f17_fall - Fluids Lecture 17 Notes 1 Floweld prediction 2...

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Fluids – Lecture 17 Notes 1. Flowfield prediction 2. Source Sheets Reading: Anderson 3.17 Flowfield Prediction Problem definition The flowfield examples used so far were used to demonstrate the basic ideas behind the method of superposition. We chose some combination of elementary flows (uniform flow, sources, vortices, etc.), and then determined the resulting flowfield. The corresponding body shape was determined from the shape of the dividing streamline. However, such an approach is not practical for engineering applications, where we want to specify the body shape, rather than have it as an outcome. The problem can therefore be stated as follows. Given : Body shape Y ( x ), Freestream velocity vector V Determine : Superposition of suitable elementary flows which produce the velocity field vector V ( x, y ) about the body. It turns out that sources, vortices, and doublets are not ideally suited to this task because of their strong singularities. The constraint that these singularities must be inside the body is difficult to meet, especially if the body is very slender. For this reason we now define slightly more elaborate elementary flows which are smoother, and therefore better suited to representing smooth bodies. Source Sheets Definition Consider a sequence of flows where a single source of strength Λ is repeatedly subdivided into smaller sources which are evenly distributed along a line segment of length . The limit of this subdivision process is a source sheet of strength λ = Λ /ℓ . Λ 2 × Λ 2 4 × Λ 4 8 × Λ 8 . . . λ The units of Λ are length 2 / time, while the units of λ are length / time (or velocity). Note that the total source strength is not changed in this process.
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