Lecture-15_(10-21_10)

Lecture-15_(10-21_10) - APPH 4200 Physics of Fluids Review...

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Unformatted text preview: APPH 4200 Physics of Fluids Review October 21, 2010 1.! ! Review 2.! Problems from old Midterms 3.! Midterm 2008 1 Review • Introduction • Tensors, vectors, symmetric and antisymmetric tensors, vector calculus, Gauss’ and Stokes’ Theorems, … • Streamlines, pathlines, convective derivative • Definitions: strain-rate tensor, vorticity, circulation, rigid rotation, stream function 2 Review • Conservation of mass, Navier-Stokes, Newtonian fluids, Stokes’ model for stress tensor, deformation work, viscous dissipation • Bernoulli’s principle for inviscid, irrotational flow • Rotating frame of reference, centrifugal force, Coriolis force 3 Review • Vorticity equation, Kelvin’s theorem, vorticity dynamics • Potential flow, 2D Euler flow, complex velocity potential, Blasius theorem • Dimensional analysis, Reynolds number • Steady, laminar flow with strong viscocity 4 Midterm 2006 • • • • • 4 problems. Closed book Conservation of mass Bernoulli’s principle Viscous flow Vorticity conservation 5 Midterm 2007 • • • • • 4 problems. Closed book Flow near a stagnation point Viscous flow (Couette) Circulation, mass flow, and energy Bernoulli’s principle 6 Midterm Exam Problem 1 Applied Pher 19, 2006 ysics E4200 Octob Physics of Fluids Midterm Exam Read the questions carefully. Notice that the fluid is viscous in some problems, inviscid in others. Inviscid means the viscosity is zero, µ = ν = 0. Good luck. 1. (20 points) October 19, 2006 (a) Determine the vertical velocity, w, such that the three-dimensional flow field whose other two components are 2 2 u = x the v = is x Read the questions carefully. Notice that y z , fluid −y viscous in some problems, inviscid in others. Inviscid means the viscosity is zero, µ = ν = 0. Good luck. satisfies the incompressible mass conservation equation. You may neglect any constants of integration. 1. (20 points) (b) An idealized velocity field is given by the formula (a) Determine the vertical velocity, w, such that the three-dimensional flow u = txy x − 2 ˆ ˆ z field whose other two components are t y y + xzˆ, where x, y, ˆ are thu = x2 yvectors,=and 2t is time. ˆˆz e unit z , v −y x component of the acceleration vector du . dt Compute the x- satisfies the incompressible mass conservation equation. You may neglect 2. (25 points) Fluid fills a tank to a level h1 above the bottom. The hose emits any constants of tegration. fluid from a level hinabove the bottom. The pressure in the atmosphere outside 2 (b) the tank is pa , while thefield is givenvby the ater surface is pt , different from pa . An idealized velocity pressure abo e the w formula The force of gravity points downward, with constant gravitational acceleration ˆ z g . Consider the fluid to beu = txy x − t2 y y + xzˆ, density, and inviscid, and incompressible,ˆ constant assume the flow to be steady. Find the following quantities in terms of the given where x, ˆ z ˆ constants: y, ˆ are the unit vectors, and t is time. Compute the x- 7 Problem 2 component of the acceleration vector du . dt (a) The height to which the water stream rises, h3 in the figure. 2. (25 p(b) ts)he velocity of the waterastreamh1 abexitsthe bottom. The hose emits oin T Fluid fills a tank to level as it ove the hose. fluid from a level h2 above the bottom. The pressure in the atmosphere outside the tank is pa , while the pressure above the water surface is pt , different from pa . The force of gravity points downward, with constant gravitational acceleration g . Consider the fluid to be incompressible, constant density, and inviscid, and assume the flow to be steady. Find the 1 wing quantities in terms of the given follo constants: (a) The height to which the water stream rises, h3 in the figure. (b) The velocity of the water stream as it exits the hose. P 6- ~ i 1 hi rT -p~ ~ ( ~-------l-_. 3. (35 points) Fluid with constant density, p, and constant, nonzero kinematic viscosity, lJ, flows between two rigid boundaries y = 0, y = h. The lower 8 boundary moves in the x-direction with speed U, the upper boundary is at rest. The boundaries are porous, and the vertical (y) velocity is -Va at each one, Va being a given positive constant, so that there is an imposed flow across the i hi rT Problem 3 3. (35 points) Fluid with constant ~ (y, ρ, and constant, nonzero kinematic -p~ densit viscosity, ν , flows between ~-------l-_. two rigid boundaries y = 0, y = h. The lower 3. (35 points)moves in the x-direction with , peed U , constant, nonzero kinematic boundary Fluid with constant densitys ρ, and the upper boundary is at rest. 3. (35 points) Fluid with constant p, and constant, viscosity, ν , flolJ, areetween two density, bertical y y )0,vy= h. Theis= v0 at eaclower ws b porous, andrigid voundaries nonzero kinematic 0, The boundaries flows between two rigid boundaries ( = y elocity y lower . The h one, the −h viscosity, = boundary a gives in ositive constant, so that theU , the an impbat rest.flow is at rest. boundary moves in the x-direction with speed eed upper boundary oun v0 being mov en p the x-direction with spU, there is upperisosed dary across the The boundaries p the system vertical (y) velocity x and , one, The boundaries yreakeporous, and the to be uniform )in is -Va at yisassume steadyone, system. You maa t areorous, and the vertical (y velocity each−v0 at each flow, Va being a given positive constant, so that there is an imposed flow across the v0 being a givenvity.take e constant, so thatin x and y, assume compflow, w acrosst: do positiv the the to w (both there and steady onen and neglect gra may Findsystemflobe uniform the x is anyimposed flots). Hin the system. You system.andouonent t(v )efirst.system to bethe x and y components). Hint: do steady flow, Y neglect ak Find the y -comp maygravity.the the flow (both uniform in x and y , assume the y-component (v) first. the flow (b oth the x and y comp onents). Hint: do and neglect gravity. Find the y -component (v ) first. l Vo -0 r. '0 c: '= 'c i= L: ~ w 0 D Cl 0 c: CJ G' l' h L l C7 Va c/ \,j . -- Vi l :: (! J ~ ç7Y ¡V 9 4. (20 points) Consider a large lake, initially at rest. In the center, the water is stirred by a solid object inserted into it (e.g., a spoon or egg beater). The object is then removed. Problem 4 (a) If the fluid is inviscid, explain why no vorticity can be generated by the stirring. 4. (20 points) Consider a large lake, initially at rest. In the center, the water is stirred by a solid ob ject inserted into it (e.g., a spoon or egg beater). The ob ject 4. (20then removed. is points) Consider a large lake, initially at rest. In the center, the water is stirred by a solid ob ject inserted in2to it (e.g., a spoon or egg beater). The ob ject is(a) If remofluid is inviscid, explain why no vorticity can be generated by the then the ved. stirring. (a) If the fluid is inviscid, explain why no vorticity can be generated by the stirring. (b) If the fluid is viscous, vorticity can be created. Assume that this occurs, but only in a small region around2the ob ject. Explain why after the removal of the ob ject, the net vorticity created must still be zero, that is, the areal 2 integral of the vertical vorticity over the entire lake must vanish, ￿ ω dA ω= ∂v ∂u − ∂x ∂y lake where is the vertical vorticity. In both parts, you may assume that the water has constant density. 10 Problem 5 11 Problem 6 12 Problem 7 13 Midterm 2008 • • • • 3 problems. Closed book 2D Potential Flow Bernoulli’s principle Poiseuille Flow 14 This is a closed book exam. Please read each question carefully. Please also show your work and explain your reasoning as this helps in understanding your answer and the award of partial credit. Formula are attached. Problem 1 1. (30 Points) For two-dimensional potential flow, the divergence and curl of the flow vanishes, i.e. ∇ · U = ∇ × U = 0. The steady flow can be represented by a ˆ velocity potential, U ≡ ∇φ, or a stream function, U ≡ −z × ∇ψ . Since φ and ψ satisfy Laplace’s equation, the flow can be represented on the complex plane, z = x + iy = reıθ , by an analytic function called the “complex velocity potential”, w(z ) ≡ φ + iψ . When w(z ) = Az n , w(z ) represents the flow around or inside a corner located at (x, y ) = (0, 0). Show that (i) when n < 1, the flow passes over the “outside” of a corner having an angle greater than 180◦ , and (ii) when n > 1, the flow passes on the “inside” of a corner having an angle less than 180◦ . For each case, n > 1 and n < 1, determine whether the fluid velocity increases or decreases when approaching the corner. J Q fl .. o0 .) .. 'i ~ .. oJ 01 .J ~ \. " 0 h IN i'~ V ii" ;:' '"'-, ,"- 3 0 ,J U. q, " :i \) - ~ l\ - " ~ Figure 1: For Question 1. Two-dimensional potential flow near corners. 15 Problem 2 1 2. (35 points) An incompressible liquid of density, ρ1 , is flowing through a tube at a volumetric rate of Q. The tube cross-sectional area as two parts. On the entrance side, the area is A1 , and on the exit side the area is A2 (with A2 > A1 .) A “U-tube” is connected to the tube as shown in the figure below. The U-tube is partially filled with a liquid of density ρ2 , which is denser than the fluid flowing through the tube, ρ2 > ρ1 . The two fluids do not mix. The differerence between the heights of the upstream and downstream sides of the U-tube is h. Your problem : Derive an expression for h in terms of Q, A1 , A2 , ρ1 , ρ2 , and g , the acceleration of gravity. [Hint: The fluid in the U-tube is in a state of hyrdostatic equilibrium, where pressure gradients balance gravitational forces. The pressure must be constant where the Utube connects to the tube with flowing fluid.] 16 Figure 2: For Question 2. Tube with U-tube connected. Problem 3 3. (35 points) A viscous fluid flows at a steady rate through a tube with an elliptical cross-section. See figure below. What is the relation between the volumetric flow rate, Q, through the tube and the axial rate of pressure drop, dP/dz , in terms of the dimensions of the tube and the fluid viscosity. (You may leave integrals unevaluated, provided they are defined fully.) Show that the pressure gradient is balanced by a viscous force. [Note: You do not have to evaluate any expressions for any integral. Just define them fully.] y z 2c x 2b 17 Figure 3: For Question 3. Tube with elliptical cross-section for steady fluid flow. 3 ...
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