An Experiment Determining Lift from Pressure Distributions

Abstract

The experiment described in this report used a pressure distribution over a NACA 0012 airfoil to determine the lift and moment coefficient about the quarter-chord on that airfoil. Lift is one of the most important and governing forces acting on a body moving in a fluid. The calculation of lift and drag is at the forefront of aerodynamics as a whole. Furthermore, understanding an airfoil’s pitching moment is of the utmost importance with regards to an aircraft’s stability. The lift and moment were calculated from the integration of the measured pressure distribution over both the upper and lower surfaces. While calculating lift and moment, the pressure distribution characteristics were shown for different angles of attack. The tests were conducted in the 2 foot by 2 foot wind tunnel, capable of Reynolds number ranging from 32,000 to 950,000.

Introduction

All aerodynamic forces and moments are generated by two main sources: pressure distribution and shear stress distribution. The experiment described in this report gives a way to determine the lift, drag and the moments (at least part of them) on a body from the pressure distribution. The pressure and shear are natures only two ways of communicating a force acting on a body [1].  The experiment does not deal with shear stress, so some of the drag, called skin friction drag should not accounted for. If all of the pressure and shear forces are summed up over the whole body, the resulting force and moment will be the total aerodynamic force and moment [1].

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 The lift generated by an aircraft’s wings is what allows the aircraft to overcome the effects of gravity. This makes determining the lift of airfoils of utmost priority to fit aircraft parameters and restrictions. The drag of an aircraft is the force that acts in the opposite direction of the free stream velocity. The drag is also important because with less drag means there will be less thrust required to achieve flight. This is very important because generating thrust is expensive, so by reducing drag the overall efficiency of the aircraft can be improved.

 The experiment used a NACA 0012 airfoil shape, which is symmetric. It has 18 static pressure               ports along the upper surface of the airfoil and these ports recorded the pressure at several different angles of attack and Re. The pressures are ran through a pressure transducers and values are stored so they can be later used to determine the lift and moment of the airfoil.

 The lift force on a body moving through a fluid is the force acting perpendicular to the free stream velocity. The drag force on a body is the force acting parallel to the free stream velocity. These two forces are due entirely to the pressure and shear stress distributions over the body [3]. If a coefficient of these forces is introduced, the calculation of these forces can be simplified when changing certain parameters. The coefficient of lift is described by

                                                         
cl=L‘q∞c
                                                                   [1]

The coefficient of lift can also be described by

                                  
cl=cos⁡α∫xc=0xc=1.0CP,lxc–CP,uxcd(xc)
                              [2]

where 
CPxc
is a function. The moment about the quarter chord is derived by
cm,le = MLE‘q∞c2= –π∝2
                                                    [3]

and

cm,c/4= cm,le + cl4
                                          [4]

However, regarding a symmetric airfoil,
π∝
= cl/2, meaning
cm,le=–cl4
                                                           [5]

Therefore, theoretically the moment coefficient about the quarter-chord should be equal to zero
cm,c/4=0
                                                          [6]

These equations create a relationship between the pressure distributions, velocity, density, lift and moment. The coefficients are merely a convenient way of holding certain things constant during the calculation of the lift and drag so the calculation does not have to be repeated. The coefficient of pressure, as is present in

the earlier formulas, can be given by

                                           
CP=Ps,local–Ps,freestreamq
                                                        [7]

The pressure coefficient is useful because once the coefficient of pressure has been determined, the pressure can be calculated for any other parameters since
CP
is independent of them [2].

 The Reynolds number is the ratio of inertial forces to viscous forces in an airflow. If the Reynolds number and Mach number are the same for two flows, as well as the model being geometrically similar, the flows can be modeled as dynamically similar [2,3] Wind tunnel testing hinges on this concept because it allows for the testing of small-scale models and controlled experiments to produce the exact same outputs as the real world flight would. The Reynolds number is given by

                                                       
Re= q∞V∞lμ∞
                                                               [8]

The higher the Re, the more the inertial forces dominate the flow.

 The pressure force is the force per unit area that the fluid molecules exert on the body from the transfer of momentum. The total pressure is comprised of the static pressure plus the dynamic pressure. The static pressure is the pressure from the random motion of the fluid particles and does not include any of the momentum from the flow velocity. The dynamic pressure is the pressure due entirely to the flow velocity

momentum [2,3]. Total pressure can then be described by the equation

                                                    
Pt=Ps+12ρ∞V∞2
                                                         [9]

where the dynamic pressure is the second term on the right hand side of the equation.

 A NACA 0012 airfoil is symmetric. This means that the flow over the upper and lower surfaces are the exact same at a 0-degree angle of attack, and the flows of any magnitude angle of attack are mirrored over the chord line. In other words, if at some
α
the flow over the upper and lower surfaces have certain characteristics, then at
–α
the flow will be a mirrored over the chord line. The pressure data can then be collected on one side at a positive and negative alpha in order the get the distribution over the upper and lower surfaces for each magnitude of
α
while only having pressure ports on one side.

 As a general predication, it is hypothesized that as the Re goes up than lift will increase as well. Also, increasing the angle of attack will similarly induce an increase in lift. That is until the airfoil reaches a critical angle of attack, causing flow separation and an immediate loss of lift. Careful examination of the different Re and angle of attack combinations and which cause flow separation will be of utmost importance during this experiment. Regarding the moment about the quarter-chord, it is hypothesized that not much fluctuation in the moment coefficient will occur. Possibly some jumps may occur while the airfoil is tested under its most extreme angle of attack. 

Description of Test Equipment and Procedure

The experiment setup is shown in Figures 1 and 2. A 2 ft. by 2 ft. open circuit wind tunnel was used to test the airfoil.

Figure 1: NACA 0012 Airfoil placed inside wind tunnel.

A monometer inside the tunnel was used to test the dynamic pressure inside the tunnel. A NACA 0012 airfoil, with chord 6 inches and a 2 ft. span that spanned the entire width of the test section. Since the airfoil spanned the whole width of the test section, the flow could be modeled as a 2D flow. 18 pressure ports on the upper surface of the airfoil were used to measure the pressure distribution for angles of attack at 2, 4, 6, 8, 10, 12, and 14 degrees, each at 30, 60, and 90 feet per second. Additionally, a 16 degrees angle of attack was also measured only at 90 feet per second. Then the process was repeated for the negatives of these angles so as to have the flow from the lower surface in part one to be on the upper surface where the pressure ports were in part two. The location of the pressure ports is shown in Table 1.

Table 1: Pressure port locations measured from leading edge [5]

The ports were connected to a scanivalve system which has a built-in pressure transducer and utilizes a PC with a data acquisition card to record the pressure data shown in Figure 2.

   

(a)                                                                (b)

Figure 2: (a) PC recording pressure distribution data with (b) scanivalve system and control system.

Results

 The atmospheric pressure in the experiment was 14.3172 psi, and the temperature was 72
°F
. These were used to calculate the density and pressures inside the testing section and determine the velocity.

Upon completion of the in-lab portion of the experiment, all data was collected and compiled. Results of the Pressure Coefficient vs. Chord Length for 30, 60, and 90 feet per second are shown in Figures 3,4 and 5, respectively. Results for the Lift Coefficient vs. Angle of Attack are shown in Figure 6. And the results of the Moment Coefficient about the Quarter-Chord vs. Angle of Attack are shown in Figure 7.

Figure 3: Pressure Coefficient vs Chord Length at 30 ft/s

Figure 4: Pressure Coefficient vs Chord Length at 60 ft/s

Figure 5: Pressure Coefficient vs Chord Length at 90 ft/s

Figure 6: Lift Coefficient vs Angle of Attack

 From studying Figure 6, the experimental data suggests the following. During the low Re value test, the airfoil produced a maximum lift coefficient of 0.9919 and a critical angle of attack of 10 degrees. The mid Re value test produced a maximum lift coefficient of 1.6617 and a critical angle of attack of 8 degrees. The high value Re test produced a maximum lift coefficient of 1.9127 and a critical angle of attack of 10 degrees.

Figure 7: Moment Coefficient about the Quarter-Chord vs. Angle of Attack

 As theoretically predicted, the moment coefficient about the quarter-chord remains approximately zero. This pattern holds until the airfoils maximum angle of attack. Stalling and flow separation around the airfoil enables the moment coefficient to drastically drop.

Discussion

 After examination and interpretation of the experiment results, it seems that the collected data coincides with the initial predictions. The purpose of this experiment was to determine the pressure coefficient distribution, lift coefficient and moment coefficient about the quarter-chord of a NACA 0012 airfoil. Each of these properties was found by analyzing the pressure distribution on the upper and lower surfaces of the airfoil. The pressure distribution was found by taking pressure readings from eighteen pressure taps placed along the surface of the airfoil. Several different trials of this experiment were conducted each at a different Reynolds number. It was found that at low Reynolds numbers, the lift coefficient was not largely affected by a change in Reynolds number. However, the published results for the larger Reynolds number did exhibit different lift coefficients. It was also found that, discounting stall, all of the calculated properties performed very closely to that of thin-airfoil theory. Stall must be discounted because thin-airfoil theory does not account for stall.

As shown in Figures 3,4 and 5, both the lower and upper pressure coefficients are very different at the leading edge of the airfoil. The pressure on the upper face of the airfoil initially experiences a large decrease due to an increase in fluid velocity around the airfoil. Vice versa, the lower face of the airfoil is subject to an increase of pressure due to the fluid flow slowing down. Then the pressures slowly converge until nearing zero at the trailing edge of the airfoil.

 As shown in figure 6, Reynolds number does seem to impact the lift coefficients. The lower Reynolds number clearly had a lower maximum lift coefficient while the higher Reynolds number had a greater maximum lift coefficient. Each experimentally determined response demonstrates the same general shape in that the lift coefficient generally increases linearly with angle of attack until it approaches the stall angle where it levels off and then begins to decrease. The experimental lift coefficient was determined to be in line with thin-airfoil theory. Thin-airfoil theory states that the lift coefficient increases by 2π units per radian [1]. Thin-airfoil theory, however, does not account for stall. This is illustrated in Figure 6 by the fact that the theoretical lift coefficient increases linearly and never levels off. The theoretical lift coefficient does behave similarly to the experimentally determined data very closely for the linear part of the response. This means that, disregarding stall, the NACA 0012 airfoil performed fairly like thin-airfoil theory predicted. Figure 6 shows the lift coefficient versus angle of attack results from several different trials of this experiment each at a different Reynolds number.

The center of pressure for a thin airfoil is defined as the position along an airfoil where the moment is equal to zero. As observed in Figure 7, all three responses exhibit similar shapes. Each center of pressure is relatively at around a value of 1.5 inches. Then the moment coefficients for each trial drastically fall and all converge at about -0.35. Overall, there is a very small change in the center of pressure throughout the experiment. This is compared to the theoretical center of pressure for thin-airfoil theory which is one-fourth of the chord length, or 0.25 in non-dimensional terms [1]. In general, the experimentally determined center of pressure is relatively close to the theoretical value if stall is discounted, which is acceptable since thin-airfoil theory does not account for stall. Also, from thin-airfoil theory it is stated that the aerodynamic center and the center of pressure are located at the same point [4].

Even though the results of this experiment lined up relatively closely with theory, there are still some unexplained anomalies that may be due to experimental error. Most importantly, the calculation of the free-stream velocity, and consequently Reynold number, may not have been accurate. The free-stream velocity was determined using the dynamic pressure reading that was assumed to be in the free-stream. It is possible that there was some kind of turbulence that effected this reading because of the nature of man-made wind. Another source of error was the angle of attack setting. The divisions on the dial were very small and difficult to see. Another source of error could be in the readings themselves. The pressure readings were read off a pressure transducer and input by hand into Excel. It is entirely possible that some of these readings could have been entered incorrectly or mislabeled. However, any major errors would have been apparent in the resulting plots, as seen in the lift coefficient vs angle of attack figure. A final source of error could be in the pressure transducer. There is concern that the pressure transducer was either calibrated incorrectly or not calibrated at all in-between tests. Also, it was fed pressure by small tubes connected to each tap on the airfoil; it is possible that there were leaks in these tubes which could have affected the pressure readings.

Conclusion

 In conclusion, this experiment performed as expected. Discounting stall, the experimental results proved to be very similar to those of thin-airfoil theory. It was also found that at low Reynolds numbers, the lift coefficient was not drastically affected by a change in Reynolds number. The moment coefficient was found to remain around zero until reaching the maximum angle.

  Lift is one of the most important and governing forces acting on a body moving in a fluid. Being able to fully study and understand this phenomenon can greatly increase performance, economic and environmental aspects for the aircrafts of the future.

References

 

[1] Anderson, J. D. Jr., “Fundamentals of Aerodynamics”, 5th ed., McGraw Hill, New York, 2011, pp. 19-22,233-234,320-360.

[2] Barlow, J. B., Rae, W. H., and Pope, A., “Low-Speed Wind Tunnel Testing”, 3rd ed., John Wiley & Sons, New York, New York, 1999, pp. 6-8, 102-109, 471-482.

 [3] Anderson, J. D., “Fundamentals of Aerodynamics”, 4th ed., McGraw Hill, New York, 2007, pp. 19-30, 210-213.

[4] Anderson, J. D., “Introduction to Flight”, 8th ed., McGraw Hill, New York, New York, 2015, pp. 322-324

[5] Ahmed, A., Van Horn, E., Khan, O. and Shah, S. H. R., “Aerodynamics Laboratory”, Auburn University, 2019.

Appendix

 

Benford’s Law and Other Surprising Distributions

Introduction
Benford’s law is a surprising mathematical concept which at first seems rather counter-intuitive. It explains the distribution of the leading digits in a large set of data. A simple example displays its initial peculiarity. Imagine we look at the current share price of every company on the FTSE 350, an index of the 350 largest UK companies. Within this set of data, each share price has the possibility of the first digit being any number between 1 and 9 ( d∈{1,…9})

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The average person would believe that each price would have an equal chance of starting with each number between 1 and 9, so if one of the 350 prices was selected at random the probability that the first digit was 1 would be about 19 (11.1%) and the probability of the first digit being a 9 would also be about 19 (11.1%). However, this is in fact not the case at all. If you were to calculate this, the probability of the first digit being 1 would actually be closer to 30% than 11%. Furthermore, the probability of the leading digit being a 9 would only be just over 4%! I initially read about this strange distribution in an economics context. I was keen to investigate the mathematics behind this and test the limits and applications of it. The physicist Frank Benford first discovered this in 1938 when he noticed that pages closer to the beginning of his log tables were increasingly becoming more worn than those closer to the end, meaning that they were searching for numbers that were starting with a 1 much more than higher numbers. Benford started to test this theory across newspapers, populations and river lengths. He found more or less the same result every single time. Numbers starting with a 1 turned up approximately 30% of the time, almost all the time. Eventually, this was formed into a mathematical law which includes an equation which displays the exact probabilities of each number between 1 and 9 occurring as the leading digit. The aim of this investigation is to explore the explanations and applications behind Benford’s Law and to touch upon other equally strange distributions and examine if they link to Benford’s law in any way. NB: For the purposes of this exploration, all logarithms will be assumed as base 10 if the base is not stated.
Why Study Leading Digits?
Before discussing the law’s explanations and applications, it is first useful to understand how leading digits are studied in the world of mathematics and their importance. Scientific notation (also known as standard form) is a pivotal part of this. The notation follows the system that a positive number x can be expressed in the form S(x)×10n in which 1≤Sx10, meaning that the number is expressed first as a value of 1 or greater but less than ten, multiplied by 10 to a given exponent which reaches the initial number. For example, in this format the number 865,000,000 would be expressed as 8.65×108 The initial number before the exponent is known as the significand.1 This system allows numbers spanning many different magnitudes to be expressed in a similar fashion, for example comparing atomic radii to planetary radii.
An Overview of the Law
After Benford’s experiments, he discovered the approximate percentages for the probability of each number occurring as the leading digit. The pattern followed in a way as shown on the graph below:

2
For the purposes of explaining the law and the derivation of its equation, the leading digit between 1 and 9 is represented as d with the probability of the digit being the leading digit is represented as P(d).
The basic explanation of the law states that the space between digit d and d+1 is proportional to the quantity of P(d) on a logathrimic scale. A logarithmic scale is one which is non-linear3 and based on orders of magnitude meaning each increasing unit on the scale is the unit on the previous value multiplied with a constant.
Deriving the Equation of the Law
By understanding logathrimic scales we can begin to better understand how the percentages in Benford’s law are derived. When we are working with many values spanning multiple orders of magnitude, as Benford’s law does, the basic explanation states that:
The leading digit d will be 1 when log⁡1≤log⁡dlog⁡2 and similarly, d will be 9 when log⁡9≤log⁡dlog⁡10. On a linear scale the difference between 2 and 1 would be equal to the difference between 10 and 9. However on a logathrimic scale the differences are as follows:   

Logarithmic Interval

Difference of Interval

log⁡2–log⁡1

0.301

log⁡3–log⁡2

0.176

log⁡4–log⁡3

0.125

log⁡5–log⁡4

0.0969

log⁡6–log⁡5

0.0792

log⁡7–log⁡6

0.0669

log⁡8–log⁡7

0.0580

log⁡9–log⁡8

0.0512

log⁡10–log⁡9

0.0458

If we apply this to all the numbers between 1 and 9 the results are as follows:
These log calculations are in fact the probabilities of each number from 1 to 9 occurring as the leading digit! This can be seen on the graph of the results below, which follows the exact same pattern as the graph shown above.

From this we can see that the probability P(d) is given by the log of the digit subtracted from the log of the digit plus one i.e:
P(d)=log10⁡d+1–log10⁡d = log10⁡(d+1d)
Further explanations and details of the Law
Aside from the initial explanation of the law and the derivation of the equation, there are more detailed explanations and perspectives to the law and how it works. One of these is the Geometric Explanation.1This approach to the law follows the idea that in a model of a number n in a constant growth rate, n will spend a greater amount of time ‘hanging around’ the lower digits than the higher ones. To better explain this, I will refer back to an economically minded example of a geometric series, compound interest. A geometric series is a series in which there is a constant ratio r between each term u and (u+1). 4Therefore the deductive rule follows as Un=U1r n-1. For this compound interest example, let us assume I invest $2000 in 2019 for my retirement in a very generous savings account with an annual 7% interest rate for the long term of 60 years. This function is modelled by the equation Un =2000×1.07n. Note the absence of the subtraction of 1 from n in the exponent. This is due to the fact that we wish to calculate the value as compounding at the end of each year so the subtraction of 1 is not useful. This model shows that the balance in the savings account at the end of the 60 years will be 2000×1.0760=$115,892.85. However, we are more interested in where the balance lies at the end of each year over the whole period rather than just the end. See the appendices for the full balance sheet at the end of each year. When we examine this table from a Benford perspective, we can see that the balance does indeed tend to stay towards low numbers for the first digit and quickly accelerates through the higher numbers. For example, the period between when the balance is $10,000 and $20,000 lasts from 10 years from 2043 to 2053 during which the first digit is 1 on the balance sheet. The table below illustrates this for the point between $10,000 and $99,000 in the account.

Value interval

Time Spent (years)

$10,000-$20,000

10

$20,000-$30,000

6

$30,000-$40,000

4

$40,000-$50,000

4

$50,000-$60,000

3

$60,000-$70,000

3

$70,000-$80,000

3

$80,000-$90,000

2

$90,000-$99,000

1

From a reflective standpoint, perhaps this is why we as humans focus so heavily on financial achievements which essentially get us ‘back to one’ such as setting targets at one million or one billion. A more recent example of this is Apple making headlines for being the first company to achieve a net worth of $1trillion. Maybe the fact that financially we spend such a long time at these lower values makes them more psychologically valued to us as humans once we reach them at the next order of magnitude.
Scale Invariance
Another aspect of Benford’s Law which adds to its uniqueness is it’s universality. What I mean by this is that if a situation follows Benford’s law, it will tend to continue to follow Benford’s law no matter what operators are imposed upon it. For example, if I took the data set used in the previous explanation, the list of the investment balance year upon year and converted it into every single commonly used currency in the world, from the Euro to the Pound and Vietnamese Dong, the chances are the data would continue to satisfy Benford’s law in almost every single currency. Since each value in the list would have the same operation applied to it, this means it is still likely to span many orders of magnitude which is the main condition for Benford’s Law to apply.
Extensions of the law: Digits beyond the first
Another aspect of Benford’s Law is that it can be extended to further digits rather than just the first digit of the number.5 It is possible to calculate the probability of a number occurring as the 2nd or 3rd digit. To do this we must manipulate the equation into a series in sigma notation which allows us to express a series of additions in one notation. If we have a digit between 0 and 9 (NB: zero can now be included as it is not possible to have zero as the first digit of a number but it is certainly possible to have it as a following digit) then the probability that this digit will be the nth digit in a number is given by the equation:
∑x=10n–210n–1log10⁡(1+110x+d)
In which d represents a number between 0 and 10 and n represents the nth digit which the probability is wanted to be calculated for. However, this is only particularly useful up to the 3rd digit as once the calculation is past the 3rd digit the numbers follow a more expected distribution and tend closer to each number appearing 10% of the time i.e truly random.
Applications of Benford’s Law
Benford’s law has one major application which makes it particularly useful, fraud detection. Due to the fact that Benford’s law is present in every aspect of life when numbers are distributed, any large sets of data which do not follow Benford’s Law could be argued to be fraudulent, particularly financial data. Programs which test for compliance with Benford’s Law are often used by tax institutions or banks during audits or to check if data submitted to them is possibly fraudulent. Benford’s Law was also used as part of fraud detection in the 2009 Iranian election6. This raises the question as to if it is moral to use mathematical laws in legal proceedings or as evidence in prosecutions. This morality debate is even more prevalent when there is a certain degree of uncertainty within the law, or limitations to the law as will be discussed below.
Limitations to Benford’s Law
Not every single set of data will be able to follow Benford’s law, for example telephone numbers, human height in meters or feet and page numbers of small documents. Benford’s law also does not apply to data which is generated by humans themselves or written within specific ranges. The chance of Benford’s Law being useful highly depends on how many orders of magnitude the data set spans. For example, the earlier example of human height in meters or feet doesn’t follow the law as it only spans one order of magnitude. In meters almost all human heights will start with a 1, possibly with a few that start with 2 or less than 1. The same applies if human height is measured in feet, there would have to be a human over 3 meters tall in order to exceed the 10ft boundary into the next order of magnitude! Also, if there an extremely large number of orders of magnitudes, then the law also may not apply. For example, Benford’s law wouldn’t apply to the data set of all real numbers, as clearly if these numbers continue to go on forever then then the probability for each digit from 1 to 9 to be the leading digit will be the same.
Further analysis: Similar Laws?
Benford’s law is surprisingly not alone in its strangeness. Contrary to what one may think after reading about the uniqueness of Benford’s law, there are a few other patterns and principles which exist through many different areas of life. Some of these have mathematical patterns which could link to Benford’s law. One of these is Ziph’s Law which relates to language and literature rather than numerical data. Ziph’s Law states that in a large set of words, if the most frequent word is taken, the second most frequent word will appear half as often as the most frequent word and the third most frequent word will appear half as often as the second most frequent word. Essentially, the frequency of a word will be inversely proportional to how often the word appears overall. For example, the most common word in the English language is the word ‘the’ which accounts for 7% of all words appears twice as much as the second most common word ‘of’ which accounts for 3.5% of all words. An equation for Ziph’s law has been created in the context of the English language which states that in a distribution of X number of words in the language, the frequency of each word occurring in relation to its rank of how common it is follows this equation:
1/k∑x=1X1/n
In which X is the number of words in the English language and k is their sequential rank of how common they are in the language. Some have argued that Benford’s law is simply a special case of Ziph’s law however I personally believe they should be held as separate laws. Ziph’s law could better be considered as literature’s version of Benford’s law. 
Conclusion
Overall, Benford’s Law is deeply rooted into the way numbers are distributed in the real world and it’s useful applications cannot be denied. The law which at first seems strange and unexplainable can indeed be explained and analysed as I have demonstrated throughout this investigative report. The geometric analysis behind Benford’s Law is key to its explanation. Understanding Benford’s Law is now extremely useful as a student deeply interested in the field of economics and finance. I had always been curious into how institutions such as HMRC are able to detect fraud and prosecute those who avoid tax or commit fraudulent actions. Through conducting this exploration, I have been able to gain a greater understanding of mathematics while also being able to explore this economic aspect of fraud detection. Overall, I now have a greater understanding of how mathematics can connect with other fields, even literature which the average person might say is the ‘furthest you can get from mathematics’ is seen to have a mathematical distribution through Ziph’s Law. This exploration continues to demonstrate how mathematics is rooted in every part of life even if we cannot notice it at first.
Bibliography

http://assets.press.princeton.edu/chapters/s10527.pdf
ttps://www.isaca.org/Journal/archives/2011/Volume-3/Pages/Understanding-and-Applying-Benfords-Law.aspx?utm_referrer=
“Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space”
Fannon, P. (2012). Mathematics for the IB Diploma Standard Level. Cambridge: Cambridge University Press. p155
https://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?referer=&httpsredir=1&article=1041&context=rgp_rsr

Benford’s law and the Iranian election


https://aclweb.org/anthology/W98-1218

Appendices
Table of financial investment:

Year

Balance

2019

$2,057.18

2020

$2,201.19

2021

$2,355.27

2022

$2,520.14

2023

$2,696.55

2024

$2,885.31

2025

$3,087.28

2026

$3,303.39

2027

$3,534.63

2028

$3,782.05

2029

$4,046.79

2030

$4,330.07

2031

$4,633.17

2032

$4,957.50

2033

$5,304.52

2034

$5,675.84

2035

$6,073.15

2036

$6,498.27

2037

$6,953.14

2038

$7,439.86

2039

$7,960.65

2040

$8,517.90

2041

$9,114.15

2042

$9,752.14

2043

$10,434.79

2044

$11,165.23

2045

$11,946.80

2046

$12,783.07

2047

$13,677.89

2048

$14,635.34

2049

$15,659.81

2050

$16,756.00

2051

$17,928.92

2052

$19,183.94

2053

$20,526.82

2054

$21,963.70

2055

$23,501.16

2056

$25,146.24

2057

$26,906.47

2058

$28,789.93

2059

$30,805.22

2060

$32,961.59

2061

$35,268.90

2062

$37,737.72

2063

$40,379.36

2064

$43,205.92

2065

$46,230.33

2066

$49,466.45

2067

$52,929.11

2068

$56,634.14

2069

$60,598.53

2070

$64,840.43

2071

$69,379.26

2072

$74,235.81

2073

$79,432.32

2074

$84,992.58

2075

$90,942.06

2076

$97,308.00

2077

$104,119.56

2078

$111,407.93

2079

$115,892.85