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Published byAnika Wiginton Modified about 1 year ago

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Experiment #5 Momentum Deficit Behind a Cylinder

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Problem Statement Find the drag force on the rod located in a 2.0 mph 40 °F water stream Use analogy with air flow normal to a model cylinder in a wind tunnel Estimate the error in the drag force Water, 40 °F 2.0 mph 0.5” rod

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Available Major Equipment 6 x 6 inch in cross section wind tunnel Hot film anemometer to measure air speed Model cylinder to place in the wind tunnel

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Approach Measure the drag on a model cylinder in the wind tunnel Use results from dimensional analysis to establish the conditions for similarity between the flow around the model and the prototype (the rod in the water stream) Use the non-dimensional numbers derived from dimensional analysis to estimate the drag force on the rod Estimate the error in the drag force on the model and the prototype

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Results from Dimensional Analysis (consult a fluids text) Drag Coefficient, C D, of a cylinder in cross-flow F D = drag force on the cylinder = air density D = cylinder diameter w = cylinder length U 1 = approach velocity of air stream = kinematic viscosity of air Similarity between model and prototype requires that C Dmodel = C Dprototype = f(Re model ) = f(Re prototype ) or that Re model = Re prototype

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Equality of Reynolds Numbers implies equality of the Drag Coefficients

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Experimental Approach for the Drag Force Estimate the Reynolds number for the prototype (rod in water), and compute the wind tunnel air speed necessary for similarity with the model cylinder –Note: account for the fact that the pressure in the wind tunnel is less than atmospheric Find the drag force on the model cylinder from measurements carried out in the wind tunnel Compute the model Reynolds number and Drag coefficient. –Compare with the values from the previously shown C D vs. Re curve. Assuming similarity (check this) use the magnitude of the experimental drag coefficient on the model cylinder to find the drag force on the prototype stack

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Static pressure inside the wind tunnel Use Bernoulli’s equation between points o (outside the tunnel) and 1 (inside the tunnel) Note that at point o the pressure is the atmospheric level, and that the velocity at point o is zero. Using the known wind tunnel speed, V 1 find the pressure p 1.

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Control Volume Force and Mass Balances ( = constant) Surface 1, rate of momentum in Surface 2,rate of momentum out Surfaces 3 and 4, rate of momentum out Force balance Mass balance Surface 1, rate of mass in Surface 1, rate of mass out Surfaces 3 and 4, rate of mass out Multiplying the second eq. by U 1 and substituting in the first eq, results in:

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Drag force on the Model Cylinder Measure the velocity profile U 1 without the cylinder Measure the velocity profile u 2 with the cylinder at two location downstream from the cylinder Compute F D for each of the two profiles using the experimental data (numerical integration needed) Take the mean of the two F D values

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Drag force on the Rod in the Water

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Error Estimation Find the error in F D of the model assuming that the only measuring error is a ±1% in the measurement of velocity. Use the RSS method. (Consult your 650:350 Measurements Book) Error in F D whereand u u2 and u U1 are the ± errors in the velocity measurement. Use ±1% of measured values Numerical integration is needed to evaluate the integrals

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Error Estimation (continued) The error in the F D of the prototype rod is then found in a similar way from:

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