2 NavierStokes Equations


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1 1 Integral analysis 1. Water enters a pipe bend horizontally with a uniform velocity, u 1 = 5 m/s. The pipe is bended at 90 so that the water leaves it vertically downwards. The input diameter d 1 = 0.1 m and the input pressure p 1 = Pa. The output diameter is d 2 = 0.05 m. The size of the bend is such that it contains 4 kg of water between its two flanges and the mass of the steel walls of the bend is 6 kg. Neglecting viscous forces and assuming steady state conditions, find the total force which acts on the bend, i.e.that force that the bolts at the bend connections must support. 2. The mass off a rocket and fuel together is m(t)= tkg where t is time (s). The crosssectional area of the rocket nozzle at the exit is m 2. The speed of the exhaust gas there is u e = 1350 m/s, relative to the nozzle. The atmospheric pressure around the rocket is p 0 = 10 5 Pa. (a) The rocket is fixed on a horizontal test bench and fired. The pressure of the gas at the exit is p e = Pa. The test bench is held in place by a dynamometer. Find the force read on the dynamometer. (b) The rocket is set with its nose upward and fired. Neglect the aerodynamic resistance and find its acceleration. 3. A singleengine jet airplane flies at the speed of V = 280 m/s. The atmospheric pressure is p a = 10 5 Pa. The welldesigned air intake of the engine has a crosssectional area of A 1 = 0.1 m 2. The exit cross section of the jet nozzle has the area of A 2 = 0.3 m 2. The gas pressure at the exit is p 2 = Pa. The fueltoair mass ratio of the engine is 1 : 20. The specific density of the air at the inlet is ρ=1.2 kg/m 3. The gas speed at the exit relative to the airplane is U = 650 m/s. What is the horizontal force transferred by the bolts connecting the engine to the airplane body? 4. A fireman hose ends with a nozzle as shown below. Measurements show that A F = 0.01 m 2, A E = m 2, u F = 8 m/s, p F = N/m 2, p E = N/m 2 and so is the atmospheric pressure. The nozzle is connected to the hose by a flange at G. Whare are the magnitude and direction of the forces acting on the flange? 1
2 5. A steel pipe of d= m, wall thickness of 6 mm and m long is used to supply water. The water flows at 10 m/s and the nominal stress allowed in the pipe wall is N/cm 2. To prevent high stresses due to water hammer, the valve at the end of the pipe must not be closed faster than a certain rate. What is the minimal time required to close the valve? 6. A floating anchor is a deviced used in lifeboats to keep the nose of the boat directed against the waves. It is made of heavy clothe and has the shape of a cone, with holes at both ends (see below). It is tied to the rear of the boat and is dragged underwater by the boat, which is itself dragged by the waves and the wind. The water is thus forced into the anchor at its larger opening (diameter d i = 1 m) and comes out at the narrower end (diameter d e = 0.4 m). Experiments show that the pressure at the exit, i.e.the narrow end, is close to the hydrostatic pressure for this depth but that the pressure at the wide end is above hydrostatic pressure by about 0.4ρV 2, where V is the speed of the anchor relative to the water. A good assumption is also that the water does not leave the anchor at a relative speed higher than V. Estimate the resistance force of this anchor to relative speeds of 1 m/s, 5 m/s and 10 m/s. 7. A jet of water has the diameter of 0.04 m and the average velocity of 8 m/s. The water hits a stationary flat vane which is tilted by the angle π/4. The pressure everywhere outside the water jet is atmospheric and so it is inside the jet well before it hits the vane. Viscous forces are neglected. Assume the flow twodimensional and find the force acting on the vane and the power extracted by the vane. 8. A round jet of water comes straight up from a nozzle in a water fountain. The jet diameter as it comes out of the nozzle is m and its velocity there is 20 m/s. A little boy places a small glass sphere in the jet and enjoys seeing it 2
3 balances there. The glass sphere has a mass of 0.01 kg. Find the diameter of the water jet just before it hits the glass sphere. 9. A laminar flow (density ρ, viscosity µ) occurs trough a slender nozzle of length l and whose radius varies with the longitudinal coordinate x according to R(x)=R 1 +(R 2 R 1 ) x l where R 1 is the radius at the inlet and R 2 < R 1 is the radius at the outlet. The mass flux Q is constant. (a) Calculate the velocity distribution inside the nozzle assuming a parabolic velocity profile with a mean velocity Ū which is half the maximum velocity. (b) Determine the force acting on the nozzle. 10. After touchdown of aircrafts, the thrust reverser blocks the jet to the rear and redirects if forward to produce reverse thrust. The exiting jet (subsonic jet p 0, ρ 0, relative velocity w 0, section A 0 ) is divided into two symmetric jets with a deflection angle of π β. Thus, the aircraft experiences a deceleration a. Neglecting body forces and viscous forces and neglecting theengine inlet momentum flux (but not the mass flux), show that the deceleration is given by a= ρ 0w 2 0 A 0 cosβ m tot where m tot is the total mass of the aircraft. 2 NavierStokes Equations 1. An experimental system consists of a long tube of radius R 0 filled with glycerin. The pressure gradient along the tube is given and the flow is fully developed. The temperature in the glycerin at the center of the tube is sought, 3
4 and a suggestion is made to stretch a thin wire along the tube axis. It is claimed that, since the wire is very thin, the flow field is only slightly modified by the presence of the wire. Assess this claim by considering the ratio of the maximum velocity and the average velocity of the annular flow to that of the tube flow. 2. A metal wire of radius R i = 2mm is pulled vertically upward with the speed V i through a long pipe of radius R 0 = 5 mm. The gap between the wire and the pipe is filled with molten plastic material of density ρ and viscosity µ. As the wire comes out of the pipe, it carries on its surface a layer of plastic which cools and solidifies. The thickness of the solid layer is 0.1 mm. Find the speed with which the wire is pulled upward. 3. Coating of electric wire with insulating material is done by drawing the wire through a tubular die as shown below. The viscosity of the coating material is 100 poise. Write the equations for this case and calculate the force F required to draw the wire. 4. Glycerin (ρ=1.26 g/cm 3, µ = 10 poise) slides down on a semiinfinite vertical wall and form a laminar layer of thickness δ. Calculate the shear stress on the wall and the flowrate of glycerin. 5. An instrument for measuring viscosity consists of a rotating inner cylinder and a stationary outer cylinder. The inner cylinder rotates at 3600 r.p.m. and the viscosity of the fluid is 100 poise. (a) What is the moment acting on the outer cylinder? (b) What is the efficiency of this instrument as a hydraulic transmission of moment? Find the moment as a function the r.p.m. of the outer cylinder and the power transmitted. 4
5 6. The radial gap of an unloaded bearing which is filled with a Newtonian fluid can be modelled by a twodimensional gap if the radial dimension of the gap h is much smaller than the internal radius R. Assume a steady plane flow induced by the rotation of the journal at the constant angular velocity Ω. The material properties ρ, µ, k are constants. Body forces are neglected. (a) Calculate the torque exerted on the journal and the necessary power. (b) Determine the dissipation function. (c) Calculate the energy per unit time dissipated in the bearing gap. (Compare with the driving power). (d) Determine the heat flux that must be rejected from the fluid in steady operation. (e) Calculate the temperature gradient at the bushing (external axis) if the total heat flux flows through the bushing alone. (f) Determine the temperature distribution in the gap when the bushing is kept at the constant temperature T B. 7. Newtonian fluid flows through a channel with the height h and large extensions in the x 1 and x 3 directions. The plane flow is steady, the density ρ and viscosity µ are assumed to be constant, and body forces are neglected. The top and bottom wall are porous such that a constant normal velocity component V W can be established at the walls. The pressure gradient is constant in the x 1 direction and zero along x 3. Calculate the velocity distribution. What happens in the limiting case V W Incompressible Newtonian fluid flows steadily over a flat plate with large extensions in x and z directions. A boundary layer develops which normally would grow with increasing x. However, succion is applied over the length L such that the boundary layer thickness remains constant. The pressure p is assumed to be constant. Far from the plate, the velocity component u(y) has the value U. (a) Give the boundary condition for the velocity field. (b) Compute the velocity field and check that the mass flux entering the control volume DC is equal to the suction mass flux. (c) Calculate the drag per unit depth and plate length L by directly inegrating the wall shear stress and by application of a momentum balance to the control volume ABCD. 3 Compressible flow 1. A large pressure vessel contains at the total (stagnation) pressure p 0 = Pa and the total temperature T 0 = 350 K. The atmospheric pressure is p a = 5
6 10 5 Pa. Design a convergentdivergent nozzle that discharges 1.5 kg/s air to the atmosphere at atmospheric pressure. Find the speed of the discharged air. The nozzle is then made to discharge into another vessel where the pressure is (a) p e = Pa ; (b) p e = Pa ; (c) p e = Pa ; (d) p e = 10 5 Pa ; (e) p e = Pa Describe the resulting flows. 2. A large pressure vessel contains gas at the stagnation properties p 0 = 400 kpa, T 0 =420 K. The gas is approximately ideal with R = 287 J/kg/K and γ = 1.4. The outside atmospheric pressure is p a = 100 kpa. A convergentdivergent nozzle is designed to pass a mass flux of 1 kg/s from the vessel to the outside. Find (a) the speed of the gas at the exit from the nozzle, (b) the exit Mach number, (c) the critical crosssection area, (d) the exit crosssection area. 3. A normal shock wave moves through a quiescent air at p=10 5 Pa, T = 300 K. The speed of the shock is 694 m/s. Find the pressure left immediately behind the shock. Is the aire immediately behind the shock quiescent? If not, what is its velocity? 6
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