1 / 87

Title : Potential Flows Advisor : Ali R. Tahavvor , Ph.D. By : Hossein Andishgar 87682175

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.

edie
Download Presentation

Title : Potential Flows Advisor : Ali R. Tahavvor , Ph.D. By : Hossein Andishgar 87682175

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. FLOWS

  3. FLOWS

  4. FLOWS

  5. STREAKLINES

  6. The Streaklines

  7. Streamtubes

  8. 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)

  9. 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.

  10. 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

  11. 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

  12. Elementary Planar Irrotational FlowsUniform Stream • In Cartesian coordinates • Conversion to cylindrical coordinates can be achieved using the transformation Proof with Mathematica

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

  14. 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

  15. 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

  16. Elementary Planar Irrotational FlowsLine Source/Sink • If source/sink is moved to (x,y) = (a,b)

  17. 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

  18. Elementary Planar Irrotational FlowsLine Vortex • If vortex is moved to (x,y) = (a,b)

  19. Two types of vortex

  20. Rotational vortex Irrotational vortex

  21. Cyclonic Vortex in Atmosphere

  22. 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 :

  23. 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.

  24. Dipole

  25. Elementary Planar Irrotational FlowsDoublet • Adding 1 and 2 together, performing some algebra, and taking a0 gives K is the doublet strength

  26. 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

  27. Doublet 2D

  28. Examples of Irrotational Flows Formed by Superposition • Superposition of sink and vortex : bathtub vortex Sink Vortex

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

  30. circular cylinder

  31. Flow Around cylinder

  32. Flow Around cylinder

  33. Flow Around a long cylinder

  34. Streaklines around cylinder & Effect of circulation and stagnation point

  35. 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.

  36. 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,

More Related