This is usually termed locally uniform It does not correspond to any real flow

This is usually termed locally uniform it does not

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location they all have the same magnitude. This is usually termed locally uniform . It does not correspond to any real flow physics, but as we will later see, it is often a very good, and useful, approximation in some circumstances. Figure 2.15(c) shows a case closer to actual flow physics, and it is nonuniform. In fact, most actual flows are nonuniform, but we will see that especially local uniformity will be an important, and often quite accurate, simplification. 2.4.4 Rotational and irrotational flows Intuitively, we can think of rotational flows as those containing many “swirls” or “vortices;” i.e. , the fluid elements are rotating. Conversely, fluids not exhibiting such effects might be considered to be irrotational. But we will see from the precise definition, and some examples that follow, that these simple intuitive notions can sometimes be quite inaccurate and misleading. It thus is preferable to rely on the rigorous mathematical definition. Definition 2.11 A flow field with velocity vector U is said to be rotational if curl U negationslash = 0 ; otherwise, it is irrotational . To understand this definition and, even more, to be able to use it for calculations, we need to consider some details of the curl of a vector field ; this is given by the following. Definition 2.12 The curl of any (3-D) vector field F = F ( x, y, z ) is given by curl F = ∇× F = parenleftbigg ∂x , ∂y , ∂z parenrightbigg × ( F 1 ( x, y, z ) , F 2 ( x, y, z ) , F 3 ( x, y, z )) . (2.13) We next need to see how to use this definition for practical calculations. In the case that is of interest in this course, the vector field will be the fluid velocity U ( x, y, z ) = ( u ( x, y, z ) , v ( x, y, z ) , w ( x, y, z )) T , and curl U is called vorticity , denoted ω . Thus, ω ≡ ∇× U , (2.14)
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2.4. CLASSIFICATION OF FLOW PHENOMENA 33 and if ω negationslash = 0 the corresponding flow field is rotational, by Def. 2.11. In Cartesian coordinate systems ω is easily calculated from the following formula: ω = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle e 1 e 2 e 3 ∂x ∂y ∂z u v w vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = parenleftbigg ∂w ∂y ∂v ∂z parenrightbigg e 1 + parenleftbigg ∂u ∂z ∂w ∂x parenrightbigg e 2 + parenleftbigg ∂v ∂x ∂u ∂y parenrightbigg e 3 . (2.15) In this expression the e i , i = 1 , 2 , 3 are the unit basis vectors for the three-dimensional Euclidean space R 3 . It is clear that, in general, vorticity is a vector field of the same dimension as the velocity field. But we note that for a 2-D velocity field with one component, say w , identically constant, vorticity collapses to a scalar; it will, however, still depend on the same two spatial coordinates as does the velocity field. Furthermore, vorticity will be either steady or unsteady according to whether the velocity field is steady or unsteady; viz. , time does not explicitly enter calculation of vorticity.
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