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Unformatted text preview: VI. Irrotational Flows • For an irrotational flow, ∇ x u = 0 ⇒ velocity potential φ exists : u = ∇φ For incompressible flow, ∇ ⋅ u = ⇒ For an irrotational incompressible flow, ∇⋅∇φ = 0 ⇒ ∇ 2 φ = 0. (6.1) 6.1 Topological Notions • In order to discuss the general properties of the velocity potential φ for commonly encountered irrotational flows and to specify the correct B.C.’s for φ , it is necessary to introduce some topological notions . Definition 1: a regime (region) R of space is connected if we can pass any point to any other point along a path, every point of which lies in R. Definition 2:Two paths are reconcilable if they can be made to coincide by continuous deformation without leaving the region R. A B A B C 1 C 2 C 3 C 1 C 2 3 (a) (b) R S S R C A B A B C 1 C 2 C 3 C 1 C 2 3 (a) (b) R S S R C C 2 and C 3 are reconcilable. C 1 and C 2 are irreconcilable. C 1 and C 3 are irreconcilable. A B C 3 y x (c) C 1 C 2 S C 2 and C 3 are reconcilable. C 1 and C 2 are irreconcilable. C 1 and C 3 are irreconcilable. A B C 3 y x (c) C 1 C 2 S Fig. 6.1 a) Reducible paths in the interior of a closed surface; b) Reducible paths in the exterior of a closed surface; c) irreconcilable paths in the region exterior of a 2D cylinder. • Consider the region interior or exterior to a finite closed surface S. • Let A and B be two arbitrary points, C 1 , C 2 , and C 3 be any 3 paths lying in R & connecting A and B. Any 2 of the 3 paths can be made to coincide with each other by continuous deformation of the paths without leaving the region R. Thus the paths are reconcilable . Definition 3: If a closed path in a region can be contracted to a point of the region without passing out of the region, the region is said to be reducible . e.g.1. The region outside a 3D object in an infinite domain is reducible. In Fig. 6.2, the close path 1 can be obviously contracted to a point without passing out of the region. For the closed path 2, one first moves the closed path away from the 3D object, say, in the z direction, and subsequently contracts the path. Definition 4:A region in which any path is reducible is termed simply connected . e.g.1. The region outside a 3D object in an infinite domain is simply connected, as shown in Fig. 6.2. 3D object closed path 1 closed path 2 Fig. 6.2 Reducible region in 3D space. x y z 3D object closed path 1 closed path 2 Fig. 6.2 Reducible region in 3D space. x y z e.g.2. The region outside a torus in a 3D space is not simply connected. Fig. 6.3 A torus in 3D space. Definition 5:A region is said to be doublyconnected if it can be made simply connected by the insertion of a barrier (or cut ) which cannot be crossed....
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 Fall '09
 RENWEIMEI
 Fluid Dynamics, Region, dφ

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