part2 - 16.540 Spring 2006 PRESSURE FIELDS AND UPSTREAM...

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16.540 Spring 200 PRESSURE FIELDS AND UPSTREAM INFLUENCE 1 6
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PLAN OF THE LECTURE 2 Pressure fields and streamline curvature – Streamwise and normal pressure gradients – One-dimensional versus multi-dimensional flows Upstream influence and component coupling – How does the pressure field vary upstream of a fluid component and when does this matter? Pressure fields and the asymmetry of real fluid motions
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NORMAL AND STREAMWISE PRESSURE GRADIENTS Inviscid flow Streamwise: •N o r m a l udu =− dp ρ or , u u l = 1 p l r c n n l dl l l l + d l d l α d u 2 r c = 1 p n 3
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STREAMLINES AND WALL STATIC PRESSURES B B Curved wall One-dimensional Straight wall C C D D A A Pressure n n 4
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ONE-DIMENSIONAL AND TWO-DIMENSIONAL DESCRIPTIONS 5 In a one-dimensional representation of a contraction, the pressure gradient is always negative (or zero) If adopt a higher fidelity (2D) description, this is not true •I s i t possible to have a contraction in which there is no location that has an adverse (non-favorable) pressure gradient? Why is this important?
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UPSTREAM INFLUENCE OF FLUID COMPONENTS Approximate equation for the static pressure field 2-D, inviscid, steady flow, constant density Velocity viewed as a uniform mean flow, , plus “small” non- uniformities, : Neglect products of small quantities in momentum equation x - momentum: u ∂′ u x x + u y u x y =− 1 ρ p x where p is the departure from uniform static pressure There is no term u x u x because u is uniform u x = u x + u x u y = u y u x , u y u X 6
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So, u x ∂′ u x x =− 1 ρ p x (a) and u x u y x 1 p y (b) Take (a) x + (b) y , yielding, using continuity 0 = u x u x x + u y y 1 2 p x 2 + 2 p y 2 2 p = 0 Laplace' s Equation Famous equation with neat properties We will apply this to see the upstream influence 7
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UPSTREAM INFLUENCE Important question in internal flow systems- – When are components coupled aerodynamcially – When can they be considered independent? Laplace’s equation gives direct and simple qualitative answer Laplace’s equation gives direct and simple quantitative answer •F i r s t p a r t : - 2 p has no intrinsic length scale - If pick a y - length => x - length is set - Consider an "unrolled" annular flow field 8
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A DIGRESSION: WHAT DO I MEAN BY “LENGTH SCALE”? What is an example of an equation with a length scale? How about the momentum equation for viscous, constant-pressure flow? – This is an idealized example but it does make the point Does this equation lead to some length scale, i.e. does a length scale naturally arise out of the structure of the equation? If so, what does it mean physically?
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This note was uploaded on 11/07/2011 for the course AERO 16.512 taught by Professor Manuelmartinez-sanchez during the Fall '05 term at MIT.

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part2 - 16.540 Spring 2006 PRESSURE FIELDS AND UPSTREAM...

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