ENGR 2343 – Fluid Mechanics
Chapter 7: Dimensional Analysis & Modeling
Homework #8 - Due 10/22/14 @ beginning of class
Problem #7.1
a.) List the seven primary dimensions.
2)
3)
b.) List the three primary purposes of dimensional analysis.
c.) List the three necessary conditions for complete similarity between a model and a prototype.
Problem #7.2
The water flowrate, Q, in an open rectangular channel can be measured by placing a plate across the channel as shown. This type of device is called a weir (pg. 742, 2nd ed). The height of the water, H, above the weir crest is referred to as the head and can be used to determine the flowrate through the channel. Assume that Q is a function of the head, H, the channel width, b, and the acceleration of gravity, g. Determine a suitable set of dimensionless variables for this problem.
Problem #7.3
From theoretical considerations it is known that for the weir described in Problem #7.2, the flowrate, Q, must be directly proportional to the channel width, b. In some laboratory tests it was determined that if b = 3 ft and H = 4 in., then Q = 1.96 ft3/s. Based on these limited data, determine a general equation for the flowrate over this type of weir.
[Q = 0.598*b*g1/2*H3/2]
Problem #7.4
The drag force, F, on a smooth sphere (pg. 606, 2nd ed.) depends on the relative speed, V, the sphere diameter, D, the fluid density, ρ, and the fluid viscosity, µ. Obtain a set of dimensionless groups that can be used to correlate experimental data.
Identify these commonly encountered dimensionless groups from Table 7-5.
Problem #7.5
The drag of a sonar transducer is to be predicted, based on wind tunnel test data. The prototype, a 1-ft diameter sphere, is to be towed at 5 knots (nautical miles per hour) in seawater at 40ºF. The model is 6 in. in diameter. Determine the required test speed in air. If the drag of the model at these test conditions is 0.60 lbf, estimate the drag of the prototype. [157 ft/s, 5.8 lbf]
Problem #7.6
A periodic Karman vortex street is formed when a uniform stream flows over a circular cylinder. Use the method of repeating variables to generate a dimensionless relationship for Karman vortex shedding frequency fk as a function of free-stream speed V, fluid density ρ, fluid viscosity µ, and cylinder diameter D. Show all your work.
Problem #7.7
Under certain conditions, the flow of fluid past a circular cylinder will produce a Karman vortex street behind the cylinder. This vortex street consists of a set of vortices (swirls) that are shed alternately from opposite sides of the cylinder and then swept downstream with the fluid. An experiment is performed in which the flowrate (Q) of water is adjusted to control the velocity (V) of the fluid flowing past four different cylinders of diameter (D). The width (b) and depth (y) of the channel are indicated below. The time (t) required for (N) vortices to be shed for each experiment is recorded below.
For each data set, calculate the vortex shedding frequency, ω = N/t, which is expressed as vortices (or cycles) per second. Also, calculate the dimensionless frequency called the Strouhl number, St = ωD/V, and the Reynolds number, Re = rVD/µ (remember... this relationship was determined in the previous problem.)
a.) On a single graph, plot the vortex shedding frequency, ω, as ordinates and the water velocity, V, as abscissas for each of the four sets of data.
b.) On another graph, plot the Strouhl number as ordinates and the Reynolds number as abscissas for each of the four sets of data. Plot the literature data included below along with your experimental data.
Experimental data:
Data from literature:
Problem#7.8
An incompressible fluid of density ρ and viscosity µ flows at average speed V through a long, horizontal section of round pipe of length L, inner diameter D, and inner wall roughness height ε. The pipe is long enough that the flow is fully developed, meaning that the velocity profile does not change down the pipe. Pressure decreases (linearly) down the pipe in order to “push” the fluid through the pipe to overcome friction. Using the method of repeating variables, develop a nondimensional relationship between pressure drop ΔP (which is P1 – P2) and the other parameters in the problem. Be sure to modify your Π groups as necessary to achieve established nondimensional parameters, and name them. (Hint: For consistency, choose D rather than L or ε as one of your repeating parameters.) [Eu = f (Re, ε/D, L/D)]
Problem #7.9
Water at 20°C flows through a long, straight pipe. The pressure drop is measured along a section of the pipe of length L = 1.3 m as a function of average velocity V through the pipe. The inner diameter of the pipe is D = 10.4 cm.
a.) Based on your analysis from Problem #7.8, non-dimensionalize the data below and use Excel to plot the Euler number as a function of the Reynolds number. Has the experiment been run at high enough speeds to achieve Reynolds number independence?
b.) Use EXCEL to extrapolate the experimental data to predict the pressure drop at an average speed of 80 m/s. [1,940,000 N/m2]