying on the x axis; If the circle is centered in (xc ,0), the circle maps in an airfoil that is symmetric with respect to the x' axis; If the circle is centered in (0,yc ), the circle maps into a curved segment; If the circle is centered in and (xc , yc ), the circle maps in an asymmetric airfoil.

Flow Over An Airfoil

Mapping of cylinders through JOUKOWSKIS transformation : (a) flat plate ; (b) elliptical section ; (c) cymmetrical wings;(d)asymmetrical wing

JOUKOWSKI

1-Fluid Mechanics M.S.Moayeri2-Introduction to Fluid Mechanics - Yasuki Nakayama3-Mechanics of Fluids - Irving Herman Shames 4-White Fluid Mechanics 5th 5-Mansun Mechanical Fluid 6-Wikipedia7- http://physicsarchives.com

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