1. Experiment #5 Momentum Deficit Behind a Cylinder

2. Problem Statement Water, 40 °F 2.0 mph 0.5” rod
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
0.5” rod

3. 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

4. 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

5. Results from Dimensional Analysis (consult a fluids text)
Drag Coefficient, CD,
of a cylinder in cross-flow
FD = drag force on the cylinder
r = air density
D = cylinder diameter
w = cylinder length
U1 = approach velocity of air stream
n = kinematic viscosity of air
Similarity between model and prototype requires that
CDmodel = CDprototype= f(Remodel) = f(Reprototype) or that
Remodel = Reprototype

6. Equality of Reynolds Numbers implies equality of the Drag Coefficients

7. 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 CD 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

8. 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, V1 find the pressure p1.

10. Control Volume Force and Mass Balances (r = constant)
Force balance
Surface 1, rate of momentum in
Surface 2,rate of
momentum out
Surfaces 3 and 4, rate of
momentum out
Mass
balance
Surfaces 3 and 4, rate of
mass out
Surface 1, rate of
mass in
Surface 1, rate
of mass out
Multiplying the second eq. by U1 and substituting in the first eq, results in:

11. Drag force on the Model Cylinder
Measure the velocity profile U1 without the cylinder
Measure the velocity profile u2 with the cylinder
at two location downstream from the cylinder
Compute FD for each of the two profiles using the
experimental data (numerical integration needed)
Take the mean of the two FD values

12. Drag force on the Rod in the Water

13. Error Estimation
Find the error in FD 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 FD
where
and
uu2 and uU1 are the ± errors in the velocity measurement. Use ±1% of measured values
Numerical integration is needed to evaluate the integrals

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