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3241.CQ4 Solution

3241.CQ4 Solution - integral is 29 n U AU y ˆ ⋅ Uy=U in...

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3241: Aerodynamics and Flight Mechanics Name: ___Solution____________ Concept Quiz #4 The purpose of this concept quiz is to examine the application momentum equation for part of a control surface analysis. The momentum equation is: ( 29 = + F dS n U U dV U dt d S V d d d d ˆ ρ What is the y-component of momentum flux across this surface? Assume steady flow, no accelerations, constant density. (1) –u 2 ( ρ A)sin( θ )sin( θ ) (2) u 2 ( ρ A)sin( θ )sin( θ ) (3) –u 2 ( ρ A)cos( θ )cos( θ ) (4) u 2 ( ρ A)cos( θ )cos( θ ) (5) –u 2 ( ρ A)sin( θ )cos( θ ) (6) u 2 ( ρ A)sin( θ )cos( θ ) (7) I have no idea how to do this The answer to this question is (5). The flow is steady, so we are only concerned with the second integral of the momentum equation. The control surface is not moving, so there is no distinction between relative and inertial velocities here. We want to write this equation in the y-direction, since we are looking for the y-component of the momentum flux in this direction. The density is constant and the integral of the surface area is A. The second

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Unformatted text preview: integral is: ( 29 n U AU y ˆ ⋅ Uy=U in the y direction. This quantity is positive in accordance with the sign convention shown to the left of the figure. Uy=Ucos( θ ). Now examine the term ( 29 n U ˆ ⋅ : This is the dot product of two vectors. First notice that U and n are in the opposite direction, this gives the minus sign and tells us that the flux is into the control surface. The dot product asks us to find the component of the vector U in the direction of n, which is –Usin( θ ). Note that the term ( 29 n U ˆ ⋅ is the same for both the x and the y directions. Combining these, we get the y-component of the momentum flux across this surface: ( 29 θ cos sin 2 A U-x U θ n A ρ y The x-component of the momentum flux across this surface is: ( 29 θ ρ 2 2 sin A U-To find the resultant momentum flux, we could that the square root of the sum of the squares of each of these quantities....
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3241.CQ4 Solution - integral is 29 n U AU y ˆ ⋅ Uy=U in...

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