Report on Momentum Deficit Laboratory

Modeling a cylinder in water flow was done using methods of dimensional analysis. Looking at the velocity profile of air in a wind tunnel allowed us to find its drag force and drag coefficient. We measured the voltage change with an anemometer that was inserted in the control volume to measure air flow around a cylinder. The cylinder was put into the wind tunnel at a certain length that allowed us to calculate velocity profile, drag force per unit length, and drag coefficient. With a Reynolds' number of 8560, our average drag force per unit length was found to be 0.512 Lbs/ft, with a drag coefficient of 0.71. Since dimensional analysis was used to relate the two models of air and water, the Reynold’s number were very close for both situations which also meant a very similar drag coefficient in either case. Hence, the drag force of water per unit length was calculated to be 7.96 Lbs/ft, with an uncertainty of 0.1835 Lbs/ft. When we compare the values of drag coefficient around a cylinder as 1.0, we get 29.0% error.

Introduction

During this lab, air flow dynamics with a cylinder in the control volume were monitored to model similar effects of water flow. With the assumptions of zero cavitation as water flows around the rod and an identical Reynold’s number shared between each model, the velocity profile experimented in the wind tunnel can show the same results as if water were flowing. Dimensional analysis allows us to relate the variables in each model and obtain this common Reynold’s number. The experimental set up conducted can be described as followed. A 0.5” OD rod (cylinder) was installed into a 6”x6” wind tunnel. The wind tunnel had a built-in hot-wire anemometer and pressure transducer that could measure the relevant air flow velocities at different points around the cylinder. Also using a pilot-static probe, very accurate velocity profiles could be taken around the cylinder. Figure 1 is a schematic of the experimental set-up. Calibration of the experimental set up was conducted by performing a linear curve fitting of the calibration data.

Two sets of data with a wind speed of 30 ft/s were collected. One without a cylinder in place and another with a cylinder of 5.5" in length. For all trials, the anemometer position was adjusted using a micrometer by increments of 0.2" between 2.0" and -2.0" with respect to the center of the tunnel to map the velocity profiles.The Reynolds number can be calculated by using Equation 1. where ρ is the fluid density, V is the free stream velocity, D is the diameter of the cylinder, µ is the dynamic viscosity and ν is the kinematic viscosity of the fluid. The momentum deficit obtained from the velocity profile of the free stream can help us calculate the drag force on the cylinder (Equation. 2) inside the wind tunnel. Furthermore, by using Equation 3 we can calculate the coefficient of drag.

We can assume the flow is flowing in one direction, and can only be related in the x and y coordinate system. As seen in Figure 2, refers to the steady stream inlet velocity while refers to the velocity at different positions inside the wake created from the cylinder. Acting against the flow going upstream, the drag force will cause the relation of > . The drag force () was found by gathering the required data of the velocity profile past the cylinder (downstream). The width of the cylinder () is 5.5", with L=2, the bounds of integration are (L,-L). () shows the relation between velocity profiles that were created from the wake of the cylinder. () is equal to 30 ft/s which is what we set the free stream velocity to be. Using excel and the data points given, it is easiest to perform this integration using the trapezoidal method. (Equation 4) Hence, with the velocity points taken at different positions according to the control volume, we can calculate the drag force in such a manner.

Results and discussion

Calibration was done by taking a linear curve fitting of the data presented in the lab manual. The calibration shows a direct linear relationship between pressure and voltage. Figure 3. Calibration Curve of Pressure/VoltageTwo sets of data at a wind speed of 30 ft/s were collected. The first without a cylinder in place. Hence, this is just a free stream flow. The second, with a cylinder of 5.5" in length in place of the control volume. During each trial, the anemometer was adjusted in 0.2” For each trial, the position of the anemometer was adjusted. The velocity profiles are seen in Figures 4 and 5. From Figure 4, the average velocity can be taken to be 1.8 V. Referring to Figure 5 of the lab manual shows us that this is almost exactly 30 ft/s which is our testing velocity. Figure 5 shows a relationship between voltage and height that we would expect do to the drag force. Directly behind the cylinder at Y=0”, the graph takes its lowest dip. Hence, the lowest velocity of the downstream is directly in the center of the wake. Taking the integral which is the area under the curve, from -2.0" to 2.0" with respect to the center, is known as inner product of the integration on Equation 2. The drag force of the model can be calculated by having the density(ρ) and width(w) of the cylinder. 5.5" Cylinder FD (Lb/ft) 0.490 Drag Coeff. 0.710 Reynolds Number 8560.27

Table 1 shows the results from our calculations with the 5.5” cylinder placed in the wind tunnel. In order to compare the experimental drag force of air flow with the real conditions given, the Reynolds numbers should match between the two scenarios. The average Reynolds number for air was 8,560 and of water was 7,345 which give a 14.2% discrepancy. We assumed that this similarity supported the assumption to use the same coefficient of drag to find force of drag per unit length. With the help of Equation 5, we found the drag force per unit length of a cylinder in 40°F water to be 7.96 Lb/ft with a Reynolds number of 7,345, as we can see in Table 2.

Lastly, the uncertainty propagation at 95% probability between all calculated values can be calculated using Equation 6. A typical linearity error of an anemometer of 10%, ∂Fad/∂U1 of 0.012 and ∂Fd/∂U3 of 0.0778, gives us a 0.1835 Lb/ft error. This is somewhat tolerable since our model let air escape through an opening at the anemometer, but it shows consistent with our experimental data.

Conclusion

Different methods to characterize properties of air flow around a cylinder were studied to model water flow. A wind tunnel was used to find values of velocity, given voltage reading from an anemometer. The velocity profile of a cylinder was obtained. A Reynolds' number in air of 6,150 gave us an average drag force per unit length to be 0.512 Lbs/ft, with a drag coefficient of 0.71. By assuming that the Reynolds number are similar, the velocity profile in a wind tunnel was modeled as water flow. We calculated the drag force per unit length of a cylinder in 40°F water to be 7.96±0.1835 Lbs/ft with a Reynolds number of 7,345. Difficulty arose from the fact that parts of this laboratory needed a better inspection of potential leakage of air. However, this analysis led us to acceptable result. During the calibration test, we obtained only six data points to calculate the wind speed. This greatly constrained our accuracy. However, it did not contribute much to our error analysis.