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# HW 5 - Professor Philip S Marcus ME106 Problem Set 5 Due in...

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10/09/09 Professor Philip S. Marcus ME106 Problem Set 5 Due in class Friday October 16, 2009 Read Munson (any Edition): Chapter on Finite Volume Control Analysis (Chapter 5 in most editions). Do problems: 3.15 and 3.16. Use Munson’s method. O1) Use Matlab to draw the streamlines at time t = 2 (by setting dy/dx = v y /v x ) of the unsteady velocity v ≡ ∇ ψ , where ψ = cos (3 t )[ ln ( x 1) 2 + y 2 ) ln ( x + 1) 2 + y 2 )] . (1) All quantities have units of the MKS system. What is the vorticity ω = ∇ × v ? (Before you begin differentiating, remember that the curl of a gradient is zero.) If the pressure at t = 0, x = 1 and y = 2 is 12, and the density of the fluid is 3 (for all locations and time), find a closed-form expression for the pressure at the origin as a function of time. Remember that Euler’s equation can be written in terms of the vorticity as v /∂t = v × ω − ∇ ( v 2 / 2) ( P ) g ˆ z (2) 1

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In the case where v ≡ ∇ ψ and where ρ and g are constant, we can moving everything to the left-hand side of the equation and inside the gradient to write Euler’s equation as: ( ∂ψ/∂t + v 2 / 2 + P/ρ + gz ) = 0 . (3) What happened to the v × ω term? How does equation (3) relate to Bernoulli’s theorem? Is the quantity ( ∂ψ/∂t + v 2 / 2 + P/ρ + gz ) analogous to the Bernoulli function that we have used for more general flows? How does this new quantity vary along streamlines? How does it vary across streamlines? The purpose of the next two problems is to show the proper way of finding the Bernoulli function across streamlines. Munson’s method, which you used for the first two problems, is not very rigorous, hard to generalize, and can get you into deep trouble. The first problem will deal with flows in Cartesian geometries, and the second problem with cylindrical geometries. O2) Consider the 3D flow between two sets off parallel walls, where the fluid velocity is v x = v y = 0 and v z = ay + bx , where a and b are constants. The walls confine the fluid to the domain −∞ <z< , L x / 2 x L x / 2, and L y / 2 y L y / 2. What is the vorticity of this flow as a function of position? Here you will have to remember that vorticity is a vector and have to use the lecture notes or a textbook to find the Cartesian components of the curl of a vector. The flow has constant density ρ and the gravity has constant magnitude g and points in the ˆ z direction. At y = z = 0 and z = 19 . 7, the pressure is P atm . Find the pressure everywhere. Use the usual formulation of Bernoulli’s theorem to find the pressure everywhere along the streamline that passes through the point (0 , 0 , 19 . 7). To find the pressure elsewhere, it will be necessary to go across streamlines. You have two choices. You 2
can use the following form of the steady-state Euler’s equation with constant density and constant gravity: 0 = v × ω − ∇ ( v 2 / 2 + P/ρ + gz ) . (4) or you can use: 0 = ( v · ∇ ) v − ∇ ( P/ρ + gz ) (5) The Bernoulli functions (the quantities that the gradient operator acts on) are dif- ferent in these two equations. (But you will get the same expression for the pressures regardless of which equation you use.) In either case, to obtain the pressures, you

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HW 5 - Professor Philip S Marcus ME106 Problem Set 5 Due in...

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