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Title : Potential Flows Advisor : Ali R. Tahavvor , Ph.D. By : Hossein Andishgar 87682175 Pouya Zarrinchang 87682177 Islamic Azad University Shiraz Branch Department of Engineering .Faculty of Mechanical Engineering Year:2012 June. FLOWS. FLOWS. FLOWS. STREAKLINES.
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Title: Potential Flows Advisor : Ali R. Tahavvor, Ph.D. By : HosseinAndishgar87682175 PouyaZarrinchang 87682177 Islamic Azad University Shiraz Branch Department of Engineering .Faculty of Mechanical Engineering Year:2012 June
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 streamfunctionand the velocity potential
Elementary Planar Irrotational FlowsUniform 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 FlowsLine Source/Sink • Potential and streamfunction are derived by observing that volume flow rate across any circle is • This gives velocity components
Elementary Planar Irrotational FlowsLine 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 FlowsLine Source/Sink • If source/sink is moved to (x,y) = (a,b)
Elementary Planar Irrotational FlowsLine 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 FlowsLine Vortex • If vortex is moved to (x,y) = (a,b)
Rotational vortex Irrotational vortex
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.
Elementary Planar Irrotational FlowsDoublet • Adding 1 and 2 together, performing some algebra, and taking a0 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
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
Streaklines around cylinder & Effect of circulation and stagnation point
V2Distribution of flow over a circular cylinder The velocity of the fluid is zero at = 0oand = 180o. Maximum velocity occur on the sides of the cylinder at = 90oand = -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,