Middle East Technical University Department of Mechanical Engineering ME 305 Fluid Mechanics I Fall 2018 Section 4 (Dr. Sert) Study Set 7 Reading Assignment R1. Read the section Common Dimensionless Groups in Fluid Mechanics section in Munson s book. R2. Read the section Correlation of Experimental Data section in Munson s book. 1. Provide one or two sentence definitions of the following terms geometric similarity Cavitation number kinematic similarity Froude number dynamic similarity Mach number incomplete similarity Cauchy number Reynolds number Weber number Euler number Strouhal number 2. (Munson) It is desired to determine the wave height when wind blows across a lake. The wave height, H, is assumed to be a function of the wind speed, V, the water density, ρ, the air density, ρ a, the water depth, d, the distance from the shore, l, and the acceleration of gravity, g. Use d, V, and ρ as repeating variables to determine π groups that could be used to describe this problem. 3. (Fox) When a small tube is dipped into a pool of liquid, surface tension causes a meniscus to form at the free surface, which is elevated or depressed depending on the contact angle at the liquidsolid-gas interface. Experiments indicate that the magnitude of this capillary effect, Δh, is a function of the tube diameter, D, liquid specific weight, γ, and surface tension, σ. Determine the π groups. 4. (Fox) The load-carrying capacity, W, of a journal bearing is known to depend on its diameter, D, length, l, and clearance, c, in addition to its angular speed, ω, and lubricant viscosity, μ. Determine the Pi groups that characterize this problem. 1
5. (Aksel) The power of a turbomachine, P, depends on the impeller diameter, D, fluid properties ρ and μ, volumetric flow rate, Q, head, h and the angular speed, ω. Determine the Pi groups. 6. (Fox) The rate dt/dt at which the temperature T at the center of a rice kernel falls during a food technology process is critical. Too high a value leads to cracking of the kernel, and too low a value makes the process slow and costly. The rate depends on the rice specific heat, c, thermal conductivity, k, and size, L, as well as the cooling air specific heat, c p, density, ρ, viscosity, μ, and speed, V. Determine the π groups. 7. (Aksel) The performance of a hydraulic torque converter can be expressed in terms of the transmitted torque, T, angular speed ω, diameter D, and density, ρ. a) Determine the nondimensional π groups. b) It is required to determine the torque transmitted at an angular speed of 1500 rpm. For this reason, a 1:4 scale model is built. The transmitted torque is measured as 10 N when the model is running at 3000 rpm. The same hydraulic oil is used both in the model and the prototype. 8. (Aksel) The pressure drop p in a Venturi meter depends on the fluid properties ρ and μ, pipe diameter, D, throat diameter, d, and the velocity of the fluid, V, in the pipe. a) Determine the π groups. b) A Venturi meter, which is used to measure the flow rate of gasoline (ρ = 700 kg/m 3, μ = 0.003 Pa s) at 0.15 m 3 /s is to be tested in the laboratory by using water (ρ = 1000 kg/m 3, μ = 0.001 Pa s). In the laboratory, the pressure drop is measured to be 5 kpa when the velocity in the pipe is 4 m/s. If the pipe diameter of the prototype flow is 0.15 m, determine the corresponding pressure drop. Also determine the required volumetric flow rate of the water. 9. (Fox) It is harder to achieve dynamic similarity in tests of large trucks and buses; models must be made to smaller scale than those for automobiles. A large scale for truck and bus testing is 1:8. To achieve complete dynamic similarity by matching Reynolds numbers at this scale would require a test speed of 700 km/h. What similarity issues, if any, will arise in such an experiment? 10. (Munson) When a fluid flows slowly past a vertical plate of height h and width b, pressure develops on the face of the plate. Assume that the pressure, p, at the midpoint of the plate is a function of plate height and width, the approach velocity, V, and the fluid viscosity, μ. Make use 2
of dimensional analysis to determine how the pressure, p, will change when the fluid velocity, V, is doubled. 11. (Munson) The spillway for the dam is 20 m wide and is designed to carry 125 m 3 /s at flood stage. A 1:15 model is constructed to study the flow characteristics through the spillway. The effects of viscosity is to be neglected. a) Determine the required model width and flow rate based on Froude number similarity. b) What operating time for the model corresponds to a 24 h period in the prototype? 12. (Munson) If an airplane travels at a speed of 1120 km/h at an altitude of 15 km, what is the required speed at an altitude of 8 km to satisfy Mach number similarity? Assume the air properties correspond to those for the U.S. standard atmosphere. 13. (Munson) The pressure drop between the entrance and exit of a 150 mm diameter 90 o elbow, through which ethyl alcohol at 20 o C is flowing, is to be determined with a geometrically similar model. The velocity of the alcohol is 5 m/s. The model fluid is to be water at 20 o C, and the model velocity is limited to 10 m/s. a) What is the required diameter of the model elbow to maintain dynamic similarity? b) A measured pressure drop of 20 kpa in the model will correspond to what prototype value? 14. (Çengel) Consider the common situation in which a researcher is trying to match the Reynolds number of a large prototype vehicle with that of a small-scale model in a wind tunnel. Is it better for the air in the wind tunnel to be cold or hot? Why? Support your argument by comparing wind tunnel air at 10 C and at 45 C, all else being equal. 15. (Çengel) Some wind tunnels are pressurized. Discuss why a research facility would go through all the extra trouble and expense to pressurize a wind tunnel. If the air pressure in the tunnel increases by a factor of 2, all else being equal (same wind speed, same model, etc.), by what factor will the Reynolds number increase? 16. (Fox) The fluid dynamic characteristics of a golf ball are to be tested using a model in a wind tunnel. Dependent parameters are the drag force, F D, and lift force, F L, on the ball. The independent parameters should include angular speed, ω, and dimple depth, d. Determine suitable dimensionless parameters and express the functional dependence among them. A golf pro can hit a ball at V = 75 m/s and ω = 8100 rpm. To model these conditions in a wind tunnel with a 3
maximum speed of 25 m/s, what diameter model should be used? How fast must the model rotate? (The diameter of a U.S. golf ball is 4.27 cm) 17. (Munson) The pressure drop per unit length, Δp l, for the flow of blood through a horizontal small-diameter tube is a function of the volume rate of flow, Q, the diameter, D, and the blood viscosity, μ. For a series of tests in which D = 2 mm and μ = 0.004 Pa s, the shown data were obtained, where the Δp listed was measured over the length, l = 300 mm. Perform a dimensional analysis for this problem, and make use of the data given to determine a general relationship between Δp l and Q (a relationship that is valid for other values of D, l, and μ). 18. (Çengel) A student team is to design a humanpowered submarine for a design competition. The overall length of the prototype submarine is 4.85 m, and its student designers hope that it can travel fully submerged through water at 0.44 m/s. The water is freshwater (a lake) at T = 15 C. The design team builds a one-fifth scale model to test in their university s wind tunnel. A shield surrounds the drag balance strut so that the aerodynamic drag of the strut itself does not influence the measured drag. The air in the wind tunnel is at 25 C and at one standard atmosphere pressure. At what air speed do they need to run the wind tunnel in order to achieve similarity? If they measure a drag force of 6 N on their model, what is the actual drag force on the prototype? 19. (Çengel) Repeat the previous problem with all the same conditions except that the only facility available to the students is a much smaller wind tunnel. Their model submarine is a one-twentyseventh scale model instead of a one-fifth scale model. At what air speed do they need to run the wind tunnel in order to achieve similarity? Do you notice anything disturbing or suspicious about your result? 4
20. (Çengel) The aerodynamic drag of a new sports car is to be predicted at a speed of 90 km/h at an air temperature of 25 C. Automotive engineers build a one-third scale model of the car to test in a wind tunnel. The temperature of the wind tunnel air is also 25 C. The drag force is measured with a drag balance, and the moving belt is used to simulate the moving ground (from the car s frame of reference). Determine how fast the engineers should run the wind tunnel to achieve similarity between the model and the prototype. 21. (Munson) In order to maintain uniform flight, smaller birds must beat their wings faster than larger birds. It is suggested that the relationship between the wingbeat frequency, ω, beats per second, and the bird s wingspan, l, is given by a power law relationship, ω ~ l n. Use dimensional analysis with the assumption that the wingbeat frequency is a function of the wingspan, the specific weight of the bird, γ, the acceleration of gravity, g, and the density of the air, ρ a, to determine the value of the exponent n. 22. (Munson) A liquid flows with a velocity V through a hole in the side of a large tank. Assume that V = f(h, g, ρ, σ) where h is the depth of fluid above the hole, g is the acceleration of gravity, ρ the fluid density, and σ the surface tension. The following data were obtained by changing h and measuring V, with a fluid having a density = 10 3 kg/m 3 and surface tension = 0.074 N/m. Plot these data by using appropriate dimensionless variables. Could any of the original variables have been omitted? 23. (Munson) The time, t, it takes to pour a certain volume of liquid from a cylindrical container depends on several factors, including the viscosity of the liquid. Assume that for very viscous liquids the time it takes to pour out two-thirds of the initial volume depends on the initial liquid depth, l, the cylinder diameter, D, the liquid viscosity, μ, and the liquid specific weight, γ. The data shown in the following table were obtained in the laboratory. For these tests l = 45 mm, D = 67 mm, and γ = 9.6 kn/m 3. Perform a dimensional analysis, and based on the data given, determine if variables used for this problem appear to be correct. Explain how you arrived at your answer. 5