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Lagrangian_Eulerian_Solution

# Lagrangian_Eulerian_Solution - MIT Department of Mechanical...

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MIT Department of Mechanical Engineering 2.25 Advanced Fluid Mechanics Particle Kinematics Lagrangian and Eulerian Frames - Material Derivatives The Eulerian velocity field ( u , v ) of a steady two-dimensional uniform flow with velocity U in the x -direction past a circle with radius a in an inviscid fluid is given by u ( x, y ) = U + Ua 2 y 2 - x 2 ( x 2 + y 2 ) 2 (1) v ( x, y ) = - 2 Ua 2 xy ( x 2 + y 2 ) 2 (2) a) Derive the ordinary differential equations that govern the particle path lines. b) Determine the stream function and use it to derive an analytical expression for the flow streamlines. c) A particle on the surface of the circle always stays on the circle. Use this property to derive the boundary condition that the velocity field must satisfy on the surface of the circle. d) Consider a particle located at ( - ξ 0 ,0) at t = 0, with ξ 0 > a . Derive and solve the differential equation governing its motion. Determine the time it will take for the particle to reach the stagnation point ( - a ,0). e) Consider the case where the ambient velocity field is absent but instead the circle translates in the positive x -direction with velocity U . Derive the flow velocity field ( u * , v * ). f) Derive the differential equations that govern the particle path lines in e). Comment on their shape. g) How would your answers in e) and f) change when the circle velocity U ( t ) is an arbitrary function of time? 2.25 Advanced Fluid Mechanics 1 Copyright c 2010, MIT

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Continuum and Kinematics Particle Kinematics Solution: First let’s plot the velocity field, in order to better visualize the flow around the circle. ! 2 ! 1 0 1 2 ! 2 ! 1 0 1 2 x/a y/a a) Here we will designate ξ ( t ) as the Lagrangian coordinate position of a fluid particle, such that ξ ( t ) = [ ξ 1 ( t ) , ξ 2 ( t )], where ξ 1 and ξ 2 are scalar quantities indicating the x and y position of the particle, respectively. Recall that a path line is the locus of points through which a particle of a fixed identity has traveled. For consistency, the velocity expressed in the Lagrangian frame should be equal to the velocity expressed in the Eulerian frame at time t . Accordingly, the governing equation for the particle path lines is
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Lagrangian_Eulerian_Solution - MIT Department of Mechanical...

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