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Unformatted text preview: 650:460 Aerodynamics Chapter 9 Prof. Doyle Knight Tel: 732 445 4464, Email: doyleknight@gmail.com Office hours: Tues and Thur, 4:30 pm  6:00 pm and by appointment Fall 2009 1 Compressible Flow Assumptions Inviscid Steady Irrotational Isentropic (and therefore barotropic) From Kelvins Theorem, the flow is irrotational if the flow upstream of any aerodynamic body is irrotational and p = p ( ). This latter condition is satisfied if we assume the flow is isentropic. This implies that there are no shock waves. 2 Compressible Flow Bernoullis Equation Conservation of momentum bracketleftbigg vector v t + ( vector v ) vector v bracketrightbigg = p where is the gradient operator p = p x vector i + p y vector j + p z vector k and vector i , vector j and vector k are the unit vectors in the x, y and zdirections. 3 Compressible Flow Bernoullis Equation Vector identity ( vector v ) vector v = ( 1 2 vector v 2 ) vector v where = vector v is the vorticity. Substitute into the conservation of momentum vector v t + ( 1 2 vector v 2 ) + 1 p = 0 since the vorticity is assumed zero everywhere. Since the fluid is assumed barotropic 1 p = parenleftbiggintegraldisplay dp parenrightbigg 4 Compressible Flow Bernoullis Equation Since the fluid is assumed irrotational vector v = and thus vector v t = t = parenleftbigg t parenrightbigg Therefore the conservation of momentum is bracketleftbigg t + 1 2 vector v 2 + integraldisplay dp bracketrightbigg = 0 and thus t + 1 2 vector v 2 + integraldisplay dp = f ( t ) where f ( t ) is a function of time only. 5 Compressible Flow Bernoullis Equation For isentropic flow p p o = parenleftbigg o parenrightbigg and thus integraldisplay dp = ( 1) p Thus, the integral of the conservation of momentum is t + 1 2 + ( 1) p = f ( t ) This is the compressible Bernoulli equation . 6 Compressible Flow Equation for Take the inner product of vector v with the conservation of momentum parenleftbigg vector v vector v t + vector v ( vector v ) vector v parenrightbigg = vector v p Identity vector v ( vector v ) vector v = vector v ( ( 1 2 vector v 2 ) vector v ) = vector v ( 1 2 vector v 2 ) since = vector v = 0 by assumption....
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This note was uploaded on 01/24/2012 for the course MECHE 343 taught by Professor Professor during the Spring '11 term at Carnegie Mellon.
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