Potential Flows Title: Advisor : Ali R. Tahavvor, Ph.D. Islamic Azad University Shiraz Potential Flows Branch Department of Engineering .Faculty of Advisor : Ali R. Tahavvor, Ph.D. Mechanical Engineering Year:2012 June By : Hossein Andishgar 87682175 Pouya Zarrinchang 87682177

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 = Uz The 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 field Velocity 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.

Irrotational Flow Approximation Irrotational 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

Irrotational Flow Approximation 2D flows For 2D flows, we can also use the stream function Recall 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

Elementary Planar Irrotational Flows Uniform Stream In Cartesian coordinates Conversion to cylindrical coordinates can be achieved using the transformation Proof with Mathematica

Elementary Planar Irrotational Flows source & sink A doublet is a combination of a line sink and source of equal magnitude Source Sink

Elementary Planar Irrotational Flows Line Source/Sink Potential and streamfunction are derived by observing that volume flow rate across any circle is This gives velocity components

Elementary Planar Irrotational Flows Line Source/Sink Using definition of (Ur, U) These can be integrated to give  and  Equations are for a source/sink at the origin Proof with Mathematica

Elementary Planar Irrotational Flows Line Source/Sink If source/sink is moved to (x,y) = (a,b)

Elementary Planar Irrotational Flows Line Vortex Vortex at the origin. First look at velocity components These can be integrated to give  and  Equations are for a source/sink at the origin

Elementary Planar Irrotational Flows Line Vortex If vortex is moved to (x,y) = (a,b)

Two types of vortex

Rotational 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): Complex potential of a sink 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 :

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 Flows Doublet Adding 1 and 2 together, performing some algebra, and taking a0 gives K is the doublet strength

THE 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 Superposition Superposition of sink and vortex : bathtub vortex Sink Vortex

Examples of Irrotational Flows Formed by Superposition Flow over a circular cylinder: Free stream + doublet Assume body is  = 0 (r = a)  K = Va2

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 Benoulli’s 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

JOUKOWSKI’S transformation INTRODUCTION A 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

C=0.8

C=0.9

C=1.0

Translation Transformations If 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 JOUKOWSKI’S transformation : (a) flat plate ; (b) elliptical section ; (c) cymmetrical wings;(d)asymmetrical wing

JOUKOWSKI

Thank you for your attention source : 1-Fluid Mechanics M.S.Moayeri 2-Introduction to Fluid Mechanics - Yasuki Nakayama 3-Mechanics of Fluids - Irving Herman Shames 4-White Fluid Mechanics 5th 5-Mansun Mechanical Fluid 6-Wikipedia 7- http://physicsarchives.com Thank you for your attention Good Luck