3 Airfoils
3.2 Aerodynamic properties of the NACA 4410 Airfoil
It now has to be decided how to obtain the aerodynamic coefficients, cl, cd, and cm. After learning about and studying the program XFOIL, available online for download and use under the GNU General Public License, this is found to be a fit solution for this task [18]. XFOIL is an “interactive program for the design and analysis of subsonic isolated airfoils” [19]. It makes it possible to, among other things, execute a viscous or inviscid aerodynamic calculation on existing airfoils, delivering lift- and drag predictions just beyond clmax. By combining high-order panel methods with advanced boundary layer and wake computations and, among other parameters, allowing variable Reynolds numbers, XFOIL has achieved recognition and a good reputation over the years.
XFOIL is now being executed for different values for AoA and utilizing the NACA 4410 airfoil, which already is stored in the program’s library. In this connection, fixed, predetermined settings for Mach-, Reynolds- and Ncrit
numbers are being used. Ncrit refers to the “log of the amplification factor of the most-amplified frequency which triggers transition” [19]. In other words is this a parameter, which specifies the disturbance level in the stream and how the transition of the stream consequently is affected. For this work, the default setting “9” for Ncrit was chosen. The value 9 is a typical value for an average wind tunnel, while a higher value, e.g. 12 -14 would be appropriate for a sailplane. A lower value would indicate e.g. a “dirty” wind tunnel [19]. Even though the blades of a wind turbine are fairly sleek and aerodynamically clean, it also has to be considered that they operate almost continuous in all sort of weather and therefore typically bear marks of this through scaring of the surfaces from dust, ice, etc. The aerodynamically cleanness of the wind turbine blades are therefor classified worse than a sailplane and the default value 9 is found to be appropriate.
For the Mach number, “0,1” was chosen, as this would be a good approximation for most of the wind speeds addressed in this work. As will be explained in more detail in the chapter covering the calculation program, is a correction factor for the Mach number utilized, if the Mach number exceeds the value “0,3” and the compressibility effect has to be taken into account. Regarding the Reynolds number was the value 107 chosen. This would represent the approximate average Reynolds number generated over the blades of a wind turbine similar to the reference wind turbine used in this work, which will be presented later.
The operating window and results from XFOIL can be seen in Figure 3.2 and Figure 3.3:
Figure 3.2: XFOIL user window
Figure 3.3: XFOIL results
The results for the coefficients cl, cd, and cm from XFOIL are now being stored in the spreadsheet along with the respective AoA, so that graphs over the coefficients may be produced:
!"#$#%&'(%')*# +,-./#$#0# ## ##
12,3#$#'&%# ## ## ##
)/+>6' =/' =#' =8'
CJ# C<=<M,<<# <=<<J><# C<=@<KT<#
CK# <=@K>,<# <=<<JKN# C<=@<,J<#
C@# <=K>T<<# <=<<J@M# C<=@<J,<#
@# <=JMTM<# <=<<J<?# C<=@<>T<#
K# <=?NNM<# <=<<JK,# C<=@<TN<#
J# @=<K@J<# <=<<>J?# C<=@<,M<#
T# @=N,NJ<# <=<<?,?# C<=@<NM<#
M# @=,,?><# <=<@<K<# C<=@<<N<#
@<# @=J,T@<# <=<@@KJ# C<=<M?N<#
@@# @=>,@><# <=<@NJ<# C<=<MJ><#
@N# @=TNMK<# <=<@K?,# C<=<M@?<#
@,# @=?J?,<# <=<@TJ,# C<=<TT<<#
@># @=MJKM<# <=<NKMK# C<=<>@M<#
@?# @=MMKJ<# <=<KTN@# C<=<,MJ<#
N<# @=M?J@<# <=<JMTT# C<=<,J?<#
NN# @=M@<K<# <=<M,K@# C<=<J@K<#
N,# @=T?T@<# <=@KMK,# C<=<>??<#
N># @=>,>,<# <=@MKM,# C<=@<,N<#
N?# @=,NJT<# <=N?T,,# C<=@T??<#
Table 3.2: Results from XFOIL for the NACA 4410 airfoil
It proved difficult to obtain data from XFOIL at AoA’s above 28°, as the program did not manage to conclude the calculations. This can be traced back to the detachment of the airflow experienced at such high AoA’s, which in turn not meet the requirements for this sort of calculation method to successfully be completed. As the calculation program is not intended to be used for extreme
settings of the variable pitch system, as e.g. for an aerodynamic brake motion of the rotor, it is decided that the interval of [-5°; 28°] for AoA is sufficient.
As explained above, these results will now be used to produce graphs for the coefficients with respect to AoA, and let Excel generate approximate functions for the graphs. To make the functions as accurate as possible, they are divided into two intervals, and each interval are assigned its own function. The intervals are adjusted individually for each coefficient through testing, to make the transition between the intervals as seamless as possible:
Figure 3.4: cl over AoA from XFOIL [-5°; 12,5°]
Figure 3.5: cl over AoA from XFOIL <12,5°; 28°]
Figure 3.6: cd over AoA from XFOIL [-5°; 7,36°]
Figure 3.7: cd over AoA from XFOIL <7,36°; 28°]
Figure 3.8: cm over AoA from XFOIL [-5°; 9,25°]
Figure 3.9: cm over AoA from XFOIL <9,25°; 28°]
The results for the approximated functions for cl, cd, and cm and their associated intervals are listed below:
cl:
!5°"!"12, 5°
[ ]
:cl
( )
! =!0, 0008!2+0,1133!+0, 4864 (3.4) 12, 5° <!!28°( ]
:cl
( )
! =!0, 0062!2+0, 231!!0,144 (3.5) cd:!5°"!"7, 36°
[ ]
:cd
( )
! =!0, 0000001!5+0, 0000007!4+0, 0000095!3+ 0, 0000223!2!0, 0000668!+0, 0051011(3.6)
7, 36° <!!28°
( ]
:cd
( )
! =0, 000001!4!0, 00002!3+ 0, 000107!2+0, 000607!+0, 004178(3.7)
cm:
!5°"!"9, 25°
[ ]
:cm
( )
! =!0, 00000161!4+0, 00001686!3+ 0, 00011477!2!0, 00056891!!0,10637366(3.8)
9, 25° <!!28°
( ]
:cm
( )
! =!0, 00000039!5+0, 00003434!4!0, 00124294!3+ 0, 02253063!2!0,19382689!+0, 52428728(3.9)
Using these functions for the coefficients, it is now possible to compare them to the results from XFOIL for evaluation:
45632## =/' =/'?@ABC'' =#' =#'?@ABC' =8' =8'?@ABC'
CJ=<<# C<=@<<@<# C<=<M,<<# <=<<JJ># <=<<J><# C<=@<KTT# C<=@<KT<#
CK=<<# <=@KMK<# <=@K>,<# <=<<JKK# <=<<JKN# C<=@<,NN# C<=@<,J<#
C@=<<# <=KTNK<# <=K>T<<# <=<<J@?# <=<<J@M# C<=@<JT@# C<=@<J,<#
@=<<# <=JM?M<# <=JMTM<# <=<<J<T# <=<<J<?# C<=@<>?@# C<=@<>T<#
K=<<# <=?@M@<# <=?NNM<# <=<<JKM# <=<<JK,# C<=@<>TN# C<=@<TN<#
J=<<# @=<KNM<# @=<K@J<# <=<<>>,# <=<<>J?# C<=@<JNJ# C<=@<,M<#
T=<<# @=N,<K<# @=N,NJ<# <=<<?M?# <=<<?,?# C<=@<N?@# C<=@<NM<#
M=<<# @=,,@K<# @=,,?><# <=<@<J<# <=<@<K<# C<=@<<,T# C<=@<<N<#
@<=<<# @=JKM,<# @=J,T@<# <=<@@N@# <=<@@KJ# C<=<MM,># C<=<M?N<#
@@=<<# @=>KJM<# @=>,@><# <=<@N@K# <=<@NJ<# C<=<MJMM# C<=<MJ><#
@N=<<# @=TK<?<# @=TNMK<# <=<@K,@# <=<@K?,# C<=<M<<<# C<=<M@?<#
@,=<<# @=?T,?<# @=?J?,<# <=<@T>T# <=<@TJ,# C<=<T,,># C<=<TT<<#
@>=<<# @=M>,?<# @=MJKM<# <=<NJJN# <=<NKMK# C<=<J?>N# C<=<>@M<#
@?=<<# N=<<JN<# @=MMKJ<# <=<K???# <=<KTN@# C<=<,JJJ# C<=<,MJ<#
N<=<<# @=MM><<# @=M?J@<# <=<><<># <=<JMTT# C<=<KT@N# C<=<,J?<#
NN=<<# @=MKTN<# @=M@<K<# <=<M@T,# <=<M,K@# C<=<KJ,T# C<=<J@K<#
N,=<<# @=?N??<# @=T?T@<# <=@KT<<# <=@KMK,# C<=<,,JJ# C<=<>??<#
N>=<<# @=>T<?<# @=>,>,<# <=@MMNM# <=@MKM,# C<=<T@><# C<=@<,N<#
N?=<<# @=,>KN<# @=,NJT<# <=N?N,J# <=N?T,,# C<=@N?>K# C<=@T??<#
Table 3.3: Comparison of functions and XFOIL
Figure 3.10: Coefficients from XFOIL
Figure 3.11: Coefficients from functions
The results from the approximate functions for the coefficients cl, cd, and cm are found to be satisfactory, and (3.4) to (3.9) will be incorporated in the calculation program to generate these values when needed in the BEM iterations.