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ME 363 - Fluid Mechanics Homework #11 Due Wednesday, April 16, 2008 Fall Semester 2008 1] A student team is to design a human-powered submarine for a design competition. The overall length of the prototype submarine is 2.24 m, and its student designers hope that it can travel fully submerged through water at 0.56 m/s. The water is freshwater (a lake) at T = 15 degrees C. The design team builds a one-eighth 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 degrees C and at atmospheric pressure. At what air speed do they need to run the wind tunnel in order to achieve similarity? 2] This problem builds on problem 1. The students measure the aerodynamic drag on their model submarine in the wind tunnel. They are careful to run the wind tunnel at conditions that ensure similarity with the prototype submarine. Their measured drag force is 2.3 N. Estimate the drag force on the prototype submarine at the conditions given in problem 1. 3] Some students want to visualize flow over a spinning baseball. Their fluids laboratory has a nice water tunnel into which they can inject multicolored dye streaklines, so they decide to test a spinning baseball in the water tunnel. Similarity requires that they match both the Reynolds number and the Strouhal number between their model test and the actual baseball that moves through the air at 80 mph and spins at 300 rpm. Both the air and the water are at 20 degrees C. At what speed should they run the water in the water tunnel, and at what rpm should they spin their baseball? no figure Problem #1 Problem #2 Problem #3 Problem 1 Solution For a scale model of a submarine being tested in air, we are to calculate the wind tunnel speed required to achieve similarity with the prototype submarine that moves through water at a given speed. Assumptions 1 Compressibility of the air is assumed to be negligible. 2 The wind tunnel walls are far enough away so as to not interfere with the aerodynamic drag on the model sub. 3 The model is geometrically similar to the prototype. Properties For water at T = 15oC and atmospheric pressure, = 999.1 kg/m3 and = 1.138 10-3 kg/ms. For o air at T = 25 C and atmospheric pressure, = 1.184 kg/m3 and = 1.849 10-5 kg/ms. Analysis Similarity is achieved when the Reynolds number of the model is equal to that of the prototype, Rem Similarity: pVp Lp mVm Lm Rep m p (1) We solve Eq. 1 for the unknown wind tunnel speed, Vm Vp m p p Lp m Lm 1.849 105 kg/m s 999.1 kg/m3 0.560 m/s 8 61.4 m/s 3 3 1.138 10 kg/m s 1.184 kg/m Discussion At this air temperature, the speed of sound is around 346 m/s. Thus the Mach number in the wind tunnel is equal to 61.4/346 = 0.177. This is sufficiently low that the incompressible flow approximation is reasonable. Problem 2 Solution We are to estimate the drag on a prototype submarine in water, based on aerodynamic drag measurements performed in a wind tunnel. Assumptions 1 The model is geometrically similar. 2 The wind tunnel is run at conditions which ensure similarity between model and prototype. Properties For water at T = 15oC and atmospheric pressure, = 999.1 kg/m3 and = 1.138 10-3 kg/ms. For o air at T = 25 C and atmospheric pressure, = 1.184 kg/m3 and = 1.849 10-5 kg/ms. Analysis Since the Reynolds numbers have been matched, the nondimensionalized drag coefficient of the model equals that of the prototype, FD ,m mVm Lm 2 2 FD ,p pVp 2 Lp 2 (1) We solve Eq. 1 for the unknown aerodynamic drag force on the prototype, FD,p, 2 p Vp Lp 999.1 kg/m s 0.560 m/s 2 FD,p FD,m 2.3 N 8 10.3 N 1.184 kg/m s 61.4 m/s m Vm Lm 2 2 where we have used the wind tunnel speed calculated in Problem 7-36. Discussion Although the prototype moves at a much slower speed than the model, the density of water is much higher than that of air, and the prototype is eight times larger than the model. When all of these factors are combined, the drag force on the prototype is much larger than that on the model. Problem 3 Solution We are to calculate the speed and angular velocity (rpm) of a spinning baseball in a water channel such that flow conditions are dynamically similar to that of the actual baseball moving and spinning in air. Properties For air at T = 20oC and atmospheric pressure, = 1.204 kg/m3 and = 1.825 10-5 kg/ms. For o water at T = 20 C and atmospheric pressure, = 998.0 kg/m3 and = 1.002 10-3 kg/ms. Analysis The model (in the water) and the prototype (in the air) are actually the same baseball, so their characteristic lengths are equal, Lm = Lp. We match Reynolds number, Rem pVp Lp mVm Lm Rep m p (1) and solve for the required water tunnel speed for the model tests, Vm, Vm Vp m p p Lp 1.002 103 kg/m s 1.204 kg/m3 80.0 mph 1 5.30 mph 5 3 m Lm 1.825 10 kg/m s 998.0 kg/m (2) We also match Strouhal numbers, recognizing that n is proportional to f, St m f p Lp f m Lm St p Vm Vp nm Lm np Lp Vm Vp (3) from which we solve for the required spin rate in the water tunnel, Lp nm np Lm Vm Vp 5.30 mph 300 rpm 1 19.9 rpm 80.0 mph Discussion Because of the difference in fluid properties between air and water, the required water tunnel speed is much lower than that in air. In addition, the spin rate is much lower, making flow visualization easier. (4)