Experiment 7

Aerodynamics of Flow Around a Cylinder

Objective
Theoretical Background

Apparatus

Experimental Procedure Questions to be Answered Download Data Sheet

Objective

The objective of this experiment is to determine the aerodynamic lift and drag forces, F_L and F_D, respectively, experienced by a circular cylinder placed in a uniform free-stream velocity, U_¥.Two different methods will be used to determine these forces.

Theoretical Background

The total drag on any body consists of skin friction drag and form drag. The skin friction drag is a result of the viscous forces acting on the body while the form drag is due to the unbalanced pressure forces on the body. The sum of the two is called total or profile drag. There are several methods that can be used to determine the drag forces, two of which will be used in this experiment and are discussed in detail below.

Method I - Prediction of Drag from Wake Measurements

By measuring the velocity profiles in the wake and using conservation of linear momentum, the drag force on the cylinder can be determined, provided that the flow is steady. With reference to Figure 1, a control surface around the body is selected as shown. Section II, the region where wake measurements will be obtained, is located a short distance behind the body, as shown.The static pressure at this location is different from the free-stream pressure, p_¥, hence there will be a net contribution to the momentum balance due to this pressure difference. In order to account for and minimize this effect, another section, Section I (an imaginary section), is chosen far behind the body such that the pressure is equal to the free-stream pressure. Therefore, the net pressure forces acting on this new control surface will be zero. The general conservation law can then be written without considering the effects contributed by pressure, as follows:

(1)

where W is the width of the body and u₁ denotes the velocity profile in the wake measured at Section I.

Unfortunately, the flow cannot relax to the free stream pressure, where p₁ = p_¥unless the measuring station is placed very far downstream of the cylinder, usually more than 100 diameters. It is therefore not realistic to obtain measurements at section I.Nevertheless, it is possible to establish the relationship between the flow quantities measured at the hypothetical section I and those measured at actual section II which is located close to the cylinder.The procedure is as follows:

Figure 1. Sketch of the test setup.

In order to find the relation between F_D and the velocity profile at the designated measuring station II, we apply the continuity equation, equation 2, along a stream tube

(2)

Hence, it follows that:

(3)

Furthermore, we make the assumption that the flow moves from Section II to I without pressure losses, i.e., the total pressure remains constant along each streamline between I and II:

(4)

Introducing the total pressure as

(5)

We can rewrite equation 3 by using equations 4 and 5.Hence, we have:

(6)

where the integral extends over cross-section II.

Introducing a dimensionless drag coefficient, C_D, as follows :

(7)

where W_d is the reference area of the body, we can rewrite Equation 6 as follows:

(8)

where

(9)

Method II - Measurement of the Normal Pressure distribution on the body

When the Reynolds number is sufficiently large, Re > 10³, the skin friction drag of a bluff body is relatively negligible compared to its form drag. Then the measurement of the drag forces due to normal pressures acting the body will be a good approximation to the total drag.

For a cylinder, the body lift and drag per unit length due to normal pressure only are given by:

(10)

(11)

where the integrations are taken around the contour of the cylinder. A cylindrical coordinate will be used instead, and we can rewrite equation 11 as:

(12)

where r is the radius of the cylinder, p is the pressure, q is the angular position, and the integration is taken around the cylinder, starting from the stagnation point.

Similarly, the lift force can be estimated as:

(13)

Apparatus

The following components will be used:

1. Wind Tunnel.

2. Pitot-static tube (see description below).

3. A cylindrical test model with circumferential pressure ports (see description below).

4. Scanivalve and scanivalve digital interface unit.

5. ADC Card on a Pentium-based PC.

6. Computer-controlled vertical drive.

Pitot-Static Tube

A pitot tube can be used in the wind tunnel to measure the velocity of the tunnel. The assumption we have to make is that the static pressure is constant everywhere in a uniform free-stream inside the wind tunnel. This is a reasonable assumption considering that there is no pressure loss, therefore, no pressure gradient, in the system. However, the situation will be very different for measurements taken inside a wake behind a bluff body where a significant amount of pressure variation exists across the wake profile. In order to accurately determine the velocity profile in the wake, a pitot-static tube should be used. The pitot-static tube, a sketch of which is shown in Figure 2, is a combination of the static tube and the pitot tube, which works in the following manner. Assume that the tube is properly aligned with the flow direction.If we further assume that the flow is steady, one-dimensional, incompressible and inviscid, all of which are very good assumptions under most conditions, we can derive the following from Bernoulli’s equation:

where V is flow velocity, r is the density of the fluid, p_stag is the stagnation pressure of the free-stream and p_stag is the static pressure.

Figure 2. Pitot-static tube

It is usually more difficult to accurately measure the static pressure. The difference between the true p_stagand measured p_stagmay be due to one or all of the following:

1.Misalignment of the tube axis and the flow velocity vector.

2.Finite tube diameter. Streamlines next to the tube must be different from those in undisturbed flow, hence the mere presence of the tube results in a static a pressure value different from the actual pressure of the undisturbed flow.

3.Influence of the tube-support leading edge.

Cylindrical Test Model

Figure 3 depicts the circumferential locations of the pressure ports around the cylindrical test model. Uniformly distributed holes are located along one half of the cylinder with an angular displacement of 15 degrees. Three reference pressure ports with angular displacements of 45 degrees, are provided along the other half.These ports are used to align the direction of the cylinder so that the free-stream is perpendicular to port 1.

Figure 3. Pressure port locations on the test model

Experimental Procedure

WARNING:Start the wind tunnel at a counter setting of 100 to 200 only. Do not start it at higher counter settings

1.Wake Measurement:

Select two downstream locations, x/D = 5 and x/D=10 for the measuring section II.Set the wind tunnel speed counter at 550.

(a)Insert the pitot, or pitot-static, tube into the wind tunnel upstream of the bluff body. Measureand record the dynamic pressure upstream of the body.

(b) Remove the pitot tube from the upstream.

(d) Using the computer, measure the ADC output. Record this value.

(e) Switch reference to P₂.

(f)Using the computer, move the pitot-static tube vertically upwards by 1 step.Each step moved

by the pitot-static tube using the computer constitutes a movement of 4 mm.

(g) Measure and record the ADC output at this point.

(h) Repeat the measurement procedure for a total of 26 steps, i.e. a total travel distance of 100 mm.

(i) Move the pitot-static tube to the x/D=10 location.Center it at the center of the cylinder.

(j) Repeat the measurement procedure followed for the x/D=5 location.

2. Normal Pressure distribution:

Two different Reynolds numbers shall be used for this part of the experiment.Wind tunnel speed counter settings of 550 and 350 will be used to obtain the two different Reynolds number.

(a) Set the wind tunnel speed counter at 550.

(b)Using the scanivalve, select the port # 2 on the scanivalve digital interface unit (SDIU).This

corresponds to the port # 1 on the Figure 3.

(d) Step through to the next port on the SDIU.

(e) Repeat Step 2c.

(f) Go through this process for all the ports, i.e., up to port 16 on the SDIU.

(g) Set the wind tunnel speed counter at 350.

(h) Repeat Step 2b through to Step 2f

Questions to be Answered

Present all of your data in a dimensionless form, that is, use C_D and C_L, instead of drag and lift forces respectively.Also indicate the Reynolds number of the flow when presenting the values for C_D and C_L.

1.Plot the non-dimensional vertical distance versus the non-dimensional velocity at the locations x/D=5 and x/D=10.Use D and U_¥as the non-dimensionalizing parameters.Discuss the wake profiles obtained at the two locations.

2.Calculate C_Dfrom the wake profiles, compare to values obtained from Fig. 4.Discuss reasons for discrepancies, if any.

3.Plot the variation of C_p as a function of the angular location and compare it to the theoretical profile for the cylinder in an inviscid flow (Refer to any Fluid Mechanics book).Explain the difference.

4.Calculate the values of C_D and C_L from surface pressure measurements? Compare C_D values to those obtained from Figure 4.

5.Compare the results obtained by the two different measuring methods and discuss reasons for any differences. Which method is more accurate ?

6.Does the pitot-static tube indicate a steady pressure differential inside the wake? How about, outside the wake? Can pitot-static tubes accurately measure turbulent fluctuations? Why?