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- 1. NACA 4412 - Force Balance, Pressure-Tapped Wing, and Wake Rake Tests Gregory Day, Mike Kellerman, Braxton Cullors, Brett T. Campbell Lab Section 2 Aerospace Engineering, California Polytechnic State University, San Luis Obispo, California, 93407 May 15, 2015 Abstract This report details a series of experiments involving the NACA 4412 airfoil. The objectives of these experiments were to determine the aerodynamic force and moment coefficients of finite and “infinite” or full-span NACA 4412 wings and sections using combinations of force balance and pressure measurements on the surface of the wings and in the wake, describe the stall characteristics of a 4412 airfoil and their relationship to Reynolds number, and finally, perform sufficient error and uncertainty analysis to better understand the limitations of the tests. The experiments were conducted in Cal Poly’s open-loop low speed wind tunnel in a variety of testing conditions. Through our series of testing, we see that the NACA 4412 is best suited for a cruiser aircraft that flies in the low angle of attack flight regime. Nomenclature Symbol Meaning Units 𝐴 C Ca CD Cn CL Axial Force Chord Length Axial Force Coefficient Drag Coefficient Normal Force Coefficient Lift Coefficient lb in None None None None Cp Pressure Coefficient None D Drag Force lb L Mc/4 N Lift Force Quarter-chord Pitching Moment Normal Force lb lb-ft lb P Static Pressure psi P∞ qi q∞ Freestream Pressure Dynamic Pressure at each panel Freestream Dynamic Pressure psi psi psi Re Reynolds Number None Δy α ρ Distance between Pitot-Tubes Angle of Attack Density in ° kg/m3
- 2. I. Introduction It is important to analyze the various design aspects and behaviors of a wing in order to meet safety requirements of the Federal Aviation Administration (FAR’s) and to accurately construct the desired flight envelope of an aircraft before deciding on an airfoil design. As an aerodynamicist, these behavioral characteristics of the wing are taken into consideration before a final design is chosen because it is important to cater the desired wing characteristics to an airfoil that can provide what is needed for the design and mission requirements. Before experimentally testing these areas, it is important to first have an understanding of the theoretical results, so that discrepancies may be recorded and analyzed. As we experiment on three different wings, we cover a series of design areas, including stall behavior, drag and maximum lift coefficients, and post-stall recovery. Stall of lifting surfaces, or more specifically, wings are caused by flow separation, which largely depends on airfoil stall characteristics, the effect of planform and twist, and taper ratio. All three of the wings that we used in this experiment have a taper ratio of 1.0 (a rectangular planform), which effectively has a larger downwash angle at the tip compared to the root. This leads to a reduced effective angle of attack at the tip, causing this region to stall later than the root. A taper ratio of 1.0 is therefore inefficient, because of the additional induced drag it creates. Although all three wings have this taper ratio, our “infinite” wing would theoretically have slightly less induced drag because the tip effects wouldn’t be as relevant for those experiments. When analyzing the correlation between Aspect Ratio and Lift for wings with low sweep angles, we find that as the wing’s aspect ratio increases, the more 2-dimensionally the wing behaves, with an exception to the wing tip region. This means that CLmax, the maximum lift coefficient, will increase with an increase of aspect ratio. All wings used in our experiment have a sweep angle of zero, meaning that this increase in CLmax is relatively slight, compared to sweep-angles that are greater than 10 degrees. We will not go in too much detail about the effect of forward and aft sweep angles as they are irrelevant for this experiment. Twist generally influences spanwise load distribution and has a positive effect on reducing tip stall. Since our wings have no twist, it is apparent that they do not receive this added benefit. Our wings, however, do receive a benefit from camber, which, when tailored, can be used to delay compressibility effects, which ultimately results in a slower drag coefficient rise. One interesting characteristic that we will find in this experiment is that angle of attack and lift have a path-dependent, non-linear relationship. We will find that, post-stall, the flow does not necessarily reattach at the same angle that it detached at, but rather at a lower angle than when it stalled. In our particular experiment, we will use 3 different wings, consisting of a NACA 4412 airfoil. Our finite wings will be mounted onto a sting during testing and our “infinite” wing will occupy the full width of the test section. Our two finite wings differ by chord length and aspect ratio, while all wings have zero sweep, zero twist, and a rectangular planform. Although we took data over a series of weeks, we matched each experiment to the prior Reynolds numbers to improve repeatability and to compare data accurately. Included in this report is our force balance, aspect ratio, infinite wing, and wake rake sections which will go into detail of how we conducted each experiment, what data we recorded, and the results from our experimentation. Ultimately, we will analyze all differences between our experimental and theoretical results and discuss the possible reasoning for certain variations between the two data sets.
- 3. II. Methodology This series of NACA 4412 experiments was conducted in Cal Poly’s open-loop low speed wind tunnel in a variety of testing conditions. The force balance experiment offered useful information on how the force and moment coefficients vary with Reynolds Number and aspect ratio. The force balance yielded a voltage difference that was converted to the corresponding lift and drag forces. Two finite wings, later described as the red and blue airfoils, were used for the experiment. The full-span pressure-tapped wing lab utilized an “infinite” wing that allowed for lift and drag calculations without interference from any undesired effects at the wingtips. The Scanivalve system took pressure readings that were then converted to lift and drag forces. The wake rake experiment took into account the change in the momentum of the flow that occurred after its interaction with the full-span wing. This information was used to determine the total drag coefficient, which accounts for both pressure and viscous drag. Testing conditions for all three experiments can be found in the tables below. A. Experiment #1: Force Balance The purpose of this experiment was to determine the aerodynamic force and moment coefficients of two finite NACA 4412 wings using a force balance. Additionally, this experiment provided useful information on how the lift, drag, and moment coefficients varied with both aspect ratio and Reynolds Number. To find the relationships between these coefficients and the aspect ratio, the Reynolds Numbers at which the two finite wings were operating were matched. Conversely, each finite wing was individually tested at different Reynolds Numbers in order to observe the effects of the Reynolds Number on the force and moment coefficients. Static pressure and temperature readings for this experiment were as follows: P = 101557.63 Pa T = 292 K Test Number Wind Tunnel Reynolds Number Angle of Attack [°] RPM [m/s] Test 1 640 25 188063.37 -5, 0, 5, 10, 13 (stall angle), 15, 17, 19, 21 Test 2 640 25 speed was decreased until separation occurred 195869.4436 12 Table 1. Force Balance Testing Conditions – Red Airfoil (c = 4.645 in) Test Number Wind Tunnel Reynolds Number Angle of Attack [°] RPM [m/s] Test 1 483 18.784 188063.37 -5, 0, 5, 10, 13, 15, 17 (stall angle), 19, 21 Table 2. Force Balance Testing Conditions – Blue Airfoil (c = 6.182 in)
- 4. Useful equations – Mc/4 = Mmc – N(l) – A(h) The equation for the pitching moment at the quarter-chord is useful in this experiment because it shows the influence that lift and drag have on the rotation of the airfoil. Length l and height h can be seen in Figure 1 below. L = Ncos(α) – Asin(α) The equation for lift is significant in this experiment because the force balance provides the user with a voltage reading that corresponds to the forces acting on the wing. By converting the outputted voltage to normal and axial forces, the lift on the wing can be determined. D = Nsin(α) + Acos(α) The equation for drag is significant in this experiment because the force balance provides the user with a voltage reading that corresponds to the forces acting on the wing. By converting the outputted voltage to normal and axial forces, the drag on the wing can be determined. B. Experiment #2: Full-span Pressure-tapped Wing The purpose of this experiment was to determine the aerodynamic force coefficients of an “infinite” or full-span NACA 4412 wing using pressure measurements on the surface of the wing. The Scanivalve system took pressure readings that could be converted to a pressure coefficient. From the pressure coefficient, the lift and drag forces on the wing were determined. This experiment was significant because a full-span wing allows for force calculations without interference from any undesired effects at the wingtips. Static pressure and temperature readings for this experiment were as follows: P = 101512.28 Pa T = 295 K Figure 1. Schematic of airfoil mounted to force balance strut (1) (2) (3)
- 5. Test Number Wind Tunnel Reynolds Number Angle of Attack [°] RPM [m/s] Test 1 315 12 195869.4436 -5, 0, 5, 10, 13, 15, 17, 19, 21 Test 2 515 20 327616.7768 -5, 0, 5, 10, 13, 15, 17, 19, 21 Table 3. Full-span Pressure-tapped Wing Testing Conditions – “Infinite” Wing (c = 9.799 in) Useful equations – Cl = Cncos(α) – Casin(α) The equation for the lift coefficient is significant in this experiment because it provides useful information on the stall point and general stall behavior of the NACA 4412. This information can be used to predict airfoil performance at various angles of attack. Cd = Cacos(α) + Cnsin(α) The equation for the drag coefficient is significant because it offers insight into the pressure drag production at various angles of attack. Additionally, a comparison between the drag and lift coefficients can be made in order to analyze the lift to drag ratio in a given configuration. C. Experiment #3: Wake rake The purpose of this experiment was to determine the total drag coefficient by taking into account the change in the momentum of the flow after it impacts the wing. Like the previous experiment, the wake rake test utilized the full-span or “infinite” wing. Static pressure and temperature readings for this experiment were as follows: P = 101613.86 Pa T = 288 K Test Number Wind Tunnel Reynolds Number Angle of Attack [°] RPM [m/s] Test 1 515 20 342758.5102 0, 2, 5, 7, 10, 13 Test 2 315 12 195869.4436 0, 2, 5, 7, 10 Test 3 515 20 342758.5102 20 Table 4. Wake Rake Testing Conditions – “Infinite” Wing (c = 9.799 in) (4) (5)
- 6. Useful equations - Cd_Total = ∑ (√ 𝑞𝑖 𝑞∞ − 𝑞𝑖 𝑞∞ ) 𝛥𝑦 𝑐 20 𝑖=1 , where Δy = 𝑟𝑎𝑘𝑒𝑠𝑝𝑎𝑛 (20−1) The equation for the total drag coefficient is significant in that it is calculated differently from the drag coefficient in the previous experiment. In this case, the dynamic pressure at each panel, as well as the rake span, must be taken into account. The summation of these elements yields the total drag coefficient, which is used in a plot to show how the total drag varies with angle of attack. Note the rake span is the distance from the centerline of the top pitot tube to the bottom pitot tube on the rake. D. Inconsistencies The most likely cause of error or inconsistency in the data was a noticeable trend in the tunnel velocity to deviate slightly during testing. It became apparent that when attempting to run the tunnel at a constant speed, say 20m/s, the actual reading displayed a range from 19.5-20.4m/s over the course of the test. Interestingly, the tunnel’s fan was set to a constant RPM, but the flow’s velocity varied fairly significantly over time. The likely cause of this variation in velocity during testing was the nature of Cal Poly’s Low Speed Wind Tunnel. The university’s wind tunnel is an open loop system as opposed to a closed loop re-circulating tunnel that a large aerospace and defense company might utilize. This implies the tunnel can be sensitive to slight changes in atmospheric pressure over time or even pressure changes due to unexpected or unintentional flow recirculation through the tunnel. These changes in pressure most likely led to the fairly significant variations in velocity during testing. Figures 2 and 3 below offer a comparison between an open loop system, such as Cal Poly’s Low Speed Wind Tunnel, and a closed loop system, such as NASA’s Transonic Wind Tunnel. III. Results and Discussion A. Force Balance Figure 4. Coefficient of Lift and CL/CD versus Angle of Attack from the Force Balance Lab Figure 2: Representation of Cal Poly’s open- loop low speed wind tunnel Figure 3: Representation of closed-loop wind tunnel used in industry (6)
- 7. The CL graphs for both the red and blue airfoil show the right trends. The coefficient of lift increases to stall but then there is a sharp drop off and flattening after separation and stall has occurred. We believe the flattening to be due to the nature of the stall effects and not taking enough angle of attack readings past stall. After stall, CL starts to decrease but at higher angles of attack, the CL starts to increases. It is possible that we captured the same CL value during the decrease and the later increase at the higher angle of attack which is why the line appears to be straight after stall. Figure 4 shows the CL over CD, which is a good indicator of the efficiency of a wing. The NACA 4412 seems to be most efficient at an angle of attack of 0 degrees which would be best suited for a cruiser aircraft that rarely travels at angles of attack. Comparing our experimental values of CL with those of JavaFoil: Table 5. Percent Difference Comparing JavaFoil and Experimental Values The reason that the percent difference gets better as the angle of attack increases has to do with the sensitivity of the strain gauges. This is explained in more detail later, but in general, the larger the value, the better the accuracy. B. Pressure Tapped Wing As part of our experimentation, we took surface pressure data multiple times in order to confirm that our initial data was correct. As you can see in figure 5 and figure 6, Figure 5 (Right), Figure 6 (Left). Coefficient of Pressure along the surface of the wing with repeatability data. the two sets of data taken a week apart are almost on top of each other. From this we can confirm that any trends we infer from our data is reliable. We can compare our results to a study done by NASA. In their study, they were trying to validate a CFD turbulence model, however you can see in figure 7 that the trend of the surface pressure seems to match. There is a suction peak near the beginning and the surface pressures converge from there. Angle of Attack Java Foil Experimental Values Difference Percent Difference -5 0.046 0.0945 -0.0485 69.03914591 5 1.104 0.9609 0.1431 13.86023536 13 1.468 1.412 0.056 3.888888889
- 8. Figure 7. Pressure in NASA experiment. We can compare the surface pressure plots (figure 6 vs. figure 8), to see that lift will increase with angle of attack. Figure 8. Surface Pressure at zero angle of attack. This can be seen by looking at the area under the curve. By looking at the area we get the coefficient of the normal force (normal to the chord line). From there we take the cosine of the angle of attack and that will give us the coefficient of lift. In figure 9
- 9. Figure 9. Experimentally obtained lift curve. we see the lift curve of the NACA 4412. Let us compare this to figure 10, Figure 10. Lift curve from “Theory of Wing Sections”. from the book “Theory of Wing Sections”. We can see that our experimental data is lower than the number provided in figure 10.
- 10. Using this method, we can also calculate some drag for the wing section. I say some drag because we can only find the pressure drag of the wing section. The method of taking pressure data does not account for the viscous drag so we can determine what type of drag dominates as the angle of the wing section increases. Another study in this series of experiments were comparing all of these effects between non-stalled and stalled condition. When the wing stalls the lift dramatically decreases. When we look at figure 11, Figure 11. Surface Pressure Pre & Post Stall. we can see the surface pressure along the upper surface increases, creating a lower pressure differential and decreasing the lift. This is manifested in the graph by a smaller area under the pressure curve. Now when we look at the Cl vs alpha graph, we can see that stall occurs between 15 & 17 degrees. This would appear to match up with previous experiments with finite wings and experiments performed by other groups where stall occurred around 17 degrees angle of attack. C. Wake Rake Experiment Figure 12. The Processed Data from both the 12 m/s and 20 m/s Wake Rake Lab
- 11. Figure 12 shows the processed pressure data from the wake rake experiment. Each line represents a different angle of attack run at the respective speed. The overlaying of each angle of attack onto one graph allows for easy comparison of two things: (1) the minimum value of CP that each case reaches, and (2) the total size of the wake for each case. Looking at the minimum value of CP of each case, the minimum value of gets smaller as angle of attack is increases. This is true for the 12 m/s and 20 m/s case. This is due to the increase of vorticity in the wake. As the angle of attack increases, the percent blockage of the flow increases, which disturbs the flow increasing and in turn creates more vorticity. The percent blockage at 2, 5, 7, 10, and 13 degrees is 0.5817, 1.4526, 2.0312, 2.8941, and 3.7492% respectively. The vorticity decreases the x-momentum of the wake. The decrease in x-momentum decreases pressure behind the airfoil creates and the imbalance of pressures from front to back of the airfoil dramatically increases drag. Looking at the size of the wake, at lower angles of attack (0 to 7) the wake seems to be about the same size; in fact, the entire wake was captured in one rake span. At higher angles of attack (10 to 13), it took a couple data collections to accurately record the entire wake because the wake was larger than just one rake span. This relates back to the fact that the higher angle of attack blocks more of the flow, creating more disturbances, and a larger wake as result. One final thing to note is the large jump in minimum value from 7 to 10 degrees at 12 m/s and 10 to 13 degrees at 20 m/s. We believe this large jump is caused by flow separation. By flow separating from the surface of the wing upstream, the recirculating air creates a wake with a lot of swirling, turbulent air which decreases CP of the wake and increases drag of the wing. It is observed that the large jump occurs at different angles for the two different speeds. At 12 m/s, the flow coming down the tunnel is more laminar than at 20 m/s. Laminar flow separates sooner than more turbulent flows. This is why the jump occurs sooner at 12 m/s than at 20 m/s. Figure 13. Total Drag Coefficient versus Angle of Attack from Wake Rake Lab Figure 13 shows the coefficient of drag for both speeds over a range of angle of attacks. At lower angles of attack (0 to 7 degrees), the values of CD are very similar (smallest difference, 0.0002 and the largest difference, 0.0009). In the low range of angles, each flow speed has fully attached flow. At 10 degrees, the 12 m/s test deviates from this trend. The CD increases because we now have separated flow from the wing, which causes an increase in drag, as explained above. The 20 degree test was done to see the post- stall characteristics of the NACA 4412. As we can see, after stall the CD increases dramatically compared to pre-stall and close-to-stall regime due to the large, low-pressure wake downstream of the airfoil. Referring to the Voiltech graph, we can use the plot digitizer to find the drag coefficient. Looking at a Reynolds number of 195,869, which correlates to the 12 m/s test, at angle of attack 10 degrees, the voiltech graph yields a CD of 0.0092. Our wake rake experiment found the CD to be 0.0086. That is a percent error of 6.74%.
- 12. IV. Conclusion The force balance, full-span pressure-tapped wing, and wake rake experiments were performed in order to determine the aerodynamic force and moment coefficients of finite and “infinite” NACA 4412 wings and sections. These tests were performed in a variety of flow conditions, which give useful information on how the force coefficients vary with Reynolds Number and aspect ratio. From the force balance test, it was determined that a higher aspect ratio wing such as the red airfoil stalls sooner than a lower aspect ratio wing like the blue airfoil. Additionally, the full-span pressure-tapped wing and wake rake experiments both demonstrated that higher angles of attack produce greater pressure drag than do lower angles. These results agree with experimental data found in literature and suggest that the tests performed on the NACA 4412 airfoil were successful. V. Appendices A. Raw v Processed Data Table 6. Raw to Fully Processed Data for Blue Wing at 18m/s and 15° Here is the process for getting to the final fully processed data. 1) Take data with just the strut at speed (18 m/s for this case). This is used to find the axial contribution of just the strut and the weight of the strut at speed. 2) Take data with the wind tunnel off and the wing attached, at each angle. (15 degrees for this case) 3) Run the tunnel at desired speed and angle of attack and collect data. 4) Since all we want is the force contribution of the airfoil and the raw data includes the contribution of the weight of the airfoil, the weight of the strut, and the drag of the strut. The weight of the strut must be subtracted from Just the Strut at 18 m/s to get just the drag contribution of the strut. That is added to the Wind Off for Blue at 15 degrees. 6) That sum is then subtracted from the Raw Data and a mean is taken to average all the data points for each column. That mean is the Processed Voltage Data. 7) Take the Processed Voltage Data and run that through the calibration functions to get Voltage Data After Calibration. 8) Finally, plug those Raw Data Normal (V) Axial (V) Pitch (V) Wind Off for Blue at 15 degrees Normal (V) Axial (V) Pitch (V) -1.234674 4.236726 -2.29993 -1.341432 4.26315 -1.84408 -1.234369 4.237651 -2.29941 -1.341103 4.263382 -1.844545 -1.235121 4.237116 -2.3 -1.341466 4.262693 -1.84418 -1.234048 4.237394 -2.29911 -1.341423 4.262911 -1.844176 -1.234597 4.237294 -2.29839 -1.341169 4.263172 -1.844766 -1.235202 4.237557 -2.29772 -1.340376 4.263495 -1.844338 -1.235426 4.237257 -2.29767 -1.341496 4.263425 -1.844279 -1.234528 4.237285 -2.29893 -1.341622 4.263375 -1.844536 -1.234859 4.237309 -2.29857 -1.341825 4.263171 -1.844125 Just Strut at 18 m/s Normal (V) Axial (V) Pitch (V) Processed Voltage Data Normal (V) Axial (V) Pitch (V) -1.335358 4.206567 -1.86182 0.1116174 -0.02592 -0.489309333 -1.334034 4.206464 -1.86269 -1.335972 4.206643 -1.86254 Voltage Data After Calibration Normal (lb) Axial (lb) Pitch (in*lb) -1.3352 4.207038 -1.86261 -4.220891 0.559732 0.559732129 -1.335864 4.206489 -1.86205 -1.334377 4.206693 -1.86205 Fully Processed Data Lift (lb) Drag (lb) M_c4 (in*lb) -1.3346 4.206544 -1.86232 4.2219368 0.551787 1.303559079 -1.335553 4.206429 -1.86305 -1.334615 4.206691 -1.86254 Blue Wing at 18 m/s and Angle of Attack 15 degrees
- 13. values into the coefficient of lift, drag, and quarter-chord moment equations to get the Fully Processed Data. B. Standard Deviation and Error i. Standard Deviation Table 7. Standard Deviation of Raw Data for just 5 Ports Table 7 compares the standard deviation of the raw, imported data of the first 5 ports at both 12 and 20 m/s for all the angles of attack for the wake rake experiment. The data acquisition program records many points for each port for each test case. The standard deviation is an indicator of the unsteadiness of the flow in the wind tunnel, in other words, the smaller the standard deviation, the more steady the flow. Examining the data, a standard deviation less than 2E-4 seems appropriate because the actual data is 4 orders of magnitude greater than the standard deviation. Looking at the standard deviation from the 20 degree, fully detached case, the standard deviation gets as large as 0.00189. This is expected because the wake behind a fully, detached airfoil is extremely unsteady, as explained above. The data is less reliable in this test due to the high standard deviation and the unsteady flow. ii. Error in Wake Rake Figure 14. Wake Pressure Distribution for Fully Detached Flow Figure 14 shows the wake pressure distribution for the fully detached test case. Focus on the first and last values of the distribution. These values should be 1 because they are supposed to be in the freestream flow, which is completely undisturbed. Yet, the values that are in the “freestream” are not exactly 1. Well, the freestream flow is not completely undisturbed. There are disturbances created due Speed 12 m/s 20 m/s 12 m/s 20 m/s 12 m/s 20 m/s 12 m/s 20 m/s 12 m/s 20 m/s 6.00E-05 1.04E-04 6.80E-05 1.12E-04 6.87E-05 1.14E-04 6.05E-05 1.12E-04 6.52E-05 1.14E-04 5.01E-05 5.49E-05 5.45E-05 5.78E-05 5.35E-05 6.06E-05 5.57E-05 6.01E-05 5.62E-05 7.61E-05 3.30E-05 7.53E-05 4.46E-05 8.24E-05 4.01E-05 7.90E-05 4.46E-05 8.10E-05 4.13E-05 8.14E-05 3.80E-05 8.91E-05 4.17E-05 1.14E-04 4.32E-05 1.13E-04 4.19E-05 1.15E-04 4.54E-05 0.000131 5.75E-05 1.06E-04 1.53E-04 1.22E-04 1.66E-04 1.31E-04 1.35E-04 1.35E-04 0.000132 1.59E-04 2.14E-05 3.87E-05 3.91E-05 4.10E-05 4.49E-05 4.17E-05 5.44E-05 4.10E-05 0.000101 4.08E-05 N/A 0.000147 N/A 0.001891 N/A 0.001794 N/A 0.001606 N/A 0.00135 3 4 5 Standard Deviation of Raw Data Port Number 10 13 20 1 2 Angle of Attack 0 2 5 7
- 14. to wing to wall interactions, boundary layer growth on the walls as the air travels down the tunnel, and the large blockage due to the airfoil (5.7003%). In fact, at this high angle of attack, the wall and airfoil interaction starts to create vortices that are comparable to wingtip vortices on a finite wing. This inaccuracy causes an increase in drag. iii. Error in Force Balance Figure 15. Coefficient of Drag versus Angle of Attack from Force Balance Lab No, we did not invent the first perpetual motion machine that uses drag to accelerate an airfoil. The negative drag is a product of the sensitivity and least scale reading of the test equipment. The strain gauges in the wind tunnel are rated from -25 to 25 pounds and the least scale reading could be on the order of ±0.5 lbs. Since the total drag is on the order of 0.5 lbs, we run into the least scale reading problem. C. Momentum Method Total momentum is always conserved. That means, the total amount of momentum of the flow upstream of the wing is equal to the total momentum of the flow downstream of the wing. Consider the situation below.
- 15. Looking at the momentum solely in the x-direction, it is observed that there is a loss in x-velocity downstream of the airfoil. The loss in x-velocity correlates to a loss in x-momentum. Wait, I thought you just said momentum conserved. This is still true. Total momentum is conserved because the flow coming in as U1 is assumed to be purely in the x-direction but, once the flow encounters the airfoil, the flow changes direction due to the disturbance. This change in direction causes an increase in y-momentum. Therefore, what is lost in x-momentum is gained in y-momentum, conserving total momentum of the flow. We can use the x-momentum loss to find the drag of the wing. Assuming no density change since we are below M=0.3, the drag equation reduces to something very simple, equation 7. Since we are interested in the dimensionless number CD and our test data outputs in dynamics pressures, we can modify equation 7 to get equation 8. Finally, equation 8 is tweaked to process our data in the form of equation 9. 𝐶 𝑑 𝑡𝑜𝑡𝑎𝑙 = ∑ 2(√( 𝑞𝑖 𝑞∞ ) 20 𝑖=1 − 𝑞𝑖 𝑞∞ ) ∆𝑦/𝑐 Where CDtotal is the total drag coefficient, qi is the dynamic pressure at each pitot tube, and Δ𝑦 = 𝑟𝑎𝑘𝑒𝑠𝑝𝑎𝑛 20 − 1 where the rake span is the distance from the centerline of the top pitot tube to the centerline on the bottom pitot tube on the rake. D. Panel Method The panel method is used to approximate the pressure distribution across and airfoil because we do not have an infinite amount of pressure ports along the wing to record pressures at every point. Rather, there are 20 ports that cover the top and bottom surface going from leading edge to trailing edge. The panel method approximated the pressures between these ports. To implement this method, we need the pressure at each port, the pressure of the freestream, and the freestream dynamic pressure. Luckily, there are ports in the freestream upstream of the airfoil and ports on the wing. Therefore, we have all the data to implement the equation below: 𝐶 𝑝𝑖 = 𝑝𝑖 − 𝑝∞ 1 2 𝑝∞ 𝑉∞ 2 = Δ𝑃𝑖 𝑞∞ where i, is the port location. Solving for CP at each port and implementing the panel method, we can obtain an approximation for the total pressure distribution across the wing. (7) (8) (9) (10) (11)

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