ChapterIII.PartII - AE 3003 Chapter III Handout#2 Sources...

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AE 3003 Chapter III - Handout #2 Sources, Sinks and Doublets - the Building Blocks of Potential Flow In the previous handout we developed the following equation for the velocity potential: 0 0 2 2 2 2 2 2 2 = = + + φ Or z y x (1) where the operator 2 is called the Laplacian operator. This equation holds for 2-D and 3-D inviscid irrotational flows. If we are only interested in 2-D irrotational inviscid flows, we may also solve for: 0 2 = ψ (2) where is the stream function. After we have solved for the velocity potential or the stream function, we can compute the velocities. In a Cartesian coordinate system, for 2-D flows, we will use: x y v y x u - = = = = (3) In a polar coordinate system, for 2-D flows we will use: r v r r v r - = = = = = = θ r 1 velocity Tangential 1 velocity Radial (4) In 3-D, the velocities are given only in terms of the velocity potential, as follows: z w y v x u Or V = = = = , (5) Once the velocity is known, we can find pressure from the Bernoulli's equation.
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In this section, we consider some simple solutions to the Laplace's equation (1 or 2). Since equation 91) and 92) are linear, we can superpose many such simple solutions to arrive at a more complex flow field. This is like building a complex configuration using Lego blocks. The individual simple solutions are the individual Lego pieces, which on their own, are not very interesting. Together, however, they can solve some very interesting flows, including flow over airfoils and wings. Building Block #1: 2-D Sources and Sinks: A source is like a lawn sprinkler. It sprays the water (or air) radially, and equally, in all the directions, at the rate of Q units per unit time. If this is a sink (e.g. a drain hole on a concrete pavement) the velocity vectors will still be radial, but directed inwards towards the center. The sign of Q will be positive for a source, and negative for a sink. Consider a circle of radius r enclosing this source. Let v
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ChapterIII.PartII - AE 3003 Chapter III Handout#2 Sources...

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