assignment5 - ME372ComputerAidedMechanicalEngineering

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ME 372 Computer-Aided Mechanical Engineering Instructor: Dr. Emre Alpman Assignment #5 Due Date: April 20 th , 2010  BOUNDARY VALUE PROBLEMS: STEADY FLUID FLOW BETWEEN CONCENTRIC CYLINDERS  1.Problem Statement Consider a steady, incompressible flow between concentric cylinders as shown in Figure 1. The fluid  between the cylinders is water.  The inner cylinder is rotating with an angular speed  , while the Ω   outer cylinder is stationary . This would cause a circumferential motion of the fluid. For this problem  we   assume   that   there   is   no   axial  motion   (axial  direction   is   out   out  of   the   paper)   and   flow  is  axisymmetric.    Figure 1. Flow between concentric cylinders.  Radius r i . Rotating with  angular speed  Ω Radius r o . Stationary Fluid:  Water Radial  direction Circumferential  direction
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Continuity equation written for this flow shows that the radial component of the velocity is zero,  leaving only the circumferential velocity as a function of radial direction,  r . The Navier-Stokes equation  written for the circumferential direction,  θ , reduces to the following ODE: 1 r d dr r d V dr = V r 2 (1) Boundary conditions are given as: V r i = r i (2) V r o = 0 (3)  2.Lab Assignment Letting the independent variable,  t  =  r , rewrite the above equations in linear BVP form: y ' = A t y b t (4) with BCs B a y r i  B b y r o = C (5) Here   y   will be the   state vector with   m   elements . For numerical solution the flow domain will be  divided into a mesh with  N  intervals and  N  + 1 points distributed as: r i = t 1 t 2 ⋯ t N 1 = r o (6) Applying Mid-Point method to equation (4) yields: y n 1 = y n h n A t n 1 / 2 y n 1 y n 2 b t n 1 / 2 ,n = 1,2, , N (7) where h n = t n 1 t n (8) The BCs are written as: B a y 1 B b y N 1 = C (9)
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This note was uploaded on 04/05/2012 for the course MECHANICAL ME372 CAME taught by Professor Emraalpman during the Spring '10 term at Yeditepe Üniversitesi.

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assignment5 - ME372ComputerAidedMechanicalEngineering

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