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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$3^2))+1)*(TAN(E
10))^2
=SQRT(F10-
3*G10*H10)
Example spreadsheet formulae for computing Beta values.
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zi
=(($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$3^4)
=(J10-
9*K10*L10*(TAN(E10))^2)/(
I10^3)
Example spreadsheet formulae for computing Beta values.
beta
beta 1
(deg)
=((4*PI())+ACOS(
M8))/3
=2*I8*COS(
N8)
=($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$3^2)
=ATAN(P10
/Q10)
=R10*18
0/PI()
Example spreadsheet formulae for computing Beta values.
M3 P03/P3 P02/P2 P2 P3
2.1
=(1+(0.2)*L10^2)^(1.4/
0.4)
=(1+(0.2)*A10^2)^(1.4/
0.4)
643.851066371
964
=O10*N10/M
10
=L10+0.0733
=(1+(0.2)*L11^2)^(1.4/
0.4)
=(1+(0.2)*A11^2)^(1.4/
0.4)
611.634177395
02
=O11*N11/M
11
=L11+0.0733
=(1+(0.2)*L12^2)^(1.4/
0.4)
=(1+(0.2)*A12^2)^(1.4/
0.4)
581.342434538
754
=O12*N12/M
12
Example spreadsheet formulae for computing Pressure values.
x
componentsside a side b side c side d sum(D) C_d
=D12*SIN($
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-$C13)
=SUM(H1
3:K13)
=L13/(0.5*101.3*1.4*9*2*
COS(10*PI()/180))
-
8/12/2019 Gas Project
19/19
19
=D14*SIN($
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.