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07%20Incompressible%20Flows%20EMG%206812%20F11

07%20Incompressible%20Flows%20EMG%206812%20F11 - EGM 6812...

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EGM 6812, F11, University of Florida, M. Sheplak 1/20 Section 7, Incompressible Flows 7 Incompressible Flows 7.1 Conditions for Incompressibility Incompressible fluid : thermodynamic term Incompressible flow : fluid mechanics term Define : incompressible flow: fractional change of density of a fluid particle is zero: 1 0 D Dt , which means that compressibility effects in the flow are negligible with respect to other effects As a result, via continuity, 0 V  “divergence free” or solenoidal velocity field “Dilatation” or rate of expansion is zero: 1 0 DV V V Dt   Why is it so important to establish a flow as incompressible? Continuity equation simplifies to a kinematic condition, 0 V  . Stokes’ hypothesis is automatically met. The energy equation is decoupled from the continuity and momentum equations. As we will see below there are five conditions for incompressibility, all of which MUST be met for the flow to be considered incompressible. Note: This is DIFFERENT than the way Panton approaches this topic in Chapter 10. This class emphasizes Batchelor’s approach 1 0 D V Dt    or 1 V D L Dt  , where V typical velocity L typical length Recall from thermodynamics , s p , (NOTE : , T p is treated in Panton) 2 0 1 p s c d dp ds p s , where 0 c is the isentropic speed of sound , 0 c RT ( a RT is also used) Replace     D d Dt
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EGM 6812, F11, University of Florida, M. Sheplak 2/20 Section 7, Incompressible Flows 2 0 changes in pressure changes in entropy 1 1 1 s p D Dp Ds Dt c Dt s Dt 1 V D L Dt  will be satisfied if and only if the conditions 2 0 1 Dp V c Dt L  AND 1 Ds V s Dt L  are met! 7.1.1 Isentropic compression terms 2 0 1 Dp V c Dt L  The variations of density of the material element due to isentropic compressions must be small: Acoustic radiation effects Mach number effects Hydrostatic pressure effects Note : Dp p V p Dt t  We need to expand p out via the mechanical energy equation: 2 2 D V f V V p Dt  We have neglected “viscous work” because we have assumed isentropic in the evaluation of 2 0 1 Dp c Dt (isentropic pressure variations) the effects of viscosity and heat conduction serve to modify the pressure distribution rather than control the pressure variation 2 2 2 2 2 0 0 0 0 1 1 1 2 b a c V D Dp p f V V c Dt c t c c Dt L  1.
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