AEROSPACE ENGINEERING DEPARTMENT Second Year - Second Term (2008-2009) Fluid Mechanics & Gas Dynamics Similitude,Dimensional Analysis &Modeling (1) [7.2R*] Some common variables in fluid mechanics include: volume flowrate, Q, acceleration of gravity, g, viscosity, μ, density, ρ, and length, L. Which of the following combinations of these variables are dimensionless? (a) Q 2 /gl 2 (b) ρq/μl (c) gl 5 /Q 2 (d) ρql/μ (2) [7.3R*] A fluid flows at a velocity V through a horizontal pipe of diameter D. An orifice plate containing a hole of diameter d is placed in the pipe. It is desired to investigate the pressure drop, Δp, across the plate. Assume that Δp = f (D, d, ρ, V) Where ρ is the fluid density. Determine a suitable set of pi terms. (3) [7.6R*] The thrust T, developed by a propeller of a given shape depends on its diameter, D, the fluid density, ρ, and the viscosity, μ, the angular speed of rotation, ω, and the advance velocity, V. Develop a suitable set of pi terms, one of which should be ρd 2 ω / μ.. (4) [7.5*] At a sudden contraction in a pipe the diameter changes from D 1 to D 2. The pressure drop Δp, which develops across the contraction, is a function of D 1 and D 2, as well as the velocity, V, in the larger pipe, and the fluid density, ρ, and the viscosity μ. Use D 1, V, and μ as repeating variables to determine a suitable set of dimensionless parameters. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable? (5) [7.6*] Assume that the power, P, required to drive a fan is a function of the fan diameter, D, the fluid density ρ, the rotational speed, ω, and the flowrate, Q. Use D, ω and ρ as repeating variables to determine a suitable set of pi terms. (6) [7.9*] The pressure rise, Δp, across a pump can be expressed as Δp = f (D, ρ, ω, Q) where D is the impeller diameter, ρ the fluid density, ω the rotational speed, and Q the flow-rate. Determine a suitable set of dimensionless parameters. 1
(7) [7.10*] The drag, D, on a washer-shaped plate placed normal to a stream of fluid can be expressed as: D = f (d 1, d 2, V, μ, ρ) where d1 is the outer diameter, d2 is the inner diameter, V the fluid velocity, μ the fluid viscosity, and ρ the fluid density. Some experiments are to be performed in a wind tunnel to determine the drag. What dimensionless parameters would you use to organize these date? (8) [7.18*] The pressure drop Δp, along a straight pipe of diameter D has been experimentally studied, and it is observed that for laminar flow of a given fluid and pipe, the pressure drop varies directly with the distance, L, between pressure taps. Assume that Δp is a function of D and L, the velocity, V, and the fluid viscosity, μ. Use dimensional analysis to deduce how the pressure drop varies with pipe diameter. (9) [7.8R*] The pressure drop per unit length in a 0.25-in diameter gasoline fuel line is to be determined from a laboratory test using the same tubing but with water as the fluid. The pressure drop at a gasoline velocity of 1.0 ft/s is of interest. (a) What water velocity is required? (b) At the properly scaled velocity from part (a), the pressure drop per unit length (using water) was found to be 0.45 psf/ft. What is the predicted pressure drop per unit length for the gasoline line? (10) [7.34*] The drag characteristics of a torpedo are to be studied in a water tunnel using a 1 : 5 scale model. The tunnel operates with freshwater at 20 o C, whereas the prototype torpedo is to be used in seawater at 15.6 o C. To correctly simulate the behavior of the prototype moving with a velocity of 30 m/s, what velocity is required in the water tunnel? (11) [7.40*] The lift and drag developed on the hydrofoil are to be determined through wind tunnel test using standard air. If full-scale tests are to be run, what is the required wind tunnel velocity corresponding to a hydrofoil velocity in seawater at 15 mph? Assuming Reynolds number similarity is required. (12) [7.44*] The drag on a 2-m-diameter satellite dish due to an 80 km/hr wind is to be determined through a wind tunnel test using a geometrically similar 0.4-m-diameter model dish. Assume standard air for both model and prototype. (a) At what air speed should the model test be run? 2
(b) With all similarly conditions satisfied, the measured drag on the model was determined to be 170 N. What is the predicted drag on the prototype dish? (13) [7.50*] The drag, F D, on a sphere located in a pipe through which a fluid is flowing is to be determined experimentally (see the figure). Assume that the drag is a function of the sphere diameter, d, the pipe diameter, D, the fluid velocity, V, and the fluid density, ρ. (a) What dimensionless parameters would you use for this problem? (b) Some experiments using water indicate that for d = 0.2 in., D = 0.5 in., and V = 2 ft/s, the drag is 1.5x10-3 lb. If possible, estimate the drag on a sphere located in a 2-ft-diameter pipe through which water is flowing with a velocity of 6 ft/s. The sphere diameter is such that geometric similarity is maintained. If it is not possible, explain why not. (14) [7.10R*] The drag on a 30-ft long, vertical, 1.25-ft diameter pole subjected to a 30 mph wind is to be determined with a model study. It is expected that the drag is a function of the pole length and diameter, the fluid density and viscosity, and the fluid velocity. Laboratory model tests were performed in a high-speed water tunnel using a model pole having a length of 2 ft and a diameter of 1 in. Some model drag data are shown in figure. Based on these data, predict the drag on the full-sized pole. 3
(15) [7.24*] The pressure rise, Δp = p 2 p 1, across the abrupt expansion shown in figure, through which a liquid is flowing can be expressed as: Δp = f (A 1, A 2, ρ, V 1 ) where A 1 and A 2 are the upstream and downstream cross-sectional areas, respectively, ρ is the fluid density, and V 1 is the upstream velocity. Some experimental data obtained with A 2 = 1.25 ft 2, V 1 = 5.00 ft/s. and using water with density ρ = 1.94 slugs/ft 2 are given in the following table. A 1 (ft 2 ) 0.10 0.25 0.37 0.52 0.61 Δp (lb/ft 2 ) 3.25 7.85 10.3 11.6 12.3 Plot the result of these tests using suitable dimensionless parameters. With the aid of a standard curve fitting program determine a general equation for Δp and use this equation to predict Δp for water flowing through an abrupt expansion with an area ratio A 1 /A 2 = 0.35 at a velocity V 1 = 3.75 ft/s. 4
(16) [7.65*] The pressure rise, Δp, across a centrifugal pump of a given shape (see the figure) can be expressed as: Δp = f (D, ω, ρ, Q) where D is the impeller diameter, ω the angular velocity of the impeller, ρ the fluid density, and Q the volume rate of flow through the pump. A model pump having a diameter of 8 in. is tested in the laboratory using water. When operated at an angular velocity of 40 π rad/s the model pressure rise as a function of Q is shown in figure. Use this curve to predict the pressure rise across a geometrically similar pump (prototype) for a prototype flowrate of 6 ft 3 /s. The prototype has a diameter of 12 in. and operates at an angular velocity of 60π rad/s. The prototype fluid is also water. *Selected problems from the text book: Fundamentals of Fluid Mechanics, 3 rd Edition, by: Munson, Young and Okiishi, 1998 5
Viscous Flow in Pipes (1) [8.2R*] A fluid flows through two horizontal pipes of equal length, which are connected together to form a pipe of length 2L.The flow is laminar and fully developed. The pressure drop of the first pipe is 1.44 times grater than it is for the second pipe. If the diameter of the first pipe is D, determine the diameter of the second pipe. (2) [8.4*] Air at 100 o F flows at standard atmospheric pressure in a pipe at a rate of 0.08 lb/s. Determine the minimum diameter allowed if the flow is to be laminar. (3) [8.5*] Carbon dioxide at 20 o C and a pressure of 550 kpa (abs) flows in a pipe at a rate of 0.04 N/s. Determine the maximum diameter allowed if the flow is to be turbulent. (4) [8.11*] Water flows in a constant diameter pipe with the following conditions measured: At section (a) P a = 32.4 psi and z a = 56.8 ft, at section (b) P b = 29.7 psi and z b = 68.2 ft. Is the flow from (a) to (b) or from (b) to (a)? Explain. (5) [8.5 R*] Water flows in a smooth plastic pipe of 200-mm diameter at a rate of 0.10 m 3 /s. Determine the friction factor for this flow. (6) [8.17 *] Glycerin at 20 o C flows upward in a vertical 75-mm-diameter pipe with a centerline velocity of 1.0 m/s. Determine the head loss and pressure drop in a 10-m length of the pipe. (7) [8.18*] A fluid flows through a horizontal 0.1-in. diameter pipe. When the Reynolds number is 1500, the head loss over a 20- ft length of the pipe is 6.4 ft. Determine the fluid velocity. (8) [8.6 R*] After a number of years of use, it is noted that to obtain a given flowrate, the head loss is increased to 1.6 times its value for the originally smooth pipe. If the Reynolds number is 10 6, determine the relative roughness of the old pipe. (9) [8.9 R*] Determine the pressure drop per 300-m length of new 0.20-mdiameter horizontal cast iron water pipe when the average velocity is 1.7 m/s. (10) [8.21*] A fluid flows in a smooth pipe with a Reynolds number of 6000. By what percent would the head loss be reduced if the flow could be maintained as laminar flow rather than the expected turbulent flow? 6
(11) [8.29*] Carbon dioxide at a temperature of 0ºC and a pressure of 600 kpa (abs) flows through a horizontal 40-mm-diameter pipe with an average velocity of 2 m/s. Determine the friction factor if the pressure drop is 235 N/m 2 per 10- m length of pipe. (12) [8.11 R*] An above ground swimming pool of 30 ft diameter and 5 ft depth is to be filled from a ground hose (smooth interior) of length 100 ft and diameter 5/8 in. If the pressure at the faucet to which the hose is attached remains at 55 psi gauge, how long will it take to fill the pool? The water exits the hose as a free jet 6 ft above the faucet. (13) [8.12 R*] Water is to flow at a rate of 1.0 m 3 /s. through a rough concrete pipe (ε = 3 mm ) that connects two ponds. Determine the pipe diameter if the elevation difference between the two ponds is 10 m and the pipe length is 1000 m. Neglect minor losses. (14) [8.13 R*] Without the pump shown in figure it is determined that the flow rate is two small. Determine the horsepower added to the fluid if the pump causes the flowrate to be doubled. Assume the friction factor remains at 0.020 in either case. (15) [8.52*] Gasoline flows in a smooth pipe of 40-mm diameter at a rate of 0.001 m 3 /s. If it were possible to prevent turbulence from occurring. What would be the ratio of the head loss for the actual turbulent flow compared to that if it were laminar flow? (16) [8.53*] A 3-ft-diameter duct is used to carry ventilating air into a vehicular tunnel at a rate of 9000 ft 3 /min. Tests show that the pressure drop is 1.5 in. of water per 1500 ft of duct, what is the value of the friction factor for this duct and the approximate size of the equivalent roughness of the surface of the duct? (17) [8.54*] Natural gas ( ρ = 0.0044 slugs/ft 3 and υ = 5.2 x 10-5 ft 2 /s) is pumped through a horizontal 6-in.-diameter cast-iron pipe at a rate of 800 lb/hr. 7
If the pressure at section (1) is 50 psi (abs), determine the pressure at section (2) 8 mi downstream if the flow is assumed incompressible. Is the incompressible assumption reasonable? Explain. (18) [8.82*] A water flowrate of 3.5 ft 3 /s is to be maintained in a horizontal aluminum pipe ( ε = 5 x 10-6 ft). The inlet and outlet pressures are 65 psi and 30 psi, respectively, and the pipe length is 500 ft. Determine the diameter of this pipe. (19) [8.89*] The pump shown in figure adds 25 kw to the water and causes a flowrate of 0.04 m 3 /s. Determine the flowrate expected if the pump is removed from the system. Assume f = 0.016 for either case and neglect minor losses. (20) [8.97*] Air, assumed incompressible, flows through the two pipes shown in figure. Determine the flowrate if the minor losses are neglected and the friction factor in each pipe is 0.015. Determine the flowrate if the 0.5-in.- diameter pipe were replaced by a 1-in.-diameter pipe. Comment on the assumption of incompressibility. *Selected problems from the text book: Fundamentals of Fluid Mechanics, 3 rd Edition, by: Munson, Young and Okiishi, 1998. 8
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One-Dimensional Isentropic Flow 1. Air flows through a variable area duct. Measurements indicate that the pressure is 80 kpa, the temperature is 5 o C, and the velocity is 150 m/s at a certain section of the duct. Estimate, assuming incompressible flow, the velocity and pressure at a second section of the duct at which the duct area is half that of the section where the measurements were made. Comment on the validity of the incompressible flow assumption in this situation. 2. Air is assumed to be perfect gas, flowing with a velocity of 200 m/s. If the static temperature and pressure are 25 o C and 1 atm respectively, calculate the total properties, the critical properties, and the non-dimensional velocity M*. 3. An airplane flies at an altitude of 15 km with a velocity of 800 km/h. Calculate the maximum possible temperature on the airplane, the maximum possible pressure intensity on the airplane, the critical velocity of the air relative to the airplane, and the maximum possible velocity of the air relative to the airplane. 4. The exhaust gases from a rocket engine can be assumed to behave as a perfect gas with a specific heat ratio of 1.3 and a molecular weight of 32. The gas is expanded from the combustion chamber through the nozzle. At a point in the nozzle where the cross-sectional area is 0.2 m 2 the pressure, temperature, and Mach number are 1500 kpa, 800 o C, and 0.2 respectively. At some other point in the nozzle, the pressure is found to be 80 kpa.find the Mach number, temperature, and cross-sectional are at this point. Assume one-dimensional, isentropic flow. 5. A convergent nozzle has an exit area of 6.5 cm 2 and total inlet conditions of 680 kpa and 370 o K. Assume isentropic flow calculate the mass flow rate if the ambient pressure is 359 kpa, 540 kpa, and 200 kpa. 6. An aircraft is flying at a Mach number of 0.95 at an altitude where the pressure is 30 kpa and the temperature is 50 o C. The diffuser at the intake to the engine decreases the Mach number to 0.3 at the inlet to the compressor. Find the pressure and temperature at the inlet to the compressor. 7. Consider a rocket engine that burns hydrogen and oxygen. The combustion chamber temperature and pressure are 3800 K and 1.5 MPa respectively, the velocity in the combustion chamber being very low. The pressure on the nozzle exit plane is 1.5 kpa. Assuming that the flow is isentropic, find the Mach 10
number and the velocity on the exit plane. Assume that the products of combustion behave as a perfect gas with γ = 1.22 and R = 519.6 J/kg/K. Normal Shock Waves 1. A blast wave created by an explosion moves with a velocity of 1384 km/s into still atmospheric air having a pressure of 1 atm and temperature of 25 o C calculate; the Mach number of the shock wave relative to the stationary air, and the static and stagnation values of the pressure and temperature behind the shock wave relative to a stationary observer. 2. A normal shock wave is propagated into still air with a velocity of 700 m/s. The still air pressure is 1 atm and its temperature is 300 o K, calculate the Mach number, the pressure, temperature, and the velocity in back of the normal shock wave, relative to a stationary observer. 3. Air at a static pressure of 150 kpa and a static temperature of 300 o K flows at 150 m/s in a pipe. The valve at the end of the pipe is suddenly closed, propagating a normal shock wave back into the pipe. Calculate the velocity of the shock wave relative to the pipe. 4. A pitot tube is mounted on a nose of an airplane. The stagnation and the free stream pressure readings, for three different flight conditions are: (0.816 atm, 0.688 atm), (1.570 atm, 0.460 atm), (3.580 atm, 0.297 atm). Calculate the free stream Mach number at which the airplane is flying for each flight. 5. A blunt-nosed missile is flying at Mach 2 at standard sea level. Calculate the temperature and pressure at the nose of the missile. One-Dimensional Flow with Heat Addition 1. Air enters a constant area duct at M 1 = 0.2, p 1 =1atm, and T 1 =273 K. Inside the duct, the heat added per unit mass is q = 1.0 x 106 J/kg. Calculate the flow properties M 2, p 2, T 2, ρ 2, T o2, and P o2 at the exit of the duct. 2. Air enters a constant area duct at M 1 = 3, p 1 =1atm, and T 1 =300 K. Inside the duct, the heat added per unit mass is q = 3 x 105 J/kg. Calculate the flow properties M 2, p 2, T 2, ρ 2, T o2, and P o2 at the exit of the duct. 3. Air enters the combustor of a jet engine at P 1 =10 atm, T 1 = 1000 R, and M 1 = 0.2. Fuel is injected and burned, with a fuel to air ratio (by mass) of 0.06. Calculate the Mach number, the static pressure, and the static temperature at the exit of the combustor. Assume one-dimensional frictionless flow with γ = 11
1.4 for the fuel air mixture. The heat released during the combustion per slug of fuel is equal to 4.5x10 8 lb.ft. One-Dimensional Flow with Friction 1. Consider the flow of air through a pipe of inside diameter = 0.15 m and length = 30 m. The inlet flow conditions are M 1 = 0.3, p 1 = 1 atm, and T 1 = 273 K. Assuming f = constant =.005, calculate the flow conditions at the exit M 2, p 2, T 2, ρ 2, T o2, and P o2. 2. Consider the flow of air through a pipe of inside diameter = 0.4 ft and length = 5 ft. The inlet flow conditions are M 1 = 3, p 1 = 1 atm, and T 1 = 300 K. Assuming f = constant = 0.005, calculate the flow conditions at the exit M 2, p 2, T 2, ρ 2, T o2, and P o2. 3. Air is flowing through a pipe of 0.02 m inside diameter and 40 m length. The exit flow conditions of the pipe are M 2 = 0.5, p 2 = 1 atm, and T 2 = 270 K. Assuming adiabatic, one-dimensional flow, with a local friction coefficient of 0.005, calculate the flow conditions at the entrance of the pipe (M 1, p 1, T 1, ρ 1, T o1, and P o1 ). Variable Area Flow 1. A convergent divergent nozzle has total conditions of 700 kpa and 330 o K. The mass flow rate is 1 kg/s. the total exit pressure is 550 kpa and the static exit pressure is 500 kpa. If the flow is isentropic except for the occurrence of a shock, calculate the throat area, the Mach number before and after the shock, the area where the shock occurs, the exit area and exit velocity and density. 2. A nozzle is to be designed to expand air isentropically from the total conditions 300 o K and 2.042 atm to the atmosphere, at 15 km altitude, with a rate of 25 kg/s. Assume that the air is a perfect gas with γ =1.4, R=287 J/kg/K and the flow is one dimensional. Calculate the throat and exit diameters of the nozzle and find the flow velocity at the exit section. What is the exit velocity and the mass flow rate when the nozzle is operated at sea level. (At sea level the ambient pressure is 101.325 kpa and at 15 km altitude the ambient pressure is 12.112 kpa) 3. A convergent divergent nozzle is designed to operate with an exit Mach number of 1.75. The nozzle is supplied from an air reservoir at 1000 psia. Assuming one dimensional flow, calculate: The maximum back pressure to choke the nozzle, 12
The range of back pressures over which a normal shock will appear in the nozzle, The back pressure for the nozzle to be perfectly expanded to the design Mach number, The range of back pressures for supersonic flow at the nozzle exit plane. 4. A convergent divergent nozzle is designed to expand air from a chamber in which the pressure is 700 kpa and the temperature is 35 o C to give a Mach number of 1.6. The mass flow rate through the nozzle under design conditions is 0.012 kg/s. Find: the throat and exit areas of the nozzle, the design back pressure and the temperature of the air leaving the nozzle with this back pressure, the lowest back pressure for which there will be no supersonic flow in the nozzle, The back pressure below which there are no shock waves in the nozzle. 5. Air flows through a convergent divergent nozzle that has an inlet area of 25 cm2. The inlet temperature and pressure are 50 o C and 550 kpa respectively, and the velocity at the inlet is 80 m/s. If the flow is assumed to be isentropic, and if the exit pressure is 120 kpa, find the throat and exit areas and the exit velocity. 6. Air enters a convergent divergent nozzle at Mach number 0.2. The stagnation pressure is 700 kpa and the stagnation temperature is 5 o C. The throat area of the nozzle is 46 cm 2. If the pressure at the exit to the nozzle is 500 kpa, determine if there is a shock in the divergent portion of the nozzle. If there is a shock wave, determine the nozzle area at which the shock occurs and the Mach number and pressure just before and just after the shock wave. 7. Air with a stagnation pressure and temperature of 100 kpa and 150 o C is expanded through a convergent divergent nozzle that is designed to give an exit Mach number of 2. The nozzle exit area is 30 cm 2. Find the exit pressure and the mass rate of flow through the nozzle when operating at design conditions. Also find the exit pressure if a normal shock wave occurs in the divergent portion of the nozzle at a section where the area is half between the throat and exit areas. Oblique Shock and Expansion Waves 1. A flat plate airfoil of chord c is in a Mach 3 at an angle of attack of 8 deg. Use the shock expansion theory calculates the lift and drag coefficient C L & C D. 13
2. A symmetric double wedge two-dimensional airfoil having a thickness to chord ratio of 0.07 is placed at an angle of attack of 7 deg in a supersonic air stream of Mach number 2.5. Calculate the lift and drag coefficients. 3. Consider a symmetric double wedge airfoil. If the semi wedge angle is 10 deg and the free stream Mach number is 2, use shock expansion theory to compute the lift and drag coefficients, the moment coefficient about the leading edge point. Make these calculations for angle of attack 10 deg. 4. A thin two dimensional flapped flat plate airfoil is placed at an angle of attack α in a stream of Mach number 4. Calculate the lift and drag coefficients for the following cases: ( α = -2.5,2.5,5,7.5,10) and ( δ = -5,0,5,10) where δ is the deflection angle of the flap whose length is 20% of the total airfoil chord. 5. Consider a two dimensional flow about an infinite wedge whose cross sectional shape is a right angle. The free stream Mach number is 5 and the wedge angle is 10 deg. The upper surface is aligned with the free stream direction. Calculate the lift and drag coefficients if (P b is set equal to zero & P b is set equal to P ) Estimate without calculations the exact values of C L & C D. M P b Prof. Dr. Mohamed Madbouli Abdelrahman 14