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Unformatted text preview: Fluids – Lecture 9 Notes 1. Momentum-Integral Simplifications 2. Applications Reading: Anderson 2.6 Simplifications For steady ﬂow, the momentum integral equation reduces to the following. circlecopyrt ρ V · ˆ vector n dA + ρvectorg d V + vector vector n V dA = circlecopyrt − p ˆ F viscous (1) Defining h as the height above ground, we note that ∇ h is a unit vector which points up, so that the gravity acceleration vector can be written as a gradient. vectorg = − g ∇ h Using the Gradient Theorem, this then allows the gravity-force volume integral to be con- verted to a surface integral, provided we make the additional assumption that ρ is nearly constant throughout the ﬂow. ρvectorg d V = − ρ g ∇ h d V = circlecopyrt − ρgh ˆ n dA We can now combine the pressure and gravity contributions into one surface integral. n dA + ρvectorg d V → circlecopyrt − ( p + ρgh ) ˆ circlecopyrt − p ˆ n dA Defining a corrected pressure p c = p + ρgh , the Integral Momentum Equation finally becomes circlecopyrt ρ V · ˆ vector ˆ F viscous (2) vector n V dA = circlecopyrt− p c n dA + vector Aerodynamic analyses using (2) do not have to concern themselves with the effects of gravity, since it does not appear explicitly in this equation. In particular, the velocity field vector V will not be affected by gravity. Gravity enters the problem only in a secondary step, when the true pressure field p is constructed from p c by adding the “tilting” bias − ρgh ....
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.
- Fall '05