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Report on the Optimization of thickness of an airfoil by computational simulation

Category: Engineering Paper Type: Assignment Writing Reference: APA Words: 5300

Table of Contents

1.     Introduction. 4

2.     Literature Review.. 7

Airfoil 7

CFD.. 8

Flow around airfoil 8

Lift and drag optimization. 10

Lift/Drag. 11

Jet Aircraft/Propeller Aircraft 11

Slow-speed flight 11

Maneuvering Performance. 12

Lateral stability and control 12

3.     Computational Modeling. 13

Methodology. 13

4.     Results. 16

5.     Discussion. 18

Influence of Angle of Attack Optimization. 19

Influence of Camber Optimization. 20

6.     Conclusion. 20

7.     References. 21

Introduction of Optimization of thickness of an airfoil by Computational simulation

In a world where we seem to have little to no time, we are always seeking out opportunities to increase efficiency. A main concern which consumes a lot of our time is travelling. Flight is the most efficient mode of transport for covering long distances in a short amount of time. Aeroplanes are the most common mode of flight transportation. Despite this, the engineering behind the plane is commonly overlooked, from the aerodynamics of the flight to the effect of the tail on the direction of the plane. The question is then raised, is there small changes in these vast engineering marvels that could make our flight more efficient.

The inspiration behind this report stems from the great works of Abbas ibn Firnas, a scholar from the Andalusia period. His scientific studies helped revolutionise the world, from designing a ‘mechanised planetarium’ to a ‘water clock’. His work on clear glass contributed massively towards the development of lenses for correcting eyesight. Despite all these accomplishments, what inspired me most is his work on aviation. Attaching a flying contraption made from bamboo to himself, he jumped off a cliff and stayed aloft for 10 minutes before crash landing. Thus, as stated by Lynn Townsend White Jr., Firnas (Wikipdia, 2010), is best known as being the first to demonstrate flight successfully. Having learnt from his mistakes, Firnas wrote a book highlighting the significance of having an appropriate landing mechanism for stability. This inspired many to develop flight.

This report, will look at how a change of thickness of an aerofoil can affect the flow around the aerofoil, it will specifically look at how the lift and drag is affected under incompressible flow. Two 4 digit NACA aerofoils will be chosen both with the same first 2 digits but different last 2 digits as this is what determines the thickness of the aerofoil. Using computational fluid dynamics to run these results, a sound conclusion will be determined.

An aerofoil is a shape mainly used for wings of an airplane, but can also be used for their tails or even on formula 1 cars. The aerofoil shape is considered the optimum shape for aerodynamic travel and its main purpose is to produce lift and drag. The official definition of an aerofoil according to (Airfoil, 2020) is “a body (such as an airplane wing or propeller blade) designed to provide a desired reaction force when in motion relative to the surrounding air”. Figure 1 shows an aerofoil and what makes an aerofoil.

 


Aerofoil’s were not always the optimum shape they are now, earlier aerofoils had a much shorter maximum thickness in when they were first used for flight, as can be seen in figure 2. From 1919, the upper surface curved out more, (Boyne, 2000) claims that during this period of development the greatest height achieved was in the first third of the chord. As time went on both the lower surface was starting to curve and then greatest thickness moved more towards the centre of the aerofoil. After this, the next problem was traveling at supersonic speed, in 1949 “The Bell X-1 had broken the so-called "sound barrier," and both the Air Force and the Navy were looking for next generation aircraft that could operate at supersonic speeds”, (Wallace, 2001), in the 1950’s Richard Whitcomb came up with the area rule, the purpose of the area rule also known as the transonic area rule is “a design technique used to reduce an aircraft’s drag at transonic and supersonic speeds” (Wikipedia, Area Rule, 2020).

In this report, I will be mainly concerned with the NACA (National Advisory Committee for Aeronautics) airfoils. These are specific airfoil shapes created for aircraft wings. Each wing is different, and the shape can be identified by the set of numbers after NACA. These numbers when inserted into an equation determine the cross section of a specific wing. There are two main series of digits that follow “NACA”, 4-digit series and 5-digit series. However, for this report I will be comparing only the 4-digit series airfoils Two 4 digit NACA aerofoils will be chosen both with the same first 2 digits but different last 2 digits as this is what determines the thickness of the aerofoil. Using computational fluid dynamics to run these results, a sound conclusion will be determined.

The 4 digits define the profile by the following:

1st digit – describes max camber as a percentage of the chord length

2nd digit – position of the maximum camber from the leading edge as percentage of the chord length divided by 10,

 3rd and 4th digit – the maximum thickness as a percentage.

As an example, I will use to explain this is NACA 6412:

The 6 tells us that the camber is 6% (0.06) of the chord length.

The 4 tells us that the maximum camber is 40% (0.4) of the chord length.

The 12 tells us that thickness is 12% (0.12) of the chord.

Literature Review of Optimization of thickness of an airfoil by Computational simulation
Airfoil

According to the author Sullivan  (2010), the aerodynamics as well as the Airfoils is the unique characteristics of the Swept Wing aerodynamic; in which airfoil is a cross section of a body which is placed in the airstream in order to create the useful aerodynamic force in an efficient manner. Examples of the airfoils are a Windmill blades, propeller blades as well as cross section of wings, hydrofoils. The subsonic flow concepts is demonstrated the speed 50 m/s of the wind tunnel. And the lift along with the drag coefficients for the symmetric airfoils is attained through an analysis of the measured pressures which distributes a various pressure taps in the surface of the airfoils (Koziel et al , 2011).

Then the test of the wind tunnel is conducted on the NACA 0012, NACA 0018, and airfoils. And the drag along with the lift forces is acting on the every of the airfoils which is successfully measured the velocity of the airfoils along with the attacks of the airfoils.

The comparison of the optimization of airfoil among the airfoils that are generated along with the expected range of the drag as well as lift forces. More lift is expected in the NACA 0012 airfoils where the correct theory is lifted to identify as well as visually confirmed (Sullivan, 2010).

CFD

According to the author Anitha et al (2018), it is conducted that by the application of CFD( computational fluid dynamic) at a time, then an absence which is reliable for the flow solver and that could also accurately predict the difference viscous effect, turbulence and it often seemed the practicality and reduced the empirical application for the optimized design of solutions. Thus the practical application for their solution is also limited through the unreliability of the CFD tools, and the studies which are reliable for flow solver also have a big issue in the airfoil designs. By the availability of the cutting edge design tools in the modern era, there are various errors among the simulated as well as empirical results which become negligible. Thus the use of the reliable flow of the slover like the XFOIL fro the single element of airfoils (Anitha et al , 2018).

According to author Günel et al  (2016), it is conducted that Analysis of CFD for performance of aerodynamics of airfoil were completed by utilizing ANSY-FLUENT. A FLUENT code explains the RANS conditions utilizing limited volume discretization. Consistent state solver, SIMPLE weight based solver as well as Green-Gauss cell based discretization were utilized in this analysis. Likewise, second system was utilized for the force and disturbance conditions discretization. While applying a CFD investigation to airfoil at low Reynolds numbers, it is hard to measure limit layer components with regular choppiness techniques. Therefore, more error has been gotten in estimation of drag power. To acquire progressively right expectation of drag power, change choppiness models are increasingly appropriate. By SST k-w progress model of turbulence, the outcomes were obtained. The assembly of the numerical arrangement was constrained by observing numerical mistake of a arrangement. O-ring type area structure was picked. An outer area is a circle which has a distance across of 25 m. It was characterized as a limit state of "Speed Inlet". Airfoil bases as well as top surfaces were characterized as "Divider" limit conditions. A space which is characterized as air which has a thickness (ρ) of 1.225 kg/m3 and dynamic consistency (μ) of 1.7894e-05 Pa s (Günel et al , 2016).

Flow around airfoil of Optimization of thickness of an airfoil by Computational simulation

The characteristic of the aerodynamic of the NACA0012 by the wing geometry at a very low Reynolds number along with the angle of attacks which is also investigated by using the numerical simulations along with the results that are validated through the experiments (Sereez et al , 2016). A lift coefficient is increased by the angle of attack where its units reach the maximum that is the stall angle. In the angel attack the further increments and the decrement of the lift coefficients until it reaches the minimum values. The fluid domain that is generated around the rectangular   by the wing geometry with respect to ratio, as shown in the below figure 1. Whereas in the below figure the C-type grid is used to generate a mesh around boundaries of the domain, and it is located the 20 chords away from the wing geometry that allows the developments of flow around a wing (Eftekhari & et al, 2019).


Figure 1: Fluid domain around the wing

The characteristics of the lift of the NACA0012 in the subsonic tunnel. Various velocities for the air which is observed in the different angles of attack by the two-degree increments. The calculate the lift coefficients which is plotted with each other along with the angel attacks and then compared the results.  In the airplanes the primary features are Airfoil, and it is also providing the lift. Then the airflow disturbances, which are caused through the moving objects in the results of fluid and the shear stress, which is acting on the said objects and the pressure distribution in this object ( Rubel & et al, 2016).      

In the flowing fluid, the body is immersed for both viscous forces and pressure. The forces sum which also acts the normal to a free stream in the direction lift as well as the sum is also acted the free stream in the direction of drag. The dynamic and geometric characteristic of the airfoil which is shown in the below figure;


Figure: dynamic and geometric characteristic of the airfoil

Lift and drag optimization of Optimization of thickness of an airfoil by Computational simulation

According to the author Ai & et al (2016), it is conducted that there are different methods to calculate the lift, whereas, in this research of the lift forces L, the airfoil would be calculated by the integrations of the measured pressure distribution over an airfoil surface. The projection of an airfoil and pressure distribution of airfoil is normally shown in the below figure;


Figure: Pressure distribution on an airfoil

Lift/Drag of Optimization of thickness of an airfoil by Computational simulation

Resultant of the aerodynamics is a sum of the forces which are acting on an airfoil which is placed in the flow. There are two components and the two forces where the Resultant of aerodynamics is broken into it is the drag as well as lift.

Lift is perpendicular to a relative of a wind that opposes the weight of the aircraft.

To a relative wind, Drag is the parallel that opposes a forward movement of an aircraft.

In the above of two forces, the action of one of two forces is to modify the aerodynamics resultant which consequently affects the other forces (Ai & et al, 2016)..

Then the relationship of the drag and a lift is shown below;

Jet Aircraft/Propeller Aircraft of Optimization of thickness of an airfoil by Computational simulation

In the Jet Aircraft, it is the type of aircraft that produced the forward forces on an account that have high speed and exhaust the gases which are released from the engine. The Jet aircraft is fixed with the wing aircraft, as well as with the propelled by the engine. At the lower speed along with the attitudes, the maximum efficiency of the engine is achieved in the propeller aircraft, whereas the aircraft engine is automatically achieved the maximum efficiency. Jet aircraft which is generally the cruise for the faster speed around about it has the 0.8 and the 981km of the altitudes by the aircrafts (Mukesh, 2014)

And in the Propeller aircraft, it is the oldest type of the aircraft which driven the piston of the engine along with the Turbo of the engine and has the various difference of the construction features. The propeller aircraft which is the aircraft that use the electrohydrodynamics that provides the thrust or the lift without requiring the moving parts or the combustion of the engine.

Slow-speed flight

In most of the aircraft's, the airplanes maintain their speed, which does not excess of the 1.3 times of the VSO of the instrument approach. Aircraft are normally used the slowed which is the landing speed, and it is the final approach, for the prior landing. As by the information of the swept wings aerodynamics, the power as well as pitch coordination which is needed due to the stability of the speed and it the comparatively neutral where speed tends to remain a new value which is not returning for an original speed.

The precise airspeed which is control, by the pilot and it’s normally changed the configuration of the aircraft through the extending of the landing flaps. And the variation of the configurations which means that the pilot and it must be alert to unwanted the changes of pitch at the low attitude (Dole, 2016).

Maneuvering Performance

Relationship and its effect on aircraft design then the performance of the Maneuverability/controllability is the aircraft quality which refers the maneuvers easily as well as withstand the different stresses which is imposed through the maneuvers. Performance of the Maneuverability/controllability which is governed through the weight of the aircrafts , along with the size , inertia as well as the location of the controls of the flights with the strength of the structural plus the power plant. Performance of the Maneuverability/controllability is includes in the design characteristics of the aircrafts.

Performance of the controllability the aircraft capability which is responded to the control of the pilot, especially by the regard of the flight path as well as the by the attitude. Quality of the aircraft which is response the performance of the controllability for the application of the pilot control when the maneuvering of the aircraft is regardless controls the characteristics of the stability (pilotsofamerica.com, 2018)

Latera          l stability and control of Optimization of thickness of an airfoil by Computational simulation

In a Longitudinal stability, which is the pitch of the stability, and its tendency of an aircraft reduces the pitching along with the return for the level of the positions, where unless it is countered through the elevators.

The roll of the stability is the Lateral stability, which tendency is the to reduce the aircraft for the return as well as for the rolling to the upright positions unless which is continually maintained due to the positions of the ailerons (Lu, 2006).

Computational Modeling of Optimization of thickness of an airfoil by Computational simulation
Methodology of Optimization of thickness of an airfoil by Computational simulation

The method which is used in this research study is the SST-k turbulence model which is the two equation eddy viscosity model and it became very popular. A shear stress which is also transports the SST formulation and combines by two worlds. The use of k formulation for the inner parts of the boundary layer also creates the model directly which is usable for all methods down and by the viscous sub layer, thus the SST- kmodel could also be used by the Low-Turbulence model (Cfd-online, 2019).

The kinematics Eddy viscosity is given below;


The segregated implicit solver, ANSYS fluent which is used to simulate the problem. An airfoil profile which is simulated the design modeler as well as boundary condition where the meshes are created. By the help with a commercial CFD software ANSYS fluent, two dimensional airfoil of aerodynamics performances is simulated numerically.

In this research we supposed the flow around the fully parameterized NACA airfoil at the various attack angles 4, 6 and 8 as well as also have the wide thickness of the 0.08 to 0.2 like the percent of chord. The optimization solver evaluates the optimal geometry which order to maximize the lift to drag ratio (ZHANG et al , 2016).  

The SS-k model is similar; 

If airfoil is positioned at a dramatically increased angle of attack, the separation can occur at a point of maximum thickness of airfoil as well as tremendous wake will grow behind a point of separation. Due to this phenomenon a list is significantly reduced and there can be condition of aerodynamic stall. In this condition, the aircraft can further lose speed due the very high pressure drag caused the wake. The wall of the boundary layers is approximately thick, where the test section is the turbulence, and it is also measured less than the 0.12% over the tunnel on the operating range. The downstream of a transition in the boundary layer development as well as Reynolds’s numbers which also capture the local effect of turbulence intensity (Sreejith & et al, 2018) . The following governing equations;

Results

 

 

 

NACA 0012

NACA 0018

AOA

CL

CD

CL

CD

4

0.010225797

0.43895841

0.011555257

0.43194301

6

0.010871912

0.64242083

0.012891825

0.64644119

8

0.012786779

0.84848094

0.014747308

0.85728051

 

Discussion of Optimization of thickness of an airfoil by Computational simulation

This study was conducted by using the ANSYS fluent simulation software for a NACA airfoil. The airfoil which is used is the asymmetrical structure as well as the airfoil is changing the thickness along with also operating the Reynolds number. The drag as well as lift coefficient is determined by using a same methods as well as graphs which is generated for the drag and lift coefficient , and lift to drag ratio , pressure as well as distribution of velocity over the airfoil surface. At the start the numerical analysis for the initial airfoil NACA 0018, and NACA 0012 is performed plus it also optimized the airfoil which is compared by the original NACA 0012 (Muftah, 2019). The effect of the roughness is maximum on lift and drag which is increases by increasing the airfoil thickness as well as also decrease slightly by increasing the maximum lift. To separate an effects of airfoil thickness as well as maximum lift coefficient, two of the airfoils (NACA 0012 and NACA 0018) have a common maximum lift coefficient and two (NACA 0012 and NACA 0018) have a common airfoil thickness (Somers , 2005).

In the above analysis experiment examine the errors in the experimental investigation, and it is also discussed some basic theory which is necessary for the understanding of the aerodynamic. Among the Sources of inaccuracy, the care is also taken as distinguish, which is also emphasized by the physical understanding rather than the analytical complexity.

By the errors the experimental measurements which inevitably influenced, practical limitation of the types of equipment like the minimum scale of the pressure gauge ( Higazy, 2019).   The accuracy of the experiment is mostly right; all the result is according to the theoretical calculation.

The drag equation is used by this where the lower of the drag coefficient is also presented the object which has the less aerodynamics and the hydrodynamics drag. Then the drag coefficient is always connected by the particular surface of area.

Now a drag on the cylinder is not zero, but it could also estimate for the measured pressure distribution as in the below; Now suppose the cylinder element of surface of cylinder where the length is  . Now a force per the unit span on a different element is due to the normal pressure; 

Influence of Angle of Attack Optimization

Ratio of drag to lift of NACA was determined. As indicated by results, Ratio of drag to lift increment with an expanding working Reynolds number. Lift coefficient addition is relative to the Reynolds number simultaneously as well as during process of optimization the most angle of attack is related with low Reynolds number yet it isn't after 10 degree. A performance gets most extreme at angle of attack is optimized and changing from 4.65 to 5.85 degree by with expanding Reynolds number. It very well may be seen that all airfoils with various Reynolds number requires a positive approach (Srinath et al , 2010). Be that as it may, with expanding Reynolds, the expansion in the enhanced lift coefficient and lift to drag proportion turns out to be moderate when it is contrasted with starting airfoils, and the drag coefficient increment simultaneously, and the limit of drag coefficients are expanded in low rate with Re = 103 to 105

 Influence of Camber Optimization

Tool for the optimization of the airfoil likewise, in contrasted as well as a past case, it should have an option to change a state of the airfoil, it must have an option to straightforwardly to alter its outside shape by change greatest thickness of airfoil to acquire an optimum  camber as well as its situation along a chord line. The research objective is to optimization of the airfoil that will be utilized. Meanwhile a most extreme camber  are mutually dependent as well as both have been improved simultaneously. As the state of the airfoil changes, a stream around it additionally changes, this prompts a rotated distribution of pressure which thus alters an optimal design property of a model. Simply, a constrain circulation are delicate to changes of geometry. As a camber expands an inclination to upper surface limit layer division become progressively noteworthy. It very well may be seen likewise as an ideal camber expanded, an airfoil takes an asymmetry structure as well as is normally used to control its zero - lift approach. In light of CFD optimization as well as modeling technique, an airfoil was enhanced legitimately with a greatest lift to drag proportion as a goal (Bu et al , 2013).

Conclusion of Optimization of thickness of an airfoil by Computational simulation

Summing up all the discussion, the report is about the optimization of airfoil of the Aerodynamics, which have the different unique characteristics. Aerodynamics of optimization of airfoil introduced the various concepts which are used in the aircraft, as it mentioned in the above discussion. For a specific aspect ratio, an optimization of airfoil structural span is lengthened by it which means that it requires a heavier structure. Aerodynamics of optimization of airfoil specifications is explained along with their role in the aircraft. The weight of optimization of airfoil is raised even more for resisting the twist of aerodynamic which is incurred during the process of bending. Finally, the numerical method is applied to calculate aerodynamic characteristics of airfoils of different relative thicknesses, to investigate the effects of relative thickness on airfoil performance. Increasing the thickness will increase the lift. Increasing the area will increase the lift.

References of Optimization of thickness of an airfoil by Computational simulation

Higazy, M. (2019). Analysis and Accuracy of Experimental Methods. InternationalJournal of Petrochemistry and Research, 3(1).

Rubel, R. I., & et al. (2016). Numerical and Experimental Investigation of Aerodynamics Characteristics of NACA 0015 Airfoil. INTERNATIONAL JOURNAL of ENGINEERING TECHNOLOGIES, 2(4).

Afs.enea.it. (2019). 4.5.2 Shear-Stress Transport (SST) $k$- $\omega$ Model. Retrieved from https://www.afs.enea.it/project/neptunius/docs/fluent/html/th/node67.htm

Ai, Q., & et al. (2016). Experimental investigation of aerodynamic performance of airfoils fitted with morphing trailing edges. Conference: 54th AIAA Aerospace Sciences Meeting, 2-12.

Airfoil. (2020, May 02). Airfoil. Retrieved from Merriem-webster: https://www.merriam-webster.com/dictionary/airfoil

Alper, A., McCall, A., Leyne , K., Fulton, N., & Laird, W. (2012, December 02). Hydrofoil. Retrieved from University of Strathclyde: http://www.esru.strath.ac.uk/EandE/Web_sites/11-12/MORE/hydrofoil/introduction.html

Anitha et al , D. (2018). Air foil Shape Optimization Using Cfd And Parametrization Methods. Materials Today Proceedings, 5(2), 5364–5373.

Boyne, J. E. (2000, January 12). Airplane. Retrieved from Brittanica: https://www.britannica.com/technology/airplane#ref64146

Bu et al , Y. (2013). Aerodynamic optimization design of airfoil based on CST parameterization method. J. Northw. Polytech. Univ, 829–835.

Century of flight. (2005, November 14). Develpment of aviation technology. Retrieved from Century of flight: http://www.century-of-flight.freeola.com/Aviation%20history/evolution%20of%20technology/Airfoils.htm

Cfd-online. (2019). SST k-omega model. Retrieved from https://www.cfd-online.com/Wiki/SST_k-omega_model

DeSena, G. (2012). Pressure Distribution on an Airfoil. United States Naval Academy.

Devenport, W., & et.al. (2019, janurary 29). Experiment 3 - FLOW PAST A CIRCULAR CYLINDER. Retrieved from http://www.dept.aoe.vt.edu/~aborgolt/aoe3054/manual/expt3/index.html

Dole, C. (2016). Flight Theory and Aerodynamics: A Practical Guide for Operational Safety. John Wiley & Sons, .

Eftekhari, S., & et al. (2019). Investigation of a NACA0012 Finite Wing Aerodynamics at Low Reynold’s Numbers and 0° to 90° Angle of Attack. 11.

Günel et al , O. (2016). COMPARISON OF CFD AND XFOIL AIRFOIL ANALYSES FOR LOW REYNOLDS NUMBER. International Journal of Energy Applications and Technologie, 3(2), 83 – 86.

Guo,, D. (2019). Cross Cylindrical Flow: Measuring Pressure Distribution and Estimating Drag Coefficients. Retrieved from https://www.jove.com/science-education/10451/cross-cylindrical-flow-measuring-pressure-distribution-estimating

Koziel et al , S. (2011). Computational Optimization, Methods and Algorithms. Springer,.

Lu, J. (2006). Patent No. US8712639B2. US.

metu, c., & et al. (n.d.). EXPERIMENT 6 CHARACTERISTICS OF AN AIRFOIL. Retrieved from http://courses.me.metu.edu.tr/courses/me410/exp6/exp6.pdf

Muftah, A. (2019). CFD Simulation and Optimization of 4-Digit NACA Airfoils. CFD Modeling in Measurement Systems.

Mukesh, R. (2014). Airfoil shape optimization using non-traditional optimization technique and its validation. Journal of King Saud University - Engineering Sciences, 191-197.

pilotsofamerica.com. (2018, November 2). Controllability vs Maneuverability? Retrieved from https://www.pilotsofamerica.com/community/threads/controllability-vs-maneuverability.114767/

Sereez et al , M. (2016). Computational Simulation of Airfoils Stall Aerodynamics at Low Reynolds Numbers. Applied Aerodynamics.

Somers , D. (2005). Effects of Airfoil Thickness and Maximum Lift Coefficient on Roughness Sensitivity. National Renewable Energy Laboratory.

Sreejith, B., & et al. (2018). Numerical study on effect of boundary layer trips on aerodynamic performance of E216 airfoil. Engineering Science and Technology, an International Journal, 21(2), 77-88.

Srinath et al . (2010). Optimal aerodynamic design of airfoils in unsteady viscous flows. Computer Methods in Applied Mechanics and Engineering, 1976-1991.

Sullivan, A. (2010, May 6). Aerodynamic forces acting on an airfoil.

Wallace, L. E. (2001). The Whitcomb Area Rule: NACA Aerodynamics Research and Innovation. In P. E. Mack, FROM ENGINEERING SCIENCE TO BIG SCIENCE (p. 396). Washington: John Henry.

Webster, M. (2020, May 02). airfoil. Retrieved from Merriem-Webster.com: https://www.merriam-webster.com/dictionary/airfoil

Wikipdia. (2010, May 1). Abbas Ibn Firnas. Retrieved from Enacademic: https://enacademic.com/dic.nsf/enwiki/768576

Wikipedia. (2010, May 21). Abbas Ibn Firnas. Retrieved from enacademic: https://enacademic.com/dic.nsf/enwiki/768576

Wikipedia. (2020, April 17). Area Rule. Retrieved from Wikipedia : https://en.wikipedia.org/wiki/Area_rule

ZHANG et al , T. t. (2016). A study of airfoil parameterization, modeling, and optimization based on the computational fluid dynamics method. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 17(8), 632-645.

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