Eulerian and Lagrangian descriptions air object 0 A cylinder-shaped object (e.g., hockey puck) is producing an...
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Eulerian and Lagrangian descriptions air object 0 A cylinder-shaped object (e.g., hockey puck) is producing an air cushion by sucking air from above and ejecting it from below. The radius of the object is L and its lower surface is at a height h above the ground. We use cylindrical coordinates and consider the flow in the region z <h and r ≤ L. a) Air is ejected with the steady velocity v (r,h) = vr(r)evoz, and we look for a velocity field in the form v (r, 2) = v,(r) + v₂ (r, 2) 2. The rigid boundary at z = 0) requires v₂ = 0 there. i) Assuming the air to be incompressible, write the mass conservation equation for a cylinder of radius r below the object, and show that v,(r) = ruo/(2h). ii) Write the local condition for incompressibility and show that v₂ (r, 2)=-zvo/h. iii) Calculate the velocity potential o. b) Calculate the equation of the streamlines. What is that of the trajectories? Check that the streamlines and equipotentials are perpendicular to each other. c) Consider a fluid clement which at t=0 is at ro= (roso, 2o). Calculate the position of this fluid element as a function of time and check that this is consistent with the equation of the trajectories obtained in question (b). d) Calculate the Lagrangian velocity V (t, ro) of the fluid element. Verify that: Dv av Ət Dt. Eulerian and Lagrangian descriptions air object 0 A cylinder-shaped object (e.g., hockey puck) is producing an air cushion by sucking air from above and ejecting it from below. The radius of the object is L and its lower surface is at a height h above the ground. We use cylindrical coordinates and consider the flow in the region z <h and r ≤ L. a) Air is ejected with the steady velocity v (r,h) = vr(r)evoz, and we look for a velocity field in the form v (r, 2) = v,(r) + v₂ (r, 2) 2. The rigid boundary at z = 0) requires v₂ = 0 there. i) Assuming the air to be incompressible, write the mass conservation equation for a cylinder of radius r below the object, and show that v,(r) = ruo/(2h). ii) Write the local condition for incompressibility and show that v₂ (r, 2)=-zvo/h. iii) Calculate the velocity potential o. b) Calculate the equation of the streamlines. What is that of the trajectories? Check that the streamlines and equipotentials are perpendicular to each other. c) Consider a fluid clement which at t=0 is at ro= (roso, 2o). Calculate the position of this fluid element as a function of time and check that this is consistent with the equation of the trajectories obtained in question (b). d) Calculate the Lagrangian velocity V (t, ro) of the fluid element. Verify that: Dv av Ət Dt.
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Solutions Step 1 The question is based on Euler and Lagrangian equations Cylindrical coordinates and consider the flow in the region hand zh and r a Air is ejected with the steady velocity vrhvrrvo an... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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