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This document discusses several methods for approximating solutions to the Navier-Stokes equations (NSE), including nondimensionalization, creeping flow, inviscid flow, irrotational flow, and potential flow. It explains how these approximations simplify the NSE by removing terms to create linear, analytically solvable forms. Elementary flows like source/sink, vortex, and doublet are introduced that can be combined using superposition to model more complex flows.

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Chapter 10: Approximate Solutions of the Navier-Stokes Equation ME 331- Fluid Dynamics Spring 2008
Objectives ,[object Object],[object Object],[object Object],[object Object]
Introduction ,[object Object],[object Object],[object Object]
Nondimensionalization of the NSE ,[object Object],[object Object],[object Object],[object Object]
Nondimensionalization of the NSE ,[object Object]
Nondimensionalization of the NSE ,[object Object],[object Object]
Nondimensionalization of the NSE ,[object Object],[object Object]
Nondimensionalization of the NSE ,[object Object],Strouhal number Euler number Inverse of Froude number squared Inverse of Reynolds number Navier-Stokes equation in nondimensional form
Nondimensionalization of the NSE ,[object Object],[object Object],[object Object],[object Object],If we have properly normalized the NSE, we can compare the relative importance of the terms in the equation by comparing the relative magnitudes of the nondimensional parameters St, Eu, Fr, and Re.
Creeping Flow ,[object Object],[object Object],[object Object],[object Object]
Creeping Flow ,[object Object],[object Object],Pressure forces Viscous forces
Creeping Flow ,[object Object],[object Object],[object Object],[object Object]
Creeping Flow ,[object Object],[object Object]
Inviscid Regions of Flow ,[object Object],~0 if Re large Euler Equation
Inviscid Regions of Flow ,[object Object],[object Object],[object Object],No-slip BC u = v = w = 0 Slip BC  w = 0, V n = 0 V n = normal velocity
Irrotational Flow Approximation ,[object Object],[object Object]
Irrotational Flow Approximation ,[object Object],[object Object],[object Object],[object Object], is a scalar potential function
Irrotational Flow Approximation ,[object Object],[object Object],[object Object],Cartesian Cylindrical
Irrotational Flow Approximation ,[object Object],[object Object],Laplace Equation
Irrotational Flow Approximation ,[object Object],[object Object],[object Object],= 0
Irrotational Flow Approximation ,[object Object],[object Object],nondimensional dimensional
Irrotational Flow Approximation ,[object Object],[object Object]
Irrotational Flow Approximation ,[object Object],Inviscid Irrotational (  = 0)
Irrotational Flow Approximation ,[object Object],[object Object],[object Object],[object Object],Valid for 3D or 2D
Irrotational Flow Approximation 2D Flows ,[object Object],[object Object],[object Object],[object Object]
Irrotational Flow Approximation 2D Flows ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Flow solution can be achieved by solving either  2  or  2  , however, BC are easier to formulate for 
Irrotational Flow Approximation 2D Flows ,[object Object],[object Object],[object Object],Planar Axisymmetric
Irrotational Flow Approximation 2D Flows
Irrotational Flow Approximation 2D Flows ,[object Object],[object Object],[object Object],[object Object]
Irrotational Flow Approximation 2D Flows ,[object Object],[object Object],[object Object],[object Object],[object Object]
Elementary Planar Irrotational Flows Uniform Stream ,[object Object],[object Object]
Elementary Planar Irrotational Flows Line Source/Sink ,[object Object],[object Object]
Elementary Planar Irrotational Flows Line Source/Sink ,[object Object],[object Object],Equations are for a source/sink at the origin
Elementary Planar Irrotational Flows Line Source/Sink ,[object Object]
Elementary Planar Irrotational Flows Line Vortex ,[object Object],[object Object],Equations are for a source/sink at the origin
Elementary Planar Irrotational Flows Line Vortex ,[object Object]
Elementary Planar Irrotational Flows Doublet ,[object Object],[object Object],[object Object]
Elementary Planar Irrotational Flows Doublet ,[object Object],K is the doublet strength
Examples of Irrotational Flows Formed by Superposition ,[object Object],Sink Vortex
Examples of Irrotational Flows Formed by Superposition ,[object Object],[object Object]

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