analysis of the rae 2822 airfoil using computational fluid...

Analysis of the RAE 2822 Airfoil using Computational

Fluid Dynamics

Saeed Al Jaberi, Saray Checo, Logan Krawchyk

Morton Lin, Christopher McElroy, Jesse Shaffer, Richard Zang

AERSP 312, Team 1

In aerodynamics, the process of accurately analyzing the Mach number, density, andpressure is necessary for modern aircraft to work efficiently. The purpose of this projectwas to gain a fundamental understanding of the relationship between an airfoil and its fluidsurroundings, by the utilization of a 2-Dimensional Reynolds-Averaged Navier Stokes codethat makes use of a Baldwind-Lomax turbulence model for calculation, and FieldView, aflow visualization program developed by Intelligent Light, in order to give a visual repre-sentation of the calculated data. Numerous noteworthy occurrences were examined usingMach numbers varying between 0.25 and 2, as well as angles of attack ranging from -2 to 10degrees, in 2 degree increments. These phenomena included fluid property variation acrossshockwaves, both attached and detached, flow separation, angles of stall, and various otherhappenings. Through the analysis of data obtained during the experiment, relations suchas a correlation between angle of attack and lift, Mach number and the formation of shock-waves, and flow separation due to airfoil orientation have been revealed. Sources of errorand differences between real world and simulated results, both expected and unexpected,as well as suggestions for improving the accuracy of the experiment are also discussed.

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American Institute of Aeronautics and Astronautics

Contents

I Introduction 4A RAE 2822 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

II Procedure 5

IIIAnalysis and Discussion 8A Lift, Drag, and Pitching Moment Coefficient Analysis . . . . . . . . . . . . . . . . . . . . . . 8

1 Lift Coefficient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Pitching Moment Coefficient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Drag Coefficient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

B Pressure Coefficient Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Mach Number M=.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Mach Number M=.50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Mach Number M=.85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

C Density Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Mach Number M=.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Mach Number M=.50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Mach Number M=.85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

D Local Mach Number Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Mach Number M=.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Mach Number M=.50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Mach Number M=.85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Bow Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

E Flow Separation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 M=.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 M=.50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 M=.85 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

F Compressibility Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Critical Mach Number, Mcr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Drag Divergent Mach Number, Mdd . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Cl/Cd vs. M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Cd vs. M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Cl vs. M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

IV Conclusion 35

V Appendix 36A Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1 M=.25, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 M=.50, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 M=.85, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

B Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 M=.25, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 M=.50, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 M=.85, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

C Local Mach Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 M=.25, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 M=.50, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 M=.85, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

D Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 M=.25, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 M=.50, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 M=.85, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

E Velocity Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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1 M=.25, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 M=.50, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 M=.85, α = -2, 0, 2, 4, 6, 8, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

VI Team Member Work Distribution 71

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I. Introduction

Fluid dynamics is a concept that governs not only the Aerospace industry, but also affects the lives ofeveryday people in forms such as commercial transport via airplane, recreational flight, and military strategy.Because of its vast array of uses, understanding the behavior of fluid flow and its effects on objects in flight,and having an especially thorough understanding of fluid/airfoil interaction is essential in contributing tothe modern Aerospace workplace. The objective of this project was to grant that critical understanding ofthe aforementioned airfoil/fluid relation.

Through the use of a 2-Dimensional Reynolds-Averaged Navier Stokes code based upon a code developedby A. Jameson and later revised by Turkel and Swanson using a Baldwind-Lomax turbulence model forcalculation, and utilization of FieldView, a flow visualization program manufactured by Intelligent Light,results for many different cases involving an RAE 2822 airfoil were modeled and analyzed. Contours and meshvisualizations were created in order to look for any noteworthy occurrences in the properties of coefficient ofpressure, Mach number, and density. Small experiments into things such as stall angle, shocks, and programlimits were also tested.

After modeling was completed, multiple comparisons, such as lift coefficient and pitching moment co-efficient versus angle of attack, coefficient of pressure along the surface length, and drag coefficient versusthe lift coefficient. Interesting results, ranging anywhere from shockwaves to pressure losses, stall angles,changes in fluid densities, changes in Mach numbers across the flow, and even a possible breakdown in thecomputer coding are examined in further detail, and causes and consequences are promptly discussed. Erroris present in any experiment, even one based on computer coding, and so probable causes of error in resultsare also addressed.

A. RAE 2822 Airfoil

The RAE 2822 airfoil, also known as the RAE2822 Transonic Airfoil, is an airfoil which has become a widelyused airfoil for turbulence modeling. The airfoil, shown below in Figure 1, is made up with a max camberof 2%, camber position of 80%, and maximum thickness/chord ratio of 22%. This particular airfoil is oftenused in computational fluid dynamics in order to model shockwaves and other phenomena in 2-Dimensionalflow.

Figure 1: RAE 2822 Airfoil Schematic from NASA Database

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II. Procedure

In this project, a computational fluid dynamics code and FieldView were used to obtain the data neededto create the necessary plots for the given Mach numbers and angles of attack. The necessary files are shownin the Figure 2, below.

Figure 2: Project Files

To obtain the total lift, drag, and mach coefficients, the input file, rae.inp, was edited to include the givenmach numbers and angles of attack. The mach number and angle of attack, marked alpha, was changedafter each computation.

Figure 3: Edited Input for Mach = 0.25 and Alpha = 0

The CFD code, labeled flomg, was run using the edited input, rae.inp, and the resulting output valuesare stored in the file raem0.25a0.out. The resulting coefficients were located at the bottom of the outputfile, shown in Figure 4 below.

Figure 4: Total Lift, Drag, and Mach Coefficient Output

The CFD code was repeated for all seven angle of attacks and the three Mach numbers until 21 sets ofcoefficient values were obtained. These values were then used to plot the required Cl vs. Cm, Cd vs. Cl,and Cm vs. alpha plots. After each successful run of the CFD code, the program produced output filesfort.10. These files were converted into a grid file (xyz) and a solution file (q) using the program, xyzq.exe.

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The xyzq.exe program converts the output files from the CFD program into FieldView readable files in theformat should in Figure 5.

Figure 5: xyzq.exe Files for Angle of Attack = -2

In order to generate the Cp, Pressure, and Mach number distributions over the given airfoil at eachnumber, the grid and solution files were input into FieldView.

Figure 6: FieldView Contour Interface

For this report, contours were used to display the distribution of Density, Cp, and Mach number over theairfoil at each angle of attack. This process was repeated for each mach number and angle of attack untilall 63 distribution plots were created.

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Figure 7: Procedure Flow Chart

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III. Analysis and Discussion

A. Lift, Drag, and Pitching Moment Coefficient Analysis

1. Lift Coefficient Analysis

For aerospace engineering and airfoil applications, it is imperative to discuss lift, drag, and pitching moment,as these values are crucial to the aerodynamic qualitiies of an airfoil, from performance to stability. The liftcurve, or Cl vs. alpha, represents the coefficient of lift at increasing angles of attack. This graph can allowus to extract additional data about the airfoil, such as airfoils’ stall angles, maximum Cl, and angle of zerolift. These values are meaningful to aerodynamicists, because they represent the limitations of the aircraft.

Another important parameter for flight is the angle of zero-lift. The value of the zero lift angle is afunction of the airfoil camber (not velocity), and so the zero lift angle for the same airfoil will be the samefor any free stream velocity. For the RAE 2822 airfoil, this angle of zero-lift can be seen as the point atwhich the Cl vs. alpha graph (as discussed in future sections) crosses the horizontal axis, as this is where Cl

=0. The stall of an aircraft wing occurs when the lift increases disproportionately with angle of attack. Theangle at which the wing stalls occurs when the Cl reaches a maximum value (Clmax) and begins to decrease.Similar to the zero lift angle, the stall angle does not depend on velocity, but rather on angle of attack.

For a 2-Dimensional airfoil, the slope of the lift curve should equal exactly 2π per radian (or, 0.106 perdegree), in the theoretical case. If the airfoil has a positive camber, the lift curve would shift up; and if theairfoil has a negative camber, the lift curve would shift down. Because of this, all airfoils with a postivecamber have a negative value for the zero lift angle. Additionally, the critical angle of attack decreaseswith increasing camber, so that the maximum lift coefficient increases less than linearly. In this section, theprocessed data of the RAE 2822 airfoil will be evaluated at Mach numbers of 0.25, 0.50, and 0.85. For thesubsonic flows, the slope is very close to the theoretical value and exhibited similar values for the coefficientof lift at each angle of attack.

Mach Number M=.25 At a Mach Number of M=.25, the RAE 2822 airfoil exhibits signs of stall atapproximately 6 degrees. A trend line can also be applied to the linear portion of the lift curve (Figure 8),which can be used to find the zero lift angle of attack. Signs of stall can be seen at an angle of attack atapproximately 6 degrees; however, as angle of attack increases, the graph appears to indicate that the airfoilwill pull out of stall, as Cl continues to increase, rather than drop off. This behavior is not expected of aconventional airfoil, but can possibly be explained by limitations of the code and its ability to accuratelymodel a perfectly realistic airfoil, that may conform to theoretical assumptions.

Figure 8: Lift Curve, M=.25

The orange points refer to the linear portion of the lift curve, which a trend line was applied to. An easyway to determine the zero lift angle of attack is to find the x-intercept, which is at -1.89 degrees.

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This analysis was applied to additional free stream mach numbers, and the data is gathered in Table 2,organized by increasing mach number.

Table 1: Lift Coefficient Analysis

Mach Number Zero Lift α (degrees) Stall Cl Stall α (degrees)

.25 -1.89 .92 6

.50 -1.91 .97 6

.85 -1.14 .81 6

2. Pitching Moment Coefficient Analysis

The pitching moment coefficient is integral to the calculations for the stability and control of an aircraft.For an aircraft to achieve longitudinal stability, it will often experience a small negative normal force actingon the wing, and an equally small positive force on the tail of the aircraft. These two forces form a couplethat will generate a nose down pitching moment, which leads to a negative pitching moment coefficient.

To further analyze the pitching moment coefficient for the RAE 2822 airfoil, the pitching moment co-efficient will be graphed against values of angle of attack. It is expected that the slope of this graph willbe negative, so that increases in angle of attack will directly cause an increase in the nose down pitchingmoment. By the same theory, a decrease in the angle of attack will decrease the nose down pitching moment,ultimately causing the aircraft to be more stable.

Figure 9: Pitching Moment Coefficient Analysis

For the majority of the mach numbers (M=.25, .5, .85), it is observed that for most angles of attack,stability is achieved. These curves are not perfect or ideal, as there are certain angles of attack that causeportions of the curve to have a positive slope (and therefore, instability in the airfoil). However, it isimportant to note that these instabilities seem to occur at angles of attack greater than 6 degrees, which iswhere it was previously observed that this is where stall occurs. Therefore, a possible explanation for theseinstabilities is the lost of lift due to stall. Ultimately, all three of these mach numbers have pitching momentcoefficients that are all negative, and have overall generally negative slopes.

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3. Drag Coefficient Analysis

Graphing the lift coefficient versus the drag coefficient allows for simple analysis of the minimum dragcharacteristics of an airfoil, in addition to its stability. The shape of this curve results in the drag bucket.

Figure 10: Drag Coefficient Analysis

Greater mach numbers reflect an overall increase in the drag coefficient, which is likely due to the higherlevel of turbulence at a higher free stream velocity. For low mach numbers (M=.25, .5), the values of thedrag coefficient do not vary much at all, and almost resemble a vertical line at Cp=.008. However, as angleof attack increases past 6 degrees, the drag coefficient increases sharply. This is further represented in thecomparison of lift to drag ratio below for M=.25, as it is evident that an angle of attack at 6 degrees returnsthe greatest lift to drag ratio.

Table 2: L/D Comparison, M=.25

α L/D

-2 -1.85

0 27.41

2 55.34

4 79.84

6 98.12

8 64.54

10 33.34

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B. Pressure Coefficient Analysis

One of the most important aspects of an airfoil to examine during testing and modeling is the pressurecoefficient, Cp. This coefficient is representative of actual pressure around the airfoil, and is important on abasic level, that is: an airplane could not fly without a pressure differential. The difference in pressure aboveand below the wing is key in the generation of lift, and therefore is critical for flight. Through observation ofthe RAE 2822 airfoil in a variety of conditions where both angle of attack and Mach number were changedas the experiment progressed, multiple trends were apparent in the data.

1. Mach Number M=.25

Beginning with a Mach number of 0.25, the pressure coefficient data was recorded for the airfoil undervarying angles of attack, ranging from -2 to 10 degrees. The lower angles of attack, -2, 0, and 2 degrees allexhibited behavior that was very similar in the sense that pressure on the top and bottom of the airfoil wasnearly equal, visible below in Figure 11, where the airfoil was examined at zero angle of attack.

Figure 11: Airfoil Cp at α = 0

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As the airfoil was tilted to the mid-level angles of 4 and 6 degrees, the pressure difference on the upperand lower halves became more apparent. Shown below in Figure 12 is the RAE 2822 at an angle of 4 degrees.


In Figure 2, the higher pressure, shown in yellow, circles underneath, leaving a lower pressure area,denoted by the green glow, to reside above the upper surface. Another noteworthy detail is that the airfoilexperiences a high pressure coefficient at the stagnation point at the front, shown by the small red bubbleto the left. Notice also that the rest of the flowfield surrounding the airfoil seems nearly uniform, as thepresence of the airfoil at such a Mach number is still relatively unnoticed by the surroundings.

As the angle of attack is increased, the pressure difference between the top and bottom surface alsoincreases. Figure 13 below illustrates this difference in a manner that is easily seen, as the airfoil at an8 degree angle of attack shows very high pressure (red) underneath, and a substantially lower pressure,denoted by the yellow, on the top of the airfoil. The Cp of the flow surrounding the airfoil is also beginningto increase, showing that the airfoil at this higher angle of attack is beginning to have a greater effect on thefluid enveloping it.


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While the above figures provide an excellent visual on the difference of pressures around the airfoil, andmay give a conceptual understanding of the relation between things such as lift and pressure, there is perhapsa better way to show the lift/pressure relationship. Pictured below, Figure 14 shows the collected Cp dataplotted against the length of the airfoil.

Figure 14: Cp vs X. Angle of Attack of 0, 4, 8

Figure 14 shows the plots of Cp vs the distance along the chord of the airfoil. The most importantobservation to be made from the data is that the airfoil placed at lower angles of attack reaches a lowermaximum pressure coefficient along the length of the chord than the cases with the higher angle of attack.This is extremely important, as the higher Cp often results in a greater bounded area in between the upperand lower curves of the graph. This area is critical, as it represents the sectional lift coefficient of the airfoil,Cl. The sectional lift coefficient is linearly representative of the lift generated by the wing. Above data makeclear the fact that up to a certain point, higher angles of attack will generate more lift. The above casesshow only the Cp vs X plots for the angles of 0, 4, and 8 degrees. Plots and models for the other angles areincluded in the appendix.


After the analysis of the airfoil at Mach 0.25 was complete, the next step was to move to a higher Machnumber, Mach 0.50.

Many of the relations discussed for the Mach number of 0.25 are still present, such as higher angles ofattack corresponding with higher pressure coefficients, as well higher angles affecting the surrounding fluidmore than the lower angles. One interesting difference between the Mach numbers of 0.25 and 0.50 is thatthe area affected by the airfoil has increased. Figure 15 shows a side by side comparison of the airfoils at 2degree angle of attack, but with Mach numbers of 0.25 and 0.50, respectively.

Figure 15: Mach 0.25 to 0.5 Flow Field Comparison

Figure 15 shows clearly that for the same angle of attack, the airfoil on the right, in a Mach 0.50 condition,generates much more interference with the surrounding flow field. This is due to the higher speed of incomingair experiencing a more violent parting along the surface of the airfoil.

While the area of the flow field interference has changed noticeably, it was not the only thing to change.Just like Figure 14, Figure 16 shows the Cp vs X plots for the RAE 2822 airfoil, this time at a Mach numberof 0.50.

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Figure 16: Cp vs X Comparison for RAE 2822 Mach 0.50

Very similar to Figure 14, Figure 16 shows the same trends as the airfoil at Mach 0.25. X = 0, or the frontof the airfoil once again shows the biggest pressure spike, which was to be expected, as stagnation pressureat the front of the airfoil is often much higher than pressure anywhere else on the airfoil. The sectional liftcoefficient remains largest in the higher angle of attack, which is also to be expected.


The last Mach number to be examined, Mach 0.85 produced some very interesting results in terms of pressurecoefficient phenomena.

First on the list is the most interesting of all observed occurrences during the experiment, the appearanceof shockwaves. A shockwave is a phenomenon that occurs when fluid must act in a quick manner in order tosatisfy certain downstream conditions. Shockwaves in air begin to occur as the speed of sound is approached,so Mach 0.85 yielded the first instance of shocks viewed in the experiment. In the case of pressure coefficients,and of total pressure, the buildup in pressure over the surface of the airfoil due to high speeds must be abruptlyreleased in order to unify once again behind the airfoil. Figure 17 shows the formation of a shockwave at anangle of attack of 10 degrees, at Mach 0.85.

Figure 17: Shock Wave Formation at 10 Degree Angle of Attack

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The shockwave in Figure 17 is easily seen as the blue wave emanating from the top of the airfoil. Thetotal pressure drops across the shock, shown by the propagation of the dark blue wave along the upperairfoil, and begins smoothing out the pressure difference between the upper and lower surfaces. Notice thatthis shock is still an attached shock, that is, it forms very near to the surface of the airfoil. At higher Machnumbers, and certain angles of attack, these attached shocks could become a bow shock.

Figure 18: Shock Wave Formation at 0 Degree Angle of Attack

The shockwave formed at an angle of attack of 10 degrees is much more violent, and much steeper, thanshocks formed at lower angles of attack. As shown in Figure 18 below, which shows the progression of ashockwave around the RAE 2822 airfoil, the shock at a lower angle of attack (in this case, 0 degrees) occurslater on the airfoil, and appears much more smooth than the shock in Figure 17. This is because the pressuredifference from the buildup is much less than the difference that occurs at higher angles of attack.

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The other area of interest at a Mach number of 0.85 is the generation of lift, which can be seen byexamination of the Cp vs X plots. Analysis of Figure 9 shows that the pressure differences on the upper andlower surfaces of the airfoil at the lower angle of attack were nearly zero near the front of the airfoil, butwith the formation of shockwaves, the pressure on the upper surface takes over and some lift is generated.At the higher angle of attack, 10 degrees, the difference in pressure coefficient is immediately apparent, asthe curves on the plot are split starting at the leading edge of the airfoil. This is due mainly to the formationof the shock that occurs solely on the top surface of the airfoil.

Figure 19: Cp vs X of RAE 2822 at Mach 0.85, Angle of Attack of 0, 10

One other intriguing detail shown by both Figures 17 and 19 is that for the higher angle of attack of 10degrees, the pressure instability, the difference in pressure coefficients continues on even after the flow hasmoved beyond the airfoil. This is most likely due to flow separation, as at high angles of attack, and withincreasing Mach number, turbulence and separated flow are more likely to occur. In both cases: that ofturbulence and separation; mixing and rotation occur in the flow surrounding the airfoil, causing instabilitiesand fluctuations in many aspects of the flow, pressure included.

Plots and models for other angles of attack and Mach number can be found in the Appendix, in the casethat further intermediate analysis is needed.

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C. Density Analysis

For an ideal gas such as air, density is proportional to pressure, now when analyzing changes in an airfoil it isonly taken into account pressure changes not density changes. This is because aerodynamicists are directlyinterested in differences in pressure and directly interested in the total density, not differences in density.

The lift depends on a pressure difference between the top and bottom of the wing. Similarly pressuredrag depends on pressure differences. Therefore the relevant differential pressures are zero plus importantterms proportional to density. Meanwhile, the relevant pressures are proportional to the total density, whichis some big number plus or minus unimportant terms proportional to density.

Consequently, flight depends directly on total density but not directly on total atmospheric pressure, justdifferences in pressure. In other words, the density-changes are small because the pressure-changes are smallcompared to the total atmospheric pressure.

After combining the conservation of momentum equation and the isentropic flow equation, the followingrelation is achieved (Mach Number: Role in Compressible Flow, Glenn Research Center-NASA),

−M2 dv

v=dρ

ρ(1)

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It is observed that behavior of density change locally or around the airfoil is considerably very small. This isagreeable with the above equation as M=0.25, which states that the change of density is equal to the changeof velocity multiplied by 0.0625. In other words, the density of air around an airfoil at low subsonic speedsremains unchanged and flow is said to incompressible.

Figure 20: Density contour lines for M=0.25 and angle of attack -2,4,6, and 10 degrees

From the figures above, the contour lines for each case on the airfoil almost keep the same color, whichimplies that density value is constant.

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The flow is still subsonic when Mach number is 0.5; however, the change of density is relatively greater andbecomes more significant than the case of Mach 0.25. The equation yields that the density change is equalto the change in velocity multiplied by 0.25.

Figure 21: Density contour lines for M=0.5 and angle of attack -2, 4, 6, and 10 degrees

The contour lines that are close to the airfoil are more distinguishable as they move further away fromthe airfoil. The difference is more apparent from the Mach 0.25 case, since the change of the density is moresignificant. Compressibility is now present and should be considered.

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The flow is transonic when Mach number is 0.85. The compressibility effects are now significant and thechange of density is much more apparent. The change of density is now equal to the change of velocitymultiplied 0.7225, which approximately equals the change of velocity.

Figure 22: Density contour lines for M=0.10 and angle of attack -2, 4, 6, and 10 degrees

The contour lines show variations over the airfoil, corresponding to pressure and density changes. Aphenomenon that can be seen clearly in Mach 0.85 are the stagnation points (the red colored sections) atthe front of the airfoils. Since the velocity at that section is essentially zero, more fluid is compressed at thatpoint, increasing the density at the stagnation points.

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D. Local Mach Number Analysis

Mach number is a dimensionless quantity used in fluid mechanics to represent the ratio of speed of an objectmoving through a fluid and the local speed of sound. It varies by the composition of the surrounding mediumand also by local conditions, especially temperature and pressure.

Aerodynamicists often use the terms subsonic, supersonic, transonic, hypersonic, high-hypersonic andre-entry speeds to talk about particular ranges of Mach values. When the regime is subsonic, the Machnumber is <0.8 and the plane has most often propeller-driven and commercial turbofan aircraft with highaspect-ratio wings. When the regime is transonic, the Mach number is between 0.8-1.2 and the aircraft nearlyalways have swept wings, delaying drag-divergence. For supersonic regimes, the Mach number is between1.2-5.0 and the aircraft is designed to have sharp edges and thin airfoil sections. For hypersonic regimesthe Mach number is between 5.0-10 and the aircraft has cooled nickel-titanium skin and small wings; samefor high-hypersonic regimes, but the Mach number is between 10-25. Lastly for re-entry speeds the Machnumber is <25 and the aircraft has ablative heat shield and small or no wings.

When and aircraft exceeds Mach 1 a large pressure difference is created just in front of the aircraft. Thisabrupt pressure difference, called a shock wave, spreads backward and outward from the aircraft in a coneshape. As the Mach number increases, so does the strength of the shock wave and the Mach cone becomesincreasingly narrow.

The Mach numbers analyzed are 0.25, 0.50 and 0.85, therefore the flow is subsonic and transonic, andthere are weak shockwaves created.


Beginning with a Mach number of 0.25, the contours data of each Mach number was recorded for the airfoilunder varying angles of attack, ranging from -2 to 10 degrees. The lower angles of attack, -2, 0, and 2 degreesall exhibited behavior that was very similar in the sense that the Mach number on the trailing edge andleading edge was nearly equal, visible below in Figure 23, where the airfoil was examined at zero angle ofattack.

Figure 23: Airfoil Mach number at Alpha = -2

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As the airfoil was tilted to the mid-level angles of 4 and 6 degrees, the Mach number difference on theupper and lower halves became more apparent. Shown below in Figure 24 is the RAE 2822 at an angle of 4degrees.

Figure 24: Airfoil Mach number at Alpha = 4

In Figure 24, the higher pressure, shown in yellow, circles above the upper surface, leaving a lower pressurearea, denoted by the green glow, to reside above the upper surface. Another noteworthy detail is that theairfoil experiences a weak shockwave at that angle of attack because it remains green glow on the top andbottom of the airfoil. Notice that the rest of the flowfield surrounding the airfoil seems nearly uniform, asthe presence of the airfoil at such a Mach number is still relatively unnoticed by the surroundings.

As the angle of attack is increased, the Mach number difference between the trailing edge and leadingedge increases too. Figure 25 below illustrates this difference in a manner that is easily seen, as the airfoilat a 10 degree angle of attack shows a decrease in the Mach number, it goes from a glow green (high) tolight blue (low). That is seen when the weak normal shock reaches the trailing edge and wants to becomean oblique shock, however because it is a very low Mach number that does not happen.


Notice that the Mach number is lower in the leading edge because the angle of attack was increased.Also, notice that the weak normal shock wave made the Mach number to decrease in the trailing edge.

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After the analysis of the airfoil at Mach 0.25 was complete, the next step was to move to a higher Machnumber, Mach 0.50.

Many of the relations discussed for the Mach number of 0.25 are still present, such as higher angles ofattack corresponding with lower Mach numbers, as well higher angles affecting the surrounding fluid morethan the lower angles. One interesting difference between the Mach numbers of 0.25 and 0.50 is that theMach number in the leading edge increases and then after hitting the shock wave decreases in the trailingedge. Figure 26 shows a side by side comparison of the airfoils at 6 degree angle of attack, but with Machnumbers of 0.25 and 0.50, respectively.

Figure 26: Airfoil Mach number 0.25 (left) and 0.50 (right) at Alpha = 6

Figure 26 shows clearly that for the same angle of attack, the airfoil on the right, in a Mach 0.50 condition,generates a stronger shock wave, converting the flow from being supersonic (notice the color red shows aMach number >1) to subsonic (notice the color light blue shows a Mach number >1). This is due to shockwave becoming oblique when it is at a transonic regime.

As the angle of attack increases very low Mach numbers are shown in the bottom of the airfoil. Noticethat Figure 26 has a Mach number between the colors green glow and light blue, however Figure 27 showsan airfoil with an angle of attack 2 with a Mach number between green glow and yellow in the bottom, andyellow and orange in the top.


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The last Mach number to be examined, Mach 0.85 produced some very interesting results in terms of Machnumber phenomena.

First on the list is the most interesting of all observed occurrences during the experiment, the appearanceof a strong shockwave. Shockwaves in air begin to occur as the speed of sound is approached, so Mach 0.85yielded the first instance of shocks viewed in the experiment. Figure 28 shows the formation of a shockwaveat an angle of attack of 10 degrees, at Mach 0.85.


As mentioned before, in order to have a shockwave, an increase in the speed is needed. Notice that inFigure 28 the zone of Mach >1 increases towards the leading edge. As M=1 is reached and passes, thenormal shock reaches the trailing edge and becomes a weak oblique shock, because the flow is subsonic Machdecreases in the trailing edge.

Figure 29: Airfoil Mach number with streamlines at Alpha = -2

The shockwave has a direct relationship with the angle of attack. Notice that the Mach number increasesin the top of the airfoil for the 10 degree angle, however for the -2 degree angle the Mach number increasesin the bottom surface. Figure 8 shows how the Mach number changes from M<1 in the leading edge to M>1in the mid-section and M<1 in the trailing edge.

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4. Bow Shock

Figure 30: Bow Shock at Mach 2

The situation above shows the RAE 2822 airfoil as a bow shock forms around the front of the object.Mach number across the shock near the front of the airfoil decreases is slower than the Mach number of theflow affected by the airfoil. This is due to the fact that the airfoil has moved to a Mach number where itis moving faster than the speed of sound, and so it is effectively separating the air and shedding the slowerfluid over the front of the airfoil. Note that the bow shock is a detached shock, as it has no contact with theairfoil surface itself. At supersonic speeds, the airfoil is able to influence fluid before it comes into contactwith them, due to movement speed faster than the fluids sonic condition.

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E. Flow Separation Analysis

All objects traveling through a fluid develop a boundary layer in the immediate vicinity of their surface. Onan aircraft wing, the boundary layer is the part of the flow where viscous forces distort the surrounding flow.The resulting boundary layer can be laminar or turbulent. Laminar flow occurs at lower Reynolds numbersthan turbulent flow and is usually characterized as smooth. Turbulent flow occurs at higher Reynoldsnumbers and speeds than laminar flow and is often undergoes irregular fluctuations. The difference betweenlaminar and turbulent flow is illustrated in the comparison of their respective velocity profiles.

Figure 31: Comparison of Laminar and Turbulent Velocity Profiles: Douglas, J. F., J. M. Gasiorek, and J.A. Swaffield. Fluid Mechanics. p327-332.

A velocity profile is one way to characterize the fluid flow, showing the magnitude of the velocity of theflow as a function of distance from the surface. As seen in Figure 31 above, both flows have a velocity ofzero at the boundary, but the turbulent velocity profile reaches higher speed after smaller distances than thelaminar boundary layer. Flow separation occurs when the boundary layer travels against an adverse pressuregradient and the speed of the boundary layer relative to the airfoil decreases to zero. The fluid detaches fromthe surface and creates eddies and vortices, ultimately resulting in reverse flow. Flow separation often resultsin an increase in drag, specifically pressure drag, which results from the differences in pressure between thefront and rear surfaces of the airfoil.

Another way to characterize the fluid flow is through the use of streamlines. Streamlines are field lineresulting from the vector field description of the flow and as a result, different streamlines at the same instantdo not intersect. Streamlines are beneficial because they trace out the path a fluid particle follows in thesurrounding flow field. A laminar flow field is shown when the streamlines are smooth, following the contourof the surface, while turbulent streamlines are random and chaotic. Streamlines show flow separation by thephysical separation of the streamlines from the airfoil. In the analysis of flow separation, the Mach numberscalar plot was placed in the background to easily see flow separation. Any areas of very low Mach numbers(blue area) was where flow separation and reverse flow of the velocity profile occurred.

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1. M=.25

Separation begins at small angles of attack, but as angle of attack increases, separation becomes moreprominent, moving farther up the airfoil surface. For flow with a low Mach number, separation is uncommon.For the Mach 0.25 flow fields no separation was present between the angles of attack of negative two and sixdegrees.

Figure 32: Streamlines at Mach = 0.25 and Alpha = 6 degrees

As seen in the figure above, the streamlines hug the body and the flow field is smooth. When the angleof attack was increased, however, the flow near the trailing edge experiences separation.


As the flow reaches the trailing edge, the Mach number of the flow decreases to zero and the streamlinesdeviate from the surface. The separation is caused by an adverse pressure gradient that exists by the trailingedge. By zooming in on the surface of the airfoil near the trailing edge, the flow separation becomes moreobvious.

Figure 34: Velocity Profile at Mach = 0.25 and Alpha = 8

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Near the surface of the airfoil, the velocity profile shows signs of reverse flow, one of the characteristicsof flow separation. This causes an increase in pressure drag as the separation bubble has a lower pressurethan the flow on the leading edge of the airfoil and will pull backwards on the surface.

As the angle of attack increases to 10 degrees, separation becomes more evident. At an angle of attackof 10 degrees, the separation bubble moves up the airfoil surface. The separation in the velocity profile alsobecomes more prominent, as the separation bubble grows larger.

Figure 35: Reverse Flow at Mach = 0.25 and Alpha = 10 degrees

Compared to the separation bubble at eight degrees angle of attack, the separation bubble at 10 degrees islarger and farther up the airfoil. The reverse flow increases in magnitude and continues for a longer distance.From these diagrams, it can be seen that the angle of attack directly influences flow separation. As the angleof attack increases, the separation bubble moves farther up the surface and the separation bubble expands.

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2. M=.50

A few more angles of attack had separation of flow for Mach 0.50 than Mach 0.25, which is intuitive sincethe flow field is moving faster, increasing the chance of adverse pressure gradients that cause separation offlow. No separation of flow occurred between angles of attack of negative two and four. The first prominentseparation of flow happens at angle of attack of six degrees for Mach 0.50.



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The separation for the angle of attack six degrees occurs near the trailing edge of the airfoil, and it isa rather mild reverse flow compared to other cases. As the angle of attack increases, the separation occurscloser to the front of the airfoil. An interesting case happens at 10 degrees of the angle of attack. Separationoccurs near the front of the airfoil.


Figure 39: Front Velocity Profile at Mach = 0.50 and Alpha = 10

The velocity profile shows reverse flow happening very abruptly at the front of the airfoil. This is dueto the high angle of attack, and a higher Mach number. The front of the airfoil essentially cuts the flowabruptly, causing separation of flow near the front. A much smoother reverse flow velocity profile can beseen further down the airfoil as can be seen below.


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3. M=.85

Separation of flow occurs at every single angle of attack for Mach 0.85. This is due to the shock wavesthat form for every angle of attack causing an adverse pressure gradient behind the shock wave. Even for 0degrees, a shockwaves form near the back of the airfoil, and separation occurs right behind the shockwaves.The blue area behind the shockwave is a sudden drop in Mach number, and an adverse pressure gradientcausing separation, as can be seen below.


At 10 degrees angle of attack, the separation of flow becoming turbulent can be readily seen in thestreamline over Mach number plot below. The blue area behind the shockwave is wavy in appearance dueto the strong vorticity that is occurring behind the trailing edge. The separation does not occur at the frontof the airfoil at 10 degrees as it did at Mach 0.50 due to the formation of a shockwave. The separation doesnot occur until after the shockwave. The corresponding velocity profile at the start of reverse flow is belowas well.



In summary, as the absolute value of angle of attack increases, the chance of separation of flow increases.The same correlation happens when the Mach number increases. And as the chance of separation of flowincreases, the starting point of separation of flow occurs closer to the front of the airfoil until the Machnumber reaches a point where shockwaves form, and then the separation occurs behind the shockwave.

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F. Compressibility Effects

1. Critical Mach Number, Mcr

Mcr, or the critical Mach number, is defined to be the lowest Mach number when somewhere along theairfoil the local Mach number is 1, i.e. reaches the speed of sound. At such Mach numbers, there are somelocal points where the airflow speed exceeds the local speed of sound. This phenomenon usually takes placeat Mach numbers lower than 1. An aircraft for example can be flying at a speed that will result in Machnumber lower than 1, and still can experience weak shock waves. We easily notice from Figure 44 that closerto the trailing edge, our airfoil experiences Mach numbers close to 1.5 even though the free stream Machnumber is 0.85. If we consider a transonic airfoil flying at speeds higher the critical Mach number, there willa dramatic increase in the drag coefficient which could eventually lead to stall.

Figure 44: Critical Mach Number, Mcr

2. Drag Divergent Mach Number, Mdd

Mdd, or the drag divergence Mach number is Mach number at which drag coefficient on a body increasesrapidly. The drag divergence Mach number is always greater than the critical Mach number. Shock waves atsuch Mach numbers induce flow separations. Interestingly enough, Prandtl-Glauert rule predicts an infiniteamount of drag when Mach number is 1, which might be the result of the lack of technology back in thedays. The following formula for calculating the coefficient of pressure in a compressible flow was developedby Prandtl and Glauert,

Cp =Cp0√

1−M2(2)

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where Cp0 is the coefficient of pressure of in an incompressible of the same point in the airfoil, as seen inFigure 45. This equation yields an infinite pressure coefficient in the case of Mach 1.

Figure 45: Drag Divergent Mach Number, Mdd

3. Cl/Cd vs. M

The ratio of lift to drag (or Cl/Cd) is an important parameter when analyzing performance. For now, wewill use it to analyze the influence of Mach numbers on in the lift to drag ratio.

Figure 46: Cl/Cd vs. Alpha

Comparing the two curves of Mach 0.25 and Mach 0.50, one will have more lift per drag at a Machnumber of 0.5 up an angle of attach of approximately 4.3 degrees. Then, flying at a 0.25 Mach number willyield a higher lift to drag ratio (better performance). When reaching a Mach number of 0.85, compressibilityeffects clearly dominates. For example, at an angle of attach of 4 degrees, Cp rapidly increases from 0.008813at Mach 0.5 to 0.0945862 which is approximately 11 times higher. It is evident that the critical and dragdivergence Mach numbers were exceeded.

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4. Cd vs. M

An interesting pattern was observed. The drag increases with Mach number until reaching the critical Machnumber and the drag divergence Mach number, respectively, where the drag rapidly increases (peak close toMach 1). Then, the drag will start to decrease after becoming supersonic (around Mach 1.2). The patterncan be seen from the plot. In the first region, Cd increases due to losses in shock waves, shock inducedseparation, and shock waves increase in strength as M∞ increases. While in the second region, even thoughshock strength and losses increase as M increase, the boundary layer separation point , moves aft, decreasingthe drag due to flow separation.

Figure 47: Cd vs. M

5. Cl vs. M

After analyzing the effects of Mach number (or compressibility) on the lift coefficient while using the angleof attach as a parameter, it is observed that the general behavior of the coefficient lift can be characterizedin three phases. Firstly, Cl increases slightly from Mach 0.2 to 0.4. Then, Cl decreases (not very dramati-cally) up to approximately a Mach number of 0.8. Between Mach 0.8 and 1.0, the lift coefficient decreasesdramatically. After exceeding Mach 1 (or sometimes Mach 1.2), Cl does not change very much for low angleof attacks, while keep decreasing almost linearly for angle of attacks 4,6,8, and 10 degrees.

Figure 48: Cl vs. M

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IV. Conclusion

After careful observation of simulated flow patterns in FieldView for differing situations; and after a morethorough analysis of the effects that slight changes in flow conditions have on the RAE 2822, multiple thingsof note appeared. The flow behaves well at lower Mach numbers; the airfoil does not have a large impacton the fluid around it. However, as the Mach number rises, the effected flow field surrounding the airfoilgrows substantially. At a certain point, roughly estimated around Mach 0.75, the flow around the airfoil isdisturbed enough that flow can no longer naturally satisfy downstream flow conditions. This effect leads towhat can be seen in the Mach 0.85 and higher figures, the formation of a shockwave on and near the airfoil.

The first shockwave that was mentioned, an attached shock for the transonic Mach number of 0.85,harbors characteristics normal to shockwaves, an increase in Mach number, slight decrease in density, andconsiderable pressure drop as the wave traverses the surface of the airfoil. Using this information, and theconcepts found in higher Mach flow, such as the formation of a bow shock around the body instead of ashock that remains attached to the airfoil, a clear relation between shockwaves and Mach number can bedrawn: as the Mach number increases, both the size of the shockwave and its effect on the surrounding flowincrease.

While the formation and development of shockwaves was an interesting phenomenon to observe, it was notthe only noteworthy experimental finding. Another interesting result that appeared during the experimentwas the separation of flow off of the back of the airfoil. As the Mach number increases, the separation seemsto occur more quickly, but the true area of interest lies within the angle of attack. At higher angles of attack,the flow separates more quickly from the airfoil than observed flow from the lower angles of attack. This ismost likely due to the need of an adverse pressure gradient for flow separation in 2-D flow. Increasing angleof attack increases the pressure gradient, making it steeper, and allowing separation much earlier. This flowseparation can also be characterized by the presence of reverse flow, shown in the velocity profiles. If a moreaccurate representation of separation was desired, the code used to map the airfoil and flow could possiblybe rewritten from a 2-D Navier Stokes code to a 3-D version.

The relation between angle of attack and pressure gradient is critical flow separation, but the relationbetween angle of attack and generation of lift is much more important, as lift is needed for flight. Data takenand analyzed regarding the angle of attack and lift relation revealed something that was both expected, andinteresting. As the angle of attack is increased, the amount of lift generated also appears to increase. Thisis expected and easily explained, as in theory, increasing the angle of attack up until stall will result in thehighest lift coefficient, and therefore, the highest amount of lift.

The three aforementioned relations involving shockwaves, flow separation, and lift generation, can becombined with other sectional concepts included in the report to tie together nearly all of the fluid propertiesand flow properties, and are fairly accurate in representing the true conditions of the airfoil in the flow. Eventhough the simulation may be a good representation of a real world situation, it is not perfect. If higherprecision was needed, a few small areas of the experiment could be rewritten. First, the use of a 2-DimensionalNavier Stokes code would be moved to a 3-Dimensional code, to allow for a more accurate simulation athigher Mach numbers and angles of attack, as the current code begins to break down where high amountsof flow separation are present. Also, flow conditions and numerical values would be more accurate if takenfrom more than one planar viewpoint. Observing fluid behavior around the front of the airfoil may providesome new information about fluid movement or disturbance by the airfoil.

Even though the simulation was not perfectly representative of the real world situations modeled, theresults that were obtained have given an accurate description of fluid properties and their relationship toairfoil orientation and respective flow properties. Overall, data taken and analyzed in this experiment hasgiven valuable insight into fluid/airfoil relations, and may be of great use for the years to come.

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V. Appendix

A. Pressure

1. M=.25, α = -2, 0, 2, 4, 6, 8, 10

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2. M=.50, α = -2, 0, 2, 4, 6, 8, 10

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3. M=.85, α = -2, 0, 2, 4, 6, 8, 10

(a) α=0

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B. Density

1. M=.25, α = -2, 0, 2, 4, 6, 8, 10

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2. M=.50, α = -2, 0, 2, 4, 6, 8, 10

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3. M=.85, α = -2, 0, 2, 4, 6, 8, 10

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C. Local Mach Number

1. M=.25, α = -2, 0, 2, 4, 6, 8, 10

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2. M=.50, α = -2, 0, 2, 4, 6, 8, 10

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3. M=.85, α = -2, 0, 2, 4, 6, 8, 10

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D. Streamlines

1. M=.25, α = -2, 0, 2, 4, 6, 8, 10

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2. M=.50, α = -2, 0, 2, 4, 6, 8, 10

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3. M=.85, α = -2, 0, 2, 4, 6, 8, 10

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E. Velocity Profiles

1. M=.25, α = -2, 0, 2, 4, 6, 8, 10

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2. M=.50, α = -2, 0, 2, 4, 6, 8, 10

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3. M=.85, α = -2, 0, 2, 4, 6, 8, 10

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VI. Team Member Work Distribution

1. Saeed Al Jaberi

(a) Initial FieldView Investigation

(b) Coefficient Plots (Cl, Cd, etc.)

(c) Compressibility Effects

(d) Density Discussion

(e) Proofread/edit

2. Saray Checo


(b) Conclusion

(c) Mach Number Analysis

(d) Density Discussion

(e) Proofread/edit

3. Logan Krawchyk


4. Morton Lin

(a) Coefficient Plot Analysis and Discussion

(b) Coefficient Plots (Cl, Cd, etc.)

(c) Appendix

(d) Compiled

(e) Edited

(f) Formatted

(g) Proofread

5. Christopher McElroy


(b) Report quality pressure, density, mach plots

(c) Procedure

(d) Velocity Profiles and Streamline plots

(e) Velocity and Profiles discussion

(f) Proofread/edit

6. Jesse Shaffer

(a) Initial FieldView investigation

(b) Pressure Analysis

(c) Introduction

(d) Conclusion

(e) Proofread/edit

7. Richard Zang

(a) Initial FieldView investigation

(b) Report quality pressure, density, mach plots

(c) Velocity Profiles and Streamline plots

(d) Velocity and Profiles discussion

(e) Density Discussion

(f) Proofread/edit

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analysis of the rae 2822 airfoil using computational fluid...

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