Lab Reports Due on Monday, 11/24/2014


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1 AE 3610 Aerodynamics I Wind Tunnel Laboratory: Lab 4  Pressure distribution on the surface of a rotating circular cylinder Lab Reports Due on Monday, 11/24/2014 Objective In this lab, students will be tasked to measure the pressure distribution around a circular cylinder and to calculate the drag force based on the local pressure measurements. Given constants ρ = [ kg m 3] µ = 1.79E 05 [Pa s] Introduction In this lab, we will be comparing the ideal potential flow theory with the flow in real life conditions. The potential flow theory allow engineers to predict the velocity and pressure distributions about the circumference of a circular cylinder. This theory also allows us to predict the lift and drag forces on the cylinder and can be applied to airfoils and even turbomachinery applications.
2 Ideally, the flow is assumed to be inviscid and incompressible. From this, we are able to assume that the flow far away from the body is uniform in nature. From the above assumptions, the flow can then be described as irrotational. This means that when the air comes into contact with the cylinder, the flow is assumed to flow tangentially, flowing smoothly without friction or separation along the surface of the cylinder. This can be seen in Figure 1 below. U r 2U Figure 1: Ideal potential flow around a circular cylinder Figure 1 depicts the cross section of a circular cylinder with radius a in a potential flow from left to right. The fluid velocity at large distances from the body is U and the corresponding static pressure is p. Note that for this experiment to work, the angle values have to be converted into radians. Also
3 note the symmetry of the flow around the circular cylinder due to our assumptions above. Theory For potential flow, the tangential velocity along the surface of the cylinder is given by U s 2U sin (1) For steady, inviscid, incompressible flow along a streamline, Bernoulli s equation can be used. (Note that for viscous flow, Bernoulli s equation can no longer be applied!) By applying Bernoulli s equation to Equation 1, it can be found that C p p s 1 2 U p 2 (2a) 2 1 4sin (2b) Where ρ is the fluid density and P s () is the surface static pressure at angle. C p is the pressure coefficient. Theoretically, this value should be 1 at both the forward and rearward stagnation points and a minimum value of 3 at = /2. This can be seen in Figure 2 below.
4 U y x p s p0 1 2 U 2 deg Figure 2: Surface pressure coefficient vs. Angle Note that the pressure measurement on the surface of the cylinder does not take into consideration drag caused by skin friction. The total drag that the cylinder experiences is obtained by integrating the static pressure force over the projected surface area A of the cylinder facing the incoming flow. This becomes F p da (3) D A s x Due to the double symmetry discussed in connection with Figure 2, the total drag acting on the cylinder should theoretically be zero. It is useful to identify that there is no lift force generated on a cylinder. However, the hypothesis that there is no drag force generated on the cylinder should be questioned. The inviscid flow assumption stated above allows for complete pressure recovery along the rear side of the cylinder. However, this does not
5 happen in real flows as will be seen from the measurements made in this experiment. With the ideal potential flow assumption, the drag force equals zero independent of the shape of the body in the flow. This outcome, referred to as D Alembert s paradox, is a result of the omission of the influence of viscosity. (Look in your textbook, it is quite interesting) Practically, the approach necessary to calculate the drag on the cylinder is to measure the pressure along the entire cylinder surface and perform the integration of Equation 3 using differential surface elements. This method is illustrated in your textbook by equations 1.8, 1.12, 1.16 and Look at figure 1.21 for a graphical representation of the integral on an airfoil. For this experiment, a circular cylinder will be used and a graphical presentation is shown in figure 3 below. For an angle of attack = 0 degrees, the axial force on a body equals the drag force. Thus, according to equation 1.8, TE TE A = D = ( p u sin θ) ds u + (p l sin θ) LE LE ds l From equation 1.12, dx = ds cos θ
6 ds = dx cos θ Substitute ds with dx in equation 1.8 to calculate the Drag force. y θ LE c R c TE x Figure 3: Graphical representation of experiment for integration Moving on to Coefficient of Drag, equation 1.16 provides the solution. Equation 1.16 states that without frictional force, 0 c C a = C d = 1 c [ (C dy u p,u dx C dy l p,l dx ) dx ] Using the formula: (x R) 2 + y 2 = R 2 where R is the radius of the cylinder, find an equation for dy. Note that x = 0 at the leading edge of the cylinder. Thus, at the trailing edge of the cylinder, x = 2R. From the above equation, you will be able to numerically integrate equation 1.16 using the Trapezoidal rule taught in lab 3. To make sure you are dx
7 on the right track for the use of the Trapezoidal rule, use f(x) = (C p,u dy u dx C p,l dy l dx ). Make sure to substitute the values found above (C p and dy dx ) before doing the integration! Experimental Procedures In this experiment, students will measure the difference between p and p s () at 10 increment along the surface of the cylinder that has been placed in the wind tunnel test section. Manometer 1 (This has been labelled for your convenience) on the test bench will measure the difference in pressure, p s () p. Manometer 2 will be measuring the dynamic pressure of the incoming flow so that you will be able to measure the velocity of the incoming free stream. section. Also, measure the span and the diameter of the cylinder in the test
8 Experimental Results and Discussion The experimental results and discussion of this lab will be split into three sections. The first section will cover what you did in this lab. The second section will take a look at labs 2, 3, and 4. The last section is optional. Section 1 1. Calculate C p as a function of angle (Equation 2a). 2. Plot calculated C p vs. for the specific speed setting. Also calculate the potential theory value of C p (Equation 2b) and plot it on the same graph. Explain the differences between the ideal potential flow theory and the measured C p values. 3. Use the steps in the theory section of this manual, compute the drag force on the cylinder based on the pressure coefficient measurements. 4. Calculate the Cd value. 5. Plot Cd vs. Re. Do your values follow the same trend as that seen in the previous labs? Why or why not? 6. Identify the limitations of the Ideal Potential Flow Theory.
9 Section 2 1. Compare your drag values from labs 2, 3, and 4. Plot them in the same graph to see their relationship. (Use speed setting in xaxis) 2. Compare your coefficient of drag values from labs 2, 3, and 4. Plot them in the same graph to see their relationship. (Use speed setting in xaxis) 3. Account for their differences. (I am especially looking for the evaluation of the different methods of drag calculations between the 3 labs. i.e. what drag force did you measure in lab 2, what drag force did you measure in lab 3, etc.) Do your values make sense? 4. Discuss which method was the most accurate for drag force measurements. Section 3 (Optional) In this section, we ask for your feedback regarding the labs. Your comments in this section will not affect your grades whatsoever. We strive for perfection, but we are all imperfect one way or another. We deeply apologise for any inconveniences we have caused you in any way. The replies in this section will be taken to heart and will help us to improve the way we teach the next batch of AE3610 students. Thank you for your understanding and participation.
10 1. How can we improve the labs in general? 2. Was there sufficient information provided in the labs? If not, how can we improve it? 3. Were the lab manuals clear and concise? If not, how can we improve it? 4. Do you have any additional comments or feedbacks? As always, feel free to contact me at to clear up any doubts about this lab.
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