16100lectre47_cj

16100lectre47_cj - Computational Methods for the Euler...

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Computational Methods for the Euler Equations Before discussing the Euler Equations and computational methods for them, let’s look at what we’ve learned so far: Method Assumptions/Flow type 2-D panel 2-D, Incompressible, Irrotational Inviscid Vortex lattice 3-D, Incompressible, Irrotational Inviscid, Small disturbance Potential method 3-D, Subsonic compressible, Irrotational, Inviscid, Prandtl-Glauert Small disturbance Euler CFD 3-D, Compressible (no M limit), Rotational, Shocks, Inviscid The only major effect missing after this week will be viscous-related effects. 2-D Euler Equations in Integral Form Consider an arbitrary area (i.e. a fixed control volume) through which flows a compressible inviscid flow: n v outward pointing normal (unit length) dS elemental (differential) surface length j dx i dy dS n v v v = Note: Path around surface is taken so that interior of control volume is on left. y x c dS C δ n v dS n v dy dx dS
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Computational Methods for the Euler Equations 16.100 2002 2 Conservation of Mass C in mass of dA dt d change of rate dA C in Mass C of out flow mass of rate C in mass of change of rate C C ∫∫ ∫∫ = = = + ρ 0 where fluid of destiny Now, the rate of mass flowing out of C : Mass flow out of = = C u dS n u C δ v v v velocity vector Conservation of x-momentum Recall that: total rate of change momentum = forces For x-momentum this gives: = + C of out momflow x of rate C in momentum x of change of rate Forces in x-direction ∫∫ = + C C dS n u u udA dt d v v Forces in x-direction Now, looking closer at x-forces, for an inviscid compressible flow we only have pressure (ignoring gravity). Recall pressure acts normal to the surface = C dS i n p x in s Force v v ∫∫ = + C C dS n u dA dt d 0 v v dS dS n p v Into surface Normal to surface Gives x-direction
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16100lectre47_cj - Computational Methods for the Euler...

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