To: Ann M. Anderson; Professor From: Lutao Xie (X.L.); Student Partners: Rebecca Knepple; Ellen Muehleck Date: 03/06/13 Subject: Aerodynamics of Mercedes CLK: Lift and Drag Effects Overview/Summary: This memo reports on the relations of drag and lift coefficients to Reynolds numbers of a Mercedes CLK model in wind tunnel testing. Under different wind frequencies, the velocity range was found from 6.584 + 0.646 m/s to 42.153 + 0.663 m/s. The slope of velocity versus wind frequency was 0.813, which was very similar to a previously calculated slope calibration of 0.869, shown in Figure 2 of Attachment A. The lift coefficient C L ranged from 0.217+0.162 to 0.388+0.019 and the drag coefficient C D ranged from 0.408+0.560 to 0.550+0.031. The corresponding Reynolds number ranged from 1.067 10 5 to 6.830 10 5. The graphs of both lift and drag coefficient versus Reynolds number are shown in Figure 1. At low values of Re, C L and C D were strong functions of the Reynolds number, however when the Reynolds increased to about 4.5 10 5 they turned independent of Re, indicating turbulent flow. Experimental Procedure A 1/8 th full-scale model of the car was pre-mounted in the wind tunnel, shown in Figure 3 (Attachment D). The pre-calibrated pressure transducer, the lift dynamometer and the drag dynamometer were connected respectively to channel 1, 2 and 3. 100 readings at a rate of about 6Hz were recorded by using the data acquisition for each channel. This was done for motor frequencies ranging from 10Hz to 54 Hz in 4 Hz increments. The wind speed was calculated using the results from Channel 1. Lift and drag coefficient, as well as the matching Reynolds number were calculated using results found respectively from Channel 2 and 3. Information about related calculations and equations can be found in Attachment C. Further details about experiment can be seen in Attachment D. Lift and Drag Coefficient Results Plots of both C L and C D versus corresponding Reynolds number are shown in the Figure below: 0.80 0.70 0.60 Lift and Drag Coefficient 0.50 0.40 0.30 0.20 0.10 Lift Drag 0.00 0 1 2 3 4 5 6 7 8 Re x 100000 Figure 1. Plots of the values of C L and C D as functions of matching Reynolds number obtained in this lab as well as their corresponding uncertainties.
Velocity Analysis and Discussion Velocities of the wind were calculated for each motor frequency setting using data collected from the pressure transducer. To find the uncertainty of the velocity, the uncertainty of pressure level and air density were used. A detailed process of these calculations is shown in Attachment C. The velocity ranged from 6.584 + 0.646 m/s to 42.153 + 0.663 m/s. A plot of velocity vs. frequency is shown in Figure 2a (Attachment A), where we found the slope to be 0.813. Comparing to the previous calibration of the wind tunnel, shown in Figure 2b (Attachment A), we found the slops of both the experimental data and that data from the manufacturer to be 0.870; therefore, our results were very close and acceptable. Analysis: Lift and Drag Forces and Coefficients The lift and drag forces were converted from voltage output by using the lift and drag calibration information provided in lab handouts [1]. The lift forces ranged from 0.420+0.102 N to 10.261+0.245 N, and the drag force ranged from 0.155+0.101N to 4.080+0.137N. Therefore, the lift and drag coefficient can be found by applying equation (13) and (14) (see Attachment C), where C L ranged from 0.217+0.162 to 0.388+0.019 and C D ranged from 0.408+0.560 to 0.550+0.031. Note that we used top area to find the lift coefficient and front area for the drag coefficient, both provided in the lab handouts. Uncertainty Analysis of C L and C D The uncertainties of forces were found by combining the SLF and calibration accuracy and the standard deviation of our data. Similar to velocity, the uncertainty percentage of C L and C D were found first by using the uncertainty percentage of pressure, density and velocity. Note that the effect of area was ignored in this calculation for uncertainty analysis. Therefore the uncertainties were found by multiplying the uncertainty percentage by corresponding coefficient values. For the detailed calculation process, see Attachment C and the uncertainties for each value calculated can be seen in Attachment B. Discussion: C L and C D VS. Re Reynolds numbers were found by using equation (17) (Attachment C). Note that the length we used in calculation was the scale length of the car model. Plots of C L and C D versus Re are shown in Figure 1(Attachment A).The plots indicate the relation of lift coefficient (red) and drag coefficient (blue) to their matching Reynolds numbers: both lift and drag coefficients were shown as strong functions of the Reynolds number at low values of Re. As Re number increased, the relations got weaker. The drag coefficient leveled off while the lift coefficient leveled up for Re above a threshold value of Re, approximately 4.5 10 5.The flow became Reynolds number independent above this threshold Re value. It generally occurred when the boundary layer and the wake were both fully turbulent [2]. Closing This experiment determined the relations of lift and drag coefficients to their corresponding Reynolds number, as well as provided a comparison of velocity results with the previous wind tunnel calibration. The lift and drag coefficient indicates strong relations to the Reynolds numbers at low values of Re. The relations weakened when Re value increased and after a threshold value of Re, approximately 4.5 10 5, the flow became Reynolds number independent, which indicates that the boundary layer and the wake were both fully turbulent at this point. The results of velocity were reasonable and matched with our wind tunnel calibration. If you have any further questions or concerns please contact me at xiel2@garnet.union.edu. I affirm that I have carried out my academic endeavors with full academic honesty Signed: Lutao Xie, Ellen Muehleck, Rebecca Knepple (03/06/13)
Acknowledgements Dan Wolfe, Tim Cameron, Rebecca Knepple, Ellen Muehleck References [1] Anderson. Ann. MER 331 Fluid Mechanics Lab. "Race Car Aerodynamics Part 2: Lift and Drag." Union College, 2013. [2 McGraw-Hill Higher Education,Boston, 2010. [3] Anderson. Ann. MER 331 Fluid Mechanics Lab. "Race Car Aerodynamics Project." Union College, 2013. Attachments A) Figures B) Data Analysis Results C) Equations and Detailed Calculations D) Experimental Procedure
Attachment A: Figures C L and C D VS. Re Lift and Drag Coefficient 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 Lift Drag 0.00 0 1 2 3 4 5 6 7 8 x 100000 Re Figure 1. Plots of the values of C L and C D as functions of matching Reynolds number obtained in this lab with corresponding uncertainties. Both lift and drag coefficients were shown as strong functions of the Reynolds number at low Re values. As the Reynolds number value increased, the relations got weaker. The flow became Reynolds number independent above a threshold value of Re, approximately 4.5 10 5. It generally occurred when the boundary layer and the wake were both fully turbulent, where drag coefficient leveled off while the lift coefficient leveled up.
Velocity VS. Frequency Velocity (m/s) 46 41 36 31 26 21 16 11 6 y = 0.8128x - 1.6068 10 20 30 40 50 60 Frequency (Hz) Velocity Linear (Velocity) Figure 2a. Plots of velocity of wind as a function of frequency settings from this experiment. The uncertainty bars are present but might not be big enough to observe due to the scales. 30 Velocity VS. Frequency 25 y = 0.8692x - 0.9016 20 Velocity (m/s) 15 10 y = 0.8739x - 1.7768 Experimental F/V Manufacture F/V 5 0 0 5 10 15 20 25 30 35 Frequency (Hz) Figure 2b. Ex r f r r w r rv fr previous experiments. The uncertainty bars are present but might not be big enough to observe due to the scales. A plot of wind velocity versus each frequency setting in this lab was created, shown in Figure 2a, to compare to the previous calibration curve, shown in Figure 2b. The slop of the graph from this experiment was found as 0.813, which is very close to that of previous calibration curves: similar slopes were found around 0.870 either in an experimental way r fr f r T indicated that our data was acceptable and reliable to be used in other calculations.
Attachment B: Data Analysis Results Table 1. Results of wind velocity, lift and drag coefficient, and uncertainty analysis F(HZ) Avg.P(Pa) δp(pa) V(m/s) δv(m/s) C L u CL δc L C D u CD δc D Re 10 25.837 5.007 6.584 0.646 0.217 74.9% 0.162 0.408 137.3% 0.560 106669 14 56.460 4.995 9.733 0.456 0.298 26.3% 0.078 0.391 66.0% 0.259 157684 18 99.294 5.054 12.907 0.385 0.314 15.2% 0.048 0.384 39.2% 0.150 209112 22 156.999 5.034 16.230 0.362 0.306 12.1% 0.037 0.547 18.1% 0.099 262946 26 227.213 5.072 19.525 0.373 0.307 8.3% 0.026 0.529 13.2% 0.070 316325 30 309.480 5.969 22.787 0.416 0.312 6.8% 0.021 0.645 8.8% 0.057 369176 34 405.840 5.255 26.095 0.439 0.320 6.1% 0.019 0.608 7.8% 0.047 422761 38 514.120 5.729 29.370 0.484 0.333 5.7% 0.019 0.596 7.2% 0.043 475827 42 634.773 5.214 32.635 0.524 0.350 5.2% 0.018 0.575 6.6% 0.038 528721 46 765.910 5.496 35.848 0.571 0.382 5.3% 0.020 0.564 6.2% 0.035 580772 50 905.219 6.942 38.972 0.623 0.394 5.1% 0.020 0.556 5.8% 0.032 631384 54 1059.036 5.554 42.153 0.663 0.388 5.0% 0.019 0.550 5.6% 0.031 682924
Attachment C: Equations and Detailed Calculations Density and Density Uncertainty Analysis To start the data analysis, we found density of the air under room temperature by using the following equation: Where P is the atmospheric pressure (1.01KPa), R is the Gas Constant (287 J/Kmol) and T is Temperature (296K). The density was found to be 1.192 kg/m 3. The uncertainty of the density was found by using the equation: Where P is atmospheric pressure (1.01KPa), R is the gas constant, is the uncertainty in temperature (10K) and is the uncertainty in atmospheric pressure. We found to be 0.037 kg/m 3. Pressure and Pressure Uncertainty Analysis Calibration for transducer provided last week was used to convert voltage outputs pressure: W r E v r f v f r fr q y ΔP v r v r r r difference [3]. The average standard deviation of pressure can be converted by using the equation: Where represents the average value of standard deviation of pressure difference and is the standard deviation of voltage outputs for each frequency setting. To find the pressure levels uncertainty, we had to include the standard deviation of our measurements as well. Therefore:
W r δ Pcalibration was a given (0.02 inches of water) and the standard deviation of the pressure ( ) was found to range from 0.002 to 0.01Pa by using equation (4). Velocity and Velocity Uncertainty Analysis The velocity of the air in the wind tunnel at different frequency setting can be found by using Equation 3 which is r v fr B r Eq Where Po is the pressure found at the pitot probe, P is the atmospheric pressure ρ y f with equation (1) W ΔP f q (3) f calculating. Then the uncertainty in velocity was found by multiplying velocity with corresponding uncertainty percentage shown as Equation 7: ( ) ( ) Where v is the velocity found per frequency with Equation 6, is the uncertainty of pressure found in Equation 5, is the pressure difference of the pitot probe (in inches of water) found in Equation 3, was found using Equation 2, and was found using Equation 1. Lift and Drag Forces and Their Uncertainty Analysis Calibration information provided in our handouts was used to convert voltage outputs pressure: Lift force: Drag force: Where E is the average of voltage outputs for each frequency and F D is the converted drag force and F L is the converted lift force [3]. The average standard deviation of lift and drag forces can be converted by using the equation: Lift: Drag:
W r δ Pcalibration was a given (0.02 inches of water) and the standard deviation of the pressure ( ) was found to range from 0.002 to 0.01Pa by using equation (4). The uncertainty of lift and drag forces were found by combining SLF and calibration accuracy and the variations in our measurements: Where is the uncertainty of lift or drag forces we want to find; is the combination of SLF and calibration accuracy provided in our hangouts, which was 0.1N for both lift and drag forces; can be obtained by using equation (10) and (11). Lift and Drag Coefficients and Their Uncertainty Analysis The lift and drag coefficient can be found by using equation 8 and 9: Lift coefficient: Drag coefficient: Where F L and F D represent lift force and drag force found by using equations (8) and (9); ρ r density obtained by using equation (1); V is the wind velocity found by using equation (6) and A is the area. Note that to calculate lift coefficient, we used top area (0.025m 2 ) of the car and for drag coefficient, we used front area (0,007m 2 ) of the car. To find the uncertainties of both lift and drag coefficients, equation (15) was used: Lift: ( ) ( ) (15) Drag: ( ) ( ) (16) Where F L and F D are lift and drag forces found by using equations (8) and (9); L and D are the lift and drag force uncertainties found by using equation (12); the air density ρ and r y r y δρ were found by using equation (1) and (2); the wind velocity and its uncer y δv f y equation (6) and (7). Note that area was ignored in this calculation.
Reynolds Number The Reynolds number was found by using the following equation: W r ρ w r ensity found by equation (1); v w w v y f y q (6) μ was the dynamic viscosity (1.8394 10 5 Ns/m 2 ) and L was the length of car model, 0.25m, provided in our handouts. Detailed Calculation Results Table 2 below shows values and results of each parameters used in calculations. Note that all the units were SI Units. Table 2. Values and Results of Each Parameters Used in Calculations Table 2.a. Constant properties Temperature T (K) Gas Constant R(J/Kmol) Density ρ (Kg/m 3 ) Density uncertainty δρ (Kg/m 3 ) Top Area (m 2 ) Front Area (m 2 ) Car Length L (m) Dynamic Viscosity μ(ns/m 2 ) 296 287 1.192 0.037 0.025 0.007 0.25 1.8394 10 5 Table 2.b. Velocities and Pressure F(Hz) Avg. STDEV. STDEV. Avg. Avg.P Velocity δp (Pa) δv (m/s) Volt. (V) Volt.(V) P ( H 2 O) P( H 2 O) (Pa) (m/s) 10 0.020 0.001 0.002 0.104 25.837 6.584 5.098 0.658 14 0.083 0.001 0.002 0.227 56.460 9.733 5.052 0.461 18 0.172 0.002 0.004 0.399 99.294 12.907 5.280 0.397 22 0.291 0.002 0.003 0.631 156.999 16.230 5.206 0.369 26 0.436 0.002 0.004 0.913 227.213 19.525 5.349 0.380 30 0.606 0.007 0.013 1.244 309.480 22.787 8.260 0.466 34 0.805 0.003 0.007 1.631 405.840 26.095 6.013 0.449 38 1.029 0.006 0.011 2.066 514.120 29.370 7.549 0.504 42 1.278 0.003 0.006 2.551 634.773 32.635 5.872 0.529 46 1.549 0.005 0.009 3.078 765.910 35.848 6.823 0.579 50 1.836 0.010 0.019 3.638 905.219 38.972 10.885 0.649 54 2.154 0.005 0.010 4.257 1059.036 42.153 7.008 0.669
Table 2.c. Lift Forces and Coefficient F(HZ) STDEV. Avg. F L δf L (N) C L u cl δc L Re F L (N) (N) 10 0.007 0.140 0.101 0.217 0.749 0.162 106669 14 0.011 0.420 0.102 0.298 0.263 0.078 157684 18 0.018 0.780 0.106 0.314 0.152 0.048 209112 22 0.041 1.200 0.129 0.306 0.121 0.037 262946 26 0.031 1.745 0.117 0.307 0.083 0.026 316325 30 0.029 2.417 0.116 0.312 0.068 0.021 369176 34 0.041 3.251 0.130 0.320 0.061 0.019 422761 38 0.056 4.283 0.150 0.333 0.057 0.019 475827 42 0.057 5.550 0.152 0.350 0.052 0.018 528721 46 0.096 7.318 0.217 0.382 0.053 0.020 580772 50 0.094 8.925 0.213 0.394 0.051 0.020 631384 54 0.112 10.261 0.245 0.388 0.050 0.019 682924 Table 2.d. Drag Forces and Coefficient F(HZ) STDEV. Avg. F D δf D (N) C D u DL δc D Re F D (N) (N) 10 0.003 0.074 0.100 0.408 1.373 0.560 106669 14 0.007 0.155 0.101 0.391 0.660 0.259 157684 18 0.012 0.267 0.103 0.384 0.392 0.150 209112 22 0.014 0.601 0.104 0.547 0.181 0.099 262946 26 0.013 0.842 0.103 0.529 0.132 0.070 316325 30 0.014 1.397 0.104 0.645 0.088 0.057 369176 34 0.021 1.726 0.108 0.608 0.078 0.047 422761 38 0.034 2.146 0.121 0.596 0.072 0.043 475827 42 0.036 2.554 0.123 0.575 0.066 0.038 528721 46 0.041 3.026 0.130 0.564 0.062 0.035 580772 50 0.043 3.520 0.132 0.556 0.058 0.032 631384 54 0.047 4.080 0.137 0.550 0.056 0.031 682924
Attachment D: Experimental Procedure Lab set up is shown in Figure 3 below. The Mercedes CLK car model was pre-mounted in the wind tunnel. The pressure transducer was the same as we used previously for calibration and it was precalibrated. Calibration information for the dynamometer was provided in the handouts [1]. Figure 3. The basic set up for this lab: The pre-mounted Mercedes CLK model in wind tunnel. Three channels were used to make measurements in this lab. transducer was connected to Channel 1, the dynamometer lift was connected to channel 2 and the dynamometer drag was connected to Channel 3 on the Daq connector box. This was connected to the computer. The stagnation and static pressure was measured through the wind tunnel pitot probe. 100 readings from each channel were recorded by running the Daq software at a rate of about 6Hz. Note that data acquisition program was started first by reading three channels when there was no flow in order to zero the lift and drag system. The wind tunnel frequency was set to 10Hz first and then up to 54 Hz in 4Hz increments to take the measurements. Note that every time after changing the wind tunnel frequency, we waited about 2 minutes for the system to get stable. Directions of three components of aerodynamic forces acting on a race car are shown in Figure 4 to help with understanding how aerodynamic forces work [1].
Figure 4. Directions of three components of aerodynamic forces acting on a car [3]. Since lift force is acting on y direction, when we were calculating the lift coefficient we used top area of the car. Drag force is acting on x direction, so we used front area of the car to calculate drag force. Since the direction of the wind aligned with x axis and the car was mounted in the center, side forces were canceled.