This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: VI. Irrotational Flows For an irrotational flow, x u = 0 velocity potential exists : u = For incompressible flow, u = For an irrotational incompressible flow, = 0 2 = 0. (6.1) 6.1 Topological Notions In order to discuss the general properties of the velocity potential for commonly encountered irrotational flows and to specify the correct B.C.s for , 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....
View
Full
Document
This note was uploaded on 09/09/2010 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.
 Fall '09
 RENWEIMEI

Click to edit the document details