Problem 5.75 - da cd bc ab s d V s d V s d V s d V From the...

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(a) show that the velocity field represents a possible incompressible flow (b) Rotation at (x, y) = (1, 1) (c) Circulation about the unit square shown above [Difficulty: 2] Problem 5.75 Γ 0.5 m 2 s Given: Two-dimensional flow field Find: Solution: We will apply the definition of circulation to the given velocity field.    0 t w z v y u x Governing Equations: (Continuity equation) V 2 1 (Definition of rotation) s d V (Definition of circulation) Assumptions: (1) Steady flow (2) Incompressible flow (3) Two dimensional flow (velocity is not a function of z) Based on the assumptions listed above, the continuity equation reduces to: x u y v 0 This is the criterion against which we will check the flow field. x u y v Ay 2B y 1 ms y 2 1 2m s y 0 This could be an incompressible flow field. k Ax By Axy z y x k j i ˆ 2 1 0 ˆ ˆ ˆ 2 1 2 s rad ˆ 5 . 0 k From the definition of rotation: At (x, y) = (1, 1)
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Unformatted text preview: da cd bc ab s d V s d V s d V s d V From the definition of circulation we break up the integral: dy By Axydx j dy i dx j By i Axy s d V 2 2 The integrand is equal to: Therefore, the circulation is equal to: x a x b x A x y d y b y c y B y 2 d x c x d x A x y d y d y a y B y 2 d A 2 x b 2 x a 2 y a y c B 3 y c 3 y b 3 y a 3 y d 3 Since y a y d and y b y c we can simplify: A 2 x b 2 x a 2 y c Substituting given values: 1 2 1 m s 1 2 2 m 2 1 m...
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