Gas Project

8/12/2019 Gas Project

1/19

UNIVERSITY AT BUFFALO

Supersonic Airfoil Design

Gas Dynamics and Compressible Flow

Casey R. Robertson

4/19/2010

A Brief Study on the effects of geometry on supersonic airfoil design with respect to lift, drag, angle of

attack, and fluid viscosity.


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Contents1. Problem Statement .............................................................................................................................. 3

1.1 Objectives............................................................................................................................................ 3

2. Methods of Solution ............................................................................................................................. 3

2.1 Design Considerations......................................................................................................................... 3

2.2 Oblique Shock Relations (--M) ........................................................................................................ 4

2.3 Prandtl-Meyer Expansion Waves ........................................................................................................ 6

2.4 Exact Beta Value Calculations ............................................................................................................. 6

3. Discussion of Results ............................................................................................................................ 7

3.1 Proposed Airfoil Design Selection ....................................................................................................... 7

3.2 Calculation of Oblique Shock Properties............................................................................................. 8

3.3 Expansion Wave Calculations ............................................................................................................. 9

3.4 Causes of Drag Force at Supersonic Speed ....................................................................................... 10

3.5 Calculation of Lift and Drag Coefficients ........................................................................................... 11

3.6 Verifying Results with Simulation ..................................................................................................... 13

3.7 Effects of Viscosity on Airfoil Design ................................................................................................. 15

4. Conclusion .......................................................................................................................................... 15

5. References .......................................................................................................................................... 16

6. Appendix ............................................................................................................................................. 16


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1. Problem StatementAirfoil design must incorporate the calculations of both lift and drag forces. Subsonic airfoil profiles

will be different than those of both supersonic and transonic flight. The conventional wing teardrop

shape used in subsonic flight is not suitable for the supersonic. The objective is to design an airfoil

which will support favorable conditions in supersonic flight while illustrating the relationships of thelift and drag coefficients on angle of attack and Mach number.

1.1 Objectives

Indicate airfoil shape with proper justification for selection. Describe physical mechanisms responsible for producing drag force at supersonic speeds. Neglecting viscosity, calculate the lift coefficient (Cl) and drag coefficient (Cd) at an angle of

attack (=5o) and a mach number (M=3).

Calculate and plot the lift coefficient (Cl) and drag coefficient (Cd) verses the angle of attack . Discuss how the effects of viscosity will change the design of the airfoil.

2. Methods of Solution2.1 Design Considerations

Before analyzing supersonic characteristics a suitable airfoil selection must be made. For the sake of

comparison 3 different types of design profiles will be examined. By comparing the differences of

intent in each design, a final selection will be made. The different styles of airfoils in question are

displayed below.

Figure 1: Shows 6 different types of basic airfoils for comparison.

Option 3

Option 2

Option 1


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Evaluation will begin with Option 1 or the later airfoilas depicted in Figure 1. This particular design

has a concave lower surface to optimize lift. Assuming this wing is of fixed design too much speed

will be sacrificed by way of lift to suffice for supersonic design purposes. Recent advancements in

aerospace engineering have made this wing type viable for high speed flight with the addition of

leading and trailing edge flaps, but that is beyond the scope of these simple analyses.

The second candidate for consideration or Option 2 is labeled the laminar flow airfoil. This option

is designed with a streamlined body for minimum drag due to the boundary layer of air being

uninterrupted. The radial or rounded leading edge of this design does not lend itself to supersonic

flight well. This radial leading edge will allow a detached bow shock to form ahead of the airfoil. One

of the main purposes for this design is to reduce flow separation over a wide range of operating

parameters.

Option 3, the double wedge airfoil, will serve supersonic needs better than the other options. The

sharp angular leading edge will help to prevent the formation of a detached bow shock upstream of

the airfoil during supersonic flow. This will decrease wave drag during supersonic flight. The sharp

leading edge will however raise issues to be addressed later such as high susceptibility to angle of

attack because of the dependency on flow separation.

2.2 Oblique Shock Relations (--M)

When calculating the properties due to oblique shocks some geometrical relationships must be

made. Mach number and velocity can be broken down into components as follows.

Figure 2: Shows the relationship of oblique shock components.

When crossing the oblique shock the tangential components of velocity, (and), are the sameon either side of the wave. Also, the changes across this shock wave can be found with the normal

vector components, (and ).


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Using the geometrical relations shown in Fig.2 and the assumption of a calorically perfect gas some

relationships can be derived as follows.

( )

These established relationships are also shown in the commonly used diagram below.

Figure 3: Shows the results of the (--M) relationships for oblique shocks.


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2.3 Prandtl-Meyer Expansion Waves

Because in some instances shock waves are turned away from themselves expansion wave theory is

necessary to account for this. So in other words these waves will be the opposite of shock waves. A

centered expansion fan can account for this in an isentropic and continuous fashion. The Prandtl-

Meyer function is listed here in separate terms to correlate exactly with the written Excel function

to predict this expanding behavior.

Where:

2.4 Exact Beta Value Calculations

Rather than read values from the Compressible Flow text, the actual value of beta can be calculated

as a function of the Mach number and theta. The base equation has 3 real roots, but one is negative

and therefore nonphysical. The 2 positive roots correspond to both weak and strong shocks where

=1, or =0 respectively. These calculations are described by the following equations.


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3. Discussion of Results3.1 Proposed Airfoil Design Selection

Figure 4: Shows a diagram of the proposed airfoil.

As discussed earlier the proposed airfoil will be of diamond shape, symmetrical about all axes, but

noting that there is a positive angle of attack (AOA) of 5 degrees. All angle, pressure, and Mach

subscripts will be labeled with respect to the particular region for which they are occurring in.

Calculations begin in the upstream region or region 1 at the leading edge of the airfoil. From region

1, an oblique shock wave is formed which must be crossed to proceed with calculations. These

oblique shock wave calculations are valid for transitions into both regions 2 and 4 on the top and

bottom of the foil respectively. Upon calculating the properties for oblique regions 2 and 4, the

properties for regions 3 and 5 are computed. This is done with Prandtl-Meyer expansion waves.

From the property values for regions 1-5, the drag and lift coefficients were computed.

5o

1

54

2

3

20o

Airfoil


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3.2 Calculation of Oblique Shock Properties

All calculations are completed using Excel whenever possible, but Beta values were initially read

from the chart of oblique shock properties in the Compressible Flow text due to the complexity of

the iteration process to find the actual values. These initial Beta values and the flow calculations

made with them will later be compared to actual Beta value computations made using a complex

Spreadsheet formula.

Region 1 to 2 Region 1 to 4

= 1.4 1.4

P1= 101.3 101.3

M1= 3 3

Half Foil Angle= 10 10

Angle of Attack= 5 5

Theta= 5 0.087266 15 0.261799

Beta= 23 0.401426 32.3 0.563741

Deg. Radians Deg. Radians

Figure 5: Shows the upper and lower leading edge oblique shock information.

Figure 5 shows the given properties at the leading edge of the airfoil. Both regions 2 and 4 share the

same upstream flow coming from region 1.

Region 1 to 2 Region 1 to 4

Mn1to 2= 1.172193 Mn1to 4= 1.603057

P2/P1= 1.436377 P4/P1= 2.831424

P2= 145.505 P4= 286.8232

Mn2= 0.859998 Mn4= 0.667519

M2= 2.783012 M4= 2.244707

Figure 6: Shows the upper and lower leading edge oblique shock calculations for regional transitions 1-2

and 1-4(all pressures in kPa.).

Figure 6 displays the results for the calculated values on the upper and lower surface of the airfoil

leading edge. All equations used to calculate the solutions (eq.1-6) are shown in section 2.2 as well

cited in the appendix. All calculated values correlate with those given in the shock tables of the

Compressible Flow text.


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3.3 Expansion Wave Calculations

To characterize the diamond shape of this airfoil expansion waves will occur at the peaks of the

airfoil where the shock waves must turn into themselves. Using the previously defined Prandtl-

Meyer equations (eq. 7-10) the streamline calculations can be continued using a continuous

centered expansion fan.

= 1.4

M2= 2.783012

3= 20

M4= 2.244707

5= 20

term 1 term 2 term 3 (radians) (deg)

(M2) 2.44949 0.814648 1.203254 0.792217 45.39071

(M4) 2.44949 0.687079 1.109072 0.573922 32.88329

Figure 7: Shows expansion wave values and calculations of the Prandtl-Meyer function.

In Figure 7 both M2 and M4 are the conditions upstream of the upper and lower expansion waves

respectively. These values were previously calculated using oblique shock wave relations. Both

theta values are obtained from geometry and the diagram. The Prandtl-Meyer functions were

calculated using the previously defined equations in section 2.3.

Using the previously calculated values of and knowing the downstream theta values from

geometry and angle of attack the following relationship can be used to find the downstream Mach

numbers in regions 3 and 5. Excel was used to interpolate values from the table in the Compressible

Flow text.

interpolation interpolation

M M

x= 52.88 y= 3.167222 x= 65.39 y= 3.970455x1= 52.57 y1= 3.15 x1= 65.12 y1= 3.95

x2= 53.47 y2= 3.2 x2= 65.78 y2= 4

Figure 8: Shows the interpolation process for finding Mach numbers 3 and 5.


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Now that the Mach numbers are known in regions 3 and 5, the pressures must be found to later be

used in calculations for the lift and drag coefficients.

To find the pressures in regions 3 and 5 some relationships will need to be used. For an isentropic

process the ratio of total to static pressures are given by:

Realizing that this process is again isentropic, therefore P0is constant and P3 for example can be

found with:

Regions 3 and 5 pressure calculations.

M2= 2.783 P02/P2= 26.44265378 P3= 26.35879981

M3= 3.9705 P03/P3= 145.9678667 P5= 69.82460708

M4= 2.2447 P04/P4= 11.46760783

M5= 3.1672 P05/P5= 47.10622543

P2= 145.51

P4= 286.82

Figure 9: Shows the pressure calculations for regions 3 and 5.

3.4 Causes of Drag Force at Supersonic Speed

Three of the leading causes of drag during supersonic flight are skin friction, drag due to lift, and

wave drag due to thickness or volume. These parameters can be accounted for in the following

equation:

The skin frictional component of drag is derived from the realization that there exists a viscous

boundary layer surrounding the airfoil. When considered at an infinitesimally small scale this

boundary layer at the outer foil wall will have a velocity of zero or a non-slip condition contributing

to frictional losses.


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Drag due to lift occurs whenever an airfoil encounters a moving fluid or redirects the air. This type of

induced drag will normally increase as the angle of attack increases.

Wave drag is related to the loss of total pressure and increase in entropy across shock waves and is

what will be considered in this analysis. This type of drag is dependent on the thickness or volume of

the foil or wing entering a supersonic flow. The body forces on the airfoil can be easily calculatedwith respect to foil geometry, angle of attack, and the upstream velocity vector.

3.5 Calculation of Lift and Drag Coefficients

Now that the static pressures are known in each region the lift and drag coefficients can be

calculated by taking a summation of forces in the x and y directions for the drag and lift respectively.

The drag and lift coefficient equations used will be as follows.

Where D and L are summations of the drag and lift on each side of the foil, represented by:

Notice also that the airfoil has been non-dimensionalized. The length of each side in question is a

function of the cord length. Therefore due to basic trigonometric relations within airfoil geometry:

Where C is the cord length and l is the length of each respective airfoil side.

Using Excel these equations can be easily handled and computed using equations 18-22 as follows in

Figure 10 below.

P2= 145.505 2= 5 M= 3

P3= 26.3588 3= 15 P= 101.3

P4= 286.823 4= 15 gamma= 1.4

P5= 69.8246 5= 5

D= 74.0091 L= 176.2

Cd= 0.05888 Cl= 0.1402

Figure 10: Shows the calculation of the drag and lift coefficients.


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Now that the lift and drag equations have been established some useful plots and relationships can

be developed-in this case plots of drag and lift verses angle of attack. This is a more difficult task

than performing only one calculation of drag and lift as in Fig.10, because of how certain properties

about the airfoil change continuously with respect to the angle of attack. As the angle of attack ()changes, both shock angles ( and ) change also. This completely changes values for computational

purposes. As the angles and change, the components of force in either the x or y directions on

that particular surface will also change- as well as Mach numbers, pressures, and values in the

Prandtl-Meyer equations. To cope with this continual updating or changes in property as the angle

of attack varies, an Excel function was written to handle changing values of using equations 11-13.

From these changing Beta values, all Mach components could be calculated and pressures

established using equation 3 for the oblique shock relationships. Using a similar approach the

expansion regions were also calculated to find pressure ratios as described by equations 15 and 16.

Figure 11: Shows the Excel plots of the drag and lift coefficients vs. the angle of attack.

In Fig.11, the continuous updating of Fig. 10 is plotted, otherwise known as lift and drag verses angle of

attack. Again using a different Excel function the results of Fig.10 can be verified on the plot at AOA=5 0.

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-25 -20 -15 -10 -5 0 5 10 15 20 25

Angle of Attack (degrees)

Lift and Drag Coefficients vs Angle of Attack

Drag Coefficient Lift Coefficient Lift to Drag Ratio*(0.1)


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3.6 Verifying Results with Simulation

For experimental purposes a virtual simulation was run to verify independent results with my

calculations via equations and written Excel functions. Figure 12 shows the proposed airfoil design

in flight with the same geometry, angle of attack, upstream Mach number, specific heat ratio, and

pressure.

Figure 12: Shows a simulation of the proposed airfoil performing in flight with all given parameters.

Simulator interface courtesy of www.hasdeu.bz.edu.

Mach Number Comparisons

M1 M2 M3 M4 M5

Calculated 3 2.783 3.9705 2.2447 3.1672

Simulation 3 2.7497 3.9182 2.2549 3.1818

% Difference 0 1.211 1.3348 0.4523 0.4589

Figure 13: Shows the close relationship and small error between calculated values (Fig.9) and

performed simulation (Figs.12 and 14).


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Figure 14: Shows graphical results of drag coefficient, lift coefficient, and L/D ratio via a simulation of the

proposed airfoil performing in flight with all given parameters. Simulator interface courtesy of

www.hasdeu.bz.edu.

Pressure Comparisons

P1 P2 P3 P4 P5

Calculated 101.3 145.51 26.36 286.82 69.82

Simulation 101.3 147.29 27.2 285.82 69.2

% Difference 0 1.2085 3.0882 0.3499 0.896

Figure 15: Shows the close relationship and small error between calculated values (Fig.10) and

performed simulation (Figs.12 and 14).


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3.7 Effects of Viscosity on Airfoil Design

One large assumption was made when designing and computing values for this airfoil- inviscid flow.

This means that for the sake of calculation skin friction could be neglected as a factor in contributing

to the drag on the airfoil. For the optimized supersonic aircraft nearly 60% of its drag is skin friction

drag, a little over 20% is induced drag, and slightly under 20% is wave drag. This means that for a

realistic approach to the design of the airfoil fluid viscosity must be accounted for to more

accurately predict behavior. In a real design situation the viscosity cannot be neglected near the

airfoil surface where the boundary layer occurs. There are 2 types of boundary layers that will cause

drag on the surface of the wing; laminar and turbulent. If the critical Reynolds number is reached

then turbulent flow will be established often within a few percent of the chord length. When the

flow becomes turbulent eddies will form and drag is increased in the layer surrounding the airfoil in

question when the slower moving eddies essentially mix with the faster moving air surrounding the

wing surface.

4. ConclusionSupersonic airfoil geometry can be quite different from both subsonic and transonic. The leading

edge is often sharp or of angled geometry rather than a teardrop shape to help prevent a detached

bow shock wave upstream as well as wave drag.

Different procedures must be taken in order to properly analyze the flow surrounding an airfoil. For

the leading edge (both top and bottom) oblique shock relations can be made which describe the

flow characteristics. Preceding across the airfoil the waves will turn into themselves and the Prandtl-

Meyer expansion theory must be used to then describe the properties when following the

streamline.

In certain instances merely reading the values for Beta from a chart as in Fig.3, although proven to

suffice for one calculation with accuracy, is not the appropriate approach. When plotting or solving

for properties which depend on each other a specific relationship should be recognized and

corresponding function or program written to solve for these relationships.

Looking at the lift and drag plots we can see some interesting observations. At an angle of attack of

50, this airfoil profile is in the range of its lowest drag coefficient, although not quite optimized.

Looking at the L/D ratio we can see that the design is not receiving the optimum amount of lift for

the amount of drag, although again it is near the optimized peak region. This compromise may be

acceptable because the drag is going to be such a critical factor at supersonic speeds.

Although wave drag was considered for the sake of analysis this would not suffice for a realistic

analysis. Due to the fact that at supersonic speeds over half of the drag is caused by skin friction, an

inviscid flow would be an invalid assumption for design or analysis purposes. The boundary layer

near the airfoil surface must be considered along with turbulent effects due to the critical Reynolds

number transition from laminar to turbulent flow for more accurate analysis. Also while it may be a


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seemingly obvious point the angle of the airfoil itself should be decreased to imitate thin wing

design-especially for supersonic flight.

5. ReferencesHow Airplanes Work How stuff works

http://static.howstuffworks.com/gif/airplane-airfoil4.gif&imgrefurl

John D. Anderson. Modern Compressible Flow

New York, NY, McGrawHill Education, 2004

The Drag Coefficient- Nasa

http://www.grc.nasa.gov/WWW/K-12/airplane/dragco.html

Supersonic Wing!Hasdes.edu

http://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/shw.gif&imgrefurl

Supersonic Airfoils- Wikipedia

http://en.wikipedia.org/wiki/Supersonic_airfoils

Airfoil Design- Aerospaceweb

http://www.aerospaceweb.org/question/airfoils/q0035.shtml

Aeronautical Knowledge Handbook - Blogspot

http://4.bp.blogspot.com/_fX9doSZqagk/SsViwiJm1jI/AAAAAAAABSI/aabBlc1vI5Q/s320/Figure%2B3-

7%2BAirfoil%2Bdesign.jpg&imgrefurl

Airfoil - MIT

http://web.mit.edu/2.972/www/reports/airfoil/airfoil.html

6. AppendixMn1to 2= =C$5*SIN(D$9)

P2/P1= =1+((((2*C$3)/(C$3+1)))*(C$12^2-1))

P2= =C$13*C$4

Mn2= =SQRT(((2/(C$3-1))+(C$12^2))/(((((2*C$3)/(C$3-1))*(C$12^2))-1)))Example spreadsheet formulae for computing oblique shock properties.

term 1 term 2

(M2) =SQRT((D$3+1)/(D$3-1)) =ATAN( SQRT((D$4^2-1)*(D$3-1)/(D$3+1)))

(M4) =SQRT((D$3+1)/(D$3-1)) =ATAN( SQRT((D$6^2-1)*(D$3-1)/(D$3+1)))
http://static.howstuffworks.com/gif/airplane-airfoil4.gif&imgrefurlhttp://static.howstuffworks.com/gif/airplane-airfoil4.gif&imgrefurlhttp://www.grc.nasa.gov/WWW/K-12/airplane/dragco.htmlhttp://www.grc.nasa.gov/WWW/K-12/airplane/dragco.htmlhttp://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/shw.gif&imgrefurlhttp://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/shw.gif&imgrefurlhttp://en.wikipedia.org/wiki/Supersonic_airfoilshttp://en.wikipedia.org/wiki/Supersonic_airfoilshttp://www.aerospaceweb.org/question/airfoils/q0035.shtmlhttp://www.aerospaceweb.org/question/airfoils/q0035.shtmlhttp://4.bp.blogspot.com/_fX9doSZqagk/SsViwiJm1jI/AAAAAAAABSI/aabBlc1vI5Q/s320/Figure%2B3-7%2BAirfoil%2Bdesign.jpg&imgrefurlhttp://4.bp.blogspot.com/_fX9doSZqagk/SsViwiJm1jI/AAAAAAAABSI/aabBlc1vI5Q/s320/Figure%2B3-7%2BAirfoil%2Bdesign.jpg&imgrefurlhttp://4.bp.blogspot.com/_fX9doSZqagk/SsViwiJm1jI/AAAAAAAABSI/aabBlc1vI5Q/s320/Figure%2B3-7%2BAirfoil%2Bdesign.jpg&imgrefurlhttp://web.mit.edu/2.972/www/reports/airfoil/airfoil.htmlhttp://web.mit.edu/2.972/www/reports/airfoil/airfoil.htmlhttp://web.mit.edu/2.972/www/reports/airfoil/airfoil.htmlhttp://4.bp.blogspot.com/_fX9doSZqagk/SsViwiJm1jI/AAAAAAAABSI/aabBlc1vI5Q/s320/Figure%2B3-7%2BAirfoil%2Bdesign.jpg&imgrefurlhttp://4.bp.blogspot.com/_fX9doSZqagk/SsViwiJm1jI/AAAAAAAABSI/aabBlc1vI5Q/s320/Figure%2B3-7%2BAirfoil%2Bdesign.jpg&imgrefurlhttp://www.aerospaceweb.org/question/airfoils/q0035.shtmlhttp://en.wikipedia.org/wiki/Supersonic_airfoilshttp://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/shw.gif&imgrefurlhttp://www.grc.nasa.gov/WWW/K-12/airplane/dragco.htmlhttp://static.howstuffworks.com/gif/airplane-airfoil4.gif&imgrefurl


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term 3 (rad) (deg)

=ATAN(SQRT(A$4^2-1)) =#REF!*A11-B11 =C$10*180/PI()

=ATAN(SQRT(A$6^2-1)) =#REF!*A12-B12 =C$11*180/PI()

Example spreadsheet formulae for computing Prandtl-Meyer expansions.

interpolation M

x= 65.39 y= =K17+(I16-I17)*(K18-K17)/(I18-I17)

x1= 65.12 y1= 3.95

x2= 65.78 y2= 4

Example spreadsheet formulae for computing interpolations.

D=

=$N23*SIN($Q23*PI()/180)+$N25*SIN($Q25*PI()/180)-$N24*SIN($Q24*PI()/180)-

$N26*SIN($Q26*PI()/180)

Cd= =N28/(2*$T24*$T25*$T23^2*0.5*COS(10*PI()/180))

L=

=-$N23*COS($Q23*PI()/180)+$N25*COS($Q25*PI()/180)-

$N24*COS($Q24*PI()/180)+$N26*COS($Q26*PI()/180)

Cl= =T28/(2*$T24*$T25*$T23^2*0.5*COS(10*PI()/180))

Example spreadsheet formulae for computing initial drag and lift coefficients.

angle of attack AOA in rad theta(deg) theta1-2(rad)

-20 =B8*PI()/180 =E8*180/PI() =$E$4-C8

-19 =B9*PI()/180 =E9*180/PI() =$E$4-C9

-18 =B10*PI()/180 =E10*180/PI() =$E$4-C10

Example spreadsheet formulae for computing Beta values.

lambda

=(($C$3^2)-

1)^2

=((($C$2-

1)/2)*($C$3^2))+1

=(((($C$2+1)/2)*($C$3^2))+1)*(TAN(E

8))^2

=SQRT(F8-

3*G8*H8)=(($C$3^2)-

1)^2

=((($C$2-

1)/2)*($C$3^2))+1

=(((($C$2+1)/2)*($C$3^2))+1)*(TAN(E

9))^2

=SQRT(F9-

3*G9*H9)

=(($C$3^2)-

1)^2

=((($C$2-

1)/2)*($C$3^2))+1

=(((($C$2+1)/2)*($C$3^2))+1)*(TAN(E

10))^2

=SQRT(F10-

3*G10*H10)



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zi

=(($C$3^2

)-1)^3

=((($C$2-

1)/2)*($C$3^2))+1

=(((($C$2-

1)/2)*($C$3^2))+1)+((($C$2+1)/4)*$

C$3^4)

=(J8-

9*K8*L8*(TAN(E8))^2)/(I8^

3)

=(($C$3^2)-1)^3

=((($C$2-1)/2)*($C$3^2))+1

=(((($C$2-

1)/2)*($C$3^2))+1)+((($C$2+1)/4)*$C$3^4)

=(J9-

9*K9*L9*(TAN(E9))^2)/(I9^3)

=(($C$3^2

)-1)^3

=((($C$2-

1)/2)*($C$3^2))+1

=(((($C$2-

1)/2)*($C$3^2))+1)+((($C$2+1)/4)*$

C$3^4)

=(J10-

9*K10*L10*(TAN(E10))^2)/(

I10^3)


beta

beta 1

(deg)

=((4*PI())+ACOS(

M8))/3

=2*I8*COS(

N8)

=($C$3^2)-

1+O8

=3*TAN(E8)*(1+($C$2-

1)/2*$C$3^2)

=ATAN(P8/

Q8)

=R8*180/

PI()

=((4*PI())+ACOS(

M9))/3

=2*I9*COS(

N9)

=($C$3^2)-

1+O9

=3*TAN(E9)*(1+($C$2-

1)/2*$C$3^2)

=ATAN(P9/

Q9)

=R9*180/

PI()

=((4*PI())+ACOS(

M10))/3

=2*I10*COS

(N10)

=($C$3^2)-

1+O10

=3*TAN(E10)*(1+($C$2-

1)/2*$C$3^2)

=ATAN(P10

/Q10)

=R10*18

0/PI()


M3 P03/P3 P02/P2 P2 P3

2.1

=(1+(0.2)*L10^2)^(1.4/

0.4)

=(1+(0.2)*A10^2)^(1.4/

0.4)

643.851066371

964

=O10*N10/M

10

=L10+0.0733

=(1+(0.2)*L11^2)^(1.4/

0.4)

=(1+(0.2)*A11^2)^(1.4/

0.4)

611.634177395

02

=O11*N11/M

11

=L11+0.0733

=(1+(0.2)*L12^2)^(1.4/

0.4)

=(1+(0.2)*A12^2)^(1.4/

0.4)

581.342434538

754

=O12*N12/M

12

Example spreadsheet formulae for computing Pressure values.

x

componentsside a side b side c side d sum(D) C_d

=D12*SIN($

C$4-$C12)

=-

E12*SIN($C$

4+$C12)

=F12*SIN($C

$4+$C12)

=-

G12*SIN($C$

4-$C12)

=SUM(H1

2:K12)

=L12/(0.5*101.3*1.4*9*2*

COS(10*PI()/180))

=D13*SIN($

C$4-$C13)

=-

E13*SIN($C$

4+$C13)

=F13*SIN($C

$4+$C13)

=-

G13*SIN($C$

4-$C13)

=SUM(H1

3:K13)

=L13/(0.5*101.3*1.4*9*2*

COS(10*PI()/180))


19/19

19

=D14*SIN($

C$4-$C14)

=-

E14*SIN($C$

4+$C14)

=F14*SIN($C

$4+$C14)

=-

G14*SIN($C$

4-$C14)

=SUM(H1

4:K14)

=L14/(0.5*101.3*1.4*9*2*

COS(10*PI()/180))

Example spreadsheet formulae for computing drag coefficient.

y

components

side a side b side c side d sum(D) C_l

=-

D12*COS($C

$4-$C12)

=-

E12*COS($C$

4+$C12)

=F12*COS($C

$4+$C12)

=G12*COS($

C$4-$C12)

=SUM(H

12:K12)

=L12/(0.5*101.3*1.4*9*2

*COS(10*PI()/180))

=-

D13*COS($C

$4-$C13)

=-

E13*COS($C$

4+$C13)

=F13*COS($C

$4+$C13)

=G13*COS($

C$4-$C13)

=SUM(H

13:K13)

=L13/(0.5*101.3*1.4*9*2

*COS(10*PI()/180))

=-

D14*COS($C

$4-$C14)

=-

E14*COS($C$

4+$C14)

=F14*COS($C

$4+$C14)

=G14*COS($

C$4-$C14)

=SUM(H

14:K14)

=L14/(0.5*101.3*1.4*9*2

*COS(10*PI()/180))

Example spreadsheet formulae for computing lift coefficient.

Documents

Gas Project