F (z) = U 0 2a. Figure 1

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1 Exam, SG2214 Fluid Mechanics 1 November 2013, at 14:00-18:00 Examiner: Anders Dahlkild (ad@mech.kth.se) SCI, Mekanik, KTH 1 Copies of Cylindrical and spherical coordinates, which will be supplied if necessary, can be used for the exam as well as a book of basic math formulas, a calculator and an English dictionary. The point value of each question is given in paranthesis and you need more than 20 points to pass the exam including the points obtained from the homework problems. 1. (10 p.) Consider the complex velocity potential F (z) = U 0 2a (z 2 + ( a2 z )2 a) Show that the coordinate axes and the circle z = a are streamlines. b) Obtain the velocity components in polar coordinates. c) Let s assume that this 2D flow field describes the flow towards a half circular cylinder resting on a plane wall at y = 0, see figure 1. Make a rough sketch of the streamlines. Under what conditions of the physical parameters is the given velocity field for z a a reasonable flow model of a real fluid? Are there any particular areas of the real flow field where you would expect large differences compared to the given velocity field? d) Now, think of the half cylinder as hollow, like a thin shell of radius a, and with a small hole in the shell at the highest stagnation point. This small pressure hole evens out the constant pressure of the stagnant fluid inside the shell to the same value as the stagnation pressure of the flow outside. Calculate the vertical component of the net force on the whole shell due to the pressures distribution on both sides of the shell obtained from the present flow model. ). Figure 1 2. (10 p.) Consider the boundary layer in a viscous, incompressible fluid around the stagnation point of a slowly curved surface where the velocity at the edge of the boundary layer is approximated by U e (x) = U 0 ( x L 2 3 ( x L )3 ), for x < L, where the velocity U 0 and the length L are given constants. a) Roughly, in which region along the surface x L should one expect separation of the boundary layer? Explain why? b) The velocity in the boundary layer can be written in the form ( x u(x, y) = U 0 L f 1(η) 2 3 ( x ( L )3 f 3(η) + O ( x L )5)), where f 1 and f 3 are given functions of the boundary layer coordinate η = y νl/u0. Neglecting the terms of order 5 or higher in x/l < 1, determine the y-component of the velocity in the boundary layer if f 1 and f 3 and their derivatives are regarded as known. Also, what boundary conditions for f 1, f 1, f 3 and f 3 should hold at η = 0 and for f 1 and f 3 as η? c) It is known from a more detailed analysis that f 1 (0) = 1 and f 3 (0) = 2.5. Determine the point of separation of the boundary layer.

2 2 SCI, Mekanik, KTH 3. (10 p.) In a Rankine vortex the fluid in the core rotates like a solid body, whereas outside the core the flow is that of an irrotational line vortex. Consider a tornado that can be idealized in 2D as a Rankine vortex with a core of radius R = 15 m. The pressure at this radius is 1.5 kpa below that of the atmospheric pressure at large distance from the tornado. a) Calculate the circulation around any circuit enclosing the whole core of the tornado. b) Consider such a tornado which center translates at a constant velocity of 20 m/s due to the generally stormy weather conditions. If the tornado is moving straight towards your position, find the time required for the pressure to drop from 0.06 kpa to 1.5 kpa below the atmospheric pressure. You can neglect effects of compressibility and assume the density to be constant at ρ = 1.2 kg/m 2. The atmospheric pressure at large distances from the tornado is now that of the wind blowing at the constant speed of 20 m/s. c) You have a manometer mounted outside your house to register the approach of the tornado. Calculate the time derivative of the measured pressure as function of the distance to the tornado s centre as it approaches. You may assume the core of the tornado has not yet reached your house. (The rapid drop of pressure in combination with the resulting excess pressure inside the walls of buildings may cause a house to virtually explode.) 4. (10 p.) Consider the 2D flow under a sloped bearing pad which slides horizontally with a constant velocity U on a thin layer of oil resting on a horizontal surface. The gap between the bearing pad and the surface is given by h(x) = h 0 (1 + αx/l) where the slope α << 1 and the lengths h 0 << L are a given constants. In a coordinate system following the bearing pad, see figure 2, one can show that the approximate momentum equations in the gap as α << 1 are given by 0 = p x + µ 2 u y 2, 0 = p y, with body forces neglected. The vertical component of the velocity, v << u, can be neglected in the following. Experimentally one has found that the pressure can be described approximately by the formula p(x) = p atm + α 3µLU h 2 0 where p atm is the exterior atmospheric pressure. x L (1 x L ), a) Derive an expression for the velocity u(x, y) in the coordinate system following the bearing pad using the approximations above. b) Calculate the stress vector acting in a position on the surface of the bearing pad. You may neglect terms that are quadratic in the slope, i.e. α 2 << α. The unit normal to the surface can be expressed e n = n/ n where n = (h(x) y). Figure 2

3 SCI, Mekanik, KTH 3 5. (10 p.) A viscous liquid film of constant thickness h flows due to gravity, g, down a plane inclined wall at an angle α with respect to the horizontal plane. For fully developed flow the velocity component along the plate is given by y u(y) = U 0 h (2 y h ), where y is the distance from the plate measured in the normal direction. The plane wall is in thermal contact with a heat reservoir at constant temperature T wall and at the free surface you can assume adiabatic conditions (i.e. there is no heat exchange). a) Determine the constant U 0 if the viscosity of the liquid, µ, and the density, ρ, are considered as constants. b) Make an energy analysis of the system at the fully developed steady state and perform the following tasks: Consider a 2D control volume of length L along the plate and with thickness h. Evaluate each of the integrals of the total energy equation and try to determine the heat flux, if any, through the plane wall of the control volume. If your result is non-zero, try to explain physically where this energy (the heat flux) comes from. The energy equation can be written d ρ(e+ 1 dt V 2 u iu i )dv + S ρ(e+ 1 2 u iu i )u k n k ds = V ρg i u i dv + u i ( pδ ik +τ ik )n k ds q k n k ds. S S ΦdV, where Also, calculate the total dissipation in the control volume, i.e. calculate V for an incompressible fluid Φ = 2µe ij e ij and e ij = 1 2 ( u i x j + u j x i ).

4 z P (r, θ, z) z y θ r x A.3 Curvilinear coordinates Cylindrical Polar Coordinates Figure A.11: Cylindrical polar coordinates The cylindrical polar coordinates are (r, θ, z), where θ is the azimuthal angle, see figure A.11. The velocity can be written as u = u r e r + u θ e θ + u z e z, (A.2) where the unit vectors are related to Cartesian coordinates as e r = e x cos θ + e y sin θ, e θ = e x sin θ + e y cos θ, (A.3) e z = e z. Non-zero derivatives of unit vectors Gradient of a scalar p Laplacian of a scalar p Divergence of a vector u Advective derivative of a scalar p Curl of a vector u u = e r θ = e θ, e θ θ = e r. p = p r e r + 1 p r θ e θ + p z e z. 2 p = 1 r u = 1 r ( r p ) p r r r 2 θ p z 2. (ru r ) r + 1 u θ r θ + u z z. p (u )p = u r r + u θ p r θ + u p z z. ( 1 u z r θ u ) ( θ ur e z r + z u ) z e r θ + 1 ( (ruθ ) u ) r e r r θ z. Incompressible Navier-Stokes equations with no body force u r t +(u )u r u2 θ r = 1 p ρ r + ν u θ t +(u )u θ + u r u θ = 1 p r ρr θ + ν ( 2 u r u r u z t +(u )u z = 1 p ρ z + ν 2 u z. r 2 2 u θ r 2 θ ( 2 u θ + 2 u r r 2 θ u θ r 2 (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) ), (A.10) ), (A.11) (A.12)

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