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Unformatted text preview: Fluids Lecture 14 Notes 1. Helmholtz Equation 2. Incompressible Irrotational Flows Reading: Anderson 3.7 Helmholtz Equation Derivation (2-D) If we neglect viscous forces, the x- and y-components of the 2-D momentum equation can be written as follows. u t + u u x + v u y = 1 p x + g x (1) v t + u v x + v v y = 1 p y + g y (2) We now take the curl of this momentum equation by performing the following operation. x braceleftbigg y-momentum (2) bracerightbigg y braceleftbigg x-momentum (1) bracerightbigg If we assume that is constant (low speed flow), the two pressure derivative terms cancel. Since the gravity components g x and g y are generally constant, these also disappear when the curls derivatives are applied. Using the product rule on the lefthand side, the resulting equation is t parenleftBigg v x u y parenrightBigg + u x parenleftBigg v x u y parenrightBigg + v y parenleftBigg v x u y parenrightBigg + parenleftBigg v x u y parenrightBiggbracketleftBigg u x + v y bracketrightBigg = 0 We note that the quantity inside the parentheses is merely the z-component of the vorticity v/x u/y , so the above equation can be more compactly written as t + u x + v y + bracketleftBigg u x + v y bracketrightBigg = 0 We further note that the quantity in the brackets is the divergence of the velocity, which in low speed flow must be zero because of mass conservation.low speed flow must be zero because of mass conservation....
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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