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f09_fall - Fluids Lecture 9 Notes 1. Momentum-Integral...

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Unformatted text preview: Fluids Lecture 9 Notes 1. Momentum-Integral Simplifications 2. Applications Reading: Anderson 2.6 Simplifications For steady ow, the momentum integral equation reduces to the following. circlecopyrt V vector n dA + vectorg d V + vector vector n V dA = circlecopyrt p 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 is nearly constant throughout the ow. vectorg d V = g h d V = circlecopyrt gh n dA We can now combine the pressure and gravity contributions into one surface integral. n dA + vectorg d V circlecopyrt ( p + gh ) circlecopyrt p n dA Defining a corrected pressure p c = p + gh , the Integral Momentum Equation finally becomes circlecopyrt V vector 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 gh ....
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f09_fall - Fluids Lecture 9 Notes 1. Momentum-Integral...

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