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Problem 2.21

# Problem 2.21 - Problem 2.21[Difficulty 3 Given Eulerian...

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Problem 2.21 [Difficulty: 3] Given: Eulerian Velocity field Find: Lagrangian position function that was at point (1,1) at t = 0; expression for pathline; plot pathline and compare to streamlines through same point at the instants t = 0, 1 and 2s Solution: Governing equations: For pathlines (Lagrangian description) u p dx dt = v p dy dt = For streamlines v u dy dx = Assumption: 2D flow Hence for pathlines u p dx dt = A = A 2 = m s v p dy dt = B t = B 2 = m s 2 So, separating variables dx A dt = dy B t dt = Integrating x A t x 0 + = x 0 1 = m y B t 2 2 y 0 + = y 0 1 = m The Lagrangian description is x t ( ) A t x 0 + = y t ( ) B t 2 2 y 0 + = Using given data x t ( ) 2 t 1 + = y t ( ) 1 t 2 = The pathlines are given by combining the equations t x x 0 A = y B t 2 2 y 0 + = B x x 0 ( ) 2 2 A 2 y 0 + = Hence y x ( ) y 0 B x x 0 ( ) 2 2 A 2 = or, using given data

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